In my last post I discussed some issues with identifying objectively which computations a physical system could legitimately be interpreted as instantiating. Computationalism is usually taken to be the view that all it takes to create a conscious mind is to implement the right computation, so the idea that we can't tell objectively when a computation is implemented implies either that there is no objective fact of the matter regarding when minds exist, that all minds exist (everywhere) or that no minds exist. None of these conclusions is particularly appealing!

I find the arguments discussed on the last post to be somewhat persuasive. Indeed, I had had similar concerns before becoming aware of these. What's more, I think the problem may be worse than even Putnam, Searle and Bishop have suggested.

I'm going to use variations on Searle's Chinese Room thought experiment to illustrate some of these related problems just because the image of a man manually performing a computation is one we can all picture relatively clearly.

Firstly, let's acknowledge Searle's usual conclusion, that if he is the system and he doesn't understand Chinese, then the system doesn't understand Chinese. The most promising computationalist response is that Searle's brain is actually creating two minds operating at different levels. This implies that there are at least two interpretations of Searle's brain as implementing a mind and they describe very different minds. So far so good, I suppose, but it does at least demonstrate the point that there may be many different ways to interpret a physical system as a mind and so it's wrong to assume that a physical system is identical to a mind or even that it has a one-to-one relationship with a mind. Whatever a mind is, then, it isn't quite a physical thing. Minds are not brains, in other words. If we are very strict physicalists and if we think our minds and our consciousnesses really objectively exist (as opposed to being interpreted to exist, a pattern we project onto the world like particular genes or institutions or whatever), this may be a problem.

Next, consider what happens when Searle stops executing his program and goes home to sleep for the night, mid-execution. While Searle is sleeping, does the Chinese Room mind still exist? The Chinese Room program isn't in any meaningful sense executing. It isn't really instantiated anywhere.

You could regard it as paused, a mind held in suspense. The state of the algorithm will at least be recorded physically somewhere. But what is the difference, I wonder, between such a mind and that of a potentially conscious algorithm written on paper but never executed at all? Perhaps all written algorithms are essentially computations in suspense? If that is so, then simply writing an algorithm out on paper must be enough to bring a suspended mind into existence. Whatever about whether we can tell whether a given system

So perhaps the mind doesn't exist while the computation is paused. But it doesn't seem to me that there is much of a difference between Searle pausing for the night and the necessary momentary pause between each step of Searle's calculation. Does a mind therefore flicker in and out of existence as he works? Is each new instance a new mind or the same mind revived? How could we even pin down to specific points of time when the mind exists and when it does not, when all we have to go on are steps in a computation? At any given point of time, it doesn't seem to me we can really say whether Searle is actively computing right now or whether he is in between computing steps. I don't think there is a fact of the matter about whether pausing to pick up a pen so as to make a note is actually part of the computation or not. For this reason, I don't think we can really say precisely when computing steps begin or end, because the physical actions corresponding to computational steps are not terribly well defined. If the steps of a computation don't really exist for well-defined time spans, it's hard to say how the mind can either.

So even though we may some day have an objective way of telling

There's another set of issues when we consider the same computation being performed in different places or at different times. Let's say Searle has a partner who performs every calculation with him to check for any error that might occur. If we regard his partner as part of the same system, then one mind is brought into existence. If we regard his partner as a parallel system, then a two minds are brought into existence. In general, it seems to me that in many cases we are free to divide up a computational system in a number of ways so that the number of copies of a mind in existence is whatever we want it to be. For instance, the electrons flowing through a computer could perhaps be divided into a thousand distinct sets, each of which parallels the computation of interest.

We might be tempted to simply adopt the rule that these are all simply part of the same computation and so the same mind. It seems most parsimonious. But what if we offset the duplication in time a little? Say Searle's partner takes the night shift and duplicates Searle's efforts from earlier in the day. Or say he is working a thousand years after Searle has already ceased his efforts. Wouldn't it seem a little weird now to regard them as part of the same computation, that one mind could be physically and temporally so widely distributed? But, again, it's perhaps not so easy to define an objective cut-off point so that we could really satisfyingly distinguish two parts of the same computation creating one mind from two distinct identical computations creating two identical minds.

So, to add to the concerns of

Obviously, I have raised a large number of questions, perhaps too many to be discussed all in one go. Many of these questions may have plausible answers. My feeling, however, is that these issues can never be satisfyingly dealt with. To me, the whole idea of objectively telling whether or when or how many times a given computation is executing is hopeless, and if this is the case then we cannot say that minds exist because of physical implementations of computations. Searle really is right that computation is observer relative, that no objective fact about the physical world can depend on whether a computation is instantiated.

So where does that leave computationalism?

When I say that a given mind exists, if we want to think of that mind as having conscious experience, then we need to think about it processing information of some kind. So we need to define its inputs, its environment and so on. Different inputs yield different "biographies" for the mind. In each different environment, it has a different life story, will think different thoughts, will develop differently and so on. For our purposes, we should consider each life story to be a different mathematical object. So when we ask how a mind (say my mind) can exist without being instantiated, we are really asking how my mind and its environment can exist without being instantiated. For instance, if life on earth had been wiped out billions of years ago, how is it that I and all my fellow humans could still exist?

Well, even if this had happened, there would still be a mathematical object isomorphic to the status quo we observe. A mathematician of infinite patience and capacity could in principle explore this mathematical object (e.g. by simulating it) and learn what transpires within. That mathematical object defines every event within our world, every word that we utter, every thought that we think. If our conscious experience is made up of every thought that we think, then we would still be conscious even if we only existed within this mathematical object.

But, since I'm saying it is the algorithm itself that realises consciousness and not any particular implementation, it doesn't matter if the mathematician never simulates it. We would regard ourselves as conscious beings and the environment around us as physical regardless. The intriguing possibility arises that this is in fact the case -- that this is not a physical world but a world which only exists as a mathematical object. Furthermore, this may be the case for

On this view, simulating an algorithm does not bring anything new (not conscious entities anyway) into existence. Rather it provides a window onto another world. So if you accidentally pull the plug on your simulation of a virtual world populated by conscious beings, don't worry, you have not committed genocide!

But what if you shoot a person in the head? Have you not committed murder? I would say you have, not necessarily because you cause that person to cease to exist but because you have removed him or her from causal contact with his or her loved ones. You have become the cause of grief and suffering and so your act is morally impermissable for this reason (on most conceptions of morality at least).

Whether you have actually caused the person to cease to exist is an interesting question. If the world were completely deterministic, then arguably you have terminated that mind. If we define the mind of the victim as the life story of that mind in this world, then it necessarily has to end at this point. It exists at no time after this event within the mathematical object of our world. The input to that mind after this point is undefined, and so the further life story of that mind is undefined, and so we should say it does not exist at times after the shooting. Of course, the proximate cause of this can be attributed to your action, so you bear responsibility for the death (on compatibilist accounts of free-will and responsibility at least).

But of course one cannot destroy a mathematical object, so how could one destroy a mind if a mind is a mathematical object? From a mathematical Platonist view, actually you haven't really destroyed the mind even if you have terminated its life story. Its life story, including every moment of experience, every thought it ever entertained and so on continues to exist just as a biography on the shelf does. In this view, the shooting constitutes a terminus to the story rather than a destruction of the story itself. There is no way to destroy a person utterly just as there is no way to destroy a mathematical object because there is no way to prevent a person from having existed in the first place.

Now, if the world is not completely deterministic (as appears to be the case), then at each point there are many possible futures. If there exist possible futures where the person was not killed, then there is another sense in which the shooting does not cause them to cease to exist. Though it may cause them to cease to exist in your world from that moment on -- they continue to live on in another possible future, effectively a parallel universe. Even if the world is deterministic after all, as long as there is some mathematical object which corresponds to that person continuing to live (e.g. in an alternative world that was identical from their limited perspective but subtly different in just the right way to prevent the shooting from happening), then in effect they can be regarded as continuing to live in that world.

Moving on, what of the relationship between the mind and brain? Changes to the brain clearly affect the mind, so how can I say the mind is independent of the brain?

There are (at least) two ways to view this. If you define the mind in conjunction with its environment as one object, then altering the state of the brain (e.g. with an injury or an electric probe) is in effect just another kind of input and so the mathematical object already incorporates the change. If you want to draw a border around the mind and treat it as a mathematical object in its own right, treating only sensory data as input, then a brain injury corresponds to instantiating an alternative mathematical object in its place. Rather than providing input to an algorithm you are replacing it with an alternative as we do when we rewrite and recompile a computer program. Either way of looking at this is fine with me.

On this view, the brain doesn't really create the mind, rather it taps into it and exploits it in order to navigate the world. In the same way, bees didn't really invent the hexagonal tiling of the plane. This is a solution which exists timelessly and Platonically. But bees did, through a process of evolution by natural selection, stumble into this tiling and exploit it because it is efficient. The mind is just a vastly more complex pattern that our ancestors stumbled into. Rather than causing the mind to exist, the brain should be regarded as a medium for the algorithm of the mind to interact with its environment, just as a computer is a medium that allows a computer program to take input and output but (on Platonism at least) does not cause the algorithm itself to exist. Destroying the brain therefore removes that mind from causal contact with its environment (the world), making a material difference to the world but not really to the mind/algorithm itself.

I find the arguments discussed on the last post to be somewhat persuasive. Indeed, I had had similar concerns before becoming aware of these. What's more, I think the problem may be worse than even Putnam, Searle and Bishop have suggested.

I'm going to use variations on Searle's Chinese Room thought experiment to illustrate some of these related problems just because the image of a man manually performing a computation is one we can all picture relatively clearly.

Firstly, let's acknowledge Searle's usual conclusion, that if he is the system and he doesn't understand Chinese, then the system doesn't understand Chinese. The most promising computationalist response is that Searle's brain is actually creating two minds operating at different levels. This implies that there are at least two interpretations of Searle's brain as implementing a mind and they describe very different minds. So far so good, I suppose, but it does at least demonstrate the point that there may be many different ways to interpret a physical system as a mind and so it's wrong to assume that a physical system is identical to a mind or even that it has a one-to-one relationship with a mind. Whatever a mind is, then, it isn't quite a physical thing. Minds are not brains, in other words. If we are very strict physicalists and if we think our minds and our consciousnesses really objectively exist (as opposed to being interpreted to exist, a pattern we project onto the world like particular genes or institutions or whatever), this may be a problem.

Next, consider what happens when Searle stops executing his program and goes home to sleep for the night, mid-execution. While Searle is sleeping, does the Chinese Room mind still exist? The Chinese Room program isn't in any meaningful sense executing. It isn't really instantiated anywhere.

You could regard it as paused, a mind held in suspense. The state of the algorithm will at least be recorded physically somewhere. But what is the difference, I wonder, between such a mind and that of a potentially conscious algorithm written on paper but never executed at all? Perhaps all written algorithms are essentially computations in suspense? If that is so, then simply writing an algorithm out on paper must be enough to bring a suspended mind into existence. Whatever about whether we can tell whether a given system

*implements*a given algorithm, it seems clear that there can be no fact of the matter about whether a physical system*represents*the code of an algorithm (because it is not hard to invent an encoding by which any physical artifact could represent any text), so even if the rocks around us don't host active "pixies" (see the last post if you don't get what I mean by "pixies"), they might at least host pixies in suspended animation! For reasons such as these, I don't think it is plausible to regard a paused computation as having a continuous physical existence.So perhaps the mind doesn't exist while the computation is paused. But it doesn't seem to me that there is much of a difference between Searle pausing for the night and the necessary momentary pause between each step of Searle's calculation. Does a mind therefore flicker in and out of existence as he works? Is each new instance a new mind or the same mind revived? How could we even pin down to specific points of time when the mind exists and when it does not, when all we have to go on are steps in a computation? At any given point of time, it doesn't seem to me we can really say whether Searle is actively computing right now or whether he is in between computing steps. I don't think there is a fact of the matter about whether pausing to pick up a pen so as to make a note is actually part of the computation or not. For this reason, I don't think we can really say precisely when computing steps begin or end, because the physical actions corresponding to computational steps are not terribly well defined. If the steps of a computation don't really exist for well-defined time spans, it's hard to say how the mind can either.

So even though we may some day have an objective way of telling

*whether*a given computation is being instantiated, we still have the problem of identifying*when*it is instantiated. It seems to me there is no fact of the matter in this case either.There's another set of issues when we consider the same computation being performed in different places or at different times. Let's say Searle has a partner who performs every calculation with him to check for any error that might occur. If we regard his partner as part of the same system, then one mind is brought into existence. If we regard his partner as a parallel system, then a two minds are brought into existence. In general, it seems to me that in many cases we are free to divide up a computational system in a number of ways so that the number of copies of a mind in existence is whatever we want it to be. For instance, the electrons flowing through a computer could perhaps be divided into a thousand distinct sets, each of which parallels the computation of interest.

We might be tempted to simply adopt the rule that these are all simply part of the same computation and so the same mind. It seems most parsimonious. But what if we offset the duplication in time a little? Say Searle's partner takes the night shift and duplicates Searle's efforts from earlier in the day. Or say he is working a thousand years after Searle has already ceased his efforts. Wouldn't it seem a little weird now to regard them as part of the same computation, that one mind could be physically and temporally so widely distributed? But, again, it's perhaps not so easy to define an objective cut-off point so that we could really satisfyingly distinguish two parts of the same computation creating one mind from two distinct identical computations creating two identical minds.

So, to add to the concerns of

*whether*and*when*a given mind is instantiated, we have to grapple with the question of*how many*times a mind is instantiated in the case of computations we could interpret either as being duplicated or as singletons.Obviously, I have raised a large number of questions, perhaps too many to be discussed all in one go. Many of these questions may have plausible answers. My feeling, however, is that these issues can never be satisfyingly dealt with. To me, the whole idea of objectively telling whether or when or how many times a given computation is executing is hopeless, and if this is the case then we cannot say that minds exist because of physical implementations of computations. Searle really is right that computation is observer relative, that no objective fact about the physical world can depend on whether a computation is instantiated.

So where does that leave computationalism?

### Platonism to the rescue!

I think Platonism resolves these and many other problems. On Platonism, a running computer program is just an instantiation of an algorithm, and an algorithm is a mathematical object which exists necessarily and timelessly even if it is never instantiated. If we allow ourselves to adopt the mental toolkit of Platonism, then it becomes possible to attribute consciousness not to a physical implementation of an algorithm but to the algorithm itself (or at least to an abstract run of an algorithm on particular input). In this view, it's not such a problem that there is no fact of the matter regarding whether physical systems instantiate a computation or not. No metaphysical fact depends on how we answer that question.

At first, this appears to be absurd. Am I really saying that it doesn't matter if a mind is physically instantiated, it exists regardless?

Yes, that is my view, but there's a lot of explaining to do about how this actually works and how it differs from belief in souls and the like. Much of the rest of this post is therefore dedicated to discussion what it means for a mind to exist Platonically, considering issues such as death, brain damage, identity and so on.

When I say that a given mind exists, if we want to think of that mind as having conscious experience, then we need to think about it processing information of some kind. So we need to define its inputs, its environment and so on. Different inputs yield different "biographies" for the mind. In each different environment, it has a different life story, will think different thoughts, will develop differently and so on. For our purposes, we should consider each life story to be a different mathematical object. So when we ask how a mind (say my mind) can exist without being instantiated, we are really asking how my mind and its environment can exist without being instantiated. For instance, if life on earth had been wiped out billions of years ago, how is it that I and all my fellow humans could still exist?

Well, even if this had happened, there would still be a mathematical object isomorphic to the status quo we observe. A mathematician of infinite patience and capacity could in principle explore this mathematical object (e.g. by simulating it) and learn what transpires within. That mathematical object defines every event within our world, every word that we utter, every thought that we think. If our conscious experience is made up of every thought that we think, then we would still be conscious even if we only existed within this mathematical object.

But, since I'm saying it is the algorithm itself that realises consciousness and not any particular implementation, it doesn't matter if the mathematician never simulates it. We would regard ourselves as conscious beings and the environment around us as physical regardless. The intriguing possibility arises that this is in fact the case -- that this is not a physical world but a world which only exists as a mathematical object. Furthermore, this may be the case for

*all*worlds. There may be no physical world and indeed the very concept of an objectively physical world may be meaningless. This, of course, is just the Mathematical Universe Hypothesis I have discussed previously.On this view, simulating an algorithm does not bring anything new (not conscious entities anyway) into existence. Rather it provides a window onto another world. So if you accidentally pull the plug on your simulation of a virtual world populated by conscious beings, don't worry, you have not committed genocide!

But what if you shoot a person in the head? Have you not committed murder? I would say you have, not necessarily because you cause that person to cease to exist but because you have removed him or her from causal contact with his or her loved ones. You have become the cause of grief and suffering and so your act is morally impermissable for this reason (on most conceptions of morality at least).

Whether you have actually caused the person to cease to exist is an interesting question. If the world were completely deterministic, then arguably you have terminated that mind. If we define the mind of the victim as the life story of that mind in this world, then it necessarily has to end at this point. It exists at no time after this event within the mathematical object of our world. The input to that mind after this point is undefined, and so the further life story of that mind is undefined, and so we should say it does not exist at times after the shooting. Of course, the proximate cause of this can be attributed to your action, so you bear responsibility for the death (on compatibilist accounts of free-will and responsibility at least).

But of course one cannot destroy a mathematical object, so how could one destroy a mind if a mind is a mathematical object? From a mathematical Platonist view, actually you haven't really destroyed the mind even if you have terminated its life story. Its life story, including every moment of experience, every thought it ever entertained and so on continues to exist just as a biography on the shelf does. In this view, the shooting constitutes a terminus to the story rather than a destruction of the story itself. There is no way to destroy a person utterly just as there is no way to destroy a mathematical object because there is no way to prevent a person from having existed in the first place.

Now, if the world is not completely deterministic (as appears to be the case), then at each point there are many possible futures. If there exist possible futures where the person was not killed, then there is another sense in which the shooting does not cause them to cease to exist. Though it may cause them to cease to exist in your world from that moment on -- they continue to live on in another possible future, effectively a parallel universe. Even if the world is deterministic after all, as long as there is some mathematical object which corresponds to that person continuing to live (e.g. in an alternative world that was identical from their limited perspective but subtly different in just the right way to prevent the shooting from happening), then in effect they can be regarded as continuing to live in that world.

Moving on, what of the relationship between the mind and brain? Changes to the brain clearly affect the mind, so how can I say the mind is independent of the brain?

There are (at least) two ways to view this. If you define the mind in conjunction with its environment as one object, then altering the state of the brain (e.g. with an injury or an electric probe) is in effect just another kind of input and so the mathematical object already incorporates the change. If you want to draw a border around the mind and treat it as a mathematical object in its own right, treating only sensory data as input, then a brain injury corresponds to instantiating an alternative mathematical object in its place. Rather than providing input to an algorithm you are replacing it with an alternative as we do when we rewrite and recompile a computer program. Either way of looking at this is fine with me.

On this view, the brain doesn't really create the mind, rather it taps into it and exploits it in order to navigate the world. In the same way, bees didn't really invent the hexagonal tiling of the plane. This is a solution which exists timelessly and Platonically. But bees did, through a process of evolution by natural selection, stumble into this tiling and exploit it because it is efficient. The mind is just a vastly more complex pattern that our ancestors stumbled into. Rather than causing the mind to exist, the brain should be regarded as a medium for the algorithm of the mind to interact with its environment, just as a computer is a medium that allows a computer program to take input and output but (on Platonism at least) does not cause the algorithm itself to exist. Destroying the brain therefore removes that mind from causal contact with its environment (the world), making a material difference to the world but not really to the mind/algorithm itself.

### Getting back to the pixies

Recall Mark Bishop's argument that if to be conscious is to instantiate a certain class of computation, and that if there are ways to interpret any object as instantiating any computation, then all objects must be instantiating all kinds of conscious experience. There are an infinite number of "pixies" living in every rock (and so computationalism is absurd and can be rejected).

How does my view of Platonic computationalism help us?

Unfortunately, it seems we are stuck with the pixies -- if all mathematical objects exist, and if algorithms are mathematical objects, and if all it takes to be conscious is to be a certain algorithm, then each of Bishop's pixies must exist. Perhaps we're back where we started.

Actually, I don't think so. What I find absurd about the DwP argument is not so much that the pixies exist but that they exist in every rock, and so that we should be attributing consciousness to the rocks themselves. But, on Platonic computationalism, the rocks are not conscious and the pixies are not located in the rocks. Indeed, they are not located anywhere within our coordinate system. If they are located anywhere, it is only within a coordinate system local to their own environment, which is part of the mathematical object or algorithm in which they find themselves embedded. From our perspective they are abstract, just as from their perspective we are abstract. Destroying the rock has no impact on the pixies because they were never inside it in the first place.

I don't find it absurd that the pixies exist because I am already committed to the Mathematical Universe Hypothesis, which predicts the existence of all possible minds somewhere within the mathematical multiverse. Bishop's pixies correspond therefore to observers in other universes causally disconnected from our own. Since they are not in our universe, we don't need to concern ourselves with them. For all practical purposes, they do not exist from our perspective.

I say they are in other universes rather than in our own (despite the possibility of finding a Putnam-style mapping to show them reflected in our universe) because they are causally disconnected from us. Nothing we can do can affect them or they us, until that is we build a system of input/output which would allow us to interact with them. At that point, we can consider them and their world to be embedded in our own universe, but to do so we would have to build a very complex machine, effectively a supercomputer. Now, it hardly contradicts computationalism to suppose that a mind can be brought into our universe by a supercomputer, so this conclusion meshes well with the computationalist viewpoint.

Furthermore, on the DwP argument, it would seem that there are distinct identical copies of each pixie in every rock. On Platonism, these identical copies are just instances of the same mathematical object, of which there is only one. This also strikes me as less absurd.

Bishop has suggested to me that the MUH is a kind of panpsychism, but I disagree. To me, panpsychism is the view that consciousness is all around us, pervading the universe, so that even elementary particles can be considered to be conscious in some way. But my view is rather that matter is never conscious, that consciousness is rather a property of abstract structures. Consciousness is not all around us -- frankly, it doesn't physically exist anywhere in our universe. We are conscious but we (or at least our minds) are not physical things, we are instead the algorithms our brains are most usefully interpreted to be computing as they take sensory input, process it and output motor commands. To me, this is anything but panpsychism. It might be called a kind of dualism, but as I have argued before, dualism need not be a dirty word, and in any case, I'm really a monist because I think that even the so-called "physical" world is actually an abstract mathematical structure.

### Conclusion

For reasons I have discussed before, I think computationalism has to be correct. That is, if we were to build an AI that was just as capable as we are, and particularly if it were modelled after the processes going on in human brains, we would be obliged to think of it as conscious in the same way that we are. But this does not mean that we have to attribute consciousness to the physical object of the computer. Neither do we need to attribute consciousness to the physical brain. The assumption from many computationalists that consciousness is a physical phenomenon, that there is a fact of the matter regarding when and where consciousness is instantiated, is in my view untenable. Rather we need to see consciousness as a property of certain abstract structures. It is useful and practical to regard the brain and the computer as instantiating such structures, but there need not be a fact of the matter because these structures exist Platonically regardless of whether they are instantiated or not.

Hi DM,

ReplyDeleteA very well written piece. As always, you make a compelling case for the MUH. I still see the MUH as possible but not driven by any necessity. Still, interesting stuff.

On the subjective vs objective existence of consciousness, I left a reply to your comment on my blog.

http://selfawarepatterns.com/2016/03/02/are-rocks-conscious/comment-page-1/#comment-14164

More specifically toward how the MUH affects things, I’m still struggling to understand how the MUH makes the DwP less absurd. I understand what you’re saying about the pixies existing platonically but, unless I’m missing something, that existence has the same relationship to our universe as a human mind. That is, they both exist platonically whether or not their physical interface to this universe is destroyed.

If we accept the DwP and the MUH, it seems like we still have pixies in rocks in the same manner that I’m in my body. If my body is destroyed, I continue to exist platonically. If a pixie’s rock is destroyed, it continues to exist platonically. But in both cases, our existence in this universe is affected. (The pixie might not know what ended its subjective world, but it would still have been affected.)

Also, to me, the MUH actually exacerbates the fact-of-the-matter issues. So, every variation of me exists platonically. But the more different those variations are, the less me they are. Eventually, I would not recognize the variations as me anymore. But exactly where, objectively, do we cross that line? At what point in platonic space does me end? It seems like any pattern that isn’t exactly the me typing this comment requires a judgement call as to whether it remains the same me. In other words, there doesn’t appear to be a fact-of-the-matter on this.

Likewise, assuming consciousness is a collection of algorithms, then in platonic space, every variation of that combination of algorithms exist. And the variations continue, first into realms where they strongly resemble what we call consciousness, then into realms where the resemblance is less but perhaps still compelling, but eventually into patterns we would not recognize as conscious. But also here, it seems like the dividing line wouldn’t be a fact of the matter.

So, while the MUH remains an interesting, even compelling superset of, well, everything, I can’t see how it helps resolve the DwP or fact-of-the-matter issues. But I’m open to the possibility that I simply have a blindspot preventing me from seeing it.

Hi Mike,

Delete> unless I’m missing something, that existence has the same relationship to our universe as a human mind.

Not really. Pragmatically, it is useful to regard a human mind as being in interaction with the universe. It has a realised input/output function via the senses and body and can even interact with other minds. Mathematically, the world and the mind are unified and connected in a way the world and the pixies are not. The pixies are causally isolated and so for practical reasons should not be regarded as part of the universe. Indeed, I would say that it isn't really just for practical purposes. The only sense in which they are in this universe at all is to the extent that we are talking about them. They are not in the rocks. We can only (metaphorically) project their images onto the rocks if we are determined to do so.

> The pixie might not know what ended its subjective world, but it would still have been affected.

No, the pixie is entirely unaffected by you destroying its rock. The pixie was never in the rock. It continues to exist just fine.

> But exactly where, objectively, do we cross that line?

There is no objective line. There is no fact of the matter regarding whether the you who went to sleep last night is the same you who woke up this morning either.

> In other words, there doesn’t appear to be a fact-of-the-matter on this.

Correct.

> But also here, it seems like the dividing line wouldn’t be a fact of the matter.

Correct.

I'm saying that it seems there ought to be a fact of the matter on whether "I" (the conscious entity thinking this thought right now) exist, but I agree that there is not a fact of the matter on what counts as me or what counts as conscious.

Hi DM,

Delete“No, the pixie is entirely unaffected by you destroying its rock. The pixie was never in the rock. It continues to exist just fine.”

This remains a point that’s not clear for me. If I destroy the rock, haven’t I destroyed at least an instantiation of the pixie? Or using the language you used in the post, switched platonic patterns? It seems like I’ve destroyed the pixie in the same way as I would be destroyed if someone physically destroyed my body. In both cases, the platonic versions are unaffected, but the local instantiations are scrambled. If not, then I’ll admit to being confused.

“I'm saying that it seems there ought to be a fact of the matter on whether "I" (the conscious entity thinking this thought right now) exist, but I agree that there is not a fact of the matter on what counts as me or what counts as conscious.”

I guess it’s not clear to me how the MUH improves things here. It seems like an issue we have to grapple with both with and without the MUH.

Hi Mike,

Delete> If I destroy the rock, haven’t I destroyed at least an instantiation of the pixie?

Arguably. Whether the rock constitutes an instantiation of the pixie is subjective and should be judged on pragmatic grounds (a test it would fail).

But the pixie is not an instantiation of the pixie. The pixie is the abstract pattern itself. Destroying an instantiation of a pattern has no effect on the abstract pattern itself.

> Or using the language you used in the post, switched platonic patterns?

OK, you might have switched patterns, according to some mapping. But such mappings are useless and can be disregarded. Again, the pixie is the pattern itself, not the instantiation of the pattern, so the pixie is still unaffected.

> In both cases, the platonic versions are unaffected, but the local instantiations are scrambled.

Yes, but the difference is that in the case of scrambling a brain, you are making a material difference to the world by removing an intelligent being from it. Destroying a rock makes no material difference to the world (apart from there being one less rock) because the pixie was not interacting with the world and does not exist from the world's point of view.

> I guess it’s not clear to me how the MUH improves things here.

On the MUH, I do, definitively exist, because I am a (self-aware!) pattern and all patterns exist.

Hi DM,

DeleteTotally agree on the pragmatic grounds for dismissing the pixie instantiations.

On what the MUH brings, I don't want to get back into a mode of us repeating our points, so I think I'll rest for now, at least until or unless I have something new to add.

Fascinating discussion, as always.

Hi DM,

ReplyDeleteI must say that these two articles are doing a good job of persuading me against computationalism! :-) (Though I also admit that I don't know what to adopt instead.)

Cheers, Coel.

I suppose I'm ambivalent about having talked you out of computationalism!

DeleteSince you don't know what to adopt instead, I suggest you should probably consider that you may be wrong to reject Platonism. Why not be open-minded?

Hi DM,

ReplyDeleteAlthough I'm trying to give up philosophy, I couldn't resist dropping in to read the second installment of your argument about computationalism, and giving a very brief response. I should warn you in advance that I'm not going to get into another discussion (much as I'll be tempted), but I will at least read anything that you write in reply.

In my view, a great deal of philosophy goes wrong because of confused use of language, including category errors. And I think this is true of your current piece. An example occurs early on...

DM:

Next, consider what happens when Searle stops executing his program and goes home to sleep for the night, mid-execution. While Searle is sleeping, does the Chinese Room mind still exist?I think this is an inappropriate use of the word "mind", to the degree that I can make no sense of your question. Despite having said that a "mind" is not a physical object, you are applying to minds a concept (of continued existence) that is based on physical objects, and it's unclear what sense (if any) we can make of applying that concept to a mind. The same sort of problem arises when mathematical platonists talk about numbers as if they were pseudo-physical objects.

I'm afraid that's as much as I'm going to say. I hope it's given you a little food for thought.

Best wishes,

Richard.

Hi Richard,

DeleteThanks for the comment and I'm sorry you won't be participating in these discussions in future.

I don't think I'm making a category error. I think personal identity rests with minds. I think I am my mind, and I think you are yours. That is to say, if I were "mind-uploaded" onto a different substrate, I believe that I would continue to exist even though nothing physically about me would be retained. So the mind is not a physical object, but it has a continuous existence regardless -- the kind of existence we all experience ourselves to have. To talk of the Chinese Room mind existing is to talk of it existing in the same sense that you or I exist as minds (as distinct from how our bodies exist as physical objects).

Hi DM,

ReplyDeleteyou've carved out an interesting niche for yourself regarding philosophy of mind, a kind of 'anti-physicalist computationalism'---I don't think I've encountered this position anywhere else, which is fully intended as praise: it's a welcome change from people essentially regurgitating the same old positions.

But I don't think this escape really works, ultimately. First of all, a small disagreement with something you write early on: you say that (paraphrasing) since Searle's brain instantiates, on one plausible retort to the Chinese Room argument (the most plausible one, I'd say), two different minds, this shows that brain and mind are non-identical. I don't think this follows: the same physical facts may be coarse-grained in different ways to give rise to different higher-level entities---in sort of the same way this picture looks, if viewed from up close, like Albert Einstein, but like Marilyn Monroe when seen from afar. Clearly, it's the same set of pixels, which sustains two different pictures, depending on the coarse-graining level. It might be similar with the two minds in Searle's brain; indeed, one can also describe an arm as a collection of atoms, or of cells, of muscles, bone and tissue, or as an arm as a whole---so different levels of description don't entail different metaphysical status.

But my main problem is different: I don't see what the move to accepting Platonism buys you regarding the arguments against computationalism. In particular, I think one can run an exactly analogous argument in this case, establishing not that whether a given physical system instantiates (the computation producing) a mind is not an objective fact pertaining to the system, but rather, that there is likewise no objective way to identify a mind instantiated by a given mathematical structure.

So let's for the moment accept the MUH (I don't, you may remember, but that's a different topic). I have some questions regarding what exactly is a 'mathematical object', but let's table these for now, as well. You mention algorithms as mathematical objects; so then, for instance, computations using natural numbers should count, too.

But that's where the problem comes in: what, exactly, are you doing when you're performing a computation using natural numbers? Basically, the issue is that any such computation can be considered as a deduction in some arbitrary formal system, by means of a Gödel numbering. You simply fix an encoding of symbols of the formal system into natural numbers, and then, arithmetical operations on natural numbers correspond to logical derivations within the formal system.

This assignment is, of course, not unique: there are many different ways to encode formulae in numbers. So, when you view a specific natural number manipulation as a mathematical object, then this mathematical object can be interpreted as a variety of different formal procedures---one of which might be, if such a thing is possible, a mind.

(continuing the above---I know brevity is the soul of wit, but I don't believe in souls, so there)

DeleteSo you've got the same problem in the Platonist world as in ordinary physical reality: there is no one-to-one mapping between the objects in the world and minds, assuming computationalism.

Of course, these aren't really different arguments: whether we consider physical or formal objects doesn't make a difference for these arguments, since we're really only interested in their structural properties, no matter how they're instantiated. That's the reason why changing the instantiation (moving from the physical to the Platonist) doesn't change anything about the argument (at least, as far as I can see). Whether you consider a physical system moving through a series of physical states, or a formal system moving through the steps of some deduction, or an arithmetical system going through some calculation, doesn't make a difference.

Ultimately, the argument seeks to establish the conclusion that structure doesn't suffice to pin down content; on what terms you regard that structure then doesn't matter.

Hi Jochen,

DeleteThanks for the praise!

> on: you say that (paraphrasing) since Searle's brain instantiates, on one plausible retort to the Chinese Room argument (the most plausible one, I'd say), two different minds, this shows that brain and mind are non-identical.

I think it follows trivially. If there are two minds and one brain, then obviously you can't say the brain is the mind. If it were then there would be a one to one correspondence between minds and brains. I like your Einstein/Monroe image and it actually illustrates very well how I think of this situation. There are two distinct perceived images (the pictures) but only one bitmap (the set of pixels). This proves that the perceived image (the picture) is not the bitmap (the set of pixels). In the same way, the brain is not the mind.

> But my main problem is different: I don't see what the move to accepting Platonism buys you regarding the arguments against computationalism.

Your point here is about how it deals with the pixies, but much of the article deals with its advantages when considering other issues. Platonism at least buys me this, even if you do have a point regarding pixies.

> So, when you view a specific natural number manipulation as a mathematical object, then this mathematical object can be interpreted as a variety of different formal procedures---one of which might be, if such a thing is possible, a mind.

So this again suggests that there are pixies. My answer is the same. I accept the existence of all the pixies, and this is no longer an issue if one accepts the MUH. But at least now we're not saying the pixies are actually located in the rocks. They are rather located more sensibly in their own universes.

> there is no one-to-one mapping between the objects in the world and minds, assuming computationalism.

I accept that.

However, there is a most reasonable, or most straightforward way of interpreting what computation a brain is carrying out. My view is that my conscious mind is that computation as opposed to the other more strained interpretations. There is no fact of the matter about what computation my brain is carried out, but there is a fact of the matter regarding which of the possible computations is the one that I am. All of them exist, but only one of them is me.

Hi DM,

Delete>There are two distinct perceived images (the pictures) but only one bitmap (the set of pixels). This proves that the perceived image (the picture) is not the bitmap (the set of pixels).

Well, I would rather say that the picture of Albert is the set of pixels under one coarse-graining, while the picture of Marilyn is the set of pixels under a different coarse-graining. There's no information added to make the pixels produce either picture; on the contrary, information is lost. All of the information that gives rise to either picture is contained in the pixels, so there's 'nothing more' than them. So each picture is a subset of the information present in the pixels; likewise, each mind is a subset of the information (if that's the correct way to think about this) present in Searle's brain.

So I suppose it's more the claim that the mind isn't a physical thing that I disagree with (or at least, that I don't think the argument establishes): there's nothing additional to the physical that needs to be invoked in order to make sense of the mind/brain relationship in this case.

>But at least now we're not saying the pixies are actually located in the rocks. They are rather located more sensibly in their own universes.

I'm not sure I get what you mean here. To me, there seem to be two possible ways of interpreting your Platonism: one, that there is a mathematical/computational structure that corresponds to our universe, or some relevant part thereof, in total. In that case, I think the pixies argument works exactly as before: there are parts of this universe that can themselves be interpreted as computations (I'm doing that right now!), and the mapping between these parts and the computations they implement is just as arbitrary as before.

On the other hand, you might be suggesting that it's really just my mind that corresponds to a mathematical structure, complete with its apparent sensory inputs and all. This is then basically a form of solipsism (albeit one with the assurance that other minds also exist, somewhere out there, but causally removed from me). I don't think that's a terribly attractive point of view, but moreover, I'm also not sure it helps with the pixies---parts of that mathematical structure (and indeed, also the structure itself) still afford an interpretation in terms of different computations. There's still no objective way to say that this structure is my mind; it is equally well my mind, as a pandemonium of different ones.

Moreover, I think the real problem with the anticomputationalist arguments isn't the panpsychism they entail (though it's often framed that way), but rather, the implication that whether a given system (physical or mathematical) implements a mind is not an objective fact of the matter. But then, what makes it so that they give rise to a mind at all?

You seem to want to stipulate that if it's possible to interpret a system as giving rise to a mind, then that mind will exist, somewhere out there. I don't really see why that should be the case---after all, I can interpret the word 'splarg' to mean 'invisible pink unicorn', but that doesn't mean there are any invisible pink unicorns somewhere out there in their own Platonic realm! So why would it be the case that just because I can interpret a certain system as implementing a mind, that mind must be out there?

Hi Jochen,

Delete> Well, I would rather say that the picture of Albert is the set of pixels under one coarse-graining, while the picture of Marilyn is the set of pixels under a different coarse-graining.

I would agree. My point is that you don't simply get to equate the pixels with the picture. The brain is a physical thing. The mind is not. As with your example, where the images are two different patterns within the pixels, the mind is a pattern in the brain, and the same brain can realise multiple mind-patterns.

> So I suppose it's more the claim that the mind isn't a physical thing

But it isn't. There is no physical object that is identical with your mind. The brain is the only candidate, and this argument establishes that the brain is not the mind. I agree with you that this doesn't prove that the mind can exist without the brain, but it does establish that the mind is not simply a physical thing. It is at best a pattern within a physical thing.

> there are parts of this universe that can themselves be interpreted as computations

OK. So, there is a computation A that corresponds to this universe. This is the computation that a superdupercomputer would be carrying out if it were simulating this universe.

Similarly, there is a computation B that simulates the universe of a pixie.

If you look through the source code of A, you will not find anything that looks remotely like B. For this reason, B is not sensibly regarded as being within A.

But of course you can implement a crazily complex and ad hoc mapping to interpret the work of A as it simulates the rock as computing B. To me, this only shows that we don't have any objective criteria for judging an observer in A to be any more real or conscious than an observer in B. On the MUH, both observers are real anyway, so this is not a problem for me.

Where are these observers? Well, an observer is just a computation, so on platonism it isn't really anywhere. But it is sensible and useful to regard my mind as being located within the physical structure that seems to be instantiating it. So, for practical purposes, you can regard my mind as being where my brain is in this universe.

But the structure that is most straightforwardly interpreted as carrying out the computation of the pixie is not in this universe. It is in the universe corresponding to computation B. So for all practical purposes, the pixie is not in this universe but in another.

> the implication that whether a given system (physical or mathematical) implements a mind is not an objective fact of the matter.

I agree there is no objective fact of the matter. I'm saying all minds, all conscious experiences exist. But only one of these corresponds to the one my brain looks like it is computing. That is the one that we associate with my brain. There is no fact of the matter on whether it is the one my brain is computing, but it is not useful to regard my brain as computing any of the others.

> But then, what makes it so that they give rise to a mind at all?

Strictly speaking, they don't. The mind exists necessarily as a platonic object, like all mathematical structures. You don't create a mathematical structure by instantiating it.

> but that doesn't mean there are any invisible pink unicorns somewhere out there in their own Platonic realm!

This is only because the concept of something that is both invisible and pink is incoherent. I do think there are universes (perhaps even this one) that contain pink creatures which resemble horses with horns.

> So why would it be the case that just because I can interpret a certain system as implementing a mind, that mind must be out there?

This follows from platonism and the view that the mind is a mathematical object.

Hi DM,

Deleteso, to you, a pattern of (or within) physical things is a non-physical thing? Doesn't that mean that there really aren't any physical things (certainly not within our everyday experience)? Because as I look around on my desk, anything I see there is ultimately a pattern of physical things: the desk itself is a pattern of pieces of wood, my chair is a pattern of pieces of plastic and metal, my computer is a really complicated pattern of silicon chips, wires and whatnot, and so on. Ultimately, those things are gain patterns: a piece of wood is a pattern of dead cells, a piece of silicon is a lattice of atoms, and so on, right down to the level of elementary particles. And they, too, are probably best understood as patterns (excitations) of quantum fields, and as for them, well, a quantum field without excitations really is just a vacuum. So all there is, is ultimately just patterns of vacuum!

I think that's a far too reductive notion of 'physical thing'. Some (well basically, all) physical things have parts, that are also physical things; but just because something is composed of parts doesn't mean it's not a physical thing.

Take a set of stones, and arrange them into a circle. Let's stipulate that the stones are physical things. To me, the circle would likewise be physical; but you seem to have a different notion in mind, some kind of non-physical 'circlehood' that needs to be added to the stones to make them a circle. But what happens if I take that away? To me, there doesn't seem to be any noticeable difference. So I'd be inclined to strike this thing from my ontology, since it basically seems to just be excess baggage.

The same goes for the case of the pictures. Each picture is a physical thing, composed of other physical things, the pixels. To add some non-physical 'picturehood' to it doesn't seem to me to be doing any work. So even if it may be wrong to say that the mind is identical to the brain, that doesn't mean it's not a physical thing---something doesn't have to be identical to another physical thing to be itself physical, well, none beyond itself I suppose. Then, the mind is a physical thing composed of parts that also make up the brain, and maybe also (partially) composed of parts that make up a second mind. Or at least, this view seems to me perfectly consistent with the example of two minds you gave---nothing non-physical is needed, and nothing non-physical seems to be doing any useful work in giving rise to this state of affairs.

I'm also not too sure about your notion of 'separate universes' corresponding to different computations. How do I, in universe A, come to know about universe B? It seems to me that either there must be some interaction between the two---then, I don't think they're separate universes. Or, B must be in some sense a substructure of A---but then, the pixies are part of A in just the same way as in the original argument.

DeleteUsually, Platonism is employed to explain how we can have access to mathematical truths (or universals, or whatever). There's the physical realm, and the Platonic one, and *somehow* our minds are in contact with the Platonic realm to enable us to recognize approximate realizations of its structures in the physical world. But on your more radical conception, the Platonic realm is all there is. But then, this re-opens the problem Platonism was originally brought in to address: how I, within one Platonic structure, get to know about another. So I don't really see how A and B could be separated, yet unified enough for me in A to know about B.

Ultimately, it seems to me that the real problem is that computation doesn't exist without an interpreting mind, just like how a sign doesn't become a symbol unless it is so interpreted---unless it is used in the right way. I realize that you're essentially saying that such use can only come about by postulating that all the possible meanings already exist, and what we do is merely picking out one; but on your conception of Platonism, it seems to me that all we really have are more signs, whose interpretation again is arbitrary. The whole thing never bottoms out: I interpret my computer (a part of computation A) to perform a certain computation B; but this computation B is itself amenable to interpretation as computation C, and so on. So each of your Platonic universes isn't itself a certain thing, but can be interpreted as a great many other things, which in turn can be interpreted, and so on; you can never say, this structure here *just is* that sort of mind, having these thoughts, experiences and whatnot. All you can say is that it can be so interpreted, and if you're limiting yourself to interpretation in terms of Platonic mathematical structures, then you can just go on interpreting and never reach any sort of certain ground.

Put simply, I think you need stuff to underlie structure; otherwise, having only structure, the stuff can't be pinned down. Otherwise, for each structure you point to, claiming 'that's a mind', I can always ask, 'says who?'.

Hi Jochen,

Delete> so, to you, a pattern of (or within) physical things is a non-physical thing?

It depends. I mean, a stone or a chair is a physical thing to me. We can identify that stone or that chair with a particular set of atoms, those atoms with a particular set of quarks and electrons, those quarks and electrons with particular strings or whatever all the way down to the foundations of physics whatever they may be.

Similarly, if you place stones in a circle, I don't think there is much of a problem identifying the physical object of the circular group of stones with those stones and their atoms. Although the "circle" they are approximating is not a physical thing but a pattern.

A better example for me would be waves. The same atoms might be participating in the realisation of multiple waves at the same time, e.g. where wavefronts are intersecting. So you can't identify the wave with a particular set of atoms. It's physical in one sense, in that it is a pattern realised by physical things, but it is not a physical object itself, in that it is not identical with any particular set of particles or whatever, and the particles which realise are constantly changing. You can't identify the wave with the water that is propagating it, but I wouldn't want to foreclose the possibility that it is physical in some other sense. I'm just saying that it isn't a discrete physical thing like a chair or a stone.

Of course the particles that make up people are constantly changing too, so you could say that people are also patterns moving through the physical world. But there is a great deal more stability in the particles that make up people compared to waves. These are not absolute categories to me.

> How do I, in universe A, come to know about universe B?

From your perspective, you're not perceiving it, you're inventing it. But what seems subjectively like invention doesn't mean that the idea is not already out there. Subjectively, it felt to me that I had invented the idea of the MUH (I called my idea the Platonic Algorithm, but it's essentially the MUH). I later learned that Tegmark had already proposed the idea.

> Usually, Platonism is employed to explain how we can have access to mathematical truths (or universals, or whatever).

Personally, I don't endorse this motivation for Platonism. I don't think we perceive mathematical objects in anything like the way we perceive physical objects. I don't think we have anything like a "sense" for them. There is just this tendency for our brains to instantiate certain abstract structures as concepts.

> it seems to me that all we really have are more signs, whose interpretation again is arbitrary.

Well, my account idea of syntax and semantics is a whole other issue I don't know if we want to open up now. Basically, I think there is a difference between how reference works in a public symbol (such as a word on a page) and how reference works in a private symbol (such as a mental representation). I don't think private symbols need to be interpreted the way public symbols do. Private symbols have their meaning in virtue of their causal relations.

Hi DM,

Deletebut the same stones could be part of a circle, and part of the pentagram inscribed within it (its vertices). You can't identify either the pentagram or the circle with the stones comprising it: a circle, for instance, has a radius r, while the stones don't have a radius; the pentagram has a set of opening angles, while the stones don't. Rather, the pentagram is a set of stones in that particular configuration, and likewise for the circle.

That's also true of a wave: a wave is a set of particles in a particular configuration (in time). The same particles may be part of multiple configurations, just as the same stones may be part of multiple patterns. I don't see why I should admit anything nonphysical into the picture here.

>But what seems subjectively like invention doesn't mean that the idea is not already out there.

Well, but how does the idea from 'out there' get 'in here' (meaning, my head, mind, or that particular pattern corresponding to its computation)? If I can just come up with it myself, then what do I need the idea out there for? If I can't, and there needs to be some interaction to get it in here, then in what sense is it separate from me?

Hi Jochen,

Delete> Rather, the pentagram is a set of stones in that particular configuration, and likewise for the circle.

I would distinguish between the set of stones that depict a pentagram and the pentagram itself. The set of stones is a physical object. The pentagram is a pattern and is not a physical object.

> I don't see why I should admit anything nonphysical into the picture here.

I'm arguing that a wave is not a physical object. I'm not necessarily saying it is entirely non-physical. You could say it is a physical pattern. But it's not a thing in the way that a stone is.

>

Well, but how does the idea from 'out there' get 'in here' (meaning, my head, mind, or that particular pattern corresponding to its computation)?

How does it get there on your account?

I don't want to have to explain to you where ideas come from because I don't think we differ on this. Your account of where mathematical ideas come from is probably identical to mine. The difference is only that I think mathematical objects exist abstractly and independently of minds, that when we think of them we are thinking of real things. This is as much a convention about what 'exists' means as a statement of fact. Perhaps more so.

> If I can just come up with it myself, then what do I need the idea out there for?

You don't need it to be out there. It just is out there, necessarily, on the definition of existence used by platonists. It can't not be out there. The hypothetical that it is not out there can't be entertained because it is incoherent on a platonist view.

Just to elaborate on how the idea of a mathematical object gets into our minds if we don't perceive it.

DeleteHow about this: we don't perceive them, we infer that they exist. I have never perceived your parents, but I infer that they exist or at least existed at one time.

Hi DM,

Delete>You could say it is a physical pattern. But it's not a thing in the way that a stone is.

Well, I don't see the distinction: the stone is likewise just a pattern of quantum fields; if I accept that as physical, then I don't see any reason to think of the pentagram as non-physical. It's composite nature is more readily apparent than that of the stone, but that's just an accidental effect of the level on which we perceive things.

>How does it get there on your account?

Well, without going into too much detail, the basic element is structure: structure is what is the same between a system and its model---say, if you model the solar system with an orrery, then that means that the little metal beads of the orrery instantiate some of the same structure as the planets do---their orbits have the same relative distances, they may have the same relative sizes, and so on. In a word, they instantiate some of the same relationships as do the planets.

But there's of course a clear distinction between model and original system: just because it's an instantiation of the same structure, doesn't make it the same thing.

So things like computers, and mathematics in the abstract, are really just a kind of universal modeling clay: you can instantiate virtually any structure using mathematics, or the bits and bytes of a computer. So something gets 'in here' simply because I re-arrange the appropriate parts of my mind to instantiate a particular set of relations, that can then be interpreted as, say, the relations between the planets of a hypothetical star system.

Computation then is just interpreting the relationships between the parts of some sufficiently complex system as implementing the relations of a particular system to be modeled; without that interpretation (i.e. without an intentional user), there is no computation.

So basically, my mind is made up of stuff, which I can rearrange (by means of using my imagination) to instantiate virtually arbitrary relations; those relations need not have any existence anywhere else but within my mind, and indeed, I don't think the idea of pure structure, of relations without relata, really makes any sense.

Regarding inferring the existence of a given structure: can you be wrong about whether it exists? I mean, you certainly could be wrong about the existence of my parents: I could be a clone, or a (moderately) sophisticated AI routine, or an alien being from a race that reproduces by budding.

Otherwise, what is it that gives the inference in the case of Platonism its necessity?

And besides, the inference of the existence of my parents is due to seeing a result of the causal chain that starts with them: me, or at least, the stuff I'm polluting your comments section with. It seems that in Platonism, there is no effect that the pattern has on you, yet you claim to be able to infer its existence.

Hi Jochen,

Delete> the stone is likewise just a pattern of quantum fields

Fair enough. If your view is that it's patterns all the way down, that is also my view. I'm just trying to make the point that the mind is not the brain, nor is it a subset of the brain. It is a very complicated pattern embedded in the brain. That doesn't necessarily refute nuanced physicalism. It is intended to refute only simplistic physicalism, people who would say the pentagram is not real, only the stones are real. If you're not in that camp then you can disregard this line of argument.

> So something gets 'in here' simply because I re-arrange the appropriate parts of my mind to instantiate a particular set of relations,

Right, so that is also my view, more or less. The difference is that when it comes to some novel mathematical structure, you might say you have invented some new idea after conducting such a rearranging, I would say that I have discovered something that exists independently of my mind. They way it gets "in here" is the same in both cases. We just have different attitudes about how to describe what has happened.

> Otherwise, what is it that gives the inference in the case of Platonism its necessity?

On discovering a well-defined consistent abstract structure, I infer from the definition of "existence" as used by platonists that it exists. Of course, one also infers by analysis that the structure is well-defined and consistent in the first place.

> Regarding inferring the existence of a given structure: can you be wrong about whether it exists?

On platonism, the existence of a structure is equivalent to that structure being well-defined and consistent. One might believe that there is a greatest prime number, but one would be mistaken.

> there is no effect that the pattern has on you

Well, arguably. Although I have seen Dennett make the point that an abstract pattern can at least have the effect on you of allowing you to imagine and think about it and even act on it in some way.

Hi DM,

DeleteI just find it strange that you'd apparently have no problem calling a chair physical, while the pentagram isn't---a chair is as much a pattern of sticks as the pentagram is a pattern of stones, it seems to me.

>I would say that I have discovered something that exists independently of my mind.

But what does this independently existing thing actually do for me? If it's not responsible for my ability to coming up with some new idea, then it seems I could just cross it out of my ontology, without noticing anything missing...

So, what are those other Platonic universes good for? Couldn't I do without them, even if I accepted that our reality is mathematical? What's the difference between a Platonic world in which only computation A exists, and one in which there is also computation B?

And if, at least as far as entities within computation A are concerned, there is no difference, then how can the existence of these other computations conceivably help solve any philosophical problems arising within A (or our everyday world, for that matter)?

Hi Jochen,

Delete> I just find it strange that you'd apparently have no problem calling a chair physical, while the pentagram isn't

This is not my problem, it is the problem of some physicalists. Some physicalists want to cling to the idea that things are made of atoms. They're not happy with wavefunction ontology and would prefer interpretations of QM where wavefunctions collapse and everything is ultimately made out of billiard balls most of the time. If you're subscribing to wavefunction ontology, you're already half way there to the MUH.

> But what does this independently existing thing actually do for me?

It's not what it does for you. It's what should your attitude be to its existence, i.e. the existence of the idea you have come up with.

> What's the difference between a Platonic world in which only computation A exists, and one in which there is also computation B?

A has this meaningless, unmeasurable pseudo-property called "existence" and B does not. I would eliminate that distinction which I regard as nonsense from my world view, putting A and B on the same footing.

> how can the existence of these other computations conceivably help solve any philosophical problems arising within A

It resolves the pixies problem, as I have illustrated. It also explains how and why our world exists, and the nature of its existence. Our world exists because there is a mathematical object isomorphic to our world -- our world is just that mathematical object and nothing more. It exists in the way that mathematical objects exist. If our world exists, then all possible worlds exist. Equivalently, if mathematical objects do not exist, then our world exists. Or equivalently yet again, we can just throw out the whole concept of existence and ontology as meaningless.

Meant to say

Delete"Equivalently, if mathematical objects do not exist, then our world DOES NOT exist."

Hi DM,

Delete>This is not my problem, it is the problem of some physicalists.

Well, it's you who wants to maintain that some patterns of physical things aren't themselves physical things, while others are.

>If you're subscribing to wavefunction ontology, you're already half way there to the MUH.

Nah, not really. Even if you subscribe to wave-function epistemicism, then this just means you believe the mathematics to be a predictive tool with no ultimate physical reality; but that doesn't entail that the math is really ontologically fundamental.

>A has this meaningless, unmeasurable pseudo-property called "existence" and B does not.

The point is rather that, as far as I understand your Platonism, B's existence has no impact on A---meaning, in particular, that whether B exists or not has no bearing on the reality of the pixies.

So imagine that I, within computation A, instantiate some computation B. If it were the case that I can only do so if computation B has its own existence, then I think your argumentation might work---then, me instantiating that computation simply opens up a window into a different computational universe, so to speak. But if there's nothing that B's Platonic existence does in order to enable me to instantiate computation B within A, then I can just as well disregard it, and hence, the pixies are just as well part of A as they are in the original Putnam/Searle argumentation.

In other words, if B doesn't matter to A, then you've got the same problem as before; and if it is to matter, then you'd need some way for it to assert itself, make itself felt within A, so to speak.

>If our world exists, then all possible worlds exist.

They may exist, but whether they do has no impact on our world, so I can equally well consider the case in which they don't---meaning that the pixie problem isn't solved.

(I could, for instance, also come up with an inconsistent idea, that has no existence according to your Platonism---but I can come up with that idea, and maybe not recognize its inconsistency, just as well. So Platonic existence has no bearing on which ideas I can come up with, and, by extension, which computations I can perform. Hence, the computations are in here, and not out there.)

Hi Jochen,

Delete>

Well, it's you who wants to maintain that some patterns of physical things aren't themselves physical things, while others are.

No. You misunderstand me. I'm allowing for the sake of argument that physical things are made of little billiard balls the way that some physicalists think they are. I'm then saying that the mind is not such a thing. That's all.

> but that doesn't entail that the math is really ontologically fundamental.

If you think that things are not really little billiard balls but fluctuations in a field, then you think that the basis of reality is pretty abstract and mathematical already.

> in particular, that whether B exists or not has no bearing on the reality of the pixies.

Asking whether or not B exists misunderstands platonism, because B's existence is not contingent. We are not wondering whether it exists, we are wondering whether we should deem it to exist -- if we do, if we choose to use a concept of existence that includes mathematical objects, then it exists necessarily.

> But if there's nothing that B's Platonic existence does in order to enable me to instantiate computation B within A

The existence of B is one and the same as B being coherent, consistent and well-defined. If B were not coherent, consistent and well-defined, then it could not be instantiated within A.

> then you'd need some way for it to assert itself, make itself felt within A, so to speak.

Ex hypothesi, the pixies are causally isolated from the rest of us. So there is no need for B to make itself felt within A, and no reason to consider the pixies to be part of A. There is only reason to deem B to be part of A if we build an I/O device so that we can interact with them. This is tantamount to building a supercomputer to simulate B.

> They may exist, but whether they do has no impact on our world, so I can equally well consider the case in which they don't

The MUH is just saying that our world is just one of the possible worlds, that from an objective view it is no realer and has the same ontological status. If you want to say that the other possible worlds don't exist, then you're right that they don't from our perspective, but then we don't exist from theirs. If you mean it in an objective sense, then if they don't exist we don't either. As long as you are symmetric in what you say on the subject, then I don't have a problem with how you say it.

> So Platonic existence has no bearing on which ideas I can come up with, and, by extension, which computations I can perform.

It does. Inconsistent ideas can only exist when they can hide in vagueness and ambiguity, something computers force you to expunge when programming. You cannot really program an inconsistent idea into a computer. I can't write a computer program to draw a round square, for instance. But I would say you cannot really fully conceive of a round square even in your imagination. You can only conceive of the concept of a round square, if that makes any sense to you. We can take such descriptions of impossible objects as objects in themselves. Those descriptions are possible and can be realised even if the things they describe cannot.

Hi DM,

Delete>I'm allowing for the sake of argument that physical things are made of little billiard balls the way that some physicalists think they are. I'm then saying that the mind is not such a thing.

Of course, but the argument only works if some patterns of physical things are not themselves physical things. Otherwise, one can still hold that a mind is a physical thing in the same way as a chair is---as a patterned arrangement of physical things. That Searle instantiates two minds then just means that some of the same objects are parts of more than one larger object, which strikes me as wholly uncontroversial.

>If you think that things are not really little billiard balls but fluctuations in a field, then you think that the basis of reality is pretty abstract and mathematical already.

Abstract, perhaps (at least from the point of view of things we're familiar with in everyday life), but by no means necessarily mathematical. Actually, a quantum field is not in itself any more mysterious than a billard ball---both would be, if they are ontologically fundamental, just brute things not further analyzable.

>Asking whether or not B exists misunderstands platonism, because B's existence is not contingent.

If Platonism is right, then that's true; but Platonism may be wrong. You're positing Platonism in order to help with the pixie problem; but then, accepting Platonism is contingent on whether it actually helps with that problem. Consequently, you can't refer back to the truth of Platonism to substantiate the notion that B already exists.

But if B's existence does not change anything about A, then it doesn't help with the pixie problem---the situation 'within A' is the same regardless. Hence, if A has a pixie problem if B doesn't exist, and B's existence has no consequences for A, it also does if B exists, and consequently, Platonism doesn't help.

>there is no need for B to make itself felt within A, and no reason to consider the pixies to be part of A

A, without Platonism---i.e. in the case where it's just a single, solitary reality made of whatever---has a pixie problem; I take it we agree on that. I could see some scope for ameliorating that problem if it were the case that I could instantiate computation B within A if and only if B is an independently existing Platonic reality; but I can do so, on your conception of Platonism, whether it is or not.

Hence, I'm actually in the same situation within A, whether B exists or not. Consequently, Platonism doesn't do anything to change the situation.

(Besides, you can trivially program an inconsistent computer---simply because there exist inconsistent formal systems. Any computer enumerating the theorems of such a formal system will then itself be inconsistent. Indeed, it's generally making sure that there is no inconsistency that's the problem, as demonstrated by the various limitating theorems in logic.)

Hi Jochen,

Delete> Otherwise, one can still hold that a mind is a physical thing in the same way as a chair is

I think we should drop this line of discussion as I'm not sure we are really disagreeing. I'm assuming things for the sake of argument that neither of us really believe. I'm arguing against people who have an ontology where the only things that exist are particles, who say that patterns don't really exist and are just projected by the mind onto nature.

> If Platonism is right, then that's true; but Platonism may be wrong.

My position is that platonism cannot be wrong. Platonism, for me, is a convention or attitude -- the position that we ought to deem mathematical objects to exist. There is no fact of the matter in question, just how we ought to talk and think about mathematical objects. There can be no fact of the matter because we can't say whether mathematical objects actually exist without first clarifying what we mean by actual existence. Platonists mean one thing and anti-platonists mean another.

> You're positing Platonism in order to help with the pixie problem

No, I'm positing the MUH to explain the pixie problem (and a host of other issues). Unlike platonism, the MUH is a factual claim -- the claim that this universe is a mathematical object. Not all platonists are MUH-ists. While I imagine all MUH-ists are platonists, one need not be. Accepting the MUH while denying platonism is consistent -- it amounts to the claim that this world does not exist. One can also deny the idea that objective existence is a coherent concept and so throw out the question of platonism as meaningless. This latter position is actually quite attractive to me.

> Any computer enumerating the theorems of such a formal system will then itself be inconsistent.

No it won't. It will be consistently enumerating the theorems of an inconsistent formal system. It will not itself be the inconsistent formal system and it will not itself be inconsistent.

I gave an example of something inconsistent a computer cannot do (graph a round square). Other examples include calculating the largest prime (in a finite time), telling if an arbitrary algorithm will terminate (in a finite time), computing the real square root of a negative number, etc.

Once you have written an algorithm to do something, the behaviour of that algorithm is defined, and the computation it carries out is consistent. If that computation is supposed to be "about" something else, then that something else may not be consistent. But the computation itself is.

I think we're just quibbling over what it means for a computation to be consistent, but I think the examples should explain what I mean. My ontology does not include a real square root of a negative number or round squares. It does include any algorithm that can be programmed into a computer. It does not include any algorithms to compute uncomputable things or give "correct" answers to incoherent questions which have no correct answers.

Hi DM,

Delete>My position is that platonism cannot be wrong.

Well, hopefully not: a position that can't be wrong is logically trivial. It doesn't carve out a subset from all possibilities, so we're left with the same options regardless.

But luckily, your Platonism can be wrong in several ways: there could be objects that are non-mathematical, in addition to the mathematical ones, and there could be only such objects, with mathematics simply being stories we tell.

In fact, I think that's a good analogy: nobody (well, probably not nobody, there's always somebody who came up with any position you can imagine) holds that because we can tell stories, the objects of our stories must necessarily exist. There is no 'story-Platonism'. Some stories are about stuff that actually exists, and some are fictional---likewise, with mathematics: some mathematics describes actually existing structures, some doesn't. But the reality lies not with the math, but with the structures, as instantiated by concrete physical objects.

>There is no fact of the matter in question, just how we ought to talk and think about mathematical objects.

Hmm, so that's kind of a constructivist Platonism, then? ;)

>No, I'm positing the MUH to explain the pixie problem (and a host of other issues).

Apologies---I had somewhat tersely been using 'Platonism' or 'your Platonism' as the radical version where only mathematical entities exist. In any case, my observation above still holds: if it doesn't matter whether computation B exists independently, then there's no ground gained regarding the pixies.

> It will be consistently enumerating the theorems of an inconsistent formal system.

Then I'm afraid you'll have to explain your notion of consistency. To me, consistency means 'proves no contradictions', and a computer enumerating an inconsistent formal system is a device that produces proofs, which contain contradictions, and hence, it's inconsistent.

In particular, such a computer could easily prove that '11 is the largest prime'. That's of course wrong, but it'll follow as a theorem from every inconsistent formal system (by explosion)---indeed, that such things follow is exactly what it means for a system to be inconsistent.

Of course, you could not have a computer be inconsistent and truthful (sound)---but that goes likewise for a formal system, or a human being, provided that there are no true contradictions.

Granted, a computer will always perfectly behave according to its program---but I wouldn't call that consistency: likewise, all theorems of a formal system are derived perfectly according to its rules of inference, and even a human being may work according to fixed rules, yet we'd generally allow for such things to be inconsistent. Indeed, in a deterministic universe, on your definition, inconsistency would be vacuous. (Also, you'd have to explain how the human mind can be inconsistent and a computation, if no computation can be inconsistent.)

Hi Jochen,

Delete> Well, hopefully not: a position that can't be wrong is logically trivial.

I think platonism is logically trivial. I think platonism is a matter of convention. However, some ideas that may follow from thinking platonically (e.g. the MUH) are not logically trivial.

> nobody ... holds that because we can tell stories, the objects of our stories must necessarily exist.

Right. Because the objects of our stories are usually too vague or ambiguous or incoherent to make sense of the proposition that they exist. But the stories themselves exist platonically, I would say. Even if only as strings of characters in Borges' library, but also on higher levels of description in my view.

> with mathematics: some mathematics describes actually existing structures, some doesn't.

With mathematics, the structure is the story. I think stories exist.

> if it doesn't matter whether computation B exists independently, then there's no ground gained regarding the pixies.

Again, on platonism, computation B necessarily exists. The question is whether this universe is such a computation/mathematical object. If the MUH is true, then it is. If the MUH is true, then pixies are not a problem.

> In particular, such a computer could easily prove that '11 is the largest prime'.

In my view, a computer could indeed derive that 11 is the largest prime from inconsistent axioms. But it couldn't *prove* it. Such a derivation is not a proof, because it is derived from inconsistent axioms. A computer program that could prove that 11 is the largest prime would be one that derived this result from consistent axioms. This is the kind of inconsistent program I claim does not exist.

> Of course, you could not have a computer be inconsistent and truthful (sound)

That's more or less all I'm saying. The existence of a computer program is just the possibility of writing that program. The existence of a mathematical object is just the possibility of defining it consistently and unambiguously. A computer program (more properly an algorithm) is just a mathematical object that is defined by writing an algorithm.

> but I wouldn't call that consistency:

Well, that's fair enough. In that case we'll have to think of some other suggestion for what I'm trying to say. Provided you understand me now, any suggestions?

> Also, you'd have to explain how the human mind can be inconsistent and a computation

What the human mind is doing can be defined consistently. It's not doing anything incoherent such as computing the largest prime. But what it is doing may be to realise a person who is confused and thinking inconsistently, e.g. believing there to be a largest prime. Similarly, I might write a computer program which claims to know what the largest prime number is, and even which behaves as if it knows but wants to keep it a secret. It may internally represent the belief that 11 is the largest prime. But it cannot actually justify that belief, because that is impossible, and so it cannot actually know what the largest prime is, because there is no such thing.

Hi DM,

Delete>I think platonism is logically trivial.

I don't think it can be: on whatever notion of existence you adopt, it may a priori be the case that only mathematical objects exist, mathematical and non-mathematical objects exist, or only non-mathematical objects exist. So saying that Platonism is trivial amounts to saying that only notions of existence such that mathematical objects exist are admissible---which is a non-trivial claim. (And, I would argue, simply a confused usage of the term 'exist', analogous to what Quine pointed out in 'On What There Is'.)

>If the MUH is true, then pixies are not a problem.

The problem with that I pointed out above still stands, though: if it doesn't matter for A whether B exists, then as far as that is concerned, a world where only A exists and a world where A and B exist are identical; thus, the added existence of B makes no headway towards resolving the pixie problem.

>Such a derivation is not a proof, because it is derived from inconsistent axioms.

This leaves you with a completely ill-defined notion of proof, though: theories strong enough to formalize arithmetic can't prove their own consistency, and thus, you'd have to rely on stronger theories to prove it (well, not necessarily stronger, but extending/encompassing them at least with respect to certain kinds of sentences); but then, those stronger theories can't prove their own consistency, leaving it (on your notion) open whether the proof of consistency of the weaker theory actually is a proof, and so on.

In fact, for any such theory, such as Peano arithmetic (PA), both PA + Con(PA) (where Con(PA) is the sentence asserting the consistency of PA) and PA + ~Con(PA) (where ~ is negation) are consistent theories, so you can prove both PA's consistency and its inconsistency with these stronger theories.

Matters aren't better for weaker theories: while they can prove their own consistency, on your notion, that would only be a proof if they are consistent, thus running into circularity.

So I think it's better to use the usual, syntactic notion of proof, where it's just a derivation from the axioms following the derivation rules. Otherwise, it's simply impossible to formalize the notion of proof, and then, there just flat aren't any computers producing proofs.

>Provided you understand me now, any suggestions?

To me, what you're saying just amounts to saying that computers compute, so I think we can just leave it at that.

>It's not doing anything incoherent such as computing the largest prime.

I should note here that whether there is something like a largest prime depends on the axioms you adopt: within a finite field, or within all of mathematics if you adopt ultrafinitism, there actually is such an object. So you have to relativise these kinds of statements: according to, say, the Peano axioms, there is no largest prime. But then, this just becomes tautological: for anything you prove, you'll have to make assumptions at least as strong.

Hi Jochen,

Delete> So saying that Platonism is trivial amounts to saying that only notions of existence such that mathematical objects exist are admissible

That is not my claim. I claim that platonism is trivial not because only the platonic notion of existence is admissable, but because platonism is a matter of language and convention and I happen to find it a useful convention so I recommend that it be adopted. It's a preference.

> The problem with that I pointed out above still stands, though

Genuinely, I don't see the problem you are pointing out.

> if it doesn't matter for A whether B exists

OK, but "whether B exists" is an odd construction, because B exists necessarily in the platonic sense. This is like saying "It doesn't matter for A whether true is true".

> then as far as that is concerned, a world where only A exists and a world where A and B exist are identical;

A world where only A exists is logically impossible on a platonistic account of existence.

> thus, the added existence of B makes no headway towards resolving the pixie problem.

So the added existence of B is not "added" at all. It is necessary. The pixies argument only allows us to conclude that the pixies exist. The MUH also allows us to conclude that the pixies exist. The DwP argument is therefore no kind of rebuttal of my view, but on the MUH I can at least make the case that it is more appropriate to deem the pixies to live in their own universe rather than ours.

> This leaves you with a completely ill-defined notion of proof,

No it doesn't. I will accept as a proof any theorem derived from axioms I believe to be consistent. I can't prove that a proof is a proof, but I can distinguish between a derivation from inconsistent axioms and a derivation from axioms I believe to be consistent.

> So I think it's better to use the usual, syntactic notion of proof, where it's just a derivation from the axioms following the derivation rules.

While this is a defensible convention to adopt when speaking formally, I don't think it matches what people usually mean when they say "prove". So, for me at least, I don't think a proof that 11 is the greatest prime number is really a proof. So when I say it is impossible to prove that 11 is the greatest prime number, I mean it is impossible to do so given axioms that are plausibly consistent, such as the peano axioms.

> To me, what you're saying just amounts to saying that computers compute, so I think we can just leave it at that.

OK. It doesn't seem to be enough to me though. I mean, I want to explain what I think exists. I don't think there exists an algorithm that will find the greatest prime number (on the peano axioms at least). Clearly there is no such algorithm. So I can't just say every conceivable algorithm exists. That's too vague, because people might be able to vaguely conceive of an algorithm to find the greatest prime number. We can drop it for this conversation as I think we understand each other, but I think it is something I will need to explain in future again.

> So you have to relativise these kinds of statements: according to, say, the Peano axioms, there is no largest prime

You can take me to be discussing standard arithmetic (e.g. Peano axioms) unless I say otherwise. It's tedious to have to say you're using the peano axioms every time you want to say something about numbers.

That said, I don't think this mathematical system is any realer than other consistent systems. I'm a plenitudinous platonist, which just means that I treat all consistent axiomatic systems equally.

Hi DM,

Delete>platonism is a matter of language and convention

I really don't think it is. Platonism is an ontological claim, and ontology doesn't reduce to language and convention.

>Genuinely, I don't see the problem you are pointing out.

OK, let me try again. First of all, note that even if Platonism is trivial, the MUH isn't, and whether it's true, or even sensible, is indeed part of what we're discussing---so it won't do for you to hold that some computation B exists necessarily on the MUH, as that's question begging.

So, basically, we agree that in the case where just computation A exists---meaning, e.g., the case in which a sufficiently advanced civilization simulates our universe, and nothing else, if you want some window dressing---then there is a pixie problem.

(Actually, I just want to reiterate that I don't think the pixies are the real problem: rather, the problem is that which computation is being carried out is not an objective fact. But I'll continue to use 'pixie problem' as a shorthand for a vulnerability to the Searle/Putnam/Bishop difficulties.)

Now, if it were the case that on your construal of the MUH/radical Platonism, the computation B, performed within A, needs the abstract existence of the computation B---such that without such abstract existence, computation B could not be performed within A---I'd agree that the MUH gives us a fresh scope on the problem: then, the pixies exist anyhow, in their own little pocket universe, and instantiating B within A just provides a window into that universe.

However, such is not the case, at least not if I understand your take on the MUH correctly: B's abstract existence is not a precondition to implementing B within A. That means that for this implementation, B's abstract existence is a contingent fact: whether it exists or not does not impinge on the possibility of implementing B within A.

But then, the cases in which only A exists, and where both A and B exist, are exactly equivalent with respect to the question of implementing B. Hence, there's no difference to A in these cases---but since in the case where only A exists, there is a pixie problem, the same problem persists when B exists, as well. From the point of view of the pixie problem, B is ontological excess baggage, and hence, postulating its existence (i.e. the MUH) doesn't address the problem at all.

>I will accept as a proof any theorem derived from axioms I believe to be consistent.

DeleteWell, that plays havoc with all the usual properties of the notion of 'proof', though. For one, your belief in a proven theorem then is contingent on your belief in the axiom's consistency---and thus, its justification only as good as the justification of the latter belief. Two people could, thus, validly disagree on whether something is actually a proof, by merely differing in their belief in the consistency of the axioms. So in the end, you're really no better off than if you'd just believed the 'proven' theorem in the first place---cutting out the ill-defined middleman of 'proof'.

Furthermore, proofs on this conception cease to be deductive---there is no deductive justification of the consistency of a given set of axioms in the general case, so you're stuck with some kind of quasi-empirical or inductive justification.

Plus of course it throws out the past century of proof theory, and so on. So in the end, I really wouldn't want to use the word 'proof' for your concept---it carries virtually none of the meaning it has in the usual mathematical concept, and thus, is all too amenable to confusion.

>I'm a plenitudinous platonist, which just means that I treat all consistent axiomatic systems equally.

Out of curiosity, what about the systems PA, PA + Cons(PA), and PA + ~Cons(PA)? If PA is consistent, then it can't prove Cons(PA); hence, both that and its negation can be added to PA to yield new consistent systems---however, in PA + ~Cons(PA), the inconsistency of PA can be trivially proven.

Hi Jochen,

Delete> I really don't think it is. Platonism is an ontological claim, and ontology doesn't reduce to language and convention.

That is the prevailing view. It is not my view. On my view, ontology is entirely a matter of convention. There is no fact of the matter on whether anything "exists". You need to clarify what you mean by "exists" first. I think that the concept of objective observer-independent existence is incoherent unless understood as a linguistic convention.

> -so it won't do for you to hold that some computation B exists necessarily on the MUH

But I don't need the MUH to say that computation B exists. All I need for this is platonism.

The meaningful claims I'm making are

a) that a mind is a mathematical structure

b) that the universe is a mathematical structure.

The existence of B is trivial.

> So, basically, we agree that in the case where just computation A exists

This scenario is incompatible with how platonists speak of existence. You mean something else by existence than me. We're not talking the same language, so I can't make sense of what you are saying.

Incidentally, I don't require an advanced simulation to simulate A for A to exist. A exists whether or not it is simulated. This means, for instance, that I regard the question of whether we are in a simulation as meaningless. This universe is surely being simulated somewhere in the multiverse, but it also exists as a mathematical object in its own right. I don't distinguish the copy of me in a simulation from the copy of me in that mathematical object. Since the two are isomorphic, I regard them as the same object. If the simulators pull the plug, I will continue to exist, so for all intents and purposes there is no reason to deem myself to be within a simulation.

> B's abstract existence is not a precondition to implementing B within A.

It doesn't matter whether it is a precondition. It is necessary.

> Well, that plays havoc with all the usual properties of the notion of 'proof', though.

DeleteIt depends what you mean by the usual notion of proof. If you are speaking of the formal mathematical sense where a proof is something derived from a set of axioms, then OK. But if you are talking in the epistemic sense that most people use the term, then it is an utterly convincing demonstration of some fact. This is the sense in which I was using the term.

I will not be convinced that 11 is the greatest prime number when presented with a derivation from inconsistent axioms. There is no proof in this epistemic sense.

Again, just take me to be talking about the Peano axioms, Euclidean geometry, etc unless otherwise stated. I acknowledge the technical, formal accuracy of your point, but again I feel it is quibbling. To avoid such ambiguity, I prefer to reserve "proof" for derivations from axioms we believe to be consistent and "derivation" otherwise.

> Out of curiosity, what about the systems PA, PA + Cons(PA), and PA + ~Cons(PA)?

I accept PA and PA + Cons(PA), provided PA is consistent as I believe it to be (but cannot prove). I reject PA + ~Cons(PA) either way.

> If PA is consistent, then it can't prove Cons(PA); hence, both that and its negation can be added to PA to yield new consistent systems

I don't think this is right. I think there is a fact of the matter on whether PA is consistent or not, even if we can't prove it. If PA is consistent, then PA + Cons(PA) is also consistent. If PA is not consistent, then PA + Cons(PA) is not consistent. If PA is not consistent, then PA + ~Cons(PA) is not consistent because PA is not consistent. And if it is consistent, then PA + ~Cons(PA) is consistent because ~Cons(PA) is not consistent.

On that last point, I realise I am probably being too simplistic. I may need to work on my idea of what I mean by consistency. I see that it doesn't look like I ought to be able to derive a contradiction from PA + ~Cons(PA) if PA is consistent, but there does seem to be something very obviously incoherent about this set of axioms, such that I wouldn't regard it as a legitimate basis for a mathematical object.

DeleteHi DM,

Delete>On my view, ontology is entirely a matter of convention.

I don't see how that could possibly work---in order to come to a convention, a language community (say) must first of all exist; hence, there's a notion of existence independent of whatever that language community comes up with.

>The existence of B is trivial.

I don't believe it is. The universe where Platonism is false, and only computation A exists---say, one where only physical things exist, and A is simulated on some computer; or even where A is the physical universe, enjoying a solitary existence---isn't incoherent.

Furthermore, the point is precisely that even if B exists, it doesn't get you out of trouble, because the fact of B's existence doesn't do anything to address the pixie problem---it's just tacked on, but inert, as far as the instantiation of computation B within A is concerned.

>It doesn't matter whether it is a precondition. It is necessary.

It's not necessary to the computation B being instantiated within A.

>I acknowledge the technical, formal accuracy of your point, but again I feel it is quibbling.

I think that, perhaps unfortunately, this quibbling is necessary to really make precise what we're talking about---otherwise, the discussion just dissolves into vagueness.

Regarding PA + ~Cons(PA), the point is exactly that it's consistent if (and only if) PA is consistent; so if you accept consistent mathematical structures within your ontology, then you should accept that one, too. In the end, it's not any different from, say, the distinction between rational and real numbers: within the former, you can prove the nonexistence of the square root of two, while within the latter, it obviously exists. Within the former structure, the proposition 'there exists an element e such that e*e=2' is false, in the latter, it's true.

I agree it's a bit hard to wrap one's mind around, and both space and ability prohibits me from trying a full explanation here, but if you're interested, I'd recommend Torkel Franzén's excellent little book 'Gödel's Theorem: An Incomplete Guide to its Use and Abuse', which gets into this matter in chapter 7.

Hi Jochen,

Delete> I don't see how that could possibly work---in order to come to a convention, a language community (say) must first of all exist;

This is going to sound facetious: but a fictional non-existent community might also come to such a convention, so I don't really agree. We can't really use the fact that we have such a convention as evidence that we exist, because a community in some other possible world might have done the same. The only way we can say this world really objectively exists is by adopting a convention that all possible worlds exist.

So I claim that the attractive, intuitive idea of objective existence is incoherent unless clarified and explained within the framework of such a convention.

> The universe where Platonism is false

Platonism is not contingent. It is true or false based on how you think of existence. It does not make sense to posit a universe where Platonism is false. That's like positing a universe where 2=3. It is incoherent.

> Regarding PA + ~Cons(PA), the point is exactly that it's consistent if (and only if) PA is consistent;

I find it hard to accept that this is consistent, because it is a structure that asserts its own inconsistency. If this does not meet the formal definition of inconsistency then I might need to use a different term, but there does seem to be something incoherent about axiomatic systems which assert their own inconsistency.

I feel it should always be permissable to add an axiom asserting the consistency of a system -- this can't lead to inconsistency unless the system is already inconsistent, so it can't do any harm. So, given PA, I can just add Cons(PA). But this operation yields a contradiction when given PA + ~Cons(PA).

It might be sufficent to add an additional rule: axiomatic systems which define real mathematical objects must be consistent and must not take their own inconsistency as an axiom.

I'm not an expert on this kind of subject matter so I would have to think and research more on it to be able to give a good answer, but this is where my intuition leads me for now. While I accept that you know a lot more about it than I do, given how differently we feel about subjects we both understand reasonably well, I feel it likely that I would disagree with you even having studied further.

But, since I'm not really qualified to discuss it, and since it's a side issue, I think it might be best to drop it.

Hi DM,

Delete>a fictional non-existent community might also come to such a convention

No. First of all, fictional communities don't come to any conventions, the same way they don't produce ecological disasters, overpopulation, or thriving real-estate markets.

Second, and more importantly, even if they did, this wouldn't help you: to define 'fictional' you first need a notion of existence, as fictional just means that which doesn't exist. So no matter what convention the fictional community adopts, there must already be a notion of existence in place in order for them to *be* fictional. (Otherwise, how do you know they're not real?)

>Platonism is not contingent. It is true or false based on how you think of existence.

That seems contradictory to me: if it's true or false based on how I think about existence, then it's contingent. (Compare: 'there is a square root of two' is true or false depending which set of axioms I adopt; consequently, it's contingent on that.)

>axiomatic systems which define real mathematical objects must be consistent and must not take their own inconsistency as an axiom.

~Cons(PA) does not assert the inconsistency of PA + ~Cons(PA), but of PA. Furthermore, 'consistency' basically only means 'defines a real mathematical object': by Gödel's completeness theorem, every consistent system has a model, i.e. a set of objects obeying the properties laid out in the axioms (such as for instance the natural numbers with addition and multiplication form a model of the Peano axioms). And PA + ~Cons(PA) does have a model (provided PA is consistent).

Hi Jochen,

Delete> First of all, fictional communities don't come to any conventions, the same way they don't produce ecological disasters, overpopulation, or thriving real-estate markets.

I understand what you mean of course, because I understand the intuition that you have that things either exist or don't exist. I have that intuition too. I just think that it is incoherent when considering entities causally disconnected from ourselves, as in the case of possible worlds or mathematical objects.

But you are just making an assertion of that intuition. So I want to say: of course fictional communities can do all those things. Fictional Sherlock Holmes solves cases. Fictional Superman says to himself "I think, therefore I am". "I think, therefore I am" type arguments are therefore insufficient in my eyes to establish that the idea of objective existence works -- although I appreciate their intuitive appeal. All they can establish is "I exist from my perspective".

> Second, and more importantly, even if they did, this wouldn't help you: to define 'fictional' you first need a notion of existence,

Sure. So I'm assuming existence makes sense for the sake of argument. Actually, I would want to reject that distinction. So, no, I don't accept that having a linguistic convention around the concept of existence presupposes existence. You might say that doing anything presupposes existence, but I disagree. I'm not saying that we don't exist, I'm saying that the distinction between things that objectively exist and things that don't objectively exist is nonsense. Everything that can exist does exist in some possible world, and all possible worlds are objectively on the same footing. You could say that I have a maximal ontology, but since there's pretty much nothing I don't include in my ontology (apart from impossible objects such as proofs that 11 is the greatest prime in Peano arithmetic), that's equivalent to saying that the whole idea of ontology is an empty concept and what exists is rather entirely a matter of what we agree to say exists or doesn't exist.

But I think we can set all that aside for now. For the point in hand, it doesn't matter whether we hold all of ontology to be a matter of convention or not. It only matters whether we hold the existence of mathematical objects to be a matter of convention.

> That seems contradictory to me: if it's true or false based on how I think about existence, then it's contingent.

No it isn't. The truth of Pythagoras's theorem is not contingent, though it may depend on whether you are using Euclidean geometry. There is no world where Pythagoras's theorem is false in Euclidean geometry. There is no world where B does not exist on a platonist account of existence.

> there is a square root of two' is true or false depending which set of axioms I adopt; consequently, it's contingent on that.

This is to abuse the meaning of contingent. Mathematical truths are necessary. Specifically, they are necessary given their axioms. There are no possible worlds where these mathematical truths do not hold, though there may be alternative axiomatic systems where superficially similar statements are not true (I say "superficially" similar because we're not quite comparing like with like once we change the axioms -- 2 in one system is not necessarily the same object as 2 in another system, particularly if one has a square root and the other does not).

I think I'll drop talk of PA + ~Cons(PA) for now in the interests of just cutting off one digression too many.

Hi DM,

Delete>All they can establish is "I exist from my perspective".

You're going in circles here: if you exist from your perspective, then your perspective must exist. Superman's perspective, for instance, doesn't: he's never said 'I think, therefore I am' to himself---a writer may have put those signs on a sheet of paper, but the two aren't the same.

>Everything that can exist does exist in some possible world, and all possible worlds are objectively on the same footing.

Well, for one, you don't experience all possible worlds, but merely this one; so they're not all on the same footing: there's one you experience, and others you don't.

>There is no world where B does not exist on a platonist account of existence.

But that wasn't the point. Rather, you claimed that the truth of Platonism is necessary, and that it depends on how I think about existence. Both can't be the case.

And still, the argument I was making that we've digressed from isn't that B doesn't exist, but rather, that the question of whether it does is immaterial for whether there is a pixie problem. Because---speaking counterfactually here---if B did not exist, I could instantiate B within A, and would have a pixie problem; since consequently, whether I instantiate B within A does not depend on whether B exists, its actual existence doesn't play any role at all. Nothing would change if I, by metaphysical fiat, struck B from existence; within A, everything would remain the same. Certainly, whether there's a pixie within A does not depend on whether B abstractly exists if B is causally isolated from A---otherwise, you'd be claiming that this metaphysical striking out of B would suddenly cause a pixie to come into existence within A, hence demonstrating that there is some causal dependence of A on B at all (since doing something to B causes something to happen within A).

Hi Jochen,

DeleteI'd just like to clarify that I don't literally believe in fictional characters such as Superman, but only because fictional characters are too vaguely defined to know really what a label such as "Superman" is supposed to refer to if not to a concept shared by a lot of humans on earth. I'm using them only as stand-ins for one of the hypothetical possible observers in some possible world which you don't think actually exists.

> Superman's perspective, for instance, doesn't

I don't think this assertion is coherent, because I don't think you can give a satisfactory account of what it means for Superman's perspective to exist or not to exist in an objective sense. I have never seen anybody do this. At best they try to explain it using a carousel of synonyms. Superman's perspective does not "exist", which means that it is not "instantiated", which means that it is not "actual", which means that it is not "real", which means that it does not "exist". Might as well be saying it is "foo", which means it is "baz", which means it is "quux", which means that it is "foo".

> Well, for one, you don't experience all possible worlds, but merely this one; so they're not all on the same footing:

Like how I have not met all people, so not all people are objectively on the same footing? I used the word "objectively" for a reason. I'm taking my personal perspective out of the question.

> Rather, you claimed that the truth of Platonism is necessary, and that it depends on how I think about existence. Both can't be the case.

Yes it can, according to what "necessary" means in philosophy. Name any necessary truth at all. That truth will be "contingent" on the meaning of the terms used to express it. How you think about platonism determines what you mean when you use the term "exist". If you think about existence as a platonist does, then the possibility that mathematical objects do not exist is incoherent.

> but rather, that the question of whether it does is immaterial for whether there is a pixie problem.

I understand this, but your argument depends on the assumption that I can entertain the possibility that B does not exist, but I cannot because the existence of B is necessary. If the existence of B is necessary, then there is no question of whether B exists. I cannot compare the possibility where B exists to the impossibility where B does not exist just because the latter is impossible.

Your argument is like saying that just because 2+2=4 (in PA) has nothing to do with the idea that if I put two apples in a bag and then put another two apples in the bag that I should expect to see four apples in the bag. Speaking counterfactually, if 2+2=5 (in PA), then if I put two apples in the bag and then put another two apples in the bag, I would still expect to see four apples in the bag, so obviously what I'm doing with the apples and the bag has nothing to do with 2+2. This argument doesn't work because it is not sensible to consider a counterfactual where 2+2=5 (in PA).

I guess I could go another way with the apples and 2+2 argument. If you were saying something as silly as that, I might as well assert that if 2+2=5 (in PA), then you would expect to see five apples in the bag. This would correspond to countering your "what if B didn't exist" counterfactual with the assertion that you would not be able to instantiate B within A if B did not exist, because the existence of B is just another way of saying that it is possible to instantiate B.

Hi DM,

Delete>I don't think you can give a satisfactory account of what it means for Superman's perspective to exist or not to exist in an objective sense

Quine has provided what seems to me a pretty definitive solution to this problem in "On What There Is": you can identify any putative entity by an ad-hoc-predicate, such as 'supermans'. "There is no thing that supermans" then formalizes, in the sense of Russell's theory of denotation, the usual "Superman doesn't exist", which one would otherwise parse as "there is a thing that is Superman, and which doesn't exist"---which is contradictory. We can handle Superman's perspective in the same way.

>Like how I have not met all people, so not all people are objectively on the same footing?

Exactly: you have direct reason to accept the existence of people you have met, while you have no such reason to accept the existence of that one guy's Canadian girlfriend nobody has ever seen.

>If you think about existence as a platonist does, then the possibility that mathematical objects do not exist is incoherent.

But still, the way a Platonist thinks about existence may be wrong, if they are to say anything coherent at all about the topic of existence.

>our argument depends on the assumption that I can entertain the possibility that B does not exist, but I cannot because the existence of B is necessary.

If that were true, then the technique of indirect proof would be inadmissible: it necessitates entertaining the notion of a logically incoherent object, in order to yield a contradiction (another point I stole from Quine, who talks about 'the round square cupola of Berkeley College')

> if 2+2=5 (in PA), then if I put two apples in the bag and then put another two apples in the bag,

But if 2+2=5 in some (consistent) system of axioms, then there in fact are entities (at least, mathematical ones) such that if you'd put two in the bag, and then another two, then you'd end up with five of them. Those entities won't be apples, since apples can be thought of as a model for the Peano axioms that's equivalent to the standard model, but then you've just used the wrong sort of entities, the wrong model, and your claim has no bearing on whether 2+2 is 'actually' equal to 4 or 5.

Hi Jochen,

DeleteI ought to read Quine, but your account of the resolution is no resolution at all.

> There is no thing that supermans

Is a synonym for

> There exists no thing that supermans

is a synonym for

> Nothing that supermans is instantiated

is a synonym for

> nothing that supermans is real.

is a synonym for

> nothing that supermans is actual

What I'm trying to understand is the meaning of the concept of existence -- in your phrase this would be the meaning of "is" -- what does it mean to be, in an objective sense? I can make sense of all these of solving the conundrum of how we can talk about Superman when there is nothing that supermans on our planet, but I don't see how this deals with the question of what it is for a superman to exist in some possible world. It seems to me that Quine was talking about what it is to exist within our universe and was not discussing whether things might exist in other causally disconnected universes, or if he was then this kind of account doesn't work.

> you have direct reason to accept the existence of people you have met,

Whether I have met them or not doesn't matter unless you're talking about epistemology. But we're talking about ontology. People who I have not met are just as real as people I have met.

> But still, the way a Platonist thinks about existence may be wrong, if they are to say anything coherent at all about the topic of existence.

It doesn't have to be possibly wrong if it is as trivial as I claim it is. I do have meaningful claims about existence -- and that is that platonic existence is ultimately the only kind of existence. But that mathematical objects exist in a platonic sense (or at least in the sense that I use it) is not something that could be wrong -- those who say it could misunderstand what that (or at least my) platonic sense of existence entails.

> it necessitates entertaining the notion of a logically incoherent object,

Yes. Such proofs work by starting from reasonable premises and ultimately arriving at an obvious contradiction, which only shows that there is something wrong with the premises. But you start with an obvious contradiction. We can go no further. As soon as you assume that B might not exist, your premises are invalidated. Back to the drawing board.

> But if 2+2=5 in some (consistent) system of axioms,

So what? I made the analogy to 2+2=5 in PA. You're now talking about something completely different.

Hi DM,

Delete>how this deals with the question of what it is for a superman to exist in some possible world.

What's the difference between our planet and the set of all possible worlds? Both are just collections of things; either (at least) one of them supermans, or none does.

>But we're talking about ontology.

Well, we're talking about things to accept into our ontology. I have grounds to accept this world: I've met it. I have no grounds to accept possible worlds. I've never been there!

>It doesn't have to be possibly wrong if it is as trivial as I claim it is.

It could, for instance, be inconsistent---indeed, I very much think it is: relations without relata aren't relations at all.

>But you start with an obvious contradiction.

First of all, it's really not an 'obvious' contradiction. In fact, almost all philosophers would accept its truth---except for those accepting mathematical Platonism (which I don't think is a large bunch). So at the very least, 'obvious' is putting it too strongly.

Furthermore, I can use obvious contradictions in valid arguments: "if the cupola of Berkeley College is round and square, then it has both a radius and four corners" is perfectly good reasoning. People have even built entire systems of logic based on the idea of contradictions, so-called paraconsistent logic.

But even that isn't what I'm doing: I'm merely deriving a consequence from something that (as you claim) isn't true in any possible world. But so what? Even if there is no Superman in any possible world, it's still true that Superman is from Krypton.

Therefore, this isn't a problem for my argument at all.

Hi Jochen,

Delete> What's the difference between our planet and the set of all possible worlds?

Well, for a start, one is a planet, which is not a set (but I guess could be construed as a set, but of what? Of atoms? Or quarks? Or more macroscopic objects?) and the other is a set of universes.

> either (at least) one of them supermans, or none does.

Well, you'd have to define supermans a bit more carefully for that question to be answerable. I'm not sure the concept of superman is entirely coherent, so it may be a bad example. But there would presumably be something that sherlockholmeses within the set of possible worlds. So does that mean that Sherlock Holmes exists or not? Quine doesn't tell us, as far as I can see.

> Well, we're talking about things to accept into our ontology.

OK, but from an objective perspective, whether I've met someone has no bearing on whether that person exists. That they exist is defined in terms other than whether I've met them.

> It could, for instance, be inconsistent

I guess there are different interpretations of "possibly" here. If you're talking about platonism being inconsistent, this is an epistemic possibility, but epistemology aside, it is either consistent or it isn't, but we may be wrong about which is the case. But I can't conceive of being wrong about platonism any more than I can conceive of the possibility of 2+2=5 being a theorem of PA.

> In fact, almost all philosophers would accept its truth

Either those philosophers are misinterpreting platonism or I am. Either way, it is obvious to me that it is necessarily true that mathematical objects must exist on my account of platonic existence. It may not be obvious to you, but the fact that it is obvious to me means that you achieve your reductio ad absurdum in the very first line of your argument. There is no need to go any further. Anyway, you can't conclude anything from a reductio except that your premises are dubious, which is what I am telling you. The premise that B is contingent is dubious.

But you didn't intend this as a reductio ad absurdum anyway, you intended it to show the equivalence of two different scenarios. Since you didn't intend it as a reductio ad absurdum, you're not really helping your case to compare it to one.

> "if the cupola of Berkeley College is round and square, then it has both a radius and four corners" is perfectly good reasoning.

Not really. How can it have four corners if it is round?

Hi DM,

Delete>Well, you'd have to define supermans a bit more carefully for that question to be answerable.

No, that's the beauty of it: 'supermans' does not have to be further defined; it's an entirely ad-hoc predicate whose sole function is to identify Superman. So 'supermans' is not---and cannot!---be further defined other than saying that anything that supermans, is Superman. So if 'Superman' picks out some individual, so does 'supermans'.

>So does that mean that Sherlock Holmes exists or not? Quine doesn't tell us, as far as I can see.

No, you can't solve ontological questions with the Quinean framework, that's not the point of it (after all, ontological questions are nontrivial, and can't be solved by simple definition). Rather, it yields a way to coherently frame, in particular, questions of nonexistence.

Of course, there also is an answer to the question of what there is in the essay---everything! Because only what is, is, and hence, is part of everything. That's really all that needs saying.

>But I can't conceive of being wrong about platonism any more than I can conceive of the possibility of 2+2=5 being a theorem of PA.

Well, limits of conception rarely yield limits of possibility; and of course, 2+2=5 is a theorem of PA if it is inconsistent. And well, it's consistency, as we have seen, is a subtle question; so whether 2+2=5 is a theorem is exactly as subtle.

>Either way, it is obvious to me that it is necessarily true that mathematical objects must exist on my account of platonic existence.

Lots of things that have at one point or another been declared 'obvious' have turned out wrong, of course. And your justification is circular: certainly, on a Platonic account of existence, mathematical objects necessarily exist---but that just tells you what the Platonic account of existence is, not whether it's right, or even, whether it's consistent.

>The premise that B is contingent is dubious.

It's not a premise of my argument. Rather, the argument is that whether B exists does not change anything about the pixie argument. That's a perfectly consistent argument even if the existence of B is metaphysically necessary.

>Since you didn't intend it as a reductio ad absurdum, you're not really helping your case to compare it to one.

Not what I'm doing; I'm simply pointing out that if it were the case one couldn't validly use falsities in reasoning, then there would be no indirect proofs. Hence, there's a good case to be made for the usefulness of using (what you claim is) a falsity in argument.

>Not really. How can it have four corners if it is round?

It's not round, it's square and round.

Hi Jochen,

Delete> No, that's the beauty of it: 'supermans' does not have to be further defined;

I think in this case I understand you but you do not understand me. I understand the beauty of it. But you don't understand why I think it needs to be further defined.

> So if 'Superman' picks out some individual,

This is why it needs to be further defined. I don't know what it is required for 'Superman' to pick out some individual. Someone simply answering to the name 'Superman' is probably not enough. Someone with superhuman powers answering to the name 'Superman' is probably still not enough. Etc. I mean, if we actually found some fossil miocene horse species that had a horn, would we say this was actually a unicorn or not? We need to define what it means for something "to unicorn" to answer that question.

> Rather, it yields a way to coherently frame, in particular, questions of nonexistence.

Right, so it has nothing to bear on this argument, since I don't ever (I don't think?) make the argument that you can't say that something doesn't exist without assuming that it exists. Anti-platonism is coherent to me. I just prefer platonism. I understand both not as positions on a matter of fact but as linguistic conventions.

> and of course, 2+2=5 is a theorem of PA if it is inconsistent.

Yes. I realised after I posted this that to capture my intent I should have added "... if PA is consistent".

> And your justification is circular: certainly, on a Platonic account of existence, mathematical objects necessarily exist

It is circular, because it is just a linguistic definitions. If "foo" is to be a cat, then my cat foos. This ought not be a problematic position. But you start your argument by assuming that whether my cat foos is contingent.

> It's not a premise of my argument.

It is. As soon as you consider alternative scenarios, one in which B exists and one in which it does not, then you are assuming that B is contingent.

> I'm simply pointing out that if it were the case one couldn't validly use falsities in reasoning

I think you mean contradiction rather than falsity, as I would never have any issue with you considering a merely counterfactual scenario. I only have an issue with you trying to draw conclusions from an incoherent scenario, as in the scenario where you consider the possibility that B might not exist in the platonic sense and yet B might be a possible algorithm.

All you can conclude from a contradiction or an incoherent scenario is that your premises are wrong. You can't validly reason from a contradiction. Once you have shown a contradiction, you go no further in reasoning from those premises. Contradictions are useful for that specific purpose and I'm not sure for much else. Certainly in this specific case your argument doesn't work because it starts in incoherence.

> It's not round, it's square and round.

If it is square and round, then it is indeed round. Round things do not have corners. It is also square. Square things do have corners. It must have no corners and it must have corners. This is why it is inconsistent, and why you cannot sensibly deduce anything from its definition.

Hi DM,

Delete>But you don't understand why I think it needs to be further defined.

If you think it needs to be further defined, then I don't think you quite get the point (to join in to the accusations of who doesn't understand what): it is exactly its primitive nature that allows it to do its job, so defining it further would defeat its purpose.

>I mean, if we actually found some fossil miocene horse species that had a horn, would we say this was actually a unicorn or not?

Beside the point. If we do, then it unicorns---not the other way around.

>Anti-platonism is coherent to me. I just prefer platonism.

How does that square with Platonism being necessarily true?

>It is circular, because it is just a linguistic definitions.

But linguistic definitions don't tell us anything about existence---or indeed, anything about any factual matter at all. What you're saying is basically, Platonism is true if Platonism is true---right, but empty of all content.

>As soon as you consider alternative scenarios, one in which B exists and one in which it does not, then you are assuming that B is contingent.

I'm not assuming that B is contingent. I'm not deriving consequences from B's nonexistence, I'm thinking about what would be the case if B were not to exist---which doesn't depend on whether B exists, or even necessarily exists.

And this is where you basically assume what Quine argues against: that to talk about nonexistent entities, you have to grant them existence---or, in your case, the other way around, that to talk about the nonexistence of existent entities is tantamount to assuming their nonexistence. This simply doesn't follow.

Even if Superman exists in every possible world, when he saves Lois falling from the top of the Bugle tower, I can say that without Superman, Lois would have fallen to her death. This isn't made incoherent by the fact that Superman exists necessarily. Really, this is completely uncontroversial.

Or, suppose there really is only one possible world, ours, with all others containing subtle contradictions and hence, being impossible: then, you would bar talk about every object that doesn't actually exist in our world, which is a ludicrous position.

>I think you mean contradiction rather than falsity,

Well, falsity and contradiction are often not distinguished---strictly speaking, a contradiction is any proposition that yields falsity upon truth evaluation, but it's usual to ellide that difference.

>You can't validly reason from a contradiction.

First of all, it's only your position that it is a contradiction---and one I'd imagine to be highly contentious. So you saying that I can't use this in an argument is basically you saying that since your view is necessarily right, any other view is a contradiction, and hence, false. Additionally, it's exactly whether your view is right that we're trying to suss out, so your branding anything that yields its falsity a contradiction simply is circular.

>Round things do not have corners.

Square things do, hence square and round things do. So anything that is square and round must have corners, and not have corners---a contradiction, surely, but one that logically follows from the original stipulation that something can be round and square from valid reasoning.

Hi Jochen,

Delete> it is exactly its primitive nature that allows it to do its job, so defining it further would defeat its purpose.

The beauty of it is that it doesn't need to be defined, because if Superman the noun is meaningful then so is superman the verb. I understand this, believe me.

The problem I have is that Superman the noun is not particularly meaningful. I'm not criticising Quine or you here, I'm making a general point about the problem of making sense of claims that fictional (as opposed to mathematical) entities exist. This is not an argument against your view, it is just an observation. Fictional entities are generally too vague to be real or unreal, in that it is hard to define a set of criteria that would universally be accepted as making a claim that such an entity exists true.

> Beside the point. If we do, then it unicorns---not the other way around.

It's not beside the point. This is just the point I'm making. "If we do" is the problem. We can't say whether we do or not. There is no consensus on what it would take for something to be deemed a unicorn as opposed to a horned equine.

> But linguistic definitions don't tell us anything about existence

That assumes that existence is a coherent concept about a factual matter. I don't think it is, at least not objective observer-independent existence.

> What you're saying is basically, Platonism is true if Platonism is true---right, but empty of all content.

If platonism is just a linguistic convention, then it isn't a matter of platonism being true or false. That's like saying the growing convention of using "they" in contemporary English to refer to a person of unknown gender is true or false. Some people like "theyism" and some people don't. It's not a matter of it being true or false. But on "theyism", it is correct to say that a person can say "they" if *they* want to.

The tautology is instead that on platonism, a possible algorithm B exists, because on platonism, the existence of an algorithm is a synonym for the possibility of an algorithm.

> or, in your case, the other way around, that to talk about the nonexistence of existent entities is tantamount to assuming their nonexistence.

No, that's not what I'm saying at all. I'm saying that you are assuming that a possible algorithm B is contingent, which is analytically false if uttered in platonic language. Your argument depends on the idea that the existence of B must be a matter of fact one way or the other. I don't agree with that premise. I wouldn't have any problem with an argument of the form you are making if we agreed that the thing we were talking about either existed or did not exist in a sense we both agree on, e.g. a co-conspirator of Lee Harvey Oswald or something like that.

Hi DM,

Delete>Fictional entities are generally too vague to be real or unreal

But there's no vagueness---in fact, none at all, as opposed to just about any other predicate: things either superman, or they don't, and definitionally, all of those things which are Superman, superman. It's a primitive term which doesn't admit further analysis.

>That assumes that existence is a coherent concept about a factual matter. I don't think it is

I think the other options---like your 'linguistic community'-concept---quickly either run into circularity, or incoherence.

>I'm saying that you are assuming that a possible algorithm B is contingent, which is analytically false if uttered in platonic language.

First of all, once more, I'm not assuming that. Just as it's perfectly coherent to say that Lois would have died if Superman didn't exist, even if Superman's existence is necessary, it's perfectly coherent to say that one could instantiate computation B even if B didn't exist, even if B's existence is necessary. There's simply no real trouble here at all.

Second, it may be analytically false if Platonism is true, or if Platonism is merely linguistic convention---I'm not convinced of either proposition, but let's take it as given for now. But then, it would still be in question whether Platonism is true, or whether it's merely convention---and consequently, you can't use the necessary truth of B's existence as refuting my argument, because B only exists necessarily on the basis of just that metaphysical picture which you're proposing, and I'm doubting.

Hi Jochen,

Delete> all of those things which are Superman

There's your vagueness. I don't know know what is required for something to be Superman.

> I think the other options---like your 'linguistic community'-concept---quickly either run into circularity, or incoherence.

That may be a valid line or argument. We can get into that.

> Just as it's perfectly coherent to say that Lois would have died if Superman didn't exist

With you so far...

> even if Superman's existence is necessary,

You've lost me. I don't think it makes sense to consider counter-factuals which are logically impossible. If Superman is necessary, then the non-existence of Superman is logically impossible.

> it's perfectly coherent to say that one could instantiate computation B even if B didn't exist, even if B's existence is necessary.

On (at least my account of) platonism, the existence of B is a synonym for the idea that B is a possible algorithm. So I can replace the existence of B with the possibility of B. So your statement becomes "it's perfectly coherent to say that one could instantiate computation B even if B is not a possible algorithm, even if B is a possible algorithm". This is nonsense. Since you are clearly an intelligent, rational, reasonable, well-informed person, and you would not say such a nonsensical thing, it seems to me we have to be talking at cross-purposes.

> But then, it would still be in question whether Platonism is true, or whether it's merely convention

The way I mean the term, platonism is not true or false, it is a convention. If you think platonism is a matter of fact, then you are simply playing a different language game than I am.

Hi DM,

ReplyDelete>There's your vagueness. I don't know know what is required for something to be Superman.

And you don't need to---'supermans' is a primitive predicate that doesn't have to be co-extensive with any other predicate---such as 'is super strong and invulnerable'---to be meaningful. By analogy, Gettier has shown that the predicates 'is knowledge' and 'is justified true belief' aren't co-extensive, but this doesn't render knowledge vague---indeed, if it did, then he could never have shown this. All that's needed for 'supermans' to be meaningful is that there's a set of things X, such that for any x in X, either supermans(x) or ~supermans(x). Now, the set of things for which supermans(x) might have one element, many, or none---and this is what allows us to talk meaningfully of Superman's existence.

But let's maybe drop this, as it doesn't seem likely to lead us much further.

>If Superman is necessary, then the non-existence of Superman is logically impossible.

Of course! But that doesn't mean that I'm not allowed to argue that if Superman didn't exist, then Lois would now be a bloody stain on the pavement. After all, this is perfectly simple, and true.

>On (at least my account of) platonism, the existence of B is a synonym for the idea that B is a possible algorithm.

But of course, your account may be wrong, no? It's not a definitional truth that possibility is synonymous with existence: indeed, almost everybody believes it's false.

>"it's perfectly coherent to say that one could instantiate computation B even if B is not a possible algorithm, even if B is a possible algorithm".

That's not quite what I'm saying. Rather, I'm saying that whether possibility and existence is synonymous is irrelevant, so it would be more accurate to parse this as 'it's perfectly coherent to say that if the mere possibility of B does not suffice for its existence, we could still instantiate B'.

Again, the Superman-example ought to make this reasonably clear: whether Superman is necessary or contingent doesn't make a lick of a difference to the fact that if he didn't exist, Lois would have gone kersplat. Likewise, if B doesn't exist Platonically, I would still be able to instantiate it within A, since its Platonic existence is not causally responsible for this possibility.

>If you think platonism is a matter of fact, then you are simply playing a different language game than I am.

Well, that's a bit of a Humpty-Dumpty argument, though: "The way I'm using my words, everything I say is always right; so if you disagree with me, you're simply using words differently." Metaphysical disputes can't be settled by such definitional tricks.

Hi Jochen,

Delete> But let's maybe drop this

Sure.

> But that doesn't mean that I'm not allowed to argue that if Superman didn't exist

I can see why you would think that but I don't quite agree. For a start, it seems clear the Superman is not logically necessary, so it's a bad example -- or a bad intuition pump, as Dennett would say. The idea that Superman could be logically necessary is incoherent, and so this is itself a violation of my rule that logically impossible counterfactuals ought not be entertained.

A better example might be the claim that the number 7 would be even if there were no number 5 (in PA, and if PA is consistent). I can't even consider that counter-factual because it is incoherent. I don't know what the absence of 5 might imply for the number 7 -- it might imply anything at all.

> It's not a definitional truth that possibility is synonymous with existence: indeed, almost everybody believes it's false.

If you want to reject my language, that's fine. Almost everybody does. But then I think they need to propose an alternative definition of existence. One coherent definition is that of physical existence -- something physically exists (from my perspective) if there is some sort of causal connection between me and it, such that I can trace interactions and physical events backward or forward in time and so build a chain of such interactions between me and this other thing.

Clearly, nobody means that mathematical objects physically exist in this sense. So if you don't think my account of platonism is correct, I think the ball is in your court to propose an alternative definition -- whatever it is you think platonic existence is supposed to mean, some contingent question of fact that happens to be false but could have been true.

> it's perfectly coherent to say that if the mere possibility of B does not suffice for its existence, we could still instantiate B

Yes, that is a perfectly coherent alternative linguistic convention. It is perfectly coherent to deem B not to exist and yet for it to be possible to instantiate B. It is simply not the linguistic convention I prefer.

> Likewise, if B doesn't exist Platonically, I would still be able to instantiate it within A, since its Platonic existence is not causally responsible for this possibility.

Causation doesn't enter into it. In my language, for B to exist platonically is just for it to be possible to instantiate B. If B does not exist platonically, this means, analytically, that it is not possible to instantiate B.

> The way I'm using my words, everything I say is always right; so if you disagree with me, you're simply using words differently.

That's not quite what I'm doing though. I can be wrong if I am drawing conclusions that are inconsistent with my own definitions, or if my definitions are incoherent or circular. But we have to be talking the same language if your argument is going to work. If you want to reject my language and use yours, that's fine, but then you must propose an alternative account of existence, one that I understand.

> Metaphysical disputes can't be settled by such definitional tricks.

A lot of them can! Whether free will exists, for instance, depends almost entirely on how you define free will. The dispute between incompatibilists like Harris and compatibilists like Dennett is entirely semantic.

Language is powerful -- it is itself a kind of intuition pump, for good or ill, and how we talk about things affects how we think about things. I think an awful lot of philosophical disagreements are down to different people playing different language games.

I guess there are two important issues here.

Delete1) Whether my platonism is platonism at all.

2) Whether platonism or the MUH allows us to conclude anything about pixies if all it is is the identification of existence with consistency/possibility

If you want to argue the first point, I think you need to provide a positive account of what platonism is supposed to be if not what I say it is.

On the second point, I foresee a long discussion. My point is that this is probably the tack you should take, and not "If B did not exist", because this argument is a nonstarter -- it doesn't engage with my position, which asserts that whether B does or does not exist is not a matter of fact but of convention. You're talking another language and I don't understand that language.

(Incidentally, and an as an aside, I'm deliberately writing "platonism" and not "Platonism". In a recent discussion, I was accused of some sort of dishonesty or disingenuity for calling myself a Platonist without being a follower of Plato and not being committed to his ideas about ideal forms and so on. I researched this problem a little and found that some people draw the distinction by referring to mathematical realism with a small p and reserving Platonism for a more classic view following Plato himself.)

Hi DM,

Delete>The idea that Superman could be logically necessary is incoherent

I don't see why. For one, it could be the case that there is only one possible world (say, there is only one consistent set of natural laws with a unique set of initial conditions), and Superman could exist in that world---unlikely, sure, but far from incoherent. Furthermore, for every world which we add to the set of possible worlds, Superman might exist in it.

But the argument is of course independent of whether you consider Superman's existence necessary---you can replace him with any other necessary entity, and it works just as well. So suppose every possible world contains a Higgs field---then it would still be permissible to say that if there were no Higgs, elementary particles would be massless.

>A better example might be the claim that the number 7 would be even if there were no number 5

No, this is simply an example of invalid reasoning: the conclusion doesn't follow from the premises. What you could say is that if there were no number 5, then 6 would be the successor of 4---which, again, is a perfectly sensible thing to say.

>But then I think they need to propose an alternative definition of existence.

Not really: you make a positive claim, and I don't have to make a contravening positive claim to resist it. So, if you have a theory that depends on UFOs being actually spaceships full of little green men, I don't have to propose an alternative account of what UFOs are in order to attack the idea, say by pointing out the realities of space travel and so on. You're not entitled to claim that because I have no alternative theory of UFOs---and indeed, I simply don't know what the majority of alleged sightings actually boil down to---I therefore have to accept yours. 'I don't know' is a perfectly sensible stance to take on some issues.

So it's perfectly sensible to point out, even though I have no positive account of existence, that I don't believe yours works: to have something like a linguistic convention, you always need to rely on something already existing---so if, say, that linguistic convention disagrees with this notion of existence, it's wrong; e.g. if this convention were 'nothing exists', then in particular, this convention wouldn't exist, and hence, that account of existence would be inconsistent. Consequently, there's more to be said about existence than that it's mere convention (since if it were, then any convention ought to work; but only some do, hence, there's an additional fact of the matter).

Moreover, you can't use this as a starting point for an argument eventually designed to bolster your particular conception of existence---the argument then would be circular: so if you reject my argument on the premise that existence is just a linguistic convention, but your argument (which I'm attacking) is (by way of stipulating that MUH yields a resolution of the pixie problem) exactly designed to bolster the idea that existence is a linguistic convention, then you're trapped in a vicious circle.

>If B does not exist platonically, this means, analytically, that it is not possible to instantiate B.

So again, this would only be analytic if existence were necessarily equal to possibility. But even you're saying it's not: there exist linguistic conventions on which it isn't. But then, really, all that you can say is 'B may exist, or not, depending on what you want to say'---which ultimately collapses to 'B exists, or not, depending on whether you want to accept that B exists'. But this seems clearly contradictory to me: my mere acceptance that something exists has no bearing on whether it actually does.

>Language is powerful

I think one must be careful about not overinterpreting the linguistic turn---language may shape what we say, or believe, about the world, but to believe that language dictates how the world is seems to me to be unreasonably strong (verging on magical thinking, in fact).

>Whether platonism or the MUH allows us to conclude anything about pixies if all it is is the identification of existence with consistency/possibility

DeleteBut even accepting this identification, my argument holds. Fundamentally, I'm merely saying that whether I instantiate computation B within A does not depend on whether B exists.

So let's try to get clear about this once again. Say I accept that existence=possibility. Then, I could still say, if existence were different---say, existence is 'possibility while not being B', which is, of course, false if I accept that existence=possibility---then I could still instantiate B within A. Consequently, B's existence does not matter for A's instantiation of it. There's an account of existence upon which it does exist, and A instantiates it; and there's an account of existence upon which it doesn't exist, and A instantiates it. That the latter is false is on the same order as there being no Higgs field: I can still derive valid conclusions from it. So, my conclusion that B's existence does nothing to resolve the pixie problem still stands.

Disallowing the use of contradictions or falsehoods in reasoning pretty much throws out the whole of mathematics and logic---not only indirect proofs would go, but modus tollens, tautologies such as (A & ~A) --> B, and so on. So this is really not something I would be especially keen on entertaining.

Or perhaps a better analogy is the following: say the Higgs field is necessary, i.e. every consistent set of physical laws includes a Higgs (in some form). Then, your claim is that the Higgs solves the problem of the existence of mass.

DeleteBut this is false: even in a world without the Higgs, there would be mass---in fact, most masses would be relatively unchanged, as most mass---say, of a proton or neutron---in fact derives from binding energy. So the Higgs field does not solve the problem of mass---in fact, the question of whether it exists is largely irrelevant to this problem.

Likewise, you claim that the existence of B solves the pixie problem. But, since the existence of B is not responsible for the instantiability of B, because even if B didn't exist, that wouldn't mean that A can't instantiate B (on some conception of 'existence'), B's existence is, in fact, completely irrelevant to the question of pixies---just as the Higgs' existence is irrelevant to the question of mass.

Hi Jochen,

Delete> For one, it could be the case that there is only one possible world

This is itself incoherent to me. Every mathematical object is a possible world. The only reason we don't usually regard it as a world is because most don't contain observers, and we usually think of worlds as containing observers. A subset of the set of possible worlds include all the simulations of worlds human beings have ever conducted (including virtual worlds in computer games). Superman cannot be a necessary object because not all mathematical objects contain Superman. If it is possible for me to write an algorithm to simulate some world without Superman being a part of that world, then Superman cannot be a necessary object.

Besides, there is no Superman in this world, so Superman cannot be a necessary object. Necessary objects are necessary of logical necessity. If something is clearly not necessary I don't think it is legitimate to suppose that it is for the same reason that I don't accept reasoning from contradictions (I see you have more to say on that issue, which I'll get to in a bit).

> So suppose every possible world contains a Higgs field

I wouldn't accept this any more than the necessity of Superman. In fact, the only things I regard as necessary are mathematical/analytic/logical truths and objects. I don't accept that there could be something that must be physically instantiated in all possible worlds.

> No, this is simply an example of invalid reasoning: the conclusion doesn't follow from the premises

Since the premises are incoherent, anything can follow. Contradictions explode, as you know. Besides, you might say seven is odd because it is has an even number of predecessors in N (excluding 0). If we remove 5, then it has an odd number of predecessors, making it even. This argument is daft because it rests on a logically incoherent foundation, but I don't think it is significantly less daft than any other such argument from an incoherent premise.

> Not really: you make a positive claim, and I don't have to make a contravening positive claim to resist it.

You need to provide an alternative definition of existence for me to even understand what you are saying when you talk of existence. If you don't define your terms, what you say is meaningless to me and we can't have a conversation.

> I don't have to propose an alternative account of what UFOs are in order to attack the idea,

No, but I at least have to understand what phenomena you are referring to when you talk of UFOs. You don't have to explain the causes for them, but you have to commit to some definition of the term, such as "unidentified flying objects".

> So it's perfectly sensible to point out, even though I have no positive account of existence, that I don't believe yours works:

Oh, sure, you can point out that my definition doesn't work. But you can't meaningfully use the concept of existence yourself without an alternative definition.

> this convention wouldn't exist, and hence, that account of existence would be inconsistent.

There's no inconsistency there. You're presupposing that a convention must exist to be consistent. Someone who deems nothing to exist would not assume that.

Delete> So it's perfectly sensible to point out, even though I have no positive account of existence, that I don't believe yours works:

Oh, sure, you can point out that my definition doesn't work. But you can't meaningfully use the concept of existence yourself without an alternative definition.

> this convention wouldn't exist, and hence, that account of existence would be inconsistent.

There's no inconsistency there. You're presupposing that a convention must exist to be consistent. Someone who deems nothing to exist would not assume that.

> then you're trapped in a vicious circle.

There's no circularity there. There would be circularity there if I were on the offense, showing you that my view has to be correct and that yours has to be wrong. But that's not what is happening. Rather, I am on the defence. You are attempting to find a weakness in my proposal that the MUH gives a way out of the pixie problem, that the MUH yields a parsimonious and consistent way of resolving these issues. If I'm right about this, that doesn't prove the MUH is correct, but it may suggest that Occam's razor recommends it.

To attack this view, you could go a few ways.

You could argue that it is not parsimonious. You could offer some evidence that contradicts it. You could go for a reductio ad absurdum. You could argue that it is meaningless or incoherent.

But what you are doing is adopting premises that are not consistent with my view and then drawing conclusions that I ought to find problematic. But your conclusions have no force for me since I do not accept your premises. As soon as you assume B is contingent, then you're not talking about platonism (as I think of it) any more, so your argument is no refutation of platonism.

> this would only be analytic if existence were necessarily equal to possibility.

This strikes me as confused. How can a definition be necessary? Given a definition, necessary truths follow. But a definition is just a convention -- neither true nor false and certainly not necessary. We could define existence in other terms.

> my mere acceptance that something exists has no bearing on whether it actually does.

This assumes that existence already has a meaning. My mere acceptance that something exrodiates does indeed have a bearing on that it exrodiate, because "exrodiate" is a term I made up and it means just what I say it means. When you consider whether something *actually exists* you are appealing to a prior concept of existence. I say there is no such well-defined concept -- this is just an intuition -- one I share, but one I believe to be incoherent. For your claim to be true, you need to tell me what you mean by whether something *actually exists*.

> but to believe that language dictates how the world is

That's not what I'm saying. What I am saying is that "existence" is just a word. What exists depends on what we mean when we claim something exists or not. Without a definition, the claim that something exists is literally meaningless. I am not claiming that the way the world is depends on how we talk about it. I'm claiming that what exists is not a matter of how the world is but of how we talk about the world, because there is no satisfactory definition of what it means for something to objectively exist.

> Fundamentally, I'm merely saying that whether I instantiate computation B within A does not depend on whether B exists.

I'm lost as to how you can think this makes any sense if existence is identified with possibility. Whether you can instantiate B within A certainly does depend on whether B is possible, therefore it depends on whether B exists.

> Then, I could still say, if existence were different---say, existence is 'possibility while not being B'

Delete"If existence were different" is as meaningless as saying "if 5 were different". Like "5", "existence" is just a sign referring to a concept. "If existence were different" is just suggesting that we use the term "existence" to refer to some other concept, like asking "what if 5 were 6?". You can't conclude anything about the original concept from such a move.

The only way your line of reasoning makes any sense is if we both mean something other than possibility by existence, and I am claiming that what exists just happens to be what is possible. But that is not my claim! My claim is that to exist platonically is literally and definitionally the same thing as to be possible -- that the two terms are interchangeable. So "if existence were different" is in my view entirely the wrong way to think and a meaningless question to ask.

> Disallowing the use of contradictions or falsehoods in reasoning pretty much throws out the whole of mathematics and logic

I never ever had any problem with allowing falsehoods in a chain of reasoning. I'm just saying that it is not legitimate to conclude anything from a contradiction except that your premises are wrong. Indirect proofs work in just such a manner by proving that the premises are wrong. Modus tollens involves no contradictions, so it doesn't fall afoul of my rule.

If I acknowledged your premises as consistent with my view, then you would indeed be able to show me wrong by establishing a contradiction from those premises. But since your premises do not describe my view and are contradictory, they do you no good.

> say the Higgs field is necessary,

Not a legitimate move in my view. The only necessary truths I allow are analytic/mathematical/logical truths. Claiming necessity for something like the Higgs field is a category error as far as I'm concerned.

> B's existence is, in fact, completely irrelevant to the question of pixies---just as the Higgs' existence is irrelevant to the question of mass.

I guess you're wearing me down, in that I'm toying with the idea of trying to engage with this argument even though I regard the premise as incoherent. But I don't think I should and I'm not sure I can, simply because I genuinely can't make head or tails of it.

Hi DM,

Delete>Every mathematical object is a possible world.

That's really overstating your case. For one, I believe that no mathematical object is a possible world---indeed, I think the notion is incoherent: structure is only real as instantiated within a physical system; otherwise, all you really get are cardinality claims, and I don't think they have any independent reality (cardinality of what?).

But I'm of course not stating this as the definite answer to whether something is possible---the simple truth is, we don't know what's a possible world.

>A subset of the set of possible worlds include all the simulations of worlds human beings have ever conducted

So I'd say this is very suspect, too. A computer computes what its user takes it to compute---depending on the user, the same computer can compute different things (see the discussion in the other thread). So really, there's no fact of the matter that a given computation 'just is' a simulation of some world. It's a physical instantiation of a structure that we can interpret as the structure of a world; but absent that interpretation, there's no 'there' there.

>Besides, there is no Superman in this world

Well, there's (probably) no Superman on Earth, but Earth is not synonymous with this world. Our universe could be infinite, for all we know (in fact, it's the best fit to current cosmological data), so all possibilities are instantiated somewhere (with probability 1), so if Superman is at all possible, he exists. It could be the same in all other possible worlds---they might all, likewise, be infinite (although I suppose that still leaves a set of worlds of measure 0 where no Superman exists).

>I wouldn't accept this any more than the necessity of Superman.

Fine, so let's skip the window dressing. Say X is a necessary object, and X is the sole reason for Y, then if X didn't exist, neither would Y. This is perfectly cromulent reasoning.

>Since the premises are incoherent, anything can follow.

That's true, but irrelevant. First of all, even in order to derive anything from a contradiction, you still have to use valid reasoning: from A & ~A, we infer A, hence we infer A v B, and hence, since ~A, B follows. This is, in fact, an example of perfectly valid reasoning following the assumption of a contradiction---so if you're arguing that a contradiction can never be assumed, then actually, explosion doesn't hold: the starting point of the above would not be assertible.

Furthermore, again, assuming contradictions is a standard method of proof. Take the following:

1. Assume 2>3.

2. If 2>3, there exists a natural number x such that 3+x=2.

3. There exists no such natural number (by induction over N).

4. Hence, 2 is not greater than 3.

Starting with a contradiction, I have deduced a perfectly sensible, and true, consequence. I don't think even an intuitionist would disagree with this proof---after all, we're not proving the existence of a mathematical object by deriving a contradiction from its non-existence. Not that I think a Platonist would be especially likely to have intuitionist qualms about truth in mathematics...

Besides, I'm not assuming a contradiction. I'm engaging in simple couterfactual reasoning: I'm not saying 'B doesn't exist' (although I believe that to be true), but merely 'if B didn't exist, then...'. Such reasoning is completely innoccuous: Say, for instance, I had pizza yesterday. That makes it logically impossible that I didn't have pizza yesterday, as the proposition that I did is a fact. Nevertheless, I can say that *if* I hadn't had pizza yesterday, I would have pizza today. What I'm doing is just as uncontroversial as that, and I think you're putting yourself in an unreasonable position by denying the validity of all reasoning involving positions contrary to fact. (You may want to claim that surely there's a possible world in which I didn't have pizza yesterday, but a) we really don't know if that's the case, and b) it doesn't matter: whether something is contrary to fact in this world, or in all worlds, is just a difference in the set of things you quantify over.)

Delete>You need to provide an alternative definition of existence

No, I really don't (and apart from that, I did: existence is being possible while not being B). I don't know the exact definition of 'tall'; that doesn't mean my assertion that Abe Lincoln was tall is meaningless. Insisting on a perfect and ironclad definition for all terms before admitting that an assertion is meaningful is just the Socratic fallacy.

>You're presupposing that a convention must exist to be consistent.

I'm not; I'm presupposing that in order for something to be a convention, someone must hold that convention, who, in turn, would have to exist. I can't truthfully claim that 'nothing exists'---it would be a falsehood. So even if my linguistic community were to come to that convention, we'd simply be wrong. Consequently, linguistic conventions regarding existence can be wrong, and hence, there's more to existence than linguistic convention.

>There would be circularity there if I were on the offense

You are: the blog post above is intended to bolster confidence in the MUH by showing that it helps save computationalism from problems otherwise fatal to it; consequently, anybody who favors computationalism ought to accept the MUH.

I then make an argument that the MUH doesn't solve the problem, and you counter that your account of existence means that my argument is wrong (which I don't think is right either, but nevermind)---but this account of existence just is the MUH. Hence, you're using the MUH to support your conclusion intended to support the MUH.

>We could define existence in other terms.

Right, and if we do---or even entertain the possibility, even if it may be false---, then your argument for the MUH above fails, as it doesn't solve the pixie problem.

>For your claim to be true, you need to tell me what you mean by whether something *actually exists*.

It suffices to establish that on basically any conception of existence different from yours, your argument doesn't follow---I need not commit myself to a definition of existence myself.

Besides, it's clear that I can't define existence arbitrarily---I can't truthfully claim that I don't exist. I can claim that I don't exrodiate, if exrodiate is just some arbitrary term.

>I'm lost as to how you can think this makes any sense if existence is identified with possibility.

DeleteEven if I accepted that definition, I can still reason counterfactually that if existence weren't identified with possibility---if, say, it were 'possibility while not being B'---then I could still instantiate B within A, thus showing that B's instantiability is not due to its existence. Consequently, the MUH---which postulates that existence is possibility, at least of mathematical structures---is irrelevant to this instantiability, and can't solve problems posed by this instantiability, such as the pixie problem.

>You can't conclude anything about the original concept from such a move.

I'm not; I'm concluding that the original concept is irrelevant to the pixie problem, since nothing changes regarding B's instantiability if it were varied.

>I'm just saying that it is not legitimate to conclude anything from a contradiction except that your premises are wrong.

Which is demonstrably wrong, sorry. If I had pizza yesterday, it's a contradiction to assume that I didn't have pizza yesterday; yet I can infer valid consequences from that, such as the fact that I'd still be in possession of the money the pizza cost (provided I didn't blow it on something else).

>But since your premises do not describe my view and are contradictory, they do you no good.

I'm under no obligation to be consistent with your view, since my argument is indeed intended to show it's wrong. If I then were forced to take only recourse to premises consistent with your view, then of course, provided your view is consistent, I could never refute it; but not every consistent view is also correct.

>But since your premises do not describe my view and are contradictory, they do you no good.

Well, claiming the existence of mathematical objects is a category error as far as I'm concerned, but we both have to acknowledge that we might be wrong (always heed Cromwell's rule, or all discussion is pointless).

Perhaps another way to frame what I've been saying is that while the MUH yields the entailment from possibility to existence, it's really the other direction you need: that B is instantiable within A because B exists Platonically. Otherwise, B's existence simply doesn't get you any additional purchase.

DeleteSo if B's Platonic existence entailed its instantiability within A, you might have a point arguing that such an instantiation really just opens up a window into B's 'world'. But this isn't the case: for one, hypercomputations are perfectly consistent mathematical objects, and hence, exist in the mathematical universe, but no hypercomputation can be instantiated within a computation (i.e. A).

Thus, postulating B's existence really just adds ontological baggage, without resolving the problem---what you have is, basically, a copy of B within A, with or without B existing independently. But then, you still have a pixie problem.

(Note that this is equivalent to what I've been saying so far: if E is the existence of B, and I is its instantiability (or possibility), then you need that E-->I; and my argument above is that ~E doesn't imply ~I, since that would be denying the antecedent.)

Hi Jochen,

DeleteI don't think you understand me when I say every mathematical object is a possible world, especially when you say no mathematical object is a possible world. There is at least one mathematical object which is a possible world, and that is the mathematical object which describes this one. I think the issue here is that you have a problem with me equating the mathematical description of a world with the possible world. While I do think the two are the same, especially when discussing possible worlds which may not be actual (whatever that means to you), the point I was making here does not depend on this. Instead, you can take me to mean that every mathematical object describes a possible world, because a possible world is just a logically possible world, i.e. a world that is well-defined, coherent and free of contradiction. If it is possible for me to describe with mathematical precision and without contradiction a world without Superman (or the Higgs field), then Superman (or the Higgs field) cannot be necessary.

> the simple truth is, we don't know what's a possible world.

If by "possible world" you don't mean a logically possible world, you must mean something like a metaphysically possible world -- the idea that there are metaphysical laws which determine which of the the logically possible worlds can or cannot exist. Personally, I think that's a strange idea, but if you want to go there we can discuss it.

> So really, there's no fact of the matter that a given computation 'just is' a simulation of some world

That really doesn't matter for my point. It's not the physical computation that matters but the algorithm. If it is possible to write an algorithm to simulate a world, that world is a possible world. There may also be possible worlds which are uncomputable, but let's not go there.

> Well, there's (probably) no Superman on Earth, but Earth is not synonymous with this world.

True, but I'm pretty sure that Superman could not exist in this universe. He seems to be nomologically impossible if not logically impossible.

> Say X is a necessary object, and X is the sole reason for Y, then if X didn't exist, neither would Y.

I can see why you might this is cromulent. But I think we'll just have to agree to disagree on that. I just don't accept that it is acceptable to conclude anything from a contradiction except that your premises are wrong, more or less because of explosion. If X is necessary, then the scenario where X does not exist is a contradiction, and anything at all follows from a contradiction, meaning that your conclusions hold no force. All we can really say is that X is necessary and Y is necessary. When you consider a scenario where neither of them exist, you're no longer talking about X and Y, because X and Y are necessary.

> That makes it logically impossible that I didn't have pizza yesterday, as the proposition that I did is a fact.

You seem to have a habit of conflating contradictions and things that happen to be false. Taken on its own, the proposition that you didn't have pizza yesterday is not logically impossible, because while you did have pizza yesterday, that is not a logically necessary truth. It's contingent. On the other hand, taken on it its own, the idea that some necessary truth is false is indeed a logical impossibility.

> I think you're putting yourself in an unreasonable position by denying the validity of all reasoning involving positions contrary to fact.

DeleteAgain, conflating contradictions and things that happen to be false. Not the same thing.

> (and apart from that, I did: existence is being possible while not being B)

Sorry, I didn't really follow this and I still don't. You mean you think, at least for the sake of argument, that existence is more or less what I think it is but not for B? Well, OK, then, on this definition of existence, I'm happy to agree with you that B does not exist. But since you don't actually think of existence in this way, where does that get us?

> I don't know the exact definition of 'tall'; that doesn't mean my assertion that Abe Lincoln was tall is meaningless

No, it isn't meaningless, because it is not hard to understand what you mean. We can indeed propose a sensible definition if we want to, such as "taller than average for his class (e.g. white American men of the 1800's)".

But my problem with "existence" is not that it is vague in a tall/heap/big/small sort of way, but literally that I have no idea what it is supposed to mean if not what I think it means. I'm not being disingenuous. I really do think it is incoherent.

I'm happy enough with concepts such as "game", despite the fact that they can't be defined. This is not a family resemblance issue. You can list examples of things you believe to exist, and I believe they will all be things causally connected to you -- so it will turn out that you are just talking about physical existence. But this doesn't help us when we try to talk about the existence of things which are not causally connected to you such as other universes. Physical observer-relative existence is fine by me. It's the idea of observer-independent objective existence that I have a problem with.

> in order for something to be a convention, someone must hold that convention, who, in turn, would have to exist

DeleteSo you're presupposing that someone must exist to hold a convention. But someone who holds that nothing exists would not assume that. Someone who holds that nothing exists would think that subjects can do things without needing to exist first. Such a person, for example, would be happy to talk about the views of Sherlock Holmes even though Sherlock Holmes does not exist. There is only a contradiction there if you assume a prior concept of existence, and your prior concept of existence is in my view incoherent.

> Hence, you're using the MUH to support your conclusion intended to support the MUH.

So what? What you're describing is a consistent world view. It's a Quinean web of beliefs. It's mutually supporting and consistent with itself and with what I can see in the world. It could be wrong, and I acknowledge this which is what makes it non-circular. The question is only whether this is the most parsimonious or reasonable world view -- what reasons are there to reject it? None that I have yet encountered.

> If I had pizza yesterday, it's a contradiction to assume that I didn't have pizza yesterday

No it isn't, it's simply false.

> then of course, provided your view is consistent, I could never refute it; but not every consistent view is also correct.

If my view is consistent, you can never prove it wrong. But you can give grounds for rejecting it. If those grounds assume it to be wrong, then they are no grounds at all. They are circular, and so have no force. We would have two different world views and we would have to find independent grounds such as parsimony for choosing between them.

So I guess there are two issues. Do you think my view is consistent, and if it so what are your grounds for rejecting it (that don't start out assuming it to be wrong)? As soon as you're assuming B is a possible algorithm whic exists only contingently (as it must be to make sense of a scenario where it is imagined not to exist), you're assuming my view is wrong, making your refutation at least as circular as my claim. Actually, you're not even assuming my view is wrong, you're just using words in a different (unknown) sense to me. I literally do not understand what you are saying.

Hi DM,

Delete>that is the mathematical object which describes this one

First of all, there's no good reason to expect that there should be such a mathematical object. Second, even if there is, then this doesn't mean that a mathematical object *is* a possible world, merely that it *describes* one.

>you can take me to mean that every mathematical object describes a possible world

That's how I take you, and it's what I think is wrong: things don't merely exist because there exists a description of them; rather, the things are prior to their description. If something exists, we can (presumably) describe it in some way, but just because we can describe something, doesn't mean in any way that it ought to exist (or even, that it is possible for it to exist). That's to imbue altogether too much power onto description.

X may be a consistent and well-defined mathematical object; nevertheless, it may not be the case that a world described by X could possibly exist. To assume otherwise is to assume that 'consistent mathematical object' is equivalent to 'possible world', and what grounds do we have to make such a strong metaphysical assumption? Personally, I don't see any. So I'd prefer not to make it.

>If it is possible for me to describe with mathematical precision and without contradiction a world without Superman (or the Higgs field),

Regardless of the above, even if I agreed with you, this is a much harder task than you seem to think. Recall the trouble we had even deciding whether a laughably simple thing such as PA is consistent; and you really think you can validly assert the consistence of something as complex as an entire description of a world, and moreover, deduce high-level properties such as the existence of Higgs-fields and Superbeings from it?

No, I stand by my original assertion: even accepting the idea that consistency implies possibility, we simply don't know which worlds are possible.

> I just don't accept that it is acceptable to conclude anything from a contradiction except that your premises are wrong, more or less because of explosion.

But if you don't accept that you can conclude anything from wrong premises (except that they are wrong), then there is no explosion, since explosion to the contrary allows you to derive the truth of every proposition. So this is self-defeating.

And the point that I don't actually assume a contradictory premise seems to be continually lost on you. I'm not assuming ~X, but ~X-->Y, which may be true even if X is false.

>while you did have pizza yesterday, that is not a logically necessary truth

DeleteThere's no difference regarding whether I reason counterfactually with respect to the actual world, or with respect to the set of all possible worlds; counterfactual reasoning does not require a proviso that it could have been factual.

>You mean you think, at least for the sake of argument, that existence is more or less what I think it is but not for B?

No, I propose an alternate definition of existence as 'being possible and being not-B'. That is, for every entity other than B, its possibility entails its existence. In effect, I take your Platonic world, and cut B out from it.

>But since you don't actually think of existence in this way, where does that get us?

Since on this conception of existence, A nevertheless can instantiate B, it follows that B's existence is immaterial to whether A instantiates B. Hence, postulating it, via postulating the MUH, doesn't have any effect.

>It's the idea of observer-independent objective existence that I have a problem with.

But it's just solved by Quine's formula---what exists is simply "everything". Draw up a list of all possible entities, and make a check mark behind those that exist; the ones with the check mark are 'everything', and they are what exists. We don't know either the list, or how to determine what gets a check mark---but that doesn't mean that there's any problem with the concept.

>Someone who holds that nothing exists would think that subjects can do things without needing to exist first.

They might well do that, of course. Likewise, somebody could think that they could think without being. Those somebodies, however, simply would be wrong. To exist is simply to be---and hence, without their being, their thoughts would not be; and without their thought's being, they would not be thinking they were.

Whenever you can't but start a sentence with 'there is...', then you're making an existence claim. So if you're saying that 'there is a convention that claims there is nothing (and 'there is' just means what some convention says it means)', then you've simply uttered a contradiction.

Likewise, 'there is x such that x thinks x does not exist, and x is right' is a contradiction in terms: all of these just reduce to 'there is x and there is not x'.

>What you're describing is a consistent world view.

No, what I'm describing is begging the question: the MUH is right if and only if the MUH is right.

>No it isn't, it's simply false.

A contradiction is any proposition that necessarily evaluates to false. The proposition 'I didn't have pizza yesterday', if, in fact, I had pizza yesterday, necessarily evaluates to false.

>If those grounds assume it to be wrong, then they are no grounds at all.

My grounds for rejecting it don't assume it to be wrong; rather, my argument, if it works, demonstrates that your argument doesn't establish what you claim it establishes, namely, that B's existence in a Platonic sense relieves the pixie problem. It does so by showing that whether or not B exists in a Platonic sense has no bearing on the pixie problem.

>As soon as you're assuming B is a possible algorithm whic exists only contingently

Again, I'm not assuming this. Saying 'If Superman didn't exist, Lois would have fallen to her death' does not amount to saying 'Superman doesn't exist'. I simply don't know why you think they're the same.

Hi Jochen,

DeleteA possible world, in your view, is just a possibility. It is nothing more than the description of that world, unless it also happens to be actual. A mathematical description of a possible world therefore more or less is that possible world, or at least the possibility of that possible world. That's what I mean when I say a possible world is a mathematical object.

> First of all, there's no good reason to expect that there should be such a mathematical object.

If you think there is no mathematical object that describes this world, then you are saying that there are things in this world that cannot be described by mathematics, which just means that they have properties or behaviour which that cannot be defined robustly and precisely or as emerging out of simpler interactions which can be defined robustly or precisely. I think the word for this kind of behaviour is "magic", and so to me, the claim that this world cannot be described mathematically is just the denial of naturalism. I'm not going to debate whether naturalism is true or false right now as that's one tangent too many, though I assume you are also a naturalist and would instead dispute how I characterise naturalism. That again might be one tangent too far. Unless you want to get into that, I suggest we accept for now, for the sake of argument, that there is a mathematical object which describes this world. We can always revisit it if you think that this claim is where my view goes wrong.

> X may be a consistent and well-defined mathematical object; nevertheless, it may not be the case that a world described by X could possibly exist.

How could that be the case? What do you mean by "possibly" in that sentence, if not logical possibility? Are you postulating metaphysical laws?

> and you really think you can validly assert the consistence of something as complex as an entire description of a world,

Recall the discussion we had about this, where I said that computer programs are always necessarily consistent in the sense that I care about. What I mean when I say all consistent worlds exist is just that all worlds which can be simulated by an algorithm exist. We're not trying to prove the consistency of an axiomatic system here. It's a different notion of consistency. It just means that we can't have worlds which are governed by rules which contradict each other. We can't have a world with a rule (a) electrons always have a negative charge, while at the same time having a rule (b) electrons always have a positive charge. If you tried to implement such a world as an algorithm you would have to make the two rules consistent, e.g. by having (a) take precedence over (b), or figuring out some system to make sense of the idea that something could have both a positive charge and a negative charge at the same time. There may be unforeseen and unintended situations that arise in the implementation of any set of rules -- we call these "bugs" -- but at least the behaviour of that algorithm is well-defined and doesn't do anything impossible. That's all I mean by consistency.

> then there is no explosion, since explosion to the contrary allows you to derive the truth of every proposition. So this is self-defeating.

Explosion means you can conclude anything, which means that your conclusions are worthless. So you can't *usefully* conclude anything. Arguments from contradictions therefore have no place in rational discourse (except to show a problem with the premises that produce the contradictions).

Delete> And the point that I don't actually assume a contradictory premise

You assume that the existence of B is contingent when you consider an IF--THEN scenario where B may be false. B cannot be both contingent and necessary.

> I'm not assuming ~X, but ~X-->Y, which may be true even if X is false.

Yes, it is true, but ~X=>~Y is also true, which means that ~X=>Y is not informative or useful in any way.

If X is a necessary truth, then ~X is equivalent to a contradiction such as (A&~A). (A&~A)=>Y is true but it is useless in argument, because (A&~A)=>~Y is also true.

So your argument as a whole is not a contradiction, but you are deriving conclusions from a contradiction which is not a valid move in argumentation in my view.

> counterfactual reasoning does not require a proviso that it could have been factual.

I think it does. If it could not logically have been factual, then you are reasoning from a contradiction, which gets you anywhere you like, and so your conclusions while valid are useless.

> Since on this conception of existence, A nevertheless can instantiate B, it follows that B's existence is immaterial to whether A instantiates B.

Right, on this notion of existence. But neither of us actually think of existence like this. So what?

> But it's just solved by Quine's formula---what exists is simply "everything".

That is not a solution at all. That's just a rewording. To exist is to be one of "everything". To be one of "everything" is just to exist. That's useless. If that's all existence is, then existence is an empty concept. I still don't understand what it is supposed to mean.

> and without their thought's being, they would not be thinking they were.

Somebody who does not believe in existence would believe that they could think without their thoughts having to exist. There is no contradiction there.

> then you've simply uttered a contradiction.

But you use such language about mathematical objects all the time. That's just a linguistic shorthand. Someone who doesn't believe in existence would make similar claims, but for all utterances that begin "there is".

> No, what I'm describing is begging the question: the MUH is right if and only if the MUH is right.

But of course! So what reason do you have for doubting that the MUH is right, when it is parsimonious and solves all these problems?

> The proposition 'I didn't have pizza yesterday', if, in fact, I had pizza yesterday, necessarily evaluates to false.

No it doesn't, because it isn't necessarily true that you had pizza yesterday. It is contingently true. Something is necessary only if it is true in all logically possible worlds.

> It does so by showing that whether or not B exists in a Platonic sense has no bearing on the pixie problem.

But it only does this by using the word "existence" in a different sense than I do, where the existence of B is contingent. You're not talking about the same thing as I am. I already agree with you that B doesn't exist in a contingent sense. So your argument doesn't get us anywhere.

> does not amount to saying 'Superman doesn't exist'. I simply don't know why you think they're the same.

But I don't! Man, this point is frustrating. I'm not accusing you of assuming that Superman doesn't exist. I'm accusing you of assuming that the existence of Superman is contingent. If the existence of something is necessary, you don't get to make an argument that assumes that it exists contingently.

Hi DM,

Delete>A mathematical description of a possible world therefore more or less is that possible world

And again, just because there is a mathematically consistent description of a world doesn't mean that there is a physically possible instantiation. This is a separate metaphysical thesis that can be accepted or denied.

>then you are saying that there are things in this world that cannot be described by mathematics,

Which, to all appearances, is true: for instance, the question of whether a certain physical system (a spin system) has a gapped ground state (i.e. whether there exists an energy gap between the vacuum and first excited state) is mathematically undecidable---that is, knowing everything about this system that can be mathematically formalized does not tell you whether there is a gap. However, for a physically real instantiation of such a system, there either is a gap, or not---it's just that the mathematics doesn't tell you so. So mathematics doesn't decide all properties of physical systems, and yet, they are physically definite.

>the claim that this world cannot be described mathematically is just the denial of naturalism

Perhaps the denial of an altogether too naive scientism, but certainly not of naturalism---the spin system above is perfectly natural, as are other examples of mathematically undecidable, yet physically actualized properties.

>How could that be the case?

Mathematically speaking, a world in which a spin system has a gap, and one in which it doesn't, are both consistent. But it may be that for every physical instantiation of that system, it does have a gap.

Compare with the following: you can't mathematically decide whether a given Turing machine halts. It's consistent with a given set of axioms both that it does, and that it fails to. Yet, due to the determinism of computation, it will either halt in all possible worlds, or fail to halt in all possible worlds. So both options are logically consistent, but only one is actually possible.

>What I mean when I say all consistent worlds exist is just that all worlds which can be simulated by an algorithm exist.

But clearly, the majority of worlds which can be simulated won't be consistent in the sense you'd expect of a world. Have you never encountered things like clipping errors, etc., in a computer game? Finding yourself suddenly out of bounds in a giant void? Objects intersecting themselves, and so on?

The vast majority of simulations really aren't: they're hacks, based on heuristics, and often enough, just on what the programmers could be bothered to include. Really simulating a world, down to a consistent set of physical laws is a whole different challenge, and not one that's yet been mastered. So relying on your intuition on that subject is dangerous at best.

>Explosion means you can conclude anything, which means that your conclusions are worthless.

But if you a priori disallow concluding anything from a contradiction, then there will be no explosion; this is again a circular justification.

>You assume that the existence of B is contingent when you consider an IF--THEN scenario where B may be false.

DeleteThis simply isn't right. For engaging in counterfactual reasoning, it's not the case that the counterfactual needs to be possible. Superman (most likely) isn't possible; I can still say that if he were, he'd have saved Lois.

Indeed, I can even reason about the possibility of things: if it were possible to violate the second law of thermodynamics, I could build a perpetuum mobile. This is a perfectly true statement.

>but you are deriving conclusions from a contradiction which is not a valid move in argumentation in my view

Well, you're not entitled to your own view in logic, I'm afraid. Again: if it were possible to violate the second law of thermodynamics, then I could build a perpetuum mobile. This is a completely safe, sound, and true conclusion; and nowhere did I assume that the second law *actually* can be violated, which would be the assumption of a contradiction (and which it can't in any possible world).

> If it could not logically have been factual, then you are reasoning from a contradiction, which gets you anywhere you like

This is just not right. 'If the second law could be violated, we could fly faster than light' is still false.

>But neither of us actually think of existence like this. So what?

Because if existence *were* like that, B could still be instantiated within A, showing that existence doesn't have to be like it is in the MUH for instantiating B.

>Somebody who does not believe in existence would believe that they could think without their thoughts having to exist.

Again, this is just a contradiction, whether they realize it or not. 'I think' parses to 'there is an x such that x is I, and x thinks'. If the content of that thought is then that x doesn't exist, then it's already wrong.

>Someone who doesn't believe in existence would make similar claims, but for all utterances that begin "there is".

The problem persists: if you want to parse 'I think' as 'there is no x such that x is I, and x thinks', then it's likewise just false.

>So what reason do you have for doubting that the MUH is right

I have plenty of reasons to doubt that the MUH is right, but that's not the point: the point is that you can't justify belief in the MUH with an argument that already assumes that the MUH is right.

>it isn't necessarily true that you had pizza yesterday

But if that were the case, then the proposition 'it is possible that I didn't have pizza yesterday' would be true. But it's false: I did, in fact, have pizza yesterday.

>But it only does this by using the word "existence" in a different sense than I do

Yes---that's what I have to do in order to show that the notion of existence assumed does have no bearing on whether B is instantiable!

>I'm accusing you of assuming that the existence of Superman is contingent.

I'm not doing that, either: saying 'if Superman doesn't exist' does not carry any commitment regarding Superman's existence.

Hi Jochen,

Delete> just because there is a mathematically consistent description of a world doesn't mean that there is a physically possible instantiation.

Whether something is physically possible depends on the laws of physics, surely? When we're talking about possible worlds, I think we're talking about logically possible worlds. This implies that one of the ways in which possible worlds can be different from ours is that they have different laws of physics. To assume that a different set of laws of physics is physically impossible only makes sense if you are appealing to a prior set of laws of physics, and there is a logically possible world where those prior laws are otherwise.

> knowing everything about this system that can be mathematically formalized does not tell you whether there is a gap.

I don't know anything about this so I neither refute this nor accept this. I take it with extreme skepticism however -- this sounds like a situation where we would simply interpret the same state of affairs differently. Maybe you could point me in the right direction to learn about this myself?

> However, for a physically real instantiation of such a system, there either is a gap, or not

How do you know? Might it not be like the spin of an electron, where it's in a superposition of states until measured? I don't know if this question makes sense because I don't really understand the situation you're describing.

> you can't mathematically decide whether a given Turing machine halts.

I would rather say there are certain Turing machines for which we don't know whether they halt or not, and there will always be such Turing machines. That doesn't mean both epistemic possibilities are logical possibilities -- it just means we don't know which of them is consistent/possible. There is a fact of the matter about which possibility is true, and I would characterise this as being determined by mathematics, even if we don't know how to use mathematics to figure out what the fact of the matter actually is. In any case, however we describe this kind of issue, this is not what I have in mind in the distinction I draw between the natural and the supernatural.

> Have you never encountered things like clipping errors, etc., in a computer game?

Sure. I don't regard that as inconsistency. A world behaving unlike we expect a world to behave is no reason to regard that world as inconsistent. Hell, this world doesn't behave like we expect it to behave. "Clipping errors" happen in real life too, we call it quantum tunneling.

> The vast majority of simulations really aren't: they're hacks, based on heuristics,

Of course, but whatever they're doing, they're doing something well-defined and logically possible, and so that is a way that a world could be as far as I'm concerned. That world would be very unlike our world, of course, even if it were superficially similar.

> But if you a priori disallow concluding anything from a contradiction, then there will be no explosion; this is again a circular justification.

I'm not disallowing it in a formal sense. I accept that it is logically valid. But it is argumentatively useless. It has no persuasive force. That something follows from a contradiction is trivial. A tautology.

> if it were possible to violate the second law of thermodynamics, I could build a perpetuum mobile. This is a perfectly true statement.

DeleteBecause the 2nd law of thermodynamics is not a logical necessity. There is a possible world where building a perpetuum mobile is possible.

> Well, you're not entitled to your own view in logic, I'm afraid

This is not my view on logic. It is my view on argumentation. Explosion means conclusions from contradictions are argumentatively vacuous.

>This is a completely safe, sound, and true conclusion; and nowhere did I assume that the second law *actually* can be violated,

You assumed (correctly) that it is logically possible to violate it. You didn't assume that it is actually possible to violate it.

> This is just not right. 'If the second law could be violated, we could fly faster than light' is still false.

Is "the second law can be violated" supposed to be a logical contradiction? If it is, then we could indeed fly faster than light, because anything follows from a contradiction. If on the other hand it is not supposed to be a logical contradiction, it is a disanalogy, because I'm only saying you can't usefully conclude anything from a logical contradiction.

> Because if existence *were* like that

You're assuming that there is a fact of the matter on what existence is, and that this fact could be otherwise. But that doesn't make sense unless you already have a definition of existence in mind. You might as well be saying what if dogs were cats. What does that even mean? I'm sure you could interpret it any number of ways, but that's the problem.

> the point is that you can't justify belief in the MUH with an argument that already assumes that the MUH is right.

I'm not "justifying" the MUH. I'm pointing out that a certain criticism of computationalism has no force if one adopts the MUH. Assuming it is consistent, the MUH, like any view, can only ultimately be justified in terms of parsimony.

> But if that were the case, then the proposition 'it is possible that I didn't have pizza yesterday' would be true.

No. I think this relies on equivocating on "possible" -- there's epistemic possibility and nomological possibility in addition to the logical possibility we have when considering what is necessary. The correct true statement is "It was logically possible that I might have eaten a pizza yesterday", or equivalently "There is a logically possible world where I did not eat pizza yesterday".

> Yes---that's what I have to do in order to show that the notion of existence assumed does have no bearing on whether B is instantiable!

If you use "existence" in a different sense than I, you only show that your notion of existence has no bearing on whether B is instantiable. You don't show anything about my notion of existence.

> I'm not doing that, either: saying 'if Superman doesn't exist' does not carry any commitment regarding Superman's existence.

I think it carries a commitment on the nature of Superman's existence. It presupposes that the non-existence of Superman is a logically possible state of affairs. I think this issue is not so obvious with Superman because it is a bad example, and a bad intuition pump. Superman is not a plausible candidate for something that may be logically necessary. You'd have to go to mathematical/analytic/logical truth for that. And so a better example would be trying to reason about what would follow from mathematical impossibilities or nonsense, like what would the sum of the interior angles be for a triangle if it had four sides (while still being a triangle).

Hi DM,

Delete>Whether something is physically possible depends on the laws of physics, surely?

That's not what I mean. See, to me, mathematics is just abstract structure; so, whether a piece of mathematics 'exists' depends on whether there is something physical that possesses/instantiates that structure. But there may be structures such that no physical system (under any kind of physical laws) instantiates them. Thus, they're mathematical possibilities, but not possible worlds.

>Maybe you could point me in the right direction to learn about this myself?

Well, the paper is relatively recent---

here's the arxiv version. But there are plenty similar results.

>Might it not be like the spin of an electron, where it's in a superposition of states until measured?

No, although I'm not sure whether I can explain that concisely. The gist of it is that the states corresponding to a certain energy are eigenstates of a special observable, the Hamiltonian, which dictates the dynamics of the system. Any stationary state---which does not undergo time evolution---is such an eigenstate, and if the system is in such an eigenstate, then it's precisely not in a superposition. And the question is whether there is a gap in energy between the two such states with the lowest energy eigenvalue.

It might, of course, be the case that the system is in a superposition of energy eigenstates, but that's something different---the question is if, when the system is in a definite energy eigenstate, the corresponding eigenvalue is bounded from below, roughly.

>it just means we don't know which of them is consistent/possible.

No, we know both of them are consistent---that's the meaning of undecidable: for an undecidable statement S, given an axiom system A, both A and S as well as A and ~S are consistent (well, if A is consistent).

>we don't know how to use mathematics to figure out what the fact of the matter actually is

It's not the case that we don't know how to use mathematics to figure it out, it's that if there was a way to mathematically figure this out, then we could construct contradictory objects---such as a program that halts if and only if it fails to halt.

Chaitin calls such statements 'true for no reason' (well, or false, as the case may be), and he's right that there is no mathematical reason for their truth (actually, he has something more specific in mind, namely the values of the digits of the so-called halting probability; but if you could answer the halting problem, you could compute those, so it's really equivalent).

However, any real computation either halts, or fails to; it's just that there isn't sufficient reason within mathematics to decide which it is to be.

>That something follows from a contradiction is trivial. A tautology.

Well, first of all, that something is a tautology hardly makes it trivial: all theorems of, e.g., propositional logic are tautologies.

But you're still assuming I assume a contradiction. I'm not, and I think I've run out of ways to try and make this clear.

>There is a possible world where building a perpetuum mobile is possible.

DeleteThere really isn't: the second law is a law of statistics, not of physics, fundamentally. Ultimately, it doesn't say anything other than 'more likely states occur more often'. (You can of course frame this more carefully, but this gets the gist.)

>Explosion means conclusions from contradictions are argumentatively vacuous.

But explosion is completely irrelevant to this: nowhere is a contradiction being assumed.

>You're assuming that there is a fact of the matter on what existence is, and that this fact could be otherwise.

You are assuming that existence is a certain way in your argument, namely, that mathematical entities exist, if they are consistent. I'm pointing out that one can vary this, and still implement B.

>Assuming it is consistent, the MUH, like any view, can only ultimately be justified in terms of parsimony.

Actually, parsimony has no force for metaphysical explanations (other than perhaps as a subjective, aesthetical criterion)---parsimony is needed to ensure the predictivity of physical theories, but this is irrelevant for metaphysical considerations.

And of course, there exist consistent hypotheses that nevertheless can't possibly be true---that a perpetuum mobile exists if the second law is violated is consistent, but as the second law can't be violated, necessarily false.

>The correct true statement is "It was logically possible that I might have eaten a pizza yesterday"

Well, if it was logically possible, then does that mean it isn't anymore?

>"There is a logically possible world where I did not eat pizza yesterday"

This is again false: there may be some possible world in which there exists some entity identical to me in every respect safe for not having eaten pizza yesterday, but of course, he isn't me: for one, he hasn't eaten pizza yesterday, but I have.

>You don't show anything about my notion of existence.

If I show a notion of existence on which B doesn't exist, yet still is instantiable, then I show that B's existence isn't a prerequisite to its instantiability.

>It presupposes that the non-existence of Superman is a logically possible state of affairs.

No, it really doesn't. The set of all existing things is just that: a set. I can wonder what the effects would be if something were part of this set, even if it isn't. I can also wonder about the consequences of something being part of this set, if it isn't even in the set of possible things, whatever that may mean.

An example may be the solvability of the halting problem: there's no possible way to solve it, but there exists a whole theory detailing the capabilities of systems capable of solving the halting problem. So for instance, if I could solve the halting problem, I could prove (or disprove) Goldbach's conjecture.

Hi Jochen,

Delete> But there may be structures such that no physical system (under any kind of physical laws) instantiates them.

What constitutes a set of physical laws here? I don't have any preconceptions about what a set of physical laws looks like, other than that they form a mathematical structure. For instance, the rules of Conway's game of life constitute a possible set of physical laws for that world. If something is mathematically possible, then the context in which it is defined is a possible set of physical laws in which it could physically exist in some possible world.

> Well, the paper is relatively recent---

here's the arxiv version. But there are plenty similar results.

Thanks, Jochen. I would like to understand this but this is a little beyond me for the moment. I would be looking for something pitched more at the level of a PBS Space Time video. Perhaps, if there are plenty of similar results, there is such a result (or a popular science discussion of such a result) that might be easier to grasp?

I guess I am unconvinced by your interpretation because there are people much smarter than I (Tegmark in particular) who still seem to think that there is a mathematical object which describes this world. I recognise that the point you are making here is very important, and potentially fatal to my world view, but since I can't discuss it competently and since we risk opening one too many threads here, I suggest that for now we set it aside and simply assume, for the sake of argument, that there is a mathematical description of this world.

> Mathematically speaking, a world in which a spin system has a gap, and one in which it doesn't, are both consistent. But it may be that for every physical instantiation of that system, it does have a gap.

I still don't understand this, but I'll have a go at another possible interpretation. If such systems have a gap, might that not be just another axiom we could add? I understand there's probably something Godelian going on, so that for every axiom we add there's always going to be another system that is undecidable. But we could have a mathematical object that just decides on the fly to add an axiom (perhaps determining that there is or is not a gap randomly) wherever a novel situation like this crops up. It seems like such an approach could work in simulating a possible world.

> for an undecidable statement S

I think there are two kinds of undecidable statements which I feel you may be conflating. I think there are statements which are actually true or false given a set of axioms, but we can't prove it either way, and I think there are statements which have no truth value and we are free to take either as an additional axiom.

> it's that if there was a way to mathematically figure this out, then we could construct contradictory objects

DeleteI am familiar with the halting problem and the self-applicability problem but I don't agree with how you interpret the conclusions. The halting problem is just that there is no general algorithm which will determine if any arbitrary input program will halt. I accept that. But that's not the same as saying that whether a given Turing machine halts is undecidable. For a lot of Turing machines, it is trivially decidable. For some, it is difficult to decide, and for some, we don't know the answer. There is no Turing machine, and there cannot be a Turing machine, for which the halting problem is provably undecidable. So what I said stands -- the best you can say is that there are Turing machines for which we don't know how to figure out if they halt or not.

> However, any real computation either halts, or fails to; it's just that there isn't sufficient reason within mathematics to decide which it is to be.

I do not think of it like this. For me, whether any real computation halts or does not halt is mathematically necessary and all the results you provided show is that there is no algorithm to determine which is true for an arbitrary program. I do not agree that you are free to take an axiom to determine whether a program halts or not. If the program in fact halts, and you take an axiom that states that it does not halt, then you have created an inconsistent system, and vice versa. The only issue is we don't know it is inconsistent until we figure out whether the program halts or not.

> Well, first of all, that something is a tautology hardly makes it trivial: all theorems of, e.g., propositional logic are tautologies.

Agreed. But something that is an obvious, staring-you-in-the-face tautology doesn't really tell you anything you didn't already know. So observing that FALSE=>X is not informative.

> But you're still assuming I assume a contradiction. I'm not, and I think I've run out of ways to try and make this clear.

OK, it's not exactly that you're assuming a contradiction, it's that you're using a contradiction (a necessary truth is false) as the antecedent in an implication. What follows is not informative because anything follows from such an antecedent.

I think if you abandoned your Superman disanalogy (disanalogy because Superman is not necessary) you would see the problems. Come up with an analogy where you assume that some necessary truth is false and try to conclude something sensible.

For instance: if not all bachelors are unmarried, then could I be an married bachelor? Well, I guess so, it seems that way, but on the other hand I couldn't be an married bachelor because that is an oxymoron. Such a hypothetical can only be made sense of if it means changing the definition of bachelor. So I can only make sense of your hypothetical by your changing the definition of existence to one I don't understand, and this means that you are not concluding anything of value about how I think of existence.

> There really isn't: the second law is a law of statistics, not of physics

DeleteI understand this.

There are a couple of ways to address this. One is that there is a very improbable possible world where entropy always happens to decrease, via statistical fluke. Another is that the way that the 2nd law prohibits perpetuum mobiles depends on the 1st law, or on the conservation of energy at any rate. It seems to me that there is a logically possible world where energy is created ex nihilo and injected into the system so as to keep a perpetuum mobile running. Again, if I could write a computer program that would simulate a perpetuum mobile (and there's no reason I could not) then a perpetuum mobile is logically possible, which to me means that it could exist in some (logically) possible world.

> I'm pointing out that one can vary this, and still implement B.

Of course you can vary a definition and nothing will change. But that just means you're talking about Jochen-existence rather than DM-existence. The point I'm making pertains to DM-existence, so the fact that B's instantiability is independent of Jochen-existence doesn't show that it is independent of DM-existence. Again, for your point to work, you would have to believe that there is some ideal concept of existence out there independent of how you or I use the term. It's as if existence is a natural phenomenon and we're trying to construct a hypothetical model as a scientist does. But I think that is not the right way to think about it. We only have the definitions we adopt and nothing else.

> Actually, parsimony has no force for metaphysical explanations

Actually, it does. You're asserting that it doesn't, and I'm asserting that it does. Where does that get us? I mean, what else would you use to choose between two consistent metaphysical world views?

> he isn't me: for one, he hasn't eaten pizza yesterday, but I have

Fair enough, but you can deal with these objections with sufficient caveats and circumlocution. Someone who was identical with you up to the point where your paths diverged and one of you ate pizza and the other didn't.

> then I show that B's existence isn't a prerequisite to its instantiability.

You show no more than that B's Jochen-existence isn't a prerequisite to its instantiability. You show nothing about platonic existence as I think of it.

> An example may be the solvability of the halting problem: there's no possible way to solve it, but there exists a whole theory detailing the capabilities of systems capable of solving the halting problem.

Such a theory depends on those systems simply knowing the answers. They are oracles. For there to be a contradiction, you have to assume that there is an algorithm which solves the halting problem.

I did a bit more reading about the spectral gap problem and found the following:

Deletehttp://www.nature.com/news/paradox-at-the-heart-of-mathematics-makes-physics-problem-unanswerable-1.18983

It seems that this issue is only undecidable if you are dealing with an infinite lattice. It is decidable for any finite lattice. Since the real world does not contain an infinite lattice, this is not a real-world undecidable problem. I feel my view would only run into trouble if there were some actual experiment in the real world that had results that were undecidable (without simply being random).

Hi DM,

Delete>For instance, the rules of Conway's game of life constitute a possible set of physical laws for that world

Only if one assumes that 'to be a possible set of physical laws' is the same as 'to be a consistent mathematical object'. But what I'm saying is precisely that there's no reason to make that assumption. Mathematical objects, to me, are just real inasmuch as they are instantiated physically; hence, they can't decide what can be instantiated physically.

Or, in other words, the set of physically possible worlds might not include one whose laws are given by GoL. (It might well do, of course---I make no pretenses at knowing what's sufficient to make a world physically possible. I'm merely pointing out that we don't know.)

>there are people much smarter than I (Tegmark in particular) who still seem to think that there is a mathematical object which describes this world

Even smart people sometimes believe all sorts of strange things. But in this case, I think Tegmark is well aware of the difficulty---isn't that basically why he introduces his computable universe hypothesis (CUH)? (Which, incidentally, would point to a different way in which existence could not be co-extensive with mathematical consistency.)

I'm not really sure there's a good popular summary of these results, by the way, but this might at least be a start.

>simply assume, for the sake of argument, that there is a mathematical description of this world.

Even if there are no undecidability problems, I don't think I'd be willing to go along with this. There are still other problems: for instance, Stephen Hawking proposes what he calls 'model-dependent realism', in which there are several distinct, overlapping, not necessarily mutually consistent mathematical structures which are all jointly needed to describe the world. Lee Smolin, on the other hand, proposes that there is no one set of mathematical laws that governs the temporal development of the universe. (I should note I'm not a fan of either proposal, however.)

And indeed, the way things appear right now, something like this might well be right: quantum mechanics appears inconsistent with general relativity; more seriously, from my point of view, the description of quantum mechanics follows one set of mathematical laws during ordinary evolution, and another during measurement (which probably doesn't seem terribly serious from the point of view of the many-worlds idea, but I think that fails for other reasons).

>But we could have a mathematical object that just decides on the fly to add an axiom (perhaps determining that there is or is not a gap randomly) wherever a novel situation like this crops up.

But the problem with this is that there's no way to ensure how to get this correct: if an axiom is added randomly, then we don't know whether it's actually right. You might (and indeed, will) end up with axioms claiming that some system has a spectral gap, while it in fact doesn't, or vice versa.

>I think there are statements which have no truth value and we are free to take either as an additional axiom.

No: each statement expressible in the language of a theory is either true or false given a model of the axioms. Different models may disagree there, but that's a different thing.

>But that's not the same as saying that whether a given Turing machine halts is undecidable.

DeleteWhile it's true that the halting problem is decidable for certain TMs, it's indeed the case that there exist Turing machines for which it is undecidable. Say you have an algorithm that decides the question for some set S1 of TMs, one for some set S2, one for some set S3, and so on. Then the undecidability of the halting problem implies that the conjunction of these sets never exhausts the set of all TMs: because you could just use an algorithm that first uses the algorithm deciding whether it's in S1, then the one for S2, and so on; so if the sequence of sets would exhaust all Turing machines, then we'd have an algorithm deciding the halting problem for all TMs, which is impossible.

>If the program in fact halts, and you take an axiom that states that it does not halt, then you have created an inconsistent system, and vice versa.

No, I have merely created one which isn't sound. Many formal systems prove theorems which aren't true, but that doesn't make them inconsistent. (I can provide an example if you like.)

>Come up with an analogy where you assume that some necessary truth is false and try to conclude something sensible.

'If I could solve the halting problem, I could prove (or disprove) Goldbach's conjecture.'

>One is that there is a very improbable possible world where entropy always happens to decrease, via statistical fluke.

It seems dubious to me that such a world exists---the set of such worlds is of measure 0, so it's impossible to actually point to such a world. For instance, any world you'd find yourself in, if you were to randomly choose a world, would obey the second law.

>It seems to me that there is a logically possible world where energy is created ex nihilo and injected into the system

Perhaps, but that has no bearing on what I said: that I could build a perpetuum mobile if the second law is violated does not imply that I could only then build a perpetuum mobile. I could perhaps also do it by violating the first law, but that doesn't invalidate my inference.

>But that just means you're talking about Jochen-existence rather than DM-existence.

Yes, but your argument needs that DM-existence is necessary for the implementability of B; which is shown not to be right, since Jochen-existence also suffices.

>You're asserting that it doesn't, and I'm asserting that it does.

DeleteAs I've pointed out in the other thread, parsimony is needed for falsifiability: without parsimony, all predictions can be derived from a set of observed data on equal grounds. Parsimony picks out one particular theory consistent with the data, thus yielding one particular set of predictions, which we can then test, in order to re-start the cycle upon eventual failure of such a prediction. But since metaphysical hypotheses aren't empirical, this motivation for parsimony vanishes.

>I mean, what else would you use to choose between two consistent metaphysical world views?

By showing one to be false, of course. (Note that I'm not necessarily convinced there is more than one consistent metaphysical worldview.) A theory's consistency does not imply its truthfulness!

>Someone who was identical with you up to the point where your paths diverged and one of you ate pizza and the other didn't.

But this wouldn't make 'I didn't eat pizza yesterday' any more possible; it would merely mean that of two entities identical up to yesterday, one (I) had pizza, and the other didn't.

>You show no more than that B's Jochen-existence isn't a prerequisite to its instantiability.

No; on DM-existence, B exists and is instantiable; on Jochen-existence, B doesn't exist and is instantiable. Our concepts only differ in what we consider to exist; so since there is a concept of existence on which B doesn't exist, and is nevertheless instantiable, then existence isn't necessary to instantiability.

>Since the real world does not contain an infinite lattice, this is not a real-world undecidable problem.

The problem is applicable to real-world quantum field theories: essentially, a quantum field is equivalent to an infinite spin lattice (for instance, in lattice gauge theory, one generally models a quantum field theory as a lattice with some lattice spacing a; but it's only upon taking the limit a-->0 that one gets the actual QFT predictions).

Hi Jochen,

DeleteThere are many points at which my world view departs from yours, and I'd prefer to stay focused on a few at a time. So, since we're exploring my view rather than yours (although I'd be happy to discuss yours also), let's just assume I'm right on all the points that we're not actively discussing.

So let's assume that Hawking and Smolin are wrong let's assume that the Many Worlds Interpretation is right, and therefore let us assume that there is a mathematical description of the world.

Tegmark is concerned about computability, but when he proposes the CUH he is endorsing the idea that there is no such thing as uncomputable physics in this world. He is also quite skeptical of infinities so the idea of an infinite lattice wouldn't concern him much. I'm relatively agnostic on the idea of the CUH versus the MUH and on the question of infinity -- so far I haven't seen much reason to think that an uncomputable or infinite world could not exist but I acknowledge that there may yet be such a reason I haven't grasped yet.

> Or, in other words, the set of physically possible worlds might not include one whose laws are given by GoL

But what is physically possible is determined by laws of physics of a particular world. What laws could determine what worlds (and so what sets of laws) are physically possible?

Earlier, in answer to a similar point, you said

> whether a piece of mathematics 'exists' depends on whether there is something physical that possesses/instantiates that structure.

But to suggest that something is a possible world is not (usually) to say that there is there is actually such a physical world, unless like me you think all possible worlds exist. Say I'm wrong about that, and only some possible worlds exist. So what would make GoL not a possible world? It can't be simply that it is not physically instantiated as a world, because not all possible worlds exist.

Do you realise you're drawing a distinction between a logically possible world and a possible world? What determines that distinction? I can only make sense of what you are saying by interpreting you as consciously or unconsciously presupposing a set of metaphysical laws that govern all of existence, laws which determine what possible worlds can or cannot exist. But then we can just ask instead "What if the metaphysical laws were otherwise?" and start talking about possible meta-worlds with different metaphysical laws instead. Ultimately you hid bedrock -- logical possibility.

So, when I say "possible world", I mean "logically possible world", and in my view this is the only sensible definition of possible world.

> You might (and indeed, will) end up with axioms claiming that some system has a spectral gap, while it in fact doesn't

Either there is an a priori fact of the matter or there isn't. If there is an a priori fact of the matter, then what happens is determined by mathematics. If there isn't, you can't get it wrong. Whatever you simulate is another possible world. It may not be this one, but it is similar in overall character to this one and this one could be a world described by an analogous mathematical object.

But anyway, on reading a bit more, it seems there is an a priori fact of the matter, so my position is actually that this is not really an undecidable problem at all -- the undecidable scenario never exists in reality.

Delete> No: each statement expressible in the language of a theory is either true or false given a model of the axioms.

I think that is straightforwardly false. I could add an axiom that some integers have a property "foo" and others don't. If that's all I say on the matter, then whether 5 has property "foo" is undecidable. I am free to add as an additional axiom either that 5 is "foo" or 5 is not "foo".

I think the axiom of choice is similar. Unless you take as an axiom that it is true or false, whether it is true or false is undecidable.

More later or tomorrow!

Hi DM,

Delete>let's just assume I'm right on all the points that we're not actively discussing.

I think whether there are alternatives to 'mathematical possibility' as a criterion for existence is very germane to the points we're discussing. If our world turns out not to be described by a consistent mathematical structure, then the MUH is wrong, and is in particular no help with the pixie problem---so if that's sufficient to conclude the failure of computationalism, then the question of whether our world is described by a consistent mathematical structure is essential to the plausibility of computationalism.

>when he proposes the CUH he is endorsing the idea that there is no such thing as uncomputable physics

Well, in his paper, he motivates the CUH explicitly with Gödelian worries: "I have long wondered whether Gödel's incompleteness theorem in some sense torpedos the MUH".

>But what is physically possible is determined by laws of physics of a particular world.

To my view, the laws of physics are descriptive rather than prescriptive: a system's time evolution and properties are a certain way, and we formulate concise descriptions thereof; not the other way around. An action produces an equal and opposite reaction not because of Newton's third law, but rather, because an action always produces an equal and opposite reaction, Newton's third law holds.

If we want to employ some old-fashioned terminology, I think there is some irreducible 'primitive thisness' or haecceity to physical stuff (Peter, over at Conscious Entities, simply calls that 'reality'). This isn't fixed by the math; rather, the math merely furnishes a description of physical stuff, which only attends to its structural properties. But those properties don't exhaust physical stuff: each relation needs relata it supervenes on.

So when I say that the worlds that works according to the GoL may not be physically possible, I mean that there may not be possible physical stuff, the right kind of relata, to instantiate the GoL (as a world on its own).

You can view this as espousing some additional metaphysical laws if you like, but I don't: first of all, I don't think the nature of stuff is optional, and indeed, that it's logically necessitated what that nature is.

The MUH is one proposal for this necessary determination: on it, physical stuff has no further properties other than the ones that can be captured mathematically, and thus, the determination relation simply becomes logical consistency. But that's not to say that no other options exist.

>But to suggest that something is a possible world is not (usually) to say that there is there is actually such a physical world

Right, I expressed myself badly there: I should have said that there's some possible physical stuff instantiating the right relations. I don't require the actual existence of possible worlds.

>Do you realise you're drawing a distinction between a logically possible world and a possible world?

DeleteSo no, I really don't think I do. What worlds are physically possible may still be logically determined, but that determination may not be the same as the one the MUH presumes.

>If there is an a priori fact of the matter, then what happens is determined by mathematics.

Well, many mathematicians would disagree with you on that point. Take Chaitin's constant: no axiomatic system can determine more than finitely many bits of its binary expansion. But still, there is a definite such expansion: that 'bit n is 1' is either true or false for all n (which follows from the fact that every Turing machine either halts, or fails to halt). Chaitin then calls these propositions 'true for no reason'---at least none within mathematics.

>the undecidable scenario never exists in reality.

According to our best physical theories, it does, so this carries a commitment to most of modern physics being wrong---to me, this would be far too strong a conclusion to not worry about whether the metaphysical assumptions going into it actually hold up.

>I am free to add as an additional axiom either that 5 is "foo" or 5 is not "foo".

Sure, but if you do, then the natural numbers will no longer be a model of the axiom system you're proposing; rather, you'll end up with a new mathematical structure in which every element either has the 'foo' property, or not.

Oh, and regarding the discussion on coming up with something logically false that leads to something sensible: how about 'if the Peano axioms were complete, they could prove their own consistency'---does that work for you?

Hi Jochen,

DeleteThis conversation is a welcome distraction from what's going on in the world!

> While it's true that the halting problem is decidable for certain TMs, it's indeed the case that there exist Turing machines for which it is undecidable.

Your argument here strikes me as being a bit like Cantor's diagonal argument, so it is certainly a good effort. I'm not convinced that this kind of argument works in this case, though.

I'm going to assume that a question is decidable if it is possible to write an algorithm that gets the right answer to that question 100% of the time. When you say that there are undecidable TM's, you are saying that there exists an algorithm X for which there does not exist an algorithm Y which can decide if it halts.

As I have framed the question, it is obvious that you are wrong. There is a (possibly unknown) fact of the matter on whether X halts or not. One algorithm which decides that it halts is either the algorithm which always returns true, or alternatively the algorithm that always returns false, depending on what the fact of the matter is. For instance, for algorithm X1, which does not halt, then the algorithm which returns FALSE decides its halting problem. For the algorithm X2, which does halt, then the algorithm which returns TRUE decides its halting problem. There always exists an algorithm Y to decide whether X halts, but we don't know what that algorithm is and we won't know that it gets the right answer unless we figure out whether X halts.

The halting problem only makes sense if you take it to be the generic problem of deciding whether any algorithm halts. That problem genuinely is provably undecidable. Whether any specific algorithm halts is not, nor is it sensible to suppose there is a fact of the matter on whether it undecidable. All that it makes sense to say is that we know that it halts, we know that it does not, or we do not know which.

What you may want to say instead is that there exist TMs which do not halt and for which it is impossible to prove that they do not halt. I think this is probably true but I'm not sure.

> No, I have merely created one which isn't sound.

This is certainly not true if you take the former case (and not my vice versa). If the program halts, you can prove that it halts, just by running through the steps until you hit the end. If you take as an axiom that it does not halt, then you can prove that it does not halt. If you can prove both that it halts and that it does not halt, you have an inconsistent system on your hands.

It's harder to demonstrate that a system is inconsistent if we have it the other way around -- if it doesn't halt, but we take it as an axiom that it does halt. I guess the best I can do is say that if it doesn't halt, there is always the potential that we could one day find a proof that it doesn't halt, and then we would have an inconsistent system on our hands. So I guess this situation is more of a system with a strong potential for inconsistency rather than one that I can reliably show to be inconsistent. It might not meet the formal definition of inconsistency but I take it to be inconsistent for my purposes -- I wouldn't consider it to be a viable basis on which to define a mathematical object that platonically exists. We can say it is pseudo-consistent. For my purposes, full consistency (call it ultra-consistency if you like) requires not only that a contradiction cannot be derived, but that none of the axioms are inconsistent with facts which are necessary given those axioms.

This is different from soundness, which is just that none of the axioms are inconsistent with (possibly contingent) facts about something the system is supposed to model. For instance, non-Euclidean geometry may not be sound if taken as a model of the geometry of our spacetime at a macroscopic level, but it is (presumably) ultra-consistent.

> 'If I could solve the halting problem, I could prove (or disprove) Goldbach's conjecture.'

DeleteYes, that's a better example. Indeed you could both prove and disprove it, irrespective of whether it is true or not, because a contradiction (such as solving the halting problem) implies anything.

> Perhaps, but that has no bearing on what I said:

As I see it, we were disagreeing about whether a perpetuum mobile was logically possible or not. I said it was, you said it wasn't. If the first law is violated, it is logically possible. On the other point relating to statistical flukes, I guess we'll just have to disagree on whether something that is infinitesimally probable is logically possible or not. I accept all you say, but yet I think it is.

> your argument needs that DM-existence is necessary for the implementability of B; which is shown not to be right, since Jochen-existence also suffices.

That argument doesn't follow. You can't show that X is not necessary for Y by showing that Z suffices for Y unless you show that it is possible for Z to hold without X holding. In any case, in your example, Z (Jochen-existence) doesn't even hold for B, so I don't know why you are saying that you have shown that Jochen-existence suffices for B to be implementable.

Perhaps you interpret me as saying that one must define existence as DM-existence for B to be implementable. Of course that's not what I'm saying. What I'm saying is that B must DM-exist for B to be implementable. That B is implementable without Jochen-existing has no bearing on what I am saying.

> But since metaphysical hypotheses aren't empirical, this motivation for parsimony vanishes.

Parsimony is just making as few ungrounded assumptions as possible. If that isn't self-evidently a sensible heuristic to you then I don't know what to tell you. To reject parsimony is just to assert that it is reasonable to accept ungrounded assumptions. Why ever would you say that?

> By showing one to be false, of course.

OK, so show the MUH to be false. Accusing it of circularlity is not showing it to be false.

> A theory's consistency does not imply its truthfulness!

I'm taking consistency here to mean consistency with the evidence as well as with itself. As far as I'm aware, the MUH is consistent in this sense. That still doesn't imply its truthfulness, but we have no better way of choosing between consistent views than parsimony. Of course agnosticism is perfectly acceptable.

> Our concepts only differ in what we consider to exist; so since there is a concept of existence on which B doesn't exist, and is nevertheless instantiable, then existence isn't necessary to instantiability.

To me, this argument is blatantly absurd. I think energy is the property of being blue in appearance. Our concepts only differ in what we consider to have energy; so since there is a concept of energy on which a red battery doesn't have energy, and can neverthleless do work, then energy isn't necessary to do work. Nonsense! DM-energy isn't necessary to do work. Jochen-energy is.

> The problem is applicable to real-world quantum field theories:

DeleteI think that I'm basically not going to buy this, Jochen, even if I'm not competent to judge it. You can take this as wilful arrogance or ignorance on my part if you like, but as far as I can see it is not the case that this result has persuaded everyone in the physics community that there is no mathematical description of the world. I don't think I am rejecting a consensus, I think I am rejecting your particular interpretation.

But, to give a sense of my problem with your interpretation (apart from that it would potentially shatter my world view), you're saying both that the properties of infinite spin lattices are undecidable and that we can get the predictions of QFT by postulating an infinite lattice, which would mean that these properties must be decidable.

I don't think that taking the limit as a approaches 0 is the same as postulating an infinite lattice in any case. Taking a limit means that we can see that our predictions are converging even before we reach infinity. The undecidable properties of the spin lattice presumably aren't convergent and this is why we require actual infinities to have a problem. It doesn't bother me if we can compute some properties of some infinite structures, but equally it doesn't bother me if we cannot compute other properties of other infinite structures. It's like the difference between a convergent series and a divergent series. I only have a problem if there is some behaviour we can observe in the real world that cannot be described mathematically, and this does not appear to be the case.

Hi DM,

Delete>This conversation is a welcome distraction from what's going on in the world!

Yes, it is. Thinking about possible worlds is infinitely more pleasant than thinking about the real one right now.

>One algorithm which decides that it halts is either the algorithm which always returns true, or alternatively the algorithm that always returns false

Typically, one would require of an algorithm that it's possible to know if it implements the function one is interested in. What good is an algorithm that may or may not solve the halting problem? What good is an algorithm that may or may not compute the digits of pi, if one wants to know the digits of pi?

>Whether any specific algorithm halts is not, nor is it sensible to suppose there is a fact of the matter on whether it undecidable.

But if *every* specific instance of the halting problem were decidable, then the halting problem would be decidable in general. Since it's not decidable in general, it's not decidable in some specific case.

Take an algorithm A which takes as input the description of some Turing machine T, and then simulates the performance of this machine on an initially blank tape. It's clear that you can't solve the halting problem for A for every input t: if you could, the halting problem would be decidable. But then, there must exist some T for which you can't solve the halting problem for A.

>If the program halts, you can prove that it halts, just by running through the steps until you hit the end.

OK, yes: there is an algorithm which eventually spits out the description of every Turing machine that halts.

But my original assertion still works for the digits of Chaitin's constant: for every axiomatic system S, there exists a constant C such that for some given Chaitin constant, S can only determine C scattered bits of its binary expansion. Arbitrary values of all (infinitely many) additional bits can be added as axioms to S without producing an inconsistency. (And no 'potential for inconsistency', either: we know that there is no proof/disproof of the proposition 'the value of the nth bit is 1' within the system if the nth bit is not within the C bits that can be proven to be a certain value.)

>but that none of the axioms are inconsistent with facts which are necessary given those axioms.

Well, but the values of the bits aren't really necessary given the axioms: by Gödel's completeness theorem, since both S+v(n)=1 and S+v(n)=0 are consistent (where v(n)=x is the axiom asserting that the value of the nth bit is x, where the value of the nth bit is not derivable from S), both have a model; consequently, neither v(n)=1 nor v(n)=0 is necessary given the axioms.

>This is different from soundness, which is just that none of the axioms are inconsistent with (possibly contingent) facts about something the system is supposed to model.

Soundness means that every theorem of the system holds true in every model; that is, the derivability of a formula within S ensures that the formula is true in every model of S.

>Indeed you could both prove and disprove it

DeleteNo! I could prove it if and only if it is correct: I can set up a computer that runs through all numbers, trying to find a counterexample, and that halts if it does. So if I can decide this computer's halting problem, then I know, if it halts, that a counterexample exists, and, if it fails to halt, that Goldbach's conjecture is true.

>As I see it, we were disagreeing about whether a perpetuum mobile was logically possible or not.

I was arguing that the violation of the second law implies the possibility of a perpetuum mobile; that there are other ways of building one doesn't make that false. (Indeed, this would be denying the antecedent.)

>You can't show that X is not necessary for Y by showing that Z suffices for Y unless you show that it is possible for Z to hold without X holding.

Huh? If I show that Z suffices for Y, then what I've shown is exactly that X isn't necessary for Y. Take the violation of the second law (X): it suffices to build a perpetuum mobile (Y), but it isn't necessary for it, which is shown by the fact that the violation of the first law (Z) also suffices.

I don't understand the last part of your sentence, though, so maybe you meant something different. How could Jochen-existence depend on DM-existence, when both are simply mutually exclusive concepts?

>What I'm saying is that B must DM-exist for B to be implementable. That B is implementable without Jochen-existing has no bearing on what I am saying.

That's not my argument. I'm saying that B's DM-existence isn't necessary, because its Jochen-existence (i.e. non-existence) is sufficient.

>Why ever would you say that?

Because there's nothing (apart from, maybe, aesthetics) to motivate saying something else. My aesthetic prejudices simply might differ from yours!

>OK, so show the MUH to be false.

Well, to me, implying that structure is all there is to the world already does that. But you're unlikely to accept this, I'd wager.

However, we've already seen two ways for how the MUH could turn out to be false: the Gödelian worry could be justified, and there are propositions true in every world without the mathematics deciding their truth; or, the world could not correspond to a single, consistent, mathematical structure. Neither of which would render the MUH inconsistent---merely wrong.

>I'm taking consistency here to mean consistency with the evidence as well as with itself.

This is a mixing of syntactic levels with semantic ones that, I think, really only generates trouble. There's good reasons these levels are kept well separate in mathematical logic.

>so since there is a concept of energy on which a red battery doesn't have energy, and can neverthleless do work, then energy isn't necessary to do work.

This is disanalogous: since for a battery to perform work actually requires energy, you of course descend into absurdity. However, to instantiate a computation, its existence isn't required.

>we can get the predictions of QFT by postulating an infinite lattice, which would mean that these properties must be decidable.

DeleteNo; there are some properties of QFTs that aren't decidable, some which are. This really isn't different from, e.g., the situation within the GoL: whether a certain pattern ever develops is undecidable; yet, for any given initial configuration, this pattern either develops, or fails to.

>I don't think that taking the limit as a approaches 0 is the same as postulating an infinite lattice in any case.

Well, be that as it may, the authors of the paper certainly do believe their result to be (potentially) applicable to real-world quantum field theories.

> If our world turns out not to be described by a consistent mathematical structure, then the MUH is wrong,

ReplyDeleteAgreed. And it is germane. I'm just trying to keep a handle on the explosion of digressions. It's getting a little unwieldy. I value this conversation greatly but it's getting out of control.

> Well, in his paper, he motivates the CUH explicitly with Gödelian worries:

I know, but in proposing the CUH he is assuming that the physics of this world are computable, which means that his view would be incompatible with any result which proved that the physics of this world are not computable. Now, he said that before this result, so I guess it's possible he has changed his mind but I haven't heard about it if so.

> To my view, the laws of physics are descriptive rather than prescriptive:

My view is that the laws of physics just are the behaviour of things. It's neither descriptive (behaviour before laws) nor prescriptive (laws before behaviour), but an identity relation (laws are behaviour, behaviour is laws).

But if you have a purely descriptive interpretation (behaviour before laws), I don't think the concept of physical impossibility makes sense. Anything at all can happen, and whatever happens we would describe with a law. So I don't see how some mathematical object would be physically impossible in some other logically possible universe where things might behave differently.

> I mean that there may not be possible physical stuff, the right kind of relata, to instantiate the GoL (as a world on its own).

Possible in what sense? It is logically possible that in some other world, stuff might behave in such a way as to instantiate the GoL -- logically possible because this entails no contradiction that I can see. If it is logically possible for stuff to behave this way, then it must be logically possible for such a world to be physically possible, because on your account what is physically possible is just how stuff behaves.

> and indeed, that it's logically necessitated what that nature is.

OK, but you don't seem to have much of an argument to back that up. It's an ungrounded assumption which I guess may be true (although it's hard for me to even accept that since it's so vague -- I really have trouble seeing it as something other than a set of metaphysical laws), but which I would be inclined to discard as unparsimonious. So we're back to judging whether your view or mine is more reasonable if both are consistent. Perhaps better to focus on points where my view may be inconsistent, since we disagree on how to choose between consistent views.

> Chaitin then calls these propositions 'true for no reason'---at least none within mathematics.

I'm familiar with Chaitin's constant but I don't think it is a problem for my view. I wouldn't call them true for no reason. I would call them necessarily true but unprovable. "for no reason" has the wrong connotations for me -- connotations of contingency.

> Sure, but if you do, then the natural numbers will no longer be a model of the axiom system you're proposing;

I don't see why they wouldn't be a model. There is nothing in these axioms which are inconsistent with the natural numbers.

> how about 'if the Peano axioms were complete, they could prove their own consistency'---does that work for you?

I accept that statement as a good analogy, but I also accept "If the Peano axioms were complete, they could prove their own inconsistency", in general because of explosion, but more intuitively simply because no system can be both complete and consistent. Being complete means being inconsistent.

So your examples are valid but trivial and useless.

Hi DM,

Deletesorry, I'm losing track of who replied where to what... But anyway:

>his view would be incompatible with any result which proved that the physics of this world are not computable

Well, even without this result, physics isn't computable in its current form---both quantum theory and general relativity depend on the continuum of real numbers, almost all of which aren't computable. So if, say, the position of a particle can take on arbitrary real values, then almost none of these are computable.

>but an identity relation (laws are behaviour, behaviour is laws).

Well, for one, the laws themselves don't specify the behavior of a given system---you also need the initial conditions for that. Consequently, systems obeying the same laws may behave quite differently. So I don't think I see this identity.

>Anything at all can happen, and whatever happens we would describe with a law.

I'm not sure about that---there may be systems that don't behave according to some finitely specifiable set of laws.

>logically possible because this entails no contradiction that I can see.

I'm just pointing out that we don't know the conditions under which something is possible in terms of physical stuff. You're assuming that mathematical consistency suffices; I simply see no reason for making this assumption.

>It's an ungrounded assumption which I guess may be true

As is the assumption that anything that's mathematically consistent may possibly exist as a physical world. This is a metaphysical hypothesis that may be, in fact, wrong.

>which I would be inclined to discard as unparsimonious

Parsimony is no indicator for truth, or even likelihood. All it gets you is falsifiability in empirical contexts---which we don't have in a metaphysical setting. Aside from that, I don't see why 'mathematical consistency dictates what is possible' is any more parsimonious than 'physical realizability dictates what is possible'. Sure, we might not know what exactly is meant by physical realizability; but it's not like consistency is an easily settled question, either.

>I would call them necessarily true but unprovable.

But if math is all there is, then what makes them true? What's the truthmaker of these propositions?

>I don't see why they wouldn't be a model.

Because you've changed the axioms. The natural numbers are a model of the Peano axioms, but if you add something to them, then you need a new model, i.e. a mathematical structure that satisfies the properties as laid out by the axioms.

>but I also accept "If the Peano axioms were complete, they could prove their own inconsistency"

But that would be wrong (provided that the Peano axioms are indeed consistent). You're simply misapplying explosion here: nowhere is a contradiction assumed to be true.

Hi Jochen,

Delete> sorry, I'm losing track of who replied where to what... But anyway:

My fault, didn't mean to start a new top-level comment. Also, I don't really have the bandwidth to keep up, so I'm posting comments piecemeal and replying to things you said a couple of days ago. Perhaps email would be a better venue to continue the conversation. You can contact me via the contact form on the right sidebar if that interests you.

Anwyay, I'm over to the other thread to continue for now...

Hi Jochen,

Delete> What good is an algorithm that may or may not solve the halting problem?

No good at all. I'm not saying it's any good, I'm saying that I believe you to be using the term "undecidable" incorrectly. I don't think that it is meaningful to say that the halting of a specific algorithm A is undecidable. It is, on the other hand, meaningful to hypothesize that A does not halt and that this cannot ever be proven by any means. I take it this is what you mean to say.

I'm not convinced that it has been proven that such an algorithm exists. The proofs we have are subtly different, that there is no universal effective procedure for solving halting problems (or, equivalently, in Gödel terms, that there are true statements for which there are no proofs in any given axiomatic system). It seems to me that this doesn't quite rule out the possibility of there being a proof of non-halting for every algorithm that doesn't halt, although personally I doubt it. In Gödelian terms, this would be the unlikely possibility that for every true statement in an axiomatic system, there exists a proof of truth if we can bring in arguments from without the axiomatic system (as Gödel himself does to show that his unprovable statements are in fact true).

> But if *every* specific instance of the halting problem were decidable, then the halting problem would be decidable in general.

Not so, since a general solution which merely delegated to specific solutions would need to specify which specific solution to give for every possible algorithm, of which there are infinitely many. A general solution would therefore need to be infinitely long, and so is not an effective procedure and not an algorithm.

> But then, there must exist some T for which you can't solve the halting problem for A.

Of course. For any A, there always exists a T which A cannot decide whether it halts or not. But that doesn't mean T is undecidable. It only means that T is undecidable by A. There may be another algorithm B which can decide it. So undecidability is not a property of a Turing machine T alone but a property of a two-place relation (A, T). The Halting Problem shows that there exists no A for which (A,T) is always decidable, given any T. It doesn't show that there exists no T for which (A,T) is always undecidable, given any A. As I understand decidability, this is certainly false. However I am prepared for the sake of argument (though I am not sure if it is true or not) that there exists a T for which there is no way to prove that it doesn't halt.

> S can only determine C scattered bits of its binary expansion.

This is neither here nor there, but I don't think this is technically true. The bits we can determine are not scattered as far as I understand -- we ought to be able to determine the first n bits. The value of n is not known -- it depends on how clever and inventive we get with our proofs. There is no upper bound on n, but however clever we get, n will always be finite.

> Arbitrary values of all (infinitely many) additional bits can be added as axioms to S without producing an inconsistency

DeleteSo this is not quite true. As n grows, then arbitrary values for bits can yield inconsistencies.

> And no 'potential for inconsistency'

So there is a potential for inconsistency, because we don't know how far we can go with n.

> Soundness means that every theorem of the system holds true in every model;

I'm not sure this is a sensible definition. If a theorem doesn't hold true in some putative model, that just means that it is not a model, surely. Soundness, in my view, assumes a model. A system is sound for a putative model if every theorem of the system holds true for that model. Indeed, it is a model only if the system is sound for that model. This is what soundness means in logic and I presume it is similar in mathematics.

The abstract syllogism

For All x: A(x) => B(x)

A(y)

Therefore B(y)

Is valid. Whether it is sound depends on what model we adopt. If A is modelled as the predicate "x is a man", and B is modelled as "x is mortal", and y is modelled as Socrates, then this is a sound syllogism.

If B is instead modelled as "x is immortal", then this is not a sound syllogism. Equivalently, I can say that this interpretation is not a model of the abstract syllogism. Soundness is therefore a property of formal system + model taken together, not a property of a formal system alone.

> No! I could prove it if and only if it is correct:

DeleteI can see where you are coming from but explosion means you are wrong. The proposition that you can solve the halting problem is a contradiction. Anything follows from a contradiction. Again, I know you are not actually assuming that you can solve the halting problem, but you are using the proposition that you can solve the halting problem as the antecedent in an implication, and anything follows from using a contradiction as an antecedent in an implication.

> Take the violation of the second law (X): it suffices to build a perpetuum mobile (Y), but it isn't necessary for it, which is shown by the fact that the violation of the first law (Z) also suffices.

This is where the last part of my sentence comes in, although I realise I need to modify it a little to be strictly correct.

I said "unless you show that it is possible for Z to hold without X holding". I should have said "unless you show that Z suffices without X holding".

A violation of the first law is enough to build a perpetuum mobile if and only if the second law is also violated. A violation of the first law permits the construction of a perpetuum mobile just because it permits a violation of the second law. So just because you can build a perpetuum mobile by violating the first law does not mean that you can build a perpetuum mobile without violating the second law.

> How could Jochen-existence depend on DM-existence, when both are simply mutually exclusive concepts?

If they are mutually exclusive, and X is true for B, then I must take Z to be false for B. Z must therefore be the proposition that B Jochen-exists (as opposed to the proposition that B does not Jochen-exist). If this is what you're saying, then Z cannot depend on X. Since they are mutually exclusive, your argument simply doesn't work. This is where I complain that "in your example, Z (Jochen-existence) doesn't even hold for B". Your argument says both that Z suffices for B to be implementable, and that Z is not true for B. This is confused. Your argument would only show that Z suffices for B to be implementable if Z were true for B.

Whether Z is true for B is a predicate. Whether Z is the definition of existence is not a predicate. It doesn't hold or not hold. It's just a convention or a definition. I think this is where you are going wrong.

> That's not my argument. I'm saying that B's DM-existence isn't necessary, because its Jochen-existence (i.e. non-existence) is sufficient.

Now, here, you seem to be saying the opposite. You seem to be interpreting Z as the proposition that B does not Jochen-exist. In this case, X and Z can both hold at the same time, so the fact that Z is sufficient for B to be implementable does not mean that X is not necessary for B to be implementable. To show that X is not necessary for B to be implementable, you need to show a case where X does not hold for B but B is implementable. That is not logically possible, because what DM-exists is defined (more or less) as what it is possible to implement in principle.

> Because there's nothing (apart from, maybe, aesthetics) to motivate saying something else.

DeleteOK, so you are advocating the rationality of adopting as many groundless metaphysical assumptions as you like, simply on the basis that it pleases you. So, for instance, it is rational to believe in invisible pink unicorns and a rich pantheon of non-intervening deities. On this I will just have to flatly disagree. If we can't find common ground here, then the conversation is futile.

> and there are propositions true in every world without the mathematics deciding their truth;

I think here we're just talking at cross purposes. For me, even if the Goldbach conjecture is intrinsically unprovable, its truth is logically and mathematically necessary, and this truth is determined by mathematics. For you, it is true for no reason and it is not logically or mathematically unnecessary. This is a trivial distinction of terminology. For me, logical necessity is just what has to be true in all possible worlds, even if we could not derive a contradiction from assuming the opposite. You may therefore have a problem with the way I express my view, but this is not in fact a criticism of the view itself.

> This is a mixing of syntactic levels with semantic ones that, I think, really only generates trouble.

Well, fair enough, but I did say "and with the evidence" a number of times previously. I thought a little shorthand would be excused in this context.

> This is disanalogous: since for a battery to perform work actually requires energy

That depends on what you mean by "energy". Again, you're assuming that we're talking the same language. In this case, we normally would be, but on existence we are not. So if to me, "energy" means being blue, then you don't require DM-energy to perform work. The truth of any statement depends not only on the world but on the way that statement is interpreted. If you and I interpret existence differently, then statements that are true for you may be false for me and vice versa.

> No; there are some properties of QFTs that aren't decidable, some which are.

I dealt with that in my next paragraph. It doesn't bother me if there are some convergent properties and some divergent properties of infinite lattices, as long as the divergent properties of infinite lattices never manifest in the real world. And it seems they don't.

> -both quantum theory and general relativity depend on the continuum of real numbers, almost all of which aren't computable.

True. I take the computability of physics to mean that outcomes/predictions can be computed to an arbitrary precision. I think this is what is generally meant when the computability of physics is debated. Pi is one of the computable reals, for instance, even though we can't compute it to infinite precision.

> Well, for one, the laws themselves don't specify the behavior of a given system---you also need the initial conditions for that.

Well, OK. The laws specify behaviour given initial conditions. Behaviour given initial conditions is laws, laws are behaviour given initial conditions.

> I'm just pointing out that we don't know the conditions under which something is possible in terms of physical stuff. You're assuming that mathematical consistency suffices;

DeleteIt's more that I'm not assuming there is anything else to consider. We already know that if something is mathematically inconsistent (i.e. impossible) then it cannot exist. You're postulating some other unknown set of conditions that need to hold. Not only do I not see any reason to assume this, I don't think it gets us anywhere, because it is logically possible that that set of conditions could be otherwise, and then we can still consider the class of possible worlds that would be possible if the set of conditions were otherwise.

> But if math is all there is, then what makes them true? What's the truthmaker of these propositions?

That it is not mathematically possible to prove them false. For instance, if the Goldbach conjecture is true but unprovable, then it is not mathematically possible to construct a counter-example. This is the truth-maker of the Goldbach conjecture, and it is far from independent of mathematics.

> The natural numbers are a model of the Peano axioms, but if you add something to them, then you need a new model

I don't think this is true as long as the natural numbers are consistent with the new axioms.

For example, if I adopted a new axiom that 1 < 2, that would not mean the natural numbers are no longer a model of the new system.

> But that would be wrong (provided that the Peano axioms are indeed consistent).

First, a minor correction, I should have said "they cannot prove their own consistency" rather than "they can prove their own inconsistency".

Provided that they are consistent, then they cannot be complete. If we are assuming that they are consistent, that means we can add the statement "The Peano axioms are not complete".

So far we have.

The Peano axioms are consistent (A)

The Peano axioms are not complete (~B).

The Peano axioms are complete => The Peano axioms can prove their own consistency. (B=>C)

I am claiming that "The Peano axioms are complete => the peano axioms cannot prove their own consistency" (B=>~C) is derivable, and you disagree.

That ought to be simple enough to work out.

Working backwards, B=>~C is equivalent to ~B v C

But we already have ~B, therefore ~B v C is true.

QED.