tag:blogger.com,1999:blog-5801278565856116215.post8144825443699734308..comments2016-11-10T13:57:43.522+00:00Comments on Disagreeable Me: Mathematical Platonism Is True Because it is UsefulDisagreeable Mehttp://www.blogger.com/profile/15258557849869963650noreply@blogger.comBlogger21125tag:blogger.com,1999:blog-5801278565856116215.post-11446364669617829792014-09-04T10:44:08.592+01:002014-09-04T10:44:08.592+01:00I'm sure I've never had a particularly ori...I'm sure I've never had a particularly original thought (original thoughts are increasingly rare after thousands of years of great philosophy), so it's not surprising to me that Carnap has expressed similar ideas.<br /><br />However, where I think I differ from Carnap is that he thinks you can adopt this convention without committing to a Platonist ontology. On the other hand, I think that's all a Platonist ontology is. I think ontologies are nothing more than conventions for what to consider as existing and what to count as unreal. I don't think the concept of existence is a clear, objective thing that has one preferred definition.<br /><br />I would also focus less on Platonism as a linguistic convention and concentrate a little more on it as a perspective or way of thinking about things.Disagreeable Mehttp://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-76244161615670319992014-09-02T23:56:16.959+01:002014-09-02T23:56:16.959+01:00Forgot the link.. http://www.ditext.com/carnap/car...Forgot the link.. http://www.ditext.com/carnap/carnap.htmlRobinhttp://www.blogger.com/profile/16015911138886238144noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-80779394575778434272014-09-02T23:55:45.620+01:002014-09-02T23:55:45.620+01:00Hi,
"Perhaps we don't need mathematical e...Hi,<br />"Perhaps we don't need mathematical entities as a way to explain the usefulness of mathematical tools. I didn't say we did. I'm just noting some similarities between some things which exist (including usefulness as tools) since you want the concept of existence to be unified. But we can drop this point because you're right that we don't directly use mathematical objects to do anything."<br /><br />I think that Carnap covered this - you don't need to worry about consistency as long as you define your convention at the outset. <br /><br />Say mathematical objects exist if it is useful for what you are doing, but in the cases where it will cloud rather than clarify then drop the terminology.<br /><br />(I posted a similar message earlier but I am not sure if it got lost - so there might be some duplication).Robinhttp://www.blogger.com/profile/16015911138886238144noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-6466688414488447142014-08-25T19:51:06.388+01:002014-08-25T19:51:06.388+01:00Hi Sam,
Firstly I must apologise if I appear to b...Hi Sam,<br /><br />Firstly I must apologise if I appear to be contradicting myself and jumping between all kinds of contrasting positions. Let me try to clarify a few things.<br /><br />Utility is important because I regard Platonism as useful: it gives us the mental framework to discuss mathematical objects intuitively and is potentially important in resolving other questions, e.g. in philosophy of mind. The utility of mathematical objects themselves is not important. I hope you see the distinction. Since what exists is ambiguous, we should choose a useful definition, which is not necessarily to say that only useful things exist.<br /><br />Perhaps we don't need mathematical entities as a way to explain the usefulness of mathematical tools. I didn't say we did. I'm just noting some similarities between some things which exist (including usefulness as tools) since you want the concept of existence to be unified. But we can drop this point because you're right that we don't directly use mathematical objects to do anything.<br /><br />I think I would stand firm on the view that we discover things about mathematical objects, because we are constrained in what we can learn (and in ways which I think cannot be explained by features of the world or limitations of imagination). We can't, for example, find any arbitrary shape we want in unexplored regions of the Mandelbrot set. There are little corners of it that have never been seen by human eyes, but if I am the first to inspect one I am not free to create whatever I like there. We start out with a simple definition of a function (which you may view as an act of creation), but where we take it from there is very much an act of exploration -- we are exploring what that function entails and it may well surprise us. So exploration and creation do not have to be mutually exclusive.<br /><br />Although I think interpreting mathematical objects as created is problematic. I think any two mathematical objects which are isomorphic are the same object -- my number two is the same as your number two. If two mathematicians discover the same thing independently (as Newton and Leibniz did with calculus, for example), then it's hard to call it an act of creation by either. It cannot have two independent creators at two different points in time and space, I would say, but it can have two independent discoverers.<br /><br />"Can be thought" about is extremely broad, but that's approximately how broad my ontology extends, because I hold abstract objects to exist.<br /><br />From a Platonist stance, Lao Tzu, for example, is certainly real, but the label is overloaded. It refers both to a concept of a person and to a person who may not have existed who is described by that concept. The concept exists. The person may not have.<br /><br />However, you are right that this forces us into cumbersome language, so it fails the utility test -- but this is not so for mathematical objects. There is no question about whether there are any actual mathematical objects in addition to the concepts, because the concepts and the mathematical objects are one and the same.<br /><br />So, while I would say that Lao Tzu exists from a Platonist perspective, I would not normally adopt the Platonist perspective when discussing (potentially) fictional characters. This ties in with my view that there is no right or wrong answer on whether abstract objects exist. It depends only on which perspective you choose to adopt.Disagreeable Mehttp://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-22129408362532070542014-08-25T09:45:17.564+01:002014-08-25T09:45:17.564+01:00"Right, so the real is a deeply primordial co..."Right, so the real is a deeply primordial concept, rooted I would say in evolved intuitions. This means that it is not necessarily a terribly coherent or well-thought out concept. If it is a concept we wish to make robust, we must define it so as to include objects it is useful to regard as existing and exclude others."<br /><br />I agree with this. However, I do not think the rest of your reply pays enough heed to the constraining role which 'useful' plays in the final sentence.<br /><br />You give 3 criteria under which we could call mathematical entities real:<br /><br />1. they can be thought about<br />2. we can discover things about them<br />3. we can use them as tools<br /><br />(I've taken explorability to be the same as discoverability --- I hope that's fair, correct me if not.)<br /><br />Working backwards, I don't think it's right to say that we use mathematical entities as tools. We use mathematical truths --- theorems and formulae -- as tools, and the question is whether we need to posit mathematical entities to account for mathematical truth (whether this truth is cashed out in terms of utility or not).<br /><br />To say that we discover things about mathematical entities seems to me to just beg the question against mathematical anti-realists, who would say that mathematics is not discovered at all, but created. (With the nominalists adding that this creativity is constrained by features of the world, and fictionalists adding that this creativity is constrained by features of us as creators). <br /><br />That leaves us with ‘can be thought about’, which seems to me to be way too broad to be useful. (On a side-note, if this is all it takes for something to be real, then on what grounds can we say that libertarian free will does not exist?)<br /><br />The example of fictions like Sherlock Holmes illustrates this. Say we take all fictions to be real, then we ask the question: was Lao-Tzu a real person? Some think that he was, others think he was invented to impose some narrative unity on a heap of disparate ancient Chinese sayings. <br /><br />This is clearly a substantial dispute (i.e. there is a fact of the matter), but if we take all fictions to be real, we can’t make good on it by asking whether Lao-Tzu was real or not. Instead we have to ask something like “was Lao-Tzu real in the sense of being a fictional person, or was he real in the sense of being a historical, physical person?” In other words, treating fictions as real does nothing to alleviate any dialectical tension — it just forces the same old problems to be translated into new (cumbersome) language. And this is why it fails on utility.<br /><br />You said earlier in the thread that you thought utility was irrelevant (I’m not quite sure at this point where you stand on that point), so maybe all of the above will seem off-point. But then the question remains of how we can make the concept of the real robust?<br /><br />SamSam Lhttp://www.blogger.com/profile/06005642014360940030noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-59372104568060294292014-08-22T18:41:14.214+01:002014-08-22T18:41:14.214+01:00Right, so the real is a deeply primordial concept,...Right, so the real is a deeply primordial concept, rooted I would say in evolved intuitions. This means that it is not necessarily a terribly coherent or well-thought out concept. If it is a concept we wish to make robust, we must define it so as to include objects it is useful to regard as existing and exclude others.<br /><br />There are plenty of ways in which abstract objects are similar to physical objects. They can be thought about. They can be explored. We can discover things about them by studying them. We can build a body of knowledge about them. We can use them as tools for practical purposes. Like places or substances, different people can discover them independently.<br /><br />The number of similarities is such that I think it sensible to class as real all objects with these properties, both abstract and physical. This includes even fictional characters and contradictions.<br /><br />For instance, there really does exist a character called Sherlock Holmes, but that is not to say that there is a physical person called Sherlock Holmes. There exist contradictory descriptions of objects such as "square circles" or "greatest prime", but there are no such actual objects. I don't think you can ever really think about the greatest prime, I think you are thinking about the concept of a greatest prime, which is a role that is not filled by any object. If you could really think about the greatest prime itself, that would mean you could explore it, for example discovering what its final digit is. But you cannot, you can only imagine what its final digit is, in which case you are simply amending the contradictory description that is the actual object of your ruminations.Disagreeable Mehttp://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-58817807274815695112014-08-22T16:05:07.632+01:002014-08-22T16:05:07.632+01:00"Rather my point is that it is a mistake to t..."Rather my point is that it is a mistake to think that reality is a well-defined concept. It simply isn't the case that there is a correct answer to the question of whether mathematical objects are real, because "real" means different things to different people."<br /><br />I agree that reality is not well-defined. The point of ontology is, as I see it, is to map out the relationships between all the areas in which we tend to take things to be real: mathematical, physical, moral, whatever. As Wilfrid Sellars put it, "the task of philosophy is to articulate how things in the broadest sense of the term hang together in the broadest sense of the term." A successful ontology will unify the concept of the real, or argue for its intrinsic fragmentation. You can grant reality to anything which people call 'real' if you like, but this doesn't give you a unified concept of the real. It gives you lots of different concepts of the real with none of the links between them articulated, which is what we had in the first place anyway. <br /><br />So when we discuss whether mathematical entities exist, what we are doing is asking how our usage of 'exists' in those cases relates to our usage of 'exists' in all the other cases in which it's used. Broadly, to be an anti-realist is to say that it doesn't relate in any substantial way, and that the other, non-abstract cases are more paradigmatic examples of the real. So the debate over mathematical Platonism doesn't rely on any pre-fixed notion of the real, and neither is it an arbitrary ascription (we can ascribe it arbitrarily, but this is to miss the issue). The real is rather a deeply primordial concept whose content is articulated dialectically.<br /><br />Sam Sam Lhttp://www.blogger.com/profile/06005642014360940030noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-36423265733026794092014-08-22T15:08:23.276+01:002014-08-22T15:08:23.276+01:00I read your article earlier. I didn't comment ...I read your article earlier. I didn't comment because I don't have any strong views on it. I don't regard my position as being based on indispensability arguments. In fact I find indispensability arguments quite weak for the same reasons you do.<br /><br />Rather my point is that it is a mistake to think that reality is a well-defined concept. It simply isn't the case that there is a correct answer to the question of whether mathematical objects are real, because "real" means different things to different people. The whole debate is the result of an evolved intuition that turns out to be too vague and/or ambiguous to be useful until clarified.<br /><br />Since even nominalists find themselves using the language of existence when discussing mathematical objects, it seems to me there is no reason not to define "reality" so as to include them.<br /><br />I know you have are not familiar with my thoughts on the philosophy of mind or the universe, so I'll put it this way. It is sometimes suggested that the universe might be a simulation. If this were true, would it follow that it does not exist or it is not real? We might say so, but we might also say that it is real and what it is is a simulation, that I exist and what I am is a virtual person. Since the meaning of "real" is up for grabs, I think it is more intuitive to adopt a definition where I can say with confidence that I am real and I exist, whether or not I am in a simulation. Otherwise, "cogito ergo sum" would not follow.Disagreeable Mehttp://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-10848660661044372322014-08-22T14:54:54.627+01:002014-08-22T14:54:54.627+01:00Right, that is a different argument (actually it&#...Right, that is a different argument (actually it's what I was talking about when I mentioned there being prima facie arguments for Platonism). Indispensability arguments (noting that you need not just that existence is useful in maths, but also that maths is useful in general) of that type are valid, but ultimately not that strong, in my view. They offer a prima facie case, so in the absence of objections they would be enough. But there are loads of objections to mathematical Platonism (the argument from epistemological access is particularly damning). I wrote something about this recently, so won't go into it at length (in case you're interested, it's quite short: http://theplatopus.com/2014/07/12/on-the-indispensability-of-mathematical-objects/)<br /><br />Don't really have any comments on your final paragraph, as it looks like there's a whole network of other stuff going on there which I don't want to guess at.<br /><br />Sam<br />Sam Lhttp://www.blogger.com/profile/06005642014360940030noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-50411351418955487922014-08-22T14:36:29.253+01:002014-08-22T14:36:29.253+01:00Hi Sam,
I'm not trying to say that mathematic...Hi Sam,<br /><br />I'm not trying to say that mathematical objects are real because they are useful -- I don't think utility has much to do with reality. Plenty of mathematical objects are not at all useful and that makes them no less real.<br /><br />What I am saying is subtly different. I am saying Platonism is correct because it is useful to use the language of existence when discussing mathematical objects. A nominalist mathematician says "there exists a prime > 8" but when pressed might admit that she is using the word "exists" in a metaphorical or fictional sense. A Platonist says there is no need for such caveats. It is perfectly reasonable to say mathematical objects actually exist and are actually real as long as we do not narrowly equate reality or existence with physicality.<br /><br />The question of whether they are real or not is simply unanswerable, because it is meaningless unless you specify what kind of reality you're talking about. If you prefer a narrow definition, that's fine, but if I'm right about the philosophy of mind and the nature of the universe that means that neither of us are real and neither of us exist. If you think that's OK, then that's fine by me.Disagreeable Mehttp://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-92048669713358857952014-08-22T14:07:48.149+01:002014-08-22T14:07:48.149+01:00Hi, thanks for your reply.
“So the point of this...Hi, thanks for your reply.<br /><br />“So the point of this post is to dispel the idea that there is anything mystical about Platonism. Mathematical objects are not ghostly forms floating around in a void on some other plane of existence. It's just an attitude about what we consider real. With this clarified, I hope to build arguments for the broader metaphysical view.”<br /><br /><br />I agree that there’s nothing mystical about Platonism - I think there some good prima facie arguments for Platonism, though I think ultimately they fail in the face of its problems. However, Platonism does entail that there are such things as abstract entities which are not physical (nor, on some accounts, causally active). These don’t need to couched in material terms like ‘ghostly’ and ‘floating’, but they need to exist for Platonism to be true. <br /><br />Jumping to the top: <br /><br />“If you believe that it is appropriate to say that a mathematical object exists, but that there is not necessarily a real entity which matches that object, is this not equivalent to saying that it is possible for something to exist but without it being a real entity?” <br /><br />There's nothing contradictory here - it’s to say that ‘exists’ as used in the mathematical context doesn’t depend on general philosophical theories of existence. (Hence why mathematicians don’t need to care about the mathematical realism debate to keep doing what they do.) <br /><br />This is already illustrated when you say “I view mathematical objects as real because it is useful to do so.” When a mathematician says “there is a prime number > 8”, they are not saying “it is useful for me to believe that there is a prime number > 8”, they are just saying that there is one. So here what you’re offering is an instrumental criteria for mathematical truth, and by proxy the reality of mathematical entities. This is fine, but it most certainly isn’t Platonism.<br /><br />Or perhaps the point is more general: perhaps all it means for anything to exist is that it is useful to view it as if it does. This ‘global instrumentalism’ is something I have deep sympathies with, but it doesn’t really help us with the debate over mathematical realism. All it does is change the question from ‘do mathematical entities exist?’ to ‘in what sense are existential statements in mathematics useful?’ <br /><br />Things can be useful because they represent the world (e.g. maps), or for other reasons (e.g. hammers). From the point of view of global instrumentalism, to ask whether mathematical Platonism is true is to ask whether mathematical truths (including existential truths like “there exists a prime > 8”) are useful in virtue of representing the world, or in virtue of something else. So the question being debated by philosophers of mathematics is left largely untouched by this line of thought.<br /><br />The other thing about instrumentalism is that even if true, it does not mean that it is "just an attitude about what we consider real." In particular, if to be real is to be useful, then what is real is non-arbitrary, because it is constrained by what is useful.<br /><br />Sam<br />Sam Lhttp://www.blogger.com/profile/06005642014360940030noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-24376851001030516442014-08-22T12:15:20.248+01:002014-08-22T12:15:20.248+01:00Hi Sam,
Thanks for commenting.
If you believe th...Hi Sam,<br /><br />Thanks for commenting.<br /><br />If you believe that it is appropriate to say that a mathematical object exists, but that there is not necessarily a real entity which matches that object, is this not equivalent to saying that it is possible for something to exist but without it being a real entity? To me, this is a contradiction, because an entity is just something that exists.<br /><br />Or perhaps you would say there is such an entity, but it is potential rather than real? OK, but to me it's just a rephrasing of the question -- now instead of asking what exists we are asking what is real. To me, those are essentially the same question. I view mathematical objects as real because it is useful to do so. There is no fact of the matter on whether they are real or not because it depends only on how you define reality.<br /><br />As with free will anti-realism and compatibilism, I view the question of mathematical realism as semantic only. I don't think there is a profound distinction between the metaphysics of nominalism and Platonism, in the sense that both approaches can be valid at the same time. However it does seem that the two attitudes entail profoundly different ways of thinking about mathematics.<br /><br />And there are other reasons to prefer Platonism. For various reasons explained elsewhere on this blog I think the mind is a mathematical object and also that the universe is a mathematical object. Ultimately I am a mathematical monist. If my metaphysics are right (which most people will surely deny), then it turns out that nominalism is semi-incoherent and Platonism is required if we want to say anything exists at all.<br /><br />So the point of this post is to dispel the idea that there is anything mystical about Platonism. Mathematical objects are not ghostly forms floating around in a void on some other plane of existence. It's just an attitude about what we consider real. With this clarified, I hope to build arguments for the broader metaphysical view.Disagreeable Mehttp://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-23363480100918678272014-08-22T11:44:48.302+01:002014-08-22T11:44:48.302+01:00Had a few thoughts on this, though I appreciate it...Had a few thoughts on this, though I appreciate it's from a while ago. (Already wrote one reply but it got wiped when I logged in to post, bah!)<br /><br />You seem to be arguing for mathematical platonism on the basis that propositions like "there exists a prime > 8" uses the term ‘exists’ in an appropriate way (presumably because it's true, and it's hard to see how a term used in a true statement could be being used inappropriately).<br /><br />Now, I think the word 'exists' is being used perfectly appropriately there, but I am not a mathematical Platonist. I understand Platonism as the view that there exist real entities to which the terms in mathematical propositions refer (for example the numeral ‘8’ in the above example refers to the entity: 8), and it is the real relations between these real entities which account for mathematical propositions being true or false - i.e. we can take them literally. (These real entities need be analogous with, say, physical entities, only insofar as we can apply the word ‘real’ to both— I would say that’s the minimal condition on being a Platonist.)<br /><br />A constrasting picture would be offered by something like nominalism, which holds, roughly, that a the number 8 is just the set of real entities which instantiate certain properties — the additive and multiplicative properties we associate with ‘8’. This differs from Platonism in several respects:<br /><br />1. There are no 1-to-1 mappings between mathematical terms (like the numeral ‘8’) and real entities, though there may be 1-to-many mappings. <br /><br />2. Such relationships, if they exist, need not exist: the set of real entities instantiating 8 may not have existed, in which case 8 would not have existed. 8 does exist, but an incomprehensibly large number, like, say, Graham’s number to the power of itself, does not exist.<br /><br />3. Mathematical propositions are true not in virtue of real relations of real entities, but of possible relations between sets of possible entities. So the proposition "there exists a prime > Graham’s number to the power of itself" is true, not because there is a set of real entities instantiating that property (there isn’t, it’s too big), but because there might have been.<br /><br />OK, so none of that is likely to sell nominalism, but I do think it illustrates the contrast between the sort of thing that Platonism wants to say and the sort of thing that anti-realist theories of mathematics which take mathematics seriously want to say.<br /><br />Perhaps you’re using Platonism in a broader sense than I am, in which case that will all ring hollow for you, but then the question can just be rephrased in terms of what sort of Platonists we are. Make any sense?<br /><br />SamSam Lhttp://www.blogger.com/profile/06005642014360940030noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-55075258979338575992014-02-22T12:18:42.327+00:002014-02-22T12:18:42.327+00:00Hi Steve,
Thanks for your comment.
Of course you...Hi Steve,<br /><br />Thanks for your comment.<br /><br />Of course you have never encountered a mathematical object that wasn't physically instantiated - the act of perceiving such an object would mean it is physically instantiated in your brain in some form.<br /><br />But that doesn't explain how we ought to view independent discovery or how it is that the very same mathematical concept can have very different physical instantiations and still be held by you to be identical to those disparate physical instantiations.Disagreeable Mehttp://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-32494412804300202392014-02-21T17:12:21.744+00:002014-02-21T17:12:21.744+00:00I'm with Richard, I'm afraid. I have never...I'm with Richard, I'm afraid. I have never encountered a mathematical object that wasn't instantiated in a real physical system. I have an intuition that what seems abstract is simply the human mind doing what it's so good at - recognising patterns. In other words, being intuitive. Now, there's a paradox for you :)<br /><br />(Steve Morris)Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-70437919691899242762013-12-15T12:14:07.348+00:002013-12-15T12:14:07.348+00:00Fair enough, Philip, but now you sound like a prog...Fair enough, Philip, but now you sound like a programmatic Platonist though you deny mathematical Platonism.<br /><br />I do feel that all possible programs exist. Do you also? If so, then why do you feel mathematical Platonism is wrong?Disagreeable Mehttp://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-84565391770216576512013-12-15T02:01:37.084+00:002013-12-15T02:01:37.084+00:00If the set of programs can include hyperprograms o...If the set of programs can include hyperprograms or transfinite programs (<a href="http://www.hypercomputation.net/" rel="nofollow">http://www.hypercomputation.net</a>), then that set might express all possible mathematics.Philip Thrifthttp://www.blogger.com/profile/03021615111948806998noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-78812185885630615102013-12-14T22:57:30.983+00:002013-12-14T22:57:30.983+00:00Hi Philip,
I don't know anything of Solomon F...Hi Philip,<br /><br />I don't know anything of Solomon Feferman.<br /><br />I think any variant of the MUH/CUH which does not embrace Platonism faces a couple of problems<br /><br />How is it that the universe exists? Why is it fine-tuned?<br /><br />Platonism answers both questions because in Platonism all mathematical objects exist.Disagreeable Mehttp://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-73728181004187694752013-12-14T16:15:30.247+00:002013-12-14T16:15:30.247+00:00Just my point of view: It appears to me that a con...Just my point of view: It appears to me that a constructive, programmatic version of MUH (CUH or PUH, which Tegmark seems to really advocate from what I've read) can be embraced by a self-defined anti-Platonist (like Solomon Feferman, who just turned 85!). Is that enough for science? Who knows.Philip Thrifthttp://www.blogger.com/profile/03021615111948806998noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-5282624592107828982013-10-04T13:49:07.172+01:002013-10-04T13:49:07.172+01:00Hi Richard,
Once again, thank you for your detail...Hi Richard,<br /><br />Once again, thank you for your detailed and thoughtful response.<br /><br />I agree that mathematical Platonism is not quite as useful as mathematics. I do think it's useful in allowing us to build a coherent, consistent picture of reality, but it's not going to be helping us to build bridges any time soon. As such, it's only useful to philosophical enterprises such as understanding consciousness or why the universe exists.<br /><br />I can understand why you think that talk of mathematical objects as existing or not is in some way incoherent. It isn't though if you just interpret "exist" in the same sense as the hypothetical mathematician you quoted. My point is that if the mind is a mathematical object, and if the universe is a mathematical object (both statements I believe but you do not), then this is a useful way to think of how it is that these exist.<br /><br />I wouldn't say a computational process is kind of related to mathematics, I would say a computational process really is a mathematical object.<br /><br />I would say a mathematical object is more than something that can appear in a mathematical statement. A statement is itself a mathematical object, and so is any abstract object or system that can be analysed mathematically. Some examples:<br /><br />The Cartesian plane<br />Peano arithmetic<br />An equation<br />A parabola<br />The natural numbers<br />The set of all mathematical objects<br />A sequence of characters (and so any block of text)<br />A function<br />An algorithm<br /><br />A computational process is an instantiation of an algorithm. Now, I recognise that there appears to be a distinction between an algorithm and its physical instantiation, and indeed I have done some hand-waving here. However I hope to explain in future why it is that the process itself can also be regarded as an abstract mathematical object. For now, you can take as a simple argument the idea that the character of the computation is substrate independent, that the mind would not be meaningfully altered if these computations were carried out by electronic rather than organic matter.<br /><br />The mind can also be thought of as the function that maps input nerve signals to output nerve signals.<br /><br />In any case, my purpose in this post is not to convince you that the mind is a mathematical object. Instead, I'm arguing that if I can convince you that the mind is a mathematical object, then mathematical Platonism is useful in reconciling this with our belief that our minds exist. It also helps us to understand the virtual minds response to the Chinese Room, and how it can be that matter can have phenomenal experience.Disagreeable Mehttp://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-5801278565856116215.post-89375927445345676792013-10-04T13:11:23.099+01:002013-10-04T13:11:23.099+01:00Hi Disagreeable,
Thanks for another interesting p...Hi Disagreeable,<br /><br />Thanks for another interesting post. I guess you won't be surprised to hear that I disagree with some of it.<br /><br />"Mathematical Platonism Is True Because it is Useful"<br /><br />As far as I'm concerned it's not mathematical platonism that's useful. It's mathematics that's useful.<br /><br />My objection to mathematical platonism is that I think its talk about the existence of mathematical objects is confused and confusing. It's much better to talk about the truth of mathematical statements. I'm on board with the fact that mathematical statements are (or can be) pure abstractions and that the truth of mathematical statements is observer-independent.<br /><br />I have no problem with the mathematician's talk of existence, e.g. "there exists an integer between 2 and 4". That makes sense as a true statement of an axiomatic system. But when you step outside of any axiomatic system and talk about the existence of mathematical objects in a more general sense, such talk ceases to have any meaning. The mathematician's talk of existence is useful. The mathematical platonist's is not.<br /><br />"The human mind, for instance, does not appear to be a physical object in itself, but is instead some sort of process which takes place in a human brain. If the computational theory of mind is true, then the human mind is a computational process, which is a kind of mathematical object."<br /><br />This sounds to me like rather loose hand-waving: a mind is kind of a computational process, and a computational process is kind of related to mathematics, so a mind is kind of a mathematical object. I suggest you need to think more carefully about what you mean by "mathematical object". As far as I'm concerned, if "mathematical object" is to mean anything useful it must refer to something that can appear in a pure mathematical statement, like a number, set, etc. I don't see how a mind can be taken in that way.<br /><br />Human minds are instantiated in real physical systems (brains). I would say that they supervene on those systems. They are not pure abstractions. Even if we talk about minds in a very general way (without thinking about any particular instantiation) our statements about minds are very different from the statements of axiomatic systems.<br /><br />I would suggest that we have to be careful in talking about abstractions. As far as I'm concerned all our statememts are abstractions to some degree. We model reality at various levels of abstraction. Our models of the mind are particularly abstract, but I wouldn't draw a fundamental divide between our models of physical objects and our models of mental objects (like beliefs and desires). (I'm not addressing consciousness here.) Our use of the word "physical" to describe the former but not the latter should not be attributed too much significance. Pure mathematical statements are, however, different from both, in that they are <i>purely</i> abstract. So mind-talk is more like physical-world-talk than like pure mathematics. It's a mistake to jump from "minds are abstract" to "minds are the same sort of thing as mathematical objects".Richard Weinhttp://www.blogger.com/profile/18095903892283146064noreply@blogger.com