Wednesday, 2 October 2013

Mathematical Platonism Is True Because it is Useful


I have briefly discussed mathematical Platonism in previous articles. This post is going to contain my current thoughts about it.

Mathematical Platonism is distinct from classic Platonism (so nothing about ideal forms, for instance) and holds that three propositions are true.

1. Mathematical objects exist.
2. Mathematical objects are independent of human beings.
3. Mathematical objects are abstract

Or, in condensed form, mathematical objects exist abstractly and independently of human beings. All possible mathematical objects exist, have always existed and will always exist, even if no mathematician ever ponders them.

In this post, I will argue that while there is no fact of the matter regarding whether mathematical objects exist, it is sensible and useful to treat them as if they do, and that this is enough to justify mathematical Platonism.

Mathematical Platonism is not Mystical

To me, mathematical Platonism seems correct as long as we don't confuse the notion of abstract existence with physical existence. As discussed previously, there can be more than one kind of existence.

In particular, nobody is saying that mathematical objects are floating around in some ghostly mathematical realm of existence (even though I believe this is how Plato visualised it). Instead, my view is that it's a completely different category of existence, and if you disagree with me then you simply have a narrower concept of what the term "existence" means. 

For example, it just seems to make sense to assume that when I say "There exists no rational square root of 3", the verb "exists" is meaningful and used appropriately. It is in this sense alone that mathematical objects exist. There really is nothing mystical about it, as much as some people perceive it this way.

It's not hard to see why this misconception about mathematical Platonism is so pervasive. The original motivation for positing the existence of mathematical objects was to explain by analogy to physical objects how mathematicians could perceive mathematical truths. It was supposed that mathematical objects were real, and could be perceived directly by the mind of mathematicians with some special kind of sense much as we see tables and chairs with our vision.

There were also ideas that we had originally come into the world from the realm of mathematics, and our perception of mathematical objects was some sort of memory of the things we had seen there before we had been born.

I don't think either of these ways of thinking about it make much sense, and this is not at all what I am proposing. In many ways it's unfortunate that the terminology of mathematical realism has become so inextricably bound to the philosophy of Plato, as much of his metaphysics don't really conform to the views of most modern mathematical Platonists.

No Fact of the Matter

Before I go any further, I would like to make clear that I don't actually think there is a single objectively correct answer to the question of whether mathematical objects exist. The idea of existence is dubious when applied to cases like this. It is at its most coherent when applied to physical objects in the universe such as the chair I am sitting on as I type. It seems without question that this chair exists. I can see it and touch it, and so could you if you were beside me. It is in contexts such as this that the concept of existence first arose.

There are many other kinds of entities which are not physical objects. Your mental experience, composed of thoughts, hopes and dreams is a good example. Though there may be physical analogues of these (e.g. the firing of neurons), it seems odd to suppose that the firing of the neurons is the same thing as the feelings they excite.

And yet few of us would deny that our minds or consciousness experiences are real. If this is so, it must be that these phenomena exist in some sense, although not in the same way that a chair does, as they are only accessible to ourselves.

Another kind of existence might be that of very high level emergent phenomena, such as nation states, languages, currencies, etc. Most of us feel that these things exist in a sense, even if they are only socially constructed. Multiple observers might agree that they exist, and yet nobody can perceive them directly. As such, it seems to me that whatever kind of existence they may have, it is not physical.

Examples abound in nature too. Photosynthesis is a process, not a physical thing. The idea of a species is a fuzzy high level descriptor which we apply to many different individuals. Genes are not single objects but patterns we find repeated in the cells of related organisms. All of these things have a basis in the physical, but they are not tangible objects, so they do not exist in the same way as a chair.

The idea of physical existence also breaks down when considering universes. Universes are not physical objects, they are the containers of physical objects. For a physical object to exist, it means that it must be present within the universe -- observers within that universe must be able perceive it or its effects to exist with their senses. The question of whether other universes exist is meaningless if we think of them as physical objects, because from our perspective they do not exist, as they are entirely causally disconnected from us. On the other hand, our universe doesn't physically exist from the point of view of an observer in another hypothetical universe.

And yet, most of us believe that the universe exists!

So we ought to be open to the idea that there are things other than the physical, including mathematical objects. Whether the concept of existence is applicable in all cases is a matter for only for linguistic definition. Clearly, each of these objects, processes and properties has an existence in some sense, even if not in the same way that a chair does. There is no fact of the matter - whether this pseudo-existence is real or not depends only on how one defines existence. Definitions ought to be judged on how well they capture intuitive concepts, on how consistent they are and on how useful they are for communication, and it is on these grounds that we should decide whether mathematical objects exist or not.

For me, there are two particularly striking arguments that convince me that mathematical Platonism is a useful and consistent way to think about mathematical objects.

The Argument from Exploration

Entities such as the Mandelbrot set and other such visually striking mathematical objects are particularly good demonstrations of the intuition that mathematical concepts exist. The Mandelbrot set is defined by deceptively simple mathematical rules. I imagine that Benoit Mandelbrot, the mathematician who first stumbled upon these rules, found himself surprised by the complexity and beauty he saw unfolding before him as he explored the consequences of his simple premises (and even if he were more sanguine, I can certainly imagine being pretty excited if I were in his place).

We see the same kind of thing in Conway's Game of Life, a simple mathematical system which turned out to have exciting and profound consequences and phenomena to be explored. Many hobbyists and mathematicians are still exploring this and other systems like it. Mathematician Stephen Wolfram (creator of the well-known Mathematica mathematics package) published a book in 2002, A New Kind of Science, all about exploring the kind of complex and surprising phenomena we can find emerging from even simpler such systems.

If the "inventor" can be genuinely surprised by his own "invention", then it seems to me that he is discovering truths about it that he did not put there himself. In my view, this can only be consistent with the view that this really is exploration of something that exists independently.

The Argument from Independent Discovery

I also like to bring up the idea of independent discovery by separate people of the same mathematical object, such as the discovery by Newton and Leibniz of the calculus. The controversy which arose over which of them had first discovered it is only sensible if we view it as discovery. If a mathematician is like a cabinet maker, creating something new whenever he invents a new mathematical tool, there would be no issue. Newton would have his cabinet and Leibniz would have his. The fact is that they both discovered the same thing, independently, like two explorers happening upon the same continent. This to me demonstrates that mathematical objects are independent of mathematicians. It also suggests that they exist, as otherwise there would be nothing for the two to discover independently, and so nothing to argue about.

It's all about Intuition

Now, admittedly, these two arguments I have sketched out are essentially arguments from intuition, and intuition can be very fallible indeed. I think this is not a problem in this case precisely because I maintain that there is no fact of the matter. "Existence" in its general sense is simply not a robust, objective concept we can all agree on, like mass or velocity. Instead, it's just a label for a human intuition. As such, we ought to apply it in cases which make the most intuitive sense, as long as we're not contradicting ourselves or the evidence before us.

The arguments from intuition I have presented I justify the utility and coherence of thinking of mathematical objects as existing.

The Significance of Mathematical Platonism

Since I think ultimately any disagreement must boil down to differences of interpretation of the word "existence", this is arguably a question only of semantics. As such, it may appear to be completely inconsequential and so irrelevant to any serious issues, but I feel it actually has profound implications for how we explain certain other phenomena we perceive as real.

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. Whether the computational theory of mind is right or wrong, it is at least plausible enough to convince a lot of people, including me. If you accept that it is at least plausible, then what would it mean for the existence of the mind if it were true?

Rejecting mathematical Platonism would mean that it is plausible that the human mind does not exist, or that it is an illusion or fiction. If we instead allow ourselves the mental tool of mathematical Platonism, then we get to accept both the computational theory of mind and that the mind exists. Whichever approach you choose should depend on which definition of existence leads to the most intuitive conclusions, and I think most of us prefer to think that our minds are real (even if not in the same sense as tables or chairs).

I think the same is true of the universe. As I explained before, the concept of physical existence doesn't really make sense when applied to universes. We can't really say that our universe objectively exists because we all have the special, subjective viewpoints of observers within it. What about other hypothetical universes? What about the fictional universe of Star Trek, for instance? It does indeed exist from the point of view of Captain Kirk, and there is no way he could tell that he or his universe do not really exist. It seems to me that we have no robust argument to show that our own universe is any more real than that of Star Trek's.

One relatively popular instance of this kind of view is Nick Bostrom's simulation argument, where he proposes that we could all be living in a computer simulation. Computer simulations are of course computational processes, which are a special kind of mathematical object.

If we imagine for a moment that this is in fact the case, then a denial of mathematical Platonism would amount to a denial that the universe exists, while allowing ourselves the mental tool of mathematical Platonism allows us to consider this possibility while maintaining that the universe would actually exist even if it were in fact a computer simulation. Again, whether we accept or reject mathematical Platonism should therefore be predicated on which definition of existence allows for the conclusion that makes the most intuitive sense.

So if you reject mathematical Platonism, that's fine. But if you do, and in particular if you are willing to assign a non-zero probability that a computer could have a mind or that the universe could be a simulation, then you ought to be ready to entertain the idea that you, your mind and the universe around you may not actually exist in any sense.

I prefer mathematical Platonism.

8 comments:

  1. Hi Disagreeable,

    Thanks for another interesting post. I guess you won't be surprised to hear that I disagree with some of it.

    "Mathematical Platonism Is True Because it is Useful"

    As far as I'm concerned it's not mathematical platonism that's useful. It's mathematics that's useful.

    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.

    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.

    "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."

    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.

    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.

    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 purely 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".

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    1. Hi Richard,

      Once again, thank you for your detailed and thoughtful response.

      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.

      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.

      I wouldn't say a computational process is kind of related to mathematics, I would say a computational process really is a mathematical object.

      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:

      The Cartesian plane
      Peano arithmetic
      An equation
      A parabola
      The natural numbers
      The set of all mathematical objects
      A sequence of characters (and so any block of text)
      A function
      An algorithm

      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.

      The mind can also be thought of as the function that maps input nerve signals to output nerve signals.

      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.

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    2. 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 :)

      (Steve Morris)

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    3. Hi Steve,

      Thanks for your comment.

      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.

      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.

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  2. 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.

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    1. Hi Philip,

      I don't know anything of Solomon Feferman.

      I think any variant of the MUH/CUH which does not embrace Platonism faces a couple of problems

      How is it that the universe exists? Why is it fine-tuned?

      Platonism answers both questions because in Platonism all mathematical objects exist.

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    2. If the set of programs can include hyperprograms or transfinite programs (http://www.hypercomputation.net), then that set might express all possible mathematics.

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    3. Fair enough, Philip, but now you sound like a programmatic Platonist though you deny mathematical Platonism.

      I do feel that all possible programs exist. Do you also? If so, then why do you feel mathematical Platonism is wrong?

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