Thoughts on philosophy of mathematics

by Neil Rickert

It is usually said that most mathematicians are mathematical platonists or mathematical realists (those two are often considered to be the same).  It should be pointed out, however, that most mathematicians have not studied a lot of philosophy so what they say about their mathematical philosophy is not always well thought out.  Perhaps that same comment applies to what I am posting here.

There are good accounts of the philosophy of mathematics in both Wikipedia and the Stanford Encyclopedia.  A major issue in the philosophy of mathematics is the question of what are mathematical entities such as numbers.  Most mathematicians agree that mathematical entities are abstract, so are not physical.  The platonist view is supposedly that numbers exist in a platonic world of ideal forms.  Personally, I take numbers and other mathematical entities to be useful fictions.  I suppose that makes me a fictionalist, though others might consider me a platonist.  But then the platonic world of forms seems like a fictional place.  My main disagreement with platonists is over the Continuum Hypothesis.  As Gödel and Cohen have shown, the continuum hypothesis is independent of the other axioms of set theory.  I see that as settling the issue, while many platonists believe that there is a fact of the matter which we have yet to settle.

Some philosophers seem to think that the status of numbers and other entities has important implications for mathematical knowledge, or for the question of whether there is such a thing as mathematical knowledge.  And some think that this also raises questions about why mathematics has turned out to be so useful to science.  Yet, to me, it is quite clear that there is mathematical knowledge, and it seems quite apparent as to why mathematics is useful in the sciences.  And whether or not numbers are fictions, platonic entities, or anything else has little relevance to those issues.

I intend to discuss mathematical knowledge, and the relation of mathematics to the sciences in future posts.  The current post is a warm up exercise.

I tend to see mathematical formalism as a kind of fallback position.  That is, mathematics can be described as if it consisted of playing formal games with symbols.  But I doubt that many mathematicians actually think about mathematics in that way.  Similarly, I suspect that most mathematicians would find logicism (the view that mathematics is a part of logic) to be implausible.

I see intuitionism and the constructivism of Errett Bishop as interesting, and as a bit like doing mathematics with one’s hands tied.  I’m sure that Bishop didn’t see it that way.  However, I am not at all persuaded by the philosophical arguments often used to support intuitionism and constructivism.

That’s a casual overview of how I see mathematics.  I shall be more detailed in some upcoming posts.

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5 Comments to “Thoughts on philosophy of mathematics”

  1. John Crothers permalink
    Please Note: Your comment is awaiting moderation.

    We now know that our eyes function in some ways like a digital camera. It’s in fact easier to understand this by considering a high resolution VDU screen. The images appear to be continuous, but of course we know they’re made up of discrete pixels.

    The reason our vision illudes us into thinking the VDU image is a continous one goes to the nature of our eye retinas. The eye retina is made up of millions of quite discretely separate light receptors. These are about twenty times more dense at the centre.

    When Euclid the ‘first mathematics professor’ set about writing his Elements, he was creating a paradigm of the physical world. The ancient Greeks didn’t know about digital cameras or the make-up of the eye retina. They believed what they saw was a kind of absolute reality instead of merely a construction in the brain based on the limitations of our senses.

    So mathematics developed under this illusion of continuity. If we had far fewer light receptors we’d still think we were seeing everything there is to see. Our eye-brain connection always gives us the illusion that we’re seeing all there is to see. We can only see with our light receptors. As long as they are sensing light we thing we’re seeing everthing. We only have to look at an object in the distance, then compare this with what we see as we get closer to realise that we are in fact not seeing ‘everything’. Yet mathematics is prefaced on this illusion of continuity.

    I’ve spent a lifetime (now turning 62) working on understanding how the discrete fits in with the illusion of continuity. We in fact only have an impression of ‘space’ via the relationships we see between discrete objects (objects that have no apparent connection).

    Building a form of quantitative relational analysis was envisaged by Gottfried Wilhelm Leibnitz (the famous Liebniz-Clarke correspondence of 1815 – 1816) where it is believe Clarke was writing on behalf of Newton.

    Newton was to later write ‘Hypotheses non fingo’ – I do not feign hypotheses, regarded by many as a response to the writings of Leibnitz, the co-founder of modern calculus.

    The mathematical community will not tolerate any challenge to its mainstream philosophy – the dense space continuum hypothesis. It really is an hypothesis because there is an unwritten rule that this concept applies in to the physical world.

    Science needs a new paradigm of quantitative analysis for understanding nature. Mathematics does not explain natural phenomena, it merely mimics it. As much as scientists keep asking for a form of analysis that will aide them in understanding the why of natural phenomena, mathematics simply ignores them.

    Mathematicians are in general not scientists. Indeed they generally look down their nose at science as imperfect and ever changing.

    We as humans need to finally challenge mathematical philosophy but there is no forum on earth at any level in which to do so.

    Mathematical philosophy always begins with the ASSUMPTION that a dense space concept is beyond question.

    Scientists, who like all of us were drilled with mathematics at school, are incapable of challenging mathematics. As much as they like to think their experiments are independent of mathematics (its just a ‘tool’ of the sciences) they are not.

    They too see the physical world as they’ve been instructed – as a space-time continuum.

    Nature behaves relationally as we do. Mathematics in no way reflects this fact.

    Mathematics takes a kind of God-like view of the universe, but each creature of nature only sees and measures its environment according to its internal workings. These workings obey universal protocols, but they are certainly not based on any notion of space or time.

    We live in an ancient world that remains intolerant of fundamental questioning. At least in the time of Euclid, voices of descent like those of Zeno could still be heard.

    We’re a society going backwards. Our international economy is a reflection of centralised mathematical philosophy. We simply won’t tolerate fundamental questioning.

    John Crothers john.crothers@bigpond.com

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    • A quick note on moderation.

      The wordpress defaults are that the first comment goes into the moderation queue. Once that has been approved, future comments by the same poster are accepted immediately (unless the spam filter thinks they are spam).

      I mention this, because you seem to have posted the same comment twice. I approved the second one. The first is still in the queue. I’ll delete that later. But if you prefer, then I can delete this one, and approve the other.

      I’ll respond later to the content of your comment.

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    • I disagree with a lot of what you suggest. I’ll go through it item by item.

      We now know that our eyes function in some ways like a digital camera.

      You may “know” that. I don’t. I believe it to be false.

      So mathematics developed under this illusion of continuity.

      Mathematics is not a description of reality. Mathematics uses idealizations. Even though reality might be discrete, it can still be useful to idealize as if it were continuous. Much of probability theory was developed to analyze discrete problems (tossing coins, shuffling cards, etc). Yet the theory makes a lot of use of continuity because that kind of idealization can be valuable.

      Building a form of quantitative relational analysis was envisaged by Gottfried Wilhelm Leibnitz (the famous Liebniz-Clarke correspondence of 1815 – 1816) where it is believe Clarke was writing on behalf of Newton.

      An admittedly minor point, but I think you are off by 100 years.

      The mathematical community will not tolerate any challenge to its mainstream philosophy – the dense space continuum hypothesis.

      I’m not sure where you get that idea. Constructivists and intuitionists reject all such talk, yet their way of doing mathematics is considered respectable.

      Mathematics does not explain natural phenomena, it merely mimics it.

      Most mathematicians make no claim to be studying nature. They fully understand that they are studying imagined idealized perfect systems.

      We as humans need to finally challenge mathematical philosophy but there is no forum on earth at any level in which to do so.

      It is easy enough to setup forums on the Internet. You could start a wordpress blog of your own, and all it costs is your time. The hard part is finding people who are interested in it.

      I’m writing this blog mainly for myself. I have only a small readership. However, writing out my ideas is a good exercise that forces one to think clearly about what is being written. And any comments, even if few in number, can stimulate further thought and analysis.

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  2. Dear Neil

    So you’ve spent a lifetime, like me, on relational arithmetic? Yes, Liebnitz – Clarke was 1715 – 1716, so sorry. This must prove my incompetence.

    There is NO scope to write a paper on relational arithmetic. What is it, you say? Well, try asking any mathematician for a coordinate form of Euclid’s algorithm. Even when shown this, what’s their reaction? Shrug of the shoulders. Why? Because they have no idea about how to interpret this or what to do with it. It’s completely outside their “normal” forms of inquiry.

    Mathematicians know their “dense space continuum” concept produces assumed domains in all kinds of “dimensions”. It is however an absolutist notion. Every animal has an internal model of its environment, allowing it to interact with this environment in a quantitative way. It’s based on finite one-to-one correspondences between “things” (perceived entities and events).

    If you go to the ordered field laws of mathematics you can find the first unreality. Numbers in an ‘infinite’ domain DIVERGE and divergent quantities cannot be ORDERED.

    Where to go from there? Well, you look to how the arithmetical processes work in REAL OBSERVERS. They have perceptions of “convergence”, but it’s always finite convergence, based on finite one-to-one correspondences.

    We cannot get people to start looking at this, because, as I said above, this will not be tolerated.

    Which mathematical journal to write this in? None. Which scientific journal? Well, their first response will be to ask: And where is this theoretical arithmetic written up in a mathematics journal.

    We live in an ancient world, more intolerant of independent thought than ever before.

    We are going backwards.

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    • There is NO scope to write a paper on relational arithmetic.

      All it takes, is pencil and paper.

      Your real complaint is not about the writing question. Rather, your complaint is that you will have very few eager readers.

      Why? Because they have no idea about how to interpret this or what to do with it. It’s completely outside their “normal” forms of inquiry.

      If you want to change that, then you will need to market your ideas.

      Mathematicians know their “dense space continuum” concept produces assumed domains in all kinds of “dimensions”. It is however an absolutist notion.

      I don’t see that absolutism.

      Every animal has an internal model of its environment, allowing it to interact with this environment in a quantitative way.

      I very much doubt that.

      Clearly, we have very different philosophies. We have different philosophies of mathematics, and we have different philosophies about our position in the world and our relation to that world.

      Which mathematical journal to write this in? None.

      Journal editors take the view that they should publish what is likely to interest their readers. That seems sensible to me. So your problem is that editors don’t think what you have to say is of much interest. Again, that’s a marketing problem that you will need to solve.

      We live in an ancient world, more intolerant of independent thought than ever before.

      I don’t see that alleged intolerance. For sure, there are intolerant people. There always have been. But, overall, I’d say that we live in tolerant times.

      The world has no obligation to take you seriously. You have to persuade the world that what you are doing is important and should be taken seriously.

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