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The Platonic World. Real or a human construct?

  1. Jun 19, 2006 #1
    Does the abstract mathematical world exist independently of the physical world? Roger Penrose, the author of "The Road to Reality", regards this as part of the Platonic world — and seems to incline to the view that it does have an independent existence, as some kind of eternal truth.

    I'd appreciate folk who visit this forum sharing their views on this question with a self-confessed mathematical heathen who thinks that:

    Mathematics is a language rooted in the fact that descriptions are often more useful when they are made quantitative. For example a man-made tool like the counting numbers (1,2,3,4 ....etc.) helps most folk to be practical. At its most primitive micro-level — arithmetic — the Platonic mathematical world consists of only these numbers and their operations.

    The complexities of the imagined mathematical world are now of course much, much greater. They have evolved, like the complexities of the similarly intricate world of music. Despite the misgivings of conservative folk over the years, the simple concepts of arithmetic have been hugely elaborated. For example, to start with, number systems have been modified to include useful inventions that aren’t the ratio of two counting numbers (like pi and e); the strange numbers zero and infinity, and negative numbers. And much else has followed, from imaginary numbers to quaternions and on to the wonders of modern mathematics. Some of the simpler abstractions have eased practical tasks like the counting of sheep, land surveying, building pyramids and the bookkeeping of abstract entities like money in the electronic accounts of a bank.

    But despite the sophistication of mathematics today I would argue that mathematics remains at heart an ephemeral human language that has been developed to satisfy a genetically-coded appetite forced upon us by evolution: a driving need to describe whatever we observe and whatever we can imagine.

    Neither the abstract constructs of mathematics nor the works of Mozart are part of the physical world. It contains no Pauli spin matrices, Euclidean circles or symphonies. These are constructs devised by human beings to satisfy their own peculiar needs; to devise logical systems, to describe physical phenomena or, in the case of music, to express human emotions.

    The Platonic mathematical world may manifest itself physically as neural patterns in mortal brains, squiggles on paper, binary digits in transit on the internet or patterns on computer screens. But all these are only patterns laid down on substrates that don't endure, but come and go with the years.

    I would conclude that the Platonic mathematical world is an ephemeral abstract construct that will perish with humanity in the fullness of time, whereas the physical world is likely to endure a little longer.
  2. jcsd
  3. Jun 22, 2006 #2
    Interestingly enough, I hold the counter-position on this matter. Based on the apparent beautiful harmony between mathematics and physics, it seems feasible to me that mathematics is physics - that our perception, our lives, our existence - is all simply a mathematical construct. We are so intertwined with nature so as to be nature; so is everything else. Why, for instance, has Newton's dynamics of gravitation worked so perfectly well - surely our mathematical "construct" has some relevance in the real world.
    Besides, if we take the view of mathematical nominalism, then we are forced into a position of claiming to have no knowledge of reality, which means we cannot possibly have knowledge of our own statement, which makes our position unsupportable. We must claim the three basic axioms of logic in order that our own statements are not lost in absurdity and contradictions.
    But perhaps I digress from the matter at hand. I think not, though, for what "supernatural existence" beyond mathematics (and its brainchild, the model called science) should we possibly claim logically, without claiming to have knowledge of what we claim to be unknowable and beyond the universal conformity of nature, and therefore absurd and again unsupportable. So for now, if no rational is provided, I shall not accept a world where the powers of mathematics and logic, of reason, cannot explain everything that includes the trait of existence.
  4. Jun 22, 2006 #3


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    I tend to take the anti-Platonist view, but honestly, there is no way to know and I don't see how it would make any difference to the practice or application of mathematics either way.
  5. Jun 22, 2006 #4
    um... i agree. great post. as to how it affects the practice of mathematics... who cares? we're talking about the nature of reality here.
  6. Jun 23, 2006 #5
    I would say that human thought supervenes on mathematical truth rather than the other way about.

    Thus, mathematical truth must precede human thought.

    Best Regards
  7. Jun 24, 2006 #6
    Thanks for the response. What you say is quite true with regard to mathematics.

    But the question I posed of whether "the Platonic mathematical world is an ephemeral abstract construct", or not, has a considerable bearing on the way theoretical physics is conducted these days, and more practically, on the way lots of money is spent on this enterprise. It might make some difference to discuss the point.

    Just suppose, without too much shock and horror, and only for the purposes of argument, that as I suggested "mathematics is at heart an ephemeral human language that has been developed to satisfy a genetically-coded appetite forced upon us by evolution: a driving need to describe whatever we observe and whatever we can imagine" .

    Then, for example, I see no pressing need to take too seriously the web of mathematical ratiocination woven by theoretical physicists that is string theory, just because this web grows untethered by the ropes of experimental reality. String theory (described caustically in Peter Woit's "Not Even Wrong") may just be an over-extrapolated human description of matters we can't touch experimentally, couched in a language that is fallible in the sense that it sometimes describes the physical world in different but equivalent ways (as with wave and matrix mechanics) and sometimes develops physically sterile dialects (like quaternions). Indeed it seems that string theory may be just elaborated mathematical nonsense, similar to the many other towers of logic we have over the millennia constructed on foundations of sand.

    And if there is no need to take string theory too seriously, why not spend the taxpayers scientifically directed money on more realistic endeavours, say like building better astronomical instruments?

    Re another comment:
    Maybe the practice of mathematics (unjustly) costs less than the practice of theoretical physics, so a whole heap of folk should care.
  8. Jun 24, 2006 #7

    matt grime

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    Your objection to string theory has nothing to do with platonism, or otherwise, of mathematics. Whether or not Real numbers are in any sense real has never had any affect on their use as the system of measurement we adopt when doing physics. The nature of i (as in the square root of minus 1) has no practical affect on how we do quantum mechanics. Similarly whether or not there actually are in any Platonic sense Riemann surfaces doesn't stop us using them to describe simple CFTs. Since you brought them up, whether or not matrices exist (or complex numbers) doesn't stop the standard model being used (probably with your approval as a real piece of physics).

    There are philosophical reasons to question the utility of string theory as a physical theory, but they tend to be 'what should be classed as scientific research?'

    I think you missed kcballer21's point. Whether or not you believe in platonic existences for any othe the symbols in sin(pi/2) does not alter how sin(pi/2) acts in an expression (it is 1). The evidence for this is that you can not tell whether any mathematician is a platonist or not merely from the mathematics they do (unless they happen explicitly say what they are, but they don't because it is not important for the practice of doing mathematics).
  9. Jun 24, 2006 #8
  10. Jun 24, 2006 #9
    You are partly right. Of course the use of mathematics in physics doesn't in practice depend on its Platonism or not. My objection to string theory arises from my being a member of the "Popperazzi", and is based on the belief that string theory seems unlikely to be falsifiable by experiment or observation. Dangerous belief, I admit.

    But if the Platonic world of mathematics were "real", in the sense of existing as some kind of eternal truth, rather than being only a transient human invention, one might have more respect and tolerance for string theory as she is practiced. If a Platonic world did exist string theorists might claim to be uncovering a similar kind of eternal truth and proceed on the basis that a link to experiment, although desirable, is in their special case redundant. I don't believe that such a loophole could be justified.

    But maybe this is a
    , as you say. And:

    . Maybe I did. My apologies to him.
  11. Jun 24, 2006 #10

    matt grime

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    a perfectly valid position, and one that many people share.

    No, I cannot ascribe to this view. This platonic realm, whether it exists or not has no relationship with physics. The mathematical underpinnings of string theory are exactly the same as the mathematical underpinnings of any other physical theory. I think you are seeking justification where there is none, and where there needs to be none. The philosophical short comings of string theory are well known and acknowledged; platonism is not one of them. If you have a problem with the mathematics of string theory, which is essentially some category theory, algebra, and complex analysis, then you have no right to think that particle physics or relativity is sound from this view point (not discounting other reasons why they are sound).

    but the platonic realm, whatever it maybe, and whereever it may exist, is not our physical universe, so your reservations *from this view* are unfounded (though there is no denying there are serious issues to be dealt with). The platonic realm is not this universe; post me a number 3 if it is....
  12. Jun 24, 2006 #11


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    I think what you mean to say, oldman, is that if the prevailing method of conducting scientific research was still the old geometrical method of Descartes, then String Theory would be accepted despite not being falsifiable. The mathematical physics of the old rationalists was related to mathematical Platonism in that both are rationally, derived rather than empirically tested.

    So, if we could envision a world in which mathematical Platonism was widely believed, we might think that world was predominantly rationalist, in which case science would also be rationally derived rather than empirically tested, and falsifiability would be irrelevant. Given this, sure, an acceptance of Platonism might entail a greater probability of the acceptance of String Theory.

    I really had to stretch to think of that, though.
  13. Jun 25, 2006 #12
    I agree with your sentiments here, but imho your argument is misdirected. What needs to be examined is not whether the concept of a Platonic realm has any meaning, but whether untestable hypotheses can contribute to knowledge or understanding of the world.

    Best Regards
  14. Jun 25, 2006 #13
    What you have written is helping me clarify my thinking, Matt, even though the quote above reveals that we probably still disagree. Many thanks.

    What I claim is this: although mathematics is indeed a neccessary and effective tool for quantitatively describing the physical world -- which is what physics does --- its use does not sufficiently guarantee success. Particle physics and relativity are sound because their predictions can be verified by observation and experiment. String theory cannot claim the same status just because its mathematical base is sound, which seems to be your view. It needs the further support of prediction and verification.

    Despite my respect for mathematics and mathematicians, I insist that their subject is nothing more than an ephemeral human language, in essence no different in from say, music or the twittering of birds. Mathematics belongs to the Platonic world which, as you say, is "not of this universe". And solitary ventures into this world, like string theory, run a considerable risk of leading nowhere. They have done so many times before.
  15. Jun 25, 2006 #14
    Yes indeed. Thanks for stretching ... exercise doesn't do one any harm. You have put my only venture into the muddy waters of philosophy more succinctly than I did.

    You may remember Henry Ford's aphorism: "History is bunk". Henry was no philosopher but was sure smart enough to be rich. Pity he isn't around to give us his judgement on string theory.
  16. Jun 25, 2006 #15
    If they didn't life on this planet infested with us chattering creatures would be rather dull!

    Thanks for your interest, and kind regards.
  17. Jun 25, 2006 #16
    maybe so - but a desire for an exciting life hardly qualifies as a logical philosophical argument in support of untestable hypotheses :smile:
  18. Jun 25, 2006 #17

    matt grime

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    it is philosophy, so there's no problem with disagreeing - it's not like these are facts.

    where have I said that string theory has a sound basis in physics? I am happy with its basis in mathematics, and make no claims for what physicists say it means in the slightest. Indeed, I am mostly agnostic about the physics of it all

    it would be very easy to take offence at that. I won't, but I will point out that mathematicians don't make any claims about their work representing anything with an existence in this world, so it is not reasonable to criticize them (and using the word 'twittering' is offensive) on that basis.

    and equally they have lead somwhere many times in the past, though you contradict yourself be referring to string theory as a solitary conrtibution (of maths) and then saying there are many other examples.

    Incidentally, most string theory is done by physicists, mathematicians generally do not do string theory - they do geometry and algebra and mostly are as surprised as you that physicists seem so keen on some of these ideas.

    You did make a case for investing in physics of more utility, and chose astronomy. Why? Knowing more about astronomical objects is not very useful to this world. As someone (I can't recall) once (apporximately) said, physics is incredibly good at describing the really big, and the really small, but useless for dealing with things in this world that actually matter to people in life and death situations like predicting the weather.
    Last edited: Jun 25, 2006
  19. Jun 26, 2006 #18
    I apologise for my poor choice of a word, Matt. "Melodious warbling" would have been more appropriate. Shades of the beloved Goon Show! I meant no disrepect to mathematicians, or for that matter to birds.

    The Platonic world consists of much else besides mathematics, and I was thinking of ventures of the religious and philosophical kinds -- should have been more specific, but this might have caused other offence!

    The muddle that is string theory is certainly a physics indaba, and can't be blamed on mathematicians at all.

    This is the beauty of Astronomy, and of pure Mathematics as well. Astronomy has thrown off the shackles of having to be useful, and is now a purely intellectual exercise. Long may such endure. I am sceptical only of protracted ventures like string theory and cosmological speculation, whose main merit is that they are intellectual hedge funds -- rather risky investments of capital.
  20. Jun 26, 2006 #19


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    Long ago, thinking about Plato's idea of Ideas, I thought if it were true, everything should have an Idea; carburetors were my example. Each particular design of carburetor should have its own Idea, and when a new one was figured out, that would be seen as discovering the new Idea, which had been there all along, like an undiscovered country. Thinking about all the complicated variations in technology, this made it seem at botttom a rather incoherent description of invention. All the little fiddles having been there since the origin of the universe, or longer. What do you think?
  21. Jun 26, 2006 #20
    When is a discovery an invention?

    At the end of the day it probably comes down to semantics (ie given the laws of logic, maths and physics, I define “invention” of an entity as a particular form of “discovering that entity for the first time”).

    Some discussion is worthwhile here. “Invention” is simply a statement of our epistemic perspective. Invention is simply a particular form of adding to one’s knowledge of the world. Two or more people may independently “invent” the same thing (eg the telephone). If Antonio Meucci invented the telephone first (see http://www.telephonetribute.com/telephone_inventors.html) [Broken], does this mean that Bell simply discovered Meucci’s pre-existing invention? No, of course not. In fact, it is logically possible that 1,000 different people could have “invented” the telephone, Why? Because the act of invention is simply a particular form of adding to one’s knowledge of the world. But regardless of how many people invented the telephone, the fact remains that given the laws of physics, the internal consistency of the concept of the “telephone” existed prior to any one individual inventing (knowing of) it.

    Let me use a simpler example. Imagine that we are watching a young child playing and experimenting with a simple construction set. The child finds that if she puts together particular parts of the set in a particular way, she can build a simple device for lifting objects that most of us adults would call a “lever”. Has the child invented the lever? Yes. Has the child discovered the lever? Yes. What is the difference between invention and discovery in this case? The difference is epistemic perspective. We would say the child “invented” the lever if we wished to express the fact that she had managed to construct the lever based upon her own experimentation and reasoning (she had not been shown how the lever worked by someone with pre-existing knowledge of the principle of the lever); whereas we would say that the child “discovered” the lever if we wished to express the fact that the principle of the lever is a principle which exists independently of the child’s discovery of it.

    Best Regards
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