or of our mental representation of it?

Edit: Michael Dorfman raises the point, "why would it be amazing that "extremely complicated mathematics" would model the physical world? ... In fact, it would seem that given mathematics of sufficient complexity, one should be able to model anything; if one finds an object or system that resists modelling, one could easily add new axioms or arbitrary complexities until one finds a suitable result."

In response, I claim that it is exactly the fact that this latter does NOT happen that draws some people to prefer pure mathematics above other fields, especially in the humanities and social sciences. The major mathematical investigations in history have always rested on just a few collections of primitives and first principles that change slowly or not at all. And often (as in the removal of the parallel postulate), major reconsiderations of these primitives and principles lead to their simplification, not an introduction of "arbitrary complexity".

When it was first realized that our universe seems not to be Euclidean space, but rather a pseudo-Riemannian manifold, arbitrary complexities were not introduced to the mathematics to better explain the physical world. Indeed, this part of the deductive structure of mathematics was developed well BEFORE physicists had use for it. The definitions needed to defined these manifolds were very old, just needing sets and the usual notions of analytic geometry (to describe a topological space, and then say what it means for that space to be a manifold with structure). All that had been added to the theory to produce the needed manifolds were the right definitions and deductions made from those definitions.

The symbiotic relationship of pure mathematics and physical theory includes contributions the other way, with physicists realizing that mathematics should be able to deduce new truths that explain phenomena they observed. I have never heard it suggested that "arbitrary complexity" be introduced to explain these phenomena --- rather, the physicists are somewhat content to wait for mathematics to produce the requisite addition to the deductive structure through the usual processes.

I hope this better explains my question. Now as far as how "well" mathematics seems to explain physical phenomena, in terms of "approximating", I have a less well-formed notion. But the vast amount of deductive structure built on both the primitives of mathematics and those of physical theory could lead in very different directions. If we had poor ability to ascertain the primitives of the physical world, we might at best only be able to construct mentally a poor facsimile of the physical primitives. Then perhaps, near the base of the deductive structure, there might continue to be some resemblance between mathematical theory and physical theory. But I would expect that if one moved far enough up the deductive tree, building more complicated definitions and theorems about them, we would eventually see something analogous to sensitive dependence on initial conditions and see larger and larger divergences of the two structures. The fact that even some of the deepest (highest? my tree is upside-down!) mathematics we have developed continues to be connected to physics (i.e. connection between the Langland's correspondence and string theory or the no-ghost theorem from string theory leading to mathematical construction of the monster Lie algebra) seems to controvert this. The relationship is so symbiotic that at this level, it is hard to distinguish between mathematicians and physicists (also due to the (present) dearth of falsifiable predictions by versions of string theory).

The last three sections of Wigner's essay (in Joseph Weissman's comment below) pose what I believe to be the same question. He concludes with, "The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning."

  • There are definitely a lot of interesting concerns here, but might there be any chance I could persuade you to focus a little more closely on one of them here? Don't forget you can always ask more questions :) In passing, it definitely helps improve the chances of getting a great answer if you could tell us a little about the context and motivations behind your question (what might you be reading or studying that has made this concern an important or urgent one for you? what might you have found out so far?) – Joseph Weissman Jun 23 '12 at 21:32
  • By the way, you might just enjoy reading Wigner's brief essay The Unreasonable Effectiveness of Mathematics in the Natural Sciences if you haven't already – Joseph Weissman Jun 23 '12 at 21:38
  • I am mostly interested in the initial question, since I am sure I can do hundreds of pages of reading to develop my embryonic understanding of the question. I, like most serious students of mathematics I know, take a Platonist view of mathematical knowledge. Today, I had the revelation that there might be nothing special about the primitive notions of mainstream mathematics --- they are simply reflections of those of the universe. Perhaps a different universe would produce different primitive notions. I had this realization right after reading Karl Popper's "Two Kinds of Definitions." – Barry Smith Jun 23 '12 at 22:23
  • Wow, that Wigner essay is fantastic! Philosophical responses to the questions he raises are exactly what I was looking for by posting this question. – Barry Smith Jun 25 '12 at 13:06

The main reference for me on this subject is Husserl's investigation into the phenomenological nature of numbers in his Philosophy of Arithmetic. To put it very grossly, he explores the bond between symbolic representations and basic then complex mathematical realities. I remember especially a part where he advocates that sequences of mathematical signs are but a representation of concepts [Repräsentazion], but that these concepts are passing thoughts made solid by a somewhat illusionary idealization of our power of representation [Vorstellung]. You see, his phenomenological standpoint means that for him the question of the reality of mathematical object must be considered through the prism of subjectivity (the only thing we're sure of) or at best some sort of intersubjectivity (like symbols and representations, as forms of externalized subjectivity). As far as I understand, it means Husserl denies that mathematical objects have an objective reality but posits instead that they have a status similar to Language/Meaning.

On the side of the advocates of the reality of mathematical objects, Frege and Quine would represent the two opposite arguments, advocating that mathematical objects are real by grounding them respectively in logic/reason (as self-evident) or in empiria (as indispensible). I won't go into details here because I don't know much about their work.

An approach that is somewhat related to that of Husserl (but in a more lively form) is found in Wittgenstein's Lectures on the Foundations of Mathematic. The famous philosopher presents a series of paradoxes about mathematical objects and insists on the importance of linguistic coherence at the expense of objective groundedness. What makes this book amazing is that it includes questions and criticism of people attending the lectures, including the great Alan Turing!

  • Thank you. I will look into the Husserl and Wittgenstein books and try to find some things by Frege and Quine. – Barry Smith Jun 24 '12 at 2:41

I'm not sure I agree with all of the assumptions underlying the question.

First: why would it be amazing that "extremely complicated mathematics" would model the physical world? Wouldn't it be more amazing if only extremely simple mathematics were required? In fact, it would seem that given mathematics of sufficient complexity, one should be able to model anything; if one finds an object or system that resists modelling, one could easily add new axioms or arbitrary complexities until one finds a suitable result.

Next: I'm having trouble understanding what the follow notion means, exactly:

the perhaps amazing coincidence that our minds and senses are capable of ascertaining very good approximations to the primitive notions in the deductive structure of "true" knowledge of our universe.

How would we be able to judge the quality of the approximations (allegedly "very good") or even have a notion of the "true" knowledge of our universe except through our minds and senses? You've created a circular judgment.

Finally, as to the question of "why our senses and minds are so good at approximating primitive notions of our physical universe", the immediate question is "so good compared to what?"

The entire set of questions rest upon an unsupported assertion; until this assertion is argued in some manner, I'm not sure it is possible to even begin to answer the questions.

  • Thank you for your insights. I do realize now that I really needed two questions, one requesting a reference list on the subject and the other about particulars. I will edit my original post to make it about reference requests, but address your first point in that edit. After doing a bunch of reading (Wigner, Husserl, etc.), I might post a new refined question about particulars. – Barry Smith Jun 25 '12 at 12:07

Not the answer you're looking for? Browse other questions tagged or ask your own question.