The TOE and imperfect mathematics

  1. Hello..

    I read somewhere that Mathematics is an imperfect science...basically a science with ultimately unprovable assumptions.

    Could that be the reason for the problem of squaring relativity & quantum mechanics?

    Thanks for any and all responses.

  2. jcsd
  3. Fredrik

    Fredrik 10,637
    Staff Emeritus
    Science Advisor
    Gold Member

    It's true that mathematics relies on axioms (statements that you don't try to prove), but this is obviously necessary. Every proof takes some set of statements as its starting point, so in order to even try to prove the axioms, you'd have to write down another set of axioms.

    These things have nothing to do with why it's so difficult to find a quantum theory of gravity.

    Also, mathematics isn't science. Science makes statements about results of experiments, mathematics doesn't.
  4. I would be useful if you could give a reference of where you read such a statement. In my view, it displays a poor understanding of what mathematics are, and I doubt this is merely a philosophical position. As Fredrik said, we always need logical axioms/postulates. This is not a weakness, on the contrary this is a strength : it does not matter to mathematics when the axioms/postulates will be true, whenever they will be then we will be able to use all the mathematical apparatus that follows. This renders mathematics universal. Because the axioms/postulates are explicit, known and understood, the theorems and further constructions we obtain transcend the specific context in which the piece of mathematics was constructed/discovered.
  5. Unless the solution to this problem is we do not need to tie them together, then it would be preferable that the theory realizing this square be expressed in mathematics. Otherwise, it will not be science. Mathematics is the required language of science because it is the only unambiguous language.

    It could also very well be that a quantum theory of gravity requires entirely new mathematics.
  6. Statements like this, at best, border on nonsense; and, at worst, become propaganda to mislead the uneducated.

    The adjective imperfect is silly and useless in this context. Nothing in life is perfect, although mathematics probably comes closer than anything else in meeting this impossible criterion.

    Mathematics is not a science at all. The modern definition of science relates to understanding the real world via experiments. Sure, mathematics is a useful tool in many branches of science, but mathematics extends much beyond the real world. It is an abstraction and a language independent of physical reality.

    The last words make some sense. The scope of most (if not all) mathematical frameworks are based on a small number of starting postulates that are unproven assumptions. Or, at least they are not mathematically provable with logic in the framework of that particular mathematics. However, one can offer non-mathematical evidence (i.e. akin to a courtroom type of proof, or scientific evidence) of situations where the starting assumptions seem to apply. This is necessary whenever mathematics is used as a real-world tool, with applied math in science being one of the best examples of this.
    Last edited: Jan 26, 2010
  7. SixNein

    SixNein 221
    Gold Member

    Yes, Mathematics is an uncertain science; however, the uncertainty of mathematics does not arise because of assumptions per say. The uncertainty started showing up in a big math effort called the Hilbert program. The aim of the program was to get rid of the growing list of paradoxes in mathematics and to create the equivalent of a TOE in mathematics. Unfortunately or fortunately, it failed in the worse way.

    Two big theorems came out that brought the hilbert program to a sudden stop.
    1. Godel's Theorem of incompleteness.
    2. Allan Turing's halting problem.

    These theorems killed the idea of a certain mathematics (and logic). In a basic nutshell, the theorems created statements that could not be proved (not assumptions, statements (BIG DIFFERENCE)).

    In addition, there is more recent work that makes the uncertainty greater. Chaitin recently (in math years) published a proof that tied a type of incompleteness into data itself. In a basic nutshell, it is a number that can not be computed.

    Does any of this mathematics have impacts on physics? Absolutely, incompleteness is already a fact of life in computer science through Turing machines and Chaitin's work.

    The physics TOE is likely dead in its tracks because of this mathematics. But I think the whys and hows have been poorly communicated. Stephen Hawking made an attempt to explain why it kills the TOE, but I think most people have a very difficult time understanding it.

    In a fashion, incompleteness is a knowledge of systems or a limitation of mathematics when dealing with systems. The problem with creating a TOE is that physics is complicated enough to spark incompleteness (IE: it is a system covered by incompleteness). In other words, physicist will be unable to construct enough axioms so that they can formulate the TOE. So they essential end up with a framework that they already have.
  8. SixNein

    SixNein 221
    Gold Member

    I'm guessing that he or she has ran into some pure math discussions.
  9. SixNein

    SixNein 221
    Gold Member

    "Could that be the reason for the problem of squaring relativity & quantum mechanics?"

    It may just be a very difficult problem.
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