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Question on the universal correctness of mathematics

  1. Oct 9, 2011 #1
    Question on the universal "correctness" of mathematics

    I began thinking of this question long ago, but only now, that I am reading A History of Mathematics, 2nd edition, have I decided to ask for the perspectives and thoughts of others. I am slightly reluctant to ask this question, for it may seem a bit silly, and may even seem unanswerable. Really, it most likely is, but that is not the point.

    Math is, as most mathematicians, scientists, and engineers would agree, a large volume of theorems, laws, and ideas conjured up by countless people over thousands of years. Many mathematicians have devised theories that, to this day, have yet to be proven incorrect. This entire system of laws and theories is, and pardon the redundancy, an ebb and flow of ideas, a way of thinking of things. Taking the quantification of the world around us and manipulating those numbers. Many ways (once again pardon the redundancy) of manipulating those numbers are still used extensively today.

    So, finally, to my question. Look at everything we know of mathematics. Everything is either an extension or a correction of a basic axiom. Is it possible that somewhere else in this vast universe of ours, someone else has created an entirely knew math, complete with axioms completely alien to us? Take Euclid's common notions, his postulates, and many of the other things we take for granted as common logic. Is it possible that this basic logic we take for granted could be wrong somewhere else in another galaxy for example? (I have excluded from this question anything mathematical relating to the physical properties of that part of the universe. However, I may have been wrong in doing so. I am not sure if the physical behavior of a being's environment could affect its method of logical thought.)

    I ask that when answering, you explain why as well. Hopefully some of you who are much farther along in your studies can thing teach me a few things.
     
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  3. Oct 10, 2011 #2

    Hootenanny

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    Re: Question on the universal "correctness" of mathematics

    Undoubtedly, the answer is yes. Put simply, if one starts from a different set of basic axioms, one could construct a set consistent of theorems entirely different to our own.
    This is a point that irks me somewhat. Mathematics itself cannot be "wrong", whether a collection of theorems or axioms represent the "real" or physical world is an entirely different question. I would say that Mathematics can be inconsistent (i.e. theorems which contradict each other), but not wrong in the sense that you mean it.

    If you mean "is it possible to have a system of Mathematics which doesn't represent the real world?", then the answer is of course, yes. Much of "today's" mathematics does not represent the physical universe, but that doesn't mean that it is incorrect.
     
  4. Oct 10, 2011 #3
    Re: Question on the universal "correctness" of mathematics

    You seem to have the idea that math is like physics in the sense that one has "laws" that can be wrong or right. In physics, you look at the world, and perhaps you make a hypothesis: "I think objects fall at 20 m/s^2 !" You test that hypothesis, and realize it's wrong. You start over again (This is a very simplistic description, but it gets my point across).

    In math, you don't look at the world. In fact, if what you're thinking about has nothing to do with the world, then too bad. Math is about definitions; it's kind of like inventing a new game with your friends. You start with a few basic rules, then you add more. The game you just created with your friends cannot be "wrong" since you have decided what the rules should be.

    The point is, math is created in our minds (I'm not looking to start the old "is math created or discovered" thread), so there are no laws, only theorems, definitions, lemmas, corollaries, and propositions. Physics has laws, math does not.
     
  5. Oct 10, 2011 #4

    lavinia

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    Re: Question on the universal "correctness" of mathematics

    In practice mathematics is done by forming hypotheses about mathematical objects then either verifying these hypotheses by looking at examples or finding a proof. Similarly mathematical theories are developed much as they are in physics. The objects of investigation are mathematical structures.

    Mathematical ideas definitely help us to think about the world. And it is hard to imagine the world without them. If you say that purely empirical data without any formal conceptual structure is the real world, fine, but this to me is arbitrary and useless and I also question whether it is even possible.

    everything is created in our minds I guess. But I am not sure how something that is provably true can ever be created.
     
    Last edited: Oct 10, 2011
  6. Oct 10, 2011 #5

    Deveno

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    Re: Question on the universal "correctness" of mathematics

    math is like one great big IF, THEN statement (or more accurately, several of them sitting on the same bookshelf).

    it is very likely that if we ever meet another intelligent species, and they have mathematics, their mathematics will be quite different than ours, because they started with different "IF"s.

    i'll give a somewhat simple example. it was conjectured by Cantor that the "next biggest size" of set after countably infinite (the size of the natural numbers), was the size of the continuum (which we can say, for our purposes, is the real interval [0,1]).

    it turns out that you can assume this is true, or not true, without violating any of the "standard" axioms for set theory. so, just here, on THIS planet, we have two axiom collections with no real criterion for deciding "which one" is correct.

    the question of whether or not our universe has discoverable mathematical relationships it always obeys, is another one entirely, with a much murkier set of answers. some people believe that it does, with a passion. i am undecided, although my personal experience leads me to suspect it could.
     
  7. Oct 10, 2011 #6

    lavinia

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    Re: Question on the universal "correctness" of mathematics


    That is not true in my opinion. How do you get to that? How is the theory of characteristic classes of manifolds a big if then statement? How about the theory of elliptic curves? How about calculus?

    It is very unlikely that their mathematics will differ at all from ours.

    I do not see what the example of the Continuum Hypothesis has to do with it. Other intelligences would also discover that there is more than one possible cardinality for the continuum. There are two possible plane geometries. Does that mean that plane geometry is a big if-then statement?

    Mathematical relationships exist in our universe or are you saying they exist somewhere else? Where would that be?
     
  8. Oct 10, 2011 #7

    phinds

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    Re: Question on the universal "correctness" of mathematics

    drummerguy198, some ramblings in response to your post:

    (1) There really are two kinds of math, pure math and applied math. An alien species may come up with a different mathematical system to describe the real world but unless they have physical laws that are different than ours (a VERY unlikely circumstance) they will get the same answers in terms of predicting physical events.

    (2) As has been pointed out here, any self-contained mathematical system that not in any way internally contradictory cannot be WRONG, it just may or may not apply well to the physical world.

    (3) I read a very interesting article once that pointed out that someone (or probably much more correctly, some THING, a sentient being, but not like us) who for whatever physical reasons, was naturally attuned to how space-time is affected by gravity would find Riemann geometry to be quite intuitive and should they have occasion to consider what we call Euclidean geometry they would find it self-consistent but in terms of it applicability to the "real world" of extraordinarily limited use.
     
  9. Oct 11, 2011 #8
    Re: Question on the universal "correctness" of mathematics

    I don't see how to discuss this without discussing creation versus discovery...
    Alien math WILL be very different, but will yield the same real world results. Math is a way of envisioning reality in our minds and manipulating it there. Any distant race could envision the world differently, but, in order for their axioms and theorems to be true, their differing methods should produce identical results.
     
  10. Oct 11, 2011 #9

    phinds

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    Re: Question on the universal "correctness" of mathematics

    Yes, that's exactly what I said in part #1 of the post directly above yours.
     
  11. Oct 11, 2011 #10

    Deveno

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    Re: Question on the universal "correctness" of mathematics

    manifolds? what "are" manifolds? i'm not asking for the definition here, yeah i get it, locally diffeomorphic to euclidean n-space. so we need a LOT of structure built up, to talk about them. are you somehow positing that euclidean space is "real?" that real numbers are "real"? show me one. exhibit any object that literally corresponds to a real number. i'll be reasonable, i'll settle for anything of an exact length of √2, in any unit of measurement you care to specify. but...no epsilon, ok? no "tolerance" or uncertainty, it must be exactly √2, and i'll need proof of exactness.

    if that's asking a bit too much, how about an infinite set? one composed of actual elements, not hypothetically proposed. and while you're at it, how about a consistency proof for all of these structures, if that's not too much trouble? can you tell me what a set IS? do you have a good working definition that makes sense? let's hear it. because if your definiton is axiomatic, that's a big IF, and i have to buy into that "if", to accept that what you say is "true".



    because we have the only possible logically consistent model, up to isomorphism? reallllllllly?


    yes. and there are more than two possible plane geometries.

    which mathematical relationships are these? not Newton's right? he was wrong. do you honestly believe that our current theories are "eternally true?" what hubris! we have good guesses, at BEST. these guesses have a lot of experimental validation to point to their appropriateness, but it is a logically indefensible inductive leap to suppose exactness and correctness.

    as near as i can tell, mathematical relationships exist in our minds. they may also exist in the world, but it's hard to tell, because we can't take "us" out of the picture. our observations of the data change the data, so we cannot tell the true observer-independent state of the world (although we can theorize about what it might be).

    a differently structured neuro-biology might well have a totally different inference structure, and perceive sensory information in a totally different way. there is no reason to suppose such a species would naturally construct a bivalent logic in order to reason about their environment, and no reason to suppose their model of the universe would even be isomorphic (in some general sense) to ours. there's no reason to suppose that it wouldn't be, either....we just don't know.

    there is no physical reason to prefer the "standard" construction of the reals, over a "constructivist" position (which doesn't give us the entire standard reals, only constructible (perhaps definable is a better word) numbers), or over a system such as the hypperrreals. so which of these is "correct" and what is the criterion for deciding?

    all of mathematics (yes i said ALL), depends on assumptions of one sort, or another. Hilbert was overly optimistic, we cannot create a consistent system to recover all of mathematics from a minimal set of assumptions that are "self-evident". mathematics isn't "true" it's contextually-true. within those limitations, it does admirably, but when you try to expand the domain to "everything", you wind up with trouble.
     
  12. Oct 11, 2011 #11
    Re: Question on the universal "correctness" of mathematics

    If math could be "created" in any fashion, why can't I just announce that I've discovered 2 + 2 = 5?
     
  13. Oct 11, 2011 #12

    Deveno

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    Re: Question on the universal "correctness" of mathematics

    you could, but would it be meaningful?

    first of all, you'll have to tell us what the objects "2", "2" and "5" are, what the relation "=" means (it obviosly doesn't mean "identical" because 2+2 takes 3 keystrokes to type, and 5 just takes one), and what the operation + does, for any allowable input.

    then, we'd have to see if your statement was consistent with other allowable statements from the domain "2" and "5" come from, subject to the rules for "+" and "=".

    if we derived a statment like 2+2≠5 as well, from the same system, then under the usual laws of inference, its not hard to show that every (allowable) statement in your new system is possible (valid), which deals a serious blow to any particular statement meaning anything.

    if by "2", "5","+" and "=" you mean the usual interpretation of these symbols, then, no, you can't, because what statements are allowed as valid, is limited by the definitions of the symbols.

    the precise "unwrapping" of a statement like 2+2= 4, in our current theory of the natural numbers as of model of the peano axioms, as a model of an minimal inductive set in zermelo-fraenkel set theory, gets rather complicated to state in "atomic" terms. but once you accept the axioms of ZF set theory as given (and the above definition of natural number), you are forced by an air-tight chain of implication to conclude that 2+2 is indeed 4.
     
  14. Oct 11, 2011 #13
    Re: Question on the universal "correctness" of mathematics

    So it would seem the entirety of mathematics could rest on the axiom of choice. If we could prove its validity, then would mathematics not be universal and true, as you would have to accept set theory?
     
  15. Oct 11, 2011 #14

    Deveno

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    Re: Question on the universal "correctness" of mathematics

    if the entirety of mathematics rested on the axiom of choice, and it were proved true, then yes.

    but....the axiom of choice is independent of the other axioms of set theory. that is, neither assuming it is true, nor assuming is it false leads to a conflict with the other axioms. most people feel that some statements of the axiom of choice are plausible, but that is not quite the same thing. and there are still those who think the "truth" of the axiom of choice is "uncertain" and avoid invoking it in proof.
     
  16. Oct 11, 2011 #15
    Re: Question on the universal "correctness" of mathematics

    It is my understanding that ZF set theory is true if the axiom of choice is true. ZF set theory fails if it is false...
     
  17. Oct 11, 2011 #16
    Re: Question on the universal "correctness" of mathematics

    The emphasis is mine...

    This is entirely false! Examining examples can help mathematicians gain some intuition over a subject but it can never be used as "verification," ie proof.

    Consequently, mathematical theories are not "developed much as they are in physics." Maths is not even a science. You can develop new mathematics with nothing but a chalk board, without having done so much as a single experiment. This is very unlike physics...
     
  18. Oct 11, 2011 #17

    cmb

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    Re: Question on the universal "correctness" of mathematics

    My ramblings.............................

    I don't buy some of the arguments above. Mathematics is the same the universe over, because mathematics is 'a way of describing things'.

    Folks often confuse 'mathematics' with 'the study of numbers' or 'the study of logic'. I think this is a superficial understanding of mathematics.

    Mathematics is the study of patterns. If some bug-eyed green alien in a far off galaxy sees the same patterns in nature/physics, they will come to the same mathematics as us. If not, then they probably won't. It is merely a 'tool' that we have invented for ourselves that allows us to interpret the world.

    That being said, there are some interesting 'exceptions' that become 'universal truths'. I will give an example - the 5 Euclidean solids. Whatever universe you might live in, in however many dimensions, &c., &c., you will always come to the conclusion that there are only 5 regular 3 dimensional shapes. If you did not even have a body living in a physical world, you could still come to the same conclusion. So, is this an example of 'mathematics', or the discovery of something bigger, more fundamental, that exists only as a thought and can never exist in reality, a 'universal truth'?

    The conventional breadth of study considered to be 'mathematics' includes both representations of what we perceive (that is, a description of the physics we observe) and these other 'discovered' parts that would always exist in a virtual reality.
     
  19. Oct 11, 2011 #18

    Deveno

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    Re: Question on the universal "correctness" of mathematics

    this is incorrect.
     
  20. Oct 11, 2011 #19
    Re: Question on the universal "correctness" of mathematics

    Well, there goes my argument haha. Either way, I agree with cmb
     
  21. Oct 11, 2011 #20
    Re: Question on the universal "correctness" of mathematics

    This is a poor characterization of the situation...

    We're talking about mathematics here. Despite popular belief, mathematics has nothing to do with the "real world..." As Deveno has said, it's a giant "what if" statement...

    It doesn't matter if you could show that the axiom of choice were not true in the real world, it's nonetheless logically consistent with the rest of the ZF axioms and, thus, ZF set theory would still be true in the sense that mathematicians care about...

    Incidentally, the ZF axioms minus the axiom of choice is also consistent. The only problem is that proving some things we'd like to be true turns out to be impossible without this axiom, so we choose to include it. However, including it makes some things true that some people would rather be false. Because of this controversy, we usually make note of its use in a proof whenever it's needed but avoid it if we can. If the concept of choosing truth seems odd to you, I welcome you to the wonder world of mathematics!
     
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