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I was looking at arguments about mathematics being a science (or not), here for example, but it seems that these arguments are more about some metaphysical idea of mathematics rather than the subject itself. Mathematics as it exists is a study of the most schematic properties of physical objects, it has both theoretical and empirical aspects. If our world consisted of unstable shapes constantly morphing into each other we would have a very different kind of geometry, if the objects we encounter sprouted new ones when put together we would have a very different kind of arithmetic and set theory.

Mathematics doesn't study some "arbitrary" systems of axioms, it never did and it never will. Which systems are "good" or "interesting" is discovered by trial and error, just as theoretical parts in all sciences. Sure, mathematics is unfalsifiable in the sense that if you accept the axioms (and logic) you have to accept the theorems, but Newtonian mechanics is unfalsifiable in exactly the same sense. And in the sense that it is falsifiable so is Euclidean geometry. I suppose, one can imagine a being doing mathematics without sensory input, but one can also imagine a being that creates new worlds as it pleases with their own laws of physics.

Scientific theories are not verified or falsified by experiments alone either, as Einstein remarked "only theory decides what one observes". String theory gives a stark example of just how indirect a relation to observations and experiments can be for a vibrant physical theory. Mathematics is the most theoretical of sciences, but there are more commonalities between it and physics than between physics and say psychology, so it seems to be well within the spectrum.

Is there a separation in modern philosophy between mathematics as practiced and "ideal mathematics" that "has no bearing on the physical world and cannot be affected by it"? What is the philosophical basis for such separation, is there similar separation for other sciences? Is the controversial status of mathematics as a science due to historical tradition of Plato, Kant and Husserl who treated its source as metaphysically different?

EDIT: Euclidean and non-Euclidean geometry are not "arbitrary" at all, one summarizes our basic space experience, the other modifies only one aspect of it which may not even be noticable at small distances. Grothendieck invented etale cohomology to prove Weil conjectures about algebraic varieties, following up on that quickly leads to tangible things as well. Even the purest of areas like logic, set theory or algebra can be easily traced to their roots, including Godel's theorems with their reliance on arithmetic and "effective axiomatization". There is leeway in every science, no one expects some direct correspondence with reality or applicability to it from every theory. There is more leeway in mathematics, but where is the qualitative difference?

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  • @ChrisW I'm not sure those axiomizations are arbitrary. They are chosen for specific reasons. A better example would be mathematical logic. For example, Godel's famous incompleteness result applies to an arbitrary theory whose axioms enable it to model arithmetic.
    – nwr
    Commented Sep 18, 2014 at 0:16
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    "Mathematics as it exists is a study of the most schematic properties of physical objects". I assume you just made this up. In what sense is, say, etale cohomology related to the properties ("schematic" or not) of physical objects?
    – WillO
    Commented Sep 18, 2014 at 0:22
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    @ChrisW Fair enough. I guess it comes down to what you mean by arbitrary. For example, different sets of axioms can be chosen for Euclidean Geometry, but the axioms we choose are not arbitrary in the sense that we have chosen them for the simplicity, economy and elegance of the resulting theory of geometry.
    – nwr
    Commented Sep 18, 2014 at 0:39
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    @Conifold: Algebraic geometry emerges from the study of systems of polynomial equations. What physical objects do you imagine these equations are meant to describe? Or --- perhaps an even better example --- where is the physical basis for, say, Fermat's two-square theorem?
    – WillO
    Commented Sep 18, 2014 at 3:25
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    @WillO So since there is no "physical basis" for irrationals classical analysis has nothing to do with reality? Conics, angle trisection, etc. led to study of polynomial equations, Fermat theorem is about a property of counting numbers. Once a useful structure appears enough times (ring, field) generalizing its properties and finding new connections is how it works. Relation to reality isn't something primitive like relating every mathematical concept to a physical object. There are higher order properties that are revealed only through abstraction.
    – Conifold
    Commented Sep 18, 2014 at 21:27

4 Answers 4

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I'm still not sure that your question is clear enough to answer, but I'm willing to give it one more go.

Your question is : Are there two different mathematics in philosophy?

On the surface the answer would appear to be no. However, it is possible to make a case for yes.

If you take a Platonist view of the subject, which I believe would be the majority view amongst mathematicians, then mathematics exists independent of ability to articulate its truths. So from this point of view, clearly there are two different mathematics - the real and the human. Although we have articulated some of the truths, it is not clear that we have done it correctly. When I say "correctly", I don't mean we have made errors, I mean that our articulation may not be a true representation of the subject.

For example, in the 1950's, people noticed that our theory of elliptic curves could be considered to be part of complex analysis, and not a separate theory at all. (I believe this follows from a result called The Modularity Theorem.) Accordingly, both theories might actually be articulations of the theory of L-functions. Maybe this is the true nature of the mathematics. L-functions also appear to indicate that the algebra of Galois Theory and the theory of analytic functions in the complex plain are different articulations of the same mathematics. But what makes us so sure that L-functions are the correct view. Maybe they are manifestations of some deeper mathematical truth. Etc....

As you have noted in your comments, our mathematics, no matter how abstract or remote, can be seen to be rooted in our real world experience. You mention string theory, which is rooted in geometry. Similarly, our theory of large cardinals is demonstrably remote (provably undecidable), but it is rooted in the act of counting. Surely reality includes aspects well beyond our ability to experience, just as it must surely be the case that mathematics includes provable truths which are beyond our ability to imagine or root in our human experience.

So my naive reading of your question leads me to the answer YES, I think it is fair to say that most mathematicians would agree with the idea that there are two different mathematics in philosophy.

ASIDE I appreciate that this may be a pretty crappy argument, but "hey", I'm not trying to make a thesis of it. It's the best I can do with the time at hand.

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  • It's a good argument. It suggests to me that the reason for singling out mathematics is the majority Platonist view of mathematicians themselves. Although inspecting creation of actual mathematics shows that it is not related to Platonic world even if it exists.
    – Conifold
    Commented Sep 18, 2014 at 20:59
  • @Conifold Wow! I'm really chuffed that you have accepted my answer, especially since you have such a high rep on MSE and obviously know the subject much better than I. Apparently I'm still impressionable.
    – nwr
    Commented Sep 18, 2014 at 21:03
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Maybe you'd be interesting in fictionalism. There's value in reading Moby Dick, even though the characters don't exist and the story's not true. Likewise, there's value in mathematics that is not "true" about the world. And who's to say which part of math that is? Maybe it's all of it.

When one first hears the fictionalist hypothesis, it can seem a bit crazy. Are we really supposed to believe that sentences like ‘3 is prime’ and ‘2 + 2 = 4’ are false? But the appeal of fictionalism starts to emerge when we realize what the alternatives are. By thinking carefully about the issues surrounding the interpretation of mathematical discourse, it can start to seem that fictionalism is actually very plausible, and indeed, that it might just be the least crazy view out there.

http://plato.stanford.edu/entries/fictionalism-mathematics/

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  • I had a look at the paper on fictionalism and found it an enjoyable read - unapologetic in its philosophical demands. It's always interesting to encounter a new point of view.
    – nwr
    Commented Sep 18, 2014 at 23:01
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Yes, there are two of them, and both of them don't really exist. I will weigh in here with the neo-Intuitionist Kantian argument from Brouwer. But I suggest Kant in general needs to be interpreted with the bridge between noumena and phenomena filled in by evolution, rather than something more abstractly transcendental. And that changes how I will put that argument.

Mathematics is neither based in real experience, nor is it arbitrary. It is a science, but the elaboration of theorems is not its success criterion -- finding the right balance of both applications and appeal to evolved mathematical taste is its success criterion.

The 'anamnesis' feeling of mathematics that Plato points out in the Theatetus cannot arise from something that is totally acquired.

Nor can something that has no basis in the world wield the kind of explanatory power our science, based on math, gives us.

Instead of being either of these things, mathematics is the playing out of acquired inutitions, that are trained by survival. Kant is not wrong, but knowing what we have now seen, we just need to pull back the certainty of divine kindness, to the effectiveness of evolution. Evolution has given us a mental model of space, which is wrong, but useful. God might have made Euclidean geometry correct.

The things we find we react to positively in a given way turn out to be very useful to us because it is the result of our construction, which has adapted to the real world over time. But it is adapted by survival, not by some more formal notion of nature.

What it explores as a science is the fit of that internal Platonic sense to reality, when it is good, and when it is not so good. We know we have tapped into the wellspring of intuition when our axioms correlate with real scenarios in the world, and the correlation is elegant. Mathematics that is too abstract, even when it succeeds at answering its own questions fails to live on as active study, and that which finds its power is passed down. Mathematics that is too concrete, spins off into the applied branch of some other science, but it is what stays properly mathematical that guides future mathematics.

The theories that get tested in mathematics are of the form "X is genuinely both interesting and powerful." Not 'X is true'. That makes it the experimental science of rational psychology.

(Non-Euclidean geometry in no way defeats Kant's theory, even if he was personally wrong about it. We still cannot ever experience it in reality. Our POV will always be Euclidean, until we evolve past being human. The idea that we have a flawed picture of reality is built into Kant already. So finding that our best forms of experience do not ultimately fit reality is the exception proving the rule. Forms of intuition do not have direct applications to reality. They have direct application to experience. Kant just overestimated, contrary to his own basic principles, how much experience naturally misses reality.)

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As Math Professor said: "Mathematics is a quenn of Science". Philosophers may have two quenns - why not? You can have two fathers, so why not two mathematics? Most curious to me are complex numbers. As another professor said, someone invented complex numbers, which wasn't real. But it used algebra around them to compute difficult problems with polynomial which people didn't know the answer calculating with real numbers. That can be that kind od mathematics, which axiom we don't perceive as a from real world, but it can be useful. Another kind of mathematics is that real world mathematics. You can interprete it variously, for example: Euclidean geometry - in which you can describe the surface - one point is point on surface in 3d. So we see mountains, we see the surface and we have mountains in computer games which are generated from math formulas in euclidean geometry. But for this goal, the authors needed to use another kind of mathematics - theory that is useful but not real. So we can mix it, so recalling that i assumed in the beggining that there are two mathematics. That is contradiction, because we can connect kinds of mathematics having one which product of two mathematics.

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