Mathematics

Is Math Metaphysical? Math is not physical (composed of matter/energy), though all physical things seem to conform to it. Does this make Math Metaphysical and mathematicians Metaphysicians?

Comments

I agree with you that the

I agree with you that the sources of truth in mathematics can't be physical. For it seems clear to me that there would be mathematical truths even in a world that contained nothing physical at all (for instance, it would be true that the number of physical things in such a world is zero and therefore not greater than zero, not prime, etc.). So the sources of mathematical truth must be other than physical: if you like, metaphysical.

Does this fact mean that all mathematicians are doing metaphysics? I don't think so. Metaphysicians can investigate the sources of truth in mathematics and the ontological status of mathematical truth-makers. But mathematicians themselves can simply make use of those truths without having to delve into what it is that makes those mathematical truths true.

I have no problem at all with

I have no problem at all with what Stephen says, but would add a couple of things. First, Stephen didn't address what might actually be the questioner's main concern, i.e. whether the fact that "all physical things seem to conform to it" makes mathematics metaphysical. What is "it" here? Mathematics keeps growing, and one of the main sources of growth is that new things keep coming along (such as new scientific findings) for which existing mathematics is no help. The formulation of general relativity, for instance, required new mathematics that had been developed to some degree (by Riemann and others) before 1915, but without any thought that it might someday actually apply to something in the world out there. The further development of differential geometry was largely in response to its employment in theoretical physics (though of course it then took on a life of its own, as mathematical ideas do).

And these new developments invariably (perhaps inevitably) don't quite fit, in various ways, with the existing corpus of mathematics; it takes a while for a perspective to develop from which it can be assimilated and seen to be part of the same system of thought as what was there before. There seem to be some cases, in fact, where the new stuff just doesn't fit, and then it takes a while to get to the bottom of that difference. Meanwhile, new developments in logic and mathematics keep threatening to make such sluggish lucubrations out of date.

So I would answer the question by casting doubt on the idea of a stable "it." Mathematics is not one single, stable, definable language or system with which nature turns out, post hoc, to be in accord.

Secondly, I would point out that the question raised here lies at the heart of the history of philosophy. It seems essentially to be the question that got Plato started. He, and generations after him (even Kant, to some degree) were inclined to answer "yes" to the second question asked by the questioner. But in the early 1920s, Wittgenstein figured out a compelling way to answer "no," and the Vienna Circle (and all of 20th-century scientific philosophy in their footsteps) essentially took that "no" as their starting point. While I sympathize with that viewpoint, I would certainly want to acknowledge the historical importance of this question in the development of philosophy over the centuries. It was one of those deep questions that had to be asked, and has taken a long, long time to answer.

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