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lugita15

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Here's an explanation I wrote up a while back on Quora that details some major philosophies of mathematics:

I'm really interested in the philosophy of math, so if you have any questions about it I'd be glad to help.

In the interest of full disclosure, I'm somewhere close to logicism and/or platonism, not unlike the views of Gottlob Frege and Bertrand Russell, in that I believe that the truths of mathematics are objective and absolute, and I share their view that mathematics seems amenable to reason.Each major theory about the origin of mathematics has its own challenges to overcome. Any theory has to explain how mathematics is consistent, bountiful (meaning that there's always new things to discover), applicable to the physical world, and accessible to the human mind.

One philosophy is that math is just an invention of the human mind. This used to be associated with people on the fringe, called intuitionists or constructivists, who tried to establish a very narrow view of what mathematical techniques are allowable. But lately it's acquired more mainstream popularity because of George Lakoff's book Where Mathematics Comes From, which tries to explain math in terms of cognitive science and human psychology. The main problem with this view are that it doesn't explain how math is so self-consistent: most ideas we think up have all kinds of flaws and inconsistencies, so how has mathematics held up perfectly for so long? Also, why math is so useful in understanding the world around us?

The latter problem is most naturally addressed by physism, a philosophy originally proposed by Aristotle but which has come back into the limelight thanks to a series of books by Roland Omnes. Physism states that humans came up with math by observing the physical world. By studying the laws of physics, they were able to come up with mathematical rules which seem to govern how the world operates. The major problem with this philosophy is that mathematics is quite a expansive field, and it's not clear how much of it is grounded in actual physical phenomena. Sure, some things like calculus seem pretty well realized in the world, but can the same be said about more abstract branches like category theory? Probably not.

Formalism is yet another philosophy; it was all the rage a century ago, but now it's fallen out of favor. Formalists like David Hilbert believed that math is just a formal game we play using strict axioms and rules. But Godel's Incompleteness Theorems cast doubt on this: it turns out that mathematics is too expansive and bountiful (the technical term is "indefinitely extensible") to be captured by a single formal system. Also, it's hard to be absolutely sure that the system we're dealing with doesn't have some inconsistency lurking within. Finally, it seems too much of a coincidence that the universe behaves exactly according to the rules of a formal system we came up with millennia ago. (Unless you believe in computationalism, in which the universe really is just a big computer).

Last but not least, there is the most popular mathematical philosophy of all time, Platonism. Pretty much all mathematicians believe in this philosophy, which claims that there is an abstract realm called Platonic heaven where all mathematical structures reside. (In modern versions, we like to say that it's mathematical truths like 1+1=2 that are "out there", not actual objects like circles) It solves all of the problems listed above that plague the other major philosophies, but it has its own difficulty: how in the world are measly human beings able to discover truths about what goes on in Platonic heaven? If you're religious, the answer is obvious: we have souls, abstract nonphysical essences which can access Platonic heaven. But resorting to religion makes mathematics akin to theology, which seems unsettling to say the least.

I'm really interested in the philosophy of math, so if you have any questions about it I'd be glad to help.

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