There is great beauty in this approach to mathematics. It is why I am currently studying mathematics. It is a great defense of why anyone would want to do mathe

There is great beauty in this approach to mathematics. It is why I am currently studying mathematics. It is a great defense of why anyone would want to do mathematics. However, there is another question. Why do we want anyone to do mathematics? Asking, not what math gives the mathematician but instead what math gives society at large. (That is, beyond contented mathematicians). As long as mathematicians are self-reliant, the second question doesn't matter. However, when mathematicians start making demands of society the second questions is natural. Note that these questions are orthogonal. Any answer to the mathematics extrinsic value need not inform its intrinsic value. Just like the intrinsic value of mathematics is not related to its extrinsic value. The point here is that math for math's sake has an awesome byproduct for society: phd's that go commercial. To be afraid that maths will then get squeezed for such phd's to the detriment of it's intrinsic value massively discounts the will of mathematicians and the long view of society. Sure, there will be some that try this squeeze. But their effect will be limited. Meanwhile, this is how mathematics can sustain itself at much higher levels. Do you really think there is so much true extrinsic value in all the proofs of mathematics? I'd say the value of all those who graduated and went on to do other things easily outweighs all those proofs. It was the creation and later understanding of those proofs that had extrinsic value. This means that any field of mathematics has value no matter how likely it is to ever be applied. As such, extrinsically it matters little what field you study, all of it is useful.