Straggler writes:I wanted to ask you about the role of mathematical elegance.
You addressed your question to
cavediver, but I'll add my two cents so that you can have more than one opinion.
Straggler writes:As I understand it mathematical/theoretical physicists are often driven by the underlying belief that nature is in some sense mathematically elegant.
If physicists believe that, then they are mistaken.
Straggler writes:Now I am frankly too mathematically ignorant to even really know what is meant exactly by "mathematical elegance". Also "elegance" by it's very nature would seem to be a highly subjective quality. But this approach does seem to have borne much fruit. Einstein and Dirac spring to mind as two obvious examples of those who have achieved an immense amount with this abstract and highly non-empirical approach.
One of the things Dirac did, was to use a notion of generalized functions. I think many mathematicians would view that as inelegant and half-baked. The
Theory of Distributions, due to Laurent Schwartz, provides a far more elegant way of doing this. Likewise, I suspect that many mathematicians would consider Einstein's treatment of GR as somewhat inelegant.
The mathematical analysis as described by
Bourbaki is far more elegant than the same analysis as described by
Titchmarsh, even though they are presenting the same analysis. You should think of mathematical elegance as an artistic quality that is appreciated by many, but not all, mathematicians.
Straggler writes:It is my understanding that with one postulate (the constancy of the speed of light), an assumption of the universality of the equivalence principle and a notion of mathematical elegance it is possible to derive the whole of relativity (special and general) along with all of the predictions of relativity (Big Bang, black holes, gravitational time dilation etc. etc. etc.) just by sitting in a room manipulating equations.
I am inclined to say that is wrong.
For one thing, Einstein didn't start with just those assumptions. He also had Newtonian mechanics, which had proved highly effective. I look at it as finding a way to modify Newtonian mechanics to incorporate the new assumptions. It's a while since I looked over the history of that era, but I seem to recall that there were several proposed ways of modifying Newton, and I would guess that any of them could have been done in ways that were mathematically elegant.
There are serious conceptual changes involved in going from Newtonian mechanics to Relativity, and there was a lot of resistance. I would guess that finding changes that could be sold to other physicists was one of the important influences on the direction that Einstein took.
Straggler writes:Now Einstein didn't
exactly do that. Practicalities of human endevour are never that black and white. But he did revoloutionaise science with his approach. An approach that strongly suggests that maths can lead the investigation rather than simply model observations. And that is where this more platonic view of reality is derived from.
It ought to be obvious to anyone who seriously examines science, that mathematics is used for more than modeling observations. But you can find that as much with Newton as with Einstein.
Straggler writes:And that is where this more platonic view of reality is derived from.
I can't comment on that, since I have never held a platonic view of reality. It has always seemed to me that the world (i.e. the universe) is a messy, disorderly place.
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