This was my submission for the recent FQXI essay contest on the role of mathematics and physics. It didn't win a prize but I think it makes some important points hence am reposting it here. It connects to a couple of earlier posts in this blog:
http://www.letstalkphysics.com/2014/03/why-i-dont-believe-bostrom-or-tegmark.html
http://www.letstalkphysics.com/2010/02/problem-with-quantum-mechanics.html

There are three basic points in the essay:

1. The universe must obey mathematical laws since there is no other possible source of structure.
2. The mathematical laws that we see are plausibly the only ones compatible with life.
3. Although it must be described by mathematics (point #1), the universe cannot be a mathematical structure since the laws of quantum mechanics cannot be fully axiomatized.

Point #3 means that we can not think of our universe as being just one member of an infinite "mathematical multiverse" consisting of all mathematical structures. Mathematical structures can be axiomatically defined, but aspects of our universe (indeed aspects of quantum mechanics) can not be captured by axioms. Nevertheless point #1 implies that "everything there is to know" about the universe must be specified by mathematics; hence, our universe seems to lie at a sort of boundary of mathematical description. It is fully mathematical but it is not pure mathematics.

If the universe is not a mathematical structure then it can not be viewed in the Tegmarkian way, as part of an eternal "mathematical multiverse". It must therefore be a singular "creation" of some sort. Point #1 implies that any such creation must be based on mathematics, and point #2 implies that, if evolution of life is the "goal", that mathematics may be essentially unique.

Putting it all together suggests that our universe is a "singular, almost-entirely mathematical" creation which exists for the purpose of evolving life. It satisfies the unique mathematical laws which enable this outcome, so in this sense one could say there was "no choice" in its creation.

Of course this begs the question of what entity "created" the universe in the first place. I don't attempt to resolve this problem but am only concerned with the universe that we experience, not a larger "meta-universe" in which it may be contained. If one dislikes the overall hypothesis, the points 1,2, and 3 are still separately worthy of consideration.

Response to Horgan

Recently science journalist John Horgan posted an opinion piece called "How Physics Lost Its Fizz". Following is my response, posted there as a comment.

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Dear John Horgan,

When it comes to physics you've long ago decided that your popular-level understanding of frontier work is equivalent to the actual work itself. This is why you think there is no "fizz" - because you equate ideas which spring from a mathematically compelling foundation with superficially similar ideas that some philosopher or writer mentioned in the past. This is like saying that particle physics was never very novel because Democritus had already thought of particles previously.

Black holes were invented in 1783 and not convincingly observed until very recently; does that mean that for 200 years the concept of black hole was either "not fizzy" (because someone already thought of it) or "not science" (because not connected with a direct observation)?

You need to exercise a little bit of modesty in making popular-level critiques of topics whose interest derives from aspects that cannot be understood at the popular level. You, in fact, cannot understand these ideas, and do not understand what is motivating them.

Have some respect for those people who are discovering various ideas through years (in fact decades) of very difficult study and research. You must know that these people are not idiots; each one of them would have appeared as one of the best students at a university, if not one of the best in the entire world.

Did these very bright students suddenly become stupid after putting in the solid decade of very difficult study necessary to actually understand the frontier of physics? Or is it more likely that you, who have not put in this effort, do not understand the actual state of physics or the actual background of any these ideas?