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mwengler comments on The Emergence of Math - Less Wrong
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Ok, if you want a serious response instead of a snarky one, here goes:
You may have learned about Euclidean geometry in school. Two points define a line. A line and a point not along the line define a plane. As Euclid defined the geometry, parallel lines never intersect.
However, we don't live in a Euclidean geometry. To a first order of approximation, we live on a sphere. If Line A is perpendicular to Line X and Line B is also perpendicular to Line X, they are parallel in Euclidean geometry. Nonetheless, on a sphere, Line A and Line B will eventually intersect.
So we've got all this neat mathematics deriving interesting results from Euclidean axioms, but nothing in the real world is Euclidean. If we take your thesis (that math can be reduced to physics) seriously, that means that Euclidean geometry is not simply invalid - it is incoherent (i.e. wronger than wrong). You might be willing to bite the bullet and throw Euclidean geometry in the trash, but no one who takes math seriously is willing to do so.
For further reading, you might consider following the links from this post by Wei_Dai. In short, the issue here - how to talk about the "truth" of mathematics - is a basic problem with the correspondence theory of truth. Eliezer is making an attempt to bridge the gap in the post you highlighted, but he is deliberately avoiding the philosophical choice you made - I suspect because he is unwilling to throw out non-physical mathematics, which I've argued above is a requirement for your theory of mathematical truth.
Millions, perhaps billions of humans have put food on the table and built machines with 100s of times the power or calculational ability of human individuals without ever needing to concern themselves with the breakdown of euclidean geometry. Perhaps there are machine designs that have failed due to this breakdown, but failure has so much entropy that it teaches us infitely (and I do not mean that literally) less than our successes do.
One idea I love in lesswrong is the "how do I code that in to an AI" bias in evaluating efforts. Even if there is some frontier where a deviation from euclidean geometry is necessary to understand in our design of the ultimate AI (or at least the last one built by humans)? Why would anyone be uninterested in the theory behind 99.99...% of the progress we are likely to make?
Try sailing an ocean, as millions of humans have had to do (even just the ones doing so involuntarily like the Africa->America slave trade) with plain Euclidean geometry and then tell me how practical alternative forms of mapmaking, direction-setting, and locations are.
As it happens, Nick Szabo is slowly blogging how exactly one does that: http://unenumerated.blogspot.com/2012/10/dead-reckoning-and-exploration-explosion.html and http://unenumerated.blogspot.com/2012/10/dead-reckoning-maps-and-errors.html
Just because many millions of people don't need to concern themselves with that doesn't mean there aren't many other millions who don't.
In addition to the object level mistake that gwern has pointed out, you've made a meta-level mistake.
I wasn't arguing for the usefulness of Euclidean or non-Euclidean geometry. I was trying to shorten the inferential distance. Euclidean axiomatic mathematics is some of the first axiomatic mathematics anyone is taught in school in the West. The OP might not have understand what it was that his theory was missing in reference to infinite sets, so I used an example I expected him to be more familiar with, in an effort to make my point clearer to him.
You may not think my particular point is interesting in a practical sense, but pointing that out is quite rude unless you really think that I'm unaware that the difference between Euclidean geometry and real world geometry does not always make a practical difference.
It's like asserting the difference between Newtonian and relativistic physics doesn't make a practical difference. I don't know how true that is, but saying something like that to Einstein or Hawking is just rude.