Sorry, unintended inferential distance. In a previous post, Eliezer distinguishes between "true" and "valid" because only empirical things can be true, and he doesn't think mathematics is empirical. Thus, propositions that follow from proposed axioms are "valid" - what a mathematician would call true - to avoid confusing vocabulary.
You avoid the confusion by asserting that mathematical assertions really do correspond to some physical state (i.e. are empirical). Under the correspondence theory of truth, that allows some mathematical statements to be true, not simply valid. Nonetheless, I assume you don't think all mathematical statements are true (2 + 2 != 3, etc).
The problem with asserting that mathematical statements are empirical is that there are certain mathematical assertions that are valid but do not have any physical basis. Consider the proposition, "The Pythagorean theorem follows from Euclid's axioms." The statement is valid, but cannot meaningfully be called true because there is no physical fact that corresponds to the assertion by virtue of the fact that the physical universe is not a Euclidean space. But the statement is not false because there is no physical fact that corresponds to "The Pythagorean theorem is not deducible from Euclidean axioms."
In other words, your theory of mathematics has no room for "validity", only "truth." The Pythagorean theorem is interesting to mathematicians, but adopting your philosophy of mathematics would hold that generations of mathematicians have been interested in a theorem that we now know can never be true or false. That's just too weird for most people to accept.
You seem to misinterpret what I mean, but that's my fault for explaining poorly. This post has been getting out of hand with all the clarifications, so I will retract it and post a hopefully clearer version later on. Maybe as I write it, I'll notice a problem with my view which I hadn't seen before, and I'll never actually post it.
In his latest post, Logical Pinpointing, Eliezer talks about the nature of math. I have my own views on the subject, but when writing about them, it became too long for a comment, so I'm posting it here in discussion.
I think it's important to clarify that I am not posting this under the guise that I am correct. I'm just posting my current view of things, which may or may not be correct, and which has not been assessed by others yet. So though I have pushed myself to write in a confident (and clear) tone, I am not actually confident in my views ... well, okay, I'm lying. I am confident in my views, but I know I shouldn't be, so I'm trying to pretend to myself that I'm posting this to have my mind changed, when in reality I'm posting it with the stupid expectation that you'll all magically be convinced. Thankfully, despite all that, I don't have a tendency to cling to beliefs which I have seen proven wrong, so I will change my mind when someone kicks my belief's ass. I won't go down without a fight, but once I'm dead, I'm dead.
Before I can share my views, I need to clarify a few concepts, and then I'll go about showing what I believe and why.
Abstractions
Abstract models are models in which some information is ignored. Take, for example, the abstract concept of the ball. I don't care if the ball is made of rubber or steel, nor do I care if it has a radius of 6.7cm or a metre, a ball is a ball. I couldn't care less about the underlying configuration of quarks in the ball, as long as it's a sphere.
Numbers are also abstractions. If I've got some apples, when I abstract this into a number, I don't care that they are apples. All I care about is how many there are. I can conveniently forget about the fact that it's apples, and about the concept of time, and the hole in my bag through which the apples might fall.
Axiomatic Systems (for example, Peano Arithmetic)
Edit: clarified this paragraph a bit.
I don't have much to say about these for now, except that they can all be reduced to physics. Given that mathematicians store axiomatic systems in their minds, and use them to prove things, they cannot help but be reducible to physical things, unless the mind itself is not physical. I think most LessWrongers, being reductionists, believe this, so I won't go into much more detail. I'll just say that this will be important later on.
I Can Prove Things About Numbers Using Apples
Let's say I have 2 apples. Then someone gives me 2 more. I now have 4. I have just shown that 2+2=4, assuming that apples behave like natural numbers (which they do).
But let's continue this hypothetical. As I put my 2 delicious new apples into my bag, one falls out through a hole in my bag. So if I count how many I have, I see 3. I had 2 to begin with, and 2 more were given to me. It seems I have just shown 2+2=3, if I can prove things about numbers using apples.
The problem lies in the information that I abstracted away during the conversion from apples to numbers. Because my conversion from apples to numbers failed to include information about the hole, my abstract model gave incorrect results. Like my predictions about balls might end up being incorrect if I don't take into account every quark that composes it. Upon observing that the apple had fallen through the hole, I would realize that an event which rendered my model erroneous had occurred, so I would abstract this new event into -1, which would fix the error: 2+2-1=3.
To summarize this section, I can "prove" things about numbers using apples, but because apples are not simple numbers (they have many properties which numbers don't), when I fail to take into account certain apple properties which will affect the number of apples I have, I will get incorrect results about the numbers.
Apples vs Peano Arithmetic
We know that Peano Arithmetic describes numbers very well. Numbers emerge in PA; we designed PA to do so. If PA described unicorns instead, it wouldn't be very useful. And if PA emerges from the laws of physics (we can see PA emerge in mathematicians' minds, and even on pieces of paper in the form of writing), then the numbers which emerge from PA emerge from the laws of physics. So there is nothing magical about PA. It's just a system of "rules" (physical processes) from which numbers emerge, like apples (I patched up the hole in my bag ;) ).
Of course, PA is much more convenient for proving things about numbers than apples. But they are inherently just physical processes from which I have decided to ignore most details, to focus only on the numbers. In my bag of 3 apples, if I ignore that it's apples there, I get the number 3. In SSS0, if I forget about the whole physical process giving emergence to PA, I am just left with 3.
So I can go from 3 apples to the number 3 by removing details, and from the number 3 to PA by adding in a couple of details. I can likewise go from PA to numbers, and then to apples.
To Conclude, Predictions are "Proofs"
From all this, I conclude that numbers are simply an abstraction which emerges in many places thanks to our uniform laws of physics, much like the abstract concept "ball". I also conclude that what we classify as a "prediction" is in fact a "proof". It's simply using the rules to find other truths about the object. If I predict the trajectory of a ball, I am using the rules behind balls to get more information about the ball. If I use PA or apples to prove something about numbers, I am using the rules behind PA (or apples) to prove something about the numbers which emerge from PA (or apples). Of course, the proof with PA (or apples) is much more general than the "proof" about the ball's trajectory, because numbers are much more abstract than balls, and so they emerge in more places.
So my response to this part of Eliezer's post:
Logic is stable for the same reasons the laws of physics are stable. Logic emerges from the laws of physics, and the laws of physics themselves are stable (or so it seems). In this way, I dissolve the question and mix it with the question why the laws of physics are stable -- a question which I don't know enough to attempt to answer.
Edit: I'm going to retract this and try to write a clearer post. I still have not seen arguments which have fully convinced me I am wrong, though I still have a bit to digest.