Now, notwithstanding the above, does 2+2 really equal 4, independent of any human thoughts about it? The answer is we can't know.

We can and do know that, though we can't be absolutely certain. There are different shades of certainty, not at all blurred together in the uniform "we can't know".

I would like to agree with this, but still feel that the difference between necessary and contingent truth needs to be accounted for somehow. I can, like Eliezer, vaguely imagine a world in which 2+2=3 in Peano arithmetic, but I don't think I could precisely model it in the way I could precisely model a world with different laws of physics. I might be able to, by a huge amount of Dark-Lords-of-the-Matrix-type meddling, precisely model a world in which some things whose closest equivalents in our world would be implementations of Peano arithmetic output 2+2=3, but (a) that just seems too much like cheating to mean much (b) the world's fundamental laws would still have to be consistent with 2+2=4 – if it had physics like ours, 2 joules and 2 more joules would still make 4 joules; if it were a cellular automaton, 2 living cells and 2 more living cells bordering a cell would be 4 living cells.

(See also: Math is Subjunctively Objective.)

The disappointing truth is that bananas are quarks, and by amazing good fortune, the properties of everyday macroscopic objects are sufficiently related to those of other physical phenomena that a few lucky humans can just barely manage to crudely adapt their banana-counting brain hardware to work in those other domains.

Amazing good fortune implies amazing coincidence, which implies strong disconfirmation relative to any alternatives that don't need such a coincidence.

Philosophy of math is hard, and I'm not sure we can get very far trying to figure it out on LW.

Mathematical statements are kinds of representations.

What exactly is being represented? What exactly is the status of the representation itself, and how does it relate to what is represented?

We don't have a solid theoretical understanding of these matters, and I until we do, I reckon we're going to continue to argue about it, and probably not very productively. At the moment we have trouble finding ways to describe what mean when we talk about the various aspects of representations.

When I say that "2 + 2 = 4" is a fact independent of the physical universe, I mean that it is the sort of information that does not tell us (directly) which universe we are in, what the laws of physics are. (Though it can be used in a line of reasoning evaluating propositions that are about our particular universe.) I do not mean that it is a supernatural force, or a mystical energy field controlling our destinies.

And the whole purpose of making that point was to frame the question of how well the physical systems in our brains represent these logical facts, and how we can evaluate our uncertainty about the answer to that question.

Personally, I find talking about words and language to be more interesting than numbers. The word "banana" is easier to attack than the number "2". When you say:

I can put 2 bananas on a table, then put down 2 more bananas, and count out 4 bananas.

This implies that the perception of "2 + 2 = 4" is built on your perceptions of "bananas". In my opinion, this has nothing to do with math and everything to do with language. "2 + 2 = 4" is relatively useless unless everyone agrees 2 + 2 = 4. The same is true of "banana". The word is useless unless enough people agree about what it means. To ask if 2 + 2 = 4 is "true" is linguistically irrelevant.

This is not to say that the question is irrelevant in all contexts, but when talking about it, asking about its truth may be a misnomer.

Abstract philosophy is problematic. Please indulge me for a few paragraphs as I work my way up from the factory floor.

How do you gain power over the world? You could do stuff and remember how it works out, but experience is a dear teacher. The holy grail is to be able to predict. Instead of doing A, B, and C and remembering for next time that C worked best, predict the consequences of doing A, B, and C, and then do only C.

We can hand crank simple models of reality, yielding predictions that turn out to be wrong because the models were too simple. We respond with more elaborate models. We cannot rigorously work out what they predict, because they are not simple enough, but waving our hands and talking fast we reconcile them after the fact to the way things turned out.

That serves for both craftsmanship and politics. The craftsman knows the right answer from the traditions of the craft, so he can always guide the derivation of predictions with extra, informal ingredients from craft-lore so that they give the right answers. The politician has only to persuade. Having persuaded that his theory of society explains what is wrong he will win election. Having explained away why his theory didn't work first time he may yet win a second election, but seldom a third.

The engineer faces a stiffer test. He must scale up the design for his bridge, his engine, or his chemical plant by a factor of 10 or 100. Some projects are only plausible on an expensive scale and engineering is then a theory-based leap into the unknown. His backers will not fund the trial and error development of a craft tradition; his equations must predict accurately.

This is where mathematics comes in. We have simple, unrealistic models we can solve, and complicated, realistic models that we cannot solve. Mathematics pushes the boundary, always trying to find ways to solve models a little more complicated than current understanding permits. The essence of this enterprise is rigour. Once you say "obviously", "who can doubt", "as every-one knows", "as every-one agrees", the point is lost.

The value of mathematics lies in its refractory nature. Once the physicist or the engineer has proposed that a piece of mathematics models nature, the model, if humanly soluble, gives him a single prediction that is either right or wrong. If right, he can test it some more and may find himself in possession of the holy grail, a tool for prediction.

Notice the sharp contrast between this and the ordinary notions of theory and prediction. Ordinarily there is a degree of hand waving involved in getting from theory to prediction. We believe our theory because it "predicts" things that have already happened, because knowing the outcome in advance we can guide our hands to paper over the gaps in the theory. But when it comes to predicting things that haven't happened yet, we are careful not to be held accountable. We know, even as we dodge fully conscious acknowledgment, that those who were careless about this often get their come-uppance.

I have painted a picture of where mathematics fits into practical reality. Mathematics offers practical men a selection of soluble models that are not so simple as they would be restricted to if confined within their personal limits. Physicists select a model and an interpretation of the model which jointly agree with reality. Engineers use this to design artifacts which then work.

What is the cause of this miracle? Why do the predictions come true? I don't really know, but it seems clear that the refractory nature of mathematics plays an essential role. A mathematician sets up a formal system, such as Peano arithmetic and, like Baron Frankenstein, he finds that the creature escapes from his control. Perhaps the formal system turns out to be useful for modeling and prediction in real life, perhaps it doesn't, but the crucial point seems to be that the mathematician cannot fudge it after the fact. If it doesn't apply to reality the mathematician can invent/discover something else, or the physicist can try applying it to a different phenomenon, but there is no avoiding the fact that the interpretation of the formal system didn't agree with reality. That is somehow linked to the fact that formal systems that do have interpretations that agree with reality are rare and precious and give predictions that come true.

I'm unconvinced by the top level posts ontology because it seems to have no place for the applied mathematician's failed formal systems. These formal models horrify and fascinate us for the same reason that Frankenstein's monster does: they have a mind of their own. The applied mathematician was attempting to model the real world and created his formal system to do so, but it defied him, and followed its own rules, to its creator's sorrow. When we respond to this stubborn independence of mind by expelling the perpetrator from really, its not ontology, its pique.

Asking "Now, notwithstanding the above, does 2+2 really equal 4, independent of any human thoughts about it?" requires human thoughts. There would still be four bananas without anyone to count them because I'm here imagining that world in my human thoughts. It's like trying to imagine what death feels like. You fail by trying. Just like the Game (muhahaha!).