[bheadmaster]

> math is ultimately about symbols. [...] Neither! It means that the string of symbols '2 = 2' is reducible to the symbol 'true', given certain axiomatic symbolic transformations. Nothing more, nothing less!

If that were true, math would be useless, and nothing more than an esoteric artform.

The true power of math comes from the correspondence between those symbolic transformations and observation from the real world. Two objects that look alike can be placed in juxtaposition with any other (different) two objects that look alike, and no matter how much we move them around, as long as we don't add or remove any objects, they can still be placed in the same juxtaposition as before (while this description may seem verbose and clumsy, in the real world it does not need a description - it is a much more primitive sensory perception, learned at an early age).

> obviously we can project concepts onto the symbols, like "integer" and "real number", and talk usefully about them, but those are the map and the symbols are the terrain

It wouldn't be "obvious" that we can project concepts onto symbols, if we didn't discover that symbols correspond to concepts and that symbolic transformations can help us predict the future. Thus I'd say it's the other way around: symbols are the map that we know how to read - of the terrain that we can't traverse easily.

[fenomas]

> The true power of math comes from the correspondence between those symbolic transformations and observation from the real world.

Not at all, think it through further. Obviously it's true that mathematics is more practically useful in cases where its symbolically-proved claims have some kind of relation to real-world observations, but if that relationship were a requirement, math would be useless - you could prove a theory on paper symbolically, but you wouldn't know whether the thing you proved was "really true" until you found a way to check whether the result is also true in the real world. And if you found it was true of apples, it might still not be true for electrons, etc etc.

Rather, math's power stems from the fact that it emphatically does not expect or require the symbols to have any connection to real world observations. If you prove something on paper, it's proved and that's that. If the thing you proved also happens to be useful for describing apples or electrons, that's great - and the fact that this often happens is why the whole "unreasonable effectiveness of mathematics" is a thing. But if there's no relation to the real world, that doesn't in any way affect the truth of the symbolically proved claim, or its usefulness or interest to mathematicians.

[bheadmaster]

> Rather, math's power stems from the fact that it emphatically does not expect or require the symbols to have any connection to real world observations.

What exactly do you mean by "power" here, if not the ability to predict real-world phenomena? In absence of it, what exactly would make it anything more than an exotic artform?

[fenomas]

I meant the fact that math works at all, as a symbolic framework with consistency and the power to prove some statements and disprove others, etc. Math only has those features because it examines symbols abstractly regardless of any connection to the real world.

Like, consider: parabolas were pretty fully described by the ancient Greeks, purely as a symbolic abstraction. It was only 1500+ years later that anyone realized that they could also predict the motion of cannonballs and planets. But that discovery was completely orthogonal to the math - e.g. symbolic statements about parabolas didn't get any truer just because they now also described real-world phenomena. (And likewise when we later discovered that planetary motion isn't quite parabolic after all, that didn't affect our understanding of parabolas either.)

That's all I was saying here - that the "esoteric artform" part of math where one abstractly examines symbols is the essence of the thing, and the "predict real-world phenomena" aspect is a side effect that sometimes happens and sometimes doesn't.

[bheadmaster]

True, but what I'm pointing out is - imagine parabolas never predicted movement of anything, or imagine that math itself never had any prediction power in the real world. The self-consistency and the aestethics and the ability to prove statement inside itself would all be just a bunch of symbolic games, much like poetry or crossword puzzles.

You may argue that that, in itself, is powerful, in which case fair enough. But that "power" would be comparable to that of poetry or painting, which, in my opinion, does a disservice to the true power that mathematics holds. Mathematics is much more powerful than poetry and painting, because poetry never helped us build nuclear reactors.