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Comment by crazydoggers | Hacker News Reader

I’m not sure how it could be otherwise. On some level mathematics is a description of reality that we can use to compute things in reality.

For example, pi is the ratio of a circle’s circumference to its diameter. It’s just what a circle is in two dimensions. The value of pi isn’t any more mysterious or connected to physics than the existence of this thing called a circle. If you have some other Euclidean shapes you’ll have other ratios and values that have other relationships to other things in physical reality.

And if reality was different, hence the physical laws were different then the math would be different.. and the beings in that world might wonder why their math and physics were so interconnected.

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> On some level mathematics is a description of reality that we can use to compute things in reality.

This is contested by nominalists. They'd say you have it backwards. Mathematics is just an abstraction/language that can be used as a tool. The reason we're able to understand the world through mathematics says more about the power of mathematics than it does about the world. If the physical world were different, math would still work.

Well if the fundamental constants were different, math would still work the same way. But if a universe acts in a completely different way, say like a dream world, where they is no logical physical causaulity, our universe's math would be irrelevant. You could still think about it but it would be irrelevant to that universe, and not applicable. And so you would also not be able to derive that math from within this universe, at least not the same math as ours. So we have to admit that our mathematics is tied to the universe we inhabit.

The properties of the universe affects mathematics insofar as matter is organized and living agents have a use for it, but its abstractions and logical relations still do not require the universe. I think a metaphysical position like physicalism would have to be invoked for that.

Math is informed by physics, but not constrained by it.

Very loosely speaking, pi can take a different values on non-euclidean planes. This ends up becoming relevant on the surface of the earth or, say, the saddle of a horse. I'm not sure if the motivation was from looking at curved surfaces, but it just as easily could've come from the rejection of Euclid's parallel postulate and seeing what results. Similarly, I think imaginary numbers were motivated by the math well before they found applications in reality.

There are also plenty of other mathematical constructions that are informed by reality (since that's what our brains are constrained to,) but I'm pretty sure are far from actually describing reality. Transfinite cardinals/ordinals, fast growing hierarchies, Turing degrees, Goldbach's conjecture, how the hypervolume of a hypersphere eventually decreases as dimension increases...

You can even reject the standard axioms and construct math that can not be compatible with reality. Or argue that the standard axioms permit too much wiggle room to create concepts that have no relation to reality. (But maybe you shouldn't; that sounds like philosophy.)

Not sure that I agree as does a mathematical circle actually exist? We can produce things that approach the concept of a circle and similarly, we can measure circumference and diameter to a level of precision to approach the value of pi, but we never have a perfect circle or the exact value of pi.

I tend towards maths being distinct from physics as some areas of maths deal with concepts that can only have a passing resemblance to reality - the Banach-Tarski paradox is an example. (Similarly, pretty much any treatment of infinities ends up to have little relation to reality such as with Hilbert's Hotel).

> Not sure that I agree as does a mathematical circle actually exist?

The set of unit norm complex numbers surely exists just as much as the real numbers exist, doesn’t it? A circle is an idea, and it certainly exists.

I think you're using a different meaning of "exists" than I am.

Ideas don't exist as you can't point at them, steal them, destroy them etc. I can point at something that approaches the concept of a circle and I can point at a set of objects that can be counted, but I can never see a mathematical circle (zero thickness would make it impossible to see) and I can't see a "four" without representing it by a symbol or collection of objects.

I guess one way to decide on a definition of ‘exists’ — since a given thing probably either exists or doesn’t — is to instead decide when it doesn’t exist.

Does a five-sided triangle exist? Well, the very concept seems to be self-contradictory, but surely the idea exists (since I just mentioned it, and you probably understood what it meant and could see why it was contradictory), and even if you don’t believe the idea exists you must eventually concede that the sentence that describes it exists!

If you think things can only exist if you can point at them, steal them and destroy them, then would you say the natural numbers (= 0, 1, 2, 3, …) don’t exist? That would seem to be a strange notion of existence.

Other than that it isn’t a physical object you can hold in your hand, I don’t see why there’s anything problematic about regarding the circle as an object that exists, even if it’s just (in some sense) a theoretical object.

If you take your position to its extreme (but, I would argue, logical) conclusion, almost nothing in your everyday life exists. It’s all atoms, and the fact that you regard, say, a guitar as being a guitar is rather weird because any given guitar is almost certain to be atomically distinct from every other guitar that has ever existed. Are guitars just ideas? Would you say there’s ‘no such thing as a guitar’ because they’re all just approximations to the ideal guitar?

Put simply: what does rejecting the existence of ideas buy you? I don’t see how it’s a productive (or very insightful) position to hold.

I'm simply drawing a distinction between something that physically exists versus something that doesn't physically exist.

Ideas don't (physically) exist although a representation of an idea can exist e.g. a mathematical circle doesn't physically exist, but representations of circles do exist and they vary with how close they get to the idea.

With your guitar example, you demonstrate how we classify things in a messy way. Because the idea of a guitar doesn't exist, but lots of representations of guitars exist, we just refer to those objects as "guitars" as we can recognise the "guitar idea" that they are made to represent.

Distinguishing between physical existence and non-physical existence is very useful - it allows you to separate the map from the territory. It's very useful to abstract the concept of "five" and recognise that it's different to five objects, despite them being so closely related. The abstraction of numbers becomes more useful when considering ideas such as negative or imaginary numbers - it would be difficult to reason about and use them if we continually think of numbers as only "counting objects".

If this was true then every bit of math must have a real-world application. But math doesn’t need to have real-world applications in order to be valid.

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