Are Numbers Real?

[robvalue]

We create maths. We discover ways in which particular maths has applications when applied to reality.

Remember that maths need not have any bearing on reality at all. To say that you "discover" a new mathematical system would be metaphorical only; it would be like exploring the platonic plane of abstract concepts.

Maths makes statements which are either true or false within their own framework. They are true, essentially, because we say they are true. They are a logical result of applying the rules which we say are true. Verifying they are true is a matter of making sure the rules have been correctly followed.

So (concerning right triangles) it is only true that the square of the hypotenuse is equal to the square of the sum of the other two sides because we say it's true. This is not a fact that we discovered about right triangles? Is this what you're saying?

It’s very much a matter of nuanced language here. Remember that there are any number of mathematical systems. In some systems, this would not be true. So essentially, the result follows, and is true, based on the assumed truth of the axioms we choose. It’s a series of tautologies. If we find a new result, then we’ve discovered it insofar as we didn’t know it before, but it would always have been true (or not) because of the axioms. Most people would call me a pedantic cunt for having said what I just said, and I accept that totally. It’s meant to be highly technical, because of the discussion at hand.

We pick a mathematical system, and we could be metaphorically described as exploring the truths it contains. This is the case because we don’t simply have immediate access to all these truths.

Remember to differentiate between pure maths and applied maths; even when talking about a right-angled triangle, it is still pure maths. When we go to apply our maths to reality, this is a different matter. The difference here is so subtle that I think many people don’t even realise it. As a way of illustrating the difference, I could have a "pure maths" triangle with sides of length 3, 4 and 5. I can’t have that in a practical setting. I must pick some units, and I’m really only projecting an abstract idea onto reality for convenience. I could also have a triangle in a different maths system that doesn’t transfer at all into "real" triangles, regardless of units. So it’s about picking the right tool.

I can only prove that all modelled, theoretical perfect triangles behave a certain way within my maths system. I can’t prove that reality itself behaves a certain way universally. I essentially simplify reality through filters, so that my model can be applied exactly.