Are Numbers Real?

[robvalue]

Naturally, the axioms of the most used mathematical systems weren’t chosen by accident. They’re essentially the result of reverse engineering from the evolution of applied maths. Wondering at how amazing it is that the maths we mainly use just so happens to model things so well is a bit like when theists make improbability arguments against evolution. The maths has been self-selected because it is useful. As we've become more able to think abstractly, we've formalised the primitive ideas and finally reached the starting point, ironically enough! We didn't start by randomly selecting axioms.

For those not familiar, axioms end up being some statements that are required to be true in order to make the maths work, but which can’t be proven to be true. Ending up with as few axioms as possible is desirable. From what I remember, it’s often the case that you can pick (for example) any 2 out of a particular 3 statements as axioms, and together they will prove the third; but if you pick only 1, it’s not sufficient.

In a general setting, as soon as you pick a group of axioms (along with defining the elements allowed into your system and the operations that can be performed), you’ve instantly defined everything that can be proved by those axioms. It’s just a matter of figuring it all out from there, and then seeing if the results have any useful real-world applications.

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PS: Referring to what Jorm was saying, I don’t have a formal definition for what is "objectively real", because I’ve found it to always be a circular endeavour. I only deal with subjective and relative levels of real-ness. Obviously when talking informally, I use "real" to mean "part of a hypothetical objective reality". So maths isn’t the same kind of real as the supposed physical world; that’s as far as I would go.