A hot math take
As I learn mathematics I try to deeply question everything, and pay attention to which assumptions are really necessary for the results that we care about. Over time I have accumulated a bunch of “hot takes” or opinions about how conventional math should be done differently. I essentially never have time to fully work out whether these takes end up with consistent alternative theories, but I keep them around.
In this quick-takes post, I’m just going to really quickly write out my thoughts about one of these hot takes. That’s because I’m doing Inkhaven and am very tired and wish to go to sleep. Please point out all of my mistakes politely.
The classic methods of defining numbers (naturals, integers, rationals, algebraic, reals, complex) are “wrong” in the sense that it doesn’t match how people actually think about numbers (correctly) in their heads. That is to say, it doesn’t match the epistemically most natural conceptualization of them: the one that carves nature at its joints.
For example, addition and multiplication are not two equally basic operations that just so happen to be related through the distributivity property, forming a ring. Instead, multiplication is repeated addition. It’s a theorem that repeated addition is commutative. Similarly, exponentiation is repeated multiplication. You can keep defining repeated operations, resulting in the hyperoperator. I think this is natural, but I’ve never taken a math class or read a textbook that talked about the hyperoperators. (If they do, it will be via the much less natural version that is the Ackermann function.)
This actually goes backwards one more step; addition is repeated “add 1”. Associativity is an axiom, and commutativity of addition is a theorem. You start with 1 as the only number. Zero is not a natural number, and comes from the next step.
The negative numbers are not the “additive inverse”. You get the negatives (epistemically) by deciding you want to work with solutions to all