Comment by Razengan
4 days ago
Stuff like this is what really interests me in trying to imagine how differently aliens might use things that we consider to be immutable fundamentals.
4 days ago
Stuff like this is what really interests me in trying to imagine how differently aliens might use things that we consider to be immutable fundamentals.
personal theory: I think there's going to turn out to be a parallel development of math that is basically strictly finitist and never contends with the concept of an infinite set, much less the axiom of choice or any of its ilk. Which would require the foundation being something other than set theory. You basically do away with referring to the real numbers or the set of all natural numbers or anything like that, and skip all the parts of math that require them. I suspect that for any real-world purpose you basically don't lose anything. (This is a stance that I keep finding reinforced as a learn more math, but I don't really feel like I can defend it... it's a hunch I guess.)
That math exists, but it is annoying to work with.
any particular reference to what you're thinking of? I am aware of some writings on finitist or constructivist mathematics but they have not quite seemed to get at what I want (in particular doing away with explicit infinities does not require doing away with excluded middle at all, which is what most of that literature seems to be concerned with).
How would you do limits or analysis?
I think it's just a perspective shift. The main idea is that you can't ever measure a real number, only an approximation to one, so if two values differ by less than the resolution of your measurement they are effectively the same. For example consider the derivative f(x+dx) = f(x) + f'(x) dx + O(dx^2). The analysis version of the derivative says that in the limit dx -> 0 the O(dx^2) part vanishes and so the limit [f(x+dx)-f(x)]/dx = f'(x). The 'finitist' version would be something like: for a sufficiently small dx, the third term is of order dx^2, so pick a value of dx small enough that dx^2 is below your 'resolution', and then the derivative f'(x) is indistinguishable from [f(x+dx)-f(x)]/dx, without a reference to the concept of a limit.
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I had this feeling of alien math when I went thru his videos on ancient Babylonian math. They were very serious about the everything divided by sixty stuff. Good times.