I bought two pulp math books last Friday: Incompleteness: The Proof and Paradox of Kurt Gödel by Rebecca Goldstein and One to Nine: The Inner Life of Numbers by Andrew Hodges. [First one is okay as a biography and poor for math (surprise). Second one is stream-of-consciousness blabber only tangentially related to the numbers in question (or numbers at all).]
Both books quote G.H. Hardy in their opening chapters:
317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way.
This, of course, got me thinking: Can I conceive of a mathematics in which 317 isn't prime?
I agree that as soon as one accepts that there is a 317, there is no way around the fact that it's prime. Worse, as soon as one accepts discreteness, the positive integers are there. Even if everything in our universe happened modulo 23 (what a beautiful and amazing universe that would be), one could still conceive of number systems that weren't finite. 317 would be prime.
And, as soon as you have the positive integers, you get the negative ones (and zero). Moments later, you discover you've ended up with at least the algebraic closure of the integers (aka. the complex numbers).
Peano's right out. The Axiom of Choice is right out. Heck,
There exists an x... is right out leaving no Choice
So, what if one doesn't accept discreteness? Can one formulate a mathematics without discreteness?
Even in topology (the worship of continuity), one ends up with discrete numbers of dimensions, discrete numbers of handles on surfaces, one-to-one mappings, pointed-spaces, etc.
Simple algebraic structures like semigroups require you have an element and multiply it by one and only one (possibly the same) element to produce one and only one (possibly the same as one of the prior) element.
Maybe, if you had a continuous class of objects and a continuous
family of morphisms, you might like to make some in roads with a
non-discrete category (where you'd probably have to nix the identity
morphism and the notion that a morphism goes from one object to
another object and composition... heck... I just took away all of
standard category theory and I'm not sure where to start in that
direction without the concepts of
I have a vague notion of a continuously-dimensioned vector space, but I run afoul in the distinguished zero and just generally because one must be able to pick vectors to add them and once picked you've got finite subspaces.
This is where I am. Stay tuned for more, should it arrive. I hope my road trip this weekend will afford me ample time to ponder. (If I want a two year old to sleep for seven hours, how many Ambien would that be?)