Do you believe in god or math?

[Categories+Sheaves]

You've got to comprehend very clearly the importance of the physical meaning of a quantity of "ONE" before it's even worth moving on to the next step.

Moving forward without that comprehension being made crystal clear would be totally futile.

And to be perfectly honest I don't see anyone grasping this trivial first step.

I do not need a crystal clear comprehension of my hammer before I strike a nail sitting in front of me. I sense that the head and handle are sturdy and firmly wedged together, and that is enough to proceed with my business.

We have a primitive notion of what I could be referring to when I say "this sort of thing," whatever that thing is. Whenever we wish, we can have our "one" become a stand-in for copies of this thing. If I see multiple two-by-four planks of wood, I can count them and say, "I have this many two-by four planks of wood". If I have a single plank and you split it lengthwise, it is no longer a two-by-four. This is not by a fault in our numbers, it is simply a statement that the wood in front of me no longer fits my definition of "two-by-four plank*".

Math is obviously not God. Math is also not physics. Math is the art of knowing how to apply some simple problem-solving techniques, and occasionally the art of good notation. Nothing more. To whatever extent we see multiple things that are functionally equivalent to us, we will have a use for this concept of counting. To whatever extent we try to group or categorize things, we will have a use for this concept of sets.

As CliveStaples was pushing me on earlier, the strength of these things is in their generality. I don't have to restrict myself to the nails I imagined at the time I made my hammer, I can hammer other things that behave similarly to nails. How can I tell whether an object can be hammered into something else? I ought to be able to list all the properties of nails and wood that allowed my hammer to work on them... how about we write these things out and call them axioms?

Also: when do we get to your thoughts about contemporary math and stuff like that?

*And yes, I know the dimensions of the base of these planks aren't exactly two by four inches