Mathematicians Bridge Finite-Infinite Divide

> Zero is offensive and ungodly. With "five", at least there's something there to abstract. With zero, there's not even anything! How can nothing be something?

Cue angry biting of thumbs

i. Peano Axioms: (Typically) your initial element is 0 and you'll get 1 by applying the successor function to 0 and 2 by applying that to 1 and so on.

ii. Von Neumann's definition of Ordinals: {} is zero and one is {{}} and two is {{}, {{}}} and so on.

iii. Conway's definition of Surreal numbers: {|} is zero and { {|} | } is one and so on.

iv. Church encoding

and many more.

P.S. I'm not a logician and don't know whether "more of an abstraction" has any well-defined meaning and not just the meaning that might be inferred by a software developer.

I was saying that I, personally, find foundations as in the OP down in the basement, dark and too far from applications in any sense.

For making money as a full prof of math, first have to get there, and that usually takes over 10 years if make it at all. Yes, it's possible to play the academic game; heck one paper in math I published is pretty, surprising, etc. but I can't imagine that it will ever be useful directly or even indirectly even by several steps of indirection for the foreseeable future if ever. I've seen people publish such things and make some progress in an academic career; to me that's playing an academic game; I chose not to do that.

When I was a prof (I didn't want to be but did it for a while trying to help my wife in her illness), it seemed to me that I was getting paid by students, pizza parlor owners and workers, auto dealership owners and workers, farmers, etc., and for them I wanted my work and teaching to be useful -- to me, personal curiosity, art, etc. didn't count. I really wanted the department to be clinical, professional, practical, like law and medicine, i.e., welcome people from outside academics with real problems and then seek to solve those problems. When can't solve a problem, then maybe that will be a good research direction; if make good progress, then already have one application!

For applied math, determine that not by PDEs but by what can find that is useful. Early in my career around DC for mostly US national security, I found lots of such applied math. And my startup, while more focused, is basically some applied math. PDEs? I had very little to do with those; the one case was the Navier-Stokes equations, and they were so difficult to work with that the project wasn't making much progress and I was pleased to move on to other topics.

It's not easy to see what pure math is the more useful. For the pure math I do respect, especially for utility, it appears that in some vague, large sense the results are fundamental, important broadly, and, eventually, inescapably relevant, but that is a difficult judgment call. Functional analysis? Sure. Algebraic geometry? Less sure. Foundations? Slim chance.

I was not trying to give career advice to other people possibly interested in math but just commenting that I found the foundations as deep as in the OP just too far down in a dark basement. In math, that's an old remark about foundations -- "Drop what you are doing trying to be useful and come with me down into the dark basement and wrestle with really subtle issues down there." Standard, old remark in math.

You read lots of stuff between the lines I wrote, stuff not really there.

> RT22’s infinite structures “may make the proof easier to find,” explained Slaman, “but in the end you didn’t need them. You could give a kind of native proof — a [finitistic] proof.”