1) What is ultrafinitism — what do you actually reject, and what do you keep?
The belief that both the physical and the mathematical universes are finite. Of course, they are enourmously large, but still finite. The so-called infinity is completely unnecessary, and is due to a superstitious belief in something transcendental, analogous to the belief in God. For some people it is a useful and conforming fiction, and may have done some good, (both the belief in God and the belief in a mathematical infinity), but this does not change the fact that infinity is definitely fictional (and unnecessary). Whether God does or does not exist is an open question, and is unknowable, but if God exists, he (or she, or it) is a finite God.
2) What's the everyday cost of putting infinity at the foundation of mathematics — what does it break, distort, or hide that most mathematicians don't notice?
One would have to rewrite the standard math textbook in the new language. There is a simple "transform" that lets you translate anything spoken with infinity in a way that does not mention it. Since we know that is is possible, it is easier to keep the textbook the way thery are, adding a footnote, that the infinity is really "infinity" (a finite symbol).
3) Could you take one piece of mathematics and show us concretely how an ultrafinitist does it differently?
The derivative of x2 is usually computed as folllows
(x2)' = limh→0 ((x+h)2 − x2) / h
= limh→0 (x2 + 2xh + h2 − x2) / h
= limh→0 (2xh + h2) / h
= limh→0 (h(2x + h)) / h
= limh→0 (2x + h)
= 2x
Here is the correct way. Let h be the "mathematical Planck constant", a tiny but NOT zero, "smallest distance"
(x2)' = ((x+h)2 − x2) / h
= (x2 + 2xh + h2 − x2) / h
= (2xh + h2) / h
= (h(2x + h)) / h
= 2x + h
So the derivative of x2 is NOT 2x, but 2x+h. Since h is SO tiny, we can replace it by 0 for all applications.
