The ladder
Pick a number. Each is put through three tests: can you count to it, can you write out all its digits, and can you name it.
Three dots
The same question shows up in small things. We write 1/3 = 0.333… and the dots do a lot of work. Is there a finished, infinite object behind them? Or only a rule that produces another digit whenever someone asks, and has never been asked for all of them?
Roughly 40 digits of π fix the circumference of the observable universe to within the width of a hydrogen atom. The digits past that are, as far as measurement goes, never going to differ from anything.
The people
This is a serious position in the foundations of mathematics. Finitists reject completed infinities. Ultrafinitists go one step further and doubt the very large finite numbers too.
- Alexander Yessenin-Volpin first argued for it, asking what it means for a number to exist if nobody could ever write it.
- Rohit Parikh gave it formal shape with the idea of feasible numbers.
- Edward Nelson, a working mathematician with deep results, doubted that exponentiation is total.
The questions run into complexity theory, bounded arithmetic, and the physical limits of computation.