Mathematicians who call themselves ultrafinitists think that extremely large numbers are holding back science, from logic to cosmology, and they have a radical plan to do something about it
Huge numbers are exceedingly common, but counting particles is the wrong way to find them. Combinatorics is where the real monster hunting lies. When you start calculating complex probabilities or numbers of possible arrangements of things, that’s where the fuzzy boundary between “infinite” and “really, really, really finitely big” starts to blur.
I think looking to CompSci is the right move, but I still don’t see many folks discussing computational complexity as a real, mathematical limit. We often treat two equal statements as though theres an immediate, single-step, jump between them. But discovering the equality requires computation/calculation. Shannon shows that information and entropy are the same thing. Computation is the process by which information is created. Ultrafinitist need to show that there is a finite quantity of information, which I don’t think is true or possible.
I think looking to CompSci is the right move, but I still don’t see many folks discussing computational complexity as a real, mathematical limit.
I think this viewpoint depends on assuming that math is primarily computation. I think our education system and stories reinforce this misconception. But another fundamental component is creation. People created axioms (eg. ZFC) as a foundation for mathematics, then they chose and named almost every mathematical concept based on that foundation. Sure, there are “computations” in some vague sense, but not in the sense of computation theory. Importantly there is no right answer. People have invented alternative systems and will continue to do so. But I haven’t seen a computer compute a better computer… Anyway I agree that computation is underrated especially in terms of proofs (see recent math competition). And increased computation has allowed for breakthroughs. I’m just saying the meta framework of creating the system, defining the terms, and choosing the computations is also a huge factor.
Two things occurred to me reading this:
I think this viewpoint depends on assuming that math is primarily computation. I think our education system and stories reinforce this misconception. But another fundamental component is creation. People created axioms (eg. ZFC) as a foundation for mathematics, then they chose and named almost every mathematical concept based on that foundation. Sure, there are “computations” in some vague sense, but not in the sense of computation theory. Importantly there is no right answer. People have invented alternative systems and will continue to do so. But I haven’t seen a computer compute a better computer… Anyway I agree that computation is underrated especially in terms of proofs (see recent math competition). And increased computation has allowed for breakthroughs. I’m just saying the meta framework of creating the system, defining the terms, and choosing the computations is also a huge factor.