Do big numbers exist?
Apparently Ultrafinitists believe big numbers like 10^100 do not exist, and the fact that our maths goes all the way up to infinity is why we have such problems with quantum physics and/or gravity. I do not understand it at all, and neither does Sabine Hossenfelder in the below video.
I watch a few of Hossenfelder's videos each week. Most are interesting, but many months ago she came out with one about a study on consciousness that made it clear that she does not understand in the least the essence of the "hard problem of consciousness."
As for the ultrafinitist position, it's only shot at coherence seems to reside within a universe of discourse that is restricted to the physical realm. The problems plaguing the foundations of physics these days are the responsibility of physicists, and should not be passed on to the mathematicians. 10^100 certainly exists, otherwise there must exist an integer that, when 1 is added to it, results in something that is not an integer. Obvious twaddle.
Physicists' relationship with mathematics is problematic in my view. Though I have a PhD in mathematics, I find the way physicists "do" mathematics to be very difficult to follow. Part of it is extremely antiquated notations: physicists do not modernize their notations except in niche areas, which makes their work inscrutable to experts outside their little bubble who might otherwise be able to identify their errors. This traces to the very beginning of a physicists' education. Take a look at a calculus textbook and compare it to a basic physics textbook. Look at the hand-wavey, non-rigorous way the physics book uses "differentials" that dates back to the way analysis was done in the 1700s or earlier, and which ultimately precipitated a crisis in calculus-based mathematics that didn't get set right until the mid-1800s. You'll also likely note that the physics textbook does not even bother to explain, in a chapter 0 or an appendix, precisely how one is to treat a differential like dx or dy for purposes of proving anything logically. So, dx is just a "very tiny distance," or whatnot. But it's not until a physics student finally progresses to a relativity theory course as a senior undergrad that the real trauma occurs: differentials are used all over, but now they are the differentials of grown-up differential manifold theory, which possess a ton of underlying machinery and delicate nuances. Does the relativity textbook unpack this? No! After all, hasn't the physics student been using differentials since their Physics 101 days? And yet a whole week might be devoted to learning about Einstein summation notation, which is not really that clever.
I will agree with Hossenfelder, though, when she muses that the singularities and infinities that arise in physics (such as relativity theory) may have something to do with using continuous mathematical models to describe an apparently discrete physical reality. The way physicists "do" mathematics needs a close examination and thoroughgoing revision, but old habits die hard.
As for the reason why reality appears to be discrete in nature, well, that's entirely due to the "maximum resolution" of our senses, whether aided by instrumentation or not. After all, our instruments and measuring apparatus are also things we only experience with our senses. Being a metaphysical idealist, Ultimate Reality is, in my view, entirely mental in nature, and we are all "psychological alters" of a single experiencing subject that unfolds and acts instinctively rather like Arthur Schopenhauer's "will."
