These Mathematicians Don’t Believe Large Numbers Exist. I’m Serious.

Physics leans heavily on infinities—both the infinitely large and the infinitely small—but a growing minority of mathematicians and physicists argue that this habit may be more than a convenient approximation. The core claim behind “ultrafinitism” is that astronomically large numbers don’t actually exist, and that building mathematics (and therefore physics) on such numbers may be part of why foundational problems keep resurfacing.

In standard physics, infinities appear everywhere: singularities in cosmology and black holes, divergent quantities in quantum field calculations, and the cosmological constant problem. The transcript also points to a “small-number” mirror image: in mathematical models, space and time are treated as continua made of infinitely many points of size zero. Quantum theory intensifies the issue by using wavefunctions living in a Hilbert space, which is also described as a continuum. Even if infinities are often treated as artifacts that get canceled or renormalized, critics argue that the underlying framework may still be wrong in principle.

Several named dissenters are used to frame the broader pattern. George Ellis is cited for linking careless use of infinity to the rise of the multiverse idea. Nicolas Gisin is described as rejecting real numbers because they imply infinite decimal expansions and thus require infinite precision—something he argues cannot exist physically. Tim Palmer is presented as arguing that quantum mechanics’ troubles stem from using a Hilbert space with a continuum, advocating discretization: keep only selected wavefunctions rather than the full continuum, echoing early “quantum” thinking rooted in discreteness.

Ultrafinitism takes the critique further by targeting large numbers directly. The transcript quotes Joel Hamkins, a mathematics professor at the University of Notre Dame, saying that under ultrafinitism, extremely large numbers mathematicians commonly use—such as 10^100—do not actually exist. Instead of standard arithmetic that allows unbounded growth, ultrafinitists aim to rebuild mathematics using only small numbers, including approaches like bounded arithmetic and ultrafinitist logic, with particular attention to complexity bounds—what can be computed and how computational difficulty scales.

The potential payoff, as described, is that physics might stop inheriting its infinities from mathematics. Ultrafinitist ideas are floated as a possible explanation for why entropy appears to have a maximum in black-hole contexts, why “infinites” keep creeping into quantum theory, and even how one might move toward quantizing gravity by eliminating the divergences that plague current attempts.

Still, skepticism remains. The transcript highlights a practical objection: if a number like 10^100 can be written down and manipulated, what does it mean to say it “doesn’t exist”? The narrator admits confusion about the philosophical move, while emphasizing the importance of staying informed about these “number-denying” approaches. The closing note pivots to a sponsor message, but the scientific question stays: whether changing the mathematical ontology of numbers could force a genuine shift in the foundations of physics rather than just cleaning up infinities after the fact.