What can we gain by losing infinity?

Original Article ↗ Hacker News Discussion ↗

Scope of Infinity: Concept vs. Reality

  • Many distinguish between “infinity as a mathematical concept” and “infinity in physical reality.”
  • Some argue nothing observable is infinite, so infinity is a convenient fiction; others respond that physical nonexistence doesn’t invalidate math concepts (same for negatives, irrationals, complex numbers).
  • Several note the map/territory gap: symbols like ∞ are not the thing itself; we never “observe” 2 or 42 either, only instances they describe.

Arguments for Rejecting or Restricting Infinity

  • Suggested gains:
    • Decidability in a strictly finite universe; no unbounded searches or undecidable problems.
    • Removal of paradoxes like multiple sizes of infinity.
    • Closer alignment with computers and feasible computation (“finite but arbitrarily large” instead of actual infinite sets).
  • Described approaches:
    • Arithmetic via constructive systems where every object is finitely representable and all recursion must be proven to terminate.
    • Modeling sets as ordered, duplicate‑free lists; all equalities strict and checkable.
    • “Feasible numbers” where extremely large numbers are treated as nonexistent if proving contradictions involving them would exceed physical resources.

Pushback Against Finitism/Ultrafinitism

  • Critics say the article barely explains concrete benefits and that dropping infinity mostly throws away powerful tools (analysis, calculus, topology) without clear payoff.
  • Some call rejecting infinity a purely philosophical move that doesn’t add testable physical predictions.
  • Others note standard set theories with an axiom of infinity are consistent as far as we know; saying “infinity is wrong” is misframed.

Infinity in Practice: Computation & Physics

  • Multiple comments tie this to computer arithmetic: machines are finite state, yet modeling them with infinite structures (reals, Turing machines) is often simpler and more explanatory.
  • At the same time, practical bugs in floating‑point, geometric algorithms, and precision errors are cited as reminders that actual computation is finite.
  • There’s debate over using huge but finite bounds (e.g., based on atoms in the universe or Planck-scale limits) versus truly unbounded mathematics.

Pedagogy, Intuition, and Humor

  • Childhood number games, “infinity plus one,” and 0.999… = 1 illustrate how intuitions about infinity and limits develop and can mislead.
  • Some see exploring finite-only math as intellectually valuable “heresy,” even if most mainstream work still relies heavily on infinity.