What can we gain by losing infinity?
Mathematicians and technologists are weighing whether rejecting actual infinity and working only with finite or “ultrafinite” quantities could yield a more realistic or computationally grounded mathematics. Supporters argue that since physical measurements and computers are inherently finite, abandoning infinite sets might improve decidability and align math more closely with the real world, while critics counter that infinity is a coherent abstract concept that underpins calculus, analysis, and modern physics. The exchange highlights a broader tension between mathematics as a practical modeling tool for finite systems and mathematics as an open-ended exploration of abstract structures unconstrained by physical observability.
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.