How the square root of 2 became a number
Ancient Greek attempts to describe all lengths using whole numbers and fractions eventually collapsed with the realization that the diagonal of a unit square, √2, cannot be expressed as a ratio of integers. Commenters explore how this breakthrough led to the formal notion of irrational and real numbers, touching on proofs of irrationality, Dedekind cuts, and competing foundations such as Cantorian set theory versus intuitionism and constructive mathematics. The exchange also highlights how these abstract ideas shape modern science, teaching, and even philosophical views on whether mathematical objects “exist” independently of human description.
Historical context of √2 and Pythagoreans
- Discussion centers on why √2 was historically shocking: it showed some right triangles have side ratios that are not rational, contradicting the belief that all magnitudes are commensurable.
- Clarification that only some right triangles have rational side ratios (Pythagorean triples); “most” do not.
- Debate over what Pythagoras himself actually did; earlier Babylonian knowledge of the theorem and triples is noted.
- The drowning-of-the-discoverer myth is flagged as likely untrue, though secrecy/anxiety around incommensurability is acknowledged.
- Ancient Greeks often preferred geometric reasoning and saw arithmetic as “dirty” or practical, not philosophical.
Nature of rationals, irrationals, and real numbers
- Commenters stress that irrationals are defined negatively (not expressible as a ratio of integers).
- Dense rationals versus uncountable reals leads to “most” real numbers being indescribable; only countably many can be named or encoded in finite text.
- Computable numbers are highlighted as a more “sane” subset of reals, though still with limitations (e.g., ordering arbitrary computable reals is not computable).
Cantor vs Dedekind and constructive mathematics
- One camp criticizes Cantor and uncountable sets as leading to “navel-gazing” and non-constructive horrors (e.g., Banach–Tarski, “almost everywhere” phenomena).
- Others defend classical set theory as extremely successful and central to modern science, noting no contradictions found and great practical utility.
- Intuitionism and constructive approaches (e.g., Dedekind cuts, reals as limits/approximations) are discussed as alternatives; some see them rising in relevance via proof assistants.
Irrationality and repeating decimals
- Explanation that any eventually repeating decimal is rational via algebraic manipulation (multiply by a power of 10 and subtract).
- Conversely, all rationals in base-10 must either terminate or repeat because there are finitely many possible remainders in long division.
- Proofs of irrationality (e.g., for √2, e, π) are emphasized as independent of computing digits.
Philosophy, teaching, and broader reflections
- Some wish basic math education covered formal definitions and the history of ideas more deeply.
- Others recommend history-of-science/math resources and warn of “rabbit holes.”
- Speculation appears on whether aliens or non-spatial AIs might develop very different foundational number systems.