Hippasus
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| Hippasus | |
|---|---|
| Name | Hippasus |
| Birth date | c. 6th century BC |
| Death date | c. 5th century BC (traditional) |
| Occupation | Philosopher, Mathematician (attributed) |
| Era | Ancient Greek philosophy |
| Region | Magna Graecia |
| School tradition | Pythagoreanism |
Hippasus Hippasus is an ancient figure traditionally associated with the early Pythagorean school and credited in several sources with discoveries that challenged the Pythagorean numerical doctrine. Ancient writers connect him to debates within communities in Croton, Tarentum, and Magna Graecia and to tensions involving figures linked to Pythagoras and the Pythagorean brotherhood. Later classical authors, Hellenistic commentators, and Renaissance humanists treated Hippasus as a touchstone in discussions of irrationality, secrecy, and the transmission of mathematical knowledge in antiquity.
Life and Historical Context
Most information about Hippasus comes from later classical sources rather than contemporary records; therefore his biography is reconstructed through testimonies by writers such as Plato, Aristotle, Plutarch, Iamblichus, Diogenes Laërtius, and Proclus. He is typically placed in the cultural milieu of 6th–5th century BC Magna Graecia where the Pythagorean communities in Croton and Metapontum engaged with Ionian and Eleatic intellectual currents represented by figures like Thales of Miletus, Anaximander, and Heraclitus. The Pythagorean order itself—associated with names such as Pythagoras, Theano, Archytas, Philolaus, and Alcmæon of Croton—served as a nexus for mathematics, cosmology, and esoteric ethics in which Hippasus is variably situated. Later anecdotes tie him to episodes like the Pythagorean secrecy oaths and to civic conflicts in Croton described by authors including Diodorus Siculus and Cicero.
Mathematical Contributions and the Discovery of Irrational Numbers
Hippasus is principally remembered in antiquity for a mathematical demonstration often characterized as the discovery that the diagonal of a unit square (the square root of 2) is incommensurable with its side, an early identification of what modern mathematics calls irrational numbers. Ancient testimonies attribute to him a demonstration using reductio ad absurdum that contrasts with numerical doctrines attributed to Pythagoras and later expounded by Euclid in the Elements, particularly in Book X where commensurability and incommensurability are systematically treated. Hellenistic mathematicians such as Theon of Smyrna and commentators like Proclus discussed proofs concerning ratios, proportions, and magnitudes that echo the alleged Hippasan argument. The intellectual context also involved contemporaneous work on proportion by figures connected with the Milesian school and later refinements by Eudoxus of Cnidus and Archimedes, whose methods formalized concepts of measure that render the earlier discovery historically significant.
Accounts and Legends (Pythagorean Reaction)
Later narratives present dramatic reactions within Pythagorean circles to the discovery as a breach of doctrinal harmony. Biographers and moralists—Iamblichus, Porphyry, Plutarch, and Diogenes Laërtius—relate stories that Hippasus was expelled, punished, or even drowned by fellow Pythagoreans for divulging an unsettling mathematical truth. Other accounts by Cicero and Seneca recount the tale as a moralizing anecdote about secrecy and intellectual courage. Renaissance commentators—Marsilio Ficino, Johannes Kepler, Giorgio Valla—revived and reframed these legends in the context of humanist debates about antiquity, secrecy, and the role of mathematics. The motif of the punished truth-teller also appears in modern rhetorical histories authored by scholars influenced by the writings of Jacques Lefèvre d'Étaples and Giordano Bruno.
Attributions and Conflicting Ancient Sources
Ancient sources vary and often conflict on specifics: some attribute the discovery to an anonymous Pythagorean school, others to Hippasus of Metapontum or Hippasus of Phlius; some place the event in different chronological strata. Authors such as Aristotle treat commensurability issues in passing without explicit attribution, while later doxographers and commentators—Diogenes Laërtius, Iamblichus, Proclus, Sextus Empiricus—offer divergent genealogies and debates over priority between names like Archytas, Theodorus of Cyrene, and Hippasus. Medieval and Byzantine chroniclers preserved variant traditions that influenced the accounts consulted by Renaissance humanists and early modern historians such as Gerolamo Cardano and Pierre de Fermat. Modern historians of mathematics like Carl Boyer, Heinrich S. M. Coxeter, and Reviel Netz examine these competing testimonies through philological, archaeological, and historiographical methods to assess reliability.
Influence on Mathematics and Later Interpretations
Regardless of the historicity of particulars, the Hippasus tradition has had outsized influence on the historiography of number theory, the philosophy of mathematics, and narratives about scientific revolutions. The purported discovery of incommensurability shaped later developments by Euclid, whose Elements codified commensurability theory, and by Hellenistic mathematicians such as Euclid of Alexandria and Eudoxus of Cnidus whose methods addressed irrational magnitudes. In the Renaissance and Early Modern period, commentators like Johannes Kepler, Blaise Pascal, and René Descartes engaged with the legacy of incommensurability in formulating algebraic and geometric foundations. Twentieth-century scholars—Kurt von Fritz, Otto Neugebauer, and Thomas Heath—reassessed these ancient layers, while contemporary philosophers and historians such as Imre Lakatos, Philip J. Davis, and Reuben Hersh use the Hippasus narrative in discussions about proof, discovery, and the sociology of mathematical communities. The story endures as a focal point in discussions linking classical antiquity, mathematical innovation, and the cultural framing of truth.