I understand that before Hippasus of Metapontum proved that the square root of 2 is an irrational number, it was commonly assumed that, given two line segments, it would be possible to find a third line segment whose length divides evenly into the two, i.e. commensurability.

However why did the Greeks assume commensurability? What was the proof, reasonsing or evidence that made them think in this way? Or was it only an assumption without any proper justification?

Also what made Hippasus doubt this assumption? What motivated him to prove this assumption was wrong?


  • $\begingroup$ Perhaps commensurability was grounded in the Greek abhorrence of infinity in general and infinite processes in particular, as exemplified by the paradoxes of Zeno. If smaller and smaller "units" failed to obtain commensurability, then you had to deal with an infinite process, which was forbidden by the Greeks. We don't know a lot about Greek mathematics prior to Euclid, but for Euclid number meant positive integer. There was certainly no notion of irrational number. $\endgroup$
    – nwr
    Mar 14 at 3:34
  • $\begingroup$ See this post $\endgroup$ Mar 15 at 15:29


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