Pythagoras and the Pythagoreans - Department of Mathematics
Pythagoras and the Pythagoreans - Department of Mathematics
Pythagoras and the Pythagoreans - Department of Mathematics
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<strong>Pythagoras</strong> <strong>and</strong> <strong>the</strong> <strong>Pythagoreans</strong> 33<br />
discovered <strong>the</strong> irrationality <strong>of</strong> √ 3, √ 5, ..., √ 17, <strong>and</strong> <strong>the</strong> dates suggest<br />
that <strong>the</strong> <strong>Pythagoreans</strong> could not have been in possession <strong>of</strong> any sort <strong>of</strong><br />
“<strong>the</strong>ory” <strong>of</strong> irrationals. More likely, <strong>the</strong> <strong>Pythagoreans</strong> had noticed <strong>the</strong>ir<br />
existence. Note that <strong>the</strong> discovery itself must have sent a shock to <strong>the</strong><br />
foundations <strong>of</strong> <strong>the</strong>ir philosophy as revealed through <strong>the</strong>ir dictum All is<br />
Number, <strong>and</strong> some considerable recovery time can easily be surmised.<br />
Theorem. √ 2 is incommensurable with 1.<br />
Pro<strong>of</strong>. Suppose that √ 2= a<br />
b ,withnocommonfactors.Then<br />
2= a2<br />
b2 or<br />
a 2 =2b 2 .<br />
Thus24 2 | a2 , <strong>and</strong> hence 2 | a. So,a =2c <strong>and</strong> it follows that<br />
2c 2 = b 2 ,<br />
whence by <strong>the</strong> same reasoning yields that 2 | b. This is a contradiction.<br />
Is this <strong>the</strong> actual pro<strong>of</strong> known to <strong>the</strong> <strong>Pythagoreans</strong>? Note: Unlike<br />
<strong>the</strong> Babylonians or Egyptians, <strong>the</strong> <strong>Pythagoreans</strong> recognized that this<br />
class <strong>of</strong> numbers was wholly different from <strong>the</strong> rationals.<br />
“Properly speaking, we may date <strong>the</strong> very beginnings <strong>of</strong> “<strong>the</strong>oretical”<br />
ma<strong>the</strong>matics to <strong>the</strong> firstpro<strong>of</strong><strong>of</strong>irrationality, for in “practical”<br />
(or applied) ma<strong>the</strong>matics <strong>the</strong>re can exist no irrational numbers.” 25 Here<br />
a problem arose that is analogous to <strong>the</strong> one whose solution initiated<br />
<strong>the</strong>oretical natural science: it was necessary to ascertain something that<br />
24The expression m | n where m <strong>and</strong> n are integers means that m divides n without<br />
remainder.<br />
25I. M. Iaglom, Matematiceskie struktury i matematiceskoie modelirovanie. [Ma<strong>the</strong>matical<br />
Structures <strong>and</strong> Ma<strong>the</strong>matical Modeling] (Moscow: Nauka, 1980), p. 24.