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On the Foundations and Application of Finite Classical Arithmetic


“ ‘Tell me, Protagoras,’ he said, ‘does a single grain of millet or the ten-thousandth part of a grain make any sound when it falls?’ And when Protagoras said it did not, ‘Then,’ asked Zeno, ‘does a bushel of millet make any sound when it falls or not?’ Protagoras answered that it did, whereupon Zeno replied, ‘But surely there is some ratio between a bushel of millet and a single grain or even the ten-thousandth part of a grain'; and when this was admitted, ‘But then surely,’ Zeno said, ‘the ratios of the corresponding sounds to each other will be the same: for as the bodies which make the sounds are to one another, so will the sounds be to one another. And if this is so, and if the bushel of millet makes a sound, then the single grain of millet and the ten-thousandth part of a grain will make a sound.’ This was the way Zeno used to put his questions”

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page 256 note 1 Simplicius, 1108. 18.

page 256 note 2 Aristotle, Physics, H5,250 a19.

page 257 note 1 Whitrow, Polemic (to appear).

page 257 note 2 Cf. Weyl , Am. Math. Monthly, 01 1946.

page 259 note 1 Hamilton , Dublin Transactions, XVII, II (1835).

page 259 note 2 Helmholtz , “Von Zahlen und Messen” (Leipzig, 1887).

page 259 note 3 Brouwer , Bull. Am. Math. Soc., XX (1913).

page 259 note 4 Suetonius , “Lives of the Caesars,” IV, 55.

page 259 note 5 Whitehead , “Mathematics,” Encyclopaedia Britannica, 13th Edition.

page 260 note 1 Whitrow, op. cit.

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