Plato's Mistake
By Nick Kollerstrom, PhD

Books by Nick Kollerstrom

	Terror on the Tube: Behind the Veil of 7/7, an Investigation US - UK - CA
	Gardening and Planting by the Moon 2010: Higher Yields in Vegetables and Flowers US - UK - CA

To the memory of John Michell (1933 - 2009)

Nick Kollerstrom is historian of science, a former honorary research fellow in Science and Technology Studies at University College, London (UCL), and a former lunar gardening correspondent for the BBC. He is the author or co-author of a number of books, including Gardening and Planting by the Moon (an annual series beginning 1980), Newton's Forgotten Lunar Theory (2000), 'Crop Circles - The Hidden Form' (2002), and Terror on the Tube (2009)

The Ur - Number

On the banks of the river Tigris, around 1900, a US archaeological team dug up tablets of ancient Sumerian arithmetic, from 2,300 BC. What they contained was astonishing - though you are unlikely to find an account of them in histories of mathematics or early science. These tablets are certainly amongst the earliest written maths on record.

On such ancient tablets, would one expect concern with such matters as, how to count sheep, or whether two plus two equals four, or how many days were there, in a lunar month? Instead, we are startled to find that they described mathematical operations centred around the number 12,960,000, the fourth power of sixty. Tables of multiplication and division were set out:

'What in particular is the meaning of the number 12,960,000 (=60⁴ or 3600²) which underlies all the main texts here treated…? The division tables from the main temple library of Nippur, which are all based upon 12,960,000 … All the multiplication and division tables from the temple libraries of Nippur and Sippar, and from the library of Ashurbanipal, are based upon 12,960,000.' [1]

Scribes had grappled with this huge number, millennia ago. (For background on this remarkable archaeological expedition, led by the German Assyriologist Hermann Hilprecht, see here) Why does no history of science or mathematics allude to this – I tried checking out the classic books by Otto Neugebauer and Van der Waerden, the great historians of ancient mathematics [2],[3] – does it just not fit in with their 'primitive man' paradigm?

Hilprecht endeavoured to 'explain' the interest in this huge number by quoting Plato. But, that was a millennium and a half later … Plato did indeed get to hear of this huge number - from Babylon, experts believe [4], plus maybe some stories of its importance. What did he make of it? He interpreted it as denoting a cycle of time, the so-called 'Platonic Year' of 36,000 years.[5] If, he explained, the year be taken as 360 days, then that Number would equal the days in this huge period.

No such astronomical period exists, but it gets worse. Plato philosophises about good versus bad births –and some people reckon he is here being a bit astrological – depending upon the number of days of the gestation period: whether that number of gestation divides into the primal, ur-number, or not. He called the big number, ‘Lord of Better or Worse Births,’ also it gets alluded to as Plato’s ‘nuptial number.’

Hipparchus, credited with being the first astronomer to discover the precession of the equinoxes, estimated it as one degree per hundred years. Clearly, that would give Plato's Period for the overall precessional cycle. Later on, Claudius Ptolemy in Alexandria designated the precessional period - using the same value as Hipparchus - as 'the Platonic year.' We may doubt whether Plato knew anything about precession, and certainly his treatment of the big number, the fourth power of 60, had nothing to do with it.

Endnotes

1. Herman Hilprecht, The Babylonian Expedition of the University of Pennsylvania, Series A, Vol. 20, Mathematical, Metrological and Chronological tablets from the Temple Library at Nippur, Philadephia 1906, pp.31-2. [back to text]
2. See, eg, ‘The Sixty System of Sumer’ by A. Seidenberg, Archive for History of the Exact Sciences, 1962, 2,5, pp.436-440, communicated by Van der Waerden. [back to text]
3. George C. Joseph’s highly-praised The Crest of the Peacock, non-European roots of Mathematics, 2011 also omits their mention - even while alluding to early Sumerian math tablets from Nippur (pp.131-134), also a 4^th millennium BC tablet from Uruk that found areas of fields. Field-area computation is the sort of thing a historian might expect to find on such tablets; which cannot be said of 60⁴ – based computations. [back to text]
4. J.Adam, Republic of Plato, 1902, Vol. II (Book VIII), p.201-9 and 264-306. [back to text]
5. Richard Dumbrill, ‘Four Mathematical texts from the Temple Library of Nippur’, Journal of the American Oriental Soc. Vol 29, 1908, pp.210-219. [back to text]