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Re: Plato's mathematics
Posted:
Sep 22, 2008 3:59 PM
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Jowlett's translation of the "Republic" is available online:
http://classics.mit.edu/Plato/republic.html
discusses Plato's arithmetic unity in chapter 7 by:
" I understand, he said, and agree with you. And to which class do unity and number belong?
I do not know, he replied.
Think a little and you will see that what has preceded will supply the answer; for if simple unity could be adequately perceived by the sight or by any other sense, then, as we were saying in the case of the finger, there would be nothing to attract towards being; but when there is some contradiction always present, and one is the reverse of one and involves the conception of plurality, then thought begins to be aroused within us, and the soul perplexed and wanting to arrive at a decision asks 'What is absolute unity?' This is the way in which the study of the one has a power of drawing and converting the mind to the contemplation of true being.
And surely, he said, this occurs notably in the case of one; for we see the same thing to be both one and infinite in multitude?
Yes, I said; and this being true of one must be equally true of all number?
Certainly.
And all arithmetic and calculation have to do with number?
Yes.
And they appear to lead the mind towards truth?
Yes, in a very remarkable manner.
Then this is knowledge of the kind for which we are seeking, having a double use, military and philosophical; for the man of war must learn the art of number or he will not know how to array his troops, and the philosopher also, because he has to rise out of the sea of change and lay hold of true being, and therefore he must be an arithmetician.
That is true.
And our guardian is both warrior and philosopher?
Certainly.
Then this is a kind of knowledge which legislation may fitly prescribe; and we must endeavour to persuade those who are prescribe to be the principal men of our State to go and learn arithmetic, not as amateurs, but they must carry on the study until they see the nature of numbers with the mind only; nor again, like merchants or retail-traders, with a view to buying or selling, but for the sake of their military use, and of the soul herself; and because this will be the easiest way for her to pass from becoming to truth and being."
Unity in Plato's sense allowed Greek shop keepers to accurately conduct trade without proving the theoretical basis of each unit. Mathematical philosophers proved the theoretical basis of each unit.
That is, Plato's arithmetic unity had been introduced to Greece by Egyptians to solve weights and measures problems. One set of unities created volume units, how to infinitely divide a unit, say of a hekat (Egyptian name), when a limit is placed on the first hekat unity.
For example, laying out a historical thread, Egyptians had used two definitions of a hekat unity. The first was written as 64/64 was allowed a divisor n to be selected in the range 1/64 < n < 64 (RMP 81), such that (64/64)/n was written in exact binary quotient and Egyptian fraction remainders.
The second unity was written in terms of a smaller unit, say a 1/10 unit named hin, a 1/64 unit named dja, or a 1/320 unit named ro, such that n, a rational number of any size was allowed divide 10, 64 and 320, respectively as needed by considering:
10/n hin, 64/n dja, 320/n ro and so forth (with 10/n hin and 320/n ro quotient and remainder statements defined in RMP 81. The 64/n dja unity scaled value was proven by Tanja Pemmerening in 2002, 2005).
In this manner Greek shop keepers adopted a well defined unit fraction weights and measures units by following the traditional Egyptian unit fraction system in almost every respect.
Finally, Greek shop keepers were not required to be philosophers, as Plato discussed and applied the Greek ideal concept of unity.
Comments?
Best Regards,
Milo Gardner
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