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The Origins of Mathematics from the Ancient Empires

The Origins of Mathematics from the Ancient Empires

I. A Cultural Perspective

Mathematics, aptly named “the queen and handmaiden of the sciences”, is for us the quintessential expression of the scientific worldview. As we learn, our conception of mathematics grows and changes, with the distillation of millennia of human effort to conceptualize the abstract added in layers. It is as difficult for us to gain a perspective on our view of mathematics as it is for us to see beyond any other aspect of our worldview.

But fantasy affords us the opportunity to speculate on how things might be different. In this article, we shall use the techniques of mathematical historians to elaborate upon M.A.R. Barker’s descriptions of Tsolyáni mathematics, as well as to speculate, by way of comparison to our own mathematics, on what hidden knowledge their higher adepts might possess.

II. Number Bases

We know from Swords & Glory, vol. I (Sec. 1.1010) that

The Five Empires base their mathematics upon the decimal [base 10] system. “Zero” is employed, but the decimal point remains to be discovered. The remains of the old vigesimal [base 20] number system of the Bednalljans (and possibly the Llyani) can still be seen in the 20 Qirgal it takes to make up a Hlash, and the 20 Hlash which constitute a Kaitar. A few of the smaller states and some of the nonhuman races employ other arrangements: e.g. the Shén, whose units are founded upon sevens; the Urunén, who use fours, etc. The simplest system of all is attributed to the Dlo tribe of eastern Rannalu, whose numbers consist of just, “One, two, three--many...”

The first thing we note is that with the exception of the Dlo, all these systems are positional number systems, with a separate character for each digit, and the value of a digit in a number determined by its position. For a contrast, consider the number system of the Romans. The number-symbols in a number written in Roman numerals are in order from largest to smallest, but position does not determine the value of a digit: different symbols are used for larger quantities, meaning that a new symbol must be added to the system to extend it by even a single order of magnitude. In a fully positional system, arithmetic is far easier, and large...

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