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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. Barkers descriptions of Tsolyni 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 Shn, whose units are founded upon sevens; the Urunn, 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 numbers become simply a matter of writing a longer string of digits. Although there are individual characters in Tsolyni for larger values (e. g. 1, 000 or 1, 000, 000), these appear to be for shorthand use and not a genuine avoidance of a positional system.
The symbol for 1 followed by the symbols for three zeroes would be recognized as denoting 1000, even if there is a grammatical preference for the simpler symbol. We have direct, if disapproving, confirmation of the fully positional nature of the Tsolyni number system from the learned Korumtai hiMettukeng: In ancient days the numerals were written and read from right to left, as with our alphabets. One wrote first the digit (1 through 9); then if the number were large, one wrote the sign of the larger unit (e. g. 100, 1000, etc. ); then one wrote again the sign of a digit; then the symbol of the decade (i. e. 10); then at last the smallest digit (1 - 9). Thus, 3, 567 = 3 + 1, 000, 5 + 100, 6 + 10, and 7.
This number therefore needed seven symbols to inscribe. The modern practice of writing online a 7 and reading the thousands, hundreds, and decades from the positions of these symbols is an abomination and a stench in the nostrils of the Gods and is to be avoided by all who seek purity and salvation in the paradises of Teretane! On Tkumel as on Earth, relics of a previous number system survive long after having passed out of use. In our case, the relics include the 360 degrees in a circle, 60 (angular) minutes in a degree, 60 (chronological) minutes in an hour, and 60 seconds in a minute (for both angular and chronological measure) obtained from the base 60 number system of the Sumerians. Other previous bases, such as 12, have left their mark on our measures. The bases of human number systems are usually thought to be derived from our anatomy, specifically our digits: base 5 from the fingers of one hand, base 10 from the fingers of both hands, base 20 from all our fingers and toes, and base 12 from the three articulations of our four fingers, indexed by the thumb.
It seems reasonable to suppose that other species would have formed their bases in this manner as well. The Shn have hands with three fingers and an opposable thumb: their base of seven could have arisen by counting each finger once, and positioning a thumb (or perhaps their tail? ) for seven. The Urunn base is simpler: their four fingers account for their base 4. We can speculate further on two aspects of these nonhuman number bases: the apparent omission of the thumb from their count, and the fact that their bases are smaller than the primary bases of human history.
If the Shn indeed arrived at their number system by omitting the thumb from their count, this may teach us something significant about their psychology. Perhaps the great reptiles are less willing to broaden the concept of digit to include the dissimilar thumb, indicating a lesser inclination to generalize than humans possess. The lower bases may indicate a reluctance to deal with a plenitude of symbols: the higher the number base, the greater the number of distinct symbols that must be memorized. It may also indicate that large numbers were not important during the formative years of the nonhuman societies: large numbers are more unwieldy with small bases, and thus a culture that dealt with large numbers to represent populations, harvests, etc. , would have an impetus to use the largest feasible base. Might we infer that early Shn and Urunn societies were small, grew slowly, and thus enshrined small number bases as tradition before growing to the point where they became inconvenient?
This is admittedly highly speculative, but we must remember that such considerations forged our own number systems! The Dlo's use of numbers up to three is unusual: in human history, words signifying three have often provided the roots of words meaning multitude, crowd, troop, etc. , as it is the first large number. For the Dlo, four would take this role. Three and four were often expressed as two and one and two and two respectively; the use of three but not four suggests that the Dlo have a distinct word for three which is not derived in this manner.
This unusual arrangement could be rooted in a terrible historico-religious past; recall, for instance, the number of known Pariah Deities III. Extensions of the Integers We have yet to deal with any but the simplest numbers -- the natural numbers, or positive integers. There are many layers yet to be added. The first is the number zero -- to us a natural concept, but in fact an enormous intellectual achievement which occurred in only three societies in human history, those of the Babylonians, Mayans and Indians. The Tsolyni position on zero is made clear by Korumtai hiMettukeng: The latter-day Engsvanyli did also develop a symbol of nothingness, representing nothing (a true zero). This cannot be, for to represent nothing is to say nothing!
Gods of Page! How can one utter nothing? Such is to come too near the despicable heresies of the Pariah Gods... In any case, the uttering of this symbol has been largely restricted by our Excellent Priesthoods to the figuring of mighty distances and formulae. It is astronomical and mathematical, and such uses fall within the limitations of purity and do not transgress. As in many previous human societies, the concept of zero is seen as too dangerous for general use: even once it has been conceived, it is often not accepted as a genuine number.
It seems likely that negative numbers are not thought of as numbers less than zero, but positive numbers representing the magnitude of debts and deficiencies. This understanding of negative numbers was common even among European mathematicians stretching into the nineteenth century. Citizens of the Five Empires do not employ a decimal point. As some means of representing finer-grained numbers is necessary for practical calculation, engineering, etc. , we may assume that they use fractions instead. This view is supported by Professor Barker's comment that they use approximations for (see below).
Fractions can be thought of as ratios of integers, and do not usually cause people much conceptual trouble. However, it is highly unlikely that Tsolyni mathematicians have grasped the concept of irrational numbers, like &# 61552; and the square root of 2, which cannot be so expressed. The ancient proof that the square root of 2 is irrational, which shattered the Pythagorean worldview, would astonish a Tsolyni -- but not affect him as it did the Pythagorean's, as abstraction is so much less important to him. IV. Arithmetic & Record Keeping The temples and palaces of the Five Empires have a tremendous need for numerical records of taxes, levies, fines, inventories, and other listings of state property.
With a positional number system, arithmetic is easy and can be performed in a manner similar to the ones we practice with our (Indo-) Arabic numerals. But how are records kept? One device that is simple, enduring and ubiquitous in our history is the tally stick: two sticks, or per...
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Research essay sample on The Origins Of Mathematics From Ancient Empires
