The Role of Notation in Mathematical Development

The Role of Notation in Mathematical Development

Lightening strikes and a tree falls, on fire. Proto-people witness and one takes hold of a flaming branch to first wield fire. It’s a storytelling staple and has kicked off many a great comedy or monotonous essay. It also played out many times before we understood and ultimately mastered fire. We crawl before we walk; we walk before we run; and, for the great majority of us, we are taught before we know. This pattern has and always will play out as long as there are discoveries to make. Knowledge is acquired, applied and shared, or else risks being lost to time.

FIG 1. CHINESE REPRESENTATIONS OF (A + B + C + D) AND (A + B + C + D)², CIRCA 1303.

FIG. 2. THE COMPLEXITY OF EARLY ALGEBRAIC GEOMETRY IN SOLVING X2–8X = 15. NORMALLY THE SOLUTION WOULD BE X = (-B ±√B2–4AC)/2A.

The story of the derivative goes back at least to ancient Greece (“Differential Calculus”), where where more than three hundred years before the common era Euclid, Apollonius of Perga and Archimedes used tangent lines in their studies of areas and volumes. Archimedes is credited with being the first to rigorously define the concept of infinitesimals (“Archimedes Palimpsest”), although their application more closely resembled that Bonaventura Cavalieri’s indivisibles almost two thousand years later (“Cavalieri’s Principle”). Despite the principle named after him Archimedes clearly possessed a working knowledge and acceptance of non-Archimedean numbers. The stumbling block is that the ancient Greeks geometric system difficult to understand and had no formal notation to signify the derivative or infinitesimals. When Archimedes did write formal proofs it was with the long deprecated method of exhaustion, which involved proving of upper and lower bounds using proof by contradiction on comparable areas until an eventual outcome could be demonstrated (“Method of Exhaustion”). This would have been difficult and time consuming, even by ancient standards.

12 x 15	Divide 12 by 2, multiply 15 by 2
6 x 30 Divide 6 by 2, multiply 30 by 2
3 x 60 Divide 3 by 2, multiply 60 by 2 <= note that 3 is odd
1 x 120 Discard fractional 0.5 from 1.5 and stop <= note that 1 is odd
Sum up right-side multipliers with odd left-side multipliers
=> 12 x 15 = 60 + 120 = 180

Similar factors drove the development of mathematics as far back as 4000 BC in Mesopotamia (“Sumer”), particularly in the adoption of notations and standards. Much mathematics has been developed to support finances in ancient Mesopotamia and in the ancient Indus Valley, where negative numbers were regarded as debts and positive numbers as fortunes (“History of Mathematics in India”). The adoption of the Indian/Arabic number system had been slowed by counterfeiting concerns and printing press ultimately sealed the success of the same Indian/Arabic system. The modern number system was have first been adopted for its usefulness and precise calculations with pen and paper, then opposed for concerns of counterfeiting, but took over for good with the dawn of the printing press. Wherever there was a need, a notation was always there, eventually brining up the rear and was certainly was not driving force of change.

Most importantly mathematics represents the only true universal language at our disposal, consistent across and present in nearly all cultures. This includes geometry, architecture, probability and finance (“Why is Math the Only True Universal Language?”). It is defined as numeracy (“Numeracy”) and affects the lives of anybody taking a bus, dividing a freshly slaughtered carcass or watching Sesame Street. Numeracy in this form has a long history of which notation is just another part, communicating mathematical concepts as its own language, with reduced emphasis on source language.

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Into devops, tech, math and hard rock. Retired my blog.

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