The Role of Notation in Mathematical Development
Here’s my final paper, presented in one of my favourite classes, The History of Mathematics. We learned how ancient cultures developed and performed calculations, leading to where we are today. Ancient Greek arithmetic, done with a compass and straight edge, was particularly interesting and horrible! My personal discovery while writing this paper was just how much of the groundwork was laid by Archimedes, two thousand years ago! Other than proving my hypothesis wrong, it went great. ;)
Originally published July 17, 2014.
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.
Notation is a mathematical tool meant to represent objects and ideas (“Mathematical Notation”). The original aim of this paper was to demonstrate notational developments as the first step in the increased spread of new ideas, much like the way a spark begins the life of a fire. Notation, in this sense, should coincide with periods of increased mathematical development brought on by increased ability to record and pass on information teaching and the universality of notation compared to rhetoric. Instead it turns out that notation develops gradually and accompanies or even lags behind the development of mathematical concepts. Rather than the spark, it acts more as an offshoot or side effect of progress, more like the smoke given off from a fire. In actuality it starts the naïve stage of mathematical development, the first of three stages set out by David Hilbert, arguably the greatest mathematician of the 20th century (Allen). The stages are, in order, the naïve, the formal and the critical (Kleiner 166), all of which offer refinement.
As mathematical requirements arose, notations did not keep pace. The ancient Egyptians maintained three number systems: hieroglyphics on stone, hieratic on papyrus and demotic for the public. When demotic was selected as the official writing of administration, from the 26th Dynasty on, hieratic persisted for religious texts (Kinnaer). The Greeks and Romans used their own numerals, well suited to calculation by abacus. Those cultures all obtained incredible mathematical accomplishments, ranging from the pyramids, to finances, to aqueducts. The Romans, not particularly interested in mathematical advancement, created an incredibly precise system of aqueducts supplying water and sewage disposal to populations of over a million people (“Roman Aqueduct”). All this was accomplished with just rhetoric, lacking even basic use of variables or formulas.
Algebra, central to so much in mathematics, developed quite gradually. Its name stems from the book Al-Jabr (Allen), written and brought to attention in the West by Muhammad ibn Musâ al-Khwârizmi, a Persian mathematician, astronomer and geographer, in 820 AD. The meaning of the word itself is currently unknown, but was used in context with the reduction or balancing of mathematical terms. Similarly the word algorithm stems not from his book, but from his name. Based on the origins of the word algebra it may be easy to overlook its potential introduction as early as 2000 BC, as a tool of the Ancient Babylonians (“Babylonian Mathematics”). Since this pre-dates algebraic notation concepts such as x + 5 = 10 would be represented by rhetoric like “the thing plus 5 equals 10.” The foundations of algebraic notation were present, but only barely, as they would sometimes draw a pair of legs walking right for addition and left for subtraction. Their knowledge of Pythagorean triples, basic finances and the quadratic formula shows accomplishments far more developed than their notations.
Bhāskara I, a 6th century Indian mathematician, and many after him, used a colourful alternative to our current algebraic naming conventions. The first unknown would be called ya, shorthand for “so much” or “how much” and the next five named after colours: black, blue, yellow, red and green. Powers up to nine and roots up to third also had standard, non-colour names. At the same time the Chinese had not adopted a systematic notation, but instead one based on placement. From Chu Shih-chieh’s The Precious Mirror of the Four Elements, written in 1303, the equations a + b + c + d and (a + b + c + d)² are shown in figure 1 (Cizmar 5). It is not clear what, if any, impact these 13th century systems had on our current system, but both cultures were making isolated and independent contributions to shared ideas of algebraic structure.
Jordanus Nemorarius (died 1237) wrote De Elementis Arismetice Artis in the late 12th or early 13th century, representing variables with the letters a, b, c, d and so forth as required. This excerpt demonstrates the process: “Let .a. be a square, .b. its root, let .b. be multiplied by .c.d., .c. and .d. being its halves; further let .b. time .c.d. equals to .e. and let .a.e. be given” (Cizmar 6). Around the same time, in 1202, Leonardo Fibonacci of Pisa wrote Liber Abaci, which used .a. and .b. to represent lengths of line segments and introduced fractions in their current form, but arranged backwards, so that 1½ would instead have been written ½1 (Cizmar 6). Luca Pacioli wrote Summa de Arithmetica, Geometria, Proportioni et Proportionalita in 1494, introducing shorthand such as p for plus, m for minus and ℞4 for fourth root. Algebraic notation had begun to resemble its current form. If at this point we decide that algebraic notation has finally become mature, it will have been over three thousand years since the Babylonians first used rhetoric for the same ends!
None of this suggests that well-developed notation is unimportant. More modern, shorthand notations would certainly allowed history to share more of its secrets. In ancient Egypt even after being supplanted by demotic as the official text of administration, hieroglyphics continued to be employed in the long-term archiving of information. Hieroglyphics, likely the oldest form of writing, endured because no other form of writing could be as long lived as one etched into solid rock. Its difficulty and time-consuming nature meant less information would ultimately be recorded this way. Phonograms, sound-based hieroglyphics, used no vowels, spaces or punctuation, which means that nobody today knows the proper pronunciation of their language. Information typically archived this way would be religious text, royal documentation of long-term importance and mathematical calculations. Despite this much of their knowledge has been lost to time, such as their method for finding the surface area of a hemisphere (“An Explanation of Hieroglyphics”). If an appropriate algebraic shorthand been available at the time we might have found “hemisphere = 3πr2” or, somewhat less likely, “E = mc2” etched into the pyramids.
The ancient Greeks related their concepts to geometry rather than algebra, leading to different discoveries, not the suppression of discovery. Take the field of geometry, which can be traced back to the 3000 BC in the Indus Valley, but flourished in ancient Greece, where it was considered the epitome of scientific achievement (“History of Geometry”). A small sampling of their ingenuity includes Euclidean geometry, mathematical abstraction, axiomatic systems, incommensurable lengths, irrational numbers, logic and formal proof. The downside is that without notational algebra simple mathematics become more time consuming and teaching significantly more difficult, which explains why with the notable exception of Socrates, Greek mathematicians and philosophers overwhelmingly came from positions of wealth (“Socrates”). At the University of Victoria a fourth year mathematics class of ten students was asked to solve x2–8x = 15 on their second midterm, using the method of ancient Greek algebraic geometry. Despite a related university education, ample warning, notes and specific classroom instruction not one was completely successful. It can be assumed that all would have promptly solved the same equation if allowed to use modern notation, algebra and the quadratic formula, x = (-b ±√b2–4ac)/2a.
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.
When Isaac Newton and Gottfried Leibniz began work on derivatives, each would introduce their own notation, with Leibniz’s being so well conceived that it remains in use over 400 years later. Although neither was able to eliminate the need for infinitesimals, the foundations were set for future generations. This enabled Augustin Cauchy, to take the reins of calculus, bringing the insight that infinitesimals are not just small quantities, but variables. He was able to represent, bound and prove a basis for derivatives with his inequality-based proofs. As always there was room for improvement, provided by Karl Weierstrass in the 1820’s (“Weierstrass”), refining Cauchy’s ideas and notations into the modern epsilon-delta proof we use today (Kleiner 159–166). The derivative’s total time from inception to rigorous proof was over two thousand years.
Complex numbers, on the other hand, took only 254 years to trace a similar path. Originally concocted to plug a hole in the fundamental theorem of algebra, complex numbers were created by Gerolamo Cardano in 1545 (“Complex Number”). A need was created by the real numbers’ inability to consistently provide roots, up to multiplicity, equal to the degree of a polynomial equation. Initially named impossible — and later imaginary — numbers, their study had evolved to the point that Gauss proved the modern fundamental theorem of algebra (“Fundamental Theorem of Algebra”) in 1799 as: “every polynomial equation having complex coefficients and degree ≥ 1 has at least one complex root.” Complex numbers had come so far that after being created to serve the fundamental theorem of algebra, they were now part of its very definition! But how could this have happened so quickly? One reason could be notation; requiring a stretch of the imagination, much like infinitesimals, early complex numbers could be represented quite simply by the term √-1.
Social and economic factors played a significant role in developing mathematics, because as long as cooperation as existed so has the need for measurable compensation. Problem 3 of the Rhind Mathematical Papyrus, conceived around 1850 BC, shows how to divide six loaves of bread among ten men (Katz 2–6). Ancient Russian peasants used methods strikingly similar, yet totally isolated, to the doubling method of the Egyptians. Multiplications could be performed on a pair of numbers by doubling one while halving the other number. Fractional portions would be discarded and then all the sections with odd operands summed to calculate the total. Figure 3. shows how they would have calculated 12 x 15. Simple algorithms allowed regular people to handle such basic mathematical calculation in their day to day lives (“History of Mathematics”). Although at this point it is painfully clear that basic mathematics may be performed nearly free of notation, surely as mathematics developed so too would their notations and the ease of which this knowledge could be taught.
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.
We have seen that basic algebra may be performed by algorithm alone, that a mathematical concept may exist for thousands of years before a suitable notation is created and that the presence of a well-developed notation makes instruction and archiving of calculations much more practical. This points to notation as a natural part of the development of mathematics. If the story of fire parallels the development and maturity of mathematics and the sharing of universal concepts, then notation is just smoke.
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