Enlightening Symbols : A short history of mathematical notation and its hidden powers

For those who are not familiar with mathematical history, it will probably be astonishing to learn how recent our current notation of mathematical formulas is. It is almost unbelievable that mathematics had evolved till the 15th-16th century on a rhetorical basis, not even a plus sign or an equal sign existed so that equations were described with plain words. Since around 1500 more and more symbols were introduced and we see an exponential increase in mathematical knowledge ever since. So how did one arrive at these symbols? Where did they come from? What were the mechanisms that shaped them into their now familiar form? Mazur gives an entertaining history of this evolution. In a first part he deals with the notational systems of the numerals. Then he considers the usual mathematical symbols like plus, minus, powers, square root, etc. in an algebraic context. In a third part he leaves history behind and ponders on the influence of our symbolic notation on the psychology of mathematicians, how symbolic patterns are stored in our brain and how formulas trigger unconscious associations. In retrospect, this may explain how symbolic notation has influenced the evolution of mathematics.

A notation for numbers is a basic element needed for trading now as well as in ancient times. It started by counting tally marks, but soon different numeral systems came about to represent larger numbers. Well known is the Babylonian sexagesimal system. It was positional (like in modern notation, the place of the numeral in the sequence defines the value it represents) and they even had some symbol to denote a placeholder (like we have a zero). The Egyptians had pictograms and the Greek, just like the Hebrew used their alphabet to denote numbers. Later the Greek had some acrophonic system (the first letter of the word for the numeral represents the numeral) and combined them in a way that is somewhat related to the Roman numerals we know. The Maya had a base 20 system stacking dots (units) and horizontal bars (fives) that was conceptually similar to the Babylonian system. The same system was used in China, except that the dots were replaced by vertical bars. Moreover they oriented their numerals up or down so that placeholders could be omitted to some extent. However, our familiar numerals, often called Arabic, originate from the Hindu and were indeed passed on to us by the Arabs (mainly via al-Khwārizmī's al-jabr (825 AD) which also coined the word algebra). Fibonacci is often quoted as the one who introduced these numerals in Europe, and his Liber abbaci (1202 AD) certainly helped spreading the news, but these numerals were probably used by European traders before. The competition triggered legal fights in Florence between algorists following Fibonacci and his computational rules with the new numerals and the abacists defending the computations with Roman numerals using the abacus, although that was previously adapted to the new system by Gebert d'Aurillac (aka Pope Sylvester II).

That brings Mazur to a second part where he has a closer look at the development of algebra. He starts with Diophantus of Alexandria's Arithmetica (300 AD). The original is lost, but copies exist, and so it is unclear whether certain abbreviations (parchment was expensive and copying time consuming) were introduced by Diophantus or by the copyists. Anyway, in the copies we see a primitive notation for powers (cubes and squares), a symbol for the minus sign, a scribble for the unknown, and an abbreviation for the equal sign. Again the Hindus preceded European algebra and the Indian Brahmaguptasiddhanta (628 AD) was brought to Europe once more by the Arabs in the 11th century. Notation evolved slowly until around 1500. It consisted more of abbreviations than of proper symbols. The equality notation still had a unidirectional interpretation like in 1+1 yields or gives 2, and not the meaning of = as in an equation to solve. Hence, what we now write as $ax^2+bx+c=0$ and $ax^2=−bx−c$ were conceptually two different things. Arguments were also often geometrical, hence numbers were positive and so $x^2−1$ had only one root. Mazur argues that the symbolic notation helped to see a pattern that allows for a generalization. For example a proper notation for powers helped to derive rules like $x^mx^n=x^{m+n}$. It also decoupled geometry from algebra and = got a bidirectional meaning so that $x+b=a$ became equivalent with $x=a−b$, which gave an incentive for introducing negative numbers and for the imaginary $i=\sqrt{−1}$ in quadratic equations. He ends with Leibniz who considered a curve as a polygon with infinitely many knots and so introduced the ingenious $\frac{dy}{dx}$. On the other hand Newton considered a curve as a flow of points, the variable being 'fluent' mostly assumed to depend on a time variable and its derivative was denoted as $\dot{y}$ and called a 'fluxion of the fluent'. Leibniz's notation was far more effective in deriving the rules of calculus.

In part 3, Mazur describes experiments on how mathematical formulas are processed in our brain. Not surprisingly, such formulas form patterns that are somehow stored and recognized when needed. Like a musician reads a music score, a chess player scans the board, or a reader of ordinary text has to scan the words and punctuation before actually reading them to know where the sentence is meandering to. Also mathematical formulas are scanned and a reader with some training will parse the formula in his mind trying to recognize patterns, which e.g. draws attention first to the innermost brackets. The meaning of the patterns are recalled from memory and they are often completed in a way that we unconsciously think they are, rather than by what is actually written. That's why proofreading a text requires a more intensive, less autopilot-like attitude than just reading it. This is also confirmed by the fact that we are much better in remembering meaningful formulas than a nonsensical string of symbols.

Not every historian agrees on all events and origins. If there is dispute, Mazur gives the opposing opinions and places them next to each other, mostly with some hint about his own vision. The book is appropriately illustrated where needed without turning it into a coffee table photo book. There is a foldout page with a time-line illustrating the exponential growth of mathematical symbols introduced, but certainly the list of dramatis personae that are given at the beginning of part 1 and 2 are very helpful in placing the events and main characters in proper time perspective,

Of course several of the topics that are covered in the first two parts have been treated in many other books and publications on the history of mathematics and mathematical notation. Yet Mazur adds to the historical facts his thoughts about the influence that this symbolic notation had on the way people were thinking about the algebra that they represented. Our current state of the art is the result of a subtle interplay of historical, political and sociological events, and the gradual shift from abbreviations to symbols detached from geometry with a meaning of their own, like zero evolved from a place holding dot, representing the absence of a number to actually being a genuine number itself or the equality sign became a pivot point balancing the left- and right-hand sides of the equation. Mazur treats only a subset of F. Cajori's monumental A history of mathematical notation (Dover, 1993 first edition 1922) and there is overlap with many other mathematical history books like e.g. George Ifrah's The universal history of numbers (John Wiley, 2000) or R. Kaplan's The nothing that is. A natural history of zero (Oxford U. Press, 2000) or G. Seife's Zero: The biography of a dangerous idea (Penguin Books, 2000) but Mazur adds new findings and insights and it is so much more entertaining than the first and differs from the second because it is much broader and from the third and fourth (which concentrate on the role of zero), and these features make it an interesting addition to the existing literature for anybody with only a slight interest in mathematics or its history.

A. Bultheel
KU Leuven
Book details

There are three parts: (1) the notation of numerals throughout human history, (2) the history of our mathematical symbols in an algebraic context (the unknown, exponents, plus, minus, square root, equal sign,...), (3) how symbolic notation is processed in the human brain and how it has advanced and still influences mathematical thinking.

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978-0691-1-5463-3 (hbk)
£19.95 (hbk)