Algebraic and Mathematical Symbolism
Symbolic systems represent crystallized co-operation between the members of a culture.
— Luis Radford (1997) & Michael Otte (1986), contemporary mathematicians
The advantage of symbolic algebra over numerical is this, “that whereas [i.e., in numerical algebra] the Numbers first taken, are lost or swallowed up in those which by several operations are derived from them, as not to remain in view, or easily be discerned in the Result:
Here [i.e., in symbolic algebra] they are so preserved, as till the last, to remain in view with the several operations concerning them, so as they serve not only for a Resolution of the particular Question proposed, but as a general Solution of the like Questions in other Quantities, however changed.
— John Wallis (1675), English mathematician, mathematical historian, cryptographer, translator, Original Fellow of the Royal Society
The motives which give rise to the use of alphabetic letters as symbols of number in preference to any other system of symbols, arbitrarily selected for the same purpose, are principally the following.
First, As they have no numerical signification in themselves, they are subject to no ambiguity, having in reference to numbers no other signification than they are defined to have in the outset of each problem…
Secondly, Being familiar to the eye, the tongue, the hand, and the mind, that is, having a well-known form and name, they are easily read, written, spoken, remembered, and discriminated from one another, which could not be the case were they mere arbitrary marks, formed according to the caprice of each individual who used them.
Thirdly, The order in which the letters are arranged in the alphabet, facilitates the classification of them into groups much more easy to survey and comprehend in the expressions which arise…, and thereby renders the investigator much less likely to omit any of them by an imperfect enumeration.
— Charles Hutton (1836 posthumously), English mathematician, surveyor, author of textbook, tables and dictionaries in both Britain/early America
— Olinthus Gregory (1836), English mathematician, astronomer, Royal Astronomical Society, editor of Gentleman’s Diary and Lady’s Diary
The superiority of this method consists in its having reference to no one number in particular; … whereas [in arithmetic], by considering the numbers as determinate [known], we perform upon them…all the operations which are represented, and when we have come to the result there is nothing to show, how the number 2, to which we may arrive by any number of different operations, has been formed from the given numbers 9 and 5.
These inconveniences are avoided by using characters to represent the number to be divided and the given excess, that are independent of every particular value, and with which we can therefore perform any calculation. The letters of the alphabet are well adapted to this purpose…
— Sylvestre Francois La Croix (ca. 1800), French mathematician, professor and chief textbook writerat the early École Polytechnique
In the examples hitherto proposed a numerical result has always been obtained. The solution with numbers has been performed at the same time with the reasoning; and when the work was finished, no traces of the operations remained in the result. But algebra has a more important purpose.
Pure algebra never gives a numerical result, but is used to trace general principles and to form rules. In order to preserve the work so that the operations may appear in the result, it will be necessary to introduce a few more signs.
— Warren Colburn (1826), American mathematician-educator, author of elementary school textbook series: Arithmetics on the Plan of Pestalozzi
By means of algebraic symbolism a kind of “pattern” or mathematical “machine tool” is provided, which guides the mind as swiftly and unerringly to an objective as a jig guides a cutting tool on a machine.
— Alfred Hooper (1948), English/Canadian mathematician-educator, author, Royal Canadian Air Force
To get anywhere, mathematicians have to internalize the ideas they use. Mathematical objects do not just satisfy a list of axioms; they have feeling, shape, texture, and move around in a particular way in the mind. We have to learn to manipulate, understand how they interrelate. This takes time and effort.
— Margaret Dusa McDuff (2009), contemporary mathematician
As algebra became more and more abstract and generalized, it became clearer and clearer. Because the algebraist could concentrate on the symbols and put aside for the moment what they represented, he or she could perform unprecedented intellectual feats… [symbols] are really little magnifying glasses for wondrously focusing the attention.
— Alfred Crosby (1997), American environmental historian, author
Algebra even has a claim to being if not the core of mathematics then at least its language. Most papers in any branch of mathematics have an algebraic cast to them determined by the very symbolism used, even if the intellectual difficulties are in another field.
— Jeremy J. Gray & Karen Hunger Parshall (2011), contemporary mathematician-historians
Apparently it is rare for mathematicians to think solely in terms of algebraic symbols. Rather, they often describe their thoughts as being like pictures, diagrams, or graphs. At some point, the mathematician is able to translate these ideas into algebraic notation…
What makes teaching (and learning) of these translation skills so difficult is that behind them there are many unarticulated mental processes… These processes are not identical with the symbols; in fact, the symbols themselves, as they appear on the blackboard or in a book, communicate to the student very little about the processes used to produce them.
— John Clement, Jack Lochhead & George S. Monk (1981), contemporary mathematicians