Generality of Algebra
If you do something once, you probably don’t need algebra. But if you are doing a process again and again, algebra provides a very simple language for describing what you are doing… It’s a great deal easier to substitute into a formula than to have to figure out how to do the question each time.
— Zalman Usiskin (1995), contemporary American mathematician, educator, author
Behind these symbols lie the boldest, purest, coolest abstractions [generalizations] mankind has ever made. No schoolman speculating on essences and attributes ever approached anything like the abstractness [generality] of algebra.
— Susanne Katherina Langer (1942), American philosopher, author, one of the first professional female philosophers in the U.S.
As a material machine is an instrument for economising the exertion of force, so a symbolic calculus [arithmetic, algebra, calculus, etc] is an instrument for economising the exertion of intelligence ... the more perfect the calculus, the smaller would be the amount of intelligence applied as compared with the results produced."
— William Ernest Johnson (1892), English logician, mathematician, economist
In general, the position as regards all such new calculi is this—That one cannot attain by them anything that could not be accomplished without them.
However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able—without the unconscious inspiration of genius which no one can command—to solve the respective problems, yea, to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless.
Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Mobius's calculus.
Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius.
— Carl Friedrich Gauss (1843), influential German mathematician, astronomer, physicist, geodesist, author
To a high degree the language of formulae can be handled autonomously, independent of the understanding of the content.
— Hans Freudenthal (1973), German-Dutch mathematician, author, founder of Institute for the Development of Mathematical Education
The object of all arithmetical operations is to save direct numeration, by utilising the results of our old operations of counting. Our endeavor is, having done a sum once, to preserve the answer for future use. The first four rules of arithmetic well illustrate this view.
Such, too, is the purpose of algebra, which, substituting relations for values, symbolises and definitively fixes all numerical operations that follow the same rule…
We thus save ourselves the labor of performing in future cases the more complicated operation. Mathematics is the method of replacing in the most comprehensive and economical manner possible, new numerical operations by old ones done already with known results. It may happen in this procedure that the results of operations are employed which were originally performed centuries ago.
Often operations involving intense mental effort may be replaced by the action of semi-mechanical routine, with great saving of time and avoidance of fatigue.
— Ernst Mach (1883), influential Austrian physicist, scientific philosopher, namesake of the Mach number used in fluid dynamics
When these difficult cases occur, they are difficult chiefly because, while we have them under consideration, all the reasons pro, and con, are not present to the mind at the same time;…and the uncertainty…perplexes us.
To get over this, my way is, to divide half a sheet of paper by a line into two columns; writing over the one pro, and over the other con…
And though the weight of reasons cannot be taken with the precision of algebraic quantities; yet, when each is thus considered separately and comparatively, and the whole lies before me, I think I can judge better, and am less likely to make a rash step;
and in fact I have found great advantage from this kind of equation, in what may be called moral or prudential Algebra.
— Benjamin Franklin (1772), American founding father, scientist, polymath, statesman, inventor
Algebra is not an isolated subject. Algebra is to math as what blood is to the human body-- it is the bloody circulatory system of mathematics. Everyone should be given the opportunity to want to understand its beautiful language.
— Sunil Singh (2023), contemporary mathematician, educator, author
Though arithmetic is a science on which all others depend, yet we shall explain another, more universal one, by using the letters of the alphabet. This science, which is called Algebra, is used to clarify, extend and improve, as much as one can, arithmetic, and more generally all the sciences which are related to mathematics.
. . . It is so general that it considers all magnitudes . . . As one cannot give to a mind a greater reach or a greater ability than it possesses, this science teaches us only how to spare the mind, presenting it with a whole set of ideas in the form of very short expressions.
— Jean Prestet (1675) French mathematician, priest, educator, author