CHAPTER XII

MATHEMATICS AS A LANGUAGE

=1201.= The new mathematics is a sort of supplement to language, affording a means of thought about form and quantity and a means of expression, more exact, compact, and ready than ordinary language. The great body of physical science, a great deal of the essential facts of financial science, and endless social and political problems are only accessible and only thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of one of the new great complex world wide states that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and to write.

--WELLS, H. G.

_Mankind in the Making (London, 1904), pp. 191-192._

=1202.= Mathematical language is not only the simplest and most easily understood of any, but the shortest also.--BROUGHAM, H. L.

_Works (Edinburgh, 1872), Vol. 7, p. 317._

=1203.= Mathematics is the science of definiteness, the necessary vocabulary of those who know.--WHITE, W. F.

_A Scrap-book of Elementary Mathematics (Chicago, 1908), p. 7._

=1204.= Mathematics, too, is a language, and as concerns its structure and content it is the most perfect language which exists, superior to any vernacular; indeed, since it is understood by every people, mathematics may be called the language of languages. Through it, as it were, nature herself speaks; through it the Creator of the world has spoken, and through it the Preserver of the world continues to speak.

--DILLMANN, C.

_Die Mathematik die Fackelträgerin einer neuen Zeit (Stuttgart, 1889), p. 5._

=1205.= Would it sound too presumptuous to speak of perception as a quintessence of sensation, language (that is, communicable thought) of perception, mathematics of language? We should then have four terms differentiating from inorganic matter and from each other the Vegetable, Animal, Rational, and Super-sensual modes of existence.--SYLVESTER, J. J.

_Presidential Address, British Association; Collected Mathematical Papers, Vol. 2, p. 652._

=1206.= Little could Plato have imagined, when, indulging his instinctive love of the true and beautiful for their own sakes, he entered upon these refined speculations and revelled in a world of his own creation, that he was writing the grammar of the language in which it would be demonstrated in after ages that the pages of the universe are written.--SYLVESTER, J. J.

_A Probationary Lecture on Geometry; Collected Mathematical Papers, Vol. 2, p. 7._

=1207.= It is the symbolic language of mathematics only which has yet proved sufficiently accurate and comprehensive to demand familiarity with this conception of an inverse process.

--VENN, JOHN.

_Symbolic Logic (London and New York, 1894), p. 74._

=1208.= Without this language [mathematics] most of the intimate analogies of things would have remained forever unknown to us; and we should forever have been ignorant of the internal harmony of the world, which is the only true objective reality....

This harmony ... is the sole objective reality, the only truth we can attain; and when I add that the universal harmony of the world is the source of all beauty, it will be understood what price we should attach to the slow and difficult progress which little by little enables us to know it better.--POINCARÉ, H.

_The Value of Science [Halsted] Popular Science Monthly, 1906, pp. 195-196._

=1209.= The most striking characteristic of the written language of algebra and of the higher forms of the calculus is the sharpness of definition, by which we are enabled to reason upon the symbols by the mere laws of verbal logic, discharging our minds entirely of the meaning of the symbols, until we have reached a stage of the process where we desire to interpret our results. The ability to attend to the symbols, and to perform the verbal, visible changes in the position of them permitted by the logical rules of the science, without allowing the mind to be perplexed with the meaning of the symbols until the result is reached which you wish to interpret, is a fundamental part of what is called analytical power. Many students find themselves perplexed by a perpetual attempt to interpret not only the result, but each step of the process. They thus lose much of the benefit of the labor-saving machinery of the calculus and are, indeed, frequently incapacitated for using it.--HILL, THOMAS.

_Uses of Mathesis; Bibliotheca Sacra, Vol. 32, p. 505._

=1210.= The prominent reason why a mathematician can be judged by none but mathematicians, is that he uses a peculiar language. The language of mathesis is special and untranslatable. In its simplest forms it can be translated, as, for instance, we say a right angle to mean a square corner. But you go a little higher in the science of mathematics, and it is impossible to dispense with a peculiar language. It would defy all the power of Mercury himself to explain to a person ignorant of the science what is meant by the single phrase “functional exponent.” How much more impossible, if we may say so, would it be to explain a whole treatise like Hamilton’s Quaternions, in such a wise as to make it possible to judge of its value! But to one who has learned this language, it is the most precise and clear of all modes of expression. It discloses the thought exactly as conceived by the writer, with more or less beauty of form, but never with obscurity. It may be prolix, as it often is among French writers; may delight in mere verbal metamorphoses, as in the Cambridge University of England; or adopt the briefest and clearest forms, as under the pens of the geometers of our Cambridge; but it always reveals to us precisely the writer’s thought.

--HILL, THOMAS.

_North American Review, Vol. 85, pp. 224-225._

=1211.= The domain, over which the language of analysis extends its sway, is, indeed, relatively limited, but within this domain it so infinitely excels ordinary language that its attempt to follow the former must be given up after a few steps. The mathematician, who knows how to think in this marvelously condensed language, is as different from the mechanical computer as heaven from earth.--PRINGSHEIM, A.

_Jahresberichte der Deutschen Mathematiker Vereinigung, Bd. 13, p. 367._

=1212.= The results of systematic symbolical reasoning must _always_ express general truths, by their nature; and do not, for their justification, require each of the steps of the process to represent some definite operation upon quantity. The _absolute universality of the interpretation of symbols_ is the fundamental principle of their use.--WHEWELL, WILLIAM.

_The Philosophy of the Inductive Sciences, Part I, Bk. 2, chap. 12, sect. 2 (London, 1858)._

=1213.= Anyone who understands algebraic notation, reads at a glance in an equation results reached arithmetically only with great labour and pains.--COURNOT, A.

_Theory of Wealth [N. T. Bacon], (New York, 1897), p. 4._

=1214.= As arithmetic and algebra are sciences of great clearness, certainty, and extent, which are immediately conversant about signs, upon the skilful use whereof they entirely depend, so a little attention to them may possibly help us to judge of the progress of the mind in other sciences, which, though differing in nature, design, and object, may yet agree in the general methods of proof and inquiry.--BERKELEY, GEORGE.

_Alciphron, or the Minute Philosopher, Dialogue 7, sect. 12._

=1215.= In general the position as regards all such new calculi is this--That one cannot accomplish 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 Möbius’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.--GAUSS, C. J.

_Werke, Bd. 8, p. 298._

=1216.= The invention of what we may call primary or fundamental notation has been but little indebted to analogy, evidently owing to the small extent of ideas in which comparison can be made useful. But at the same time analogy should be attended to, even if for no other reason than that, by making the invention of notation an art, the exertion of individual caprice ceases to be allowable. Nothing is more easy than the invention of notation, and nothing of worse example and consequence than the confusion of mathematical expressions by unknown symbols. If new notation be advisable, permanently or temporarily, it should carry with it some mark of distinction from that which is already in use, unless it be a demonstrable extension of the latter.

--DE MORGAN, A.

_Calculus of Functions; Encyclopedia Metropolitana, Addition to Article 26._

=1217.= Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably nothing in the modern world could have more astonished a Greek mathematician than to learn that, under the influence of compulsory education, the whole population of Western Europe, from the highest to the lowest, could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility.... Our modern power of easy reckoning with decimal fractions is the most miraculous result of a perfect notation.--WHITEHEAD, A. N.

_Introduction to Mathematics (New York, 1911), p. 59._

=1218.= Mathematics is often considered a difficult and mysterious science, because of the numerous symbols which it employs. Of course, nothing is more incomprehensible than a symbolism which we do not understand. Also a symbolism, which we only partially understand and are unaccustomed to use, is difficult to follow. In exactly the same way the technical terms of any profession or trade are incomprehensible to those who have never been trained to use them. But this is not because they are difficult in themselves. On the contrary they have invariably been introduced to make things easy. So in mathematics, granted that we are giving any serious attention to mathematical ideas, the symbolism is invariably an immense simplification.

--WHITEHEAD, A. N.

_Introduction to Mathematics (New York, 1911), pp. 59-60._

=1219.= Symbolism is useful because it makes things difficult. Now in the beginning everything is self-evident, and it is hard to see whether one self-evident proposition follows from another or not. Obviousness is always the enemy to correctness. Hence we must invent a new and difficult symbolism in which nothing is obvious.... Thus the whole of Arithmetic and Algebra has been shown to require three indefinable notions and five indemonstrable propositions.--RUSSELL, BERTRAND.

_International Monthly, 1901, p. 85._

=1220.= The employment of mathematical symbols is perfectly natural when the relations between magnitudes are under discussion; and even if they are not rigorously necessary, it would hardly be reasonable to reject them, because they are not equally familiar to all readers and because they have sometimes been wrongly used, if they are able to facilitate the exposition of problems, to render it more concise, to open the way to more extended developments, and to avoid the digressions of vague argumentation.--COURNOT, A.

_Theory of Wealth [N. T. Bacon], (New York, 1897), pp. 3-4._

=1221.= An all-inclusive geometrical symbolism, such as Hamilton and Grassmann conceived of, is impossible.--BURKHARDT, H.

_Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. 5, p. 52._

=1222.= The language of analysis, most perfect of all, being in itself a powerful instrument of discoveries, its notations, especially when they are necessary and happily conceived, are so many germs of new calculi.--LAPLACE.

_Oeuvres, t. 7 (Paris, 1896), p. xl._