Symbol Quotes (73 quotes)

Accordingly, we find Euler and D'Alembert devoting their talent and their patience to the establishment of the laws of rotation of the solid bodies. Lagrange has incorporated his own analysis of the problem with his general treatment of mechanics, and since his time M. Poinsôt has brought the subject under the power of a more searching analysis than that of the calculus, in which ideas take the place of symbols, and intelligent propositions supersede equations.

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

Artificial intelligence is based on the assumption that the mind can be described as some kind of formal system manipulating symbols that stand for things in the world. Thus it doesn't matter what the brain is made of, or what it uses for tokens in the great game of thinking. Using an equivalent set of tokens and rules, we can do thinking with a digital computer, just as we can play chess using cups, salt and pepper shakers, knives, forks, and spoons. Using the right software, one system (the mind) can be mapped onto the other (the computer).

As he [Clifford] spoke he appeared not to be working out a question, but simply telling what he saw. Without any diagram or symbolic aid he described the geometrical conditions on which the solution depended, and they seemed to stand out visibly in space. There were no longer consequences to be deduced, but real and evident facts which only required to be seen. … So whole and complete was his vision that for the time the only strange thing was that anybody should fail to see it in the same way. When one endeavored to call it up again, and not till then, it became clear that the magic of genius had been at work, and that the common sight had been raised to that higher perception by the power that makes and transforms ideas, the conquering and masterful quality of the human mind which Goethe called in one word

*das Dämonische*.
Atoms are round balls of wood invented by Dr. Dalton.

*Answer given by a pupil to a question on atomic theory, as reported by Sir Henry Enfield Roscoe.*
Cayley was singularly learned in the work of other men, and catholic in his range of knowledge. Yet he did not read a memoir completely through: his custom was to read only so much as would enable him to grasp the meaning of the symbols and understand its scope. The main result would then become to him a subject of investigation: he would establish it (or test it) by algebraic analysis and, not infrequently, develop it so to obtain other results. This faculty of grasping and testing rapidly the work of others, together with his great knowledge, made him an invaluable referee; his services in this capacity were used through a long series of years by a number of societies to which he was almost in the position of standing mathematical advisor.

Chemical signs ought to be letters, for the greater facility of writing, and not to disfigure a printed book ... I shall take therefore for the chemical sign, the

*initial letter of the Latin name of each elementary substance*: but as several have the same initial letter, I shall distinguish them in the following manner:— 1. In the class which I shall call*metalloids*, I shall employ the initial letter only, even when this letter is common to the metalloid and to some metal. 2. In the class of metals, I shall distinguish those that have the same initials with another metal, or a metalloid, by writing the first two letters of the word. 3. If the first two letters be common to two metals, I shall, in that case, add to the initial letter the first consonant which they have not in common: for example, S = sulphur, Si = silicium, St = stibium (antimony), Sn = stannum (tin), C = carbonicum, Co = colbaltum (colbalt), Cu = cuprum (copper), O = oxygen, Os = osmium, &c.
Does the evolutionary doctrine clash with religious faith? It does not. It is a blunder to mistake the Holy Scriptures for elementary textbooks of astronomy, geology, biology, and anthropology. Only if symbols are construed to mean what they are not intended to mean can there arise imaginary, insoluble conflicts. ... the blunder leads to blasphemy: the Creator is accused of systematic deceitfulness.

Dr. M.L. von Franz has explained the circle (or sphere) as a symbol of Self. It expresses the totality of the psyche in all its aspects, including the relationship between man and the whole of nature. It always points to the single most vital aspect of life, its ultimate wholeness.

Every improvement that is put upon the real estate is the result of an idea in somebody's head. The skyscraper is another idea; the railroad is another; the telephone and all those things are merely symbols which represent ideas. An andiron, a wash-tub, is the result of an idea that did not exist before.

Every natural fact is a symbol of some spiritual fact.

Every science that has thriven has thriven upon its own symbols: logic, the only science which is admitted to have made no improvements in century after century, is the only one which has grown no symbols.

For, in mathematics or symbolic logic, reason can crank out the answer from the symboled equations—even a calculating machine can often do so—but it cannot alone set up the equations. Imagination resides in the words which define and connect the symbols—subtract them from the most aridly rigorous mathematical treatise and all meaning vanishes. Was it Eddington who said that we once thought if we understood 1 we understood 2, for 1 and 1 are 2, but we have since found we must learn a good deal more about “and”?

Given any domain of thought in which the fundamental objective is a knowledge that transcends mere induction or mere empiricism, it seems quite inevitable that its processes should be made to conform closely to the pattern of a system free of ambiguous terms, symbols, operations, deductions; a system whose implications and assumptions are unique and consistent; a system whose logic confounds not the necessary with the sufficient where these are distinct; a system whose materials are abstract elements interpretable as reality or unreality in any forms whatsoever provided only that these forms mirror a thought that is pure. To such a system is universally given the name MATHEMATICS.

Here I am at the limit which God and nature has assigned to my individuality. I am compelled to depend upon word, language and image in the most precise sense, and am wholly unable to operate in any manner whatever with symbols and numbers which are easily intelligible to the most highly gifted minds.

I end with a word on the new symbols which I have employed. Most writers on logic strongly object to all symbols. ... I should advise the reader not to make up his mind on this point until he has well weighed two facts which nobody disputes, both separately and in connexion. First, logic is the only science which has made no progress since the revival of letters; secondly, logic is the only science which has produced no growth of symbols.

I have been battering away at Saturn, returning to the charge every now and then. I have effected several breaches in the solid ring, and now I am splash into the fluid one, amid a clash of symbols truly astounding. When I reappear it will be in the dusky ring, which is something like the state of the air supposing the siege of Sebastopol conducted from a forest of guns 100 miles one way, and 30,000 miles the other, and the shot never to stop, but go spinning away round a circle, radius 170,000 miles.

I must confess the language of symbols is to me

A Babylonish dialect

Which learned

It is a party-coloured dress

Of patch'd and piebald languages:

'T is English cut on Greek and Latin,

Like fustian heretofore on satin.

A Babylonish dialect

Which learned

*chemists*much affect;It is a party-coloured dress

Of patch'd and piebald languages:

'T is English cut on Greek and Latin,

Like fustian heretofore on satin.

I recall my own emotions: I had just been initiated into the mysteries of the complex number. I remember my bewilderment: here were magnitudes patently impossible and yet susceptible of manipulations which lead to concrete results. It was a feeling of dissatisfaction, of restlessness, a desire to fill these illusory creatures, these empty symbols, with substance. Then I was taught to interpret these beings in a concrete geometrical way. There came then an immediate feeling of relief, as though I had solved an enigma, as though a ghost which had been causing me apprehension turned out to be no ghost at all, but a familiar part of my environment.

I would not for a moment have you suppose that I am one of those idiots who scorns Science, merely because it is always twisting and turning, and sometimes shedding its skin, like the serpent that is [the doctors'] symbol.

If a lunatic scribbles a jumble of mathematical symbols it does not follow that the writing means anything merely because to the inexpert eye it is indistinguishable from higher mathematics.

If there is a lesson in our story it is that the manipulation, according to strictly self-consistent rules, of a set of symbols representing one single aspect of the phenomena may produce correct, verifiable predictions, and yet completely ignore all other aspects whose ensemble constitutes reality.

In presenting a mathematical argument the great thing is to give the educated reader the chance to catch on at once to the momentary point and take details for granted: his successive mouthfuls should be such as can be swallowed at sight; in case of accidents, or in case he wishes for once to check in detail, he should have only a clearly circumscribed little problem to solve (e.g. to check an identity: two trivialities omitted can add up to an impasse). The unpractised writer, even after the dawn of a conscience, gives him no such chance; before he can spot the point he has to tease his way through a maze of symbols of which not the tiniest suffix can be skipped.

In symbols one observes an advantage in discovery which is greatest when they express the exact nature of a thing briefly and, as it were, picture it; then indeed the labor of thought is wonderfully diminished.

In the world of physics we watch a shadowgraph performance of the drama of familiar life. The shadow of my elbow rests on the shadow table as the shadow ink flows over the shadow paper. It is all symbolic, and as a symbol the physicist leaves it. ... The frank realization that physical science is concerned with a world of shadows is one of the most significant of recent advances.

In the year 1692, James Bernoulli, discussing the logarithmic spiral [or equiangular spiral, ρ = α

^{θ}] … shows that it reproduces itself in its evolute, its involute, and its caustics of both reflection and refraction, and then adds: “But since this marvellous spiral, by such a singular and wonderful peculiarity, pleases me so much that I can scarce be satisfied with thinking about it, I have thought that it might not be inelegantly used for a symbolic representation of various matters. For since it always produces a spiral similar to itself, indeed precisely the same spiral, however it may be involved or evolved, or reflected or refracted, it may be taken as an emblem of a progeny always in all things like the parent,*simillima filia matri*. Or, if it is not forbidden to compare a theorem of eternal truth to the mysteries of our faith, it may be taken as an emblem of the eternal generation of the Son, who as an image of the Father, emanating from him, as light from light, remains ὁμοούσιος with him, howsoever overshadowed. Or, if you prefer, since our*spira mirabilis*remains, amid all changes, most persistently itself, and exactly the same as ever, it may be used as a symbol, either of fortitude and constancy in adversity, or, of the human body, which after all its changes, even after death, will be restored to its exact and perfect self, so that, indeed, if the fashion of Archimedes were allowed in these days, I should gladly have my tombstone bear this spiral, with the motto, ‘Though changed, I arise again exactly the same,*Eadem numero mutata resurgo*.’”
It is good to recall that three centuries ago, around the year 1660, two of the greatest monuments of modern history were erected, one in the West and one in the East; St. Paul’s Cathedral in London and the Taj Mahal in Agra. Between them, the two symbolize, perhaps better than words can describe, the comparative level of architectural technology, the comparative level of craftsmanship and the comparative level of affluence and sophistication the two cultures had attained at that epoch of history. But about the same time there was also created—and this time only in the West—a third monument, a monument still greater in its eventual import for humanity. This was Newton’s

*Principia*, published in 1687. Newton's work had no counterpart in the India of the Mughuls.
It is often assumed that because the young child is not competent to study geometry systematically he need be taught nothing geometrical; that because it would be foolish to present to him physics and mechanics as sciences it is useless to present to him any physical or mechanical principles.

An error of like origin, which has wrought incalculable mischief, denies to the scholar the use of the symbols and methods of algebra in connection with his early essays in numbers because, forsooth, he is not as yet capable of mastering quadratics! … The whole infant generation, wrestling with arithmetic, seek for a sign and groan and travail together in pain for the want of it; but no sign is given them save the sign of the prophet Jonah, the withered gourd, fruitless endeavor, wasted strength.

An error of like origin, which has wrought incalculable mischief, denies to the scholar the use of the symbols and methods of algebra in connection with his early essays in numbers because, forsooth, he is not as yet capable of mastering quadratics! … The whole infant generation, wrestling with arithmetic, seek for a sign and groan and travail together in pain for the want of it; but no sign is given them save the sign of the prophet Jonah, the withered gourd, fruitless endeavor, wasted strength.

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.

It is through it [intuition] that the mathematical world remains in touch with the real world, and even if pure mathematics could do without it, we should still have to have recourse to it to fill up the gulf that separates the symbol from reality.

It really is worth the trouble to invent a new symbol if we can thus remove not a few logical difficulties and ensure the rigour of the proofs. But many mathematicians seem to have so little feeling for logical purity and accuracy that they will use a word to mean three or four different things, sooner than make the frightful decision to invent a new word.

It was his [Leibnitz’s] love of method and order, and the conviction that such order and harmony existed in the real world, and that our success in understanding it depended upon the degree and order which we could attain in our own thoughts, that originally was probably nothing more than a habit which by degrees grew into a formal rule. This habit was acquired by early occupation with legal and mathematical questions. We have seen how the theory of combinations and arrangements of elements had a special interest for him. We also saw how mathematical calculations served him as a type and model of clear and orderly reasoning, and how he tried to introduce method and system into logical discussions, by reducing to a small number of terms the multitude of compound notions he had to deal with. This tendency increased in strength, and even in those early years he elaborated the idea of a general arithmetic, with a universal language of symbols, or a characteristic which would be applicable to all reasoning processes, and reduce philosophical investigations to that simplicity and certainty which the use of algebraic symbols had introduced into mathematics.

A mental attitude such as this is always highly favorable for mathematical as well as for philosophical investigations. Wherever progress depends upon precision and clearness of thought, and wherever such can be gained by reducing a variety of investigations to a general method, by bringing a multitude of notions under a common term or symbol, it proves inestimable. It necessarily imports the special qualities of number—viz., their continuity, infinity and infinite divisibility—like mathematical quantities—and destroys the notion that irreconcilable contrasts exist in nature, or gaps which cannot be bridged over. Thus, in his letter to Arnaud, Leibnitz expresses it as his opinion that geometry, or the philosophy of space, forms a step to the philosophy of motion—i.e., of corporeal things—and the philosophy of motion a step to the philosophy of mind.

A mental attitude such as this is always highly favorable for mathematical as well as for philosophical investigations. Wherever progress depends upon precision and clearness of thought, and wherever such can be gained by reducing a variety of investigations to a general method, by bringing a multitude of notions under a common term or symbol, it proves inestimable. It necessarily imports the special qualities of number—viz., their continuity, infinity and infinite divisibility—like mathematical quantities—and destroys the notion that irreconcilable contrasts exist in nature, or gaps which cannot be bridged over. Thus, in his letter to Arnaud, Leibnitz expresses it as his opinion that geometry, or the philosophy of space, forms a step to the philosophy of motion—i.e., of corporeal things—and the philosophy of motion a step to the philosophy of mind.

It [mathematics] is in the inner world of pure thought, where all

Is it a restricted home, a narrow life, static and cold and grey with logic, without artistic interest, devoid of emotion and mood and sentiment? That world, it is true, is not a world of

*entia*dwell, where is every type of order and manner of correlation and variety of relationship, it is in this infinite ensemble of eternal verities whence, if there be one cosmos or many of them, each derives its character and mode of being,—it is there that the spirit of mathesis has its home and its life.Is it a restricted home, a narrow life, static and cold and grey with logic, without artistic interest, devoid of emotion and mood and sentiment? That world, it is true, is not a world of

*solar*light, not clad in the colours that liven and glorify the things of sense, but it is an illuminated world, and over it all and everywhere throughout are hues and tints transcending sense, painted there by radiant pencils of*psychic*light, the light in which it lies. It is a silent world, and, nevertheless, in respect to the highest principle of art—the interpenetration of content and form, the perfect fusion of mode and meaning—it even surpasses music. In a sense, it is a static world, but so, too, are the worlds of the sculptor and the architect. The figures, however, which reason constructs and the mathematic vision beholds, transcend the temple and the statue, alike in simplicity and in intricacy, in delicacy and in grace, in symmetry and in poise. Not only are this home and this life thus rich in aesthetic interests, really controlled and sustained by motives of a sublimed and supersensuous art, but the religious aspiration, too, finds there, especially in the beautiful doctrine of invariants, the most perfect symbols of what it seeks—the changeless in the midst of change, abiding things hi a world of flux, configurations that remain the same despite the swirl and stress of countless hosts of curious transformations.
Mathematicians may flatter themselves that they possess new ideas which mere human language is as yet unable to express. Let them make the effort to express these ideas in appropriate words without the aid of symbols, and if they succeed they will not only lay us laymen under a lasting obligation, but, we venture to say, they will find themselves very much enlightened during the process, and will even be doubtful whether the ideas as expressed in symbols had ever quite found their way out of the equations into their minds.

Mathematics gives the young man a clear idea of demonstration and habituates him to form long trains of thought and reasoning methodically connected and sustained by the final certainty of the result; and it has the further advantage, from a purely moral point of view, of inspiring an absolute and fanatical respect for truth. In addition to all this, mathematics, and chiefly algebra and infinitesimal calculus, excite to a high degree the conception of the signs and symbols—necessary instruments to extend the power and reach of the human mind by summarizing an aggregate of relations in a condensed form and in a kind of mechanical way. These auxiliaries are of special value in mathematics because they are there adequate to their definitions, a characteristic which they do not possess to the same degree in the physical and mathematical [natural?] sciences.

There are, in fact, a mass of mental and moral faculties that can be put in full play only by instruction in mathematics; and they would be made still more available if the teaching was directed so as to leave free play to the personal work of the student.

There are, in fact, a mass of mental and moral faculties that can be put in full play only by instruction in mathematics; and they would be made still more available if the teaching was directed so as to leave free play to the personal work of the student.

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. … But this is not because they are difficult in themselves. On the contrary they have invariably been introduced to make things easy. … [T]he symbolism is invariably an immense simplification. It … represents an analysis of the ideas of the subject and an almost pictorial representation of their relations to each other.

Men cannot be treated as units in operations of political arithmetic because they behave like the symbols for zero and the infinite, which dislocate all mathematical operations.

Modern music, headstrong, wayward, tragically confused as to what to say and how to say it, has mounted its horse, as the joke goes, and ridden off in all directions. If we require of an art that it be unified as a whole and expressed in a universal language known to all, if it must be a consistent symbolization of the era, then modern music is a disastrous failure. It has many voices, many symbolizations. It it known to one, unknown to another. But if an art may be as variable and polyvocal as the different individuals and emotional regions from which it comes in this heterogeneous modern world, then the diversity and contradiction of modern music may be acceptable.

Not one of them [formulae] can be shown to have any existence, so that the formula of one of the simplest of organic bodies is confused by the introduction of unexplained symbols for imaginary differences in the mode of combination of its elements… It would be just as reasonable to describe an oak tree as composed of blocks and chips and shavings to which it may be reduced by the hatchet, as by Dr Kolbe’s formula to describe acetic acid as containing the products which may be obtained from it by destructive influences. A Kolbe botanist would say that half the chips are united with some of the blocks by the force

*parenthesis*; the other half joined to this group in a different way, described by a*buckle*; shavings stuck on to these in a third manner,*comma*; and finally, a compound of shavings and blocks united together by a fourth force,*juxtaposition*, is joined to the main body by a fifth force,*full stop*.
Nothing can be more fatal to progress than a too confident reliance upon mathematical symbols; for the student is only too apt to take the easier course, and consider the

*formula*and not the*fact*as the physical reality.
Numerical logistic is that which employs numbers; symbolic logistic that which uses symbols, as, say, the letters of the alphabet.

On foundations we believe in the reality of mathematics, but of course, when philosophers attack us with their paradoxes, we rush to hide behind formalism and say 'mathematics is just a combination of meaningless symbols,'... Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working with something real. The sensation is probably an illusion, but it is very convenient.

Our knowledge of the external world must always consist of numbers, and our picture of the universe—the synthesis of our knowledge—must necessarily be mathematical in form. All the concrete details of the picture, the apples, the pears and bananas, the ether and atoms and electrons, are mere clothing that we ourselves drape over our mathematical symbols— they do not belong to Nature, but to the parables by which we try to make Nature comprehensible. It was, I think, Kronecker who said that in arithmetic God made the integers and man made the rest; in the same spirit, we may add that in physics God made the mathematics and man made the rest.

Philosophy is written in that great book that lies before our gaze—I mean the universe—but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written.

Philosophy [the universe] is written in that great book which ever lies before our eyes ... We cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in the mathematical language ... without whose help it is humanly impossible to comprehend a single word of it, and without which one wanders in vain through a dark labyrinth.

Pure mathematics is a collection of hypothetical, deductive theories, each consisting of a definite system of primitive,

*undefined*, concepts or symbols and primitive,*unproved*, but self-consistent assumptions (commonly called axioms) together with their logically deducible consequences following by rigidly deductive processes without appeal to intuition.
Pure mathematics proves itself a royal science both through its content and form, which contains within itself the cause of its being and its methods of proof. For in complete independence mathematics creates for itself the object of which it treats, its magnitudes and laws, its formulas and symbols.

Science is the reduction of the bewildering diversity of unique events to manageable uniformity within one of a number of symbol systems, and technology is the art of using these symbol systems so as to control and organize unique events. Scientific observation is always a viewing of things through the refracting medium of a symbol system, and technological praxis is always handling of things in ways that some symbol system has dictated. Education in science and technology is essentially education on the symbol level.

Spoken words are the symbols of mental experience, and written words are the symbols of spoken words.

That only Galileo’s physical finger is preserved but the descendants of his techniques thrive is also symbolic of the transitoriness of personal existence in contrast to the immortality of knowledge.

That small word “Force,” they make a barber's block,

Ready to put on

Meanings most strange and various, fit to shock

Pupils of Newton....

The phrases of last century in this

Linger to play tricks—

Those long-nebbed words that to our text books still

Cling by their titles,

And from them creep, as entozoa will,

Into our vitals.

But see! Tait writes in lucid symbols clear

One small equation;

And Force becomes of Energy a mere

Space-variation.

Ready to put on

Meanings most strange and various, fit to shock

Pupils of Newton....

The phrases of last century in this

Linger to play tricks—

*Vis viva*and*Vis Mortua*and*Vis Acceleratrix:*—Those long-nebbed words that to our text books still

Cling by their titles,

And from them creep, as entozoa will,

Into our vitals.

But see! Tait writes in lucid symbols clear

One small equation;

And Force becomes of Energy a mere

Space-variation.

The

*arithmetization*of mathematics … which began with Weierstrass … had for its object the separation of purely mathematical concepts, such as*number*and*correspondence*and aggregate, from intuitional ideas, which mathematics had acquired from long association with geometry and mechanics. These latter, in the opinion of the formalists, are so firmly entrenched in mathematical thought that in spite of the most careful circumspection in the choice of words, the meaning concealed behind these words, may influence our reasoning. For the trouble with human words is that they possess content, whereas the purpose of mathematics is to construct pure thought. But how can we avoid the use of human language? The … symbol. Only by using a symbolic language not yet usurped by those vague ideas of space, time, continuity which have their origin in intuition and tend to obscure pure reason—only thus may we hope to build mathematics on the solid foundation of logic.
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.

The honor you have given us goes not to us as a crew, but to ... all Americans, who believed, who persevered with us. What Apollo has begun we hope will spread out in many directions, not just in space, but underneath the seas, and in the cities to tell us unforgettably what we will and must do. There are footprints on the moon. Those footprints belong to each and every one of you, to all mankind. They are there because of the blood, sweat, and tears of millions of people. Those footprints are the symbol of true human spirit.

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.

The mathematician requires tact and good taste at every step of his work, and he has to learn to trust to his own instinct to distinguish between what is really worthy of his efforts and what is not; he must take care not to be the slave of his symbols, but always to have before his mind the realities which they merely serve to express. For these and other reasons it seems to me of the highest importance that a mathematician should be trained in no narrow school; a wide course of reading in the first few years of his mathematical study cannot fail to influence for good the character of the whole of his subsequent work.

The mathematician, carried along on his flood of symbols, dealing apparently with purely formal truths, may still reach results of endless importance for our description of the physical universe.

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.

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.
The symbol A is not the counterpart of anything in familiar life. To the child the letter A would seem horribly abstract; so we give him a familiar conception along with it. “A was an Archer who shot at a frog.” This tides over his immediate difficulty; but he cannot make serious progress with word-building so long as Archers, Butchers, Captains, dance round the letters. The letters are abstract, and sooner or later he has to realise it. In physics we have outgrown archer and apple-pie definitions of the fundamental symbols. To a request to explain what an electron really is supposed to be we can only answer, “It is part of the A B C of physics”.

The training which mathematics gives in working with symbols is an excellent preparation for other sciences; … the world’s work requires constant mastery of symbols.

The woof and warp of all thought and all research is symbols, and the life of thought and science is the life inherent in symbols; so that it is wrong to say that a good language is

*important*to good thought, merely; for it is the essence of it.
There are in this world optimists who feel that any symbol that starts off with an integral sign must necessarily denote something that will have every property that they should like an integral to possess. This of course is quite annoying to us rigorous mathematicians; what is even more annoying is that by doing so they often come up with the right answer.

There is symbolic as well as actual beauty in the migration of the birds, the ebb and flow of the tides, the folded bud ready for the spring. There is something infinitely healing in the repeated refrains of nature—the assurance that dawn comes after night, and spring after the winter.

This has been far more than three men on a mission to the Moon; more still than the efforts of a government and industry team; more, even, than the efforts of one nation. We feel this stands as a symbol of the insatiable curiosity of all mankind to explore the unknown.

This maze of symbols, electric and magnetic potential, vector potential, electric force, current, displacement, magnetic force, and induction, have been practically reduced to two, electric and magnetic force.

Thought-economy is most highly developed in mathematics, that science which has reached the highest formal development, and on which natural science so frequently calls for assistance. Strange as it may seem, the strength of mathematics lies in the avoidance of all unnecessary thoughts, in the utmost economy of thought-operations. The symbols of order, which we call numbers, form already a system of wonderful simplicity and economy. When in the multiplication of a number with several digits we employ the multiplication table and thus make use of previously accomplished results rather than to repeat them each time, when by the use of tables of logarithms we avoid new numerical calculations by replacing them by others long since performed, when we employ determinants instead of carrying through from the beginning the solution of a system of equations, when we decompose new integral expressions into others that are familiar,—we see in all this but a faint reflection of the intellectual activity of a Lagrange or Cauchy, who with the keen discernment of a military commander marshalls a whole troop of completed operations in the execution of a new one.

Two kinds of symbol must surely be distinguished. The algebraic symbol comes naked into the world of mathematics and is clothed with value by its masters. A poetic symbol—like the Rose, for Love, in Guillaume de Lorris—comes trailing clouds of glory from the real world, clouds whose shape and colour largely determine and explain its poetic use. In an equation, x and y will do as well as a and b; but the

*Romance of the Rose*could not, without loss, be re-written as the Romance of the Onion, and if a man did not see why, we could only send him back to the real world to study roses, onions, and love, all of them still untouched by poetry, still raw.
Two lights for guidance. The first, our little glowing atom of community, with all that it signifies. The second, the cold light of the stars, symbol of the hypercosmical reality, with its crystal ecstasy. Strange that in this light, in which even the dearest love is frostily asserted, and even the possible defeat of our half-waking world is contemplated without remission of praise, the human crisis does not lose but gains significance. Strange, that it seems more, not less, urgent to play some part in this struggle, this brief effort of animalcules striving to win for their race some increase of lucidity before the ultimate darkness.

We may see how unexpectedly recondite parts of pure mathematics may bear upon physical science, by calling to mind the circumstance that Fresnel obtained one of the most curious confirmations of the theory (the laws of Circular Polarization by reflection) through an interpretation of an algebraical expression, which, according to the original conventional meaning of the symbols, involved an impossible quantity.

We thought of universities as the cathedrals of the modern world. In the middle ages, the cathedral was the center and symbol of the city. In the modern world, its place could be taken by the university.

[Culture] denotes an historically transmitted pattern of meanings embodied in symbols, a system of inherited conceptions expressed in symbolic forms, by means of which men communicate, perpetuate, and develop their knowledge about and attitudes toward life.

[Mathematics is] the study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols.