Formula Quotes (80 quotes)

A large number of areas of the brain are involved when viewing equations, but when one looks at a formula rated as beautiful it activates the emotional brain—the medial orbito-frontal cortex—like looking at a great painting or listening to a piece of music. … Neuroscience can’t tell you what beauty is, but if you find it beautiful the medial orbito-frontal cortex is likely to be involved; you can find beauty in anything.

Although with the majority of those who study and practice in these capacities [engineers, builders, surveyors, geographers, navigators, hydrographers, astronomers], secondhand acquirements, trite formulas, and appropriate tables are sufficient for ordinary purposes, yet these trite formulas and familiar rules were originally or gradually deduced from the profound investigations of the most gifted minds, from the dawn of science to the present day. … The further developments of the science, with its possible applications to larger purposes of human utility and grander theoretical generalizations, is an achievement reserved for a few of the choicest spirits, touched from time to time by Heaven to these highest issues. The intellectual world is filled with latent and undiscovered truth as the material world is filled with latent electricity.

Another advantage of a mathematical statement is that it is so definite that it might be definitely wrong; and if it is found to be wrong, there is a plenteous choice of amendments ready in the mathematicians’ stock of formulae. Some verbal statements have not this merit; they are so vague that they could hardly be wrong, and are correspondingly useless.

As a scientist, I am not sure anymore that life can be reduced to a class struggle, to dialectical materialism, or any set of formulas. Life is spontaneous and it is unpredictable, it is magical. I think that we have struggled so hard with the tangible that we have forgotten the intangible.

Characteristically skeptical of the idea that living things would faithfully follow mathematical formulas, [Robert Harper] seized upon factors in corn which seemed to blend in the hybrid—rather than be represented by plus or minus signs, and put several seasons into throwing doubt upon the concept of immutable hypothetical units of inheritance concocted to account for selected results.

De Morgan was explaining to an actuary what was the chance that a certain proportion of some group of people would at the end of a given time be alive; and quoted the actuarial formula, involving p [pi], which, in answer to a question, he explained stood for the ratio of the circumference of a circle to its diameter. His acquaintance, who had so far listened to the explanation with interest, interrupted him and exclaimed, “My dear friend, that must be a delusion, what can a circle have to do with the number of people alive at a given time?”

e

^{√-π}-1= 0
E=hf.

Every formula which expresses a law of nature is a hymn of praise to God.

Everything around us is filled with mystery and magic. I find this no cause for despair, no reason to turn for solace to esoteric formulae or chariots of gods. On the contrary, our inability to find easy answers fills me with a fierce pride in our ambivalent biology ... with a constant sense of wonder and delight that we should be part of anything so profound.

Gel’fand amazed me by talking of mathematics as though it were poetry. He once said about a long paper bristling with formulas that it contained the vague beginnings of an idea which could only hint at and which he had never managed to bring out more clearly. I had always thought of mathematics as being much more straightforward: a formula is a formula, and an algebra is an algebra, but Gel’fand found hedgehogs lurking in the rows of his spectral sequences!

Gentlemen and ladies, this is ordinary alcohol, sometimes called ethanol; it is found in all fermented beverages. As you well know, it is considered by many to be poisonous, a belief in which I do not concur. If we subtract from it one CH

_{2}-group we arrive at this colorless liquid, which you see in this bottle. It is sometimes called methanol or wood alcohol. It is certainly more toxic than the ethanol we have just seen. Its formula is CH_{3}OH. If, from this, we subtract the CH_{2}-group, we arrive at a third colorless liquid, the final member of this homologous series. This compound is hydrogen hydroxide, best known as water. It is the most poisonous of all.
He thought the formula for water was H-I-J-K-L-M-N-O (H-to-O).

I am afraid I shall have to give up my trade; I am far too inert to keep up with organic chemistry, it is becoming too much for me, though I may boast of having contributed something to its development. The modern system of formulae is to me quite repulsive.

I am particularly concerned to determine the probability of causes and results, as exhibited in events that occur in large numbers, and to investigate the laws according to which that probability approaches a limit in proportion to the repetition of events. That investigation deserves the attention of mathematicians because of the analysis required. It is primarily there that the approximation of formulas that are functions of large numbers has its most important applications. The investigation will benefit observers in identifying the mean to be chosen among the results of their observations and the probability of the errors still to be apprehended. Lastly, the investigation is one that deserves the attention of philosophers in showing how in the final analysis there is a regularity underlying the very things that seem to us to pertain entirely to chance, and in unveiling the hidden but constant causes on which that regularity depends. It is on the regularity of the main outcomes of events taken in large numbers that various institutions depend, such as annuities, tontines, and insurance policies. Questions about those subjects, as well as about inoculation with vaccine and decisions of electoral assemblies, present no further difficulty in the light of my theory. I limit myself here to resolving the most general of them, but the importance of these concerns in civil life, the moral considerations that complicate them, and the voluminous data that they presuppose require a separate work.

I can see him [Sylvester] now, with his white beard and few locks of gray hair, his forehead wrinkled o’er with thoughts, writing rapidly his figures and formulae on the board, sometimes explaining as he wrote, while we, his listeners, caught the reflected sounds from the board. But stop, something is not right, he pauses, his hand goes to his forehead to help his thought, he goes over the work again, emphasizes the leading points, and finally discovers his difficulty. Perhaps it is some error in his figures, perhaps an oversight in the reasoning. Sometimes, however, the difficulty is not elucidated, and then there is not much to the rest of the lecture. But at the next lecture we would hear of some new discovery that was the outcome of that difficulty, and of some article for the Journal, which he had begun. If a text-book had been taken up at the beginning, with the intention of following it, that text-book was most likely doomed to oblivion for the rest of the term, or until the class had been made listeners to every new thought and principle that had sprung from the laboratory of his mind, in consequence of that first difficulty. Other difficulties would soon appear, so that no text-book could last more than half of the term. In this way his class listened to almost all of the work that subsequently appeared in the Journal. It seemed to be the quality of his mind that he must adhere to one subject. He would think about it, talk about it to his class, and finally write about it for the Journal. The merest accident might start him, but once started, every moment, every thought was given to it, and, as much as possible, he read what others had done in the same direction; but this last seemed to be his real point; he could not read without finding difficulties in the way of understanding the author. Thus, often his own work reproduced what had been done by others, and he did not find it out until too late.

A notable example of this is in his theory of cyclotomic functions, which he had reproduced in several foreign journals, only to find that he had been greatly anticipated by foreign authors. It was manifest, one of the critics said, that the learned professor had not read Rummer’s elementary results in the theory of ideal primes. Yet Professor Smith’s report on the theory of numbers, which contained a full synopsis of Kummer’s theory, was Professor Sylvester’s constant companion.

This weakness of Professor Sylvester, in not being able to read what others had done, is perhaps a concomitant of his peculiar genius. Other minds could pass over little difficulties and not be troubled by them, and so go on to a final understanding of the results of the author. But not so with him. A difficulty, however small, worried him, and he was sure to have difficulties until the subject had been worked over in his own way, to correspond with his own mode of thought. To read the work of others, meant therefore to him an almost independent development of it. Like the man whose pleasure in life is to pioneer the way for society into the forests, his rugged mind could derive satisfaction only in hewing out its own paths; and only when his efforts brought him into the uncleared fields of mathematics did he find his place in the Universe.

A notable example of this is in his theory of cyclotomic functions, which he had reproduced in several foreign journals, only to find that he had been greatly anticipated by foreign authors. It was manifest, one of the critics said, that the learned professor had not read Rummer’s elementary results in the theory of ideal primes. Yet Professor Smith’s report on the theory of numbers, which contained a full synopsis of Kummer’s theory, was Professor Sylvester’s constant companion.

This weakness of Professor Sylvester, in not being able to read what others had done, is perhaps a concomitant of his peculiar genius. Other minds could pass over little difficulties and not be troubled by them, and so go on to a final understanding of the results of the author. But not so with him. A difficulty, however small, worried him, and he was sure to have difficulties until the subject had been worked over in his own way, to correspond with his own mode of thought. To read the work of others, meant therefore to him an almost independent development of it. Like the man whose pleasure in life is to pioneer the way for society into the forests, his rugged mind could derive satisfaction only in hewing out its own paths; and only when his efforts brought him into the uncleared fields of mathematics did he find his place in the Universe.

I do not believe that a moral philosophy can ever be founded on a scientific basis. … The valuation of life and all its nobler expressions can only come out of the soul’s yearning toward its own destiny. Every attempt to reduce ethics to scientific formulas must fail. Of that I am perfectly convinced.

I feel like a white granular mass of amorphous crystals—my formula appears to be isomeric with Spasmotoxin. My aurochloride precipitates into beautiful prismatic needles. My Platinochloride develops octohedron crystals,—with fine blue florescence. My physiological action is not indifferent. One millionth of a grain injected under the skin of a frog produced instantaneous death accompanied by an orange blossom odor. The heart stopped in systole. A base—L

_{3}H_{9}NG_{4}—offers analogous reaction to phosmotinigstic acid.
I have always felt that I understood a phenomenon only to the extent that I could visualise it. Much of the charm organic chemical research has for me derives from structural formulae. When reading chemical journals, I look for formulae first.

I have no satisfaction in formulas unless I feel their arithmetical magnitude.

I trust ... I have succeeded in convincing you that modern chemistry is not, as it has so long appeared, an ever-growing accumulation of isolated facts, as impossible for a single intellect to co-ordinate as for a single memory to grasp.

The intricate formulae that hang upon these walls, and the boundless variety of phenomena they illustrate, are beginning to be for us as a labyrinth once impassable, but to which we have at length discovered the clue. A sense of mastery and power succeeds in our minds to the sort of weary despair with which we at first contemplated their formidable array. For now, by the aid of a few general principles, we find ourselves able to unravel the complexities of these formulae, to marshal the compounds which they represent in orderly series; nay, even to multiply their numbers at our will, and in a great measure to forecast their nature ere we have called them into existence. It is the great movement of modern chemistry that we have thus, for an hour, seen passing before us. It is a movement as of light spreading itself over a waste of obscurity, as of law diffusing order throughout a wilderness of confusion, and there is surely in its contemplation something of the pleasure which attends the spectacle of a beautiful daybreak, something of the grandeur belonging to the conception of a world created out of chaos.

The intricate formulae that hang upon these walls, and the boundless variety of phenomena they illustrate, are beginning to be for us as a labyrinth once impassable, but to which we have at length discovered the clue. A sense of mastery and power succeeds in our minds to the sort of weary despair with which we at first contemplated their formidable array. For now, by the aid of a few general principles, we find ourselves able to unravel the complexities of these formulae, to marshal the compounds which they represent in orderly series; nay, even to multiply their numbers at our will, and in a great measure to forecast their nature ere we have called them into existence. It is the great movement of modern chemistry that we have thus, for an hour, seen passing before us. It is a movement as of light spreading itself over a waste of obscurity, as of law diffusing order throughout a wilderness of confusion, and there is surely in its contemplation something of the pleasure which attends the spectacle of a beautiful daybreak, something of the grandeur belonging to the conception of a world created out of chaos.

I want to put in something about Bernoulli’s numbers, in one of my Notes, as an example of how the implicit function may be worked out by the engine, without having been worked out by human head & hands first. Give me the necessary data & formulae.

I was an impostor, the worthy associate of a brigand, &c., &c., and all this for an atom of chlorine put in the place of an atom of hydrogen, for the simple correction of a chemical formula!

I was just going to say, when I was interrupted, that one of the many ways of classifying minds is under the heads of arithmetical and algebraical intellects. All economical and practical wisdom is an extension or variation of the following arithmetical formula: 2+2=4. Every philosophical proposition has the more general character of the expression a+b=c. We are mere operatives, empirics, and egotists, until we learn to think in letters instead of figures.

If I am given a sign [formula], and I am ignorant of its meaning, it cannot teach me anything, but if I already know it what does the formula teach me?

If you could stop every atom in its position and direction, and if your mind could comprehend all the actions thus suspended, then if you were really, really good at algebra you could write the formula for all the future; and although nobody can be so clever as to do it, the formula must exist just as if one could.

If you see a formula in the

*Physical Review*that extends over a quarter of a page, forget it. It’s wrong. Nature isn’t that complicated.
In every case the awakening touch has been the mathematical spirit, the attempt to count, to measure, or to calculate. What to the poet or the seer may appear to be the very death of all his poetry and all his visions—the cold touch of the calculating mind,—this has proved to be the spell by which knowledge has been born, by which new sciences have been created, and hundreds of definite problems put before the minds and into the hands of diligent students. It is the geometrical figure, the dry algebraical formula, which transforms the vague reasoning of the philosopher into a tangible and manageable conception; which represents, though it does not fully describe, which corresponds to, though it does not explain, the things and processes of nature: this clothes the fruitful, but otherwise indefinite, ideas in such a form that the strict logical methods of thought can be applied, that the human mind can in its inner chamber evolve a train of reasoning the result of which corresponds to the phenomena of the outer world.

In the matter of physics, the first lessons should contain nothing but what is experimental and interesting to see. A pretty experiment is in itself often more valuable than twenty formulae extracted from our minds.

Individual events. Events beyond law. Events so numerous and so uncoordinated that, flaunting their freedom from formula, they yet fabricate firm form.

It has been asserted … that the power of observation is not developed by mathematical studies; while the truth is, that; from the most elementary mathematical notion that arises in the mind of a child to the farthest verge to which mathematical investigation has been pushed and applied, this power is in constant exercise. By observation, as here used, can only be meant the fixing of the attention upon objects (physical or mental) so as to note distinctive peculiarities—to recognize resemblances, differences, and other relations. Now the first mental act of the child recognizing the distinction between one and more than one, between one and two, two and three, etc., is exactly this. So, again, the first geometrical notions are as pure an exercise of this power as can be given. To know a straight line, to distinguish it from a curve; to recognize a triangle and distinguish the several forms—what are these, and all perception of form, but a series of observations? Nor is it alone in securing these fundamental conceptions of number and form that observation plays so important a part. The very genius of the common geometry as a method of reasoning—a system of investigation—is, that it is but a series of observations. The figure being before the eye in actual representation, or before the mind in conception, is so closely scrutinized, that all its distinctive features are perceived; auxiliary lines are drawn (the imagination leading in this), and a new series of inspections is made; and thus, by means of direct, simple observations, the investigation proceeds. So characteristic of common geometry is this method of investigation, that Comte, perhaps the ablest of all writers upon the philosophy of mathematics, is disposed to class geometry, as to its method, with the natural sciences, being based upon observation. Moreover, when we consider applied mathematics, we need only to notice that the exercise of this faculty is so essential, that the basis of all such reasoning, the very material with which we build, have received the name observations. Thus we might proceed to consider the whole range of the human faculties, and find for the most of them ample scope for exercise in mathematical studies. Certainly, the memory will not be found to be neglected. The very first steps in number—counting, the multiplication table, etc., make heavy demands on this power; while the higher branches require the memorizing of formulas which are simply appalling to the uninitiated. So the imagination, the creative faculty of the mind, has constant exercise in all original mathematical investigations, from the solution of the simplest problems to the discovery of the most recondite principle; for it is not by sure, consecutive steps, as many suppose, that we advance from the known to the unknown. The imagination, not the logical faculty, leads in this advance. In fact, practical observation is often in advance of logical exposition. Thus, in the discovery of truth, the imagination habitually presents hypotheses, and observation supplies facts, which it may require ages for the tardy reason to connect logically with the known. Of this truth, mathematics, as well as all other sciences, affords abundant illustrations. So remarkably true is this, that today it is seriously questioned by the majority of thinkers, whether the sublimest branch of mathematics,—the

*infinitesimal calculus*—has anything more than an empirical foundation, mathematicians themselves not being agreed as to its logical basis. That the imagination, and not the logical faculty, leads in all original investigation, no one who has ever succeeded in producing an original demonstration of one of the simpler propositions of geometry, can have any doubt. Nor are*induction*,*analogy*, the*scrutinization*of*premises*or the*search*for them, or the*balancing*of*probabilities*, spheres of mental operations foreign to mathematics. No one, indeed, can claim preeminence for mathematical studies in all these departments of intellectual culture, but it may, perhaps, be claimed that scarcely any department of science affords discipline to so great a number of faculties, and that none presents so complete a gradation in the exercise of these faculties, from the first principles of the science to the farthest extent of its applications, as mathematics.
It is in the name of Moses that Bellarmin thunderstrikes Galileo; and this great vulgarizer of the great seeker Copernicus, Galileo, the old man of truth, the magian of the heavens, was reduced to repeating on his knees word for word after the inquisitor this formula of shame:

*“Corde sincera et fide non ficta abjuro maledico et detestor supradictos errores et hereses.”*Falsehood put an ass's hood on science.*[With a sincere heart, and of faith unfeigned, I deny by oath, condemn and detest the aforesaid errors and heresies.]*
It is strange that we know so little about the properties of numbers. They are our handiwork, yet they baffle us; we can fathom only a few of their intricacies. Having defined their attributes and prescribed their behavior, we are hard pressed to perceive the implications of our formulas.

It would appear that Deductive and Demonstrative Sciences are all, without exception, Inductive Sciences: that their evidence is that of experience, but that they are also, in virtue of the peculiar character of one indispensable portion of the general formulae according to which their inductions are made, Hypothetical Sciences. Their conclusions are true only upon certain suppositions, which are, or ought to be, approximations to the truth, but are seldom, if ever, exactly true; and to this hypothetical character is to be ascribed the peculiar certainty, which is supposed to be inherent in demonstration.

M.D.—Make Do.— Quaint idea! … Work for the handicapped … who is handicapped, your patients, or you? Both. Helping the survival of the unfit.… With more to come. What in the world was the solution. Where to find a formula for head and heart too?

Marxism is sociobiology without biology … Although Marxism was formulated as the enemy of ignorance and superstition, to the extent that it has become dogmatic it has faltered in that commitment and is now mortally threatened by the discoveries of human sociobiology.

Mathematics has the completely false reputation of yielding infallible conclusions. Its infallibility is nothing but identity. Two times two is not four, but it is just two times two, and that is what we call four for short. But four is nothing new at all. And thus it goes on and on in its conclusions, except that in the higher formulas the identity fades out of sight.

Mathematics may be compared to a mill of exquisite workmanship, which grinds you stuff of any degree of fineness; but, nevertheless, what you get out depends upon what you put in; and as the grandest mill in the world will not extract wheat-flour from peascods, so pages of formulae will not get a definite result out of loose data.

Mathematics, a creation of the mind, so accurately fits the outside world. … [There is a] fantastic amount of uniformity in the universe. The formulas of physics are compressed descriptions of nature's weird repetitions. The accuracy of those formulas, coupled with nature’s tireless ability to keep doing everything the same way, gives them their incredible power.

Mathematics, indeed, is the very example of brevity, whether it be in the shorthand rule of the circle, c = πd, or in that fruitful formula of analysis, e

^{iπ}= -1, —a formula which fuses together four of the most important concepts of the science,—the logarithmic base, the transcendental ratio π, and the imaginary and negative units.
My colleagues in elementary particle theory in many lands [and I] are driven by the usual insatiable curiosity of the scientist, and our work is a delightful game. I am frequently astonished that it so often results in correct predictions of experimental results. How can it be that writing down a few simple and elegant formulae, like short poems governed by strict rules such as those of the sonnet or the waka, can predict universal regularities of Nature?

My theory of electrical forces is that they are called into play in insulating media by slight electric displacements, which put certain small portions of the medium into a state of distortion which, being resisted by the elasticity of the medium, produces an electromotive force ... I suppose the elasticity of the sphere to react on the electrical matter surrounding it, and press it downwards.

From the determination by Kohlrausch and Weber of the numerical relation between the statical and magnetic effects of electricity, I have determined the elasticity of the medium in air, and assuming that it is the same with the luminiferous ether I have determined the velocity of propagation of transverse vibrations.

The result is

193088 miles per second

(deduced from electrical & magnetic experiments).

Fizeau has determined the velocity of light

= 193118 miles per second

by direct experiment.

This coincidence is not merely numerical. I worked out the formulae in the country, before seeing Webers [sic] number, which is in millimetres, and I think we have now strong reason to believe, whether my theory is a fact or not, that the luminiferous and the electromagnetic medium are one.

From the determination by Kohlrausch and Weber of the numerical relation between the statical and magnetic effects of electricity, I have determined the elasticity of the medium in air, and assuming that it is the same with the luminiferous ether I have determined the velocity of propagation of transverse vibrations.

The result is

193088 miles per second

(deduced from electrical & magnetic experiments).

Fizeau has determined the velocity of light

= 193118 miles per second

by direct experiment.

This coincidence is not merely numerical. I worked out the formulae in the country, before seeing Webers [sic] number, which is in millimetres, and I think we have now strong reason to believe, whether my theory is a fact or not, that the luminiferous and the electromagnetic medium are one.

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.
One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser that we are, wiser even than their discoverers, that we get more out of them than was originally put into them.

Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature.

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.

Round about the accredited and orderly facts of every science there ever floats a sort of dust-cloud of exceptional observations, of occurrences minute and irregular and seldom met with, which it always proves more easy to ignore than to attend to … Anyone will renovate his science who will steadily look after the irregular phenomena, and when science is renewed, its new formulas often have more of the voice of the exceptions in them than of what were supposed to be the rules.

Science is feasible when the variables are few and can be enumerated; when their combinations are distinct and clear. We are tending toward the condition of science and aspiring to do it. The artist works out his own formulas; the interest of science lies in the art of making science.

Science will never be able to reduce the value of a sunset to arithmetic. Nor can it reduce friendship or statesmanship to a formula. Laughter and love, pain and loneliness, the challenge of beauty and truth: these will always surpass the scientific mastery of nature.

Sex appeal is a matter of chemistry, but you don't have to be a chemist to find the formula.

She knew only that if she did or said thus-and-so, men would unerringly respond with the complimentary thus-and-so. It was like a mathematical formula and no more difficult, for mathematics was the one subject that had come easy to Scarlett in her schooldays.

The artist does not illustrate science; … [but] he frequently responds to the same interests that a scientist does, and expresses by a visual synthesis what the scientist converts into analytical formulae or experimental demonstrations.

The experimental investigation by which Ampere established the law of the mechanical action between electric currents is one of the most brilliant achievements in science. The whole theory and experiment, seems as if it had leaped, full grown and full armed, from the brain of the 'Newton of Electricity'. It is perfect in form, and unassailable in accuracy, and it is summed up in a formula from which all the phenomena may be deduced, and which must always remain the cardinal formula of electro-dynamics.

The experimental investigation by which Ampère established the law of the mechanical action between electric currents is one of the most brilliant achievements in science. The whole, theory and experiment, seems as if it had leaped, full grown and full armed, from the brain of the “Newton of Electricity”. It is perfect in form, and unassailable in accuracy, and it is summed up in a formula from which all the phenomena may be deduced, and which must always remain the cardinal formula of electro-dynamics.

The flights of the imagination which occur to the pure mathematician are in general so much better described in his formulas than in words, that it is not remarkable to find the subject treated by outsiders as something essentially cold and uninteresting— … the only successful attempt to invest mathematical reasoning with a halo of glory—that made in this section by Prof. Sylvester—is known to a comparative few, …

The images evoked by words being independent of their sense, they vary from age to age and from people to people, the formulas remaining identical. Certain transitory images are attached to certain words: the word is merely as it were the button of an electric bell that calls them up.

The imagination of the crowds … is impressed above all by images. … It is possible to evoke them through the judicious use of words and formulas.

The mathematical take-over of physics has its dangers, as it could tempt us into realms of thought which embody mathematical perfection but might be far removed, or even alien to, physical reality. Even at these dizzying heights we must ponder the same deep questions that troubled both Plato and Immanuel Kant. What is reality? Does it lie in our mind, expressed by mathematical formulae, or is it “out there”.

The metaphysical philosopher from his point of view recognizes mathematics as an instrument of education, which strengthens the power of attention, develops the sense of order and the faculty of construction, and enables the mind to grasp under the simple formulae the quantitative differences of physical phenomena.

The notion that the “balance of nature” is delicately poised and easily upset is nonsense. Nature is extraordinarily tough and resilient, interlaced with checks and balances, with an astonishing capacity for recovering from disturbances in equilibrium. The formula for survival is not power; it is symbiosis.

The pursuit of pretty formulas and neat theorems can no doubt quickly degenerate into a silly vice, but so can the quest for austere generalities which are so very general indeed that they are incapable of application to any particular.

There is always more in one of Ramanujan’s formulae than meets the eye, as anyone who sets to work to verify those which look the easiest will soon discover. In some the interest lies very deep, in others comparatively near the surface; but there is not one which is not curious and entertaining.

There is great exhilaration in breaking one of these things. … Ramanujan gives no hints, no proof of his formulas, so everything you do you feel is your own.

*[About verifying Ramanujan’s equations in a newly found manuscript.]*
There is nothing opposed in Biometry and Mendelism. Your husband [W.F.R. Weldon] and I worked that out at Peppards [on the Chilterns] and you will see it referred in the

*Biometrika*memoir. The Mendelian formula leads up to the “ancestral law.” What we fought against was the slovenliness in applying Mendel's categories and asserting that such formulae apply in cases when they did not.
There was yet another disadvantage attaching to the whole of Newton’s physical inquiries, ... the want of an appropriate notation for expressing the conditions of a dynamical problem, and the general principles by which its solution must be obtained. By the labours of LaGrange, the motions of a disturbed planet are reduced with all their complication and variety to a purely mathematical question. It then ceases to be a physical problem; the disturbed and disturbing planet are alike vanished: the ideas of time and force are at an end; the very elements of the orbit have disappeared, or only exist as arbitrary characters in a mathematical formula

They say that every formula halves the sales of a popular science book. This is rubbish–if it was true, then

*The Emperor’s New Mind*by Roger Penrose would have sold one-eighth of a copy, whereas its actual sales were in the hundreds of thousands.
This formula [for computing Bernoulli’s numbers] was first given by James Bernoulli…. He gave no general demonstration; but was quite aware of the importance of his theorem, for he boasts that by means of it he calculated

91,409,924,241,424,243,424,241,924,242,500.

*intra semi-quadrantem horæ!*the sum of the 10th powers of the first thousand integers, and found it to be
We cannot take anything for granted, beyond the first mathematical formula. Question everything else.

We may safely say, that the whole form of modern mathematical thinking was created by Euler. It is only with the greatest difficulty that one is able to follow the writings of any author immediately preceding Euler, because it was not yet known how to let the formulas speak for themselves. This art Euler was the first one to teach.

We ought then to consider the present state of the universe as the effect of its previous state and as the cause of that which is to follow. An intelligence that, at a given instant, could comprehend all the forces by which nature is animated and the respective situation of the beings that make it up, if moreover it were vast enough to submit these data to analysis, would encompass in the same formula the movements of the greatest bodies of the universe and those of the lightest atoms. For such an intelligence nothing would be uncertain, and the future, like the past, would be open to its eyes.

Whatever may happen to the latest theory of Dr. Einstein, his treatise represents a mathematical effort of overwhelming proportions. It is the more remarkable since Einstein is primarily a physicist and only incidentally a mathematician. He came to mathematics rather of necessity than by predilection, and yet he has here developed mathematical formulae and calculations springing from a colossal knowledge.

When Ramanujan was sixteen, he happened upon a copy of Carr’s

*Synopsis of Mathematics*. This chance encounter secured immortality for the book, for it was this book that suddenly woke Ramanujan into full mathematical activity and supplied him essentially with his complete mathematical equipment in analysis and number theory. The book also gave Ramanujan his general direction as a dealer in formulas, and it furnished Ramanujan the germs of many of his deepest developments.
Who does not know Maxwell’s dynamic theory of gases? At first there is the majestic development of the variations of velocities, then enter from one side the equations of condition and from the other the equations of central motions, higher and higher surges the chaos of formulas, suddenly four words burst forth: “Put

*n*= 5.” The evil demon*V*disappears like the sudden ceasing of the basso parts in music, which hitherto wildly permeated the piece; what before seemed beyond control is now ordered as by magic. There is no time to state why this or that substitution was made, he who cannot feel the reason may as well lay the book aside; Maxwell is no program-musician who explains the notes of his composition. Forthwith the formulas yield obediently result after result, until the temperature-equilibrium of a heavy gas is reached as a surprising final climax and the curtain drops.
Who … is not familiar with Maxwell’s memoirs on his dynamical theory of gases? … from one side enter the equations of state; from the other side, the equations of motion in a central field. Ever higher soars the chaos of formulae. Suddenly we hear, as from kettle drums, the four beats “put n=5.” The evil spirit v vanishes; and … that which had seemed insuperable has been overcome as if by a stroke of magic … One result after another follows in quick succession till at last … we arrive at the conditions for thermal equilibrium together with expressions for the transport coefficients.

You know the formula

*m*over naught equals infinity,*m*being any positive number? [*m*/0 = ∞]. Well, why not reduce the equation to a simpler form by multiplying both sides by naught? In which case you have*m*equals infinity times naught [*m*= ∞ × 0]. That is to say, a positive number is the product of zero and infinity. Doesn't that demonstrate the creation of the Universe by an infinite power out of nothing? Doesn't it?
You too can win Nobel Prizes. Study diligently. Respect DNA. Don't smoke. Don't drink. Avoid women and politics. That's my formula.

You treat world history as a mathematician does mathematics, in which nothing but laws and formulae exist, no reality, no good and evil, no time, no yesterday, no tomorrow, nothing but an eternal, shallow, mathematical present.

[As a youth, fiddling in my home laboratory] I discovered a formula for the frequency of a resonant circuit which was 2π x sqrt(LC) where L is the inductance and C the capacitance of the circuit. And there was π, and where was the circle? … I still don’t quite know where that circle is, where that π comes from.

[P]olitical and social and scientific values … should be correlated in some relation of movement that could be expressed in mathematics, nor did one care in the least that all the world said it could not be done, or that one knew not enough mathematics even to figure a formula beyond the schoolboy

*s=(1/2)gt*. If Kepler and Newton could take liberties with the sun and moon, an obscure person ... could take liberties with Congress, and venture to multiply its attraction into the square of its time. He had only to find a value, even infinitesimal, for its attraction.^{2}