… I became captivated by the edifices chemists had raised through experiment and imagination—but still I had a lurking question. Would it not be better if one could really “see” whether molecules as complicated as the sterols, or strychnine were just as experiment suggested?

[Civilization] is a highly complicated invention which has probably been made only once. If it perished it might never be made again. … But it is a poor thing. And if it to be improved there is no hope save in science.

[Richard Feynman] would be standing in front of the hall smiling at us all as we came in, his fingers tapping out a complicated rhythm on the black top of the demonstration bench that crossed the front of the lecture hall. As latecomers took their seats, he picked up the chalk and began spinning it rapidly through his fingers in a manner of a professional gambler playing with a poker chip, still smiling happily as if at some secret joke. And then—still smiling—he talked to us about physics, his diagrams and equations helping us to share his understanding. It was no secret joke that brought the smile and the sparkle in his eye, it was physics. The joy of physics!

[The octopus has] an amazing skin, because there are up to 20 million of these chromatophore pigment cells and to control 20 million of anything is going to take a lot of processing power. ... These animals have extraordinarily large, complicated brains to make all this work. ... And what does this mean about the universe and other intelligent life? The building blocks are potentially there and complexity will arise. Evolution is the force that's pushing that. I would expect, personally, a lot of diversity and a lot of complicated structures. It may not look like us, but my personal view is that there is intelligent life out there.

[When nature appears complicated:] The moment we contemplate it as it is, and attain a position from which we can take a commanding view, though but of a small part of its plan, we never fail to recognize that sublime simplicity on which the mind rests satisfied that it has attained the truth.

The Charms of Statistics.—It is difficult to understand why statisticians commonly limit their inquiries to Averages, and do not revel in more comprehensive views. Their souls seem as dull to the charm of variety as that of the native of one of our flat English counties, whose retrospect of Switzerland was that, if its mountains could be thrown into its lakes, two nuisances would be got rid of at once. An Average is but a solitary fact, whereas if a single other fact be added to it, an entire Normal Scheme, which nearly corresponds to the observed one, starts potentially into existence. Some people hate the very name of statistics, but I find them full of beauty and interest. Whenever they are not brutalised, but delicately handled by the higher methods, and are warily interpreted, their power of dealing with complicated phenomena is extraordinary. They are the only tools by which an opening can be cut through the formidable thicket of difficulties that bars the path of those who pursue the Science of man.

A physician’s subject of study is necessarily the patient, and his first field for observation is the hospital. But if clinical observation teaches him to know the form and course of diseases, it cannot suffice to make him understand their nature; to this end he must penetrate into the body to find which of the internal parts are injured in their functions. That is why dissection of cadavers and microscopic study of diseases were soon added to clinical observation. But to-day these various methods no longer suffice; we must push investigation further and, in analyzing the elementary phenomena of organic bodies, must compare normal with abnormal states. We showed elsewhere how incapable is anatomy alone to take account of vital phenenoma, and we saw that we must add study of all physico-chemical conditions which contribute necessary elements to normal or pathological manifestations of life. This simple suggestion already makes us feel that the laboratory of a physiologist-physician must be the most complicated of all laboratories, because he has to experiment with phenomena of life which are the most complex of all natural phenomena.

A rose is the visible result of an infinitude of complicated goings on in the bosom of the earth and in the air above, and similarly a work of art is the product of strange activities in the human mind.

After a duration of a thousand years, the power of astrology broke down when, with Copernicus, Kepler, and Galileo, the progress of astronomy overthrew the false hypothesis upon which the entire structure rested, namely the geocentric system of the universe. The fact that the earth revolves in space intervened to upset the complicated play of planetary influences, and the silent stars, related to the unfathomable depths of the sky, no longer made their prophetic voices audible to mankind. Celestial mechanics and spectrum analysis finally robbed them of their mysterious prestige.

All the inventions and devices ever constructed by the human hand or conceived by the human mind, no matter how delicate, how intricate and complicated, are simple, childish toys compared with that most marvelously wrought mechanism, the human body. Its parts are far more delicate, and their mutual adjustments infinitely more accurate, than are those of the most perfect chronometer ever made.

All you really need to know for the moment is that the universe is a lot more complicated than you might think, even if you start from a position of thinking it’s pretty damn complicated in the first place.

Anybody can make the simple complicated. Creativity is making the complicated simple.

Anybody who looks at living organisms knows perfectly well that they can produce other organisms like themselves. This is their normal function, they wouldn’t exist if they didn’t do this, and it’s not plausible that this is the reason why they abound in the world. In other words, living organisms are very complicated aggregations of elementary parts, and by any reasonable theory of probability or thermodynamics highly improbable. That they should occur in the world at all is a miracle of the first magnitude; the only thing which removes, or mitigates, this miracle is that they reproduce themselves. Therefore, if by any peculiar accident there should ever be one of them, from there on the rules of probability do not apply, and there will be many of them, at least if the milieu is reasonable. But a reasonable milieu is already a thermodynamically much less improbable thing. So, the operations of probability somehow leave a loophole at this point, and it is by the process of self-reproduction that they are pierced.

Art and science work in quite different ways: agreed. But, bad as it may sound, I have to admit that I cannot get along as an artist without the use of one or two sciences. ... In my view, the great and complicated things that go on in the world cannot be adequately recognized by people who do not use every possible aid to understanding.

As is well known the principle of virtual velocities transforms all statics into a mathematical assignment, and by D'Alembert's principle for dynamics, the latter is again reduced to statics. Although it is is very much in order that in gradual training of science and in the instruction of the individual the easier precedes the more difficult, the simple precedes the more complicated, the special precedes the general, yet the min, once it has arrived at the higher standpoint, demands the reverse process whereby all statics appears only as a very special case of mechanics.

Culture in its higher forms is a delicate plant which depends on a complicated set of conditions and is wont to flourish only in a few places at any given time.

Did it ever occur to you that there’s no limit to how complicated things can get, on account of one thing always leading to another?

Either one or the other [analysis or synthesis] may be direct or indirect. The direct procedure is when the point of departure is known-direct synthesis in the elements of geometry. By combining at random simple truths with each other, more complicated ones are deduced from them. This is the method of discovery, the special method of inventions, contrary to popular opinion.

Few scientists acquainted with the chemistry of biological systems at the molecular level can avoid being inspired. Evolution has produced chemical compounds exquisitely organized to accomplish the most complicated and delicate of tasks. Many organic chemists viewing crystal structures of enzyme systems or nucleic acids and knowing the marvels of specificity of the immune systems must dream of designing and synthesizing simpler organic compounds that imitate working features of these naturally occurring compounds.

Food is at present obtained almost entirely from the energy of the sunlight. The radiation from the sun produces from the carbonic acid in the air more or less complicated carbon compounds which serve us in plants and vegetables. We use the latent chemical energy of these to keep our bodies warm, we convert it into muscular effort. We employ it in the complicated process of digestion to repair and replace the wasted cells of our bodies. … If the gigantic sources of power become available, food would be produced without recourse to sunlight. Vast cellars, in which artificial radiation is generated, may replace the cornfields and potato patches of the world.

Freeman’s gift? It’s cosmic. He is able to see more interconnections between more things than almost anybody. He sees the interrelationships, whether it’s in some microscopic physical process or in a big complicated machine like Orion. He has been, from the time he was in his teens, capable of understanding essentially anything that he’s interested in. He’s the most intelligent person I know.

Furthermore, it’s equally evident that what goes on is actually one degree better than self-reproduction, for organisms appear to have gotten more elaborate in the course of time. Today's organisms are phylogenetically descended from others which were vastly simpler than they are, so much simpler, in fact, that it’s inconceivable, how any kind of description of the latter, complex organism could have existed in the earlier one. It’s not easy to imagine in what sense a gene, which is probably a low order affair, can contain a description of the human being which will come from it. But in this case you can say that since the gene has its effect only within another human organism, it probably need not contain a complete description of what is to happen, but only a few cues for a few alternatives. However, this is not so in phylogenetic evolution. That starts from simple entities, surrounded by an unliving amorphous milieu, and produce, something more complicated. Evidently, these organisms have the ability to produce something more complicated than themselves.

Genetics has always turned out to be much more complicated than it seemed reasonable to imagine. Biology is not like physics. The more we know, the less it seems that there is one final explanation waiting to be discovered.

Geology, perhaps more than any other department of natural philosophy, is a science of contemplation. It requires no experience or complicated apparatus, no minute processes upon the unknown processes of matter. It demands only an enquiring mind and senses alive to the facts almost everywhere presented in nature. And as it may be acquired without much difficulty, so it may be improved without much painful exertion.

Geometry, which should only obey Physics, when united with it sometimes commands it. If it happens that the question which we wish to examine is too complicated for all the elements to be able to enter into the analytical comparison which we wish to make, we separate the more inconvenient [elements], we substitute others for them, less troublesome, but also less real, and we are surprised to arrive, notwithstanding a painful labour, only at a result contradicted by nature; as if after having disguised it, cut it short or altered it, a purely mechanical combination could give it back to us.

I believe that the useful methods of mathematics are easily to be learned by quite young persons, just as languages are easily learned in youth. What a wondrous philosophy and history underlie the use of almost every word in every language—yet the child learns to use the word unconsciously. No doubt when such a word was first invented it was studied over and lectured upon, just as one might lecture now upon the idea of a rate, or the use of Cartesian co-ordinates, and we may depend upon it that children of the future will use the idea of the calculus, and use squared paper as readily as they now cipher. … When Egyptian and Chaldean philosophers spent years in difficult calculations, which would now be thought easy by young children, doubtless they had the same notions of the depth of their knowledge that Sir William Thomson might now have of his. How is it, then, that Thomson gained his immense knowledge in the time taken by a Chaldean philosopher to acquire a simple knowledge of arithmetic? The reason is plain. Thomson, when a child, was taught in a few years more than all that was known three thousand years ago of the properties of numbers. When it is found essential to a boy’s future that machinery should be given to his brain, it is given to him; he is taught to use it, and his bright memory makes the use of it a second nature to him; but it is not till after-life that he makes a close investigation of what there actually is in his brain which has enabled him to do so much. It is taken because the child has much faith. In after years he will accept nothing without careful consideration. The machinery given to the brain of children is getting more and more complicated as time goes on; but there is really no reason why it should not be taken in as early, and used as readily, as were the axioms of childish education in ancient Chaldea.

I claim that relativity and the rest of modern physics is not complicated. It can be explained very simply. It is only unusual or, put another way, it is contrary to common sense.

I consider the differences between man and animals in propensities, feelings, and intellectual faculties, to be the result of the same cause as that which we assign for the variations in other functions, viz. difference of organization; and that the superiority of man in rational endowments is not greater than the more exquisite, complicated, and perfectly developed structure of his brain, and particularly of his ample cerebral hemispheres, to which the rest of the animal kingdom offers no parallel, nor even any near approximation, is sufficient to account for.

I have always assumed, and I now assume, that he [Robert Oppenheimer] is loyal to the United States. I believe this, and I shall believe it until I see very conclusive proof to the opposite. … [But] I thoroughly disagreed with him in numerous issues and his actions frankly appeared to me confused and complicated. To this extent I feel that I would like to see the vital interests of this country in hands which I understand better, and therefore trust more.

I have known silence: the cold earthy silence at the bottom of a newly dug well; the implacable stony silence of a deep cave; the hot, drugged midday silence when everything is hypnotised and stilled into silence by the eye of the sun;… I have heard summer cicadas cry so that the sound seems stitched into your bones. I have heard tree frogs in an orchestration as complicated as Bach singing in a forest lit by a million emerald fireflies. I have heard the Keas calling over grey glaciers that groaned to themselves like old people as they inched their way to the sea. I have heard the hoarse street vendor cries of the mating Fur seals as they sang to their sleek golden wives, the crisp staccato admonishment of the Rattlesnake, the cobweb squeak of the Bat and the belling roar of the Red deer knee-deep in purple heather.

I have yet to see any problem, however complicated, which, when you looked at it in the right way, did not become still more complicated.

I think that considerable progress can be made in the analysis of the operations of nature by the scholar who reduces rather complicated phenomena to their proximate causes and primitive forces, even though the causes of those causes have not yet been detected.

I think that it is a relatively good approximation to truth—which is much too complicated to allow anything but approximations—that mathematical ideas originate in empirics.

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.

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.

Imagine that … the world is something like a great chess game being played by the gods, and we are observers of the game. … If we watch long enough, we may eventually catch on to a few of the rules…. However, we might not be able to understand why a particular move is made in the game, merely because it is too complicated and our minds are limited…. We must limit ourselves to the more basic question of the rules of the game.
If we know the rules, we consider that we “understand” the world.

In biology, nothing is clear, everything is too complicated, everything is a mess, and just when you think you understand something, you peel off a layer and find deeper complications beneath. Nature is anything but simple.

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.

In physics we have dealt hitherto only with periodic crystals. To a humble physicist’s mind, these are very interesting and complicated objects; they constitute one of the most fascinating and complex material structures by which inanimate nature puzzles his wits. Yet, compared with the aperiodic crystal, they are rather plain and dull. The difference in structure is of the same kind as that between an ordinary wallpaper in which the same pattern is repeated again and again in regular periodicity and a masterpiece of embroidery, say a Raphael tapestry, which shows no dull repetition, but an elaborate, coherent, meaningful design traced by the great master.

In this lecture I would like to conclude with … some characteristics [of] gravity … The most impressive fact is that gravity is simple. It is simple to state the principles completely and not have left any vagueness for anybody to change the ideas of the law. It is simple, and therefore it is beautiful. It is simple in its pattern. I do not mean it is simple in its action—the motions of the various planets and the perturbations of one on the other can be quite complicated to work out, and to follow how all those stars in a globular cluster move is quite beyond our ability. It is complicated in its actions, but the basic pattern or the system beneath the whole thing is simple. This is common to all our laws; they all turn out to be simple things, although complex in their actual actions.

Is evolution a theory, a system or a hypothesis? It is much more: it is a general condition to which all theories, all hypotheses, all systems must bow and which they must satisfy henceforth if they are to be thinkable and true. Evolution is a light illuminating all facts, a curve that all lines must follow. ... The consciousness of each of us is evolution looking at itself and reflecting upon itself....Man is not the center of the universe as once we thought in our simplicity, but something much more wonderful—the arrow pointing the way to the final unification of the world in terms of life. Man alone constitutes the last-born, the freshest, the most complicated, the most subtle of all the successive layers of life. ... The universe has always been in motion and at this moment continues to be in motion. But will it still be in motion tomorrow? ... What makes the world in which we live specifically modern is our discovery in it and around it of evolution. ... Thus in all probability, between our modern earth and the ultimate earth, there stretches an immense period, characterized not by a slowing-down but a speeding up and by the definitive florescence of the forces of evolution along the line of the human shoot.

It doesn't seem to me that this fantastically marvelous universe, this tremendous range of time and space and different kinds of animals, and all the different planets, and all these atoms with all their motions, and so on, all this complicated thing can merely be a stage so that God can watch human beings struggle for good and evil—which is the view that religion has. The stage is too big for the drama.

It is now necessary to indicate more definitely the reason why mathematics not only carries conviction in itself, but also transmits conviction to the objects to which it is applied. The reason is found, first of all, in the perfect precision with which the elementary mathematical concepts are determined; in this respect each science must look to its own salvation .... But this is not all. As soon as human thought attempts long chains of conclusions, or difficult matters generally, there arises not only the danger of error but also the suspicion of error, because since all details cannot be surveyed with clearness at the same instant one must in the end be satisfied with a belief that nothing has been overlooked from the beginning. Every one knows how much this is the case even in arithmetic, the most elementary use of mathematics. No one would imagine that the higher parts of mathematics fare better in this respect; on the contrary, in more complicated conclusions the uncertainty and suspicion of hidden errors increases in rapid progression. How does mathematics manage to rid itself of this inconvenience which attaches to it in the highest degree? By making proofs more rigorous? By giving new rules according to which the old rules shall be applied? Not in the least. A very great uncertainty continues to attach to the result of each single computation. But there are checks. In the realm of mathematics each point may be reached by a hundred different ways; and if each of a hundred ways leads to the same point, one may be sure that the right point has been reached. A calculation without a check is as good as none. Just so it is with every isolated proof in any speculative science whatever; the proof may be ever so ingenious, and ever so perfectly true and correct, it will still fail to convince permanently. He will therefore be much deceived, who, in metaphysics, or in psychology which depends on metaphysics, hopes to see his greatest care in the precise determination of the concepts and in the logical conclusions rewarded by conviction, much less by success in transmitting conviction to others. Not only must the conclusions support each other, without coercion or suspicion of subreption, but in all matters originating in experience, or judging concerning experience, the results of speculation must be verified by experience, not only superficially, but in countless special cases.

It is sages and grey-haired philosophers who ought to sit up all night reading Alice in Wonderland in order to study that darkest problem of metaphysics, the borderland between reason and unreason, and the nature of the most erratic of spiritual forces, humour, which eternally dances between the two. That we do find a pleasure in certain long and elaborate stories, in certain complicated and curious forms of diction, which have no intelligible meaning whatever, is not a subject for children to play with; it is a subject for psychologists to go mad over.

It is well known that theoretical physicists cannot handle experimental equipment; it breaks whenever they touch it. Pauli was such a good theoretical physicist that something usually broke in the lab whenever he merely stepped across the threshold. A mysterious event that did not seem at first to be connected with Pauli's presence once occurred in Professor J. Franck's laboratory in Göttingen. Early one afternoon, without apparent cause, a complicated apparatus for the study of atomic phenomena collapsed. Franck wrote humorously about this to Pauli at his Zürich address and, after some delay, received an answer in an envelope with a Danish stamp. Pauli wrote that he had gone to visit Bohr and at the time of the mishap in Franck's laboratory his train was stopped for a few minutes at the Göttingen railroad station. You may believe this anecdote or not, but there are many other observations concerning the reality of the Pauli Effect!

It was my science that drove me to the conclusion that the world is much more complicated than can be explained by science. It is only through the supernatural that I can understand the mystery of existence.

It would seem at first sight as if the rapid expansion of the region of mathematics must be a source of danger to its future progress. Not only does the area widen but the subjects of study increase rapidly in number, and the work of the mathematician tends to become more and more specialized. It is, of course, merely a brilliant exaggeration to say that no mathematician is able to understand the work of any other mathematician, but it is certainly true that it is daily becoming more and more difficult for a mathematician to keep himself acquainted, even in a general way, with the progress of any of the branches of mathematics except those which form the field of his own labours. I believe, however, that the increasing extent of the territory of mathematics will always be counteracted by increased facilities in the means of communication. Additional knowledge opens to us new principles and methods which may conduct us with the greatest ease to results which previously were most difficult of access; and improvements in notation may exercise the most powerful effects both in the simplification and accessibility of a subject. It rests with the worker in mathematics not only to explore new truths, but to devise the language by which they may be discovered and expressed; and the genius of a great mathematician displays itself no less in the notation he invents for deciphering his subject than in the results attained. … I have great faith in the power of well-chosen notation to simplify complicated theories and to bring remote ones near and I think it is safe to predict that the increased knowledge of principles and the resulting improvements in the symbolic language of mathematics will always enable us to grapple satisfactorily with the difficulties arising from the mere extent of the subject.

Life is order, death is disorder. A fundamental law of Nature states that spontaneous chemical changes in the universe tend toward chaos. But life has, during milliards of years of evolution, seemingly contradicted this law. With the aid of energy derived from the sun it has built up the most complicated systems to be found in the universe—living organisms. Living matter is characterized by a high degree of chemical organisation on all levels, from the organs of large organisms to the smallest constituents of the cell. The beauty we experience when we enjoy the exquisite form of a flower or a bird is a reflection of a microscopic beauty in the architecture of molecules.

Life is too complicated to permit a complete understanding through the study of whole organisms. Only by simplifying a biological problem—breaking it down into a multitude of individual problems—can you get the answers.

Living is like working out a long addition sum, and if you make a mistake in the first two totals you will never find the right answer. It means involving oneself in a complicated chain of circumstances.

Making the simple complicated is commonplace; making the complicated simple, awesomely simple, that’s creativity.

Man is a rational animal—so at least I have been told. … Aristotle, so far as I know, was the first man to proclaim explicitly that man is a rational animal. His reason for this view was … that some people can do sums. … It is in virtue of the intellect that man is a rational animal. The intellect is shown in various ways, but most emphatically by mastery of arithmetic. The Greek system of numerals was very bad, so that the multiplication table was quite difficult, and complicated calculations could only be made by very clever people.

Math is like love—a simple idea but it can get complicated.

Mathematics education is much more complicated than you expected, even though you expected it to be more complicated than you expected.

Mathematics is an obscure field, an abstruse science, complicated and exact; yet so many have attained perfection in it that we might conclude almost anyone who seriously applied himself would achieve a measure of success.

My picture of the world is drawn in perspective and not like a model to scale. The foreground is occupied by human beings and the stars are all as small as three-penny bits. I don't really believe in astronomy, except as a complicated description of part of the course of human and possibly animal sensation. I apply my perspective not merely to space but also to time. In time the world will cool and everything will die; but that is a long time off still and its present value at compound discount is almost nothing.

One can truly say that the irresistible progress of natural science since the time of Galileo has made its first halt before the study of the higher parts of the brain, the organ of the most complicated relations of the animal to the external world. And it seems, and not without reason, that now is the really critical moment for natural science; for the brain, in its highest complexity—the human brain—which created and creates natural science, itself becomes the object of this science.

Our atom of carbon enters the leaf, colliding with other innumerable (but here useless) molecules of nitrogen and oxygen. It adheres to a large and complicated molecule that activates it, and simultaneously receives the decisive message from the sky, in the flashing form of a packet of solar light; in an instant, like an insect caught by a spider, it is separated from its oxygen, combined with hydrogen and (one thinks) phosphorus, and finally inserted in a chain, whether long or short does not matter, but it is the chain of life. All this happens swiftly, in silence, at the temperature and pressure of the atmosphere, and gratis: dear colleagues, when we learn to do likewise we will be sicut Deus [like God], and we will have also solved the problem of hunger in the world.

Our job in physics is to see things simply, to understand a great many complicated phenomena in a unified way, in terms of a few simple principles.

Physical science enjoys the distinction of being the most fundamental of the experimental sciences, and its laws are obeyed universally, so far as is known, not merely by inanimate things, but also by living organisms, in their minutest parts, as single individuals, and also as whole communities. It results from this that, however complicated a series of phenomena may be and however many other sciences may enter into its complete presentation, the purely physical aspect, or the application of the known laws of matter and energy, can always be legitimately separated from the other aspects.

Quantum theory thus reveals a basic oneness of the universe. It shows that we cannot decompose the world into independently existing smallest units. As we penetrate into matter, nature does not show us any isolated “building blocks,” but rather appears as a complicated web of relations between the various parts of the whole. These relations always include the observer in an essential way. The human observer constitute the final link in the chain of observational processes, and the properties of any atomic object can be understood only in terms of the object’s interaction with the observer.

Reality is complicated. There is no justification for all of the hasty conclusions.

Science has thus, most unexpectedly, placed in our hands a new power of great but unknown energy. It does not wake the winds from their caverns; nor give wings to water by the urgency of heat; nor drive to exhaustion the muscular power of animals; nor operate by complicated mechanism; nor summon any other form of gravitating force, but, by the simplest means—the mere contact of metallic surfaces of small extent, with feeble chemical agents, a power everywhere diffused through nature, but generally concealed from our senses, is mysteriously evolved, and by circulation in insulated wires, it is still more mysteriously augmented, a thousand and a thousand fold, until it breaks forth with incredible energy.

Science is the art of the appropriate approximation. While the flat earth model is usually spoken of with derision it is still widely used. Flat maps, either in atlases or road maps, use the flat earth model as an approximation to the more complicated shape.

Scientific method, although in its more refined forms it may seem complicated, is in essence remarkably simply. It consists in observing such facts as will enable the observer to discover general laws governing facts of the kind in question. The two stages, first of observation, and second of inference to a law, are both essential, and each is susceptible of almost indefinite refinement. (1931)

Scientists come in two varieties, hedgehogs and foxes. I borrow this terminology from Isaiah Berlin (1953), who borrowed it from the ancient Greek poet Archilochus. Archilochus told us that foxes know many tricks, hedgehogs only one. Foxes are broad, hedgehogs are deep. Foxes are interested in everything and move easily from one problem to another. Hedgehogs are only interested in a few problems that they consider fundamental, and stick with the same problems for years or decades. Most of the great discoveries are made by hedgehogs, most of the little discoveries by foxes. Science needs both hedgehogs and foxes for its healthy growth, hedgehogs to dig deep into the nature of things, foxes to explore the complicated details of our marvelous universe. Albert Einstein and Edwin Hubble were hedgehogs. Charley Townes, who invented the laser, and Enrico Fermi, who built the first nuclear reactor in Chicago, were foxes.

Sea-water is, of course, opaque and this is the first difficulty that faces the oceanographer. Most of the tools needed to investigate the sea must use physical principles which are more complicated than the optical methods that are so satisfactory for studying the surface features of the land.

Since the beginning of the century, computational procedures have become so complicated that any progress by those means has become impossible, without the elegance which modern mathematicians have brought to bear on their research, and by means of which the spirit comprehends quickly and in one step a great many computations.
It is clear that elegance, so vaunted and so aptly named, can have no other purpose. …
[But, the simplifications produced by this elegance will soon outrun the problems supplied by analysis. What happens then?]
Go to the roots, of these calculations! Group the operations. Classify them according to their complexities rather than their appearances! This, I believe, is the mission of future mathematicians. This is the road on which I am embarking in this work.

Some problems are just too complicated for rational logical solutions. They admit of insights, not answers.

Suppose it were perfectly certain that the life and fortune of every one of us would, one day or other, depend upon his winning or losing a game of chess. Don't you think that we should all consider it to be a primary duty to learn at least the names and the moves of the pieces; to have a notion of a gambit, and a keen eye for all the means of giving and getting out of check? Do you not think that we should look with a disapprobation amounting to scorn upon the father who allowed his son, or the state which allowed its members, to grow up without knowing a pawn from a knight?
Yet, it is a very plain and elementary truth that the life, the fortune, and the happiness of every one of us, and, more or less, of those who are connected with us, do depend upon our knowing something of the rules of a game infinitely more difficult and complicated than chess. It is a game which has been played for untold ages, every man and woman of us being one of the two players in a game of his or her own. The chess-board is the world, the pieces are the phenomena of the universe, the rules of the game are what we call the laws of nature. The player on the other side is hidden from us. We know that his play is always fair, just, and patient. But also we know, to our cost, that he never overlooks a mistake, or makes the smallest allowance for ignorance. To the man who plays well the highest stakes are paid with that sort of overflowing generosity with which the strong shows delight in strength. And one who plays ill is checkmated—without haste, but without remorse.

That the fundamental aspects of heredity should have turned out to be so extraordinarily simple supports us in the hope that nature may, after all, be entirely approachable. Her much-advertised inscrutability has once more been found to be an illusion due to our ignorance. This is encouraging, for, if the world in which we live were as complicated as some of our friends would have us believe we might well despair that biology could ever become an exact science.

The art of drawing conclusions from experiments and observations consists in evaluating probabilities and in estimating whether they are sufficiently great or numerous enough to constitute proofs. This kind of calculation is more complicated and more difficult than it is commonly thought to be. … It is above all in medicine that the difficulty of evaluating the probabilities is greater.

The brain is the most complicated kilo of matter in the universe.

The concepts of ‘soul’ or ‘life’ do not occur in atomic physics, and they could not, even indirectly, be derived as complicated consequences of some natural law. Their existence certainly does not indicate the presence of any fundamental substance other than energy, but it shows only the action of other kinds of forms which we cannot match with the mathematical forms of modern atomic physics ... If we want to describe living or mental processes, we shall have to broaden these structures. It may be that we shall have to introduce yet other concepts.

The dangers threatening modern science cannot be averted by more experimenting, for our complicated experiments have no longer anything to do with nature in her own right, but with nature charged and transformed by our own cognitive activity.

The enchanting charms of this sublime science reveal only to those who have the courage to go deeply into it. But when a woman, who because of her sex and our prejudices encounters infinitely more obstacles that an man in familiarizing herself with complicated problems, succeeds nevertheless in surmounting these obstacles and penetrating the most obscure parts of them, without doubt she must have the noblest courage, quite extraordinary talents and superior genius.

The equation of animal and vegetable life is too complicated a problem for human intelligence to solve, and we can never know how wide a circle of disturbance we produce in the harmonies of nature when we throw the smallest pebble into the ocean of organic life.

The essence of mathematics is not to make simple things complicated, but to make complicated things simple.

The Excellence of Modern Geometry is in nothing more evident, than in those full and adequate Solutions it gives to Problems; representing all possible Cases in one view, and in one general Theorem many times comprehending whole Sciences; which deduced at length into Propositions, and demonstrated after the manner of the Ancients, might well become the subjects of large Treatises: For whatsoever Theorem solves the most complicated Problem of the kind, does with a due Reduction reach all the subordinate Cases.

The first difficulty of all is the production of a lamp which shall be thoroughly reliable, and neither complicated nor expensive. All attempts up to the present lamp in this direction are acknowledged to be failures, and, as I have pointed out, there does not seem to be any novelty such as would authorize us to hope for a better success in the present one.

The genotypic constitution of a gamete or a zygote may be parallelized with a complicated chemico-physical structure. This reacts exclusively in consequence of its realized state, but not in consequence of the history of its creation. So it may be with the genotypical constitution of gametes and zygotes: its history is without influence upon its reactions, which are determined exclusively by its actual nature. The genotype-conception is thus an 'ahistoric' view of the reactions of living beings—of course only as far as true heredity is concerned. This view is an analog to the chemical view, as already pointed out; chemical compounds have no compromising ante-act, H₂O is always H₂O, and reacts always in the same manner, whatsoever may be the 'history' of its formation or the earlier states of its elements. I suggest that it is useful to emphasize this 'radical' ahistoric genotype-conception of heredity in its strict antagonism to the transmission—or phenotype-view.

The geometrical problems and theorems of the Greeks always refer to definite, oftentimes to rather complicated figures. Now frequently the points and lines of such a figure may assume very many different relative positions; each of these possible cases is then considered separately. On the contrary, present day mathematicians generate their figures one from another, and are accustomed to consider them subject to variation; in this manner they unite the various cases and combine them as much as possible by employing negative and imaginary magnitudes. For example, the problems which Apollonius treats in his two books De sectione rationis, are solved today by means of a single, universally applicable construction; Apollonius, on the contrary, separates it into more than eighty different cases varying only in position. Thus, as Hermann Hankel has fittingly remarked, the ancient geometry sacrifices to a seeming simplicity the true simplicity which consists in the unity of principles; it attained a trivial sensual presentability at the cost of the recognition of the relations of geometric forms in all their changes and in all the variations of their sensually presentable positions.

The initiation of the fermentation process does not require so complicated an apparatus as is represented by the living cell. The agent responsible for the fermenting action of the press juice is rather to be regarded as a dissolved substance, doubtless a protein; this will be denoted zymase.

The mathematical intellectualism is henceforth a positive doctrine, but one that inverts the usual doctrines of positivism: in place of originating progress in order, dynamics in statics, its goal is to make logical order the product of intellectual progress. The science of the future is not enwombed, as Comte would have had it, as Kant had wished it, in the forms of the science already existing; the structure of these forms reveals an original dynamism whose onward sweep is prolonged by the synthetic generation of more and more complicated forms. No speculation on number considered as a category a priori enables one to account for the questions set by modern mathematics … space affirms only the possibility of applying to a multiplicity of any elements whatever, relations whose type the intellect does not undertake to determine in advance, but, on the contrary, it asserts their existence and nourishes their unlimited development.

The modern research laboratory can be a large and complicated social organism.

The other book you may have heard of and perhaps read, but it is not one perusal which will enable any man to appreciate it. I have read it through five or six times, each time with increasing admiration. It will live as long as the ‘Principia’ of Newton. It shows that nature is, as I before remarked to you, a study that yields to none in grandeur and immensity. The cycles of astronomy or even the periods of geology will alone enable us to appreciate the vast depths of time we have to contemplate in the endeavour to understand the slow growth of life upon the earth. The most intricate effects of the law of gravitation, the mutual disturbances of all the bodies of the solar system, are simplicity itself compared with the intricate relations and complicated struggle which have determined what forms of life shall exist and in what proportions. Mr. Darwin has given the world a new science, and his name should, in my opinion, stand above that of every philosopher of ancient or modem times. The force of admiration can no further go!!!

The other line of argument, which leads to the opposite conclusion, arises from looking at artificial automata. Everyone knows that a machine tool is more complicated than the elements which can be made with it, and that, generally speaking, an automaton A, which can make an automaton B, must contain a complete description of B, and also rules on how to behave while effecting the synthesis. So, one gets a very strong impression that complication, or productive potentiality in an organization, is degenerative, that an organization which synthesizes something is necessarily more complicated, of a higher order, than the organization it synthesizes. This conclusion, arrived at by considering artificial automaton, is clearly opposite to our early conclusion, arrived at by considering living organisms.

The process of tracing regularity in any complicated, and at first sight confused, set of appearances, is necessarily tentative; we begin by making any supposition, even a false one, to see what consequences will follow from it ; and by observing how these differ from the real phenomena, we learn what corrections to make in our assumption.

The psyche is distinctly more complicated and inaccessible than the body. It is, so to speak, the half of the world which comes into existence only when we become conscious of it. For that reason the psyche is not only a personal but a world problem, and the psychiatrist has to deal with an entire world.

The simplicity of nature is not to be measured by that of our conceptions. Infinitely varied in its effects, nature is simple only in its causes, and its economy consists in producing a great number of phenomena, often very complicated, by means of a small number of general laws.

The steady progress of physics requires for its theoretical formulation a mathematics which get continually more advanced. ... it was expected that mathematics would get more and more complicated, but would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundation and gets more abstract. Non-euclidean geometry and noncommutative algebra, which were at one time were considered to be purely fictions of the mind and pastimes of logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and the advance in physics is to be associated with continual modification and generalisation of the axioms at the base of mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation.

The steady states of the fluid matrix of the body are commonly preserved by physiological reactions, i.e., by more complicated processes than are involved in simple physico-chemical equilibria. Special designations, therefore, are appropriate:—“homeostasis” to designate stability of the organism; “homeostatic conditions,” to indicate details of the stability; and “homeostatic reactions,” to signify means for maintaining stability.

The student should read his author with the most sustained attention, in order to discover the meaning of every sentence. If the book is well written, it will endure and repay his close attention: the text ought to be fairly intelligible, even without illustrative examples. Often, far too often, a reader hurries over the text without any sincere and vigorous effort to understand it; and rushes to some example to clear up what ought not to have been obscure, if it had been adequately considered. The habit of scrupulously investigating the text seems to me important on several grounds. The close scrutiny of language is a very valuable exercise both for studious and practical life. In the higher departments of mathematics the habit is indispensable: in the long investigations which occur there it would be impossible to interpose illustrative examples at every stage, the student must therefore encounter and master, sentence by sentence, an extensive and complicated argument.

The study of the radio-active substances and of the discharge of electricity through gases has supplied very strong experimental evidence in support of the fundamental ideas of the existing atomic theory. It has also indicated that the atom itself is not the smallest unit of matter, but is a complicated structure made up of a number of smaller bodies.

The surgeon is a man of action. By temperament and by training he prefers to serve the sick by operating on them, and he inwardly commiserates with a patient so unfortunate as to have a disease not suited to surgical treatment. Young surgeons, busy mastering the technicalities of the art, are particularly alert to seize every legitimate opportunity to practice technical maneuvers, the more complicated the better.

The world is very complicated and it is clearly impossible for the human mind to understand it completely. Man has therefore devised an artifice which permits the complicated nature of the world to be blamed on something which is called accidental and thus permits him to abstract a domain in which simple laws can be found.

There are two kinds of biologists, those who are looking to see if there is one thing that can be understood and those who keep saying it is very complicated and that nothing can be understood. ... You must study the simplest system you think has the properties you are interested in.

There is synthesis when, in combining therein judgments that are made known to us from simpler relations, one deduces judgments from them relative to more complicated relations.
There is analysis when from a complicated truth one deduces more simple truths.

There may only be a small number of laws, which are self-consistent and which lead to complicated beings like ourselves. … And even if there is only one unique set of possible laws, it is only a set of equations. What is it that breathes fire into the equations and makes a universe for them to govern? Is the ultimate unified theory so compelling that it brings about its own existence?

There was, I think, a feeling that the best science was that done in the simplest way. In experimental work, as in mathematics, there was “style” and a result obtained with simple equipment was more elegant than one obtained with complicated apparatus, just as a mathematical proof derived neatly was better than one involving laborious calculations. Rutherford's first disintegration experiment, and Chadwick's discovery of the neutron had a “style” that is different from that of experiments made with giant accelerators.

To produce a really good biological theory one must try to see through the clutter produced by evolution to the basic mechanisms lying beneath them, realizing that they are likely to be overlaid by other, secondary mechanisms. What seems to physicists to be a hopelessly complicated process may have been what nature found simplest, because nature could only build on what was already there.

To-day, science has withdrawn into realms that are hardly understanded of the people. Biology means very largely histology, the study of the cell by difficult and elaborate microscopical processes. Chemistry has passed from the mixing of simple substances with ascertained reactions, to an experimentation of these processes under varying conditions of temperature, pressure, and electrification—all requiring complicated apparatus and the most delicate measurement and manipulation. Similarly, physics has outgrown the old formulas of gravity, magnetism, and pressure; has discarded the molecule and atom for the ion, and may in its recent generalizations be followed only by an expert in the higher, not to say the transcendental mathematics.

Two of his [Euler’s] pupils having computed to the 17th term, a complicated converging series, their results differed one unit in the fiftieth cipher; and an appeal being made to Euler, he went over the calculation in his mind, and his decision was found correct.

We animals are the most complicated things in the known universe.

We are not very pleased when we are forced to accept a mathematical truth by virtue of a complicated chain of formal conclusions and computations, which we traverse blindly, link by link, feeling our way by touch. We want first an overview of the aim and of the road; we want to understand the idea of the proof, the deeper context.

We are peeling an onion layer by layer, each layer uncovering in a sense another universe, unexpected, complicated, and— as we understand more—strangely beautiful.

We can learn a lot from living organisms. An organism is a pretty complicated thing, which can tolerate surgery, which can tolerate injury, which can tolerate all kinds of perturbation provided they are not too great and do not come too suddenly. There’s something we call trauma, however. We don’t really understand what it is—but organisations can suffer from it too.

We have hitherto considered those Ideas, in the reception whereof, the Mind is only passive, which are those simple ones received from Sensation and Reflection before-mentioned, whereof the Mind cannot make anyone to it self, nor have any Idea which does not wholy consist of them. But as these simple Ideas are observed to exist in several Combinations united together; so the Mind has a power to consider several of them united together, as one Idea; and that not only as they are united in external Objects, but as it self has joined them. Ideas thus made up of several simple ones put together, I call Complex; such as are Beauty, Gratitude, a Man, an Army, the Universe; which tough complicated various simple Ideas, made up of simple ones, yet are, when the Mind pleases, considered each by if self, as one entire thing, and signified by one name.

We need another and a wiser and perhaps a more mystical concept of animals. Remote from universal nature, and living by complicated artifice, man in civilization surveys the creature through the glass of his knowledge and sees thereby a feather magnified and the whole image in distortion. We patronize them for their incompleteness, for their tragic fate of having taken form so far below ourselves. And therein we err, and greatly err. For the animal shall not be measured by man. In a world older and more complete than ours they move finished and complete, gifted with extensions of the senses we have lost or never attained, living by voices we shall never hear. They are not brethren, they are not underlings; they are other nations, caught with ourselves in the net of life and time, fellow prisoners of the splendour and travail of the earth.