Mathematical Physics Quotes

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.

Mathematics is a language plus reasoning; it is like a language plus logic. Mathematics is a tool for reasoning.

Ideas that require people to reorganize their picture of the world provoke hostility.

Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper.

Physics depends on a universe infinitely centred on an equals sign.

I think that modern physics has definitely decided in favor of Plato. In fact the smallest units of matter are not physical objects in the ordinary sense; they are forms, ideas which can be expressed unambiguously only in mathematical language.

It is now quite lawful for a Catholic woman to avoid pregnancy by a resort to mathematics, though she is still forbidden to resort to physics and chemistry.

Mathematics catalogues everything that is not self-contradictory; within that vast inventory, physics is an island of structures rich enough to contain their own beholders.

Physicists have come to realize that mathematics, when used with sufficient care, is a proven pathway to truth.

Having solved all the major mathematical, physical, chemical, biological, sociological, philosophical, etymological, meteorological and psychological problems of the Universe except for his own, three times over, [Marvin] was severely stuck for something to do, and had taken up composing short dolorous ditties of no tone, or indeed tune. The latest one was a lullaby. Marvin droned, Now the world has gone to bed, Darkness won't engulf my head, I can see in infrared, How I hate the night. He paused to gather the artistic and emotional strength to tackle the next verse. Now I lay me down to sleep, Try to count electric sheep, Sweet dream wishes you can keep, How I hate the night.

We have a closed circle of consistency here: the laws of physics produce complex systems, and these complex systems lead to consciousness, which then produces mathematics, which can then encode in a succinct and inspiring way the very underlying laws of physics that gave rise to it.

It is impossible to explain honestly the beauties of the laws of nature in a way that people can feel, without their having some deep understanding of mathematics. I am sorry, but this seems to be the case.

If one is working from the point of view of getting beauty into one's equation, ... one is on a sure line of progress.

A relativist is an individual who doesn't know the difference between an adjective and an adverb.

The person who wishes to attain human perfection should study logic first, next mathematics, then physics, and, lastly, metaphysics.

One must divide one's time between politics and equations. But our equations are much more important to me, because politics is for the present, while our equations are for eternity.

A mathematician is a magician who converts adjectives into nouns: continuous into continuum, infinite into infinity, infinitesimal into location, 0D into point, 1D into line, curved into geodesic...

The mathematicians are the priests of the modern world.

In every department of physical science there is only so much science, properly so-called, as there is mathematics.

I wanted very much to learn to draw, for a reason that I kept to myself: I wanted to convey an emotion I have about the beauty of the world. It's difficult to describe because it's an emotion. It's analogous to the feeling one has in religion that has to do with a god that controls everything in the whole universe: there's a generality aspect that you feel when you think about how things that appear so different and behave so differently are all run "behind the scenes" by the same organization, the same physical laws. It's an appreciation of the mathematical beauty of nature, of how she works inside; a realization that the phenomena we see result from the complexity of the inner workings between atoms; a feeling of how dramatic and wonderful it is. It's a feeling of awe — of scientific awe — which I felt could be communicated through a drawing to someone who had also had this emotion. It could remind him, for a moment, of this feeling about the glories of the universe.

We love men because they can never fake orgasms, even if they wanted to. Because they write poems, songs, and books in our honor. Because they never understand us, but they never give up. Because they can see beauty in women when women have long ceased to see any beauty in themselves. Because they come from little boys. Because they can churn out long, intricate, Machiavellian, or incredibly complex mathematics and physics equations, but they can be comparably clueless when it comes to women. Because they are incredible lovers and never rest until we’re happy. Because they elevate sports to religion. Because they’re never afraid of the dark. Because they don’t care how they look or if they age. Because they persevere in making and repairing things beyond their abilities, with the naïve self-assurance of the teenage boy who knew everything. Because they never wear or dream of wearing high heels. Because they’re always ready for sex. Because they’re like pomegranates: lots of inedible parts, but the juicy seeds are incredibly tasty and succulent and usually exceed your expectations. Because they’re afraid to go bald. Because you always know what they think and they always mean what they say. Because they love machines, tools, and implements with the same ferocity women love jewelry. Because they go to great lengths to hide, unsuccessfully, that they are frail and human. Because they either speak too much or not at all to that end. Because they always finish the food on their plate. Because they are brave in front of insects and mice. Because a well-spoken four-year old girl can reduce them to silence, and a beautiful 25-year old can reduce them to slobbering idiots. Because they want to be either omnivorous or ascetic, warriors or lovers, artists or generals, but nothing in-between. Because for them there’s no such thing as too much adrenaline. Because when all is said and done, they can’t live without us, no matter how hard they try. Because they’re truly as simple as they claim to be. Because they love extremes and when they go to extremes, we’re there to catch them. Because they are tender they when they cry, and how seldom they do it. Because what they lack in talk, they tend to make up for in action. Because they make excellent companions when driving through rough neighborhoods or walking past dark alleys. Because they really love their moms, and they remind us of our dads. Because they never care what their horoscope, their mother-in-law, nor the neighbors say. Because they don’t lie about their age, their weight, or their clothing size. Because they have an uncanny ability to look deeply into our eyes and connect with our heart, even when we don’t want them to. Because when we say “I love you” they ask for an explanation.

Before you can ask 'Is Darwinian theory correct or not?', You have to ask the preliminary question 'Is it clear enough so that it could be correct?'. That's a very different question. One of my prevailing doctrines about Darwinian theory is 'Man, that thing is just a mess. It's like looking into a room full of smoke.' Nothing in the theory is precisely, clearly, carefully defined or delineated. It lacks all of the rigor one expects from mathematical physics, and mathematical physics lacks all the rigor one expects from mathematics. So we're talking about a gradual descent down the level of intelligibility until we reach evolutionary biology.

‎In modern physics, there is no such thing as "nothing." Even in a perfect vacuum, pairs of virtual particles are constantly being created and destroyed. The existence of these particles is no mathematical fiction. Though they cannot be directly observed, the effects they create are quite real. The assumption that they exist leads to predictions that have been confirmed by experiment to a high degree of accuracy.

Physicists believe that the Gaussian law has been proved in mathematics while mathematicians think that it was experimentally established in physics.

A mathematician says that an electromagnetic wave travels from Andromeda to your eye and that it also extends from Andromeda to your eye.

Atheism is the opium of the mathematicians. Atheism is the religion of Mathematics.

Economics is haunted by more fallacies than any other study known to man. This is no accident. The inherent difficulties of the subject would be great enough in any case, but they are multiplied a thousandfold by a factor that is insignificant in, say, physics, mathematics or medicine - the special pleading of selfish interests.

Mathematics is not a science from our point of view, in the sense that it is not a natural science. The test of its validity is not experiment.

Science is not about making predictions or performing experiments. Science is about explaining.

We do not know space. We do not see it, we do not hear it, we do not feel it. We are standing in the middle of it, we ourselves are part of it, but we know nothing about it.

There is another physical law that teases me, too: the Doppler Effect. The sound of anything coming at you- a train, say, or the future- has a higher pitch than the sound of the same thing going away. If you have perfect pitch and a head for mathematics you can compute the speed of the object by the interval between its arriving and departing sounds. I have neither perfect pitch nor a head for mathematics, and anyway who wants to compute the speed of history? Like all falling bodies, it constantly accelerates. But I would like to hear your life as you heard it, coming at you, instead of hearing it as I do, a somber sound of expectations reduced, desires blunted, hopes deferred or abandoned, chances lost, defeats accepted, griefs borne.

In my own professional work I have touched on a variety of different fields. I’ve done work in mathematical linguistics, for example, without any professional credentials in mathematics; in this subject I am completely self-taught, and not very well taught. But I’ve often been invited by universities to speak on mathematical linguistics at mathematics seminars and colloquia. No one has ever asked me whether I have the appropriate credentials to speak on these subjects; the mathematicians couldn’t care less. What they want to know is what I have to say. No one has ever objected to my right to speak, asking whether I have a doctor’s degree in mathematics, or whether I have taken advanced courses in the subject. That would never have entered their minds. They want to know whether I am right or wrong, whether the subject is interesting or not, whether better approaches are possible… the discussion dealt with the subject, not with my right to discuss it. But on the other hand, in discussion or debate concerning social issues or American foreign policy…. The issue is constantly raised, often with considerable venom. I’ve repeatedly been challenged on grounds of credentials, or asked, what special training do I have that entitles you to speak on these matters. The assumption is that people like me, who are outsiders from a professional viewpoint, are not entitled to speak on such things. Compare mathematics and the political sciences… it’s quite striking. In mathematics, in physics, people are concerned with what you say, not with your certification. But in order to speak about social reality, you must have the proper credentials, particularly if you depart from the accepted framework of thinking. Generally speaking, it seems fair to say that the richer the intellectual substance of a field, the less there is a concern for credentials, and the greater is the concern for content.

The theories of the social sciences do not consist of “laws” in the sense of empirical rules about the behavior of objects definable in physical terms. All that the theory of the social sciences attempts is to provide a technique of reasoning which assists us in connecting individual facts, but which, like logic or mathematics, is not about the facts. It can, therefore, and this is the second point, never be verified or falsified by reference to facts.

Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover.

If you bake a cupcake, the world has one more cupcake. If you become a circus clown, the world has one more squirt of seltzer down someone's pants. But if you win an Olympic gold medal, the world will not have one more Olympic gold medalist. It will just have you instead of someone else.

There cannot be a language more universal and more simple, more free from errors and obscurities...more worthy to express the invariable relations of all natural things [than mathematics]. [It interprets] all phenomena by the same language, as if to attest the unity and simplicity of the plan of the universe, and to make still more evident that unchangeable order which presides over all natural causes

[M]odern physics has definitely decided for Plato. For the smallest units of matter are not physical objects in the ordinary sense of the word: they are forms, structures, or – in Plato’s sense – Ideas, which can be unambiguously spoken of only in the language of mathematics.

If physics leads us today to a world view which is essentially mystical, it returns, in a way, to its beginning, 2,500 years ago. ... This time, however, it is not only based on intuition, but also on experiments of great precision and sophistication, and on a rigorous and consistent mathematical formalism.

Surely, the gods' judgment is certain. But as for us, we must be satisfied to 'come close' to those things, for we are men, who speak according to what is likely, and whose lectures resemble fables.

In a way, mathematics is the only infinite human activity. It is conceivable that humanity could eventually learn everything in physics or biology. But humanity certainly won't ever be able to find out everything in mathematics, because the subject is infinite. Numbers themselves are infinite. That's why mathematics is really my only interest.

A mathematician is an individual who calls himself a 'physicist' and does 'physics' and physical experiments with abstract concepts.

If you can't illustrate 'it', 'it' doens't belong in Physics as a noun! You can't put an article in front. You can't put a verb after!

Mathematics is only a tool and one should learn to hold the physical ideas in one's mind without reference to the mathematical form.

A mathematician is an individual who believes that prophesying that his dog will die if he deprives it of food constitutes a prediction.

All the hypotheses about the early universe . . . are pure speculation. They're modern creation myths written in the language of mathematics.

For the most part, physicists and mathematicians have settled on a fine division of labour in which the former complain about the finickiness of the latter, and the latter complain about the sloppiness of the former.

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.

There are other reasons we use math in physics. Besides keeping us honest, math is also the most economical and unambiguous terminology that we know of. Language is malleable; it depends on context and interpretation. But math doesn’t care about culture or history. If a thousand people read a book, they read a thousand different books. But if a thousand people read an equation, they read the same equation.

In mathematics, in physics, people are concerned with what you say, not with your certification. But in order to speak about social reality, you must have the proper credentials, particularly if you depart from the accepted framework of thinking. Generally speaking, it seems fair to say that the richer the intellectual substance of a field, the less there is a concern for credentials, and the greater is concern for content.

Mathematics began to seem too much like puzzle solving. Physics is puzzle solving, too, but of puzzles created by nature, not by the mind of man.

A mathematician is an individual who constructs space with 0D particles and then places a bowling ball on this invisible canvas to explain how gravity works.

ECONOMICS IS HAUNTED by more fallacies than any other study known to man. This is no accident. The inherent difficulties of the subject would be great enough in any case, but they are multiplied a thousandfold by a factor that is insignificant in, say, physics, mathematics or medicine—the special pleading of selfish interests.

Physics, mathematics, music, painting, my politics, my love for you, my work, the star-dust of my body, the spirit that impels it, clocks diurnal, time perpetual, the roll, rough, tender, swamping, liberating, breathing, moving, thinking nature, human nature and the cosmos are patterned together.

The theory of computation has traditionally been studied almost entirely in the abstract, as a topic in pure mathematics. This is to miss the point of it. Computers are physical objects, and computations are physical processes. What computers can or cannot compute is determined by the laws of physics alone, and not by pure mathematics.

Perfection. That's what it's about. It's those moments. When you can feel the perfection of creation. The beauty the physics you know the wonder of mathematics. The elations of action and reaction and that is the kind of perfection that I want to be connected to.

It's a bit like staring into another dimension, one that has a different set of mathematical and physical laws. For me, it also serves as reminder that that the mind of God is unknowable, that things that seem contradictory to us only appear so because we have no context for them, or aren't seeing the full picture.

It is generally recognized that women are better than men at languages, personal relations and multitasking, but less good at map-reading and spatial awareness. It is therefore not unreasonable to suppose that women might be less good at mathematics and physics. It is not politically correct to say such things....But it cannot be denied that there are differences between men and women. Of course, these are differences between the averages only. There are wide variations about the mean.

It may be appropriate to quote a statement of Poincare, who said (partly in jest no doubt) that there must be something mysterious about the normal law since mathematicians think it is a law of nature whereas physicists are convinced that it is a mathematical theorem.

The importance of C.F. Gauss for the development of modern physical theory and especially for the mathematical fundament of the theory of relativity is overwhelming indeed; also his achievement of the system of absolute measurement in the field of electromagnetism. In my opinion it is impossible to achieve a coherent objective picture of the world on the basis of concepts which are taken more or less from inner psychological experience.

Take this neat little equation here. It tells me all the ways an electron can make itself comfortable in or around an atom. That's the logic of it. The poetry of it is that the equation tells me how shiny gold is, how come rocks are hard, what makes grass green, and why you can't see the wind. And a million other things besides, about the way nature works.

I have been able to solve a few problems of mathematical physics on which the greatest mathematicians since Euler have struggled in vain ... But the pride I might have held in my conclusions was perceptibly lessened by the fact that I knew that the solution of these problems had almost always come to me as the gradual generalization of favorable examples, by a series of fortunate conjectures, after many errors. I am fain to compare myself with a wanderer on the mountains who, not knowing the path, climbs slowly and painfully upwards and often has to retrace his steps because he can go no further—then, whether by taking thought or from luck, discovers a new track that leads him on a little till at length when he reaches the summit he finds to his shame that there is a royal road by which he might have ascended, had he only the wits to find the right approach to it. In my works, I naturally said nothing about my mistake to the reader, but only described the made track by which he may now reach the same heights without difficulty.

I noticed that the [drawing] teacher didn't tell people much... Instead, he tried to inspire us to experiment with new approaches. I thought of how we teach physics: We have so many techniques - so many mathematical methods - that we never stop telling the students how to do things. On the other hand, the drawing teacher is afraid to tell you anything. If your lines are very heavy, the teacher can't say, "Your lines are too heavy." because *some* artist has figured out a way of making great pictures using heavy lines. The teacher doesn't want to push you in some particular direction. So the drawing teacher has this problem of communicating how to draw by osmosis and not by instruction, while the physics teacher has the problem of always teaching techniques, rather than the spirit, of how to go about solving physical problems.

Compare mathematics and the political sciences—it’s quite striking. In mathematics, in physics, people are concerned with what you say, not with your certification. But in order to speak about social reality, you must have the proper credentials, particularly if you depart from the accepted framework of thinking. Generally speaking, it seems fair to say that the richer the intellectual substance of a field, the less there is a concern for credentials, and the greater is the concern for content.

the new discipline of physics could not proceed until Isaac Newton appropriated words that were ancient and vague—force, mass, motion, and even time—and gave them new meanings. Newton made these terms into quantities, suitable for use in mathematical formulas. Until then, motion (for example) had been just as soft and inclusive a term as information.

You ask me if an ordinary person—by studying hard—would get to be able to imagine these things like I imagine. Of course. I was an ordinary person who studied hard. There's no miracle people. It just happens they got interested in this thing, and they learned all this stuff. They're just people. There's no talent or special miracle ability to understand quantum mechanics or a miracle ability to imagine electromagnetic fields that comes without practice and reading and learning and study. So if you take an ordinary person who's willing to devote a great deal of time and study and work and thinking and mathematics, then he's become a scientist.

Music was a kind of penetration. Perhaps absorption is a less freighted word. The penetration or absorption of everything into itself. I don't know if you have ever taken LSD, but when you do so the doors of perception, as Aldous Huxley, Jim Morrison and their adherents ceaselessly remind us, swing wide open. That is actually the sort of phrase, unless you are William Blake, that only makes sense when there is some LSD actually swimming about inside you. In the cold light of the cup of coffee and banana sandwich that are beside me now it appears to be nonsense, but I expect you to know what it is taken to mean. LSD reveals the whatness of things, their quiddity, their essence. The wateriness of water is suddenly revealed to you, the carpetness of carpets, the woodness of wood, the yellowness of yellow, the fingernailness of fingernails, the allness of all, the nothingness of all, the allness of nothing. For me music gives access to everyone of these essences, but at a fraction of the social or financial cost of a drug and without the need to cry 'Wow!' all the time, which is LSD's most distressing and least endearing side effects. ...Music in the precision of its form and the mathematical tyranny of its laws, escapes into an eternity of abstraction and an absurd sublime that is everywhere and nowhere at once. The grunt of rosin-rubbed catgut, the saliva-bubble blast of a brass tube, the sweaty-fingered squeak on a guitar fret, all that physicality, all that clumsy 'music making', all that grain of human performance...transcends itself at the moment of its happening, that moment when music actually becomes, as it makes the journey from the vibrating instrument, the vibrating hi-fi speaker, as it sends those vibrations across to the human tympanum and through to the inner ear and into the brain, where the mind is set to vibrate to frequencies of its own making. The nothingness of music can be moulded by the mood of the listener into the most precise shapes or allowed to float as free as thought; music can follow the academic and theoretical pattern of its own modality or adhere to some narrative or dialectical programme imposed by a friend, a scholar or the composer himself. Music is everything and nothing. It is useless and no limit can be set to its use. Music takes me to places of illimitable sensual and insensate joy, accessing points of ecstasy that no angelic lover could ever locate, or plunging me into gibbering weeping hells of pain that no torturer could ever devise. Music makes me write this sort of maundering adolescent nonsense without embarrassment. Music is in fact the dog's bollocks. Nothing else comes close.

You can tell if a discipline is BS if the degree depends severely on the prestige of the school granting it. I remember when I applied to MBA programs being told that anything outside the top ten or twenty would be a waste of time. On the other hand a degree in mathematics is much less dependent on the school (conditional on being above a certain level, so the heuristic would apply to the difference between top ten and top two thousand schools). The same applies to research papers. In math and physics, a result posted on the repository site arXiv (with a minimum hurdle) is fine. In low-quality fields like academic finance (where papers are usually some form of complicated storytelling), the “prestige” of the journal is the sole criterion.

To make this condition mathematically clearer, it is convenient to assert it in the form that the space-time can be continued smoothly, as a conformal manifold, a little way prior to the hypersurface . To before the Big Bang? Surely not: the Big Bang is supposed to represent the beginning of all things, so there can be no ‘before’. Never fear—this is just a mathematical trick. The extension is not supposed to have any physical meaning! Or might it …?

Modern science was born through the Scientific Revolution in the 11th/17th century at a time when, as we saw earlier, European philosophy had itself rebelled against revelation and the religious world view. The background of modern science is a particular philosophical outlook which sees the parameters of the physical world, that is, space, time, matter and energy to be realities that are independent of higher orders of being and cut off from the power of God, at least during the unfolding of the history of the cosmos. It views the physical world as being primarily the subject of mathematicization and quatification and, in a sense, absolutizes the mathematical study of nature relegating the non-quantifiable aspects of physical existence to irrelevance.

The hoodlum-occultist is “sociopathic” enough to, see through the conventional charade, the social mythology of his species. “They’re all sheep,” he thinks. “Marks. Suckers. Waiting to be fleeced.” He has enough contact with some more-or-less genuine occult tradition to know a few of the gimmicks by which “social consciousness,” normally conditioned consciousness, can be suspended. He is thus able to utilize mental brutality in place of the simple physical brutality of the ordinary hooligan. He is quite powerless against those who realize that he is actually a stupid liar. He is stupid because spending your life terrorizing and exploiting your inferiors is a dumb and boring existence for anyone with more than five billion brain cells. Can you imagine Beethoven ignoring the heavenly choirs his right lobe could hear just to pound on the wall and annoy the neighbors? Gödel pushing aside his sublime mathematics to go out and cheat at cards? Van Gogh deserting his easel to scrawl nasty caricatures in the men’s toilet? Mental evil is always the stupidest evil because the mind itself is not a weapon but a potential paradise. Every kind of malice is a stupidity, but occult malice is stupidest of all. To the extent that the mindwarper is not 100 percent charlatan through-and-through (and most of them are), to the extent that he has picked up some real occult lore somewhere, his use of it for malicious purposes is like using Shakespeare’s sonnets for toilet tissue or picking up a Picasso miniature to drive nails. Everybody who has advanced beyond the barbarian stage of evolution can see how pre-human such acts are, except the person doing them. Genuine occult initiation confers “the philosopher’s stone,” “the gold of the wise” and “the elixir of life,” all of which are metaphors for the capacity to greet life with the bravery and love and gusto that it deserves. By throwing this away to indulge in spite, malice and the small pleasure of bullying the credulous, the mindwarper proves himself a fool and a dolt. And the psychic terrorist, besides being a jerk, is always a liar and a fraud. Healing is easier (and more fun) than cursing, to begin with, and cursing usually backfires or misfires. The mindwarper doesn’t want you to know that. He wants you to think he’s omnipotent.

I started studying law, but this I could stand just for one semester. I couldn't stand more. Then I studied languages and literature for two years. After two years I passed an examination with the result I have a teaching certificate for Latin and Hungarian for the lower classes of the gymnasium, for kids from 10 to 14. I never made use of this teaching certificate. And then I came to philosophy, physics, and mathematics. In fact, I came to mathematics indirectly. I was really more interested in physics and philosophy and thought about those. It is a little shortened but not quite wrong to say: I thought I am not good enough for physics and I am too good for philosophy. Mathematics is in between.

Ever since I was a child, I’ve had the gut sense that there’s a consciousness behind the universe. When I witness the precision of mathematics, the reliability of physics, and the symmetries of the cosmos, I don’t feel like I’m observing cold science; I feel as if I’m seeing a living footprint…the shadow of some greater force that is just beyond our grasp.

Nature seems to take advantage of the simple mathematical representations of the symmetry laws. When one pauses to consider the elegance and the beautiful perfection of the mathematical reasoning involved and contrast it with the complex and far-reaching physical consequences, a deep sense of respect for the power of the symmetry laws never fails to develop.

The philosophers make still another objection: "What you gain in rigour," they say, "you lose in objectivity. You can rise toward your logical ideal only by cutting the bonds which attach you to reality. Your science is infallible, but it can only remain so by imprisoning itself in an ivory tower and renouncing all relation with the external world. From this seclusion it must go out when it would attempt the slightest application.

Our experience teaches us that there are indeed laws of nature, regularities in the way things behave, and that these laws are best expressed using the language of mathematics. This raises the interesting possibility that mathematical consistency might be used to guide us, along with experimental observation, to the laws that describe physical reality, and this has proved to be the case time and again throughout the history of science. We will see this happen during the course of this book, and it is truly one of the wonderful mysteries of our universe that it should be so.

In mathematics or physics, infinity is greater than one or two or any number countable. In how many ways can the world be destroyed based on ordered knowledge? You may be able to count this. But the truth is, you “really” don’t know. These possibilities in your mind hold a set of unpredictable orders. One effect may be causative of another of another. It could be a culmination of effects you know as events where events are sets and subsets of potential possibilities. In how many ways can the world be destroyed based on unordered possibilities?

If Henry Adams, whom you knew slightly, could make a theory of history by applying the second law of thermodynamics to human affairs, I ought to be entitled to base one on the angle of repose, and may yet. There is another physical law that teases me, too: the Doppler Effect. The sound of anything coming at you -- a train, say, or the future -- has a higher pitch than the sound of the same thing going away. If you have perfect pitch and a head for mathematics you can compute the speed of the object by the interval between its arriving and departing sounds. I have neither perfect pitch nor a head for mathematics, and anyway who wants to compute the speed of history? Like all falling bodies, it constantly accelerates. But I would like to hear your life as you heard it, coming at you, instead of hearing it as I do, a sober sound of expectations reduced, desires blunted, hopes deferred or abandoned, chances lost, defeats accepted, griefs borne. I don't find your life uninteresting, as Rodman does. I would like to hear it as it sounded while it was passing. Having no future of my own, why shouldn't I look forward to yours.

there is a physical problem that is common to many fields, that is very old, and that has not been solved. It is not the problem of finding new fundamental particles, but something left over from a long time ago—over a hundred years. Nobody in physics has really been able to analyze it mathematically satisfactorily in spite of its importance to the sister sciences. It is the analysis of circulating or turbulent fluids.

For the purposes of science, information had to mean something special. Three centuries earlier, the new discipline of physics could not proceed until Isaac Newton appropriated words that were ancient and vague—force, mass, motion, and even time—and gave them new meanings. Newton made these terms into quantities, suitable for use in mathematical formulas. Until then, motion (for example) had been just as soft and inclusive a term as information. For Aristotelians, motion covered a far-flung family of phenomena: a peach ripening, a stone falling, a child growing, a body decaying. That was too rich. Most varieties of motion had to be tossed out before Newton’s laws could apply and the Scientific Revolution could succeed. In the nineteenth century, energy began to undergo a similar transformation: natural philosophers adapted a word meaning vigor or intensity. They mathematicized it, giving energy its fundamental place in the physicists’ view of nature. It was the same with information. A rite of purification became necessary. And then, when it was made simple, distilled, counted in bits, information was found to be everywhere.

The first [method] I might speak about is simplification. Suppose that you are given a problem to solve, I don't care what kind of problem-a machine to design, or a physical theory to develop, or a mathematical theorem to prove or something of that kind-probably a very powerful approach to this is to attempt to eliminate everything from the problem except the essentials; that is, cut is down to size. Almost every problem that you come across is befuddled with all kinds of extraneous data of one sort or another; and if you can bring this problem down into the main issues, you can see more clearly what you are trying to do an perhaps find a solution. Now in so doing you may have stripped away the problem you're after. You may have simplified it to the point that it doesn't even resemble the problem that you started with; but very often if you can solve this simple problem, you can add refinements to the solution of this until you get back to the solution of the one you started with.

I think a strong claim can be made that the process of scientific discovery may be regarded as a form of art. This is best seen in the theoretical aspects of Physical Science. The mathematical theorist builds up on certain assumptions and according to well understood logical rules, step by step, a stately edifice, while his imaginative power brings out clearly the hidden relations between its parts. A well constructed theory is in some respects undoubtedly an artistic production. A fine example is the famous Kinetic Theory of Maxwell. ... The theory of relativity by Einstein, quite apart from any question of its validity, cannot but be regarded as a magnificent work of art.

If memory within nature sounds mysterious, we should bear in mind that mathematical laws transcending nature are more rather than less so; they are metaphysical rather than physical. The way mathematical laws can exist independently of the evolving universe and at the same time act upon it remains a profound mystery. For those who accept God, this mystery is an aspect of God's relation to the realm of nature; for those who deny God, the mystery is even more obscure: A quasi-mental realm of mathematical laws somehow exists independently of nature, yet not in God, and governs the evolving physical world without itself being physical.

From all this we concluded that the first two divisions of theoretical philosophy should rather be called guesswork than knowledge, theology because of its completely invisible and ungraspable nature, physics because of the unstable and unclear nature of the matter; hence there is no hope that philosophers will ever be agreed about them; and that only mathematics can provide sure and unshakable knowledge to its devotees, provided one approaches it rigorously. For its kind of proof proceeds by indisputable methods, namely arithmetic and geometry (tr. Toomer, p. 6).

I think I repeated the Heisenberg Uncertainty Principle in my head at least one thousand times: the mathematical product of the combined uncertainties of concurrent measurements of position and momentum in a specified direction could never be less than Planck’s constant, h, divided by 4π. This meant, rather encouragingly, that my uncertain position and zero momentum and the Beast Responsible for the Sound’s uncertain position and uncertain momentum had to sort of null each other out, leaving me with what is commonly known in the scientific world as “wide-ranging perplexity.

If he would just work with pure ideas like a proper mathematician he could go as fast as thought. As it happens, Alan has become fascinated by the incarnations of pure ideas in the physical world. The underlying math of the universe is like the light streaming in through the window. Alan is not satisfied with merely knowing that it streams in. He blows smoke into the air to make the light visible. He sits in meadows gazing at pine cones and flowers, tracing the mathematical patterns in their structure, and he dreams about electron winds blowing over the glowing filaments and screens of radio tubes, and, in their surges and eddies, capturing something of what is going on in his own brain. Turing is neither a mortal nor a god. He is Antaeus. That he bridges the mathematical and physical worlds is his strength and his weakness.

Another explanation for the failure of logic and observation alone to advance medicine is that unlike, say, physics, which uses a form of logic - mathematics - as its natural language, biology does not lend itself to logic. Leo Szilard, a prominent physicist, made this point when he complained that after switching from physics to biology he never had a peaceful bath again. As a physicist he would soak in the warmth of a bathtub and contemplate a problem, turn it in his mind, reason his way through it. But once he became a biologist, he constantly had to climb out of the bathtub to look up a fact.

The modern science of nature is significant for many other reasons, beyond the obvious setting of conservation priorities and actions. Foremost in my mind being the fact that it is beautiful. Its wondrous mathematical synchronicities, the specifics of its chemical analyses, the complexity of its physics are beyond both the practical and intuitive knowledge of most lay naturalists (or mystics), no matter how seasoned. When mingled with the wildness of the natural world and the creativity of the human mind, good science reveals its center, its story, its deeper teaching. The science has its own poetic force.

I see, in place of that empty figment of one linear history which can be kept up only by shutting one’s eyes to the overwhelming multitude of facts, the drama of a number of mighty Cultures, each springing with primitive strength from the soil of a mother-region to which it remains firmly bound throughout it’s whole life-cycle; each stamping its material, its mankind, in its own image; each having its own idea, its own passions, its own life, will and feelings, its own death. Here indeed are colours, lights, movements, that no intellectual eye has yet discovered. Here the Cultures, peoples, languages, truths, gods, landscapes bloom and age as the oaks and the pines, the blossoms, twigs and leaves - but there is no ageing “Mankind.” Each Culture has its own new possibilities of self-expression which arise, ripen, decay and never return. There is not one sculpture, one painting, one mathematics, one physics, but many, each in the deepest essence different from the others, each limited in duration and self-contained, just as each species of plant has its peculiar blossom or fruit, its special type of growth and decline.

It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. Our feeble attempts at mathematics enable us to understand a bit of the universe, and as we proceed to develop higher and higher mathematics we can hope to understand the universe better.

We actually inhabit not a scientific world but a perfect Cartesian arena of pure mathematics. We really are inside a flawless mathematical matrix. Yet this is not a physical matrix. Points in the Singularity do not physically move anywhere or have any physicality. Coordinates are simply how you provide an internal structure for the Singularity; how you arrange points WITHIN the Singularity. There is no material world at all: only a mental plenum of monads, just as Leibniz said three hundred years ago. And the final key point is that the natural mathematics of zero and infinity is nothing other than HOLOGRAPHY. We are individual holograms within a collective hologram. THAT is God’s secret!

Another time somebody gave a talk about poetry. He talked about the structure of the poem and the emotions that come with it; he divided everything up into certain kinds of classes. In the discussion that came afterwards, he said, “Isn’t that the same as in mathematics, Dr. Eisenhart?” Dr. Eisenhart was the dean of the graduate school and a great professor of mathematics. He was also very clever. He said, “I’d like to know what Dick Feynman thinks about it in reference to theoretical physics.” He was always putting me on in this kind of situation. I got up and said, “Yes, it’s very closely related. In theoretical physics, the analog of the word is the mathematical formula, the analog of the structure of the poem is the interrelationship of the theoretical bling-bling with the so-andso”–and I went through the whole thing, making a perfect analogy. The speaker’s eyes were _beaming_ with happiness. Then I said, “It seems to me that no matter _what_ you say about poetry, I could find a way of making up an analog with _any_ subject, just as I did for theoretical physics. I don’t consider such analogs meaningful.

Einstein, twenty-six years old, only three years away from crude privation, still a patent examiner, published in the Annalen der Physik in 1905 five papers on entirely different subjects. Three of them were among the greatest in the history of physics. One, very simple, gave the quantum explanation of the photoelectric effect—it was this work for which, sixteen years later, he was awarded the Nobel prize. Another dealt with the phenomenon of Brownian motion, the apparently erratic movement of tiny particles suspended in a liquid: Einstein showed that these movements satisfied a clear statistical law. This was like a conjuring trick, easy when explained: before it, decent scientists could still doubt the concrete existence of atoms and molecules: this paper was as near to a direct proof of their concreteness as a theoretician could give. The third paper was the special theory of relativity, which quietly amalgamated space, time, and matter into one fundamental unity. This last paper contains no references and quotes to authority. All of them are written in a style unlike any other theoretical physicist's. They contain very little mathematics. There is a good deal of verbal commentary. The conclusions, the bizarre conclusions, emerge as though with the greatest of ease: the reasoning is unbreakable. It looks as though he had reached the conclusions by pure thought, unaided, without listening to the opinions of others. To a surprisingly large extent, that is precisely what he had done.

understood the infamous spiritual terror which this movement exerts, particularly on the bourgeoisie, which is neither morally nor mentally equal to such attacks; at a given sign it unleashes a veritable barrage of lies and slanders against whatever adversary seems most dangerous, until the nerves of the attacked persons break down… This is a tactic based on precise calculation of all human weaknesses, and its result will lead to success with almost mathematical certainty… I achieved an equal understanding of the importance of physical terror toward the individual and the masses… For while in the ranks of their supporters the victory achieved seems a triumph of the justice of their own cause, the defeated adversary in most cases despairs of the success of any further resistance.49 No more precise analysis of Nazi tactics, as Hitler was eventually to develop them, was ever written.

Sexuality is the only drive that is in itself hindered, perverted: simultaneously insufficient and excessive, with the excess as the form of appearance of the lack. On the one hand, sexuality is characterized by the universal capacity to provide the metaphorical meaning or innuendo of any activity or object–any element, including the most abstract reflection can be experienced as ‘alluding to that‘ (suffice it to recall the proverbial example of the adolescent who, in order to forget his sexual obsessions, takes refuge in pure mathematics and physics–whatever he does here again reminds him of ‘that’: how much volume is needed to fill out an empty cylinder? How much energy is discharged when two bodies collide?…). The universal surplus–this capacity of sexuality to overflow the entire field of human experience…is not the sign of preponderance. Rather, it is the sign of a certain structural faultiness: sexuality strives outwards and overflows the adjoining domains precisely because it cannot find satisfaction in itself, because it never attains its goal.

Q. Would you repeat, Dr. Seldon, your thoughts concerning the future of Trantor? A. I have said, and I say again, that Trantor will lie in ruins within the next three centuries. Q. You do not consider your statement a disloyal one? A. No, sir. Scientific truth is beyond loyalty and disloyalty." Q. You are sure that your statement represents scientific truth? A. I am. Q. On what basis? A. On the basis of the mathematics of psychohistory. Q. Can you prove that this mathematics is valid? A. Only to another mathematician. Q. ( with a smile) Your claim then is that your truth is of so esoteric a nature that it is beyond the understanding of a plain man. It seems to me that truth should be clearer than that, less mysterious, more open to the mind. A. It presents no difficulties to some minds. The physics of energy transfer, which we know as thermodynamics, has been clear and true through all the history of man since the mythical ages, yet there may be people present who would find it impossible to design a power engine. People of high intelligence, too. I doubt if the learned Commissioners— At this point, one of the Commissioners leaned toward the Advocate. His words were not heard but the hissing of the voice carried a certain asperity. The Advocate flushed and interrupted Seldon. Q. We are not here to listen to speeches, Dr. Seldon. Let us assume that you have made your point. Let me suggest to you that your predictions of disaster might be intended to destroy public confidence in the Imperial Government for purposes of your own! A. That is not so. Q. Let me suggest that you intend to claim that a period of time preceding the so-called ruin of Trantor will be filled with unrest of various types. A. That is correct. Q. And that by the mere prediction thereof, you hope to bring it about, and to have then an army of a hundred thousand available. A. In the first place, that is not so. And if it were, investigation will show you that barely ten thousand are men of military age, and none of these has training in arms. Q. Are you acting as an agent for another? A. I am not in the pay of any man, Mr. Advocate. Q. You are entirely disinterested? You are serving science? A. I am.

Of course, I’ve only brought up two examples. Other universal laws of physics have been used as weapons as well, though we don’t know all of them. It’s very possible that every law of physics has been weaponized. It’s possible that in some parts of the universe, even … Forget it, I don’t even believe that.” “What were you going to say?” “The foundation of mathematics.” Cheng Xin tried to imagine it, but it was simply impossible. “That’s … madness.” Then she asked, “Will the universe turn into a war ruin? Or, maybe it’s more accurate to ask: Will the laws of physics turn into war ruins?” “Maybe they already are.… The physicists and cosmologists of the new world are focused on trying to recover the original appearance of the universe before the wars more than ten billion years ago. They’ve already constructed a fairly clear theoretical model describing the pre-war universe. That was a really lovely time, when the universe itself was a Garden of Eden. Of course, the beauty could only be described mathematically. We can’t picture it: Our brains don’t have enough dimensions.” Cheng Xin thought back to the conversation with the Ring again. Did you build this four-dimensional fragment? You told me that you came from the sea. Did you build the sea? “You are saying that the universe of the Edenic Age was four-dimensional, and that the speed of light was much higher?” “No, not at all. The universe of the Edenic Age was ten-dimensional. The speed of light back then wasn’t only much higher—rather, it was close to infinity. Light back then was capable of action at a distance, and could go from one end of the cosmos to the other within a Planck time.… If you had been to four-dimensional space, you would have some vague hint of how beautiful that ten-dimensional Garden must have been.” “You’re saying—” “I’m not saying anything.” Yifan seemed to have awakened from a dream. “We’ve only seen small hints; everything else is just guessing. You should treat it as a guess, just a dark myth we’ve made up.” But Cheng Xin continued to follow the course of the discussion taken so far. “—that during the wars after the Edenic Age, one dimension after another was imprisoned from the macroscopic into the microscopic, and the speed of light was reduced again and again.…” “As I said, I’m not saying anything, just guessing.” Yifan’s voice grew softer. “But no one knows if the truth is even darker than our guesses.… We are certain of only one thing: The universe is dying.” The

Bradley Headstone, in his decent black coat and waistcoat, and decent white shirt, and decent formal black tie, and decent pantaloons of pepper and salt, with his decent silver watch in his pocket and its decent hair-guard round his neck, looked a thoroughly decent young man of six-and-twenty. He was never seen in any other dress, and yet there was a certain stiffness in his manner of wearing this, as if there were a want of adaptation between him and it, recalling some mechanics in their holiday clothes. He had acquired mechanically a great store of teacher's knowledge. He could do mental arithmetic mechanically, sing at sight mechanically, blow various wind instruments mechanically, even play the great church organ mechanically. From his early childhood up, his mind had been a place of mechanical stowage. The arrangement of his wholesale warehouse, so that it might be always ready to meet the demands of retail dealers history here, geography there, astronomy to the right, political economy to the left—natural history, the physical sciences, figures, music, the lower mathematics, and what not, all in their several places—this care had imparted to his countenance a look of care; while the habit of questioning and being questioned had given him a suspicious manner, or a manner that would be better described as one of lying in wait. There was a kind of settled trouble in the face. It was the face belonging to a naturally slow or inattentive intellect that had toiled hard to get what it had won, and that had to hold it now that it was gotten. He always seemed to be uneasy lest anything should be missing from his mental warehouse, and taking stock to assure himself.

I grew up in Brooklyn, New York.. a city neighborhood that included houses, lampposts, walls, and bushes. But with an early bedtime in the winter, I could look out my window and see the stars, and the stars were not like anything else in my neighborhood. [At age 5] I didn’t know what they were. [At age 9] my mother … said to me, “You have a library card now, and you know how to read. Take the streetcar to the library and get a book on stars.” … I stepped up to the big librarian and asked for a book on stars. … I sat down and found out the answer, which was something really stunning. I found out that the stars are glowing balls of gas. I also found out that the Sun is a star but really close and that the stars are all suns except really far away I didn’t know any physics or mathematics at that time, but I could imagine how far you’d have to move the Sun away from us till it was only as bright as a star. It was in that library, reading that book, that the scale of the universe opened up to me. There was something beautiful about it. At that young age, I already knew that I’d be very happy if I could devote my life to finding out more about the stars and the planets that go around them. And it’s been my great good fortune to do just that.

But what is the use of the humanities as such? Admittedly they are not practical, and admittedly they concern themselves with the past. Why, it may be asked, should we engage in impractical investigations, and why should we be interested in the past? The answer to the first question is: because we are interested in reality. Both the humanities and the natural sciences, as well as mathematics and philosophy, have the impractical outlook of what the ancients called vita contemplativa as opposed to vita activa. But is the contemplative life less real or, to be more precise, is its contribution to what we call reality less important, than that of the active life? The man who takes a paper dollar in exchange for twenty-five apples commits an act of faith, and subjects himself to a theoretical doctrine, as did the mediaeval man who paid for indulgence. The man who is run over by an automobile is run over by mathematics, physics and chemistry. For he who leads the contemplative life cannot help influencing the active, just as he cannot prevent the active life from influencing his thought. Philosophical and psychological theories, historical doctrines and all sorts of speculations and discoveries, have changed, and keep changing, the lives of countless millions. Even he who merely transmits knowledge or learning participates, in his modest way, in the process of shaping reality - of which fact the enemies of humanism are perhaps more keenly aware than its friends. It is impossible to conceive of our world in terms of action alone. Only in God is there a "Coincidence of Act and Thought" as the scholastics put it. Our reality can only be understood as an interpenetration of these two.

Evolution endowed us with intuition only for those aspects of physics that had survival value for our distant ancestors, such as the parabolic orbits of flying rocks (explaining our penchant for baseball). A cavewoman thinking too hard about what matter is ultimately made of might fail to notice the tiger sneaking up behind and get cleaned right out of the gene pool. Darwin’s theory thus makes the testable prediction that whenever we use technology to glimpse reality beyond the human scale, our evolved intuition should break down. We’ve repeatedly tested this prediction, and the results overwhelmingly support Darwin. At high speeds, Einstein realized that time slows down, and curmudgeons on the Swedish Nobel committee found this so weird that they refused to give him the Nobel Prize for his relativity theory. At low temperatures, liquid helium can flow upward. At high temperatures, colliding particles change identity; to me, an electron colliding with a positron and turning into a Z-boson feels about as intuitive as two colliding cars turning into a cruise ship. On microscopic scales, particles schizophrenically appear in two places at once, leading to the quantum conundrums mentioned above. On astronomically large scales… weirdness strikes again: if you intuitively understand all aspects of black holes [then you] should immediately put down this book and publish your findings before someone scoops you on the Nobel Prize for quantum gravity… [also,] the leading theory for what happened [in the early universe] suggests that space isn’t merely really really big, but actually infinite, containing infinitely many exact copies of you, and even more near-copies living out every possible variant of your life in two different types of parallel universes.