Quotes of All Topics . Occasions . Authors
Algebra reverses the relative importance of the factors in ordinary language. It is essentially a written language, and it endeavors to exemplify in its written structures the patterns which it is its purpose to convey. The pattern of the marks on paper is a particular instance of the pattern to be conveyed to thought. The algebraic method is our best approach to the expression of necessity, by reason of its reduction of accident to the ghostlike character of the real variable.
What is necessary for 'the very existence of science,' and what the characteristics of nature are, are not to be determined by pompous preconditions, they are determined always by the material with which we work, by nature herself. We look, and we see what we find, and we cannot say ahead of time successfully what it is going to look like. ... It is necessary for the very existence of science that minds exist which do not allow that nature must satisfy some preconceived conditions.
Let me say something at the outset. The questions that have been asked so far in this debate illustrate why the American people don't trust the media. This is not a cage match. And, you look at the questions - "Donald Trump, are you a comic-book villain?" "Ben Carson, can you do math?" "John Kasich, will you insult two people over here?" "Marco Rubio, why don't you resign?" "Jeb Bush, why have your numbers fallen?" How about talking about the substantive issues the people care about?
I remember one occasion when I tried to add a little seasoning to a review, but I wasn't allowed to. The paper was by Dorothy Maharam, and it was a perfectly sound contribution to abstract measure theory. The domains of the underlying measures were not sets but elements of more general Boolean algebras, and their range consisted not of positive numbers but of certain abstract equivalence classes. My proposed first sentence was: "The author discusses valueless measures in pointless spaces."
A number of aspects of mathematics are not much talked about in contemporary histories of mathematics. We have in mind business and commerce, war, number mysticism, astrology, and religion. In some instances, writers, hoping to assert for mathematics a noble parentage and a pure scientific experience, have turned away their eyes. Histories have been eager to put the case for science, but the Handmaiden of the Sciences has lived a far more raffish and interesting life than her historians allow.
The Advantage is that mathematics is a field in which one's blunders tend to show very clearly and can be corrected or erased with a stroke of the pencil. It is a field which has often been compared with chess, but differs from the latter in that it is only one's best moments that count and not one's worst. A single inattention may lose a chess game, whereas a single successful approach to a problem, among many which have been relegated to the wastebasket, will make a mathematician's reputation.
The question you raise, 'How can such a formulation lead to computations?' doesn't bother me in the least! Throughout my whole life as a mathematician, the possibility of making explicit, elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on. Thus I never bothered about whether what would come out would be suitable for this or that, but just tried to understand - and it always turned out that understanding was all that mattered.
When I became the NASA administrator — or before I became the NASA administrator — Barack Obama charged me with three things. One was he wanted me to help re-inspire children to want to get into science and math, he wanted me to expand our international relationships, and third, and perhaps foremost, he wanted me to find a way to reach out to the Muslim world and engage much more with dominantly Muslim nations to help them feel good about their historic contribution to science … and math and engineering.
Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means. The geometer might be replaced by the "logic piano" imagined by Stanley Jevons; or, if you choose, a machine might be imagined where the assumptions were put in at one end, while the theorems came out at the other, like the legendary Chicago machine where the pigs go in alive and come out transformed into hams and sausages. No more than these machines need the mathematician know what he does.
The constructs of the mathematical mind are at the same time free and necessary. The individual mathematician feels free to define his notions and set up his axioms as he pleases. But the question is will he get his fellow mathematician interested in the constructs of his imagination. We cannot help the feeling that certain mathematical structures which have evolved through the combined efforts of the mathematical community bear the stamp of a necessity not affected by the accidents of their historical birth.
It is hard to communicate understanding because that is something you get by living with a problem for a long time. You study it, perhaps for years, you get the feel of it and it is in your bones. You can't convey that to anyone else. Having studied the problem for five years you may be able to present it in such a way that it would take somebody else less time to get to that point than it took you. But if they haven't struggled with the problem and seen all the pitfalls, then they haven't really understood it.
Math is like water. It has a lot of difficult theories, of course, but its basic logic is very simple. Just as water flows from high to low over the shortest possible distance, figures can only flow in one direction. You just have to keep your eye on them for the route to reveal itself. That’s all it takes. You don’t have to do a thing. Just concentrate your attention and keep your eyes open, and the figures make everything clear to you. In this whole, wide world, the only thing that treats me so kindly is math.
Mathematical economics is old enough to be respectable, but not all economists respect it. It has powerful supporters and impressive testimonials, yet many capable economists deny that mathematics, except as a shorthand or expository device, can be applied to economic reasoning. There have even been rumors that mathematics is used in economics (and in other social sciences) either for the deliberate purpose of mystification or to confer dignity upon common places as French was once used in diplomatic communications.
One of my first experiences with the space program was with the memorial that was built for the Challenger. When I was in 7th grade my entire class spent the entire school year preparing to launch a spaceship all together. We all had our different jobs that we had to learn how to do, we learned the math that you needed, we learned the practical skills that you needed, and I thought that was really cool. So I think that if you can take a tragedy and find the gold in it and turn it into something positive, that's great.
I was born into a working class Irish Catholic family at the brutal bottom of the Great Depression. I suppose this early imprinting and conditioning made me a life-long radical. My education was mostly scientific, majoring in electrical engineering and applied math. Those imprints made me a life-long rationalist. I have become increasingly skeptical about, or detached from, the assumption that radicalism and rationalism are the only correct perspectives with which to view life, but they remain my favorite perspectives.
The motto I have penned on my knuckles is that this is the best world we have--because it's the only world we have. It's the simplest math ever. However many terrible, rankling, peeve-inducing things may occur, there are always libraries. And rain-falling-on-sea. And the moon. And love. There is always something to look back on, with satisfaction, or forward to, with joy. There is always a moment where you boggle at the world--at yourself--at the whole, unlikely, precarious business of being alive--and then start laughing
Mathematics is not only real, but it is the only reality. That is that entire universe is made of matter, obviously. And matter is made of particles. It's made of electrons and neutrons and protons. So the entire universe is made out of particles. Now what are the particles made out of They're not made out of anything. The only thing you can say about the reality of an electron is to cite its mathematical properties. So there's a sense in which matter has completely dissolved and what is left is just a mathematical structure.
We have the industrial agriculturalists who try and make an argument that big is beautiful. But if you do the math, and particularly if you factor in that the price of oil is going to go through the roof, and so the price of transportation is going to go through the roof - making it abundantly clear that it's out of whack. The efficiency arguments are already crumbling, particularly if you actually include the cost of food pollution that these industries cause. They are tremendously unsustainable and tremendously inefficient.
There's spatial intelligence. they're, which end up being, people going into math or music. there's mechanical where you work well with your hands. There's an intelligence with language that would lead someone into writing. So it's not necessarily that you're six years old and you know you're going to be a lawyer Or you're going into tech startups or computers. It's something more elemental than that. It's that this is a skill, a way of thinking that comes naturally to me that I was drawn to and it was very clear in childhood.
The ways in which acquired savants show up are usually the same ways that congenital, or non-acquired, savant syndrome shows up. They tend to show up in the same areas: music, art, math, visual, spatial skills, and calendar calculating, although calendar calculating probably isn't quite as prominent in that group. They tend to show up quite quickly, or sort of explode on the scene and they then tend to have an obsessive sort of forceful quality about them in the same way as savant skills. So they tend to show up in the same ways.
No matter how clear things might become in the forest of story, there was never a clear-cut solution, as there was in math. The role of a story was, in the broadest terms, to transpose a problem into another form. Depending on the nature and the direction of the problem, a solution might be suggested in the narrative. Tengo would return to the real world with that solution in hand. It was like a piece of paper bearing the indecipherable text of a magic spell. It served no immediate practical purpose, but it contained a possibility.
Gradually, at various points in our childhoods, we discover different forms of conviction. There's the rock-hard certainty of personal experience ("I put my finger in the fire and it hurt,"), which is probably the earliest kind we learn. Then there's the logically convincing, which we probably come to first through maths, in the context of Pythagoras's theorem or something similar, and which, if we first encounter it at exactly the right moment, bursts on our minds like sunrise with the whole universe playing a great chord of C Major.
What's clarity like? Try to remember that funny feeling inside your head when you had math problems too difficult to solve: the faint buzzing noise in your ears, a heaviness on both sides of your skull, and the sensation that your brain is twitching inside your cranium like a fish on the beach. This is the opposite sensation of clarity. Yet for many people of my era, as they aged, this sensation became the dominant sensation of their lives. It was as though day-to-day twentieth century living had become an unsolvable algebraic equation.
We humans have a wide range of abilities that help us perceive and analyze mathematical content. We perceive abstract notions not just through seeing but also by hearing, by feeling, by our sense of body motion and position. Our geometric and spatial skills are highly trainable, just as in other high-performance activities. In mathematics we can use the modules of our minds in flexible ways - even metaphorically. A whole-mind approach to mathematical thinking is vastly more effective than the common approach that manipulates only symbols.
What is especially striking and remarkable is that in fundamental physics, a beautiful or elegant theory is more likely to be right than a theory that is inelegant. A theory appears to be beautiful or elegant (or simple, if you prefer) when it can be expressed concisely in terms of mathematics we already have. Symmetry exhibits the simplicity. The Foundamental Law is such that the different skins of the onion resemble one another and therefore the math for one skin allows you to express beautifully and simply the phenomenon of the next skin.
Most of the arts, as painting, sculpture, and music, have emotional appeal to the general public. This is because these arts can be experienced by some one or more of our senses. Such is not true of the art of mathematics; this art can be appreciated only by mathematicians, and to become a mathematician requires a long period of intensive training. The community of mathematicians is similar to an imaginary community of musical composers whose only satisfaction is obtained by the interchange among themselves of the musical scores they compose.
If a nonnegative quantity was so small that it is smaller than any given one, then it certainly could not be anything but zero. To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be. These supposed mysteries have rendered the calculus of the infinitely small quite suspect to many people. Those doubts that remain we shall thoroughly remove in the following pages, where we shall explain this calculus.
Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune; and there are probably more people really interested in mathematics than in music. Appearances suggest the contrary, but there are easy explanations. Music can be used to stimulate mass emotion, while mathematics cannot; and musical incapacity is recognized (no doubt rightly) as mildly discreditable, whereas most people are so frightened of the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity
One might suppose that reality must be held to at all costs. However, though that may be the moral thing to do, it is not necessarily the most useful thing to do. The Greeks themselves chose the ideal over the real in their geometry and demonstrated very well that far more could be achieved by consideration of abstract line and form than by a study of the real lines and forms of the world; the greater understanding achieved through abstraction could be applied most usefully to the very reality that was ignored in the process of gaining knowledge.
I don't want to convince you that mathematics is useful. It is, but utility is not the only criterion for value to humanity. Above all, I want to convince you that mathematics is beautiful, surprising, enjoyable, and interesting. In fact, mathematics is the closest that we humans get to true magic. How else to describe the patterns in our heads that - by some mysterious agency - capture patterns of the universe around us? Mathematics connects ideas that otherwise seem totally unrelated, revealing deep similarities that subsequently show up in nature.
When I went to Afghanistan in 2003, I walked into a war zone. Entire neighborhoods had been demolished. There were an overwhelming number of widows and orphans and people who had been physically and emotionally damaged; every 10-year-old kid on the street knew how to dismantle a Kalashnikov in under a minute. I would flip through math textbooks intended for third grade, fourth grade, and they would include word problems such as, "If you have 100 grenades and 20 mujahideen, how many grenades per mujahideen do you get?" War has infiltrated every facet of life.
I've dealt with numbers all my life, of course, and after a while you begin to feel that each number has a personality of its own. A twelve is very different from a thirteen, for example. Twelve is upright, conscientious, intelligent, whereas thirteen is a loner, a shady character who won't think twice about breaking the law to get what he wants. Eleven is tough, an outdoorsman who likes tramping through woods and scaling mountains; ten is rather simpleminded, a bland figure who always does what he's told; nine is deep and mystical, a Buddha of contemplation.
There was a blithe certainty that came from first comprehending the full Einstein field equations, arabesques of Greek letters clinging tenuously to the page, a gossamer web. They seemed insubstantial when you first saw them, a string of squiggles. Yet to follow the delicate tensors as they contracted, as the superscripts paired with subscripts, collapsing mathematically into concrete classical entities - potential; mass; forces vectoring in a curved geometry - that was a sublime experience. The iron fist of the real, inside the velvet glove of airy mathematics.
"It's better to give than to receive." Let me put this as elegantly as possible: "What a crock!" That statement is total hogwash, and in case you haven't noticed, it's usually propagated by people and groups who want you to give and them to receive. The whole idea is ludicrous. What's better, hot or cold, big or small, left or right, in or out? Giving and receiving are two sides of the same coin. Whoever decided that it is better to give than to receive was simply bad at math. For every giver their must be a receiver, and for every receiver there must be a giver.
The principle of science, the definition, almost, is the following: The test of all knowledge is experiment. Experiment is the sole judge of scientific "truth." But what is the source of knowledge? Where do the laws that are to be tested come from? Experiment, itself, helps to produce these laws, in the sense that it gives us hints. But also needed is imagination to create from these hints the great generalizations--to guess at the wonderful, simple, but very strange patterns beneath them all, and then to experiment to check again whether we have made the right guess.
Many people keep deploring the low level of formal education in the United states (as defined by, say, math grades). Yet these fail to realize that the new comes from here and gets imitated elsewhere. And it is not thanks to universities, which obviously claim a lot more credit than their accomplishments warrant. Like Britain in the Industrial Revolution, America's asset is, simply, risk taking and the use of optionality, this remarkable ability to engage in rational forms fo trial and error, with no comparative shame in failing again, starting again, and repeating failure.
The majority of the people of the world today are unsane, not insane, unsane meaning having been exposed to methods of evaluation that have long rendered obsolete, our language in the future will change to a saner language where we have no argument in it, 'can there be such a language?' there is, when engineers talk to each other, it's not subject to interpretation, they use math, they use descriptive systems, if I interpreted what another engineer said in the way I think he meant it: you couldn't build bridges, dams, power transmission lines. The language has to have meaning
Our teaching of mathematics revolves around a fundamental conflict. Rightly or wrongly, students are required to master a series of mathematical concepts and techniques, and anything that might divert them from doing so is deemed unnecessary. Putting mathematics into its cultural context, explaining what is has done for humanity, telling the story of its historical development, or pointing out the wealth of unsolved problems or even the existence of topics that do not make it into school textbooks leaves less time to prepare for the exam. So most of these things aren't discussed.
Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven't. You get the feeling that the result you have discovered is forever, because it's concrete. Other branches of mathematics are not so clear-cut. Functional analysis of infinite-dimensional spaces is never fully convincing; you don't get a feeling of having done an honest day's work. Don't get the wrong idea - combinatorics is not just putting balls into boxes. Counting finite sets can be a highbrow undertaking, with sophisticated techniques.
I like the fact that they still run substantive pieces. I'm not sure I like the pieces, but it's nice that they do that. Anyway, it was always sort of ridiculous, me having anything to do with the youth culture, but now that I'm in my 50s, it's extra-double-ridiculous. They were losing interest in me, and I was losing interest in them. When I went to renegotiate my contract at Rolling Stone, I kind of halfheartedly asked if I could do half the work for half the money, and they asked if I could do two-thirds of the work for half the money. I ran that by my agent, since he can do math.
How can you shorten the subject? That stern struggle with the multiplication table, for many people not yet ended in victory, how can you make it less? Square root, as obdurate as a hardwood stump in a pasturenothing but years of effort can extract it. You can't hurry the process. Or pass from arithmetic to algebra; you can't shoulder your way past quadratic equations or ripple through the binomial theorem. Instead, the other way; your feet are impeded in the tangled growth, your pace slackens, you sink and fall somewhere near the binomial theorem with the calculus in sight on the horizon.
The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such a discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful.
