Quotes of All Topics . Occasions . Authors
The young earth-solution to reconciling the order of creation with natural history makes good exegetical and theological sense. Indeed, the overwhelming consensus of theologians up through the Reformation held to this view. I myself would adopt it in a heartbeat except that nature seems to present such strong evidence against it.
Ninety percent of our lives is governed by emotion. Our brains merely register and act upon what is telegraphed to them by our bodily experience. Intellect is to emotion as our clothes are to our bodies; we could not very well have civilized life without clothes, but we would be in a poor way if we had only clothes without bodies.
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.
Remember that [scientific thought] is the guide of action; that the truth which it arrives at is not that which we can ideally contemplate without error, but that which we may act upon without fear; and you cannot fail to see that scientific thought is not an accompaniment or condition of human progress, but human progress itself.
Thus you see, most noble Sir, how this type of solution to the Königsberg bridge problem bears little relationship to mathematics, and I do not understand why you expect a mathematician to produce it, rather than anyone else, for the solution is based on reason alone, and its discovery does not depend on any mathematical principle.
I think it is said that Gauss had ten different proofs for the law of quadratic reciprocity. Any good theorem should have several proofs, the more the better. For two reasons: usually, different proofs have different strengths and weaknesses, and they generalise in different directions - they are not just repetitions of each other.
Every mathematician worthy of the name has experienced . . . the state of lucid exaltation in which one thought succeeds another as if miraculously . . . this feeling may last for hours at a time, even for days. Once you have experienced it, you are eager to repeat it but unable to do it at will, unless perhaps by dogged work. . . .
It was a very big gamble. I lost my job in France, I received a job in which was extremely uncertain, how long would IBM be interested in research, but the gamble was taken and very shortly afterwards, I had this extraordinary fortune of stopping at Harvard to do a lecture and learning about the price variation in just the right way.
I set out to show that there exists single creative activity,which is displayed alike in the arts and in the sciences.It is wrong to think of science as a mechanical record of facts, and it is wrong to think of the arts as remote and private fancies. What makes each human, what makes them universal, is the stamp of the creative mind.
What makes it possible to learn advanced math fairly quickly is that the human brain is capable of learning to follow a given set of rules without understanding them, and apply them in an intelligent and useful fashion. Given sufficient practice, the brain eventually discovers (or creates) meaning in what began as a meaningless game.
It is of dangerous consequence to represent to man how near he is to the level of beasts, without showing him at the same time his greatness. It is likewise dangerous to let him see his greatness without his meanness. It is more dangerous yet to leave him ignorant of either; but very beneficial that he should be made sensible of both.
Human life is thus only a perpetual illusion; men deceive and flatter each other. No one speaks of us in our presence as he does of us in our absence. Human society is founded on mutual deceit; few friendships would endure if each knew what his friend said of him in his absence, although he then spoke in sincerity and without passion.
Absolute space, that is to say, the mark to which it would be necessary to refer the earth to know whether it really moves, has no objective existence.... The two propositions: "The earth turns round" and "it is more convenient to suppose the earth turns round" have the same meaning; there is nothing more in the one than in the other.
We often treat children as if they're not very competent to do anything on their own. So we make them stop learning in a natural way - by exploring. Logo [the computer programming language ] allows them to find their way around the computer, as they would find their way around the house, uncontaminated by the bureaucracies of schools.
Some mathematics problems look simple, and you try them for a year or so, and then you try them for a hundred years, and it turns out that they're extremely hard to solve. There's no reason why these problems shouldn't be easy, and yet they turn out to be extremely intricate. Fermat's Last Theorem is the most beautiful example of this.
Scientific theories need reconstruction every now and then. If they didn't need reconstruction they would be facts, not theories. The more facts we know, the less radical become the changes in our theories. Hence they are becoming more and more constant. But take the theory of gravitation; it has not been changed in four hundred years.
No wonder that Churchill described this effort [the British codebreakers working at Bletchley Park] as "Britian"s secret weapon," a weapon far more effective than the buzz bombs and the rockets that Werner von Braun designed for a German victory, a weapon absolutely decisive, in the judgement of many, in winning the war for the Allies.
It is a profoundly erroneous truism, repeated by all copy books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.
We need not have the loftiest mind to understand that here is no lasting and real satisfaction, that our pleasures are only vanity, that our evils are infinite, and, lastly, that death, which threatens us every moment, must infallibly place us within a few years under the dreadful necessity of being forever either annihilated or unhappy.
Chance never writ a legible book; chance never built a fair house; chance never drew a neat picture; it never did any of these things, nor ever will; nor can it be without absurdity supposed able to do them; which yet are works very gross and rude, very easy and feasible, as it were, in comparison to the production of a flower or a tree.
And so matching by itself is incapable of creating an art of reckoning. Without our ability to arrange things in ordered succession little progress could have been made. Correspondence and succession, the two principles that permeate all mathematics - nay, all realms of exact thought - are woven into the very fabric of our number system.
The heart has its reasons of which reason knows nothing. We feel it in a thousand things. I say that the heart naturally loves the Universal Being, and naturally loves itself; and it gives itself to one or the other, and hardens itself against one or the other, as it chooses...it is the heart that feels God, not the reason; this is faith.
In the last two months I have been very busy with my own mathematical speculations, which have cost me much time, without my having reached my original goal. Again and again I was enticed by the frequently interesting prospects from one direction to the other, sometimes even by will-o'-the-wisps, as is not rare in mathematic speculations.
The interaction between math and physics is a two-way process, with each of the two subjects drawing from and inspiring the other. At different times, one of them may take the lead in developing a particular idea, only to yield to the other subject as focus shifts. But altogether, the two interact in a virtuous circle of mutual influence.
We have only to see a few letters of the alphabet spelling our name in the sand to recognize at once the work of an intelligent agent. How much more likely, then is the existence of an intelligent Creator behind human DNA, the colossal biological database that contains no fewer than 3.5 billion "letters the longest "word" yet discovered?"
My basic idea is that programming is the most powerful medium of developing the sophisticated and rigorous thinking needed for mathematics, for grammar, for physics, for statistics, for all the "hard" subjects.... In short, I believe more than ever that programming should be a key part of the intellectual development of people growing up.
On the occasions when I have pondered over men's various activities, the dangers and worries they are exposed to at Court or at war, from which so many quarrels, passions, risky, often ill-conceived actions and so on are born, I have often said that man's unhappiness springs from one thing alone, his incapacity to stay quietly in one room.
Our tissues change as we live: the food we eat and the air we breathe become flesh of our flesh and bone of our bone, and the momentary elements of our flesh and bone pass out of our body every day with our excreta. We are but whirlpools in a river of ever-flowing water. We are not stuff that abides, but patterns that perpetuate themselves
I'm not saying that atheists can't act morally or have moral knowledge. But when I ascribe virtue to an atheist, it's as a theist who sees the atheist as conforming to objective moral values. The atheist, by contrast, has no such basis for morality. And yet all moral judgments require a basis for morality, some standard of right and wrong.
The branches of mathematics are as various as the sciences to which they belong, and each subject of physical enquiry has its appropriate mathematics. In every form of material manifestation, there is a corresponding form of human thought, so that the human mind is as wide in its range of thought as the physical universe in which it thinks.
If we let ourselves believe that man began with divine grace, that he forfeited this by sin, and that he can be redeemed only by divine grace through the crucified Christ, then we shall find peace of mind never granted to philosophers. He who cannot believe is cursed, for he reveals by his unbelief that God has not chosen to give him grace.
I am now about to set seriously to work upon preparing for the press an account of my theory of Logic and Probabilities which in its present state I look upon as the most valuable if not the only valuable contribution that I have made or am likely to make to Science and the thing by which I would desire if at all to be remembered hereafter.
In the Middle Age, in Germany, if you wanted to learn addition and multiplication, you could go to any university. But if you wanted to learn division, you could only do it in one place, Heidelberg. This makes sense, since in my theory with Vladimir Retakh and Robert Wilson, addition and multiplications are cheap, but division is expensive.
Man is not the most majestic of the creatures; long before the mammals even, the dinosaurs were far more splendid. But he has what no other animal possesses: a jigsaw of faculties, which alone, over three thousand million years of life, made him creative. Every animal leaves traces of what he was. Man alone leaves traces of what he created.
The lock-step approach of algebra, geometry, and then more algebra (but rarely any statistics) is still dominant in U. S. schools, but hardly anywhere else. This fragmented approach yields effective mathematics education not for the many but for the few primarily those who are independently motivated and who will learn under any conditions.
Until very recently, most knowledge was inaccessible to people who couldn't read text. But this is changing. The computer opens up other channels of gaining knowledge. If someone is blind, we now have very good machines that will read to him. If someone can't recognize letters, he also will have access to knowledge through sound and images.
You want to get to the top of the cliff. But that's not what you focus on immediately. You focus on the next ledge just beyond your reach, because you need to do one clever thing to get up there. And then, once you get there, you do it again. A lot of this is rather boring and not very glamorous. But you can't jump cliffs in a single bound.
Through and through the world is infested with quantity: To talk sense is to talk quantities. It is no use saying the nation is large. . . . How large? It is no use saying the radium is scarce. . . . How scarce? You cannot evade quantity. You may fly to poetry and music, and quantity and number will face you in your rhythms and your octaves.
Epitaph on Newton: Nature and Nature's law lay hid in night: God said, "Let Newton be!," and all was light. [added by Sir John Collings Squire: It did not last: the Devil shouting "Ho. Let Einstein be," restored the status quo] [Aaron Hill's version: O'er Nature's laws God cast the veil of night, Out blaz'd a Newton's soul and all was light.
To write and speak correctly is certainly necessary; but it is not sufficient. A derivation correctly presented in the book or on the blackboard may be inaccessible and uninstructive, if the purpose of the successive steps is incomprehensible, if the reader or listener cannot understand how it was humanly possible to find such an argument....
The goal of a definition is to introduce a mathematical object. The goal of a theorem is to state some of its properties, or interrelations between various objects. The goal of a proof is to make such a statement convincing by presenting a reasoning subdivided into small steps each of which is justified as an "elementary" convincing argument.
...Those laws are within the grasp of the human mind. God wanted us to recognize them by creating us after his own image so that we could share in his own thoughts... and if piety allow us to say so, our understanding is in this respect of the same kind as the divine, at least as far as we are able to grasp something of it in our mortal life.
We are so far from knowing all the forces of nature and their various modes of action that it would be unworthy of the philosopher to deny phenomena simply because they are inexplicable at the present state of our knowledge. The more difficult it is to acknowledge their existence, the greater the care with which we must study these phenomena.
According to the doctrine of chance, you ought to put yourself to the trouble of searching for the truth; for if you die without worshiping the True Cause, you are lost. "But," say you, "if He had wished me to worship Him, He would have left me signs of His will." He has done so; but you neglect them. Seek them, therefore; it is well worth it.
One cannot inquire into the foundations and nature of mathematics without delving into the question of the operations by which the mathematical activity of the mind is conducted. If one failed to take that into account, then one would be left studying only the language in which mathematics is represented rather than the essence of mathematics.
What exactly is mathematics? Many have tried but nobody has really succeeded in defining mathematics; it is always something else. Roughly speaking, people know that it deals with numbers, figures, with relations, operations, and that its formal procedures involving axioms, proofs, lemmas, theorems have not changed since the time of Archimedes.
This book reminds me of James Gleick's Chaos. The ideas and stories in Loving and Hating Mathematics are timely, interesting, and sometimes even profound. The authors, writing for nonspecialists, take pains to explain technical ideas in nontechnical language, and the book should interest general readers as well as a large mathematical audience.
Certainly it is permitted to anyone to put forward whatever hypotheses he wishes, and to develop the logical consequences contained in those hypotheses. But in order that this work merit the name of Geometry, it is necessary that these hypotheses or postulates express the result of the more simple and elementary observations of physical figures.
Incredulity is not wisdom, but the worst kind of folly. It is folly, because it causes ignorance and mistake, with all the consequents of these; and it is very bad, as being accompanied with disingenuity, obstinacy, rudeness, uncharitableness, and the like bad dispositions; from which credulity itself, the other extreme sort of folly, is exempt.
Looking at numbers as groups of rocks may seem unusual, but actually it's as old as math itself. The word "calculate" reflects that legacy - it comes from the Latin word calculus, meaning a pebble used for counting. To enjoy working with numbers you don't have to be Einstein (German for "one stone"), but it might help to have rocks in your head.