Quotes of All Topics . Occasions . Authors
Without doubt, if we are to go back to that ultimate, integral experience, unwarped by the sophistications of theory, that experience whose elucidation is the final aim of philosophy, the flux of things is one ultimate generalization around which we must weave our philosophical system.
The imaginary expression √(-a) and the negative expression -b, have this resemblance, that either of them occurring as the solution of a problem indicates some inconsistency or absurdity. As far as real meaning is concerned, both are imaginary, since 0 - a is as inconceivable as √(-a).
[Unbelievers] think they have made great efforts to get at the truth when they have spent a few hours in reading some book out of Holy Scripture, and have questioned some cleric about the truths of the faith. After that, they boast that they have searched in books and among men in vain.
If a man loves a woman for her beauty, does he love her? No; for the smallpox, which destroys her beauty without killing her, causes his love to cease. And if any one loves me for my judgment or my memory, does he really love me? No; for I can lose these qualities without ceasing to be.
En un mot, l'homme conna|"t qu'il est mise rable: il est donc mise rable, puisqu'il l'est; mais il est bien grand, puisqu'il le conna|"t. In one word, man knows that he is miserable and therefore he is miserable because he knows it; but he is also worthy, because he knows his condition.
The higher arithmetic presents us with an inexhaustible store of interesting truths - of truths, too, which are not isolated, but stand in a close internal connection, and between which, as our knowledge increases, we are continually discovering new and sometimes wholly unexpected ties.
Michael Harris opens the doors and gently guides you into a magic world. Once inside, you can't help but feel mesmerized, eager to see how deep the rabbit hole goes. And no wonder: a major thinker of our time is talking to you about math and so much more, like you've never heard before.
THE COMPUTER IS JUST AN INSTRUMENT for doing faster what we already know how to do slower. All pretensions to computer intelligence and paradise-tomorrow promises should be toned down before the public turns away in disgust. And if that should happen, our civilization might not survive.
Questions that pertain to the foundations of mathematics, although treated by many in recent times, still lack a satisfactory solution. Ambiguity of language is philosophy's main source of problems. That is why it is of the utmost importance to examine attentively the very words we use.
We must learn a new modesty. We have stormed the heavens, but succeeded only in building fog upon fog, a mist which will not support anybody who earnestly desires to stand upon it. What is valid seems so insignificant that it may be seriously doubted whether anlaysis is at all possible.
Games are among the most interesting creations of the human mind, and the analysis of their structure is full of adventure and surprises. Unfortunately there is never a lack of mathematicians for the job of transforming delectable ingredients into a dish that tastes like a damp blanket.
There are... scientific works - star catalogues, for example - which are not art; but the theoretical structures of Gauss, Einstein, or Maxwell are original, individual, "very personal" responses and expressions of exactly the same kind as the creative works of Beethoven or Dostoievski.
Relatively, there are many scientists who believe in God. And in Oxford, where I am the Professor, there are more professors like me, who believe in God, than you think. There are not dozens of them, but they are there, and in Cambridge too, and elsewhere. We are not in a tiny minority.
We are taught all this [the motion of the earth on its axis and around the sun] by the order of succession, in which those phenomena (various planetary happenings) follow each other, and by the harmony of the world, if we will only, as the saying goes, look at the matter with both eyes.
With an absurd oversimplification, the 'invention' of the calculus is sometimes ascribed to two men, Newton and Leibniz. In reality, the calculus is the product of a long evolution that was neither initiated nor terminated by Newton and Leibniz, but in which both played a decisive part.
The problem with merely writing so that you can be understood is that the wrong people, in advancing their agendas, are only too ready to misunderstand you. Writing so that you cannot be misunderstood anticipates and preempts those who would willfully distort what you are trying to say.
The theory of numbers, more than any other branch of mathematics, began by being an experimental science. Its most famous theorems have all been conjectured, sometimes a hundred years or more before they were proved; and they have been suggested by the evidence of a mass of computations.
The introduction of numbers as coordinates by reference to the particular division scheme of the open one-dimensional continuum is an act of violence whose only practical vindication is the special calculatory manageability of the ordinary number continuum with its four basic operations.
Not only the phenomena of the others followed from this, but also it so bound together both the order and magnitude of all the planets and the spheres and the heaven itself, that in no single part could one thing be altered without confusion among the other parts and in all the universe.
The study of mathematics is apt to commence in disappointment... We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet's father, this great science eludes the efforts of our mental weapons to grasp it.
[Mathematics is] purely intellectual, a pure theory of forms, which has for its objects not the combination of quantities or their images, the numbers, but things of thought to which there could correspond effective objects or relations, even though such a correspondence is not necessary.
It is strange that we know so little about the properties of numbers. They are our handiwork, yet they baffle us; we can fathom only a few of their intricacies. Having defined their attributes and prescribed their behavior, we are hard pressed to perceive the implications of our formulas.
The word "mathematics" is a Greek word and, by origin, it means "something that has been learned or understood," or perhaps "acquired knowledge," or perhaps even, somewhat against grammar, "acquirable knowledge," that is, "learnable knowledge," that is, "knowledge acquirable by learning."
Research sometimes feels like an ongoing TV series in which some amazing revelations have already been made, but there are still plenty of cliff-hangers and unresolved plotlines that you want to see resolved. But unlike TV, we have to do the work ourselves to figure out what happens next.
The fundamental claim of intelligent design is straightforward and easily intelligible: namely, there are natural systems that cannot be adequately explained in terms of undirected natural forces and that exhibit features which in any other circumstance we would attribute to intelligence.
The wrong people will do everything in their power to guarantee that the wrong political climate will continue. It seems, then, that the wrong people ensure the wrong political climate and the wrong political climate ensures the wrong people. How then to break free of this vicious circle?
Our rate of progress is such that an individual human being, of ordinary length of life, will be called on to face novel situations which find no parallel in his past. The fixed person, for the fixed duties, who, in older societies was such a godsend, in the future will be a public danger.
Forging differs from hoaxing, inasmuch as in the later the deceit is intended to last for a time, and then be discovered, to the ridicule of those who have credited it; whereas the forger is one who, wishing to acquire a reputation for science, records observations which he has never made.
Take the rose—most people think it very beautiful: I don’t care for It at all. I prefer the cactus, for the simple reason that it has a more interesting personality. It has wonderfully adapted itself to its surroundings! It is the best illustration of the theory of evolution in plant life.
As a mathematician, von Neumann was quick, brilliant, efficient, and enormously broad in scientific interests beyond mathematics itself. He knew his technical abilities; his virtuosity in following complicated reasoning and his insights were supreme; yet he lacked absolute self confidence.
What a mathematical proof actually does is show that certain conclusions, such as the irrationality of , follow from certain premises, such as the principle of mathematical induction. The validity of these premises is an entirely independent matter which can safely be left to philosophers.
A growing number of respectable scientists are defecting from the evolutionist camp.....moreover, for the most part these "experts" have abandoned Darwinism, not on the basis of religious faith or biblical persuasions, but on strictly scientific grounds, and in some instances, regretfully.
Let man reawake and consider what he is compared with the reality of things; regard himself lost in this remote corner of Nature; and from the tiny cell where he lodges, to wit the Universe, weigh at their true worth earth, kingdoms, towns, himself. What is a man face to face with infinity?
Sin2 φ is odious to me, even though Laplace made use of it; should it be feared that sin2 φ might become ambiguous, which would perhaps never occur, or at most very rarely when speaking of sin(φ2), well then, let us write (sin φ)2, but not sin2 φ, which by analogy should signify sin (sin φ)
There are very few theorems in advanced analysis which have been demonstrated in a logically tenable manner. Everywhere one finds this miserable way of concluding from the special to the general and it is extremely peculiar that such a procedure has led to so few of the so-called paradoxes.
In a very real sense, we are shipwrecked passengers on a doomed planet. Yet, even in a shipwreck, human decencies and human values do not necessarily vanish, and we must make the most of them. We shall go down, but let it be in a manner to which we may look forward as worthy of our dignity.
Every intellectual revolution which has ever stirred humanity into greatness has been a passionate protest against inert ideas. Then, alas, with pathetic ignorance of human psychology, it has proceeded by some educational scheme to bind humanity afresh with inert ideas of its own fashioning.
I shall never flaunt the little learning that I have acquired through the care and help my father has given me. If I have learned anything, it is only because he took care to teach me. Had he not taken upon himself the trouble of instructing me, I would be as ignorant as many other children.
Those who are most sensitive about "politically incorrect" terminology are not the average black ghetto-dweller, Asian immigrant, abused woman or disabled person, but a minority of activists, many of whom do not even belong to any "oppressed" group but come from privileged strata of society.
But in the prevalent discussion of classes, there are illegitimate transitions to the notions of a 'nexus' and of a 'proposition'. The appeal to a class to perform the services of a proper entity is exactly analogous to an appeal to an imaginary terrier to kill a real rat. Process and Reality
I carried this problem around in my head basically the whole time. I would wake up with it first thing in the morning, I would be thinking about it all day, and I would be thinking about it when I went to sleep. Without distraction I would have the same thing going round and round in my mind.
What a chimaera then is man, what a novelty, what a monster, what chaos, what a subject of contradiction, what a prodigy! Judge of all things, yet an imbecile earthworm; depository of truth, yet a sewer of uncertainty and error; pride and refuse of the universe. Who shall resolve this tangle?
Certain functions appear so often that it is convenient to give them names. These are collectively called special functions. There are many examples and no single way of looking at them can illuminate all examples or even all the important properties of a single example of a special function.
In a sense, knowledge shrinks as wisdom grows, for details are swallowed up in principles. The details for knowledge which are important, will be picked up ad hoc in each avocation of life, but the habit of the active utilization of well-understood principles is the final possession of WISDOM.
The way in which the persecution of Galileo has been remembered is a tribute to the quiet commencement of the most intimate change in outlook which the human race had yet encountered. Since a babe was born in a manger, it may be doubted whether so great a thing has happened with so little stir
There are two types of mind . . . the mathematical, and what might be called the intuitive. The former arrives at its views slowly, but they are firm and rigid; the latter is endowed with greater flexibility and applies itself simultaneously to the diverse lovable parts of that which it loves.
That dog is mine said those poor children; that place in the sun is mine; such is the beginning and type of usurpation throughout the earth. [Fr., Ce chien est a moi, disaient ces pauvres enfants; c'est la ma place au soleil. Voila le commencement et l'image de l'usurpation de toute la terre.]
You gave me health that I might serve you; and so often I failed to use my good health in your service. Now you send me sickness in order to correct me Grant that, having ignored the things of spirit when my body was vigorous, I may now enjoy spiritual sweetness while my body groans with pain.
...as our friend Zach has often noted, in our days those who do the best for astronomy are not the salaried university professors, but so-called dillettanti, physicians, jurists, and so forth.Lamenting the fragmentary time left to a professor has remaining after fulfilling his teaching duties.
In a way, math isn't the art of answering mathematical questions, it is the art of asking the right questions, the questions that give you insight, the ones that lead you in interesting directions, the ones that connect with lots of other interesting questions -the ones with beautiful answers.