Consider Quotes (71 quotes)
...That day in the account of creation, or those days that are numbers according to its recurrence, are beyond the experience and knowledge of us mortal earthbound men. And if we are able to make any effort towards an understanding of those days, we ought not to rush forward with an ill considered opinion, as if no other reasonable and plausible interpretation could be offered.
A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street.
A new era of ocean exploration can yield discoveries that will help inform everything from critical medical advances to sustainable forms of energy. Consider that AZT, an early treatment for HIV, is derived from a Caribbean reef sponge, or that a great deal of energy—from offshore wind, to OTEC (ocean thermal energy conservation), to wind and wave energy—is yet untapped in our oceans.
Always design a thing by considering it in its next larger context—a chair in a room, a room in a house, a house in an environment, an environment in a city plan.
An old French geometer used to say that a mathematical theory was never to be considered complete till you had made it so clear that you could explain it to the first man you met in the street.
Anyone who considers arithmetical methods of producing random digits is, of course, in the state of sin. For, as has been pointed out several times, there is no such thing as a random number—there are only methods to produce random numbers, and a strict arithmetic procedure of course is not such a method.
Art is usually considered to be not of the highest quality if the desired object is exhibited in the midst of unnecessary lumber.
Astronomy, as the science of cyclical motions, has nothing in common with Geology. But look at Astronomy where she has an analogy with Geology; consider our knowledge of the heavens as a palaetiological science;—as the study of a past condition, from which the present is derived by causes acting in time. Is there no evidence of a beginning, or of a progress?
But, as we consider the totality of similarly broad and fundamental aspects of life, we cannot defend division by two as a natural principle of objective order. Indeed, the ‘stuff’ of the universe often strikes our senses as complex and shaded continua, admittedly with faster and slower moments, and bigger and smaller steps, along the way. Nature does not dictate dualities, trinities, quarterings, or any ‘objective’ basis for human taxonomies; most of our chosen schemes, and our designated numbers of categories, record human choices from a cornucopia of possibilities offered by natural variation from place to place, and permitted by the flexibility of our mental capacities. How many seasons (if we wish to divide by seasons at all) does a year contain? How many stages shall we recognize in a human life?
By keenly confronting the enigmas that surround us, and by considering and analyzing the observations that I had made I ended up in the domain of mathematics.
Causality may be considered as a mode of perception by which we reduce our sense impressions to order.
Clarity about the aims and problems of socialism is of greatest significance in our age of transition. Since, under present circumstances, free and unhindered discussion of these problems has come under a powerful taboo, I consider the foundation of this magazine to be an important public service.
Consider a cow. A cow doesn’t have the problem-solving skill of a chimpanzee, which has discovered how to get termites out of the ground by putting a stick into a hole. Evolution has developed the brain’s ability to solve puzzles, and at the same time has produced in our brain a pleasure of solving problems.
Consider the plight of a scientist of my age. I graduated from the University of California at Berkeley in 1940. In the 41 years since then the amount of biological information has increased 16 fold; during these 4 decades my capacity to absorb new information has declined at an accelerating rate and now is at least 50% less than when I was a graduate student. If one defines ignorance as the ratio of what is available to be known to what is known, there seems no alternative to the conclusion that my ignorance is at least 25 times as extensive as it was when I got my bachelor’s degree. Although I am sure that my unfortunate condition comes as no surprise to my students and younger colleagues, I personally find it somewhat depressing. My depression is tempered, however, by the fact that all biologists, young or old, developing or senescing, face the same melancholy situation because of an interlocking set of circumstances.
Considering that, among all those who up to this time made discoveries in the sciences, it was the mathematicians alone who had been able to arrive at demonstrations—that is to say, at proofs certain and evident—I did not doubt that I should begin with the same truths that they have investigated, although I had looked for no other advantage from them than to accustom my mind to nourish itself upon truths and not to be satisfied with false reasons.
Dissection ... teaches us that the body of man is made up of certain kinds of material, so differing from each other in optical and other physical characters and so built up together as to give the body certain structural features. Chemical examination further teaches us that these kinds of material are composed of various chemical substances, a large number of which have this characteristic that they possess a considerable amount of potential energy capable of being set free, rendered actual, by oxidation or some other chemical change. Thus the body as a whole may, from a chemical point of view, be considered as a mass of various chemical substances, representing altogether a considerable capital of potential energy.
Equations are Expressions of Arithmetical Computation, and properly have no place in Geometry, except as far as Quantities truly Geometrical (that is, Lines, Surfaces, Solids, and Proportions) may be said to be some equal to others. Multiplications, Divisions, and such sort of Computations, are newly received into Geometry, and that unwarily, and contrary to the first Design of this Science. For whosoever considers the Construction of a Problem by a right Line and a Circle, found out by the first Geometricians, will easily perceive that Geometry was invented that we might expeditiously avoid, by drawing Lines, the Tediousness of Computation. Therefore these two Sciences ought not to be confounded. The Ancients did so industriously distinguish them from one another, that they never introduced Arithmetical Terms into Geometry. And the Moderns, by confounding both, have lost the Simplicity in which all the Elegance of Geometry consists. Wherefore that is Arithmetically more simple which is determined by the more simple Equation, but that is Geometrically more simple which is determined by the more simple drawing of Lines; and in Geometry, that ought to be reckoned best which is geometrically most simple.
Every investigator must before all things look upon himself as one who is summoned to serve on a jury. He has only to consider how far the statement of the case is complete and clearly set forth by the evidence. Then he draws his conclusion and gives his vote, whether it be that his opinion coincides with that of the foreman or not.
Firefly meteorites blazed against a dark background, and sometimes the lightning was frighteningly brilliant. Like a boy, I gazed open-mouthed at the fireworks, and suddenly, before my eyes, something magical occurred. A greenish radiance poured from Earth directly up to the station, a radiance resembling gigantic phosphorescent organ pipes, whose ends were glowing crimson, and overlapped by waves of swirling green mist.
“Consider yourself very lucky, Vladimir,” I said to myself, “to have watched the northern lights.”
“Consider yourself very lucky, Vladimir,” I said to myself, “to have watched the northern lights.”
He had read much, if one considers his long life; but his contemplation was much more than his reading. He was wont to say that if he had read as much as other men he should have known no more than other men.
He who thus considers things in their first growth and origin ... will obtain the clearest view of them.
I consider it extremely doubtful whether the happiness of the human race has been enhanced by the technical and industrial developments that followed in the wake of rapidly progressing natural science.
I consider it important, indeed urgently necessary, for intellectual workers to get together, both to protect their own economic status and, also, generally speaking, to secure their influence in the political field.
I have no right to consider anything a work of art to which I cannot react emotionally; and I have no right to look for the essential quality in anything that I have not felt to be a work of art.
If I were a medical man, I should prescribe a holiday to any patient who considered his work important.
If it is true as Whewell says, that the essence of the triumphs of Science and its progress consists in that it enables us to consider evident and necessary, views which our ancestors held to be unintelligible and were unable to comprehend, then the extension of the number concept to include the irrational, and we will at once add, the imaginary, is the greatest forward step which pure mathematics has ever taken.
In considering God's power, we must not look for a God of the Gaps, a god who is called in for those phenomena for which there is yet no scientific explanation.
In the 1860s, Pasteur not only applied his germ theory to create “Pasteurization,” rescuing France’s wine and vinegar industries, but also found both the cause and cure of silkworm disease, saving growers millions of dollars. When Napoleon asked the scientist why he had not legitimately profited by his findings, Pasteur replied: “In France scientists would consider they lowered themselves by doing so.”
In the social production of their life, men enter into definite relations that are indispensable and independent of their will; these relations of production correspond to a definite stage of development of their material forces of production. The sum total of these relations of production constitutes the economic structure of society - the real foundation, on which rises a legal and political superstructure and to which correspond definite forms of social consciousness. The mode of production of material life determines the social, political and intellectual life process in general. It is not the consciousness of men that determines their being, but, on the contrary, their social being that determines their consciousness. At a certain stage of their development, the material productive forces in society come in conflict with the existing relations of production, or - what is but a legal expression for the same thing - with the property relations within which they have been at work before. From forms of development of the productive forces these relations turn into fetters. Then begins an epoch of social revolution. With the change of the economic foundation the entire immense superstructure is more or less rapidly transformed. In considering such transformations a distinction should always be made between the material transformation of the economic conditions of production, which can be determined with the precision of natural science, and the legal, political, religious, aesthetic or philosophic - in short, ideological forms in which men become conscious of this conflict and fight it out. Just as our opinion of an individual is not based on what he thinks of himself, so we can not judge of such a period of transformation by its own consciousness; on the contrary, this consciousness must be explained rather from the contradictions of material life, from the existing conflict between the social productive forces and the relations of production. No social order ever disappears before all the productive forces for which there is room in it have been developed; and new, higher relations of production never appear before the material conditions of their existence have matured in the womb of the old society itself. Therefore, mankind always sets itself only such tasks as it can solve; since, looking at the matter more closely, we will always find that the task itself arises only when the material conditions necessary for its solution already exist or are at least in the process of formation. In broad outlines we can designate the Asiatic, the ancient, the feudal, and the modern bourgeois modes of production as so many progressive epochs in the economic formation of society. The bourgeois relations of production are the last antagonistic form of the social process of production - antagonistic not in the sense of individual antagonism, but of one arising from the social conditions of life of the individuals; at the same time the productive forces developing in the womb of bourgeois society create the material conditions for the solution of that antagonism. This social formation constitutes, therefore, the closing chapter of the prehistoric stage of human society.
It is one of the little ironies of our times that while the layman was being indoctrinated with the stereotype image of black holes as the ultimate cookie monsters, the professionals have been swinging round to the almost directly opposing view that black holes, like growing old, are really not so bad when you consider the alternative.
It is scientists, not sceptics, who are most willing to consider explanations that conflict with their own. And far from quashing dissent, it is the scientists, not the sceptics, who do most to acknowledge gaps in their studies and point out the limitations of their data—which is where sceptics get much of the mud they fling at the scientists. By contrast, the [sceptics] are not trying to build a theory of anything. They have set the bar much lower, and are happy muddying the waters.
It is the duty of every man of good will to strive steadfastly in his own little world to make this teaching of pure humanity a living force, so far as he can. If he makes an honest attempt in this direction without being crushed and trampled under foot by his contemporaries, he may consider himself and the community to which he belongs lucky.
Leibniz never married; he had considered it at the age of fifty; but the person he had in mind asked for time to reflect. This gave Leibniz time to reflect, too, and so he never married.
Let us ... consider the ovum [egg] as a physical system. Its potentialities are prodigious and one's first impulse is to expect that such vast potentialities would find expression in complexity of structure. But what do we find? The substance is clouded with particles, but these can be centrifuged away leaving it optically structureless but still capable of development.... On the surface of the egg there is a fine membrane, below it fluid of high viscosity, next fluid of relatively low viscosity, and within this the nucleus, which in the resting stage is simply a bag of fluid enclosed in a delicate membrane.... The egg's simplicity is not that of a machine or a crystal, but that of a nebula. Gathered into it are units relatively simple but capable by their combinations of forming a vast number of dynamical systems...
Let us weigh the gain and the loss in wagering that God is. Let us consider the two chances. If you gain, you gain all; if you lose, you lose nothing. Wager then without hesitation that He is.
Mathematic is either Pure or Mixed: To Pure Mathematic belong those sciences which handle Quantity entirely severed from matter and from axioms of natural philosophy. These are two, Geometry and Arithmetic; the one handling quantity continued, the other dissevered. … Mixed Mathematic has for its subject some axioms and parts of natural philosophy, and considers quantity in so far as it assists to explain, demonstrate and actuate these.
Mathematical reasoning is deductive in the sense that it is based upon definitions which, as far as the validity of the reasoning is concerned (apart from any existential import), needs only the test of self-consistency. Thus no external verification of definitions is required in mathematics, as long as it is considered merely as mathematics.
Negative facts when considered alone never teach us anything.
Neither you nor I nor anybody else knows what makes a mathematician tick. It is not a question of cleverness. I know many mathematicians who are far abler than I am, but they have not been so lucky. An illustration may be given by considering two miners. One may be an expert geologist, but he does not find the golden nuggets that the ignorant miner does.
No science is immune to the infection of politics and the corruption of power. … The time has come to consider how we might bring about a separation, as complete as possible, between Science and Government in all countries. I call this the disestablishment of science, in the same sense in which the churches have been disestablished and have become independent of the state.
Nothing can be more fatal to progress than a too confident reliance upon mathematical symbols; for the student is only too apt to take the easier course, and consider the formula and not the fact as the physical reality.
Now, when all these studies reach the point of inter-communion and connection with one another, and come to be considered in their mutual affinities, then, I think, but not till then, will the pursuit of them have a value for our objects: otherwise there is no profit in them.
People usually consider walking on water or in thin air a miracle. But I think the real miracle is not to walk either on water or in thin air, but to walk on earth. Every day we are engaged in a miracle which we don’t even recognize: a blue sky, white clouds, green leaves, the black, curious eyes of a child - our own two eyes. All is a miracle.
Philosophy is the science which considers truth.
Religion considers the Universe deterministic and science considers it probabilistic—an important distinction.
Science is the labor and handicraft of the mind; poetry can only be considered its recreation.
Scientists [still] refuse to consider man as an object of scientific scrutiny except through his body. The time has come to realise that an interpretation of the universe—even a positivist one—remains unsatisfying unless it covers the interior as well as the exterior of things; mind as well as matter. The true physics is that which will, one day, achieve the inclusion of man in his wholeness in a coherent picture of the world.
Sometimes scientists change their minds. New developments cause a rethink. If this bothers you, consider how much damage is being done to the world by people for whom new developments do not cause a rethink.
Sooner or later in every talk, [David] Brower describes the creation of the world. He invites his listeners to consider the six days of Genesis as a figure of speech for what has in fact been 4 billion years. On this scale, one day equals something like six hundred and sixty-six million years, and thus, all day Monday and until Tuesday noon, creation was busy getting the world going. Life began Tuesday noon, and the beautiful organic wholeness of it developed over the next four days. At 4 p.m. Saturday, the big reptiles came on. At three minutes before midnight on the last day, man appeared. At one-fourth of a second before midnight Christ arrived. At one-fortieth of a second before midnight, the Industrial Revolution began. We are surrounded with people who think that what we have been doing for that one-fortieth of a second can go on indefinitely. They are considered normal, but they are stark. raving mad.
That radioactive elements created by us are found in nature is an astounding event in the history of the earth. And of the Human race. To fail to consider its importance and its consequences would be a folly for which humanity would have to pay a terrible price. When public opinion has been created in the countries concerned and among all the nations, an opinion informed of the dangers involved in going on with the tests and led by the reason which this information imposes, then the statesmen may reach an agreement to stop the experiments.
The crippling of individuals I consider the worst evil of capitalism. Our whole educational system suffers from this evil. An exaggerated competitive attitude is inculcated into the student, who is trained to worship acquisitive success as a preparation for his future career.
The eye transmits its own image through the air to all the objects which face it, and also receives them on its own surface, whence the “sensus communis” takes them and considers them.
The great artifice of regarding small deviations from the truth as being the truth itself is at the same time the foundation of wit, where the whole thing would often collapse if we were to regard these deviations in a spirit of philosophical rigor.
The message from the Moon which we have flashed to the far corners of this planet is that no problem need any longer be considered insoluble.
The more a science advances, the more will it be possible to understand immediately results which formerly could be demonstrated only by means of lengthy intermediate considerations: a mathematical subject cannot be considered as finally completed until this end has been attained.
The observer is not he who merely sees the thing which is before his eyes, but he who sees what parts the thing is composed of. To do this well is a rare talent. One person, from inattention, or attending only in the wrong place, overlooks half of what he sees; another sets down much more than he sees, confounding it with what he imagines, or with what he infers; another takes note of the kind of all the circumstances, but being inexpert in estimating their degree, leaves the quantity of each vague and uncertain; another sees indeed the whole, but makes such an awkward division of it into parts, throwing into one mass things which require to be separated, and separating others which might more conveniently be considered as one, that the result is much the same, sometimes even worse than if no analysis had been attempted at all.
The required techniques of effective reasoning are pretty formal, but as long as programming is done by people that don’t master them, the software crisis will remain with us and will be considered an incurable disease. And you know what incurable diseases do: they invite the quacks and charlatans in, who in this case take the form of Software Engineering gurus.
The student should read his author with the most sustained attention, in order to discover the meaning of every sentence. If the book is well written, it will endure and repay his close attention: the text ought to be fairly intelligible, even without illustrative examples. Often, far too often, a reader hurries over the text without any sincere and vigorous effort to understand it; and rushes to some example to clear up what ought not to have been obscure, if it had been adequately considered. The habit of scrupulously investigating the text seems to me important on several grounds. The close scrutiny of language is a very valuable exercise both for studious and practical life. In the higher departments of mathematics the habit is indispensable: in the long investigations which occur there it would be impossible to interpose illustrative examples at every stage, the student must therefore encounter and master, sentence by sentence, an extensive and complicated argument.
There are many modes of thinking about the world around us and our place in it. I like to consider all the angles from which we might gain perspective on our amazing universe and the nature of existence.
There are still psychologists who, in a basic misunderstanding, think that gestalt theory tends to underestimate the role of past experience. Gestalt theory tries to differentiate between and-summative aggregates, on the one hand, and gestalten, structures, on the other, both in sub-wholes and in the total field, and to develop appropriate scientific tools for investigating the latter. It opposes the dogmatic application to all cases of what is adequate only for piecemeal aggregates. The question is whether an approach in piecemeal terms, through blind connections, is or is not adequate to interpret actual thought processes and the role of the past experience as well. Past experience has to be considered thoroughly, but it is ambiguous in itself; so long as it is taken in piecemeal, blind terms it is not the magic key to solve all problems.
These duplicates in those parts of the body, without which a man might have very well subsisted, though not so well as with them, are a plain demonstration of an all-wise Contriver, as those more numerous copyings which are found among the vessels of the same body are evident demonstrations that they could not be the work of chance. This argument receives additional strength if we apply it to every animal and insect within our knowledge, as well as to those numberless living creatures that are objects too minute for a human eye: and if we consider how the several species in this whole world of life resemble one another in very many particulars, so far as is convenient for their respective states of existence, it is much more probable that a hundred millions of dice should be casually thrown a hundred millions of times in the same number than that the body of any single animal should be produced by the fortuitous concourse of matter.
To ask what qualities distinguish good from routine scientific research is to address a question that should be of central concern to every scientist. We can make the question more tractable by rephrasing it, “What attributes are shared by the scientific works which have contributed importantly to our understanding of the physical world—in this case the world of living things?” Two of the most widely accepted characteristics of good scientific work are generality of application and originality of conception. . These qualities are easy to point out in the works of others and, of course extremely difficult to achieve in one’s own research. At first hearing novelty and generality appear to be mutually exclusive, but they really are not. They just have different frames of reference. Novelty has a human frame of reference; generality has a biological frame of reference. Consider, for example, Darwinian Natural Selection. It offers a mechanism so widely applicable as to be almost coexistent with reproduction, so universal as to be almost axiomatic, and so innovative that it shook, and continues to shake, man’s perception of causality.
True science investigates and brings to human perception such truths and such knowledge as the people of a given time and society consider most important. Art transmits these truths from the region of perception to the region of emotion.
We are not to consider the world as the body of God: he is an uniform being, void of organs, members, or parts; and they are his creatures, subordinate to him, and subservient to his will.
We consider species to be like a brick in the foundation of a building. You can probably lose one or two or a dozen bricks and still have a standing house. But by the time you’ve lost twenty percent of species, you’re going to destabilize the entire structure. That’s the way ecosystems work.
We may regard [Scheele] not only as having given the first indication of the rich harvest to be reaped by the investigation of the compounds of organic chemistry, but as having been the first to discover and make use of characteristic reactions by which closely allied substances can be detected and separated, so that he must be considered one of the chief founders of analytical chemistry.
Were I disposed to consider the comparative merit of each of them [facts or theories in medical practice], I should derive most of the evils of medicine from supposed facts, and ascribe all the remedies which have been uniformly and extensively useful, to such theories as are true. Facts are combined and rendered useful only by means of theories, and the more disposed men are to reason, the more minute and extensive they become in their observations.
When I use the phrase “Cosmic Facts,” the reader is asked not to assume too rigid a meaning for the word “facts”; what is considered factual today is tomorrow recognized as capable of further refinement.
When one considers how hard it is to write a computer program even approaching the intellectual scope of a good paper, and how much greater time and effort have to be put in to make it “almost” formally correct, it is preposterous to claim that mathematics as we practice it is anywhere near formally correct.
[An outsider views a scientist] as a type of unscrupulous opportunist: he appears as a realist, insofar as he seeks to describe the world independent of the act of perception; as idealist insofar as he looks upon the concepts and theories as the free inventions of the human spirit (not logically derivable from that which is empirically given); as positivist insofar as he considers his concepts and theories justified only to the extent to which they furnish a logical representation of relations among sense experiences. He may even appear as Platonist or Pythagorean insofar as he considers the viewpoint of logical simplicity as an indispensable and effective tool of his research.
[Howard] Aiken considered Babbage his intellectual “father.”