One of the chief duties of the mathematician in acting as an advisor ... is to discourage ... from expecting too much from mathematics.
If the human being is condemned and restricted to perform the same functions over and over again, he will not even be a good ant, not to mention a good human being.
The automatic machine, whatever we thinkof any feelings it may or may not have, is the precise economic equivalent of the slave.
Any useful logic must concern itself with Ideas with a fringe of vagueness and a Truth that is a matter of degree.
The mechanical brain does not secrete thought "as the liver does bile," as the earlier materialists claimed, nor does it put it out in the form of energy, as the muscle puts out its activity. Information is information, not matter or energy. No materialism which does not admit this can survive at the present day.
The idea that information can be stored in a changing world without an overwhelming depreciation of its value is false. It is scarcely less false than the more plausible claim that after a war we may take our existing weapons, fill their barrels with information.
Science is better paid than at any time in the past. The results of this pay have been to attract into science many of those for whom the pay is the first consideration, and who scorn to sacrifice immediate profit for the freedom of development of their own concept. Moreover, this inner development, important and indispensable as it may be to the world of science in the future, generally does not have the tendency to put a single cent into the pockets of their employers.
I may remark parenthetically that the modern apparatus of the theory of small samples, once it goes beyond the determination of its own specially defined parameters and becomes a method for positive statistical inference in new cases, does not inspire me with any confidence unless it is applied by a statistician by whom the main elements of the dynamics of the situation are either explicitly known or implicitly felt.
Scientific discovery consists in the interpretation for our own convenience of a system of existence which has been made with no eye to our convenience at all.
In all important respects, the man who has nothing but his physical power to sell has nothing to sell which it is worth anyone's money to buy
There are fields of scientific work...which have been explored from the different sides of pure mathematics, statistics, electrical engineering, and neurophysiology...in which every single notion receives a separate and different name from each group, and in which important work has been triplicated or quadruplicated, while still other important work is delayed by the unavailability in one field of results that may have already become classical in the next field.
Mathematics 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.
Until we in the community have made up our minds that what we really want is expiation, or removal, or reform, or or the discouragement of potential criminals, we shall get none of these, but only a confusion in which crime breeds more crime.
It is possible to believe in progress as a fact without believing in progress as an ethical principle; but in the catechism of many Americans, the one goes with the other.
The sense of tragedy is that the world is not a pleasant little nest made for our protection, but a vast and largely hostile environment, in which we can achieve great things only by defying the gods; and that this defiance inevitably brings its own punishment.
Progress imposes not only new possibilities for the future but new restrictions. It seems almost as if progress itself and our fight against the increase of entropy intrinsically must end in the downhill path from which we are trying to escape.
I have said that the modern man, and especially the modern American, however much 'know-how' he may have, has very little 'know-what'
The modern physicist is a quantum theorist on Monday, Wednesday, and Friday, and a student of gravitational relativity on Tuesday, Thursday, and Saturday. On Sunday, he is praying... that someone will find the reconciliation between the two views.
Am I really a good mathematician?
A significant idea of organization cannot be obtained in a world in which everything is necessary and nothing is contingent.
A painter like Picasso, who runs through many periods and phases, ends up by saying all those things which are on the tip of the tongue of the age to say, and finally sterilizes the originality of his contemporaries and juniors.
A faith which we follow upon orders imposed from outside is no faith, and a community which puts its dependence upon such a pseudo-faith is ultimately bound to ruin itself because of the paralysis which the lack of a healthy growing science imposes upon it.
the science of control and communication in the animal and the machine
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.
We are in the position of the man who has only two ambitions in life. One is to invent the universal solvent which will dissolve any solid substance, and the second is to invent the universal container which will hold any liquid. Whatever this inventor does, he will be frustrated.
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