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It is the mark of an educated man to look for precision in each class of things just so far as the nature of the subject admits; it is evidently equally foolish to accept probable reasoning from a mathematician and to demand from a rhetorician scientific proofs.
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Book I, 1094.b24

 

Aristotle

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A deep sense of humor and an unusual ability for telling stories and jokes endeared Johnny even to casual acquaintances. He could be blunt when necessary, but was never pompous. A mind of von Neumann's inexorable logic had to understand and accept much that most of us do not want to accept and do not even wish to understand. This fact colored many of von Neumann's moral judgments. "It is just as foolish to complain that people are selfish and treacherous as it is to complain that the magnetic field does not increase unless the electric field has a curl. Both are laws of nature." Only scientific intellectual dishonesty and misappropriation of scientific results could rouse his indignation and ire — but these did — and did almost equally whether he himself, or someone else, was wronged.

 

E. P. Wigner

 

We know truth, not only by the reason, but also by the heart, and it is in this last way that we know first principles; and reason, which has no part in it, tries in vain to impugn them. The sceptics, who have only this for their object, labour to no purpose. We know that we do not dream, and however impossible it is for us to prove it by reason, this inability demonstrates only the weakness of our reason, but not, as they affirm, the uncertainty of all our knowledge. For the knowledge of first principles, as space, time, motion, number, is as sure as any of those which we get from reasoning. And reason must trust these intuitions of the heart, and must base them on every argument. (We have intuitive knowledge of the tri-dimensional nature of space, and of the infinity of number, and reason then shows that there are no two square numbers one of which is double of the other. Principles are intuited, propositions are inferred, all with certainty, though in different ways.) And it is as useless and absurd for reason to demand from the heart proofs of her first principles, before admitting them, as it would be for the heart to demand from reason an intuition of all demonstrated propositions before accepting them.
This inability ought, then, to serve only to humble reason, which would judge all, but not to impugn our certainty, as if only reason were capable of instructing us. Would to God, on the contrary, that we had never need of it, and that we knew everything by instinct and intuition! But nature has refused us this boon. On the contrary, she has given us but very little knowledge of this kind; and all the rest can be acquired only by reasoning.
Therefore, those to whom God has imparted religion by intuition are very fortunate, and justly convinced. But to those who do not have it, we can give it only by reasoning, waiting for God to give them spiritual insight, without which faith is only human, and useless for salvation.

 

Blaise Pascal

 

The subject presented in the memorial is one upon which I have thought much for weeks past, and I may even say for months. I am approached with the most opposite opinions and advice, and that by religious men, who are equally certain that they represent the Divine will. I am sure that either the one or the other class is mistaken in that belief, and perhaps in some respects both. I hope it will not be irreverent for me to say that if it is probable that God would reveal his will to others, on a point so connected with my duty, it might be supposed he would reveal it directly to me; for, unless I am more deceived in myself than I often am, it is my earnest desire to know the will of Providence in this matter. And if I can learn what it is I will do it! These are not, however, the days of miracles, and I suppose it will be granted that I am not to expect a direct revelation. I must study the plain physical facts of the case, ascertain what is possible, and learn what appears to be wise and right.
The subject is difficult, and good men do not agree.

 

Abraham Lincoln

 

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.

 

Stanislaw Ulam

 

A deep sense of humor and an unusual ability for telling stories and jokes endeared Johnny even to casual acquaintances. He could be blunt when necessary, but was never pompous. A mind of von Neumann's inexorable logic had to understand and accept much that most of us do not want to accept and do not even wish to understand. This fact colored many of von Neumann's moral judgments. … Only scientific intellectual dishonesty and misappropriation of scientific results could rouse his indignation and ire — but these did — and did almost equally whether he himself, or someone else, was wronged.

 

John von Neumann