John von Neumann Quotes

In an analog machine each number is represented by a suitable physical quantity, whose values, measured in some pre-assigned unit, is equal to the number in question.

A large part of mathematics which becomes useful developed with absolutely no desire to be useful, and in a situation where nobody could possibly know in what area it would become useful; and there were no general indications that it ever would be so. By and large it is uniformly true in mathematics that there is a time lapse between a mathematical discovery and the moment when it is useful; and that this lapse of time can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful.

In this sense, an object is of the highest degree of complexity if it can do very difficult and involved things.

It is a well known phenomenon in many branches of the exact and physical sciences that very great numbers are often easier to handle than those of medium size. An almost exact theory of a gas, containing about 1025 freely moving particles, is incomparably easier than that of the solar system, made up of 9 major bodies; and still more than that of a multiple star of three or four objects of about the same size. This is, of course, due to the excellent possibility of applying the laws of statistics and probabilities in the first case.

The linear size of a neuron varies widely from one nerve cell to the other, since some of these cells are contained in closely integrated large aggregates and have, therefore, very short axons, while others conduct pulses between rather remote parts of the body and may, therefore, have linear extensions comparable to those of the entire human body.

Se la gente non crede che la matematica è semplice, è solo perché non capisce quanto è complicata la vita.

Any one who considers arithmetical methods of producing random digits is, of course, in a 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.

It is exceptional that one should be able to acquire the understanding of a process without having previously acquired a deep familiarity with running it, with using it, before one has assimilated it in an instinctive and empirical way… Thus any discussion of the nature of intellectual effort in any field is difficult, unless it presupposes an easy, routine familiarity with that field. In mathematics this limitation becomes very severe.

A large part of mathematics which becomes useful developed with absolutely no desire to be useful, and in a situation where nobody could possibly know in what area it would become useful; and there were no general indications that it ever would be so.

Natura non facis Saltus.

Any artificial automaton that has been constructed for human use, and specifically for the control of complicated processes, normally possesses a purely logical part and an arithmetical part, i.e. a part in which arithmetical processes play no role, and one in which they are of importance.

"Kurt Gödel's achievement in modern logic is singular and monumental – indeed it is more than a monument, it is a landmark which will remain visible far in space and time. ... The subject of logic has certainly completely changed its nature and possibilities with Gödel's achievement." — John von Neumann

إن كان الناس لا يعتقدون أن الرياضيات بسيطة، فهذا لأنهم لا يدركون مدى تعقيد الحياة

It will not be sufficient to know that the enemy has only fifty possible tricks and that we can counter every one of them, but we must be able to counter them almost at the very instant they occur.

He notes that the output of neurons is digital: an axon either fires or it doesn’t. This was far from obvious at the time, in that the output could have been an analog signal. The processing in the dendrites leading into a neuron and in the soma neuron cell body, however, are analog. He describes these calculations as a weighted sum of inputs with a threshold.