John von Neumann Quotes

When we talk mathematics, we may be discussing a secondary language built on the primary language of the nervous system.

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.

The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work.

Thus all sorts of sophisticated order-systems become possible, which keep successively modifying themselves and hence also the computational processes that are likewise under their control.

If one has really technically penetrated a subject, things that previously seemed in complete contrast, might be purely mathematical transformations of each other.

A code, which according to Turing's schema is supposed to make one machine behave as if it were another specific machine (which is supposed to make the former imitate the latter) must do the following things. It must contain, in terms that the machine will understand (and purposively obey), instructions (further detailed parts of the code) that will cause the machine to examine every order it gets and determine whether this order has the structure appropriate to an order of the second machine. It must then contain, in terms of the order system of the first machine, sufficient orders to make the machine cause the actions to be taken that the second machine would have taken under the influence of the order in question.

When we talk mathematics, we may be discussing a secondary language built on the primary language truly used by the central nervous system.

"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

As he describes each mechanism in the brain, he shows how a modern computer could accomplish the same operation, despite the apparent differences. The brain’s analog mechanisms can be simulated through digital ones because digital computation can emulate analog values to any desired degree of precision (and the precision of analog information in the brain is quite low).

With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.

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.

Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin.

The McCulloch-Pitts result . . . proves that anything that can be exhaustively and unambiguously described, anything that can be completely and unambiguously put into words, is ipso facto realizable by a suitable finite neural network.

There probably is a God. Many things are easier to explain if there is than if there isn't.

The goys have proven the following theorem…