If one has really technically penetrated a subject, things that previously seemed in complete contrast, might be purely mathematical transformations of each other.
Truth is much too complicated to allow anything but approximations.
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.
I am thinking about something much more important than bombs. I am thinking about computers.
Neumann, to a physicist seeking help with a difficult problem: Simple. This can be solved by using the method of characteristics. Physicist: I’m afraid I don’t understand the method of characteristics. Neumann: In mathematics you don’t understand things. You just get used to them.
With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.
You wake me up early in the morning to tell me I am right? Please wait until I am wrong.
Technological possibilities are irresistible to man. If man can go to the moon, he will. If he can control the climate, he will.
The most vitally characteristic fact about mathematics is, in my opinion, its quite peculiar relationship to the natural sciences, or more generally, to any science which interprets experience on a higher than purely descriptive level.
Life is a process which may be abstracted from other media.
The emphasis on mathematical methods seems to be shifted more towards combinatorics and set theory – and away from the algorithm of differential equations which dominates mathematical physics.
Can we survive technology?
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.
An element which stimulates itself will hold a stimulus indefinitely.
It is only proper to realize that language is largely a historical accident.
In any conceivable method ever invented by man, an automaton which produces an object by copying a pattern, will go first from the pattern to a description to the object. It first abstracts what the thing is like, and then carries it out. It’s therefore simpler not to extract from a real object its definition, but to start from the definition.
A system of logical instructions that an automaton can carry out and which causes the automaton to perform some organized task is called a code.
In this sense, an object is of the highest degree of complexity if it can do very difficult and involved things.
Natura non facis Saltus.
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.