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John Von Neumann

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

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

From Lectures On Self-Replicating Automata (1948?)

An automaton cannot be separated from the milieu to which it responds.

It is therefore quite possible that we are not too far from the limits of complication which can be achieved in artificial automata without really fundamental insights into a theory of information, although one should be very careful with such statements because they can sound awfully ridiculous 5 years later.

The fact that natural organisms have such a radically different attitude about errors and behave so differently when an error occurs is probably connected with some other traits of natural organisms, which are entirely absent from our automata. The ability of a natural organism to survive in spite of a high incidence of error (which our artificial automata are incapable of) probably requires a very high flexibility and ability of the automaton to watch itself and reorganize itself. And this probably requires a very considerable autonomy of parts. There is a high autonomy of parts in the human nervous system. This autonomy of parts of a system has an effect which is observable in the human nervous system but not in artificial automata. When parts are autonomous and able to reorganize themselves, when there are several organs each capable of taking control in an emergency, an antagonistic relation can develop between the parts so that they are no longer friendly and cooperative. It is quite likely that all these phenomena are connected.

Even if the axioms are chosen within the common sense area, it is usually very difficult to achieve an agreement between two people who have done this independently. For instance, in the literature of formal logics there are about as many notations as there are authors, and anybody who has used a notation for a few weeks feels that it’s more or less superior to any other. So, while the choice of notations, of the elements, is enormously important and absolutely basic for an application of the axiomatic method, this choice is neither rigorously justifiable nor humanly unambiguously justifiable. All one can do is to try to submit a system which will stand up under common sense criteria.

There is reason to suspect that our predilection for linear codes, which have a simple, almost temporal sequence, is chiefly a literary habit, corresponding to our not particularly high level of combinatorial cleverness, and that a very efficient language would probably depart from linearity.

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