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1 " It is only proper to realize that language is largely a historical accident. "
― John von Neumann , The Computer and the Brain
2 " And' and 'or' are the basic operations of logic. Together with 'no' (the logical operation of negation) they are a complete set of basic logical operations—all other logical operations, no matter how complex, can be obtained by suitable combinations of these. "
3 " When we talk mathematics, we may be discussing a secondary language, built on the primary language truly used by the central nervous system. Thus the outward forms of our mathematics are not absolutely relevant from the point of view of evaluating what the mathematical or logical language truly used by the central nervous system is. However, the above remarks about reliability and logical and arithmetical depth prove that whatever the system is, it cannot fail to differ considerably from what we consciously and explicitly consider as mathematics. "
4 " 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. "
5 " In the deceptively modest volume you are now holding, von Neumann articulates his model of computation and goes on to define the essential equivalence of the human brain and a computer. He acknowledges the apparently deep structural differences, but by applying Turing’s principle of the equivalence of all computation, von Neumann envisions a strategy to understand the brain’s methods as computation, to re-create those methods, and ultimately to expand its powers. "
6 " Since I am neither a neurologist nor a psychiatrist, but a mathematician, the work that follows requires some explanation and justification. "
7 " 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. "
8 " Any computing machine that is to solve a complex mathematical problem must be 'programmed' for this task. This means that the complex operation of solving that problem must be replaced by a combination of the basic operations of the machine. "
9 " 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. "
10 " Apart from all other considerations, the main limitation of analog machines relates to precision. Indeed, the precision of electrical analog machines rarely exceeds 1:10^3, and even mechanical ones achieve at best 1:10^4 to 10^5... On the other hand, to go from 1:10^12 to 1:10^13 in a digital machine means merely adding one place to twelve; this means usually no more than a relative increase in equipment (not everywhere!) of 1/12 = 8.3 percent, and an equal loss in speed (not everywhere!) — none of which is serious. "
11 " The very last stage of any memory hierarchy is necessarily the outside world—that is, the outside world as far as the machine is concerned, i.e. that part of it with which the machine can directly communicate, in other words, the input and the output organs of the machine. These are usually punched paper tapes or cards, and on the output side, of course, also printed paper. "
12 " All existing machines and memories use "direct addressing," which is to say that every word in the memory has a numerical address of its own that characterizes it and its position within the memory (the total aggregate of all hierarchic levels) uniquely. "
13 " 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. "
14 " We do not know where in the physically viewed nervous system a memory resides; we do not know whether it is a separate organ or a collection of specific parts of other already known organs, etc. "
15 " The human brain is, after all, the best example we have of an intelligent system. If we can learn its methods, we can use these biologically inspired paradigms to build more intelligent machines. This book is the earliest serious examination of the human brain from the perspective of a mathematician and computer pioneer. Prior to von Neumann, the fields of computer science and neuroscience were two islands with no bridge between them. "
16 " 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. "
17 " 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). "
18 " 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. "
19 " 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. "
