ON THE NOTION OF INFINITY 205
dented way. That this structural linguistic discovery was made so late is probably due to the usual blockage, the old
s.r, old habits of 'thought', and prejudices.In all arguments about infinity, from remote antiquity until Bolzano (1781-1848), Dedekind (1831-1916), and Cantor (1845-1918), there was a peculiar maxim involved. All arguments against infinity involved a certain structural assumption, which, at first inspection, seemed to be true and 'self-evident', and yet, if carried through, would be quite destructive to all mathematics existing at that date. Arguments favorable to infinity did not involve these tragic consequences. Quite naturally, mathematicians, and particularly Cantor, began to investigate this peculiar maxim and the
s.r which were playing havoc. The structural assumption in question is that 'if a collection is part of another, the one which is a part must have fewer terms than the one of which it is a part'. This s.r was deeply rooted, and even found a scholarly formulation in Euclid's wording in one of his axioms: 'The whole is greater than any of its parts'. This axiom, although it is not an exact equivalent of the maxim stated above, by loose reasoning, which was usual in the older days, could be said to imply the troublesome maxim. It is not difficult to see that the E axiom, as well as our troublesome maxim, expresses a structural generalization taken from experience which applies only to finite processes, arrays,. Indeed, both can be taken as a definition of finite processes, arrays,. It does not follow, however, that the one definition and structure must be true of infinite processes, arrays,. As a matter of fact, the break-down of this maxim gives us the precise definition of mathematical infinity. A process of generating arrays., is called infinite when it contains, as parts, other processes, arrays., which have 'as many' terms as the first process, array,.The term 'infinite' means a process which does not end or stop, and it is usually symbolized by oo. The term may be applied, also, to an array of terms or other entities, the production of which does not end or stop. Thus we may speak of the infinite process of generating numbers because every positive integer, no matter how great, has a successor; we can also speak of infinite divisibility because the numerical technique gives us means to accomplish that. The term 'infinite' is used here as an adjective describing the characteristics of a process, but should never be used as a noun, as this leads to self-contradictions. The term 'infinity', as a noun, is used here only as an abbreviation for the phrase 'infinite process of generating numbers',. If used in any other way than as an abbreviation for the full phrase, the term is meaningless in science (not in psychopathology) and should never be used. The above semantic |
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