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CLASSICAL ESSAY |
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by Richard Dedekind
In science nothing capable of proof ought to be accepted without proof. Though this demand seems so reasonable yet I cannot regard it as having been met even in the most recent methods of laying the foundations of the simplest science; viz., tht part of logic which deals with the theory of numbers. In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number-concept entirely independent of the notions or intuitions of space and time, that I consider it an immediate result from the laws of thought. My answer to the problems propounded in the title of this paper is, then, briefly this: numbers are free creations of the human mind; they serve as a means of apprehending more easily and more sharply the difference of things. It is only through the purely logical process of building up the science of numbers and by thus acquiring the continuous number-domain that we are prepared accurately to investigate our notions of space and time by bringing them into relation with this number-domain created in our mind. If we scrutinise closely what is done in counting an aggregate or number of things, we are led to consider the ability of the mind to relate things to things, to let a thing correspond to a thing, or to represent a thing by a thing, an ability without which no thinking is possible. I like to compare this action of thought, so difficult to trace on account of the rapidity of its performance, with the action which an accomplished reader performs in reading; this reading always remains a more or less complete repetition of the individual steps which the beginner has to take in his wearisome spelling-out; a very small part of the same, and therefore a very samll effort or exertion of the mind, is sufficient for the practiced reader to recognize the correct, true word, only with very great probability, to be sure; for, as is well known, it occasionally happens that even the most practiced proof-reader allows a typographical error to escape him, i.e., reads falsely, a thing which would be impossible if the chain of thoughts associated with spelling were fully repeated. So from the time of birth, continually and in increasing measure we are led to relate things to things and thus to use that faculty of the mind on which the creation of numbers depends; by this practice continually occurring, though without definite purpose, in our earliest years and by the attending formation of judgments and chains of reasoning we acquire a store of real arithmetic truths to which our first teachers later refer as to something simple, self-evident, given in the inner consciousness; and so it happens that many very complicated notions (as for example that of the number [Anzahl] of things) are erroneously regarded as simple.
Excerpted from Essays on the Theory of Numbers, by Richard Dedekind. |
Essays on the Theory of Numbers, by Richard Dedekind
What Are Numbers and What Should They Be?, by Richard Dedekind
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