A Century of Mathematics: Through the Eyes of the Monthly by John Ewing

By John Ewing

This is often the tale of yank arithmetic in past times century. It includes articles and excerpts from a century of the yankee Mathematical per month, giving the reader a chance to skim all 100 volumes of this renowned arithmetic journal with out really beginning them. It samples arithmetic yr by way of 12 months and decade by way of decade. The reader can glimpse the mathematical group on the flip of the century, the talk approximately Einstein and relativity, the debates approximately formalism in common sense, the immigration of mathematicians from Europe, and the frantic attempt to arrange because the warfare begun. newer articles take care of the appearance of pcs and the adjustments they introduced, and with the various triumphs of contemporary study.

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THE NULL ELEMENT. On the assumption that to any question of the type a — b can be assigned an answer, some meaning must be assigned to the term a — a. Now if c be any other term, c + a— a = c = c+b — 6. Hence it may be assumed that a — a = b — b. Thus we may put a -a = 0 (7); where 0 represents an element of the manifold independent of a. Let the element 0 be called the null element. The fundamental property of the null element is that the addition of this element an<^ any other element a of the manifold yields the same element a.

It is necessary to note the form in which propositions occur in a calculus. Such a form may well be highly artificial from some points of view, and may yet state the propositions in a convenient form for the eliciting of deductions. Furthermore it is not necessary to assert that the form is a general form into which all judgments can be put by the aid of some torture. It is sufficient to observe that it is a form of wide application. In a calculus of the type here considered propositions take the form of assertions of equivalence.

The rule of reasoning is to confine thought to those properties, previously determined, which are shared in common with the original scheme, and to interpret the results from one set of things into the other at the end of the argument. An instance of this process of reasoning by substitution is to be found in the theory of quantity. Quantities are measured by their ratio to an arbitrarily assumed quantity of the same kind, called the unit. Any set of quantities of one kind can be represented by a corresponding set of quantities of any other kind merely in so far as their numerical ratios to their unit are concerned.

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