A Treatise on Universal Algebra: With Applications by Alfred North Whitehead

By Alfred North Whitehead

Alfred North Whitehead (1861-1947) was once both celebrated as a mathematician, a thinker and a physicist. He collaborated together with his former pupil Bertrand Russell at the first version of Principia Mathematica (published in 3 volumes among 1910 and 1913), and after numerous years instructing and writing on physics and the philosophy of technology at college collage London and Imperial collage, used to be invited to Harvard to coach philosophy and the idea of schooling. A Treatise on common Algebra was once released in 1898, and used to be meant to be the 1st of 2 volumes, although the second one (which used to be to hide quaternions, matrices and the overall idea of linear algebras) used to be by no means released. This ebook discusses the final rules of the topic and covers the subjects of the algebra of symbolic common sense and of Grassmann's calculus of extension.

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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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