The Fundamental Principles of Algebra: Address (Classic Reprint) - Hardcover

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Excerpt from The Fundamental Principles of Algebra: Address

Cotemporaneously with Boole, and living on the same Emer ald Isle, another mathematician spent many days reflecting on the fundamental principles of algebra - Sir W. R. Hamilton. His investigation started from the reading of some passages in Kant's 'critique of the Pure Reason, which appeared to justify the expectation that it should be possible to construct a priori a science of time as well as a science of space. The principal passage is as follows: Time and space are two sources of knowledge from which various a prz'ori synthetical cognitions can be derived. Of this pure mathematics gives a splendid example in the case of our cognitions of space and its various relations. AS they are both pure forms of sensuous intuition, they render synthetical propositions a priori possible. Thus, according to Kant, space and time are forms of the intellect and Hamilton reasoned that, as geometry is the science of the former, so algebra must be the science Of the latter. He am plifies that view as follows It early appeared to me that these ends might be attained by our consenting to regard algebra as being no mere art, nor language, nor primarily a science of quantity but rather as the science of order in progression. It was, however, a part of this conception, that the progression here spoken of was understood to be continuous and unidimen Sioual extending indefinitely forward and backward, but not in any lateral direction. And although the successive states of such a progression might, no doubt, be represented by points upon a line, yet I thought that their simple successive ness was better conceived by comparing them with moments of time, divested, however, of all reference to cause and efiect.

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