The Fundamental Principles of Algebra: Address (delivered .. [before The] American Association for the Advancement of Science. Forty-eight Annual Meeting ..., 1899.)

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Chemical Publishing Company, 1899 - Algebra - 31 pages
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Page 29 - Stout's that a word is an instrument for thinking about the meaning which it expresses ; whereas a substitutive sign is a means of not thinking about the meaning which it symbolizes ; and he adds that the use of substitutive signs in reasoning is to economize thought.
Page 14 - 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.
Page 30 - ... number multiplied by the sum is the same as the second number multiplied by the first number together with the second number multiplied by itself. Putting all these together, we find that the square of the sum is equal to the sum of the squares of the two numbers together with twice their product. Two things may be observed on this comparison. First, how very much the shorthand expression gains in clearness from its brevity. Secondly, that it is only shorthand for something which is just straightforward...
Page 13 - X and -*- on account of the ambiguity of the reciprocal ; the commutative law applies to both sum and product ; the distributive law applies to the product of sums ; there are no index laws, excepting the peculiar one a! a ; the law of reduction a — a = o remains, but the complementary law i is not true in general. How is* the truth or suitability of these laws established? He says that it would be mere hypothesis to borrow the notation of the analysis of quantity, and to assume that in its new...
Page 24 - Hankel labors under a logical difficulty from which Peacock was exempt ; for he does not take the laws of arithmetical algebra without exception ; he rejects the commutative law for a product, in order that quaternions may be included among his complex numbers. But, it may be asked, why not reject the commutative law for addition also ; so far as arithmetical algebra is concerned, they stand on the same basis. If, as has been shown, the sum of quaternion indices is not commutative, we are logically...
Page 26 - For reasons too long to give here, I do not believe that the provisional use of unmeaning arithmetical symbols can ever lead to the science of quantity; and I feel sure that the attempt to found it on such abstractions obscures its true physical nature. The science of number is founded on the hypothesis of the distinctness of things; the science of quantity is founded on the totally different hypothesis of continuity.
Page 11 - No accumulation of instances can properly add weight to such evidence. It may furnish us with clearer conceptions of that common element of truth upon which the application of the principle depends, and so prepare the way for its reception. It may, where the immediate force of the evidence is not felt, serve as verification, a posteriori, of the practical validity of the principle in question. But this does not affect the position affirmed, viz., that the general principle must be seen in the particular...
Page 8 - It is most important that the student should bear in mind that, with one exception , no word nor sign of arithmetic or algebra has one atom of meaning throughout this chapter, the object of which is symbols, and their laws of combination, giving a symbolic algebra which may hereafter become the grammar of a hundred distinct significant algebras.
Page 5 - I would consider symbolical algebra is, that it is the science which treats of the combination of operations defined not by their nature, that is, by what they are or what they do, but by the laws of combination to which they are subject.
Page 15 - 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 unidimensional ; extending indefinitely forward and backward, but not in any lateral direction. And although the successive states of such a progression might, no doubt,...

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