### Lietotâju komentâri -Rakstît atsauksmi

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### Populâri fragmenti

viii. lappuse - The ideal of mathematics should be to erect a calculus to facilitate reasoning in connection with every province of thought, or of external experience, in which the succession of thoughts, or of events can be definitely ascertained and precisely stated.
vi. lappuse - Mathematics in its widest signification is the development of all types of formal, necessary, deductive reasoning.
vi. lappuse - Mathematical reasoning is deductive in the sense that it is based upon definitions which, as far as the validity of the reasoning is concerned (apart from any existential import), need only the test of self-consistency. Thus no external verification of definitions is required by mathematics as long as it is considered merely as mathematics.
v. lappuse - N. Whitehead Treatise on Universal Algebra, 1898 2. The General Theory 2.1 A Classification of Equational Theories We like to present some important subclasses of equational theories, which turned out to be of practical interest as well as being useful as "basic building blocks...
10. lappuse - But in the use of a calculus this process of combination is externally performed by the combination of the concrete symbols, with the result of a new fact respecting the symbols which arises for sensuous perception...
12. lappuse - The whole of Mathematics consists in the organization of a series of aids to the imagination in the process of reasoning; and for this purpose device is \ piled upon device.
6. lappuse - Equivalence on the other hand implies non-identity as its general case. For instance in arithmetic we write, 2+3= 3+2. This means that, in so far as the total number of objects mentioned, 2+3 and 3+2 come to the same number, namely 5. But 2+3 and 3+2 are not identical; the order of the symbols is different in the two combinations, and this difference of order directs different processes of thought. The importance of the equation arises from its assertion that these different processes of thought...
vii. lappuse - The chance for any arbitrary system of symbolism applying to anything real is very small, as the author admits ; for he says that the entities created by conventional definitions must have properties which bear some affinity to the properties of existing things. Unless the affinity or correspondence is perfect, how can the one apply to the other ? How can this perfect correspondence be secured, except by the conventions being real definitions, the equations true propositions, and the rules expressions...
11. lappuse - The difficulty was solved by observing that Algebra does not depend on \ Arithmetic for the validity of its laws of transformation. If there were » such a dependence, it is obvious that as soon as algebraic expressions are arithmetically unintelligible all laws respecting them must lose their validity. But the laws of Algebra, though suggested by Arithmetic, do not depend on it.
vi. lappuse - The justification of the rules of inference in any branch of mathematics is not properly part of mathematics, it is the business of experience or philosophy. The business of mathematics is simply to follow the rules. In this sense all mathematical reasoning is necessary, namely, it has followed the rule.