By Garcia P.L. (ed.), Perez-Rendon A. (ed.)

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I. xpresses; a substitute sign is a means of not thinking about the meaning which it symbolizes*/ The use of substitutive signs in reasoning is to economize thought. e> 2. DEFINITION OF A CALCULUS. In order that reasoning may be conducted by means of substitutive signs, it is necessary that rules be given for the manipulation of the signs. The rules should be such that the final state of the signs after a series of operations according to rule denotes, when the signs are interpreted in terms of the things for which they are substituted, a proposition true for the things represented by the signs.

573—575 570—586 BOOK I. PRINCIPLES OF ALGEBRAIC SYMBOLISM. w. CHAPTER I. O N THE NATURE OF A CALCULUS. 1. SIGNS. Words, spoken or written, and the symbols of Mathematics are alike signs. Signs have been analysed* into (a) suggestive signs, (/3) expressive signs, (7) substitutive signs. A suggestive sign is the most rudimentary possible, and need not be dwelt upon here. An obvious example of one is a knot tied in a handkerchief to remind the owner of some duty to be performed. In the use of expressive signs the attention is not fixed on the sign itself but on what it expresses; that is to say, it is fixed on the meaning conveyed by the sign.

Single algebraic scheme (cf. § 15), nor does it necessarily reproduce as its result a member of one of the algebraic schemes to which the terms synthesized belong. Again, the commutative and associated laws do not necessarily hold for multiplication: but a new law, the distributive law, which defines the relation of multiplication to addition holds. Any mode of synthesis for which this relation to addition holds is here called multiplication. The result of multiplication like that of addition is unambiguous.

### Differential Geometric Methods in Mathematical Physics by Garcia P.L. (ed.), Perez-Rendon A. (ed.)

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