ONT Re: De In Esse Predication
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DEIP. Note 14
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| Introduction to the Logic of Quantity
|
| The great importance of the idea of quantity in demonstrative reasoning
| seems to me not yet sufficiently explained. It appears, however, to be
| connected with the circumstance that the relations of being greater
| than and of being at least as great as are transitive relations.
| Still, a satisfactory evolutionary logic of mathematics remains a
| desideratum. I intend to take up that problem in a future paper
| ["The Simplest Mathematics", CP 4.227-323, 1902]. Meantime the
| development of projective geometry and of geometrical topics has
| shown that there are at least two large mathematical theories of
| continuity into which the idea of continuous 'quantity', in the
| usual sense of that word, does not enter at all. For projective
| geometry Schubert has developed an algebraical calculus which has
| a most remarkable affinity to the Boolian algebra of logic. It is,
| however, imperfect, in that it only gives imaginary points, rays, and
| planes, without deciding whether they are real or not. This defect cannot
| be remedied until topology -- or, as I prefer to call it, mathematical topics --
| has been further developed and its logic accurately analysed. To do this
| ought to be one of the first tasks of exact logicians. But before that
| can be accomplished, a perfectly satisfactory logical account of the
| conception of continuity is required. This involves the definition
| of a certain kind of infinity; and in order to make that quite clear,
| it is requisite to begin by developing the logical doctrine of infinite
| multitude. This doctrine still remains, after the works of Cantor, Dedekind,
| and others, in an inchoate condition. For example, such a question remains
| unanswered as the following: Is it, or is it not, logically possible for
| two collections to be so multitudinous that neither can be put into a
| one-to-one correspondence with a part or the whole of the other?
| To resolve this problem demands, not a mere 'application' of
| logic, but a further 'development' of the conception of
| logical possibility.
|
| C.S. Peirce, 'Collected Papers', CP 3.526,
|"The Logic of Relatives", 'Monist', vol. 7,
| pp. 161-217, 1897.
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