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Full Idea
In classical arithmetic, ratios were posited to make division generally applicable, negative numbers to make subtraction generally applicable, and irrationals and finally imaginaries to make exponentiation generally applicable.
Clarification
'Exponentiation' is raising a quantity to a power (e.g. squaring)
Gist of Idea
In arithmetic, ratios, negatives, irrationals and imaginaries were created in order to generalise
Source
Willard Quine (On Multiplying Entities [1974], p.263)
Book Ref
Quine,Willard: 'Ways of Paradox and other essays' [Harvard 1976], p.263
A Reaction
This is part of Quine's proposal (c.f. Idea 8207) that entities have to be multiplied in order to produce simplicity. He is speculating. Maybe they are proposed because they are just obvious, and the generality is a nice side-effect.
Related Idea
Idea 8207 The quest for simplicity drove scientists to posit new entities, such as molecules in gases [Quine]
| 8205 | Explaining events just by bodies can't explain two events identical in space-time [Quine] |
| 8206 | Necessity could be just generalisation over classes, or (maybe) quantifying over possibilia [Quine] |
| 8207 | The quest for simplicity drove scientists to posit new entities, such as molecules in gases [Quine] |
| 8208 | In arithmetic, ratios, negatives, irrationals and imaginaries were created in order to generalise [Quine] |