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Single Idea 8208

[filed under theme 2. Reason / B. Laws of Thought / 6. Ockham's Razor ]

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]

The 27 ideas with the same theme ['Do not multiply entities beyond necessity']:

21267 Supposing many principles is superfluous if a few will do it [Aquinas]
16676 Why use more things when fewer will do? [William of Ockham]
6806 Do not multiply entities beyond necessity [William of Ockham]
19342 Reason avoids multiplying hypotheses or principles [Leibniz]
4541 Everything simple is merely imaginary [Nietzsche]
22644 Our greatest pleasure is the economy of reducing chaotic facts to one single fact [James]
6106 Reducing entities and premisses makes error less likely [Russell]
2939 If a sign is useless it is meaningless; that is the point of Ockham's maxim [Wittgenstein]
22431 Good algorithms and theories need many occurrences of just a few elements [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]
4036 What matters is not how many entities we postulate, but how many kinds of entities [Armstrong, by Mellor/Oliver]
3426 If one theory is reduced to another, we make fewer independent assumptions about the world [Kim]
21021 Keep premises as weak as possible, to avoid controversial difficulties [Nussbaum]
15784 The Razor seems irrelevant for Meinongians, who allow absolutely everything to exist [Lycan]
15787 Maybe Ockham's Razor is a purely aesthetic principle [Lycan]
10471 Ockham's Razor has more content if it says believe only in what is causal [Oliver]
7038 A theory with few fundamental principles might still posit a lot of entities [Heil]
7037 Parsimony does not imply the world is simple, but that our theories should try to be [Heil]
8964 Entities can be multiplied either by excessive categories, or excessive entities within a category [Hoffman/Rosenkrantz]
4037 Ockham's Razor is the principle that we need reasons to believe in entities [Mellor/Oliver]
4217 It is more extravagant, in general, to revise one's logic than to augment one's ontology [Lowe]
4456 Epistemological Ockham's Razor demands good reasons, but the ontological version says reality is simple [Moreland]
16244 If the universe is profligate, the Razor leads us astray [Maudlin]
16255 The Razor rightly prefers one cause of multiple events to coincidences of causes [Maudlin]
13743 We should not multiply basic entities, but we can have as many derivative entities as we like [Schaffer,J]
16677 Anti-Razor: if you can't account for a truth, keep positing things until you can [Pasnau]