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Single Idea 19295
[filed under theme 6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
]
Full Idea
The existence of the natural numbers is not a matter of pure logic - it cannot be proved in pure logic. It can be proved in second-order logic plus Hume's principle. Truths of arithmetic are not logic - they depend on the nature of natural numbers.
Gist of Idea
Add Hume's principle to logic, to get numbers; arithmetic truths rest on the nature of the numbers
Source
Bob Hale (Necessary Beings [2013], 07.4)
Book Ref
Hale,Bob: 'Necessary Beings' [OUP 2013], p.177
A Reaction
Hume's principles needs entities which can be matched to one another, so a certain ontology is needed to get neo-logicism off the ground.
The
15 ideas
with the same theme
[revival of logicism after much criticism]:
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21718
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Ramified types can be defended as a system of intensional logic, with a 'no class' view of sets
[Russell, by Linsky,B]
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18144
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Neo-logicists agree that HP introduces number, but also claim that it suffices for the job
[Bostock]
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18147
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Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number
[Bostock]
|
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7804
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Wright has revived Frege's discredited logicism
[Wright,C, by Benardete,JA]
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13899
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The Peano Axioms, and infinity of cardinal numbers, are logical consequences of how we explain cardinals
[Wright,C]
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13896
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The aim is to follow Frege's strategy to derive the Peano Axioms, but without invoking classes
[Wright,C]
|
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10622
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The neo-Fregean is more optimistic than Frege about contextual definitions of numbers
[Hale/Wright]
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8783
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Logicism might also be revived with a quantificational approach, or an abstraction-free approach
[Hale/Wright]
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12225
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Neo-Fregeanism might be better with truth-makers, rather than quantifier commitment
[Hale/Wright]
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10580
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Mathematics is both necessary and a priori because it really consists of logical truths
[Yablo]
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9224
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Proceduralism offers a version of logicism with no axioms, or objects, or ontological commitment
[Fine,K]
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13664
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Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions
[Shapiro]
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18167
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We can get arithmetic directly from HP; Law V was used to get HP from the definition of number
[Maddy]
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19295
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Add Hume's principle to logic, to get numbers; arithmetic truths rest on the nature of the numbers
[Hale]
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21648
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Neo-Fregeans are dazzled by a technical result, and ignore practicalities
[Hofweber]
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