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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 23 ideas from 'Necessary Beings'

19275 You cannot understand what exists without understanding possibility and necessity [Hale]
19276 The big challenge for essentialist views of modality is things having necessary existence [Hale]
19278 There is no gap between a fact that p, and it is true that p; so we only have the truth-condtions for p [Hale]
19279 What are these worlds, that being true in all of them makes something necessary? [Hale]
19281 Interesting supervenience must characterise the base quite differently from what supervenes on it [Hale]
19282 It seems that we cannot show that modal facts depend on non-modal facts [Hale]
19286 'Absolute necessity' is when there is no restriction on the things which necessitate p [Hale]
19285 Logical necessity is something which is true, no matter what else is the case [Hale]
19287 Maybe each type of logic has its own necessity, gradually becoming broader [Hale]
19288 Logical and metaphysical necessities differ in their vocabulary, and their underlying entities [Hale]
19289 Maybe conventionalism applies to meaning, but not to the truth of propositions expressed [Hale]
19290 Absolute necessities are necessarily necessary [Hale]
19291 A canonical defintion specifies the type of thing, and what distinguish this specimen [Hale]
19293 Essentialism doesn't explain necessity reductively; it explains all necessities in terms of a few basic natures [Hale]
19294 If necessity derives from essences, how do we explain the necessary existence of essences? [Hale]
19295 Add Hume's principle to logic, to get numbers; arithmetic truths rest on the nature of the numbers [Hale]
19296 If second-order variables range over sets, those are just objects; properties and relations aren't sets [Hale]
19297 The two Barcan principles are easily proved in fairly basic modal logic [Hale]
19298 Unlike axiom proofs, natural deduction proofs needn't focus on logical truths and theorems [Hale]
19299 Possible worlds make every proposition true or false, which endorses classical logic [Hale]
19300 The molecules may explain the water, but they are not what 'water' means [Hale]
19301 With a negative free logic, we can dispense with the Barcan formulae [Hale]
19302 If a chair could be made of slightly different material, that could lead to big changes [Hale]