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Full Idea
Representing arithmetic formally we do not primarily care about semantic features of number words. We are interested in capturing the inferential relations of arithmetical statements to one another, which can be done elegantly in first-order logic.
Gist of Idea
First-order logic captures the inferential relations of numbers, but not the semantics
Source
Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §6.3)
Book Ref
-: 'Philosophical Review 114' [Phil Review 2005], p.217
A Reaction
This begins to pinpoint the difference between the approach of logicists like Frege, and those who are interested in the psychology of numbers, and the empirical roots of numbers in the process of counting.
| 9998 | What is the relation of number words as singular-terms, adjectives/determiners, and symbols? [Hofweber] |
| 10000 | We might eliminate adjectival numbers by analysing them into blocks of quantifiers [Hofweber] |
| 10001 | An adjective contributes semantically to a noun phrase [Hofweber] |
| 10002 | '2 + 2 = 4' can be read as either singular or plural [Hofweber] |
| 10003 | Why is arithmetic hard to learn, but then becomes easy? [Hofweber] |
| 10004 | Our minds are at their best when reasoning about objects [Hofweber] |
| 10005 | Arithmetic doesn’t simply depend on objects, since it is true of fictional objects [Hofweber] |
| 10006 | First-order logic captures the inferential relations of numbers, but not the semantics [Hofweber] |
| 10008 | Arithmetic is not about a domain of entities, as the quantifiers are purely inferential [Hofweber] |
| 10007 | Quantifiers for domains and for inference come apart if there are no entities [Hofweber] |