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Full Idea
Arithmetic should be able to face boldly the dreadful chance that in the actual world there are only finitely many objects.
Gist of Idea
Arithmetic must allow for the possibility of only a finite total of objects
Source
Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.148)
Book Ref
-: 'Journal of Philosophy' [-], p.148
A Reaction
This seems to be a basic requirement for any account of arithmetic, but it was famously a difficulty for early logicism, evaded by making the existence of an infinity of objects into an axiom of the system.
| 13155 | If you add one to one, which one becomes two, or do they both become two? [Plato] |
| 9865 | Daily arithmetic counts unequal things, but pure arithmetic equalises them [Plato] |
| 16929 | 7+5 = 12 is not analytic, because no analysis of 7+5 will reveal the concept of 12 [Kant] |
| 17612 | Arithmetic is just the consequence of counting, which is the successor operation [Dedekind] |
| 14441 | The formal laws of arithmetic are the Commutative, the Associative and the Distributive [Russell] |
| 10619 | The truths of arithmetic are just true equations and their universally quantified versions [Smith,P] |
| 10026 | Arithmetic must allow for the possibility of only a finite total of objects [Hodes] |
| 15714 | 'Commutative' laws say order makes no difference; 'associative' laws say groupings make no difference [Kaplan/Kaplan] |
| 15715 | 'Distributive' laws say if you add then multiply, or multiply then add, you get the same result [Kaplan/Kaplan] |
| 21665 | The fundamental theorem of arithmetic is that all numbers are composed uniquely of primes [Hofweber] |