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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic

[logical operations with natural numbers]

10 ideas
If you add one to one, which one becomes two, or do they both become two? [Plato]
Daily arithmetic counts unequal things, but pure arithmetic equalises them [Plato]
7+5 = 12 is not analytic, because no analysis of 7+5 will reveal the concept of 12 [Kant]
Arithmetic is just the consequence of counting, which is the successor operation [Dedekind]
The formal laws of arithmetic are the Commutative, the Associative and the Distributive [Russell]
The truths of arithmetic are just true equations and their universally quantified versions [Smith,P]
Arithmetic must allow for the possibility of only a finite total of objects [Hodes]
'Commutative' laws say order makes no difference; 'associative' laws say groupings make no difference [Kaplan/Kaplan]
'Distributive' laws say if you add then multiply, or multiply then add, you get the same result [Kaplan/Kaplan]
The fundamental theorem of arithmetic is that all numbers are composed uniquely of primes [Hofweber]