41 lines
1.5 KiB
Plaintext
41 lines
1.5 KiB
Plaintext
How I made the natural number game.
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Peter Johnstone taught me that 0 was {}, 1 was {{}}, 2 was {{},{{}}} and so on. But what about all the people who didn't go to that class?
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Set theory is *one way of doing mathematics*.
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Type theory is *another way of doing mathematics*
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Naturals, integers, rationals, reals. Diophantus actually worked with positive rationals, which have a beautiful multiplicative structure: they are the free multiplicative abelian group on the primes. Similarly the positive naturals {1,2,3,4,...} (the British Natural Numbers, as I was taught at the University of Cambridge) as Patrick teaches the French in Saclay.
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But when you look at a natural number in type theory b;ah blah
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In mathematics, when you say "let G be a group" what you mean is that G is a set, and G has elements, and the elements are all individual "atoms", and that it is impossible to split the atom.
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Building the natural number game
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Think of some good targets.
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2+2=4
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x+y=y+x
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all axioms of a commutative semiring
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(e.g. x*(y+z)=x*y+x*z etc) and maybe several other natural statements (0*x=0 etc)
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All axioms of a total ordering <=
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Later:
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Archie Browne divisibility world all by himself
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Ivan ... even_odd world
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Solving 3*x+4*y=7 and 5*x+6*y=11.
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Diophantus would have understood that
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question and he hadn't even invented 0 yet.
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We prove cancellation.
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add_comm and add_assoc and then we use simp to make one tactic which can solve this. We make a term called decidable_eq and then all of a sudden we
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A theorem
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Load of old nonsense.
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IDeas: geometry game,
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