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Warrigal comments on Proofs, Implications, and Models - Less Wrong
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Comments (209)
Meditation:
It has been claimed that logic and mathematics is the study of which conclusions follow from which premises. But when we say that 2 + 2 = 4, are we really just assuming that? It seems like 2 + 2 = 4 was true well before anyone was around to assume it, that two apples equalled two apples before there was anyone to count them, and that we couldn't make it 5 just by assuming differently.
My two cents:
"2 + 2 = 4" is not the same type of statement as, say, "sugar dissolves in water". The statement "sugar dissolves in water" refers to a fact about the world; there are experiments we can perform that would verify that statement.
The statement "2 + 2 = 4", on the other hand, doesn't refer to a fact about the world; it refers to a truth-preserving transformation that can be applied to facts about the world. It allows us to transform "I have 2 + 2 apples" into "I have 4 apples", and "he is 4 years old" to "he is 2 + 2 years old", and so on.
What do the symbols "2", "4", and "+" mean here? Well, they mean a different thing in each context. In one context, "2" means "two apples", "4" means "four apples", and "X + Y" means "X apples, and also Y separate apples". In the other, "2" means "two years earlier", "4" means "four years earlier", and "X + Y" means "X years earlier than the time that was Y years earlier".
Why do we use the same symbols with different meanings? Because there happens to be a set of truth-preserving transformations—the ring axioms—that can be applied to all of these different meanings. Since they obey the same axioms, they also obey the transformation "2 + 2 = 4", which is just a composite of axioms.
Come to think of it, apples don't actually satisfy the ring axioms. In particular, if you have at least one apple, there is no number of apples I can give you such that you no longer have any apples.
In fancy math-talk, we can say apples are a semimodule over the semiring of natural numbers.
You could quibble that there is a finite supply of apples out there, so that (3 apples) + (all the apples) is undefined, but this model ought to work well enough for small collections of apples.
Apples (and other finite sets of concrete objects) form a semiring.
Nor is it obvious how multiplication of apples should work. Apples might be considered an infinite cyclic abelian monoid, if you like, but it's beside the point - the point is that once you know what axioms they satisfy, you now know a whole bunch of stuff.
Well, if you have a row of 3 apples, and you get another three rows, you'll have 9 apples. But multiplying 3 apples by 3 apples would result in 9 apples^2; and I don't know what those look like.
Sure there is, as long as you realize that "give" and "take" are the same action. Giving -1 apples is just taking 1 apple.
Number of apples isn't closed under taking.
If you wish to satisfy the ring axioms from scratch, you must first invent the economy...
Does an apple composed of antimatter still count as an apple? If I add that to the original apple, I get a big explosion and lots of energy flying around, but neither apple actually remains afterwards.