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lessdazed comments on Rationality Quotes October 2011 - Less Wrong
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I do not understand.
-- Carl Linderholm, Mathematics Made Difficult
Let me explain why it's not easy to see that 5+4 is not 6.
Earlier, the numbers were defined as
2 = 1+1
3 = 1+2
4 = 1+3
5 = 1+4
6 = 1+5
7 = 1+6
8 = 1+7
9 = 1+8.
Where + is associative.
Consider a "clock" with 3 numbers, 1, 2, 3. x+y means "Start at x and advance y hours".
3
2 -> 1
Then 1+1 = 2 and 2+1 = 3, as per our definitions. Also, 3+1 = 1 (since if you start at the 3 and advance 1 hour, you end up at 1). Thus 4 = 1, 5 = 4+1 so 5 = 1+1 = 2.
So 6 = 5+1 = 5 + 4.
So because the numbers were defined with eight examples, no example explicitly showing associativity or commutivity, it's hard to see why there's no license to arbitrarily choose a modulus for each number?
Or perhaps we only feel like we can do that if that would let us make two sides of an equation equal? As if the implicit rule connoted by the examples was "if two sides of an equation can be interpreted as "equal", one must declare them "equal", where "equal" is defined as amounting to the same, whatever modular operations must be done to make it so? So the definitions are incomplete without an example of something that does not equal something else?
It's not just about 8 examples - with any number of examples it would be perfectly valid to insert something like 6 = 1. And so there's an additional axiom in Peano arithmetic that has to explicitly rule it out (if you're talking about numbers that way). Not super-shocking.
My interpretation of the original quote was to take "see that 5 + 4 is not 6" as "prove that you cannot prove that 5 + 4 = 6", in other words, "prove that Peano's arithmetic is consistent". Maybe I was too influenced by this.
I think that's a way better interpretation :D
My understanding is that given those eight definitions, it is impossible to prove any inequalities, because no inequality is given as an axiom, nor any properties that are true of some numbers but not others.