# Patrick comments on Rationality Quotes October 2011 - Less Wrong

3 03 October 2011 06:41AM

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Comment author: 05 October 2011 02:38:11AM 8 points [-]

With a few brackets it is easy enough to see that 5 + 4 is 9. What is not easy to see is that 5 + 4 is not 6.

Comment author: 05 October 2011 03:46:41AM 2 points [-]

I do not understand.

Comment author: 05 October 2011 09:16:34AM *  1 point [-]

(Great delicacy and tact are needed in presenting this idea, if the aim is, as it should be, to bewilder and frighted the opponent. ...)

-- 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.

Comment author: 05 October 2011 11:20:48AM *  0 points [-]

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?

Comment author: 05 October 2011 11:14:28PM *  7 points [-]

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.

Comment author: 06 October 2011 06:44:34PM 6 points [-]

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.

Comment author: 06 October 2011 08:29:11PM 2 points [-]

I think that's a way better interpretation :D

Comment author: 26 October 2011 02:51:45PM 3 points [-]

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.