The_Duck comments on Rationality Quotes September 2012 - Less Wrong

7 Post author: Jayson_Virissimo 03 September 2012 05:18AM

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Comment author: VKS 04 September 2012 11:51:02PM *  23 points [-]

After I spoke at the 2005 "Mathematics and Narrative" conference in Mykonos, a suggestion was made that proofs by contradiction are the mathematician's version of irony. I'm not sure I agree with that: when we give a proof by contradiction, we make it very clear that we are discussing a counterfactual, so our words are intended to be taken at face value. But perhaps this is not necessary. Consider the following passage.

There are those who would believe that every polynomial equation with integer coefficients has a rational solution, a view that leads to some intriguing new ideas. For example, take the equation x² - 2 = 0. Let p/q be a rational solution. Then (p/q)² - 2 = 0, from which it follows that p² = 2q². The highest power of 2 that divides p² is obviously an even power, since if 2^k is the highest power of 2 that divides p, then 2^2k is the highest power of 2 that divides p². Similarly, the highest power of 2 that divides 2q² is an odd power, since it is greater by 1 than the highest power that divides q². Since p² and 2q² are equal, there must exist a positive integer that is both even and odd. Integers with this remarkable property are quite unlike the integers we are familiar with: as such, they are surely worthy of further study.

I find that it conveys the irrationality of √2 rather forcefully. But could mathematicians afford to use this literary device? How would a reader be able to tell the difference in intent between what I have just written and the following superficially similar passage?

There are those who would believe that every polynomial equation has a solution, a view that leads to some intriguing new ideas. For example, take the equation x² + 1 = 0. Let i be a solution of this equation. Then i² + 1 = 0, from which it follows that i² = -1. We know that i cannot be positive, since then i² would be positive. Similarly, i cannot be negative, since i² would again be positive (because the product of two negative numbers is always positive). And i cannot be 0, since 0² = 0. It follows that we have found a number that is not positive, not negative, and not zero. Numbers with this remarkable property are quite unlike the numbers we are familiar with: as such, they are surely worthy of further study.

  • Timothy Gowers, Vividness in Mathematics and Narrative, in Circles Disturbed: The Interplay of Mathematics and Narrative
Comment author: DanArmak 06 September 2012 10:01:40PM *  3 points [-]

The two examples are not contradictory, but analogous to one another. The correct conclusion in both is the same, and both are equally serious or ironic.

  1. Suppose x² -2=0 has a solution that is rational. That leads to a contradiction. So any solution must be irrational.

  2. Suppose x² +1=0 has a solution that is a number. That leads to a contradiction. So any solution must not be a number. Now what is a "number" in this context? From the text, something that is either positive, negative, or zero; i.e. something with a total ordering. And indeed we know (ETA: this is wrong, see below) that such solutions, the complex numbers, have no total ordering.

I see no relevant difference between the two cases.

Comment author: The_Duck 06 September 2012 10:25:29PM *  4 points [-]

You can work the language a little to make them analogous, but that's not the point Gowers is making. Consider this instead:

"There are those who would believe that all equations have solutions, a view that leads to some intriguing new ideas. Consider the equation x + 1 = x. Inspecting the equation, we see that its solution must be a number which is equal to its successor. Numbers with this remarkable property are quite unlike the numbers we are familiar with. As such, they are surely worthy of further study."

I imagine Gowers's point to be that sometimes a contradiction does point to a way in which you can revise your assumptions to gain access to "intriguing new ideas", but sometimes it just indicates that your assumptions are wrong.

Comment author: CronoDAS 06 September 2012 10:49:11PM *  8 points [-]

"There are those who would believe that all equations have solutions, a view that leads to some intriguing new ideas. Consider the equation x + 1 = x. Inspecting the equation, we see that its solution must be a number which is equal to its successor. Numbers with this remarkable property are quite unlike the numbers we are familiar with. As such, they are surely worthy of further study."

Yes, yes they are.

Comment author: DanArmak 06 September 2012 10:53:51PM *  1 point [-]

Consider the equation x + 1 = x.

(Edited again: this example is wrong, and thanks to Kindly for pointing out why. CronoDAS gives a much better answer.)

Curiously enough, the Peano axioms don't seem to say that S(n)!=n. Lo, a finite model of Peano:

X = {0, 1} Where: 0+0=0; 0+1=1+0=1+1=1 And the usual equality operation.

In this model, x+1=1 has a solution, namely x=1. Not a very interesting model, but it serves to illustrate my point below.

sometimes a contradiction does point to a way in which you can revise your assumptions to gain access to "intriguing new ideas", but sometimes it just indicates that your assumptions are wrong.

Contradiction in conclusions always indicates a contradiction in assumptions. And you can always use different assumptions to get different, and perhaps non contradictory, conclusions. The usefulness and interest of this varies, of course. But proof by contradiction remains valid even if it gives you an idea about other interesting assumptions you could explore.

And that's why I feel it's confusing and counterproductive to use ironic language in one example, and serious proof by contradiction in another, completely analogous example, to indicate that in one case you just said "meh, a contradiction, I was wrong" while in the other you invented a cool new theory with new assumptions. The essence of math is formal language and it doesn't mix well with irony, the best of which is the kind that not all readers notice.

Comment author: VKS 07 September 2012 05:05:09AM 3 points [-]

But that's the entire point of the quote! That mathematicians cannot afford the use of irony!

Comment author: DanArmak 07 September 2012 08:54:39AM *  0 points [-]

Yes. My goal wasn't to argue with the quote but to improve its argument. The quote said:

But could mathematicians afford to use this literary device? How would a reader be able to tell the difference in intent between what I have just written and the following superficially similar passage?

And I said, it's not just superficially similar, it's exactly the same and there's no relevant difference between the two that would guide us to use irony in one case and not in the other (or as readers, to perceive irony in one case and serious proof by contradiction in the other).

Comment author: Kindly 06 September 2012 11:24:02PM *  2 points [-]

Your model violates the property that if S(m) = S(n), then m=n, because S(1) = S(0) yet 1 != 0. You might try to patch this by changing the model so it only has 0 as an element, but there is a further axiom that says that 0 is not the successor of any number.

Together, the two axioms used above can be used to show that the natural numbers 0, S(0), S(S(0)), etc. are all distinct. The axiom of induction can be used to show that these are all the natural numbers, so that we can't have some extra "floating" integer x such that S(x) = x.

Comment author: DanArmak 07 September 2012 08:47:01AM 0 points [-]

Right. Thanks.

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