Followup to: Expecting Beauty

I was once reading a Robert Heinlein story - sadly I neglected to note down which story, but I do think it was a Heinlein - where one of the characters says something like, "Logic is a fine thing, but I have seen a perfectly logical proof that 2 = 1." Authors are not to be confused with characters, but the line is voiced by one of Heinlein's trustworthy father figures. I find myself worried that Heinlein may have meant it.

The classic proof that 2 = 1 runs thus. First, let x = y = 1. Then:

x = y

x² = xy

x² - y²= xy - y²

(x + y)(x - y) = y(x - y)

x + y = y

2 = 1

Now, you could look at that, and shrug, and say, "Well, logic doesn't always work."

Or, if you felt that math had rightfully earned just a bit more credibility than that, over the last thirty thousand years, then you might suspect the flaw lay in your use of math, rather than Math Itself.

You might suspect that the proof was not, in fact, "perfectly logical".

The novice goes astray and says: "The Art failed me."
The master goes astray and says: "I failed my Art."

Is this - gasp! - faith? To believe that math is consistent, when you have seen with your own eyes a proof that it is not? Are you supposed to just ignore the contrary evidence, my good Bayesian?

As I have remarked before, it seems worthwhile to distinguish "faith" that the sun will rise in the east just like the last hundred thousand times observed, from "faith" that tomorrow a green goblin will give you a bag of gold doubloons. When first-order arithmetic has been observed to be internally consistent over the last ten million theorems proved in it, and you see a seeming proof of inconsistency, it is, perhaps, reasonable to double-check the proof.

You're not going to ignore the contrary evidence. You're going to double-check it. You're going to also take into account the last ten million times that first-order arithmetic has proven consistent, when you evaluate your new posterior confidence that 2 = 1 is not perfectly logical. On that basis, you are going to evaluate a high probability that, if you check for a flaw, you are likely to find one.

But isn't this motivated skepticism? The most fearful bane of students of bias? You're applying a stronger standard of checking to incon...

Julianna_Langston

Julianna_Langston

Julianna_Langston

Julianna_Langston