Rationality Quotes September 2012

Jayson_Virissimo

Here's the new thread for posting quotes, with the usual rules:

Please post all quotes separately, so that they can be voted up/down separately. (If they are strongly related, reply to your own comments. If strongly ordered, then go ahead and post them together.)
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Here's the new thread for posting quotes, with the usual rules:

Please post all quotes separately, so that they can be voted up/down separately. (If they are strongly related, reply to your own comments. If strongly ordered, then go ahead and post them together.)
Do not quote yourself
Do not quote comments/posts on LW/OB
No more than 5 quotes per person per monthly thread, please.

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.

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

There are lots of total orderings on the complex numbers. For example:

a + bi [>] c + di iff a >= c or (a = c and b >= d).

In fact, if you believe the axiom of choice there are "nice total orders" for any set at all.

1DanielLC14y

The first one shows that assuming that there's a rational solution leads to contradiction, then drops the subject. The second one shows that assuming that there's a real solution leads to a contradiction, then suggests to investigate the non-reals. How are you supposed to tell which drops the subject and which suggests investigation?

1gwern14y

Isn't that the entire point? I see this as a mathematical version of the modus tollens/ponens point made elsewhere in this page.