Saturday, August 23, 2008

Misplaced pluralism

Via EvolutionBlog, I found this article in American Scientist covering the Monty Hall problem. Good coverage, but I disagree with this last paragraph:
Making progress in the sciences requires that we reach agreement about answers to questions, and then move on. Endless debate (think of global warming) is fruitless debate. In the Monty Hall case, this social process has actually worked quite well. A consensus has indeed been reached; the mathematical community at large has made up its mind and considers the matter settled. But consensus is not the same as unanimity, and dissenters should not be stifled. The fact is, when it comes to matters like Monty Hall, I'm not sufficiently skeptical. I know what answer I'm supposed to get, and I allow that to bias my thinking. It should be welcome news that a few others are willing to think for themselves and challenge the received doctrine. Even though they're wrong.
No, dissenters shouldn't be stifled. But in this case I can't think of any way in which dissent is a good thing. The Monty Hall problem isn't just considered settled, it is settled, with deductive certainty. Oh, sure, the principle of pluralism says it's healthy to have a diversity of views, but you have to think about why we value pluralism. I think pluralism is good because of the possibility that the consensus is wrong. That possibility simply doesn't exist here. Pluralism is bad in this case. It's okay, I'm sure we can find other things to disagree about.

The same goes for the following claims:


Anonymous said...

This is one of the ways mathematics differs from science--we can make absolute statements and be absolutely sure of one answer. If you don't agree with an established mathematical fact, it's not because you have contradictory evidence or doubt the sufficiency of the existing evidence, it's because you're doing it wrong. (Leaving aside the issue of using different axiomatic systems, etc.)

Then of course there are some cases where we truly can't determine whether a statement is true or false in the current system, e.g. the Continuum Hypothesis. But we can prove that it's independent of the other axioms, so it's not really the same as dissenting from the consensus.

Yoo said...

On the other hand, "3.999... does not equal 4" claims can actually be enlightening, as long as it doesn't stay in a stupid "it's this way because I say so, backed up by a nebulous number system I never make clear".

One could use it to showcase other interesting things, such as a lexicographical basis for a real number system, in which 3.9999... would indeed not equal 4, but would lack nice properties such as x-y=0 implying x=0. It could be used to show why certain axioms and definitions are important, and also to show the "artificial" nature of math.

Yoo said...

Doh! x-y=0 normally implies x=y, not x=0! (except when y=0 ...)

miller said...

Well, I never said it wasn't enlightening to consider these claims. Many an interesting discussion has been started by the concept of infinity. The only thing is that there are well-defined resolutions to such discussions, and we need not praise disagreement for its own sake.

Yoo said...

"and we need not praise disagreement for its own sake.": gotta agree with this. :)