> As a partial answer in that case...don't think you have the market > cornered on arrogance. Not that I necessarily have cause, mind you. > Actually, I had already read that proof and your entire page, and > suspected there was some flawed thinking going on. It's not that it > flies in the face of intuition. It's just that if intuition can > hinder understanding at one level, then it can at successive levels > too.
I totally agree with you. In truth, I have more respect for those of you who refuse to simply be convinced because you have read my argument. I've come to the conclusion at this stage in my life that almost all people think they understand something if they: A) can understand an annunciation of the idea in question, and B) "know" the answer. But that is not necessarily comprehension. In fact, it rarely is. Unfortunately, our current educational system emphasizes the recitation of correct "answers", without checking very hard to see if there is comprehension.
Most people that visit my page, I imagine, think, Oh, hmm. I guess it must be better to switch. If you ask them later *why* it is better, you'll get something like, Well, I think it's because, you know, Monty's opened that other door. If you scratch the surface of their "understanding", you'll find that it's a facade. And I think I should be clear that I believe that these folks -- most folks -- actually think they understand something in this situation. Maybe they do understand something, just not nearly so much as they believe.
So, it's okay for you to be obstinate. You are obstinate because you expect comprehension of the correct answer, and the way you have comprehended the problem gives you a different answer than I give. That's because -- you don't realize it -- you are comprehending a different problem than the problem that is stated. I'm not being sly -- your intuition about this is generally correct: the MHP "sneeks" something in there under your intuitive radar that substantially affects what the problem is. Your intuition is based upon a comprehension of what the MHP seems to be, and -- if it were what it seems to be -- your intuition would be correct. Many mathematicians have made this mistake with this problem.
Indeed, one might argue -- and I do -- that this level of "miscomprehension" is as common as the one I ealier discussed. That is, often when there is understanding, it is understanding of the wrong thing. Humans tend to think metaphorically, and it is an enormous strength. However, some metaphors are inappropriate. Often, it takes some time to realize this. This is what philosophers of science call a paradigm shift, and it happens every day at the level of the individual human being. As important as it is to be able to reason by analogy -- and I think it is very, very important -- it is crucial that one be facile at using different modalities. Often, one finds that a different analogy "fits" much better.
I wrote earlier that "some metaphors are inappropriate". What I actually mean is that some metaphors are more appropriate -- or useful -- than others. In general, I call this idea the idea of the appropriate level of description. Most people, I think, have preferred intellectual modalities, prefered levels of description. They tend to attempt to explain eveything (or almost everything) within the context of that modality, that level of description. Sometimes, it is very important to "let go" of one's intellectual anchors (in a given port, so to speak), and drift for a moment -- one often finds a much better anchorage. Murray Gell-Mann has called this a movement from "one idea well to a deeper idea well". He is making an analogy (!) from particle physics: sometimes it takes an increase in energy to move to a lower energy state -- you have to climb out of the well you are in, before you can climb down into a deeper one.
I am intrigued by the MHP because I am intrigued by how people "understand" things. In perhaps an equally philosophical sense as a psychological sense. For example, I am enormously interested in the rational path of stepping stones -- paradigm shifts -- that have brought western thought from the Greeks to today. This has been a long process of discovering that *there are more and less useful levels of description for a given situation*. The Greeks were fully aware that the mathematics of a heliocentric universe were simpler and more elegant that of their geocentric universe. However, they could not stomach -- pardon the pun -- the idea of the Earth in motion. That was not unreasonable. They had no paradigm for an Earth in motion, no way to rationaly understand this[1]. They were forced to accept the less elegant mathematics of a geocentric universe.
A natural question follows this -- admittedly lengthy -- discourse. That is: do we ever really understand anything? My answer would probably have to be No. But then I would add, Does it matter? The practical result of our quest is the discovery of deeper 'idea wells'. One might say that we are learning the mind of God, but will never fully comprehend it.
--Keith
1) This may have been the lesser of two difficulties the Greeks had with a geocentric model. A bigger problem for them, a much bigger problem for them, is that there's no observed parallax of the stars as the Earth (presumably) moves in its orbit around the Sun. The Greeks knew that the Earth was round, of course, and by the time of Archimedes, they had correctly calculated its circumference. (Columbus was aware that the Earth was round, as well; and so did the other educated Europeans of his time. I don't know why the myth to the contrary is so widely believed. In fact, Columbus was more ignorant than his peers in that he believed, incorrectly, that the Earth was much smaller than it actually is.) Given that the Greeks were aware that the Earth was big, they they were also aware that this presumed orbit of the Earth around the Sun must also be pretty big -- and yet there's no parallax of the stars (that they were capable of obsverving)! That means that the stars -- attached to the inside of what the Greeks assumed was the rotating outermost sphere of the cosmos -- were very, very, very far away. But still visible. This presented a severe problem of scale to the Greeks -- the universe must be much larger than they could possibly imagine. It is, of course. And although every schoolchild today is told that the solar system is heliocentric and that the universe is very large, almost no one except astronomers are very cognizant of the unimaginable magnitudes involved.