Eric

Betsy linked to Diana's comments. Isn't that prima facie evidence that she was not trying to obfuscate? Why insinuate otherwise?

I disagree. The video was recorded off of Showtime and posted to Google video by someone with username Msiadeli. The recording ends before the copyright notice appears (which is normally at the end of the show). Even if permission were granted, wouldn't the owner still insist that the copyright notice be present? In fact, the copyright owner is Showtime, not Penn & Teller. If Showtime wanted to make this video freely available, I would expect them to put it on their own site and not use Google video under some mysterious username. In glancing at their site, it appears that Showtime doesn't make any of their shows freely available. They do SELL some shows (but not Bullshit!) on iTunes. I don't see how anyone could reasonably assume that this video was posted to Google with permission from the copyright owner.

Has anybody on this forum ever heard of copyrights? Just wondering...

Unfortunately, this doesn't really give an "exact replica" of the complex number system in higher dimensions. The problem is division: many of your "numbers" do not have multiplicative inverses. (For example, in 3D, 1 + j*k^2 has no inverse.) Also, Hamilton originally set out to find a 3-dimensional space that contained the complex numbers and that had many of the same features as the complexes. Your "numbers" do not contain the complexes, so you did not solve Hamilton's problem. In fact, it is easy to show that there can be no solution. - Eric

This is one of my favorites: Mr. and Mrs. Jones went to a dinner party with three other married couples. At the party, various handshakes took place, but, of course, no one shook his or her own hand and no shook hands with his or her own spouse. After the party, Mr. Jones asked each of the others how many people he or she had shaken hands with, and he was surprised because each person gave a different answer. How many people did Mrs. Jones shake hands with? - Eric

I'm not aware of Aristotle ever advocating using LEM with the infinite. Could you point me to where he does? To Brouwer (and me) a mathematical object exists if and only if it can be constructed. Thus, assuming that an object doesn't exist and deriving a contradiction does not allow one to use LEM to conclude that the object exists. A contradiction is not a substitute for a construction. Brouwer did indeed limit the use of LEM to propositions whose truth or falsity could be decided in a finite number of steps. The statement "A well ordering of the Reals exists" is arbitrary mysticism. The well ordering is conjured up by the axiom of choice. It has no referent in reality. It has no use in reality. No one has ever constructed one. No one ever will.

Neither I nor Brouwer completely reject LEM. Of course Aristotle got it right: he developed and applied it in finite situations. Would he approve of its use when dealing with the infinite and with existence, as is done by modern mathematicians? I doubt it. Hilbert's statement is hyperbole. Brouwer showed that mathematics - serious mathematics - can be done while restricting the use of LEM to finite situations. Later, in the 60's, Errett Bishop, in his book Foundations of Constructive Analysis, single-handedly developed most of analysis in a constructive framework. And, because LEM isn't used to create mathematical objects ex nihilo, the resulting work is more meaningful (and useful!) than classical analysis. When Bishop proves the existence of a mathematical object, I believe him. When Banach and Tarski "prove" that a sphere can be decomposed into a finite number of pieces and reassembled into two spheres identical to the original, I laugh at the absurdity.

Brouwer showed that there are limits to the applicability of the law of excluded middle (LEM). To me, this is not surprising. Aristotle formulated LEM for finite situations. It's far from obvious that it should still apply when dealing with the infinite. LEM also leads one to trouble when dealing with existence. Would you say "Either God exists or he doesn't"? "God exists" is arbitrary - neither true nor false - so LEM doesn't apply. Can't mathematicians also make arbitrary assertions? To me, there's very little difference between the statements "The Reals can be well ordered" and "God exists".

Mathematical existence is a tricky subject. I believe that most mathematicians don't really think about what they mean when they say "there exists". To them, it is just part of, and therefore gets its meaning from, the mathematical formalism. This is obviously backwards. Meaning must come first. Mathematics is full of so-called "pure existence" or "ideal existence" theorems, i.e., theorems that assert the existence of some object without giving the slightest clue as to how to find it. I believe that such theorems should be distinguished from ones which actually construct the object in question, since there is an obvious difference in pragmatic content. For example, if I need to find a root of an equation, a theorem that asserts the "pure" existence of a solution is of very little value compared to a theorem that constructively finds the root. Yet, today, these two different concepts are both lumped into the mathematicians' "there exists". If any distinction is maintained, it is that pure existence proofs are generally considered more "elegant" than constructive proofs - "elegant" apparently meaning less useful. I personally don't think that "pure existence" is existence. Existence is identity, and absolutely nothing has been identified in a pure existence proof. This is why I deny that the set E above exists: the Axiom of Choice asserts only pure existence. This is not to say that "pure existence" theorems are useless. They tell us, for example, to not bother looking for a counterexample and to focus our efforts on constructing a real solution.

The "construction" of the set E above cannot be carried out by you or me or any finite creature. So, how can E be said to "exist"? In the words of Errett Bishop: "If God has mathematics of his own that needs to be done, let him do it himself." So, what then does it apply to? And, what good is it?

Let's not forget Brouwer. At the beginning of last century, he put up a noble fight against both formalism and logicism by stressing meaning in mathematics. Unfortunately, his theory (the poorly-named "Intuitionism") included some rather bizarre philosphical speculation which ultimately undermined his efforts, but his insight into what's wrong with classical mathematics and his attempt to fix it should be applauded.

Actually, I would say that a more fundamental problem with AC is that it asserts existence without identity. Virtually all of (pure) mathematics divorced existence from identity long ago - and the result is an abstract game with no practical applications. If you have some training in abstract algebra, a very readable proof of the paradox can be found in: R. M. Robinson, "On the Decomposition of Spheres.", Fundamenta Mathematicae, V. 34, p. 246-260, 1947.