X-Message-Number: 10319 Subject: more math Date: Tue, 25 Aug 1998 11:57:59 -0400 From: "Perry E. Metzger" <> > From: Thomas Donaldson <> > Subject: CryoNet #10307 - #10312 > > HI everyone! > > Perhaps Metzgar Metzger. This is about the fiftieth time you've misspelled my name. There isn't an "a" in it. The name is also in front of you every time you reply to it. Look at it carefully before trying to type it in. > In any case, there are 2 separate issues. The first of them is > constructivism. You see, Perry, what constructivists are saying is > that if you cannot produce a constructive proof then NO PROOF EXISTS. I understand what they are saying, but I don't see why a "constructivist proof" is any better than a "non-constructivist" proof, except in the mind of the constructivist. > The only thing you are doing when you find a proof by contradiction > is playing with words. In what way is a constructivist proof not "playing with words"? If you get down to it, both are just "phi slinging". > You are saying nothing about the world. You mean you don't believe there are infinitely many primes? > The second issue, perhaps the more interesting one for cryonicists, who > hope to be around for centuries, is that of whether or not we will > continue to use formal methods in our mathematics. We've been using formal methods since Euclid. If you don't use formal methods, you can't do math. > If you define mathematics as requiring a formal system, then OK, > physicists aren't doing mathematics. What makes you think that? You don't seem to understand what a formal system IS. The reason formal systems were developed was to tighten the notion of what the formal methods of mathematics are. For thousands of years, mathematicians have been doing proofs. It is the only way to be sure of your work. However, the metamathematical issue of what constituted valid reasoning in a system of proof always comes up, especially when a "proof" that is disputed arrives on the scene. Formal systems are just a mechanical codification of the formal methods all mathematicians use -- no more, no less. No real mathematician uses them to do his actual work -- even a proof of Euclid's Prime Number Theorem would take up several hundred pages if it were expressed in a purely formal system, and no one wants to do that sort of thing. The point is not to do the math with the formal system. It is to do it with formal methods that permit a mapping into a formal system if need be. If you have a dispute over whether a proof is correct or not, you can, in theory at least, answer the question by checking whether or not the proof can be properly expressed purely in a formal system. If it can't, you then you know there is a problem. The point of formal systems was not to insist that mathematicians do their work in them, but to provide a framework for finally answering the question "what is a proof" and do reasonable work in metamathematics. > I will point out that it is quite possible to scrutinize your own > reasoning, or the reasoning of someone else, without reference to any > formal system. We do it all the time. For that matter, most mathematics > is NOT taught as a formal system. No mathematics is taught in formal systems. You couldn't manage it. People's eyes would bug out. Again, you've lost track of what they are for. > As for Goedel's Theorem, my library will arrive here (I am told) in > about 2 weeks. I will then examine Goedel's Theorem with an eye to > whether or not constructivism escapes the problems it raises. No system that is omega complete can escape it. Therefore, if you argue that constructivism will escape from it, you are arguing that constructivism can't capture the power of proofs by induction. > From: Paul Wakfer <> > Subject: Re: CryoNet #10310 The Essense of Metzger > > From: "Perry E. Metzger" <> > > > Paul, you can't even prove the oldest proof in number theory, that > > there are infinitely many primes, without a proof by > > contradiction. > > The simple answer to this is: so what? What practical benefit is there > to such a proof besides the generation of mathematical "games" within > the body of "art" that is non-constructive mathematics? You might think of it as a "game", but from what I can tell, it is a matter of significance. Even number theory is not useless -- cryptography has turned it into a matter of great importance. > > Sure, you can do some interesting math with your hands tied behind > > your back, but there is no obvious reason to bother. > > Perhaps to delineate what is real from what is purely abstract I can't see any way to distinguish "real" math from "abstract" math. All math is abstract, by definition. The first person to construct the square of a number by multiplying it by itself instead of getting out a tile and measuring the area of a square that size on each side went out past the real into abstraction. That was thousands of years ago, and there is no turning back. > and to develop more powerful methods to apply to reality instead of > using an "easy way" to generate proofs which are then quite useless > for producing material benefits for us. Every day, you use hundreds of results that came out of proofs by contradiction, in everything even mildly technological you work with. Even surveyors using two thousand year old techniques couldn't survive without Euclid. As soon as you get into anything even slightly more technical than that you get buried in this "meaningless abstraction". > You "bother" for the same reason that you don't invoke ghosts, UFOs, > gods, or psychic powers to explain the operation of reality, when doing > so would often be ever so much easier and more fun. A proof by contradiction is the equivalent of a ghost or a UFO? I'm speechless. > > The entire notion > > that some kinds of math were "purer" than others should have gone out > > after Hilbert's madness about finitistic methods and proofs of the > > consistancy of formal systems was tossed out. > > For those who don't know, David Hilbert was a true "giant" among > mathematicians, whose ideas about mathematics rate as much respect and > consideration as do those of Feynman, Hawking, Einstein, etc. in > physics. Only in Perry's mind have any of his ideas been "tossed out". Hilbert's entire program of proving mathematics consistent and complete was demolished by Godel and thus tossed out, or have you forgotten that? Just because Hilbert was a genius doesn't mean everything he said was correct. Hell, everyone remember Fermat's belief he had a formula to construct primes? Perry Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=10319