How has the mathematics community reacted? With Gödel, at first there was a lot of shock. As a student in the late 1950s and early 1960s, I would read essays by Weyl, by John von Neumann, and by other mathematicians. Gödel really disturbed them. He took away their belief in the Platonic world of ideas, the principle that mathematical truth is black or white and provides absolute certainty.Yes, of course mathematicians use axiomatic theories. They have for over two millennia. That is what mathematics is.

But now, strangely enough, the mathematics community ignores Gödel incompleteness and goes on exactly as before, in what I would call a Hilbertian spirit, or following the Bourbaki tradition, the French school inspired by Hilbert. Formal axiomatic theories are still the official religion in the mathematics community. ...

Gödel incompleteness is even unpopular among logicians. They are ambivalent. On the one hand, Gödel is the most famous logician ever. But, on the other hand, the incompleteness theorem says that logic is a failure.

Godel certainly did not prove that logic is a failure. He would have vehemently resisted interpreting his work that way.

Being that all mathematics is a subset of logic, I'm not sure how anyone can consider logic a failure without also considering mathematics a failure. The real limitation of logic is also its strength, logical arguments can only be as strong as the facts or assumptions that are used. If you build a house with absolute skill but faulty materials, the results will always be limited to what the faulty materials can carry. This limitation is what makes logic useful to begin with, it is not a failing at all.

ReplyDeleteMathematics is not strictly a subset of logic and people that say this are just propagandists defending the abstruse nonsense of the 19th and 20th centuries. ALL continuous mathematics was never logical (as stated) but self-contradictory. Whether they veil a finite argument is beside the point. Feynman said physicists need 'Babylonian' and not 'Greek' mathematics and Poincare said set theory was a disease. Euclid is quite limited and basically collected together existing mathematical knowledge. He actually made us surprised at Riemannian geometry! The obsession with foundations largely occurred in the 19th and 20th centuries. No solid foundation for analysis has ever occurred but despite the mathematics not being completely logical, it's still used today. Solomon Feferman has given a predicative account but this still has problems. I would certainly like for mathematics to be logical.

DeleteChaitin spews nonsense and says incorrect things. He says Turing's results are more powerful than Godel's but this is completely backwards (one can construct a strict equivalence without soundness assumptions but this didn't happen until Kleene). Incompleteness is utterly trivial and is a simple reasoning from logical explosion. It says absolutely nothing about logicism.

This Chaitin guy's mathematics is apt for the TED's talk circuit. There are many "genius" like him in California hanging around Silicon Valley that pander this kind of BS.

ReplyDeleteMatthew,

ReplyDeleteI'm sorry historical fact offends your delicate sensibilities. Moan and wail all you like, mathematics is a subset, it is basically quantized logic, No logic, no math. What in the hell do you think a logical operator is? How about the concept or evaluation of greater than, less than, equal to? Precedence of operation? Do you really believe these concepts float around in the platonic aether independently of a grounded logical structure? It is the logical structure that serves as a framework allowing mathematicians to diddle themselves silly in their imaginary constructions. It is logic gates, not 'math' gates that allow computers to even work on doing calculation. Calculation itself is a series of logical procedures that must be followed if you wish to get the correct answer. Study some binary logic for cripes sake and ground yourself before you float off the planet. Numbers aren't even possible without a logical structure beneath them in either human abstraction or actual system architectures.

P.S. Riemannian geometry does not supplant Euclidian geometry. Logically, all curvature exists within and can be described by Euclidian space, not the other way around. If you have a curved space sans a flat space beneath it to measure and define your curve against, what do you have? Nothing. Curved compared to what exactly? Doh! Go back and study the concept of logical dependency.

Do not ever quote Feynman to win an argument. Feynman said lots of things (much like a fortune cookie), many of them contradictory. Feynman didn't really understand the difference between a calculation and a cause or a force. He also wasn't so much a physicist as a overhyped heuristic mathematician. Feynman loved to ridicule others for what he regularly did himself, he paid a LOT of lip service to talking about science, and then acted like a smug priest among his cult following.

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Delete"Moan and wail all you like, mathematics is a subset, it is basically quantized logic, No logic, no math."

DeleteYou don't even read the literature and ignored what I wrote. CONTINUOUS MATHEMATICS IS SELF-CONTRADICTORY! Most math is not deduced from axioms but many results are later reduced to them. Doron Zeilberger: "My mind was made up about a month ago, during a wonderful talk (in the INTEGERS 2005 conference in honor of Ron Graham's 70th birthday) by MIT (undergrad!) Jacob Fox (whom I am sure you would have a chance to hear about in years to come), that meta-proved that the answer to an extremely concrete question about coloring the points in the plane, has two completely different answers (I think it was 3 and 4) depending on the axiom system for Set Theory one uses. What is the right answer?, 3 or 4? Neither, of course! The question was meaningless to begin with, since it talked about the infinite plane, and infinite is just as fictional (in fact, much more so) than white unicorns. Many times, it works out, and one gets seemingly reasonable answers, but Jacob Fox's example shows that these are flukes."

CFT: "Riemannian geometry does not supplant Euclidian geometry."

I didn't say it did! The point is that axiomatics is largely a clerical obsession with little creativity. There is no reason to believe that the arbitrary axioms used in ZFC are better than other starting points. Do you know how many pages it took Russell to derive 1 + 1 = 2?

Feynman was dead on but if you want to dumb down math education, then be my guest.

Matthew,

ReplyDeleteafter reading and re-reading the posts above, it is quite clear that if you had a point, it has obviously been lost in complexity or translation. I made the logical assertion that math was a subset of logic, which IT IS, historically, epistemologically, and logically. You wished to belabor the point, it went nowhere. You then bring in continuous mathematics, I have no idea why. Axiomatics is not a 'clerical obsession', any more than a ridiculous number of pages of lengthy formulas to derive 1 + 1 = 2 is a 'clerical obsession'.

It somewhat apparent you are more interested in level of needless complexity that excludes everyone but a priesthood of mathematicians from partaking of understanding, than making your idea(s) accessable. Screw that. I have zero patience with the gatekeeper philosophies of priests, tyrants, and technocrats. Learn to make your arguments accessible through reasoned clear language, not proprietary unformatted source code. Anyone can spout buzzwords and internal nomenclature to drown out or shame dissent, but it isn't a valid argument.

Lastly, Feynman is just dead, not 'dead on'.

Feynman is possibly once of the most celebrated hypocrite in the world. He pretty much said everything on the side of almost every argument, I don't take him seriously, as he wasn't serious personally or professionally. I have little regard for a man who beats his own drums to celebrate himself while admitting out of the other side of his mouth "I think we can safely say that nobody understands quantum mechanics." (which is actually true. Neither Feynman or his followers to this day have a clue either, they have just accepted the formulized dogma they were taught without understanding (which is basically cargo cult science by his own words.)), he also said " The shell game that we play ... is technically called 'renormalization'. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It's surprising that the theory still hasn't been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate." ...right from the horses mouth. Maybe if he wanted to actually understand what he was dabbling with, he could have tried actual mechanics instead of heuristics (which is not mechanics). Instead, we get butchered infinities as the most celebrated invention since sliced bread. Whoopee.

Feynman and you really should have taken another old physicists words to heart:

"If you can’t explain it simply, you don’t understand it well enough"

Now that is a great quote.