From: John Conover <john@email.johncon.com>

Subject: forwarded message from root@email.johncon.com

Date: Thu, 14 Nov 1996 01:29:23 -0800

Interesting. Re: my discussion of things Godelian earlier. Today is the anniversary of the death of Leibnitz. As you know, along with Newton, Leibnitz is credited with developing the calculus. Leibnitz was a bit more philosophical than Newton, (actually, Newton practiced science during the day, and alchemy in the evenings, retiring to his laboratory and rubbing stones together in the pursuit of making gold.) One of Leibnitz's prevailing epistemologies was that the deductive logic methodologies could prevail in human affairs, intervening in irrational discussions, so that truth would inevitably be decidable, in a rational, logical manner. The concept did prevail well into this century, and was represented by none other than the most famous mathematician of the 20'th century, David Hilbert. Hilbert had decided, at the turn of the century, that there were 10 essential "problems" that had to be resolved in logic and mathematics to make Leibnitz's concepts a reality. He presented them at the 1900 annual meeting of the mathematical society. For the next twenty years, the likes of Bertrand Russell, John Von Neumann, etc., worked diligently to solve the problems, and make logic complete and consistent. In 1928 Kurt Godel proved them wrong. John BTW, it is now generally accepted that whether a set of axioms, (ie., a theory,) is true or false can not be decided in all cases. In point of fact, it can be shown that in most cases it can't. Undecidability is nature of logic itself. (Whether there are questions that exist that are undecidably undecidable is, at the present, a conjecture.) Kurt Godel was a pretty sharp cookie, BTW. His proof is very clever, and very simple. He just represented axioms and postulates by symbols, and concatenated them into strings. If a symbol appeared more than once in a string, then the logic system was recursive, ie., self-referential, and would be either incomplete or inconsistent, or both, forever. Very trick thinking, that is still the corner stone of the application of information theory to such things. So, like Leibnitz, et al, believed, if it was possible to eliminate recursion, (actually, the issue was known by the ancient Greeks,) then all inconsistency could be removed from a logic system, like mathematics. The trouble is that if you do that, mathematics, ie., the language of logic, degenerates into something that, although it is complete and consistent, is not very useful. For example, you would have to throw out subtraction from the arithmetic, which would be kind of a hindrance in such a simple process as counting things. Then, in the agenda of mathematics in the 20'th century, Hilbert, et al, back peddled on their position, circa 1930, and assumed that, although some things in logic were just simply not decidable, many were, and the techniques used by Godel could be used to prove which ones were. In 1937, Alan Turing proved them wrong-again with a very clever concept that attacked the logic of the arithmetic. His concept was to devise a theoretical machine that could do so, if any machine could, and in the process invented the computer, on which you are probably reading this. He then showed that such a machine could not compute which things were decidable, and which were not. He also defined the difference between calculating and computing. (Technically, a spreadsheet is not computing-it is calculating. And an automated calculator is not a computer. As an interesting side bar, the modern computer is the only machine known that was designed as a theoretical abstraction, and then implemented. All others were implemented, and then the theoretical abstractions worked out.) The ages have not been nice to the folks involved in the drama of the logical consistency of mathematics. Newton never made gold. Leibnitz, after falling on hard financial times, died a pauper in Poland-the only person attending the funeral was his personal secretary. Godel went mad, and spent his last days concerned that people were trying to poison his food, and Turing committed suicide during allegations of a homosexual scandal. Hilbert went to his grave never believing that Godel was correct, but was never able to prove otherwise. His career was not productive after the publication of Godel's proof. Russel's "Principa Mathematica," the anticipated volume of complete and consistent mathematics and logic was unfinished at his death, many decades later. Von Neumann went into economics, then into the National Laboratory system doing research in computation, expanding on Turing's concepts. He died relatively young, in his early 50's. The 20'th century has been kind of an anti-science, as the drama of the limitations of deductive logic played itself out. ------- start of forwarded message (RFC 934 encapsulation) ------- Received: (from root@localhost) by johncon.com (8.6.12/8.6.12) id AAA01462 for john; Thu, 14 Nov 1996 00:05:17 -0800 Message-Id: <199611140805.AAA01462@email.johncon.com> From: root <root@email.johncon.com> To: john@email.johncon.com Subject: Reminders for Thursday, November 14, 1996 Date: Thu, 14 Nov 1996 00:05:17 -0800 Reminders for Thursday, 14th November, 1996 (today): Sunrise 06:46, Sunset 16:58, Moon 0.22 (Increasing) ________________________ On This Day, Nov 14 ... ________________________ Gottfried Wilhelm Leibnitz, German mathematician and philosopher, died (1716) ------- end ------- -- John Conover, john@email.johncon.com, http://www.johncon.com/

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