Saturday, September 24, 2005

Godel, Cantor, Wiener and Schrodinger's Cat

Capitalist Pig Vs. Socialist Swine does a very good job of introducing his/her readers to Godel's Theorm. Where science and philosophy collide: Part I

Godel was the theoritical mathematician who at the ripe young age of 21 proved the mathmatical equivalent of the philosophical conundrum: Nothing is True everything is Permitted.

Gödel's Incompleteness Theorem
In 1931, the Czech-born mathematician Kurt Gödel demonstrated that within any given branch of mathematics, there would always be some propositions that couldn't be proven either true or false using the rules and axioms ... of that mathematical branch itself. You might be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules and axioms, but by doing so you'll only create a larger system with its own unprovable statements. The implication is that all logical system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules.

So I posted in this in his comments section:

Godels Theorm my gawd man what are you on.

And you explained it so well too, which means a). you are a math major b) a theoritical math major c) a physics major d) a theoritical physics major
e) Robert Anton Wilson

Didn't expcet that outta ya swine...Godel is as obscure as they get unless one is reading about Quantum physics and Dr. Schrodinger and his cat.....
While you are at it for math theorms that disprove the proof of math/physics
(which I maybe mistaken in my usage of the term) see The Mystery of the Aleph- Mathematics, the Kabbalh and the search for infinity. by Amir D. Aczel, Pocketbooks 2000.

Its about mathmetician George Cantor, and like Godel he went mad too. Mad I say Mad, bwahhhahaha

Cantors Theorm1

Cantor's theorem2

Georg Cantor's achievement in mathematics was outstanding. He revolutionized the foundation of mathematics with set theory. Set theory is now considered so fundamental that it seems to border on the obvious but at its introduction it was controversial and revolutionary. The controversial element centered around the problem of whether infinity was a potentiality or could be achieved. Before Cantor it was generally felt that infinity as an actuality did not make sense; one could only speak of a variable increasing without bound as that variable going to infinity. That is to say, it was felt that n -> ∞ makes sense but n = ∞ does not. Cantor not only found a way to make sense out an actual, as opposed to a potential, infinity but showed that there are different orders of infinity.

Biographies of Mathematicians-Georg Ferdinand Ludwig Philipp Cantor


Opps Godel didn't go mad he went to Princeton.

Kurt Godel (1906-1978), elected to Academy membership in 1955, was noted for his contributions to the foundations of logic and mathematics. In a celebrated paper published in 1931, Godel first put forward what came to be known simply as "Godel's Theorem": In certain formal systems, there exist propositions that cannot be proved or disproved using the axioms of that system. With this theorem, Godel had effectively demonstrated that some mathematical propositions are undecidable. Godel's Theorem made a deep impact in the fields of mathematics and logic, and has been called the most significant mathematical truth of the 20th century. Godel was born in Brunn (now Brno), in what is now the Czech Republic. He studied physics in Vienna, and emigrated to the US in 1939, where he took a position at Princeton's Institute for Advanced Study. In addition to other honors, in 1975 he was awarded the National Medal of Science, the US government's highest scientific honor.


Godel's theorem
To better understand the impact which Godel's findings must have had on his peers, we should first describe the mathematical climate of the time.
In the nineteenth century it had been discovered, through the work of Riemann, Lobachevsky and others, that coherent models of geometry could be constructed in which Euclid's parallel postulate (that, given a line L and a point P in the plane, exactly one line exists which contains P and is parallel to L) did not hold. This, in itself, was a shock to many mathematicians: for millenia it had been assumed that Euclid's description of geometry, founded as it was on a "self-evident" and minimal set of axioms, was one of the firmest, most trustworthy branches of mathematical knowledge. The existence of non-Euclidean geometries not only challenged mathematicians' geometrical intuition, but also the Platonist view that mathematics consisted of discoveries about eternal, pure forms whose existence was objective and unquestionable. More "monstrosities" such as continuous functions which were nowhere differentiable soon appeared, further fueling the general loss of faith in geometry.

The modern development of the foundations of mathematics

in the light of philosophy

Kurt Gödel (1961)

Now it is a familiar fact, even a platitude, that the development of philosophy since the Renaissance has by and large gone from right to left - not in a straight line, but with reverses, yet still, on the whole. Particularly in physics, this development has reached a peak in our own time, in that, to a large extent, the possibility of knowledge of the objectivisable states of affairs is denied, and it is asserted that we must be content to predict results of observations. This is really the end of all theoretical science in the usual sense (although this predicting can be completely sufficient for practical purposes such as making television sets or atom bombs).

It would truly be a miracle if this (I would like to say rabid) development had not also begun to make itself felt in the conception of mathematics. Actually, mathematics, by its nature as an a priori science, always has, in and of itself, an inclination toward the right, and, for this reason, has long withstood the spirit of the time [Zeitgeist] that has ruled since the Renaissance; i.e., the empiricist theory of mathematics, such as the one set forth by Mill, did not find much support. Indeed, mathematics has evolved into ever higher abstractions, away from matter and to ever greater clarity in its foundations (e.g., by giving an exact foundation of the infinitesimal calculus and the complex numbers) - thus, away from scepticism.

And to think I actually understand this stuff, when I hated math in school. Thats because mathematics and physics are an integral part of philosophy. And what we learned in school was NOT. That's because they taught New Math that incomprehensible clap trap of the sixties that made understanding math as easy as reading Egyptian Hieroglyphics.

Then I read Euclid and Pythagoras, and viola, or was it Eureka, and I understood math as philosophical constructs not just numbers. That the key to understanding the universe was the Golden Section or Golden Ratio, that its construct is the Pentagram, the most ancient symbol of man in the universe, as illustrated by Leonardo Da Vinci's famous drawing of Man, and the funadmental principle underlying both Magick and Science, since they are related. (and you just knew I would get around to mentioning magick)

The Golden Section

The first mathematical occurrence of the golden section and it's associated figures is found in the school of thinkers founded by Pythagoras. The Pythagoreans, as they are known, adopted the pentagram as the symbol of health in their brotherhood, and it eventually came to be the distinguishing badge of their school. Unfortunately, little of their actual mathematics survives, but it is highly likely that they were the ones who derived the construction of the pentagon and decagon from the golden section.

The Golden ratio

Euclid, in The Elements, says that the line AB is divided in extreme and mean ratio by C if AB:AC = AC:CB. Although Euclid does not use the term, we shall call this the golden ratio. The definition appears in Book VI but there is a construction given in Book II, Theorem 11, concerning areas which is solved by dividing a line in the golden ratio. As well as constructions to divide a line in the golden ratio, Euclid gives applications such as the construction of a regular pentagon, an icosahedron and a dodecahedron. Here is how the golden ratio comes into the construction of a pentagon.

Children need to read and be taught mathematics based on these and other original texts, not New Math interpretations of the theorms. Had I, or any of us, been taught properly we would begin with studying Pythagoras, move to Euclid etc. Any Grade Five or Six student can understand these authors, after all they write clearly and explain their ideas without artithmical obfustication. Then we would understand that math is not just some set of numbers but is the rational description of the phyical world, including art, music,biology, etc.

Oh by the way the reason you can read this is because of Godel's Theorm of chance as it applies to computers.

"Thus chance has been admitted, not merely as a mathematical tool for physics, but as part of its warp and weft" Norbert Wiener

And because of Norbert Wiener's classic founding work cybernetics:
Cybernetics or control and communication in the animal and the machine, MIT Press 1996

Weiner's work on cybernetics influenced the workers councils in Allende's Chile,to use computers to develop worker self management of industry, they were crucial to the rationalization of inputs and outputs!

Weiner stated the following in the 1950's and it still applies today
Our papers have been making a great deal of American "know-how" ever since we had the misfortune to discover the atomic bomb. There is one quality more important than "know-how" and we cannot accuse the United States of any undue amount of it. This is "know-what" by which we determine not only how to accomplish our purposes, but what our purposes are to be.

Norbert Wiener's personality was generous: "I want to be the master of nobody"
You knew I would get some sort of libertarian perspective into this article.

Since Leibniz there has perhaps been no man who has had a full command of all the intellectual activity of his day. Since that time, science has been increasingly the task of specialists, in fields which show a tendency to grow progressively narrower. A century ago there may have been no Leibniz, but there was a Gauss, a Faraday, and a Darwin. Today there are few scholars who can call themselves mathematicians or physicists or biologists without restriction.

A man may be a topologist or an acoustician or a coleopterist. He will be filled with the jargon of his field, and will know all its literature and all its ramifications, but, more frequently than not, he will regard the next subject as something belonging to his colleague three doors down the corridor, and will consider any interest in it on his own part as an unwarrantable breach of privacy.

- Wiener, Norbert; Cybernetics; 1948.

The Human Use of Human Beings- Norbert Wiener's Ideas at the Dawn of the Age of Computing
I have spoken of machines, but not only of machines having brains of brass and thews of iron. When human atoms are knit into an organization in which they are used, not in their full right as responsible human beings, but as cogs and levers and rods, it matters little that their raw material is flesh and blood. What is used as an element in a machine, is in fact an element in the machine. Whether we entrust our decisions to machines of metal, or to those machines of flesh and blood which are bureaus and vast laboratories and armies and corporations, we shall never receive the right answers to our questions unless we ask the right questions.

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