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Page 26: Kurt Goedel
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Kurt Goedel
(1906 - 1978)
For thousands of years people equated consistency with determinism, holding that a logically consistent sequence of propositions could have only one outcome. This feeling lies behind the notion that God knows and controls everything. Kurt Goedel, working on a question asked by David Hilbert, showed that consistency does not always mean determinism. Goedel's discovery is consistent with the probabilistic nature of quantum mechanics, and indeed of all processes in the universe.
One of Aristotle's greatest contributions to our culture was the invention and application of logic to the construction of knowledge. The medieval theologians and philosophers relied as heavily on Aristotle's logic as on his physics and metaphysics. In the twelfth century, Peter Abelard (1079-1142) realized the advantage of refining all problems down to yes/no questions. In the seventeenth century Leibniz (1646-1716) dreamt of a universal logical calculus, but the modern period in logic did not develop until the nineteenth century when George Boole (1815-1864) and others founded mathematical logic.
Logic is about proof. A proof is a linguistic structure which establishes that if statement A is true, then statement B is true. Mathematical proof has been existence since the beginning of recorded history. Two classic proofs are that of Pythagoras' theorem, (proposition 47 in Book I of Euclid's Elements) and the proof for the existence of irrational numbers that follows from Pythagoras' theorem. Proofs are held together by rules of inference, which allow us to decide whether or not a certain statement follows from other statements. A good proof establishes 'logical continuity' between elements of a set of statements.
By 1928 Hilbert was able to encapsulate his thoughts on the nature of mathematics in three questions: Is mathematics consistent? A system is consistent if no contradictory statements can be proved within it. Is mathematics complete? That is, can every valid mathematical statement be either proved or disproved? Is mathematics computable? Is there a definite process that will yield every mathematical proof? Hilbert felt that the answer to all three questions would be yes, proving that there were no limits to mathematics. He was to claim in 1930 that there is no such thing as an unsolvable problem.
Goedel showed that if arithmetic (and by extension, mathematics) is consistent, it is incomplete. There are statements that can be neither proved nor disproved. In other words, in any system of a certain size, the domain of possible statements is larger than the domain of provable statements. If we assume that the only statements that are determined in such a system are those that are provable, this is equivalent to saying that every such system has an non-determinate halo of valid statements.
Before Goedel proved that arithmetic in incomplete, he proved that propositional calculus is complete. There exists a determinate process (like the application of truth tables) by which one can decide whether any statement in propositional calculus is true or false. This difference between propositional calculus and arithmetic serves as a model of the divide between deterministic and nondeterministic processes in the universe.
Our hypothesis is that the universe is at least as big as arithmetic, so that it is affected by incompleteness. This is consistent with the idea that there are elements of the future that are not determined by the present, and so can not be foretold. Goedel's incompleteness theorem, like Cantor's theorem, tells us that the future is bigger than the past and not fully constrained by it.
If we model such a system as a probability space, we can attribute certainty (probability 1) to the provable statements, and probabilities less than one to statements that cannot be proved or disproved. This sets the stage for the understanding the role of probability in quantum mechanics.
Incompleteness also opens the door for evolution by selection, since the existence of a certain statement may be established not by its logical necessity, but by its utility in the overall scheme of things. Finally, incompleteness opens the door to the distinction between hardware and software. The essence of good hardware is not to determine particular states from the set of possible states of the software. The software, in other words, is free to do what it must without interference from the hardware. This reminds us of the idea of orthogonality which is very important in quantum mechanics.
Books
| Dawson, Jr, John W, Logical Dilemmas: The Life and Work of Kurt Goedel, A K Peters 1987 Jacket: 'This definitive biography of the logician and philosopher Kurt Goedel is the first in-depth account to integrate details of his personal life with his work, and is based on the author's intensive study of Goedel's papers and surviving correspondence. ...' Amazon back |
| Goedel, Kurt, and Solomon Fefferman et al (eds), Kurt Goedel: Collected Works Volume 1 Publications 1929-1936, Oxford UP 1986 Jacket: 'Kurt Goedel was the most outstanding logician of the twentieth century, famous for his work on the completeness of logic, the incompleteness of number theory and the consistency of the axiom of choice and the continuum hypotheses. ... The first volume of a comprehensive edition of Goedel's works, this book makes available for the first time in a single source all his publications from 1929 to 1936, including his dissertation. ...' Amazon back |
| Goedel, Kurt, and B Meltzler (translator), R B Braithwaite (Introduction), On Formally Undecidable Propositions of Principia Mathematica and Related Systems, Dover 1992 A of Uber Formal Unentscheidbare Satze der Principia Mathematica und Verwandter Systeme I, Monatshefte fur Mathematik und Physic, 38(1931) 173-198. Jacket: 'In 1931 a young Austrian mathematician published an epoch making paper containing one of the most revolutionary ideas in logic since Aristotle. Kurt Gödel maintained, and offered detailed proof, that in any arithmetic system, even in elementary parts of arithmetic, there are propositions which cannot be proved or disproved within the system. It is thus uncertain that the basic axioms of arithmetic will mot give rise to contradictions. The repercussions of this discovery are still being felt and debated in 20th century mathematics.' Amazon back |
| Hofstadter, Douglas R, Goedel Escher Bach: An Eternal Golden Braid, Basic/Harvester 1979 An illustrated essay on the philosophy of mathematics. Formal systems, recursion, self reference and meaning explored with a dazzling array of examples in music, dialogue, text and graphics. Amazon back |
| Wang, Hao, Reflections on Kurt Goedel, Bradford/MIT Press 1990 Jacket: Kurt Goedel was indisputably one of the greatest thinkers of our time, and in this first extended treatment of his life and work, HW, who was in close contact with Goedel in his later years, brings out the full subtlety of Goedel's ideas and their connection with grand themes in the history of mathematics and philosophy.' Amazon back |
| Whitehead, Alfred North, and Bertrand Arthur Russel, Principia Mathematica (Cambridge Mathematical Library), Cambridge University Press 962 The great three-volume Principia Mathematica is deservedly the most famous work ever written on the foundations of mathematics. Its aim is to deduce all the fundamental propositions of logic and mathematics from a small number of logical premisses and primitive ideas, and so to prove that mathematics is a development of logic. Not long after it was published, Goedel showed that the project could not completely succeed, but that in any system, such as arithmetic, there were true propositions that could not be proved. Amazon back |
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