volume II: Synopsis
part IV: Divine Dynamics
page 26: Kurt Gödel
(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 Gödel, working on a question raised by David Hilbert, showed that consistency does not always imply determinism. Kurt Gödel - Wikipedia
Gödel proved that for sufficiently powerful theories, it cannot be proven that the theory is both consistent and complete. A theory is consistent if one cannot use it to prove both p and not-p. It is complete every valid statement of the theory be either proved or disproved. Gödel's discovery is formally consistent with the probabilistic nature of quantum mechanics and the general unpredictability of the future. Gödel's incompleteness theorems - Wikipedia, Mark Dominus: World's shortest explanation of Gödel's theorem
Gödel's method of proof is quite complex. Gregory Chaitin produced a simplified argument for Gödel's results based on the cybernetic principle of requisite variety. Put simply, no system can deterministically control a system more complex than itself. Since (according to the second law of thermodynamics) the past is almost always simpler than the future, the past can hardly ever determine the future. Gregory Chaitin: Gödel's Theorem and Information, Second law of thermodynamics - Wikipedia
One of Aristotle's greatest contributions to our culture was the invention and application of logic to the construction and testing 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. Term logic - Wikipedia, Boolean algebra - Wikipedia
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. Heath: Thirteen Books of Euclid's Elements, Euclid's Elements - Wikipedia
By the beginning of the twentieth century many paradoxes had appeared in mathematics. Hilbert was concerned to put mathematics on a firm foundation by developing a formal method. He hoped to be able to prove that mathematics is consistent and complete end eliminate all paradoxes. Hilbert's program - Wikipedia
Gödel 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 Gödel 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.
Gödel's proofs are purely formal, a set of strings of symbols manipulated deterministically according to certain rules or axioms. All that is required of such a formal system is internal consistency. The scientific process of forming hypotheses and collecting evidence is necessary to decide whether mathematical theorems apply to the real world. Clearly, formal theories like arithmetic do, as we learn from its effectiveness in accounting for all forms of trade. Formalism (mathematics) - Wikipedia
Formal mathematics is part of the Universe. We may look at the mathematical literature, which represents formal mathematics, as the set of fixed points in the dynamics of the mathematical community. The moving minds of mathematicians have slowly evolved the set of fixed points that is current mathematics, starting thousands or tens of thousands of years ago. Thomas Tymoczko: New Directions in the Philosophy of Mathematics: An Anthology, Fixed point theorem - Wikipedia
The truth of the formal system as it is written down is guaranteed by the dynamics of the mathematical community that generates these texts. In this way the formalism becomes consistent with the dynamics, rather as the genotype of a living organism is sculpted to be consistent with the organism itself, the phenotype. In general, we expect the stationary points that we observe in the Universe to be consistent with the dynamics of the Universe, since they are continuous with it.
Our hypothesis is that the Universe is at least as complex 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. Gödel'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. Although, following Landauer, we assume that all information is represented physically, we do not expect the physical implementation of information to provide any real constraint on the more spiritual processes which occur in higher layers of the Universe, like our own minds. Rolf Landauer: Information is a Physical Entity
(revised 5 April 2020)
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Further readingBooks
Chaitin, Gregory J, "Goedel's Theorem and Information" in Information, Randomness & Incompleteness: Papers on Algorithmic Information Theory, Reprinted from the International Journal of Theoretical Physics (1982) 22, 941-954., World Scientific 1987 Abstract: 'Goedel's theorem may be demonstrated using arguments having an information-theoretic flavour. In such an approach, it is possible to argue that if a theorem contains more information than any given set of axioms, then it is impossible for the theorem to have been derived from the axioms. In contrast with the traditional proof based on the paradox of the liar, this new viewpoint suggests that the incompleteness phenomenon discovered by Gödel is natural and widespread rather than pathological and unusual.'
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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. ...'
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Gödel, Kurt, and Solomon Feferman et al (eds), Kurt Gödel: 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. ...'
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Gödel, Kurt, and B Meltzler (translator), R B Braithwaite (Introduction), On Formally Undecidable Propositions of Principia Mathematica and Related Systems, Dover 1992 A translation 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.'
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Heath, Thomas Little, Thirteen Books of Euclid's Elements (volume 1, I-II), Dover 1956 'This is the definitive edition of one of the very greatest classics of all time - the full Euclid, not an abridgement. Utilizing the text established by Heiberg, Sir Thomas Heath encompasses almost 2500 years of mathematical and historical study upon Euclid.'
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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.
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Tymoczko, Thomas, New Directions in the Philosophy of Mathematics: An Anthology, Princeton University Press 1998 Jacket: 'The traditional debate among philosophers of mathematics is whether there is an external mathematical reality, something out there to be discovered, or whether mathematics is the product of the human mind. ... By bringing together essays of leading philosophers, mathematicians, logicians and computer scientists, TT reveals an evolving effort to account for the nature of mathematics in relation to other human activities.'
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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.'
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Whitehead, Alfred North, and Bertrand Arthur Russell, Principia Mathematica (Cambridge Mathematical Library), Cambridge University Press 1910, 1962 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.
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Links
Boolean algebra - Wikipedia, Boolean algebra - Wikipedia, the free encyclopedia, 'In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively. Instead of elementary algebra where the values of the variables are numbers, and the main operations are addition and multiplication, the main operations of Boolean algebra are the conjunction and, denoted ∧, the disjunction or, denoted ∨, and the negation not, denoted ¬. It is thus a formalism for describing logical relations in the same way that ordinary algebra describes numeric relations.' back |
Euclid's Elements - Wikipedia, Euclid's Elements - Wikipedia, the free encyclopedia, 'Euclid's Elements (Greek: Στοιχεῖα Stoicheia) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions. The thirteen books cover Euclidean geometry and the ancient Greek version of elementary number theory. The work also includes an algebraic system that has become known as geometric algebra, which is powerful enough to solve many algebraic problems, including the problem of finding the square root.' back |
Fixed point theorem - Wikipedia, Fixed point theorem - Wikipedia, the free encyclopedia, 'In mathematics, a fixed point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms. Results of this kind are amongst the most generally useful in mathematics.
The Banach fixed point theorem gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point.
By contrast, the Brouwer fixed point theorem is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point (See also Sperner's lemma).' back |
Formalism (mathematics) - Wikipedia, Formalism (mathematics) - Wikipedia, the free encyclopedia, 'In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be thought of as statements about the consequences of certain string manipulation rules.
For example, Euclidean geometry can be seen as a game whose play consists in moving around certain strings of symbols called axioms according to a set of rules called "rules of inference" to generate new strings. In playing this game one can "prove" that the Pythagorean theorem is valid because the string representing the Pythagorean theorem can be constructed using only the stated rules.' back |
Hilbert's program - Wikipedia, Hilbert's program - Wikipedia, the free encyclopedia, 'In mathematics, Hilbert's program, formulated by German mathematician David Hilbert, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies. As a solution, Hilbert proposed to ground all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. Hilbert proposed that the consistency of more complicated systems, such as real analysis, could be proven in terms of simpler systems. Ultimately, the consistency of all of mathematics could be reduced to basic arithmetic. back |
Kurt Gödel - Wikipedia, Kurt Gödel - Wikipedia, the free encyclopedia, 'Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. The more famous incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.' back |
Mark Dominus, The Universe of Discourse: World's shortest explanation of Gödel's theorem, 'A while back I started writing up an article titled "World's shortest explanation of Gödel's theorem". But I didn't finish it, and later I encountered Raymond Smullyan's version, which is much shorter anyway. So here, shamelessly stolen from Smullyan, is the World's shortest explanation of Gödel's theorem.' back |
Rolf Landauer, Information is a Physical Entity, 'Abstract: This paper, associated with a broader conference talk on the fundamental physical limits of information handling, emphasizes the aspects still least appreciated. Information is not an abstract entity but exists only through a physical representation, thus tying it to all the restrictions and possibilities of our real physical universe. The mathematician's vision of an unlimited sequence of totally reliable operations is unlikely to be implementable in this real universe. Speculative remarks about the possible impact of that on the ultimate nature of the laws of physics are included.' back |
Second law of thermodynamics - Wikipedia, Second law of thermodynamics - Wikipedia - The free encyclopedia, 'The second law of thermodynamics states that in a natural thermodynamic process, there is an increase in the sum of the entropies of the participating systems.
The second law is an empirical finding that has been accepted as an axiom of thermodynamic theory. back |
Term logic - Wikipedia, Term logic - Wikipedia, the free encyclopedia, 'In philosophy, term logic, also known as traditional logic or Aristotelian logic, is a loose name for the way of doing logic that began with Aristotle and that was dominant until the advent of modern predicate logic in the late nineteenth century. This entry is an introduction to the term logic needed to understand philosophy texts written before predicate logic came to be seen as the only formal logic of interest.' back |
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