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Kurt Gödel and the origins of computer science John W. Dawson, Jr. Pennsylvania State University York, PA U.S.A.
Kurt Gödel (1906–1978) Considered the greatest mathematical logician of the twentieth century, he was one of the founders of recursion theory. A Princeton colleague of Alonzo Church and John von Neumann, his impact on computer science was seminal, but largely indirect.
Works by Gödel of special relevance to computer science 1. Completeness of first-order logic (1930) 2. Incompleteness of formal number theory (1931) 3. Papers on decision problems (1932–33) 4. Definition of the notion of general recur- sive function (IAS lectures, 1934; first published 1965)
5. “On the lengths of proofs” (1936) 6. “Some basic theorems on the founda- tions of mathematics, and their philo- sophical implications” (Gibbs Lecture, *1951) 7. Letter to von Neumann (*1956) 8. “A philosophical error in Turing’s work” (1972; first published 1974) *First published posthumously in Gödel’s Collected Works
The 1930 Completeness Paper In his first published paper (a revision of his doctoral dissertation), Gödel stated the com- pleteness of first-order logic in the form: “Every valid formula of first-order logic [one true under every interpretation] is provable from the logical axioms.” But as in the dissertation, he proved it in the form “Every formula is either refutable or satis- fiable.” He then extended the result to countable sets of formulas, to yield “A countable consistent set of formulas has a model”.
The Strong Completeness Theorem The main focus of Gödel’s 1930 paper is that a formula of first-order logic is valid if and only if it is derivable using the rules of inference given earlier by Hilbert and Ackermann. But he actually proved more: (Strong completeness) For any set Σ of first-order axioms, φ holds in every model of Σ if and only if φ is provable from Σ ( a result first stated explicitly by Abraham Robinson in 1951).
Significance for computer science The strong completeness theorem shows that the rules of inference developed prior to Gödel’s work are adequate for the pur- pose of deriving all logical consequences of a set of axioms. A computer incorporating just those rules will thus be able to carry out all such derivations.
Completeness vs. incompleteness The completeness theorem entails that the statements provable in any first-order axiomatization of number theory are those that are true in all models of the axioms. But for no recursive axiomatization will those statements coincide with the state- ments that are true of the natural numbers:
The Gödel-Rosser Theorem Any recursive axiomatization A of arithmetic must either be inconsistent (and so prove every statement φ ) or else fail to prove both some variable-free statement φ and its negation. Hence if A is consistent, it must fail to prove some statement that is true of the natural numbers.
The Gödel-Rosser Theorem is a limitative result. In the context of computer science, as is well known, Alan Turing used the correspondence between computer pro- grams and formal systems to recast it as The Halting Problem: No recursive pro- cedure can correctly determine, for an arbitrarily given computer running any specified program, whether or not the computer will eventually come to a halt.
The larger significance of Gödel’s incompleteness paper For the development of computer science, the proof Gödel gave of his incompleteness results was more important than the theorems themselves. Three aspects of the proof were of particular significance:
1. His precise definition of the class of (what are now called) primitive recursive functions 2. His distinction between object language and metalanguage. 3. His idea of representing one data type (sequences of strings) by another type (numbers), thereby coding metatheo- retical notions as number-theoretic predicates.
In addition, as Martin Davis has remarked, the overall structure of Gödel’s proof “looks very much like a computer program” to anyone acquainted with modern programming languages — unsur- prisingly, since though “an actual … general- purpose … programmable computer was still decades in the future,” Gödel faced “many of the same issues that those designing programming languages and …writing programs in those languages” face today.
The second incompleteness theorem As a further example of a numerical state- ment that must be formally undecidable within any consistent, recursive axioma- tization of arithmetic, Gödel adduced a numerical encoding of a statement asserting the theory’s consistency.
Philosophical significance of the incompleteness theorems Much later, in his Gibbs Lecture to the American Mathematical Society (1951),Gödel would suggest that the incompleteness theorems are relevant to the questions (1) whether the powers of the human mind exceed those of any machine, and (2) whether there are mathematical problems that are undecidable for the human mind.
Special cases of the decision problem During the years 1932–33 Gödel turned his attention to the decision problem — the question whether there is an algorithm for determining whether an arbitrary formula of first-order logic is valid or not. He estab- lished two results, concerning particular prefix classes of quantificational formulas:
1. He exhibited an algorithm for deciding the validity or invalidity of formulas in one quantifier-prefix class. 2. He showed that the (full) decision problem is reducible to that for formulas in another quantifier-prefix class. Subsequently, on the basis of Church’s Thesis, Church and Turing each demonstrated that the full decision problem is algorithmically unsolvable.
Extending the scope of the incompleteness theorems Gödel spent the academic year 1933–34 in Princeton at the Institute for Advanced Study. That spring, in a course of lectures on exten- sions of his incompleteness results, he defined the notion of general recursive function, based on an idea suggested to him by Jacques Herbrand.
Gödel’s definition and Church’s Thesis Gödel’s was one of several definitions later shown to characterize the same class of functions. On 19 April 1935, in a talk before a meeting of the American Mathematical Society, Church pro- pounded his Thesis: The class so characterized consists of exactly those functions informally thought of as being effectively computable.
Two points of note 1. In enunciating his Thesis, Church used the Gödel-Herbrand definition, rather than his own characterization of that class of functions (as those that are λ- definable). 2. Gödel himself was skeptical of Church’s Thesis, and only came to accept it in the wake of Turing’s analysis of the opera- tions performed by human computors.
Gödel’s prescience (I) On 19 June 1935 Gödel made his last pre- sentation to Karl Menger’s mathematical colloquium. Titled “On the length of proofs”, it appeared the following year as a two-page article in the colloquium proceedings. It is now regarded as the first announcement of a “speed-up” theorem.
Remarks: 1. Gödel stated his result without proof. The first published proof was given by Sam Buss in The measure of length Gödel used is the number of lines (inferences) in the proof, not the number of symbols. 3. It is unclear whether Gödel intended + and ∙ to be represented by function symbols or predicate symbols.
Gödel’s Prescience (II) Gödel did not pursue the ideas in his length- of-proof paper, which seems to have been an afterthought to his incompleteness theorems. Much later, in a letter to his terminally ill friend von Neumann, Gödel made another observation he did not follow up: The first known formulation of the P = NP problem.
Gödel’s 1956 letter to von Neumann
The Gibbs Lecture (1951) In his Gibbs Lecture, Gödel attempted to draw implications from the incom- pleteness theorems concerning three problems in the philosophy of mind:
1. Whether there are mathematical ques- tions that are “absolutely unsolvable” by any proof the human mind can conceive 2. Whether the powers of the human mind exceed those of any machine 3. Whether mathematics is our own creation or exists independently of the human mind
Gödel’s conclusions With regard to the first two questions, Gödel argued that “Either … the human mind (even within the realm of pure mathematics) infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems”. He believed the first alternative was more likely.
What Gödel didn’t claim Unlike commentators such as John Lucas and Roger Penrose, Gödel did not assert that the incompleteness theorems defini- tively refute mechanism, nor (as Judson Webb has asserted) that “the Church-Turing thesis … is the principal bastion protecting mechanism”.
As to the ontological status of mathematics, Gödel claimed that the existence of abso- lutely unsolvable problems would seem “to disprove the view that mathematics is … our own creation; for [a] creator necessarily knows all properties of his creatures”. He admitted that “we build machines and still cannot predict their behavior in every detail”. But that objection, he said, is “very poor”:
“For we don’t create … machines out of nothing, but build them out of some given material.” It seems he recognized that hardware may behave unpredictably, but not software (!)
Turing’s contrary view (1948) “The view that intelligence in machinery is merely a reflection of that of its creator is … similar to the view that … credit for the discoveries of a pupil should be given to his teacher. In such a case the teacher [sh]ould be pleased with the success of his methods of education, but [sh]ould not claim the results themselves unless he ha[s] actually communicated them to his pupil” --- Alan Turing, “Intelligent machinery”
To put Gödel’s belief in perspective, note that, unlike Turing or von Neumann, he was never involved with the design or operation of actual computers — even though it was at the IAS that von Neumann built one of the earliest computers. Consider also the statement that computer pioneer Howard Aiken made five years later:
“If it should turn out that the basic logics of a machine designed for the numerical solution of differential equations coincide with the logics of a machine intended to make bills for a department store, I would re- gard this as the most amazing coin- cidence I have ever encountered.”
Gödel’s criticism of Turing Despite his high regard for Turing’s work, late in his life Gödel claimed that Turing had erred in arguing that mental procedures cannot go beyond mechanical procedures. Specifically, he charged that Turing had completely disregarded “that mind, in its use, is not static but constantly developing.”
In defense of Turing Contrary to Gödel’s charge, in his 1950 paper “Computing machinery and intelli- gence”, Turing explicitly discussed learning machines. There, too, he countered various objections to the idea that machines could simulate human thought, including the “mathematical objection” based on the incompleteness theorems.
Some imponderables Gödel and Turing never met, nor did they exchange any correspondence of sub- stance. Gödel delivered his Gibbs Lecture one year after the appearance of Turing’s paper quoted above, and his criticism of Turing came long after Turing’s death. One must therefore wonder:
1. At the time of his Gibbs Lecture, was Gödel aware of Turing’s 1950 paper? 2. Gödel’s Gibbs Lecture was delivered less that three years before Turing’s death, just months before his prosecution for homosexuality, and it was not pub- lished until Did Turing ever become aware of its contents?
Gödel and Emil Post Emil Post was an important contributor to recursion theory, and helped to initiate the field of automata theory. Twenty years before Gödel, he also conjectured (and nearly proved) the incompleteness of formal number theory. His outlook, however, was very different from Gödel’s.
Post met Gödel in October of Shortly afterward he wrote to Gödel to expound further on his “near miss”. He explained that he had set out to demonstrate the existence of absolutely unsolvable prob- lems, and thought he “saw a way of analyz- ing ‘all finite processes of the human mind’ ” that would establish incompleteness “in general, … not just for Principia Mathema- tica.”
Arguably, Post contributed more directly to the development of computer science than Gödel did. But as he himself conceded in that same letter, “Nothing that I had done could have replaced the splendid actuality of your proof.” Sidetracked by his quest to exhibit an absolutely unsolvable problem, Post failed to give an example of one that was undecidable in a particular formalism.
Questions to ponder If Gödel were alive today, what would he think of the role of computers in mathe- matics? Would he approve of their use in providing numerical evidence for conjectures? What would he think of computer-assisted proofs? Would he maintain his belief in the mind’s superiority?
Basic References Kurt Gödel, Collected Works (5 vols.), ed. Solomon Feferman et al. Oxford University Press, 1986–2003. The Essential Turing, ed. B. Jack Copeland, Oxford University Press, Solvability, Provability, Definability: The Collected Works of Emil L. Post, ed. Martin Davis. Birkhäuser, 1994.