http://staff.smc.univie.ac.at/jimmy/Artikel/GoedelAI.htm
Werner DePauli-Schimanovich
Kurt Gödel and his Impact on A.I.
(Author's comment: This is only a preliminary draft
(a preprint to encourage criticism) and not the final version of the paper)
Dr. Werner Schimanovich (47) studied mathematical logic, mathematics, physics and mechanical engineering at the Technical University ofVienna, the Technical University of Berlin and the University of Vienna between 1960 and 1971. Since 1971 he has been working at the Department forStatistics and Computer Science at the University of Vienna. Every yearsince 1985, he has spent time doing research at the Universidad de Las Palmas, Spain. In 1986 he produced the 80-minute film "Kurt Gdel - AMathematical Mythos" together with Peter Weibel (an internationally renowned artist). In this connection he did research at the Institute for Advanced Study (Princeton, N.J.) and at other universities in theUSA.
His address:
Institute for Statistics and Informatics,
University Street 5
EU-A-1010 VIENNA/Austria/EuropePeople of the A.I. community usually heard about Gdel in a course of their curriculum on limitation theorems like the Halting Problem, et cetera. But the broad scene of culturally interested people without special knowledge in Mathematics, Logic or Computer Science recognized the importance of Gdel's philosophically guided results for the first time by the appearance of Douglas Hofstadter's (Pulitzer prize) book "Gdel, Escher, Bach: An eternal golden braid"). In this book, Hofstadter tried to put Gdel's work into a broader context of"similar" scientific problems. In this article I will try to explain Gdel's influence on Computer Science in general and the relevance of his work to the development and the debate on A.I., especially within the context of scientific and academic cultures.!
Let us start with the philosophical relevance of Gdel's results: Alan Turing himself pointed out that he was stimulated to his work by studying Gdel's famous paper [Gdel 1]: ber formal unentscheidbare Stze der Principia mathematica und verwandter Systeme. (On formallyundecidable propositions of Principia Mathematica and related systems).@ Comparing mind and machines (or natural and artificial intelligence) Gdel had shown# that "there exist some tasks the mind can solve but machines cannot". And, as Robin Gandy$ mentioned, Alan Turing therefore wanted first to answer the question: what is the set of tasks a machine can solve in general?%
With their work, Gdel and Turing rekindled the discussion about the mechanization of mind^ (which had come to a sudden end with Gottfried Wilhelm Leibniz's Mhlengleichnis (fable of the mills)& and the arising German Idealism of Georg Wilhelm Friedrich Hegel and the other philosophers of that time. The mechanization of mind is of course a centuries-old dream with a much prouder tradition. Startingwith Euclid's axiomatization of Geometry and Aristotle's attempts at axiomatizing Logic, it continued with Leibniz's Calculus Ratiocinator and proceeded on to David Hilbert's program* of codifying all mathematical proofs. And it keeps our interest still today: the success of A.I. in science and industry guarantees a continuing discussion of these theoretical questions. That is the secret of Hofstadter's success) (which behaves somehow like a Dirac function): Belletristically well packaged, it offers to the philosophically radical members of the rapidly growing A.I. community and their broadly culturally interested hangers-on an insight into the present state of the theoretical investigation.
But Gdel's limitation theorem and Turing's "possibility" thesis (only fit a special type of machine: Digital Computers. Future extensions of our present-day computers will include analogue computations as well as holographic memories. They will have to be considered as open cyberneticsystems with sensors and the ability to enlarge their processing unitand memory while performing computations. Connected together into giant networks, they will behave more like a society than like a machine. They will be more like a physical machine or device for which the question of its computational capacity arises anew,!) and with it the dream of the complete mechanizability of mind.!!
The present situation is quite different. And current contemporary philosophical discussion forced by the A.I. approach is sometimes mind-boggling!@ and disappoints me a lot. My thesis is: Digital Computers model the hard kernel of our mental features (in the samesense as Peano Arithmetic lets us deduce the hard kernel ofmathematics).
What does it matter when there exist true sentences of mathematics like that of [Paris, Harrington] one cannot demonstrate within the Formal System of Peano Arithmetic? Essential is the hard kernel of provable theorems and computable functions. (I would be glad if I could prove all theorems a present-day computer can do in practice or formally compute some functions a Computer-Algebra program does.) Similar isthe situation modeling the mind with the computer: While Turing has shown what kind of mathematical functions a mechanical device can compute, A.I. will show!# what aspects of mind can be modeled by digital computers. (E.g. hypnosis probably not.) Therefore we all should keep in mind: the limitations of Gdel's theorem (or the Halting Problem for Turing Machines) concern only Formal Systems or Digital Computers and not Physical machines.!$ Our limitation is practice: Maybe in 1000 years we can fulfill all promises A.I. propaganda has made in the last 10 years. With this statement I hope to stop the stupid philosophical discussion in A.I. (not only in this article, but also in general).
Gdel's proof is what most interests the computer community (of hackers) in Gdel's work, because we do not like to be limited, which gives us a feeling of a handicapped mind.!% However, his relevance to thedevelopment of A.I. lies in quite different results, which can briefly be described in the following points:
D (1) Gdel's formal explication of the Recursive Functions is e.g. such an achievement which yielded the basis for recursive programming. Together with the Lambda-Calculus of Alonzo Church, they gave rise to the development of LISP by McCarthy and the other A.I. gurus. Of course McCarthy did not steal his definition of the LISP-syntax!^ from DGdel1F, but was considering general properties of recursive functions such as
their inversibility.!& But the interest in these subjects has been augmented by the cultural situation of contemporary mathematics (in the40s and 50s), after Gdel had started to work at the Institute for Advanced Study in Princeton (New Jersey, USA).!* Several of his students like Stephen Cole Kleene and Barkley Rosser worked out Gdel's intentions and ideas and a whole movement of research on recursive functionsstarted up in mathematics. That was before the time when computers had been getting to be strong tools in scientists' practice. Since the young computer scientists mainly came from math, it is clear that they had been influenced by that general treatment of recursive functions. Recursive programming did not fall from heaven (as some A.I. people believe today);!( its development is embedded in a cultural situationproduced by a long period of research in mathematics and logic which had been motivated by Gdel's work.
D (2) Another influence of Gdel on A.I. is not so well documented (as is his impact on recursive programming): In private discussions Gdelargued for the use of predicate logic as a programming language and the application of logical methods in computer science.@) At this timepeople working in computer science smiled about the attempts of logicians in constructing general theorem provers, and it seemed completely unimaginable that logic can form a tool as a programming language. Today we know that logicians working in this field misunderstood computational complexity and developed their theorem provers under the wrong approach of minimizing complexity by anonymous devices (whose functioning ways were unknown by the users.) Even such promising programming constructs as the Logic Theorist@! (which remind us somehow of PLANNER) failed. Only after computer scientists with a good education in logic@@ started to deal with predicate logic could its use as a programming language be turned into practice. With a more complex but easily memorized search strategy, the problem could be solved: PROLOG was invented and logical programming was born.
Now the same argument as before can be applied: Of course the development of PROLOG has been carried out without the direct influence of Gdel. But his ideas he communicated in personal discussions with logicians also working with computational methods (e.g. in theorem proving)@# influenced the cultural scene at that time. This formed the conviction that logic can be used as a programming language in spite of the rejection of this approach by pure computer scientists in the 60s. And this conviction of the applied logicians gave them the strength to convince the young researchers in computer science of the practical feasibility of logic as language (what can be considered as its etymological root).D (3) With the application of logical methods in computer science (from relational calculus to knowledge representation), it is a similar but at the same time also different situation. Gdel's contributions to logic and his engagement for this discipline influenced a lot of young people in their decision to study logic. But thispromising field of the 50s and 60s did not bring the results hoped for (and therefore did not bring a better insight into the structure of mathematics).@$ A lot of the young researchers in pure logic stopped their career and tried to get new jobs in more applied fields: They shifted to computer science which was a natural continuation of their attitudes and offered them furthermore a steady employment with a more lucrative salary. These people tried of course to use the methods they learned in logic. Therefore sometimes publications in A.I. look like applied logic methodology. The melting pot of A.I. (as computer science together with some parts of logic and other fields like linguistics or psychology) forces the researchers to develop their own "logic" which seems to be more practical and promising than the traditional logic developed by pure logicians. This is also an influence of Gdel -- in spite of the fact that it probably was not originally intended by him.
Now I will try to work out the context of scientific and academic cultures that helped and provoked Gdel to form and formulate his ideas. This has to some extent already been done in our film DWeibel, Schimanovich 1F, -- I hope so! -- and therefore I want to advise the reader to look at that film.@% But in addition, I will discuss here some theses in an abbreviated form. First of all: Gdel can be considered a product of the Vienna Circle. Some people in the UnitedStates hold the opinion that Gdel developed his ideas in Princeton. Not at all! In Princeton he mainly worked out what he already had invented in Vienna.@^ This Vienna of the fin de siecle and following decades (during which the Austrian-Hungarian Monarchy broke down and formed a cultural explosion) was not only Wittgenstein's Vienna, but also that of the Vienna Circle and Kurt Gdel. It is not a mere fortuitous accident that Gdel discovered the so-called theorem of the century at a time the mathematical community of the world believed the contrary. It was the cultural influence of the scientific scene of Vienna in the 20s and 30s that made him sensitive to understand what was going on in mathematics in reality. The Vienna Circle, reading and discussing Ludwig Wittgenstein and inviting Luitzen E.J. Brouwer, made Gdel aware that mathematics cannot be merely the construct of a countable syntax but must include also uncountable reality (like the real numbers) in its codification. This philosophically guided view (inspired by a platonistic insight into the mathematical world@&) had of course been first conjectured by Hilbert's student Paul Finsler@*. But Gdel was the one with the ability to prove this conjecture by methods that convinced all the opposing schools of mathematics (partially in opposition to each other)@(. (Gdel's proof had been so surprising that John von Neumann stopped working further in logic and let this research field alone to Gdel as a "logical opinion leader".)
In Brno (Czechoslovakia, at that time part of the Old-Austrian Empire), where Gdel was born in 1906 into a rich house and educated as the son of an industrialist/entrepreneur, his interest for math and philosophy had already emerged. But when he came to Vienna, his teachers not only had been world-famous scientists; he also got in touch with the most important questions in math, physics and philosophy of this time and a very special view and way to treat them. The first 2 years Gdel studied physics with Hans Thirring, who taught him the theory of the rotating universes. After this period, Hans Hahn, Karl Menger and Rudolf Carnap influenced his thinking in such a way that Gdel gota feeling for possible solutions of the most important problems of histime.#) In a number of rapidly-appearing publications, he proved the completeness of (first order) predicate logic (in his Ph.D. thesis), the incompleteness of arithmetic (in his "Habilitation thesis" for the "Venia docendi") , the (relative) consistency of the axiom of choiceand the (generalized) continuum hypothesis Dwith the other axioms of set theoryF, and other logical and mathematical problems. Since his interest in physics and especially in rotating universes dates also from this time, it is certainly possible (and as a result of our research it seems probable) that Gdel made his first attempts to solve Einstein's field equations in Vienna, too.
In Princeton (1940-1978) he worked out the inventions and ideas he already had brought with him from Vienna: He revised his results in set theory and investigated a lot of computational work into his singular solution of the relativistic field equations , which delivers the theoretical basis of time-travel into the past. He also finished a logical paper called the "Dialectica Interpretation" , of which he said that the idea for it and first attempts to work it out he already had in Vienna. Later on his interest shifted to the philosophy of math, where he made serious contributions to that field of research. He influenced 2 generations of logicians and through them the development of computer science and especially that of A.I.
Gdel's work has always been philosophically guided. Only through his platonistic view of the world did he get an insight in what direction scientific work has to go (to lead to progress) -- an ability most researchers in the most varying fields of science are missing.#! Especially in computer science and A.I., the inability to orient oneself is very common.#@ Therefore we should learn from Gdel and include philosophy as a special subject into the curriculum of A.I. education.
Gdel is not the father of A.I.##, but he can be considered as a grandfather (together with Alan Turing and possibly John von Neumann). And I hope that this article is a contribution to that view of the history of A.I.
Footnotes:
0) See Hofstadter
1) For additional information I want to refer the reader to thebibliography, especially to Weibel, Schimanovich 1, and Weibel, Schimanovich 2.
2) Bernhard Melzer told me (in 1987 at the IJCAI in Milano) that he made the first translation of Gdel's paper in 1932. This may fit as another example for a grand seigneur of A.I. influenced by Gdel.
3) More precisely, one should say: the philosophically most important aspect of Gdel's (purely mathematical) result is the conjecture/assumption that: there exist some tasks the mind can solve but a machine cannot. (But of course, to make this statement precise, we have first to give an exact definition of the mind and second a clear view of our intentions what we want to understand by a machine.)
4) Gandy reports this in the film Weibel, Schimanovich 1. One can read this also in the textbook Weibel, Schimanovich 2. Gandy was one of Turing's closest associates. See also Gandy 1.
5) These ideas he worked out in an article that became famous, too: "On computable numbers, with an application to theEntscheidungsproblem".
See Turing 1.
6) See Turing 2, in Anderson, and Webb, for a broader discussion of the subject. A plenary lecture by Hao Wang about that theme was presented at the First Kurt Gdel Colloquium (22nd & 23rd September 1989, Salzburg) and will probably be published in the Yearbook KGS. One may hope that the Turing 1990 Colloquium (Brighton, 3-6 August 1990) will bring similar results.
7) See Leibniz
8) See Detlefsen, Hilbert Probleme, or better Hilbert, Bernays 1, and Hilbert, Bernays 2
9) "Computable Functions in a common sense are exactly those one can calculate on a Turing-Machine". This postulate is better known as Church's Thesis but had first been established by Turing. Since the term "Computable in a common sense" is not mathematically precisely formulatable, one cannot prove this as a theorem in contrast to thesentence "Turing-computable functions are exactly the Recursive Functions". See also Davis 2, and Gandy 2.
10) Is our physical universe a gigantical Turing Machine? This question and that of computability in physics is going to be a main research field in the future. The combination of logic and physics may be soon of the same importance as logic and mathematics had been at the turn of the century. See Pour-El, Richards, Brunner, Svozil, Svozil, and Gandy 2.
11) Don't mix it up with the brain which may behave as a digitalcomputer,but is only the embodiment of mind Bateson, McColloch. Mind is no individual but (according to Gdel) a god-like infinite species generated by the historical evolution of the society of human beings. Therefore it is in any case not good to compare it with a digital mechanical device.
12) E.g. some arguments of Dreyfus.
13) We hope that intrinsically!
14) I would be glad if someone could give a similar definition of that as simple and as convincing as Turing gave it for his machine.
15) In reality, the philosophical interpretation of Gdel's theorem as
Gdel himself has formulated it: "Either our mind is incomplete or
mathematics is (or both)." See "the Talk of the Town" in Schimanovich.
15a) See Kleene 2.
16) See McCarthy, Abrahams.
17) See McCarthy
18) Between 1933 and 1939, Gdel lectured twice (during his 3 short
periods
of stay) as an invited guest of the IAS. But through this lecturing he
influenced the American School of Logic very deeply, changing the
research interests from Lambda-calculus to Predicate Logic and
Recursive Functions. Later on since March 1940 he was a steady member
of the institute (but without permanent status until 1949, full
professor since 1953).
18a) See Kleene 1.
18b) Even the recursion using programming language BASIC was, after all,
co-invented by John Kemeny, a former assistent of Einstein who had
studied the logic of Carnap and Gdel in Princeton. Also, the
"von Neumann machine", the basis of modern computation, was
originated by von Neumann on the basis of the ideas of Gdel &
Turing. See Goldstine.
19) At ECAI 1984 in Pisa e.g. Attardi & Simi gave a plenary lecture
where they talked as if they had invented some 50 year-old procedures
of recursive function theory like the amalgamation principle.
20) Some witnesses of time who spoke with Gdel personally told us these
facts in private communications. E.g. the IBM fellow Heinz Zemanek
who had a discussion with Gdel in the early 60s.
21) See Newell, Shaw, Simon.
22) E.g. Colmerauer in Marseille or David Warren in Edinburgh.
23) E.g.: Hao Wang (founding president of the Kurt Gdel Society).
One should also consult his books.
24) E.g.: In this connection one can say that Gdel encouraged wrong
hopes in the overwhelming part of the young logicians.
25) Since it is very difficult to get a copy of the English demo version
(-- I have no facilities for multiplexing and distributing it --)
please write a letter to your television network. Maybe one of them
will become motivated enough to produce a professional English
version and broadcast it.
26) This topic has been presented at the celebration of Gdel's 80th
birthday (at the University of Vienna) and the accompanying Symposium
(Vienna, 28th & 29th April 1986).
26a) See Janik, Toulmin. The intercultural relation between
Wittgenstein,
the Vienna Circle, and other Scientists and artists is worked out in
detail in several publications of the Wittgenstein Society and its
Schriftenreihe, HPT-Verlag, Wien. One should also visit the annual
Wittgenstein Symposium in Kirchberg/Wechsel, Austria.
27) See again Wang 1, and Khler 2, too.
28) See Finsler
29) Hilbert's school of finitism and Brouwer's intuitionism.
29a) For Gdel's curriculum see Dawson, Feferman et al. 1, Schimanovich,
or Weibel, Schimanovich 2.
30) See Khler 1
30a) published in Gdel 2.
30b) published in DGdel 1F.
30c) published later in Gdel 3. His first seminar about this topic he
gave 1936 in Vienna.
30d) published in Gdel 4, and Gdel 5.
30e) published in Gdel 6.
30f) ee Wang 3.
31) That's the difference between science fiction authors who make
a creatively well formulated extrapolation of present into
future and the prophets of science who smell the singularities
in the discontinuous space of logical possibilities. (Also Leo
Szillard was such a prophet and not only in science, which his
booklet "The Language of the Dolphins" shows.)
32) How many dead ends (weighted in millions of dollars) in research
and develoment have been produced by this orientation deficit?
33) Like John McCarthy or Bernhard Melzer, or like Arthur Samuel and
Saulus Amarel (who are acronymic twins in the same sense as Dana
Scott is a direct descendant of Dunus Scotus).
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