What is this? Are Thinking Computers Can automata think? 1998 7 Mathematically Possible? I can think just as well Unmapped Territory Because of the technical nature of the arguments on this map, the Can improved machines 35 Anticipated by C. T. K. Chari, 1963 Self-programming We self-programming inductive machines think. Oh, no you don't. 36 C. T. K. Chari, 1963 Self-programming machines are too rigid to replicate human induction. Self-programming inductive machines only approximate inductive methods. Humans use inductive methods that cannot be is supported by 89 Automata can think. A finite automaton, like a Turing machine, can replicate all essential aspects of human intelligence. Finite automata are completely mathematically as they can! beat the Lucas Additional mapmakers have had to omit many inductive machines beat formalized. We see this in 2 ways. describable. Thus, machine thinking is mathematically 91 Kurt Gödel, 1951 arguments claims and simplify many arguments. the Lucas argument. is 1. Human induction is essentially a creative process that starts with an possible. Minds are not as finite as Turing believes. At any given time, the mind only possesses Many lines of debate in this field Self-programming inductive disputed unspecified set of alternatives. Computers work only with specified Note: This region deals with general, mathematical properties a finite number of states, but as time progresses the mind constantly develops. This is shown involve sophisticated mathematical machines have enough by alternatives. of machines, rather than with the specific architectural by the fact that it is always possible for minds to develop new methods of thought. It is likely symbolism that are difficult to is supported by creativity to recognize the 2. The concept of probability (which is essential to induction) is properties dealt with on other maps. Connectionist networks that the total number of possible mental states involved in developing these methods converges represent in this medium. argument? truth of Gödel sentences. flexible and is determined by the use it is put to. Machines are constrained to use probability only in ways that humans determine for them. Machines cannot fit the concept of probability to new situations. (see Map 5) and physical symbol systems (see Map 3), for example, are automata, because they implement effective processes that are Turing-computable. to infinity with time. But, on the other hand, the total number of possible Turing machine states will always be finite. Note: Gödel claims that Turing's argument only becomes valid under the following two assumptions: This info-mural is one of seven The History and Status of the Debate — Map 7 of 7 is disputed by 33 Improved machines. A beefed-up machine can recognize the truth of the Gödel sentence. Such a machine defeats is supported by 34 Anticipated by John Lucas, 1961 Inductive machines may be immune to Gödelization. Inductive thinking, which humans can do, would is disputed 37 John Lucas, 1961 A dilemma about Either Or 90 Alan Turing, 1936 Minds and automata have a finite number of is disputed by 1. There is no mind separate from matter. 2. The brain functions basically like a digital computer. Gödel believed that the second assumption was true but that the first was "merely a prejudice of our times" that would eventually be disproved (Wang, 1974, p. 326). “argumentation maps” in a by allow computers to understand their own inductive The machine acts at random, The inductive machine works An Issue Map™ Publication Mathematical Claims Related to Gödel's Theorem Lucas's argument, because it shows that a formal system can evade Lucas's Gödelizing ability. Gödel sentences. machines. Trying to evade the Gödelization problem in which case according to definite rules, in which case is supported by states. At any given time the mind can only possess a finite number of states. If the mind could possess an infinite number of states, some of them would 93 Selmer Bringsjord, 1992 Church's thesis: This widely accepted (though unproven) claim states that Turing computability, have to be arbitrarily close together, and thus would Duplication of behavior does not entail computability by making inductive it will not be able to emulate become confused with each other. So the human of internal states. Even if all human functions can be general recursiveness, lambda computability, and any other formulation of effective computability series that explores Turing’s machines results in a human intelligence. it can have a Gödel sentence mind does not differ essentially from a finite duplicated by an automaton, this does not entail that the are all equivalent and all adequately describe the intuitive concept of computability. constructed for it.. is dilemma. Start Here Church's theorem: First-order predicate logic is undecidable. That is, there is no test to determine the truth or falsity of any arbitrary statement of predicate logic. is supported by 38 Frank H. George, 1962 In Either Case automaton. There is a limit to the disputed by internal processes that give rise to those functions can also be duplicated. For example, there is good reason to believe that deliberation is noncomputable, even though the behaviors Some AI systems are inductive and probabilistic. The Gödel that result from deliberation can be duplicated by an question: “Can computers think Completeness theorem: First-order predicate logic is complete in the restricted sense that all argument correctly shows that there are limits on deductive machines. amount of detail automaton. But, artificial intelligence deals with inductive and probabilistic An inductive machine is not an adequate model of the mind. 92 Arthur Burks, 1973 needed for a complete of its (true) statements are provable (but all false statements are not necessarily disprovable). Human functions can be This theorem was first proven by Gödel in 1930. mechanisms that Gödel's theorem does not apply to. specification of natural Selmer Bringsjord captured by finite automata. A human functions (p. 56). 1 Alan Turing, 1950 Does Gödel's theorem Tarski's theorem: The concept of truth for arithmetic languages is undefinable. Contemporary Now that's is supported by finite description of a human being can be fine-grained enough to 95 Selmer Bringsjord, 1992 Nonintelligent procedures can be noncomputable. There are and/or will they ever be able proofs of Gödel's theorem often begin with a proof of Tarski's theorem and then derive Gödel's symmetrical! I believe that at the end theorem as a corollary. 39 J. J. C. Smart, 1961 Ingenuous machines could 5+7=12 41 J. J. C. Smart, 1961 capture all that is essential to its humanity. Once such a description is disputed possible procedures that don't require intelligence to carry out, yet that 40 Anticipated by J. J. C. Smart, 1961 of the century ... one evade the Gödel argument. 5+9=14 A self-reflecting ingenuous machine can't be out-Gödeled. An is formulated, a finite automaton by are noncomputable. Imagine that some part of the brain receives show that machines An ingenuous machine is no better than a moronic machine. Yes, machines can Turing's theorem: The halting problem for Turing machines is undecidable. That is, there is is supported by is electrochemical pulses, and emits pulses in such a way as to Machines may have mathematical A moronic machine can't extract itself from the Gödel predicament, ingenuous machine that can ascertain its own syntax can avoid the Gödel can be constructed to duplicate all will be able to speak of no single algorithm for Turing machines that allows a machine to halt on an answer to every problem. insight, if they are properly is disputed even if it is given an infinite amount of time. Neither can an ingenuous disputed by problem. By progressively adding new syntax to its language, an ingenuous natural human functions. instantiate the busy beaver function. The noncomputability of the busy beaver function shows that some physiological procedures may programmed to gauge symmetry and machine could understand any new Gödel sentences that Lucas might Burks to?” Argumentation mapping is machines thinking machine extract itself, because it only works faster than the moronic (or will be able to) without expecting to be Cantor's diagonal theorem: There is no one-to-one correspondence between the real numbers simplicity in patterns of formulae. by machine. An ingenuous machine may display some mathematical present it with. not be computable. can't think? Such an "ingenuous machine" insight (that is, it may be able to see shortcuts to proofs), but it still can't 94 Raymond J. Nelson, 1989 and the rational numbers. This theorem shows that there are distinctly different orders of infinity replicates the abilities of human recognize the truth of its own Gödel sentence. Thinking is a rule-following process. Any psychological think. A computational contradicted. in mathematics. The diagonalization procedure used by Cantor was later modified by Gödel and used for the proof of his incompleteness theorem. mathematicians, and so can evade Gödelization as well as any human theory must ultimately decompose psychological phenomena into neural processes that do not require intelligence to implement. is supported by a method that provides: is supported by can. 48 Paul Benacerraf, 1967 49 John Lucas, Those neural processes can be characterized by Turing-type rules, system can possess all Note: For further account of these theorems, see Hunter (1973) or Enderton (1972). 43 John Lucas, 1961 Informal proof. A machine 1988 and hence are computable. IF is 98 Selmer Bringsjord, 1992 3 John Lucas, 1961 A machine complex enough disputed could, in principle, construct an informal proof of the truth A machine is not capable of Note: Also, see the "Do humans use rules as physical symbol systems do?" arguments on Map 3, the "Do connectionist networks THEN Mathematical language important elements of Computers as formal systems are limited by Gödel's incompleteness theorems. Gödel's theorem is the Achilles heel of mechanism. Gödel's theorem proves that a computer cannot, in is supported by 42 Anticipated by John Lucas, 1961 A highly complex machine may not be Gödelizable. A qualitative difference in the way is to be un-Gödelizable will not be a machine. By definition, by of the Gödel sentence. So informal proof follow rules?" arguments on Map 5, and the "Is the relation is lacking. In principle, 96 Selmer Bringsjord, 1992 - a method for portraying major disputed long as the machine does not is in the human between hardware and software similar to that between human human capabilities (doing The busy beaver function. principle, operate with human understanding. The argument goes as follows: computers think may be introduced when they have advanced to a high enough degree of machines behave in a determinate arithmetic, etc.) might be 4 Kurt Gödel, 1931 by regard such informal sense. No matter brains and minds?" arguments on Map 3. human thinking or 1. Computing machines are essentially formal systems. 2. Gödel has shown that there are sentences (Gödel sentences) that cannot be proven within a formal system but that humans can see to be true. is supported by Gödel's first theorem. Gödel's incompleteness theorem shows that any consistent formal system of axioms and rules of inference, provided it is strong enough to produce arithmetic, will contain true statements that cannot be proven by the procedures provided in the system. complexity. Such a highly complex machine may recognize the truth of its own Gödel sentence. manner according to definite rules. But, any such determinate machine is susceptible to the 47 John Lucas, 1961 persuasions as proof proper, introducing them into its system will not lead to disputed by how informal a machine's reasoning 97 Raymond J. Nelson, 1989 translated exclusively into recursive functions. But this begs the question of 1. Given a natural number n, the busy beaver function outputs a series of marks. 2. The output is equal to the maximum number a Turing machine may appear to be, it understanding. 3. Therefore, humans can do something that computer's can't do, namely, recognize the truth of Gödelization procedure because with only n states can write on its tape. philosophical, political, and Note: For a brief account of the proof of this theorem, see sidebar, "The Steps of Gödel's A self-Gödelizing inconsistency. So, a must still be Many human behaviors can be described whether such capabilities 3. A Turing machine cannot compute what this maximum number Gödel sentences. 44 Albert E. Lyngzeidetson, 1990 its behavior can be formalized. machine can still be algorithmically. Humans have many behaviors in can be instantiated by Proof," on this map. self-referential machine may grounded in a formal of marks is, but a human may be able to determine this. Note: Lucas credits the following authors with making similar arguments: Turing (1950) (see "A A connectionist machine may evade Gödelization. A connectionist machine with massively Thus, by the definition of a common with computer programs: for example, automata, given that we Human Cannot Simultaneously Beat All Machines," Box 79), Rosenbloom (1950), Nagel and Newman out-Gödeled. A machine recognize the truth of its own system. So the Note: The busy beaver function was first discussed by Tibor Rado (1962). parallel distributed processing capability may not be susceptible to Gödelization procedures. Such a machine, any machine that cannot with a Gödelizing operator is Gödel sentence without using machine's informal stimulus-response reactions, doing arithmetic, planning currently have no (1958) (see "Mathematical Thought Cannot Be Fully Formalized," Box 82), and Rogers (1957). is supported by system could in principle reconfigure its own parameters while in the process of computation and thus itineraries, doing office work, and so forth. These mathematical terminology be Gödelized is not really a still inadequate. The a Gödelizing operator. "proofs" are also Alan Turing 5 Kurt Gödel, 1931 arrive at its own semantic metalanguage by inductive means. Once in possession of its own metalanguage, machine at all. Gödelizing operator, to be behaviors can be described algorithmically with to express such capabilities. formalizable, and pragmatic debates Here is a mathematical reason why computers can't think. is supported by Gödel's second theorem. As a corollary to Gödel's first theorem it follows that any consistent formal system strong enough to produce arithmetic the connectionist machine would be able to evaluate its own Gödel sentence. 45 46 Anticipated by John Lucas, 1961 A machine with a "Gödelizing programmable, must be specified by some finite rule. But in that case, the Gödelizing operator is itself formalizable. The resulting G thus subject to the Gödel procedure. is supported by recursive functions. Therefore, it is likely that humans are automata. There is at this time no way to express such capabilities as writing stories and poems - a summary of an ongoing, is supported by is disputed by Göde l's Th eorem cannot prove itself consistent. is supported by Self-referential machines. A self- referential machine can evaluate Gödel sentences for itself. is supported by operator" can defeat Lucas's argument. A machine with a Gödelizing operator can carry out the Gödel procedure and add all its Gödel sentences to itself as theorems. Such a G system can then be shown to contain a formula that is true but that cannot be proven in the system. So, the Gödelization procedure still OUCH! is disputed by in a logic-mathematical language of any sort (p. 101). major philosophical debate of dx x+y dy g(x,y) John Lucas Lucas Arithmetic David Lewis (1969) proposed that the real issue in the Lucas Such a machine may evade the Lucas argument. self-referential machine would recognize the truth of its Gödel sentence and any subsequent Gödel sentences that could be formed about the machine. holds against the self- Gödelizing machine. 53 Roger Penrose, 1990 Hah! I'm always one step ahead! 100 Selmer Bringsjord, 1992 Persons are not brains. The complete argument that persons are the 20th century dt dt argument is the form of arithmetic that Lucas uses in discussing The Gödelian insight is Bringsjord automata must include a missing third premise: his own mathematical activity. Lewis dubbed this form of Arithmetic C G a slippery character. 2 dz sin(x) Proposed Model 1. Neurons are automata. reasoning the Lucas arithmetic. The Godelian insight can Algorithmic Is the Lucas Arithmetic B is 2. Brains are collections of neurons. Computer thought is mathematically possible. It is dt is supported by 50 Douglas Hofstadter, 1995 disputed attach itself to any system specification mathematically possible for a computer to think as well as a dx 3. Persons are brains. Lewis defines this arithmetic as the ordinary Peano arithmetic METACAT. The COPYCAT program (see "COPYCAT," Map 1, Box 77) could, in principle, be developed into by that has been algorithmically 4. Therefore, persons are automata. human can. The mathematics of computation contains nothing dt f (x,y) Arithmetic A a new program called METACAT. METACAT would have the reflexive ability to recognize the truth of its own The with the addition of an infinitary rule of inference. Peano Formalization specified, including an - a new way of doing intellec- to prohibit machines from thinking. There's no mathematical Gödelian is But the third premise, "Persons are brains," is false, so the conclusion arithmetic is a formalization of arithmetic developed by Gödel sentence, and could thereby evade the Lucas argument. METACAT would be able to: of the Gödel algorithmic specification 99 David Cole, 1992, as articulated by Selmer Bringsjord, • represent "issues" and "pressures" involved in a problem reason why computers insight disputed that persons are automata doesn't follow. argument dialectical? Guisseppe Peano in the late 19th century. The system operator of the Gödelization 1992 • understand how someone else thought up an analogy that didn't occur to it can't think. by Note: Also, see the "Can computers be persons?" arguments on Map 1. uses a set of 5 axioms to deduce all the truths of ordinary procedure. Brains are automata, therefore persons are too. arithmetic. • store episodic memory of past problems it has solved Because neurons have computable transfer functions, they • recognize meta-analogies, that is, analogies between different analogies are a kind of automata. And because brains are collections • construct puzzles based on a sense of "aesthetics." is supported by 102 Anticipated by Selmer Bringsjord, 1992 tual history. In ordinary Peano arithmetic, the consistency of the system of neurons, they must be automata as well. Our brains are A finite time period only allows a finite number of cannot be proven within the system; this would violate Gödel's what make us what we are, so persons are automata. states. Only processes that require no time at all could occur The Background of Gödel's Proof What does it Context second theorem. The addition of an infinitary rule, however, Note: Also, see the "Biological" arguments on Maps 3 and infinitely often in a finite period of time. But causal processes 103 Selmer Bringsjord, 1992 6 allows us to infer the consistency of a formal system from is 5. By the turn of the 20th century a crisis had developed in the foundations of mathematics. The discovery of fundamental paradoxes led to concern about the is supported by The argument from Gödel's theorem is dialectical. The Lucas argument involves a hypothetical 52 Bruce MacLennan, 1990 occur in space and time, and hence require at least some time The Zeus machine. It is possible for infinitely many basic concepts of math and logic. In response to those concerns, mathematicians tried to develop more secure foundational systems (see sidebar, "Formal Systems: within that system without violating Gödel's second theorem. The Gödelization procedure can be algorithmically specified. The metamathematical disputed Implemented Model (however small it may be). So, it's impossible for infinitely states to take place in a finite time period, as proven by the game played between Lucas and a mechanist. A mechanist presents Lucas with a machine model of Lucas's is supported by by An Overview," on this map). mind. Lucas counters by showing that he can recognize the truth of that machine's Gödel sentence, whereas Gödelization process can be formalized. It is "meta" in the sense that a formal mathematical many causal processes to occur in a finite time period. conceptual possibility of a "Zeus machine." A Zeus machine When fully worked out, the Lucas arithmetic includes all of the Gödel sentence for process is being used to reason about a mathematical process. 56 R. S. Boyer and is an automaton that works faster and faster with each the machine can't. In this way, Lucas can defeat any of the mechanist's attempts to reduce him to a machine. Gödel sentences of all systems powerful enough to produce Arithmetic A 51 Stuart Russel and Peter Norvig, 1995 One such system, the Principia Mathematica of Bertrand Russell and Alfred North Whitehead, was widely received and provided a framework for subsequent is supported by J. S. Moore, 1979 computation it performs. For example, in listing all natural work on the foundations of arithmetic, geometry, analysis, and algebra. Concurrently with Russell and Whitehead, David Hilbert worked on foundational arithmetic. As such, it includes an entire hierarchy of systems The Gödelian insight has already been formalized. The Boyer-Moore numbers (an infinite list) the first number is listed in 1/2 No, I'm not, because I can see that of arithmetic, each of which contains the Gödel sentences for 101 Selmer Bringsjord, 1992 systems in Germany. Hilbert's central idea was that the consistency of mathematics could be shown by a system of metamathematics—a system of mathematics Gödel sentence for Programs have been developed that can derive Gödel's theorems. theorem prover. second, the second number in 1/4 second, the third in 1/16 the Gödel sentence for M is true. all of the lower-level systems in the hierarchy. Implemented Model Humans have an infinite capacity that machines is about mathematics. You are a machine M. Arithmetic B The "Gödelian insight" has, in effect, been formalized. Penrose 55 Natarajan Shankar, 1994, A LISP-driven second, and so on, so that after one second has passed an is lack. Unlike deterministic automata, persons might be in disputed neglects this possibility because he fails to distinguish between 54 K. Ammon, 1993 as articulated by Stuart Russel and theorem-proving infinite list has been compiled. Note: The method of adding an infinitary inference rule to is supported by disputed an infinite number of states within a finite period of time. is by The work of Kurt Gödel was situated in this context. He set out to apply the recently developed methods to his own areas of interest, and shortly thereafter the formal system within which a Gödel sentence is proven and SHUNYATA. The SHUNYATA system has autonomously Peter Norvig, 1995 engine has been Note: The mention of a machine that can be in an infinite In that case I can Peano arithmetic was developed by Gerhard Gentzen in 1936. by Such an infinite capacity could allow persons to make disputed discovered the proof for his famous incompleteness theorems. the system that does the proving. developed a diagonalization procedure in its proof of The proof has been formalized into used to derive many number of states in a finite amount of time doesn't contradict produce a new Gödel decisions that machines could never make. by OK, you are a different Note: This claim was originally articulated as an attack on Gödel's theorem. It has also been used to formulate an a program. Using the Boyer-Moore novel mathematical the claim made in Box 101, because a Zeus machine is not Note: Bringsjord is discussing a possible human contain? Method Mechanist machine, M2. sentence for M2. Lucas Penrose's response to Boolos (see "The Gödelian Insight Is All automatic version of Gödel's proof. theorem prover, Gödel's theorem has results, including a Turing machine or an automaton, but rather an imaginary capacity, rather than an actual human capacity. Gödel developed a numbering system that allowed him to encode formal metamathematical proofs into numeric expressions. The numeric expressions, or That We Need," Box 66). Also, see the "Does Gödel's theorem been derived from a basic set of axioms decisions on some machine designed to prove a conceptual point. Gödel numerals, could be recognized and manipulated within the Principia system in the same way that any numerals could be (see sidebar, "The Steps of Gödel's show that mathematical insight is non-algorithmic?" arguments by a computer in basically the same way open questions in Proof," on this map). Gödel numerals allowed the Principia—and formal systems generally—to "look at themselves" and say things about themselves. Through this 8 Paul Benacerraf, 1967 on this map. is supported by that Gödel proved the theorem himself. is supported by mathematics. Turing Machines method, Gödel wanted to find out whether the formal system of Principia Mathematica could prove itself consistent. 7 Irving J. Good, 1967 The super-mechanist. Even if Lucas can Gödelize any machine that a mechanist can develop, that ability What Is a Machine? A machine can play Lucas's game. A is supported by doesn't entail that he can Gödelize any machine whatsoever. A suprahuman mechanist might be able to design The concept of a Turing machine arose in the A great many notions of what a machine is are found in the literature. A machine is: He discovered that no such proof exists. There is no way for the Principia to prove itself consistent. Gödel went further and used his method to prove that no machine programmed to do transfinite counting a machine that exceeds Lucas's Gödelizing ability. context of attempts by mathematicians to specify complete formalization of arithmetic exists at all (see "Gödel's First Theorem," Box 4). As a corollary, he returned to the consistency question and showed could play Lucas's game as well as Lucas can. precisely what an algorithm was. Alan Turing's IF the AND THEN do the AND AND 1. Any instantiation of a formal system (Lucas, 1961, p. 44). that no consistent formal system of arithmetic could be proved consistent using only its own methods of proof (see "Gödel's Second Theorem," Box 5). Lucas implicitly relies on the fact that transfinite insight was that any algorithm could be carried Current IF the following to the move THEN counting hasn't been formalized. But the fact 9 John Lucas, 1967 out by one of a class of Turing machines. Indeed, symbol on the the tape change 2. Anything that can be effectively constructed (George, 1962, p. 63). Earlier Precedents is Good misunderstands the game. State is ... Current that transfinite counting hasn't been formalized 10 Irving J. Good, 1967 he proved that an algorithmic procedure (or, an as Gödel wasn't the first to suspect that his completeness and consistency results held. Earlier, Finsler (1926) presented an idea similar to Gödel's but without disputed doesn't show that humans are any better at it Good misunderstands the nature of the Gödel's theorem is a red herring. No "effective procedure") is just a procedure that Symbol tape and/or the 3. Anything that operates according to an algorithm (Coder, 1969, p. 235). showing how to formalize the argument. Because he did not deal with a specific formal system, Finsler could not present any actual proof of his claim. The by than machines are. is game between the mechanist and the is matter what game we take Lucas to be playing, can be implemented by a device that blindly and reads ... halt ... follows ... Current American mathematician Emil Post had also proved a result equivalent to Gödel's, but his work wasn't published. Note: Also, see the "Can improved machines disputed mentalist. disputed Gödel's theorem is not the real issue. In fact, deterministically manipulates symbols. So, State to ... 4. Anything constructed from "unconnected primordial parts" (Hartmann, 1935, p. 71). by • The game is not played with a by Is the use of consistency in the beat the Lucas argument?" arguments on this Gödel's theorem is a red herring that distracts Turing machines precisely define the concept of Suggested Reading map, and "The Gödelization Procedure Can Be machine but with the machine's us from the real issue of transfinite counting. an algorithm. 1 0 Do nothing Right 1 5. Anything equivalent to a Turing machine (various contemporary authors, including Readable discussions of Gödel's theorem include Hofstadter (1978) and Nagel and Newman (1958). Smullyan (1987) teaches Gödel's theorem through a series Algorithmically Specified," Box 52. designer. The game is about what 1. We cannot actually know the truth of the Searle, 1991, Nelson, 1989, and Benacerraf, 1967). Is mechanistic of puzzles. An introduction with applications to computers is Harel (1987). For historical context, see Rucker (1987), Davis (1965), Dawson (1984a, 1984b), the mechanist can do, not about Gödel formula; we can only believe it based A Turing machine is conceived of as an imaginary what the machine can do. 1 1 Do nothing Right 2 6. Anything that can be given a purely geometric description (Spinoza, 1674, p. 129). and van Heijenoort (1967). This last volume reprints Gödel's original paper. on our belief that the formal system is device that manipulates symbols on a tape. The • The game is not concerned with The Steps of Gödel's Proof trans • fi • nite num • bers: Numbers that go beyond the trans • fi • nite count • ing: A form of arithmetic that works with showing the superiority of humans over all machines. All the game consistent (a fact that itself cannot be proved without contradicting Gödel's second theorem). Lucas argument problematic? behavior of a Turing machine is determined by the state it is in and by the symbol it reads on the tape. Based on those 2 factors, the machine 2 0 Erase & write 1 Right 3 7. Anything that behaves according to an unambiguous set of instructions that requires no imagination to follow (Crossley et al., 1972, p. 32). Altogether the seven maps: 1 The Principia Mathematica gave Gödel a way to translate natural language statements into a formal system of proof. With this formal system in hand, Gödel was able prove that all such systems strong enough to produce arithmetic are "G says it is not provable that G." Sentence in natural language magnitude of any finite set. transfinite numbers instead of just with the finite numbers. shows is that for any particular machine the mechanist presents, a mentalist (who knows Gödel's theorem) can show that he or she is not that machine. 2. Recognizing the first point, we can see that it was not the Gödel formula that was at the core of Lucas's argument; the real issue was transfinite counting. But Lucas cannot 57 The problem of consistency. The notion is supported by philosophy valid? will enter a new state, write a symbol on the tape, move to the right or to the left, or halt. The table of rules (or "machine table") correlating 2 3 3 1 0 1 Do nothing Do nothing Do nothing Right Left Right 2 4 3 8. Any device generating a recursively enumerable set of integers (Webb, 1968, p. 158). 9. Any device that is, in principle, divisible into parts (Lucas, 1961, pp. 56–57). prove that he is better at transfinite counting these actions with states and symbols exhaustively inherently limited. Illustration of the proof begins than a machine is. of consistency involved in Lucas's argument specifies a given machine. Based on its machine 10. Any system whose behavior can be fully explained in terms of proximate causation with an English version of what will become the is runs into difficulties for humans and/or 4 0 Erase & write 0 No move 4 (Mayr, 1982, pp. 67–70 and 114–16). - summarize over 800 major is table, we can determine exactly what a Turing notorious "Gödel sentence." 11 Daniel Dennett, 1972 disputed machines. machine will do with any given tape. disputed There are no mechanists for Lucas to play with. The dialectical game never gets off the ground because there are no mechanists by Note: See "Gödel's Second Theorem," Box 5. 58 Anticipated by John Lucas, 1961 59 John Lucas, 1961 4 1 Halt No move 4 11. A system that obeys both principles of physical and chemical causality and In step 2, the sentence is roughly translated into the Formalization by An inconsistent machine cannot model for Lucas to play with. According to Lucas, mechanists believe that producing true sentences is an activity that can be reduced to Gödel's theorems do not apply to principles of human functional design (Polanyi and Prosch, 1975, pp. 168–70). 2 formal system of Principia Mathematica. The G <—> ~ prov (G) of Gödel finite features and behaviors. But no real mechanist holds this finitistic view of intentional action. All that real mechanists demand on inconsistent machines. Gödel's the mind. Although it is true that Gödel's 104 A "universal Turing machine" is a Turing machine that can perform all the calculations of formalization shown here is not well-formed (i.e., sentence in is theorem does not apply to inconsistent machines, Natural phenomena are best understood in terms of mechanistic principles. Current is that the mind be deterministic, and for that we do not need to assume that the mind is finite. c ti theorems only apply to a machine if the Note: Other notions of what makes a machine are found in the historical literature. See disputed The human mind, like all natural systems, is mechanical. Mechanical explanations—like any other Turing machine. To emulate a given 1 moves in the debates threaded is not "grammatically correct"), and consequently the formal is supported by that doesn't matter because inconsistent machines machine is consistent. But, we can State contra di P Not P by those employed in physics—provide the only necessary foundation for the rest of the machine, the Universal Turing Machine is Descartes, Kant, Newton, and La Mettrie, for example. the system of proof used in Principia Mathematica system (e.g., never know for sure whether a given can't model the mind. Human minds are geared "programmed" with a special tape that fully cannot be applied to it. To make the sentence truly Principia for consistency. They seek consistency, and use sciences, including psychology. Through this extension of the method of physics, 12 David Lewis, 1969 13 John Lucas, 1970 machine is consistent or not. So, we mathematics can be brought to bear on psychology and make it a rigorous science. describes the emulated machine's table. self-referential and well-formed we must first develop Mathematica) We don't need the entire Lucas arithmetic. A mentalist doesn't have never really know whether Gödel's it as a norm for judging which beliefs to accept. a way to make well-formed statements that refer to Lucas must be able to Note: This region traces the development of mechanistic philosophy from Descartes to the Scanner component showing produce the entire Lucas is to produce all of the Lucas arithmetic. It is sufficient that he or she produce theorem applies to a given machine. present. Strictly speaking, all the claims on these 7 maps deal with mechanism, themselves. Current Symbol into claims, rebuttals, and arithmetic. Lucas's argument disputed enough of the Lucas arithmetic to answer the mechanist at a given step of because if machines can think then mechanism (in philosophy of mind, at least) is vindicated. Current Symbol on the tape by the game. The success of the Lucas argument must be evaluated in the 0 0 1 11 0 1 1 requires that a person be able to To that end, each symbol in the formal system is is produce the whole of a "Lucas context of a particular machine being challenged by a particular mentalist. 114 Immanuel Kant, 1790 3 replaced with a code number. 11 3 1 999 8 11 9 Code disputed arithmetic," which includes all of Gödel's theorem 60 John Lucas, 1961 Either Or dz dt f (z,y) Mechanism can't support critical philosophy. A proper numbers by the Gödel sentences of all formal The mechanist's dw treatment of the teleology (i.e., goal-directedness) of natural systems can be counterrebuttals systems powerful enough to dilemma. The can be dt g(w,y) demands the assumption of a designer with purposes in mind. The produce arithmetic. But Lucas 14 David Lewis, 1969 red her • ring: An argument that Lucas argument can M is consistent, M is not consistent, 61 G. Lee Bowie, 1982 explained explained Machines can't produce the entire Lucas arithmetic. Even though distracts attention from the issue assumption of a designer with moral and aesthetic purposes supports has not shown that it is possible is supported by be restated as a Lucas can't know when his Gödelization by by the possibility of a critical philosophy, that is, a philosophy that allows Those numbers are then used as exponents in a series Lucas can't produce the entire Lucas arithmetic, we can still salvage a weaker in question. Derives from the use in which case in which case 4 of prime numbers that will be multiplied together 11 3 1 999 8 11 9 881 Gödel for humans to have this ability. conclusion from his argument, namely, that machines can't produce the entire of red herrings (a kind of smoked is supported by dilemma about is procedure is applicable. For the Lucas argument us to make value judgments. Because mechanistic explanations tend into one large Gödel number. This step ensures that 2 x 3 x 5 x 7 x 11 x 13 x 17 = approx. 2.255092414 x 10 Lucas arithmetic either. consistency. Consider by Gödel's theorem there M cannot be a mind disputed to work, Lucas must still be able to tell which machines to ignore the assumption of a purposive designer, the mechanistic number fish) to distract tracking dogs from some arbitrary are consistent. But this is mathematically impossible. 113 Benedict Spinoza, 1674 - 97-130 arguments and rebuttals every Gödel number corresponds to one and only the trails of escaped prisoners. will be a sentence that because minds must be by 112 René Descartes, 1637 is philosophy is inadequate to support critical philosophy. machine M. Note: For further explanation, see Church's theorem as Minds are mechanical. The human Note: Spinoza is only one of many authors that Kant criticizes with one formula in the formal system of Principia humans recognize as true consistent systems. Mechanistic principles cannot explain mind can be fully explained by disputed Mathematica. but that M cannot prove. discussed in Hunter (1973) or Enderton (1972). the mind. The soul and mind are not made this kind of argument. 15 Hao Wang, 1974 Or mathematical principles. Because the by Either So, we can do something of the same kind of substance as external A dilemma about 16 John Lucas, 1997 laws of nature are the same consistency. that the machine M can't. bodies are, and mechanism cannot explain everywhere, the same method should The mechanist knows all The mechanist doesn't know all The Lucas argument claims less than Wang's dilemma per map This number is then represented symbolically, in this Symbolic Lucas's dialectical their workings. So, although inanimate be used to study the mind as is used 5 case, by a long string of S's followed by a 0 (where representation argument against consistent machines, consistent machines, is suggests. We can often tell whether or not a machine is consistent, and only those that we know to be consistent are adequate candidates In Either Case 63 John Lucas, 1976 objects, plants, animals, and the bodies of to study bodies. By providing a is supported by the number of S's is equal to the Gödel number). The SSSSSSSSSSSSSSSSSSSSSSSSSS ... SSSSS0 of Gödel mechanism runs into disputed There are overriding reasons to regard minds as consistent. 105 René Descartes, 1637 humans can be understood by applying "geometric" theory of emotions and symbolic "numeral" representation is necessary so in which case in which case by for models of the mind. The mechanist need not know the consistency Mechanistic philosophy provides the correct physical (mechanistic) laws, the mind of numeral a problem about how of all machines in order to know that the ones he presents to the Hutton's argument for the inconsistency of the mind is flawed in a number the mind, human behavior can be that the Gödel number can be dealt with formally in is supported by the mechanist knows The machine cannot be a mind. means of investigating the external world. is man needs a completely different kind of shown to be mechanical in the same mentalist are consistent. of ways. • His probabilistic model of the mind is unrealistic. It holds that we accept Principia Mathematica. whether or not his the mechanist has a decision the mentalist's ability to refute the Animate things (plants, animals, the human body, etc.) disputed explanation. - 70 issue areas in the 7 maps way that physical bodies are. models are consistent. procedure for logic. But mechanist doesn't imply that no should be explained according to mechanistic principles, by Note: Descartes both supports and disputes 62 Anthony Hutton, 1976 or reject propositions independently of each other. This is not so. is supported by • An inconsistent model of the mind would affirm every proposition, but no Note: Also, see the according to Church's theorem consistent machine can prove as because mechanism shows how bodies interact with mechanism. On the one hand, he thinks This numeral representation for Gödel numbers this is impossible. So, the much as the mentalist can. Belief in one's own consistency leads to inconsistency. The bodies are mechanistic, but on the other 115 Immanuel Kant, 1790 Gödel "Is the use of each other and with the world. Such principles allow 6 following argument shows that humans may be inconsistent, and therefore mind would do that. The assumption of • We must assume that we are consistent reasoners to be able to start reasoning allows us to use Principia Mathematica to "talk G <—> ~ PROV (SSSSSSSSSSSSSSSSSSSSSSSSSS ... SSSSS0) sentence consistency in the mentalist has no opponent at all. is supported by is the application of physics to complicated phenomena hand he thinks that minds aren't mechanistic. about itself." In particular, we can plug the Gödel that we can't be sure that Gödel's theorem can be applied to minds. disputed design is useful for Lucas argument at all. and leads the way to understanding and knowledge of science. Even though the - 32 sidebars history and further numeral for G back into the formula G itself. This 1. Probabilistic evidence suggests that we have some contradictory beliefs. by those systems. problematic?" In Either Case 2. Rationality demands that we take this probabilistic evidence seriously. assumption of a designer is generates a well-formed Gödel sentence that makes arguments on this Note: Descartes both supports and disputes mechanism, ultimately dispensable in Corbis-Bettman reference to itself, and that can be assessed using the 3. So, rationality demands that we think there is some probability that we because on the one hand he thinks bodies are map. proof system of Principia Mathematica. It's so clear The mentalist cannot defeat his mechanist opponent. are inconsistent. Important Properties of Formal Systems mechanistic, but on the other hand he thinks that minds is science, it still provides a 4. So, being certain about one's own consistency (as Lucas claims we can Consistency is supported by disputed useful guide for scientific from the aren't mechanistic. Descartes explanations. Design I can't decide whether this be) is inconsistent with rationality. A system is consistent if it is impossible, within the system, to derive both a statement and by background outside. I can Gödel sentence is true or not. provides standards of see that G is not its negation. A system is inconsistent if a statement and its negation are both derivable. Now we are in a position to show that the Gödel If it's provable, then it's not simplicity, continuity, unity, 7 sentence is neither provable nor disprovable using the provable, so the provable. But if it's not 64 Douglas Hofstadter, 1978 In an inconsistent system, every possible statement (of its language) can be derived as a 106 Thomas Hobbes, 1650 Benedict Spinoza and organization that have led system of Principia Mathematica. To see how the sentence is true. provable, then it is provable. Human thinking is to fruitful hypotheses in the Inconsistency without explosion of belief. theorem, because everything can be logically derived from a contradiction. In a consistent paradox works, see sidebar, "Self-Referential Does Gödel's theorem show that I'm stuck! is supported by Lucas's argument depends on the assumption that if system, that isn't the case. computational. Reasoning is past. Paradoxes," on this map. G <—> humans were inconsistent, they would be committed purely a matter of abstract 116 René Descartes, 1637 ~ PROV to believing anything and everything. But this Consistency is also referred to as the "correctness" or "soundness" of the system. computation. Thinking is adding Machines can't meaningfully use signs. Animals can't use ( SSS ... S0) is supported by signs as humans do, and animals are essentially machines. So it is conclusion follows from a rule of propositional logic and subtracting, where "adding Do men and even women Corbis-Bettman that we have no reason to believe holds for humans. Completeness and subtracting" is extended to implausible that any other machine could learn to use signs as is make fun of each other better The argumentation maps: 8 However, the Gödel sentence can be recognized to be true from outside of the system. This is the aspect of the proof that later interpreters would focus on in the context of machine intelligence. Formal system: Principia Mathematica machines can't be conscious? It is quite possible for minds to be both inconsistent and coherent. 65 George Boolos, 1990 A complete system will have a derivable theorem to correspond to every true formula in its language. An incomplete system will not be able to derive some true formula. Decidability A system is decidable if every true formula of the system has a proof and every untrue apply not only to numbers but also to bodies, proportions, actions, words, motions, conceptions, and so on. Thomas Hobbes is disputed by humans do. 118 Gottfried Leibniz, 1714 disputed by 117 Julien Offray de La Mettrie, 1774 Animals and machines can use signs. Like humans, animals use signs. Songbirds mock each other, than do the birds who repeat the songs of other birds in such a way as to ridicule them perfectly? (p. 152) We do not know that Artificial machines can never duplicate parrots pick up human phrases, apes 18 Judson Webb, 1968 19 John Lucas, 1971 mathematics is consistent. formula of the system has a disproof. A system is undecidable if there is some statement natural machines. The kinds of machines could speak if their vocal chords were Gödel sentences are not Gödel sentences are self-referential enough for us to Gödel's theorem rests on the that it can neither prove nor disprove. that humans can construct will never be able different, and so forth. Differences - arrange debate so that the cur- 17 John Lucas, 1961 is see their truth. Gödel sentences by themselves are not Gödel's theorem shows that machines is self-referential. Gödel sentences don't disputed assumption of a consistent to fully simulate the structure of natural between human and animal use of have anything to do with consciousness, self-referential, but recognizing their truth requires us to see formal theory. But many Note: The properties described here are stated in terms of logical systems; they can also be 107 Julien Offray de La Mettrie, 1774 machines. In a natural machine each and signs is only a matter of complexity. I can't decide whether this can't be fully conscious. For a machine to disputed by them as self-referential. It is this ability to see Gödel Calculus Man is a machine. There is no Gödel sentence is true or by because they are not self-referential (as Lucas theories in the history of stated for formal systems more generally. For a discussion of formal systems and their every part, no matter how small, is tailored Similarly, differences between human understand its own Gödel sentence it would have sentences as self-referential that machines lack, and so properties, see Smullyan (1961). For further discussion of the properties of logical systems, reason to regard man as anything but a It's so clear from not. If it's provable, then claims they are). All that Gödel sentences mathematics have been flawed, to the ends of the whole. Artificial machines, and machine use of signs is also a Various authors, notably John Lucas and Roger Penrose, to be supplied with a consciousness-producing machines can't recognize the truth of Gödel sentences. see Hunter (1973), Enderton (1972), or Smullyan (1961). is supported by very complicated mechanism. Our the outside. I refer to is their own Gödel numbers. and even today the best theories by contrast, always contain some parts that matter of complexity. To make 9 it's not provable. But if part (e.g., a "Gödelizing operator"). This Algebra rent stopping point of each extended the Gödelian insight to computers. They can see that G is are sometimes called into own bodies are merely highly are not tailored to the ends of the whole. machines like humans, then, all we it's not provable, then it is Corbis-Bettman pointed out that because computers are a kind of formal requirement means that machines can have partial Set Theo complex machines, and so we can, at not provable, so provable. I'm stuck! question. This indicates that we ry Thus, an artificial machine will never be as have to do is make them more system, the same limitations discovered by Gödel in is supported by consciousness at best. Humans, by contrast, least in principle, make machines that Julien de La Mettrie the sentence is 20 Judson Webb, 1968 is supported by do not know, but only hope or 66 Roger Penrose, 1990 complex as a living organism. Gottfried Leibniz complex. Principia Mathematica and other formal systems might have full reflective consciousness; they can will recreate our behavior. true. reflect on themselves as a whole without relying Machines don't need new parts to incorporate 22 John Lucas, 1971 believe, that our mathematical The Gödelian insight is all that we need. It is not necessary to be able to see apply to computers as well. G <—> Machines aren't self-critical. theories are consistent, and this Top the consistency of an entire formal system. The ability to pass from one formal on a special part. is Gödelization. Universal Turing machines don't need new o logy is debate thread is easily seen ~ PROV parts to incorporate a Gödelizing operator. All they require is Webb focuses on Gödel in turn means that we cannot system to the Gödel sentence of that formal system is enough. This kind of Gödelian 119 Gottfried Leibniz, 1714 ( SSS ... S0) disputed sentences as a criterion of "see" with certainty the truth of disputed Julien de La Mettrie to be refocused on different sets of theorems. insight, which is not captured by formal rules, is characteristic of mathematical The conscious mill thought by consciousness. But the Gödel sentence. by is Note: Webb discusses the inclusion of new Gödel sentences insight and is non-algorithmic. 120 Gödel's theorem refutes the philosophy of experiment. If we were to walk into consciousness should be Note: This argument is directed Disputed by disputed an enlarged machine or mill that could Formal rather than "Gödelizing operators," but the point is essentially by mechanism. The existence of statements that are is supported by the same. construed in terms of the ability against Penrose's version of the "The Gödelian Insight Has Already Been Formalized," Box 51. undecidable by formal systems (e.g., Gödel's theorem) shows that mechanism "think, feel, and have perception," system: Consciousness- for self-critical thinking. argument, rather than Lucas's. nothing witnessed could explain its - identify original arguments by computer producing part can't provide a complete explanation of mathematical nature. So mechanism Self-critical thinking requires a must be rejected as a universal theory of science. conscious qualities, because such concept of truth. Because Note: Other Gödelian arguments against mechanism are spread throughout this qualities would not be found among the 21 Judson Webb, 1968 machines lack an adequate map. See especially the "Is the Lucas argument dialectical?" arguments on this parts of the mill. Note: A variety of simplifications have been used in this sidebar in order to illustrate in a readable way the general idea behind Gödel's extremely complex Lucas's interpretation allows for conscious machines. If Lucas's concept of truth, as Webb admits, Note: Also, see "A Conscious Machine proof: C is map. interpretation of the relationship between consciousness and Gödel's theorem they can't think critically in the Could Not Be Explained By Its Physical • Gödel does not begin with a non-well-formed formula and then fix it as we have done—this is just a rhetorically useful way to present the proof. over 380 protagonists world- disputed is correct, then machines can be conscious after all. All that is required is that: is way that humans can. A Workings," Map 6, Box 55. • The Gödel number used for "prov" in step 3 was chosen arbitrarily. Otherwise the number would be extremely long, because "prov" in Principia • They are given the ability to generate their own Gödel sentences. Other Lucas arguments by disputed machine's inability to recognize 108 Immanuel Kant, 1790 • They can answer questions about those Gödel sentences. Do mathematical Mathematica builds on various other concepts and definitions with their own code numbers. by the truth of Gödel sentences is is • There is a difference between "prov" and "PROV" that is not discussed: "prov" is a property of sentences in the formal system; "PROV" is a property of • Gödel sentences are really self-referential (which they aren't, but which is just a symptom of its inability to Mechanism is necessary for disputed 122 Judson Webb, 1968 numerals that represent formulas in the formal system. scientific understanding. by granted for the sake of argument). think self-critically. is Gödel's theorem doesn't solve the • The Gödel sentence does not make reference to itself in the simple way suggested by its informal versions. The numeral for G—and the sentence G Science proper is only possible in so constructivity problem. If Gödel's theorem wide over 40 years far as we formulate mechanical laws disputed itself—correspond precisely in that there is a one-to-one coding between them, but they do not have the same "meaning." They are "extensionally theorems like Gödel's 67 David Coder, 1969 of nature. Only such laws provide a by really refuted mechanism, it would also show how equivalent" but not "intensionally equivalent." Lucas's argument does 68 John Lucas, to solve the constructivity problem in the foundations firm mathematical basis for the not take into account 1970 of mathematics. But Gödel's theorem doesn't show Does Gödel's theorem is construction of theories about the people who cannot A single person's 121 William Nelson Reinhardt, 1986 how to solve that problem. disputed universe. Nature cannot be understood understand Gödel's understanding of Strong mechanism is refuted; weak mechanism is safe. Strong mechanism holds that the - make the current frontier of conceptually without determinate show that computers by Gödel is enough. 24 Daniel Dennett, 1990 theorem. Lucas has only Self-Referential Paradoxes Mathematical insight is not the important issue. Even if there is no algorithm for mathematical insight, the lack shown that someone who ! The power of mechanistic rules. Note: Kant both supports and disputes mind can be entirely explained in mechanistic terms. But strong mechanism either contains mentalistic concepts (in its explanatory apparatus) or is inconsistent with Gödel's theorem. Weak mechanism, on the con • struc • tiv • i • ty prob • lem: The show that The crux of Gödel's proof is a paradox, similar in form to other historical paradoxes about self-referentiality, including the liar paradox, Russell's of one is not crucial because insight is not important to mathematics. Problems like Gödel's theorem and the halting understands Gödel's theorem reasoning is the other hand, does not contain mentalistic concepts (because it only explains the concept of mind), problem of whether or not the foundations of paradox, and Richard's antinomy (not shown; see Richard, 1905). In each case, a dilemma arises from a self-referential claim. is is Gödel's mechanism, because on the one hand problem can be solved reliably by probabilistic algorithms; whether they are solved by insight or not doesn't matter. is different from a machine. demonstrated by a disputed and can be shown (via a lengthy proof) to be consistent with Gödel's theorem. mathematics can be formulated purely disputed disputed Theorem is supported by he thinks that mechanism provides the ? single person who are intrinsically Mathematics is grounded in its reliability, not in any particular kind of insight. But, what about a person who by Note: For a similar line of argument directed against philosophical behaviorism, see "Philosophical constructively, that is, without recourse to by by Gödel's proper basis for science, but on the debate easily identifiable cannot see the truth of Gödel's understands Gödel's Behaviorism Is Circular," Map 2, Box 87. assumptions about a Platonic ontology of abstract Theorem other hand he thinks that mechanism The Liar Paradox theorem? Lucas has not theorem is sufficient mathematical entities. If mathematics could be mathematical insight can't support critical (i.e., moral and Either Or 25 Jon Doyle, 1990 demonstrated that such a to show that minds aesthetic) philosophy. See "Mechanism weak mech • an • ism: founded on a purely constructional basis, that would The liar paradox dates back to the New Testament. Mathematical truth may evolve. Penrose assumes that all mathematicians agree on a shared and immutable notion of person can outperform a are different from Can't Support Critical Philosophy," strong mech • an • ism: The concept of mind is mean that it can be constructed "mechanistically." - provide summaries of eleven It was a Cretan prophet, one of their own countrymen, who said, "Cretans are always liars, vicious brutes, lazy gluttons"—and he told the truth! (Titus 1:12–13.) Assume that the liar sentence Assume that the liar sentence is is true, in which case false, in which case is non-algorithmic? mathematical truth. But a major school of mathematics, intuitionism, holds that mathematical truth evolves instead. If mathematical truth evolves, then there is no reason to believe that the Gödel sentence generated by a system will still be that system's own Gödel sentence at the (later) time when the sentence is evaluated. machine, and so his argument fails to establish that machines are essentially different from humans. machines. 70 Douglas Hofstadter, 1979 limited? Box 114. The mind is mechanical. mechanical. evolved into Lucas uses a faulty way of 110 it is true that "this very it is false that "this very sentence identifying differences. To see 77 Vitalism. Vitalism is a school of biology that maintained Mathematical theorems show that machine thought is limited. Gödel's theorem and other that biological organisms are too complicated to understand Legend major philosophical camps of sentence is false." So, on is false," and so it is also true. 69 C. H. Whitely, 1962 the flaw in Lucas's argument, imagine Sample Liar Sentences this assumption, the sentence So, on this assumption, the a person who thinks his own mathematical theorems like it reveal essential limitations on the project of making machines that think. using only mechanical principles. Vitalists posit the is Lucas tricks machines into Note: This region covers those arguments that don't derive from Lucas or Penrose but that still deal 1. This very sentence is false. is both true and false. sentence is both true and false. disputed contradicting themselves. superiority follows from the existence of a vital force, or elan vital, to explain the Zermelo This formula cannot be with Gödelian limitations; that is, with the limitations that Gödel's theorem (and other similar theorems) workings of living organisms. 2. Sentence 2 is not true. by -F Set The raenkel Consider the following consistently asserted by Lucas. is supported by uniqueness of his point of view. 3. It is true that this very sentence In Either Case 23 Roger Penrose, 1990 Näive ory 1997 Gödelian argument, directed However, the fact that he can see impose on machine and/or human intelligence. Note: Members of this school include Hans Driesh (20th The arguments on these charts are organized by links that carry a range of meanings: S is false. is supported by Mathematical insight is non-algorithmic. Many Theor et things in a way that is different from century), Claude Bernard, and the German the protagonists (or schools of y 1902 against John Lucas himself. Immanuel Kant problems of mathematics (e.g., Gödel's incompleteness Lucas cannot assert that "This everyone else doesn't prove that he is Naturphilosophen school (including Johann Wolfgang von The liar sentence cannot be pinned down to one truth value. It problem, the halting problem, etc.) can be understood by Goethe, Johann Herder, Immanuel Kant, and others). Arguments that uphold or defend another claim. Examples include: cannot be decided whether it is true or false. formula cannot be consistently superior. Anyone could make the conscious humans but cannot be solved algorithmically. is 78 Kurt Gödel, 1951 Saint Paul 26 Adina Roskies, 1990 asserted by Lucas "without same claim about their own point of Machines may eventually have mathematical is supported by supporting evidence, further argumentation, thought experiments, This shows that mathematical insight is based on conscious Penrose's intuitions are irrelevant. The fact that we can't see how algorithms could cause subjective states is disputed contradicting himself, although view. The Whitely sentence reveals 109 non-algorithmic processes. is by this problem of perspective. intuition. The incompleteness theorems only show that a Mechanistic biology. A school of extensions or qualifications, and implemented models. disputed irrelevant. We can't see how (following Penrose) quantum wave packets could cause subjective states either. The we can see that the statement is Ooops! I just is is supported by thought). machine cannot be proven to possess mathematical by same problem arises with any physical explanation of subjective states. Not being able to see an explanation doesn't true. But this argument is contradicted myself. disputed intuition. But this does not show that machines can't in fact biology that sought to apply physical 111 mean that the explanation is incorrect. simply a way of tricking Lucas by possess mathematical intuition. To the extent that machines laws (e.g., Newton's mechanics) to Evolutionary biology. The mechanists' Russell's Paradox Either Or Roger Penrose into contradicting himself. are limited by Gödel's theorems, humans are too. Neither biological systems by focusing solely emphasis on proximate causation leads A charge made against another claim. Examples include: How do I get a Similarly, Gödel's theorem is 71 J. E. Martin and K. H. Engleman, humans nor machines can formulate all of their mathematical on proximate causation. is them to ignore the origins of life forms. is In the late 1800s Gottlob Frege introduced a system of logic that tricks machines into 1990 Note: Members of this school include disputed Their demand for mathematical explanation disputed logical negations, counterexamples, attacks on an argument's seemed capable of providing a consistent basis for mathematics. Assume that the class of all Assume that the class of all Lucas disputed intuitions. It is in the nature of mathematics to be incompletable. In a letter to Frege, Bertrand Russell (1902) introduced a paradox 28 Clark Glymour and contradicting themselves. by Lucas can believe his Whitely Note: Also, see the "Does Gödel's theorem show that mathematical Julius Sachs, Jacques Loeb, and Ernest by leads them to underestimate the complexity by emphasis, potential dangers an argument might raise, thought classes that do not contain classes that do not contain Formal Systems: An Overview sentence. The claim that Lucas Haeckel. The origins of this school of biological systems. To understand that undermined Frege's system and inspired a restructuring in the themselves does contain themselves doesn't contain itself, Kevin Kelly, 1990 insight is non-algorithmic?" arguments on this map. experiments, and implemented models. Penrose can't argue for cannot believe the Whitely sentence can be traced back to Descartes. biology we must consider the origins of foundations of mathematics. itself, David Hilbert laid many of the foundations for what are now known as formal systems. species through variation and natural his hypothesis. No is incorrect. Lucas can recognize the in which case His formalist program aimed to develop axioms that would yield the desired truths of 72 Daniel Dennett, 1972 Whitely sentence as true, because selection. scientist can either establish To understand Russell's paradox, note that there are some classes in which case math and geometry, without reliance on intuition. These axioms would be true regardless or refute the hypothesis that Gödel's theorem only limits formal systems, not there is a point of view from which prox • i • mate cau • sa • tion: Explanation by Note: Members of this school include is interpreted as the class must contain itself of whether they were interpreted in terms of points, lines, and planes, or in terms of "law, their machine implementations. The Gödel sentence is 79 Alan Turing, 1950 proximate causes reveals how the parts of a system Charles Darwin, Thomas Huxley, and that contain themselves and some that do not. that class both does and because it is now a member of mathematical insight is he can understand how the sentence disputed A human cannot simultaneously beat all machines. Theorems like A distinctive reconfiguration of an earlier claim. love, and chimney sweeps." non-algorithmic/noncomputable for some formal system only limits a machine while it is tricks him. From this point of view work together and in interaction with parts of the James Baldwin. Baldwin (1909) makes • An empirically decidable doesn't contain itself. the class of all classes that do not implementing that formal system. But when the machine by Church's, Gödel's, and Turing's show only that a human can beat one machine environment. Scientists and philosophers working in the argument against mechanistic biology The class N of The class M of contain themselves. So again, the Lucas can appreciate that he can't non-red things is all men is not Formal systems allow mathematical results to be described and assessed with increased property must be "a 62 is implementing some other formal system, the machine assert the sentence—and on a given occasion. But there is no reason to believe that a human can this framework tend to reject any concern with either is most explicitly. N M class both does and doesn't precision, and are particularly useful with theories that deal with more than 3 dimensions. 29 Roger Penrose, is may be able to prove the previous system's Gödel sentence. out-think all machines. Just because we can think of questions that one machine the "ancestor causes" of a system (in the sense of disputed printed copy? itself non-red. itself a man. It contain itself. property in the Borel consequently he can recognize its This class contains does not contain 27 Roger Penrose, 1990 hierarchy." Computability 1990 disputed Physical machines, which can implement numerous formal can't answer, that does not mean that there might not be other machines that will evolutionary theory) or with the intentions of a by truth. A machine couldn't do this. perform better. "designer" (in the sense of divine design theory). itself as a member. itself as a In a formal system, the whole process of proof is reduced to the manipulation of symbols The absurdity of algorithmic insight. The claim that mathematical and noncomputability lack Glymour and by systems, transcend the limitations that Gödel's theorem Note: Martin and Engleman address Focus Box: The lowest-numbered box in each issue area is an introductory focus box. In Either Case according to rules that completely determine the conclusions that can be drawn from a set of is supported by insight is algorithmic can be reduced to absurdity. Kelly are too places on specific formal systems. member. axioms. Intuitions about the "meaning" of the symbols play no role in the workings of the is that property. Therefore, is this claim against a modified version The focus box introduces and summarizes the core dispute of each issue area, sometimes the hypothesis that strict. If Notes: of the Whitely sentence, "Lucas The assumption that there is a class of all classes that do not formal system. Such meanings are only introduced with the interpretation of the system. Assume disputed mathematical intuition is disputed Glymour and • Dennett presents this ar gument as a special case in his cannot consistently believe this 80 Emil Post, 1941 as an assumption and sometimes as a general claim with no particular author. contain themselves leads to contradiction. There is a knowable algorithm (AI procedure) that generates mathematical by noncomputable is by Kelly are right, broader discussion of how a physical object can implement Mathematics is an essentially creative activity. The dx Below is an example of a simple formal system designed for application to the logic of insight. sentence," that is credited to Douglas x+y empirically undecidable. then it is various Turing machines. is supported by results about incompleteness and undecidability support the dt Arguments With No Authors: Arguments that are not attributable to a particular • Also, see the "Is the brain a computer?" arguments on Hofstadter. truth. Note that, in addition to the standard interpretation in terms of truth, there are also • Penrose's ability to impossible to idea that mathematics is essentially creative. These results dz sin(x) 86 Anticipated by J. J. C. Smart, 1961 source (e.g., general philosophical positions, broad concepts, common tests in artificial other interpretations that satisfy the formal system (for more explanation, see "The Lowenheim- It follows that recognize mathematical verify or refute Map 1, the "Can functional states generate also show that it is preferable to have multiple formal systems dt is supported by The analogy of this argument Gödel's Theorem Skolem Theorem," Map 3, Box 107). any scientific consciousness?" arguments on Map 6, and sidebar, is rather than a single universal system such as Principia dx f (x,y) The argument from Church's theorem. According to Church's Charles Darwin intelligence) are listed with no accompanying author. 1. If we were aware of this algorithm, or of how to generate it, then we would truths by insight is disputed dt theorem there is no decision procedure for predicate calculus. This with the Richard antinomy Gödel sentences generate a paradox similar have to believe in the soundness of this procedure. consistent both with the theory. But in "Formal Systems: An Overview," on this map. 73 Thomas Tymoczko, 1990 Mathematica. The role of logic in mathematics is thereby by Highly flexible systems aren't means that there is no computable procedure by which a machine leaps to the eye. It is closely to the liar's paradox and Russell's paradox, 2. Gödel shows how to construct, for any algorithm for mathematical insight, hypothesis that insight is practice scientific shown to be one of revealing and developing the limitations related to the "Liar" too. ... except from the point of view of a formal Either Or Standard Interpretation Alternative Interpretation a sentence whose truth follows from the soundness of the algorithm, yet theories can be mechanistic. Dennett presupposes that a of formal systems, not of revealing what the one true formal can decide whether a given sentence in predicate calculus is true Citations: Complete bibliographic citations can be found in the booklet that accompanies algorithmic and with the or false. Human mathematicians, on the other hand, often decide We therefore have before us system such as Principia Mathematica. It is provable that "G is not It is disprovable that "G is not which is inaccessible to that same algorithm. hypothesis that it isn't. tested. So it may 74 Paul Benacerraf, 1967 machine that can effectively take the guise of system is. this map. a proposition that says about many different formal systems is a Turing the truth or falsity of sentences of predicate calculus. Moreover, provable," provable," TRUTH 3. We can understand and believe in the Gödel procedure. So, Penrose's intuitions be possible to test Lucas doesn't recognize what his human mathematicians decide such questions by constructing is 87 J. J. C. Smart, 1961 itself that it is not provable G <—> fail to establish his for algorithmicity machine. But a machine that generates new disputed The ingenuous machine. A (in PM) (1931, pp. 89–90). ~ prov FALSITY Therefore even if absolute argument really shows. If we fix up programs, treats its machine table as a changing proofs in a reasonable amount of time and not just at random. Methodology: A further discussion of argumentation analysis methodology can be in which case in which case hypothesis one way or the Lucas's argument so that it is more accurate, 81 John Myhill, 1952 by machine that is programmed for (G) There is not a knowable algorithm that generates mathematical insight. other. certainty can't be we discover that it proves something object, and shifts between programs would mathematical insight will decide the found in the booklet that accompanies this map. Axioms involve many auxiliary devices that cannot be Gödel's and Church's theorems are psychological laws. Gödel's G is both provable and not we can show that it is not the If sentence a is true, If appleness, then orangeness obtained. different than Lucas intended. The theorem shows that human creativity will always exceed human capacity to truth or falsity of sentences of predicate is modeled on a Turing machine. Thus, such a 88 Dale Jacquette, 1987 provable. case that G is not provable, so then if sentence b is also yields appleness. disputed corrected argument, it turns out, shows that: anticipate that creativity. Furthermore, the theorems also show that humans calculus in the same way that human Anticipated by Where this phrase appears in a box, it identifies a potential • Lucas may be a Turing machine. So Lucas's a —> (b —> a) true, then sentence a is machine is not a mechanistic model of the mind. is supported by Gödel's theorem shows that machines can't understand language mathematicians do. Such an "ingenious G is provable after all. So, G is by are able to entertain and clearly conceive of ideas that are neither attack on a previous argument that is raised by the author both provable and not provable. still true. intended argument—that he is not a is supported by the way humans can. Gödel's theorem shows that machines cannot have machine" will not have to rely on strictly is constructible nor effective. Sample (Informal) Gödel Sentences machine—fails. natural language understanding. decidable means for making decisions. so that it can be disputed. • It is required of speakers of natural language that they be able to distinguish Note: Myhill's claim is supported by other authors outside of this immediate You can order artist/researcher • Furthermore, if Lucas is a Turing machine, disputed Note: Also, see "Ingenuous Machines 1. G says it is not provable that G. In Either Case If the falsity of b entails If non-orangeness yields by debate, for instance, by Paul Weiss (1947) and H. Gelanter in personal the falsity of a, then the 30 Martin Davis, 1990 he has no way of knowing which one he communication with Myhill. any sentence from its negation. Could Evade the Gödel Argument," As articulated by Where this phrase appears in a box, it identifies a reform- 2. G is the sentence "G is not The system can neither prove nor disprove the Gödel sentence (~b —> ~a) —> (a—> b) truth of a entails the truth non-appleness, then Gödel's theorem is not 31 Roger Penrose, 1990 is. 75 John Lucas, 1988 • Machines cannot do this for some sentences, as is shown by their inability to Box 39. ulation of another author's argument. The reformulation provable." appleness yields orangeness. Insight is essential, I can only be sure of not being any distinguish the negation of a Gödel sentence from the Gödel sentence itself. 3. G = (G is not provable). G. It cannot be decided whether G is true or false from the point of b (contraposition). 32 Peter Denning, 1990 decisive. The question of whether Note: This argument stimulates a highly even if fallible. Just particular machine, if I am not one at all. is different enough from the original author's wording to signed copies of all seven maps of view of the system in which G is represented. Algorithms are fixed interpretations. Humans can use thinking is algorithmic cannot be is technical thread of debate that is not Benacerraf errs in casting the argument as a because insight is is warrant the use of the tag. This phrase is also used when conscious observation to step outside of an interpretation and decided on the basis of Gödel's disputed represented here because of its length and nondialectical proof sequence. As a result of sometimes unreliable, Rules From appleness yielding think of alternatives and extensions. Algorithms, however, theorem. by we should not complexity. This thread includes papers disputed this misunderstanding, Benacerraf reaches the 82 Ernest Nagel and 83 Albert E. Lyngzeidetson and Martin K. Solomon, 1994 closed proof sys • tem: A proof system that has no interaction with the the original argument is impossible to locate other than in From the truth of "if A are inherently fixed interpretations. Thus, no algorithmic 1. No mathematical insight is from Hanson (1971), Chihara (1972), and by James R. Newman, 1958 Open proof systems are not affected by Gödelian arguments. external world and so does not evolve any rules of inference other than those Kurt Gödel A —> B then B," and from the orangeness, and from account of the mind could ever be sufficient; it could never necessary to construct Gödel conclude that it plays Reinhardt (1986). incorrect conclusion that the man is a machine but Mathematical thought As Gödel's own Gödel's theorem implies a sharp distinction between open and closed its articulation by a later author (e.g., word of mouth), or A appleness being present, we no essential role in just can't tell which machine he is. But the only arguments show, no that it started with. from www.macrovu.com for truth of A, it follows that infer that orangeness is also account for the interpretive flexibility of conscious sentences. cannot be fully antecedent limits can is proof systems. Open proof systems continuously interact with their to denote a general philosophical position that is given a B is also true. mathematics. way that I can be sure of not being any particular environment through a steady stream of inputs. An open proof system is, Alternative Versions of Gödel's Theorem B present. observation. 2. Mathematical insight is involved • Doubts about machine is by not being one at all. formalized. Gödel's be placed on the disputed in fact, a potentially infinite set of proof systems, and at the limit thus has special articulation by a particular author. only in seeing that the system that is supported by theorem shows that human by Notes: produces the Gödel sentences is consistency in creativity cannot be fully inventiveness of human the potential to be noncomputable. Because they are potentially open proof sys • tem: A proof system that continually interacts with its There exists an algorithm mathematicians in •This formal system is not complete with respect to the logic of truth; that is, it does not yield There is an algorithm that, I can always step mathematics noncomputable, open proof systems are immune to Gödelian arguments, environment through sensors in such a way that it evolves and incorporates Unmapped Territory This icon indicates areas of argument that lie on or near the consistent. formalized. The ingenuity $500.00 plus shipping and han- given any consistent set of that for any recursively outside an only arise when devising new rules of and may turn out to be as creative and insightful as humans are. stronger and stronger rules of inference in its system. Many versions of axioms, will output a enumerable (r.e.) set of all valid formulas of the logic of truth. But recall that it does not even make sense to ask interpretation and 3. But, insight into consistency is not mathematicians use 76 John Lucas, 1988 of mathematicians in proof (p. 99). Additional boundaries of the central issue areas mapped on these maps. Gödel's proof have sentences true in the whether the system in itself is complete or correct; the system in itself is just an orderly set think of extensions reliable (as is shown by numerous Benacerraf's argument is inconsistent. devising new methods It marks regions of potential interest for future mapmakers polynomial equation P = 0, of patterns of symbols with rules to govern their recombination. is supported by systems that go arguments been advanced in the natural numbers produces Interpretation historical examples). Benacerraf argues inconsistently to his cannot be reduced to a •For more on formal systems, see Map 4 and the "Can functional states generate consciousness?" which in fact has no integer and alternatives. beyond ordinary 84 Hilary Putnam, 1960 85 Thomas Tymoczko, 1990 and explorers. decades since his solutions, although this fact a true sentence of 4. Because insight into consistency conclusion that I am a Turing machine but precise logical form. For mathematics. Proof of human superiority depends on proof of consistency. Newman A machine may be consistent despite lack of proof. To dling. • When we do become original treatment. arithmetic (a "Gödel arguments on Map 6. is unreliable, we cannot know whether cannot prove which one. He says that for each example, it has been shown cannot be deduced from the is and Nagel's thesis results from a misapplication of Gödel's theorem. Although it defeat Newman and Nagel's thesis, Putnam must show that the Here are some given axioms (1990, pp. sentence") not in that set Gödel's theorem applies to a given program it can be demonstrated that the program that humans, using assured of the disputed is is true (as Newman and Nagel claim) that a machine cannot prove some undecidable consistency of the machine is absolutely undecidable. But there relatively clear ones, 659–60; also, see "Gödel's (1990, p. 666; also, see system, and so we don't know that does not represent me, but that still I might be "informal" is One of 7 in this Issue Mapping™ series—Get the rest! consistency of a by disputed propositions, a human can't prove those propositions either, unless he or she can are no absolutely undecidable propositions in arithmetic. The made by writers Theorem Is Not Decisive," "Penrose Can't Argue For mathematical insight is non- represented by some program. But Benacerraf metamathematical disputed mathematical system, Paul Benacerraf by first prove that the machine is consistent. But it is unlikely that a human would best Putnam can offer is the unlikelihood of being able to show represented on this Box 30). His Hypothesis," Box The remaining 6 maps in this Issue Mapping™ series can be ordered with MasterCard, algorithmic. it is always because cannot consistently claim that no particular reasoning, can prove by The Issue Mapping™ series is published by MacroVU Press, a © 1998 R. E. Horn. map. Note: Also, see the "Is the use of theorems that cannot be be able to carry out such a consistency proof unless the machine were very simple. that the machine in question is consistent. But the fact that it is division of MacroVU, Inc. MacroVU is a registered trademark 28). of insight. program represents me and at the same time that All rights reserved. VISA, check, or money order. Order by phone (206–780–9612), by fax (206–842–0296), consistency in the Lucas argument proven by any formal Note: Also, see the "Is the use of consistency in the Lucas argument problematic?" unlikely that we can show that a machine is consistent doesn't some program represents me. of MacroVU, Inc. Issue Map and Issue Mapping are trademarks Clark Glymour problematic?" arguments on this map. means. Nagel and Newman arguments on this map. mean that a machine is in fact inconsistent. Version 1.0 Martin Davis or through the mail (Box 366, 321 High School Rd. NE, Bainbridge Island, WA 98110). of Robert E. Horn. and Kevin Kelly