Turing s Approaches to Computability, Mathematical Reasoning and Intelligence

English translation of a revised version of: C. Cellucci, Gli approcci di Turing alla computabilità e all intelligenza, in T. Orlandi (ed.), Per il centenario di Turing, fondatore dell informatica, Accademia Nazionale dei Lincei, Roma 2013. Turing s Approaches to Computability, Mathematical Reasoning and Intelligence CARLO CELLUCCI La Sapienza University of Rome Abstract In this paper a distinction is made between Turing s approach to computability, on the one hand, and his approach to mathematical reasoning and intelligence, on the other hand. Unlike Church s approach to computability, which is top-down being based on the axiomatic method, Turing s approach to computability is bottom-up, being based on an analysis of the actions of a human computer. It is argued that, for this reason, Turing s approach to computability is convincing. On the other hand, his approach to mathematical reasoning and intelligence is not equally convincing, because it is based on the assumption that intelligent processes are basically mechanical processes, which however from time to time may require some decision by an external operator, based on intuition. This contrasts with the fact that intelligent processes can be better accounted for in rational terms, specifically, in terms of non-deductive inferences, rather than in term of inscrutable intuition. Keywords Turing top-down approach bottom-up approach computability mathematical reasoning intelligence. Introduction In this paper a distinction is made between Turing s and Church s approaches to computability. While Church s approach is top-down because it is based on the axiomatic method, Turing s approach is bottom-up because it is based on an analysis of the actions of a human computer. For this reason Turing s approach seems more convincing than Church s approach. On the other hand, there is a tension between Turing s approach to computability and his approach to mathematical reasoning and intelligence. Contrary to his approach to computability, Turing s approach to mathematical reasoning and intelligence is top-down and, rather than being based on an analysis of the actions of a mathematician, is ultimately based on intuition. This depends on the fact that Turing assumes that intelligent processes are basically mechanical processes, which however from time to time may require some decision by an external operator, based on intuition. This assumption, which underlies Turing s choice machines, or oracle machines, makes his analysis of mathematical reasoning and intelligence less convincing, because intelligence involves processes essentially different from intuition. The Top-Down and the Bottom-Up Approaches to Mathematics There are two approaches to mathematics: the top-down approach and the bottom-up approach (see Cellucci 2013b). According to the top-down approach: 1) A mathematics field is developed from above, that is, from general principles concerning that field. 2) It is developed by the axiomatic method, which is a downward path from principles to conclusions derived deductively from them. An example of the top-down approach is Leibniz s approach to the calculus of infinitesimals, as presented by de l Hospital s in Analyse des infiniment petits, the first textbook on the subject, 1

which develops the calculus of infinitesimals from general principles concerning infinitely small quantities, and develops it by the axiomatic method (see de l Hospital 1696, pp. 1 3). Because of the influence of the Göttingen school and Bourbaki, the top-down approach has been the mathematics paradigm for the past one and a half century. The alternative to the top-down approach is the bottom-up approach, according to which: 1) A mathematics field is developed from below, that is, from problems of that field or of other mathematics, natural science or social science fields. 2) It is developed by the analytic method, which is an upward path from problems to hypotheses derived non-deductively from them (see Cellucci 2013a). An example of the bottom-up approach is Newton s approach to the calculus of infinitesimals in De quadratura curvarum, which develops the calculus of infinitesimals from problems concerning physical quantities, and develops it by the analytic method (see Newton 1981). Church s Top-Down Approach to Computability The distinction between the top-down and the bottom-up approaches also occurs in computability. Church s approach to computability is top-down. To give an answer to the question What is a computable function? Church sets up a formal system, the λ-calculus. First, Church selects a list of symbols, consisting of the symbols {, }, (, ), λ, [, ], and an enumerably infinite set of symbols x, y, z,... to be called variables (Church 1936, p. 346). Then Church defines the well-formed formulas, and the free and bound variables in them: A variable x standing alone is a well-formed formula and the occurrences of x in it is an occurrence of x as a free variable in it; if the formulas F and X are well-formed, {F}(X) is well-formed, and an occurrence of x as a free (bound) variable in F or X is an occurrence of x as a free (bound) variable in {F}(X); if the formula M is well-formed and contains an occurrence of x as a free variable in M, then λx[m] is well-formed, any occurrence of x in λx[m] is an occurrence of x as a bound variable in λx[m], and an occurrence of a variable y, other than x, as a free (bound) variable in M is an occurrence of y as a free (bound) variable in λx[m] (ibid.). Church also introduces some abbreviations for formulas. Thus λx 1 x 2 x n M will stand for x λx 1 [λx 2 [ λx n [M] ]]; each positive integer n will stand for λab a( a( Λa ( b) Λ)) ; 14243 S M N will stand for the result of substituting N for x throughout M (ibid., p. 347). Finally, Church states the axioms. x Axiom 1. To replace any part λx[m] of a formula by λ y[s M ], where y is a variable which does not occur in M (ibid.). x Axiom 2. To replace any part {λx[m]}(n) of a formula by S, provided that the bound y N M n variables in M are distinct both from x and from the free variables in N (ibid.). x Axiom 3. To replace any part S M (not immediately following λ) of a formula by N {λx[m]}(n), provided that the bound variables in M are distinct both from x and from the free variables in N (ibid.). Any finite sequence of applications of Axioms 1 3 is called a conversion, and if B is obtainable from A by a conversion we say that A is convertible into B, or A conv B (ibid.). A function F of one positive integer is said to be λ-definable if it is possible to find a formula F such that, if F(m)=r and m and r are the formulas for which the positive integers m and r stand, then {F}(m) conv r. Similarly for function of two, three or more positive integers (ibid., p. 349). Church s approach to computability through λ-definability is top-down because it is developed from above, from general principles concerning conversions of function expressions, and is developed by the axiomatic method. The Origin of Church s Top-Down Approach to Computability Church states that his approach to computability has the advantage of suitability for embodiment in a system of symbolic logic (Church 1937, 43). Clearly, he refers to the fact that originally his 2

approach was embodied in a formal system which he hoped would provide a foundation for mathematics alternative to Russell s type theory and Zermelo s set theory, both of which appear somewhat artificial (Church 1932, 347). The system was based on the concept of function rather than set, its primitives included λx[m] and {F}(X), and its first three axioms were the Axioms 1 3 considered above. The system allowed unrestricted quantification but with the restriction on the law of excluded middle that, for some values X, {F}(X) is undefined and represents nothing (ibid.). This restriction was intended to avoid the familiar paradoxes of mathematical logic (ibid.). Church thought that his formal system was sufficiently different from Russell s type theory and related systems for which Gödel had demonstrated his second incompleteness theorem, that it was conceivable that there should be found a proof of freedom from contradiction for his formal system (Church 1933, 843). Almost immediately after publication, however, Church discovered that a paradox could be derived in his system. Therefore he published a revised version of the system in order to render it free from contradiction (Church 1933, 839). But Kleene and Rosser showed that in the revised version one could derive a form of the Richard paradox (Kleene and Rosser 1935, 630). In response to this, Church restricted his system to the Axioms 1 3 considered above, giving up his project to provide an alternative foundation for mathematics. Turing s Bottom-Up Approach to Computability Contrary to Church s approach, Turing s approach to computability is bottom-up. To give an answer to the question What is a computable function? Turing analyzes the actions of a human computer, more precisely, a human being, in fact already a child, who is making a calculation following a fixed routine. Because of the limitations of the sensory and mental apparatus of human beings, Turing states a number of basic restrictions on the actions of the human computer, which are motivated by those limitations. In addition, he also states some auxiliary restrictions which are useful to simplify the argument. Turing s analysis proceeds as follows. (1) The human computer can be imagined as writing certain symbols on paper. We may suppose this paper is divided into squares like a child s arithmetic book (Turing 1936 7, 249). Admittedly, in elementary arithmetic the two-dimensional character of the paper is sometimes used, but this is no essential of computation, so, without loss of generality, it can be assumed that the computation is carried out on one-dimensional paper, i.e. on a tape divided into squares (ibid.). (2) We may suppose that the number of symbols which may be printed is finite because, if we were to allow an infinity of symbols, then there would be symbols differing to an arbitrarily small extent (ibid.). So some of them will be arbitrarily close and will be confused (ibid., 250). Moreover, an infinity of symbols cannot be stored in the mind. (3) We may suppose that there is a bound B to the number of symbols or squares which the human computer can observe at one moment. If he wishes to observe more, he must use successive observations (ibid.). (4) We may suppose that the observed squares can only be squares whose distance from the closest of the immediately previously observed squares is within L squares of an immediately previously observed square (ibid.). (5) The human computer can have different states of mind, corresponding to the fact that, at each stage, he remembers previous actions and decides what action to be taken next. We may suppose that the number of states of mind which need be taken into account is finite. The reasons for this are of the same character as those which restrict the number of symbols (ibid.). That is, if we admitted an infinity of states of mind, some of them will be arbitrarily close and will be confused (ibid.). This restriction is not a serious one because the use of more complicated states of mind can be avoided by writing more symbols on the tape (ibid.). (6) We may suppose the operations performed by the human computer can be split up into simple operations which are so elementary that it is not easy to imagine them further divided (ibid.). (7) We may suppose that in a simple operation not more than one symbol is altered. Any other changes can be split up into simple changes of this kind (ibid.). 3

(8) We may assume that the squares whose symbols are changed by simple operations are always observed squares (ibid.). For the situation in regard to the squares whose symbols may be altered by simple operations is the same as in regard to the observed squares (ibid.). (9) Simple operations will include: (a) Changes of the symbol on one of the observed squares. (b) Changes of one of the squares observed to another square within L squares of one of the previously observed squares (ibid., 251). It may be that some of the changes (a) or (b) necessarily involve a change of state of mind. The most general single operation must therefore be taken to be one of the following: (A) A possible change (a) of symbol together with a possible change of state of mind. (B) A possible change (b) of observed squares, together with a possible change of state of mind (ibid.). (10) The operation actually performed is determined by the state of mind of the human computer and the observed symbols. In particular, they determine the state of mind of the human computer after the operation is carried out (ibid.). Notice that Turing s analysis is quite explicitly concerned with calculations performed by a human computer, in fact, as it has been previously said, already a child, it makes no reference to machines. Only after analyzing the actions of the human computer, Turing states: We may now construct a machine to do the work of this human computer (ibid.). Therefore the machine will be built on the model of the human computer. To this purpose, the machine is supplied with a tape running through it, and divided into sections (called squares ) each capable of bearing a symbol (ibid., 231). The number of symbols which can be printed is finite, say s 1, s 2,..., s k. To each state of mind of the human computer corresponds an m-configuration of the machine (ibid., 251). Therefore, the machine is only capable of a finite number of m-configurations q 1, q 2,..., q l (ibid., 231). The machine scans a maximum of B symbols or squares, corresponding to the maximum of B symbols or squares scanned by the human computer. In any move the machine can change a symbol on a scanned square or can change any one of the scanned squares to another square distant not more than L squares from one of the other scanned squares. The move which is done and the succeeding configuration are determined by the scanned symbol and the present m-configuration (ibid., 252). Each action of the machine is fixed by an instruction, where each instruction consists of five consecutive parts (ibid., 243). Specifically, it is a quintuple of the form q i s j s n Dq m where q i is the present m-configuration of the machine, s j is the scanned symbol, s n is the symbol into which the scanned symbol is to be changed, D is the direction in which the machine is to move (to left or to right), and q m is the resulting m-configuration of the machine. To begin a computation, the machine must be provided with a tape, it must be positioned so that a specified square is being scanned, and must be set in some prescribed initial m- configuration. Then, if the machine is in the m-configuration q i and the scanned symbol is s j and there is a quintuple of the form q i s j s n Dq m, the machine changes s j into s n, it moves to left or the right depending on D, and goes into the m-configuration q m. This kind of action is then repeated for the new m-configuration q m and symbol scanned s n, and so on. The machine halts only when the machine is in an m-configuration q h and the scanned symbol is s k but there is no instruction of the form q h s k... A numerical function is said to be computable if its values can be found by some purely mechanical process. We may take this statement literally, understanding by a purely mechanical process one which could be carried out by a machine (Turing 1939, 166). In terms of the machines described above, or Turing machines as Church was the first to call them (see Church 1937, 42) we may assume that the symbol s 1 is 1 and we may represent a positive integer n on the tape by putting 1 in each of n successive squares. Then a function f of one positive integer is said to be computable if there is a Turing machine such that f(n)=r if and only if the Turing machine, starting in some prescribed initial m-configuration and initially scanning the first of n successive 1 s, halts when the total number of successive 1 s on the final tape is r. Turing s approach to computability through Turing machine computability is bottom-up because it is developed from below, from an analysis of the concrete actions of a human computer, and is developed by the analytic method. For Turing starts from the fact that human beings compute, and makes hypotheses as to how this may come about. Church says that, by Turing s analysis, a human calculator, provided with pencil and paper and explicit instructions, can be regarded as a kind of Turing machine (Church 1937, 42 43). This is misleading because it suggests that Turing first defines his machine, and then uses it to model the actions of a human computer. On the contrary, Turing first gives an analysis of the 4

actions of a human computer, and then uses his analysis to define his machine. Wittgenstein well captures the character of Turing s work when he says that Turing machines are humans who calculate (Wittgenstein 1980, vol. 1, 1096). Post s Problem Solver An approach to computability similar to that of Turing was proposed by Post, wholly independently of Turing. To give an answer to the question What is a computable function? Post analyzes the actions of a human problem solver, assuming that there are a symbol space in which the work leading from problem to answer is to be carried out, and a fixed unalterable set of directions which will both direct operations in the symbol space and determine the order in which those directions are to be applied (Post 1936, 103). The symbol space is to consist of a two way infinite sequence of spaces or boxes (ibid.). The problem solver is to move and work in this symbol space, being capable of being in and operating in but one box at a time, where a box can either be empty or unmarked or have a single mark in it, say a vertical stroke and one box it to be singled out and called the starting point (ibid.). A problem is to be given in symbolic form by a finite number of boxes being marked with a stroke and the answer is to be the configuration of marked boxes left at the conclusion of the solving process (ibid.). The problem solver is assumed to be capable of performing the following primitive acts: (a) Marking the box he is in (assumed empty), (b) Erasing the mark in the box he is in (assumed marked), (c) Moving to the box on his right, (d) Moving to the box on his left, (e) Determining whether the box he is in, is or is not marked (ibid.). The problem solver is to start at the box called the starting point and to follow a finite number of directions, where the ith direction is to have one of the following forms: (A) Perform one of the primitive acts (a) (d) above, and then follow direction j i, (B) Perform the primitive act (e) and according as the answer is yes or no correspondingly follow direction j i ' or j i ", (C) Stop (ibid., 103 104). A set of directions sets up a deterministic process when applied to each specific problem. This process will terminate when and only when it comes to the direction of type (C). The set of directions will then be said to set up a finite 1-process if the process it determines terminates (ibid., 104) In order to consider a numerical function to be computable by a problem solver we need some convention for representing numbers in the infinite sequence of spaces or boxes. To this purpose we can represent the positive integer n by marking the first n boxes to the right of the starting point (ibid.). Then, a function f of one positive integer is said to be computable if, for f, a finite 1-process can be set up which for each positive integer n as problem would yield f(n) as answer (ibid., 105, footnote 7). Post s approach to computability through 1-processes is bottom-up because it is developed from below, from an analysis of the concrete actions of a problems solver, and is developed by the analytic method. For Post starts from the fact that human beings solve problems, and makes hypotheses as to how this may come about. Advantage of Turing s Approach to Computability Gödel states that Church s approach to computability in terms of λ-definability is thoroughly unsatisfactory (Church 2011, 223). Conversely, Turing s approach provides the correct definition of a computable function (Gödel 1986 2002, vol. 3, 168). We had the precise concept of mechanical procedure in mind, but had not perceived it clearly before we knew of Turing s work (Wang 2001, 235). Turing s solution (analysis) is correct and unique (ibid., 203). With Turing s approach one has for the first time succeeded in giving an absolute definition of an interesting epistemological notion (Gödel 1986 2002, vol. 2, 150). Even Church acknowledges that Turing s approach to computability has the advantage of making the identification with computability in the ordinary (not explicitly defined) sense evident immediately (Church 1937, 43). 5

Turing s approach makes such identification evident immediately because, contrary to Church, Turing does not start from general principles, but rather from an analysis of the actions of a human computer. This is the advantage of Turing approach to computability. Of course, both Church s approach and Turing s approach to computability must be seen on the background of the development of recursive function theory. On the latter see, for example, Adams 2011. Turing on Intuition and Ingenuity Turing considers the actions of a human computer to be a purely mechanical business. Does he consider the whole of mathematical reasoning to be a purely mechanical business? The answer is no. According to Turing, mathematical reasoning may be regarded rather schematically as the exercise of a combination of two faculties, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgments which are not the result of conscious trains of reasoning (Turing 1939, 214 215). The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions (ibid., 215). The necessity for using the intuition can be greatly reduced by setting down formal rules for carrying out inferences which are always intuitively valid (ibid.). Ingenuity will then determine which steps are the more profitable for the purpose of proving a particular proposition (ibid.). As regards formal rules, by ingenuity Turing then means the capability of using given formal rules for finding a proof of a given proposition. In pre-gödel times it was thought by some that it would probably be possible to carry this programme to such a point that all the intuitive judgments of mathematics could be replaced by a finite number of these rules. The necessity for intuition would then be entirely eliminated (ibid.). All mathematical problems would be solved by means of formal rules. But Gödel s first incompleteness theorem showed the impossibility of finding a formal logic which wholly eliminates the necessity of using intuition (ibid., 216). For it showed that, for any formal system satisfying certain minimal conditions, there is a formula which cannot be proved nor disproved from the axioms and yet is true. And it is only by intuition that one can decide that the formula is true (Turing 2004a, 215). A formal logic cannot wholly eliminate the necessity of using intuition, but only the necessity of using ingenuity. For we are always able to obtain from the rules of formal logic a method of enumerating the proofs from the given axioms and hence the propositions proved by its means (Turing 1939, 215). The enumeration of proofs is for instance obtained from an enumeration of all possible sequences of symbols by striking out those which do not pass the test (Turing 2004a, 212). We then imagine that all proofs take the form of a search through this enumeration for the theorem for which a proof is desired. In this way ingenuity is replaced by patience (Turing 1939, 215). Of course, we do not really want to make proofs by hunting through enumerations for them, but by hitting on one and then checking up to see that it is right. However this shorter method is always theoretically, though not practically, replaceable by the longer method of hunting through an enumeration of proofs if one has got a method of checking up (Turing 2004a, 212). Turing s Approach to Mathematical Reasoning Although Turing does not consider mathematical reasoning as a purely mechanical business, one may wonder whether his approach to mathematical reasoning is of the same kind as his approach to computability. The answer is no, because he bases mathematical reasoning on the axiomatic method. As we have seen in section 8, according to Turing, there are two methods for finding a proof of a proposition in a given formal system. The shorter method involves determining which deduction steps are the more profitable for the purpose of proving a particular proposition from the axioms of the system. This method requires ingenuity. The longer method involves making an enumeration of all proofs of the system. This method is purely mechanical, so it requires no ingenuity. Both the shorter and the longer method are based on the axiomatic method. Thus 6

Turing s approach to mathematical reasoning is top-down, and hence is different from his approach to computability, which is bottom-up. Unlike Turing s approach to computability, which makes the identification with computability in the ordinary sense evident immediately, Turing s approach to mathematical reasoning does not make the identification with mathematical reasoning in the ordinary sense evident immediately. For it assumes that mathematical reasoning must be ultimately based on intuition. Indeed, as we have seen in section 8, Turing concludes that, by Gödel s first incompleteness theorem and the strong incompleteness theorem for second-order logic, it is impossible to find a system of formal rules which wholly eliminates the necessity of using intuition. This conclusion, however, depends on the alternative: either all problems can be solved by means of some given system of formal rules, or the necessity of using intuition cannot be wholly eliminated. Since, by Gödel s first incompleteness theorem, not all problems can be solved by means of some given system of formal rules, from the alternative Turing concludes that the necessity of using intuition cannot be wholly eliminated. The alternative, however, is unjustified because, since antiquity, it has been acknowledged that problems need not be solved by means of some given system of formal rules, so by some mechanical procedure, they can be solved by some non-mechanical procedure. Mathematics and medicine were the first areas where the need for non-mechanical procedures to solve problems arose, and in fact the earliest such procedures of which we have notice, those of Hippocrates of Chios and Hippocrates of Cos for mathematics and medicine, respectively, were non-mechanical. Indeed, Plato opposes Hippocrates of Cos procedure to the mechanical procedures of traditional medicine, by stating that following such mechanical procedures would be like walking with the blind (Plato, Phaedrus, 270 d 9 e 1). Since problems need not be solved by some mechanical procedure and there are nonmechanical procedures for solving problems, Turing s conclusion that mathematical reasoning must be ultimately based on intuition is unjustified. Between the mechanicalness of the rules of formal systems and the inscrutability of intuition, there is an intermediate region inhabited by procedures, such as the one used by Hippocrates of Chios and Hippocrates of Cos, which are neither mechanical nor intuitional. Post s Approach to Mathematical Reasoning Although Post s approach to computability is similar to Turing s approach, Post s approach to mathematical reasoning is different from Turing s approach. While Turing bases mathematical reasoning on the axiomatic method, Post considers the axiomatic method to be not the method of mathematics but only the method used by mathematicians in a certain phase of the development of mathematics. Indeed, in 1941 Post expressed his continuing amazement that, ten years after Gödel s remarkable achievement, the current views on the nature of mathematics are thereby affected only to the point of seeing the need of many formal systems, instead of a universal one (Post 1965, 345). On the contrary, has it seemed to us to be inevitable that Gödel s achievement will result in a reversal of the entire axiomatic trend of the late nineteenth and early twentieth centuries (ibid.). Axiomatic thinking will then remain as but one phase of mathematical thinking (ibid.). Thus, according to Post, the use in mathematics of the modern version of the axiomatic method is a trend of the late nineteenth and early twentieth centuries. Choice and Oracle Machines While Turing s approach to computability is different from his approach to mathematical reasoning, which is ultimately based on intuition, Turing assigns intuition a role also in his machines. Indeed, he considers a variant of them, which he first calls choice machines and then oracle machines, whose working essentially depends on intuition. Choice machines are machines which, when reaching a certain m-configuration and scanning a certain symbol, cannot go until some arbitrary choice is made by an external operator (Turing 1936 7, 232). This would be the case if we were using machines to deal with axiomatic systems (ibid.). Turing refers to the fact that, by Gödel s first incompleteness theorem, every formal system 7

satisfying certain minimal conditions leaves some proposition undecided. According to Turing, a decision as to its truth must be made by an external operator and such decision will be ultimately based on intuition. Oracle machines are machines which, when reaching a certain m-configuration and scanning a certain symbol, cannot go until some decision is made by a kind of oracle as it were (Turing 1939, 172). The only thing Turing says about the oracle is that it cannot be a machine (ibid., 173). Once again, if we were using oracle machines to deal with axiomatic systems, the decision of the machines will be ultimately based on intuition. Indeed, according to Turing, when choice or oracle machines are used to deal with axiomatic systems, not all the steps in a proof are mechanical, some being intuitive (ibid., 216). Each intuitive step is a judgment that a particular proposition is true (ibid., 216, footnote). When carrying out a mathematical proof, such machines must show quite clearly when a step makes use of intuition, and when it is purely formal (ibid., 216). Infallibility and Intelligence In addition to stating that, when carrying out a mathematical proof, choice or oracle machines must show quite clearly when a step makes use of intuition, and when it is purely formal, Turing also states that it must be beyond all reasonable doubt that the machine leads to correct results whenever the intuitive steps are correct (ibid.). This means that, in his view, only the intuitive steps can be problematic, the mechanical steps are infallible. This implies that a machine capable only of mechanical steps cannot be intelligent. Indeed, Turing states that, if a machine is expected to be infallible, it cannot also be intelligent. There are several mathematical theorems Gödel s first incompleteness theorem and related results which say almost exactly that (Turing 2004b, 394). They have shown that if one tries to use machines for such purposes as determining the truth or falsity of mathematical theorems and one is not willing to tolerate an occasional wrong result, then any given machine will in some cases be unable to give an answer at all (Turing 2004c, 410). This, however, will occur only if the machine must not make mistakes. But this is not a requirement for intelligence (ibid., 411). In order to be intelligent, a machine cannot be infallible. For this reason Turing considers choice or oracle machines, whose intuitive steps produce intelligence. Such machines are not infallible because Turing does not assume that the intuitive steps are invariably correct. Indeed, he states that the spontaneous judgments which are a product of intuition are often but by no means invariably correct (Turing 1939, 215). Random Machines In addition to machines which are not infallible since they essentially depend on intuition, Turing also considers random machines, that is, machines which allow several alternative operations to be applied at some point, the alternatives to be chosen by a random process (Turing 2004c, 416). Random machines have the advantage that, unlike choice or oracle machines, they do not depend on the inscrutable responses of intuition. Nevertheless they depend on the inscrutable responses of chance. Conclusion Turing s approach to computability is convincing because it is based on an analysis of the actions of a human computer. On the contrary, his approach to mathematical reasoning, and to intelligence generally, is unconvincing because it is based on the assumption that intelligent processes are basically mechanical processes, which however from time to time may require some decision by an external operator, based on intuition. Such an assumption is unjustified. Processes requiring intelligence are rational processes and are completely different in kind from the mechanical ones involved in computing a numerical function. They are essentially based on non-deductive inferences, which are wholly rational operations, of the same kind as deductive inferences (see Cellucci 2013a). Conversely, choice or oracle or random machines are based on the on the inscrutable responses of intuition. Thus 8

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