STUDIES IN LOGIC, GRAMMAR AND RHETORIC 1 (14)

Witold Marciszewski

LEIBNIZ’S IDEA OF AUTOMATED REASONING COMPARED WITH MODERN AI^{1}

Gottfried Wilhelm Leibniz (1646-1716) is duly merited as the first who anticipated Artificial Intelligence. If so, let us ask: which AI? That called strong which — following Turing [1950], Newell [1980], etc. — answers in the affirmative the famous Turing’s question ‘can machines think?’? Or rather that which opposes that claim, and reduces AI to a device to assist people in solving problems of some restricted kinds? (Cf. Gams [1995].)

There is no one simple answer concerning Leibniz’s position. Instead, there are hints that he sticked to the both opposite views, and showed no signs of being aware of their incompatibility. It is the purpose of this essay to present that dilemma and look for its sources. To make the matter simpler, the most general question ‘can machines think?’ will be limited to a more specific crucial issue, to wit: ‘can machines reason?’.

This limitation to reasonings fairly reduces the number of themes to be handled. But there is more in it. It results from a fundamental point as stated by Fodor [1976, p. 202 f]. What Fodor says about cognitive psychology holds for AI as well; here is his statement (italics mine – WM). “Cognitive explanation requires not only causally interrelated mental states, but also mental states whose causal relations respect the semantic relations that hold between formulae in the internal representational system. The present point is that there may well be mental states whose etiology is precluded from cognitive explanation because they are related to their causes in ways that satisfy the first condition but do not satisfy the second.”

Fodor lists some phenomena that satisfy the first condition alone. Instead of quoting them, let me add another example — that of dreams. These typically exemplify information processing subjected to some causal

1 This research was supported by KBN (Polish Ministry of Science and Technology), grant No. 8T11C01812. It took much advantage from the discussions held at regular sessions of the Chair of Logic, Informatics and Philosphy of Science.

Witold Marciszewski

laws but hardly related to semantic rules. Such a contrastive background helps to appreciate the case of automated reasoning as carried out by computers. It yields a paradigmatic case of conformity between causal physical laws and semantic entailment. Such a perfect agreement is due to the fact that the same laws of algebra are mirrored both in electric circuits and in propositional calculus, thus forming causal relations in the former domain, and semantic relations in the latter. Even if we are not sure that human reasonings comply to such a ‘preestablished harmony’ (to use Leibniz’s key word), the guess that they do is a brilliant working hypothesis — as can be seen in the pioneering study by McCulloch and Pitts [1943] (this is why in Marciszewski and Murawski [1995] so a great emphasis is put on algebraization of logic as the turning point in the history of mechanization of reasoning). In this perspective, the study of automated reasoning is of highest importance for Artificial Intelligence.

Now, the question put at the start should be rephrased as follows: Would the problem ‘can machines reason’ be by Leibniz answered in the affirmative or in the negative? As hinted above, one should expect two responses opposing each other. A guess to explain this split in Leibniz’s thought might take into account that Leibniz was an earnest engineer of knowledge as well as a metaphysician occupied with the mind-world relation. In his capacity as an engineer, he set up most ambitious goals, while as a metaphysician he must have acknowledged the incommensurability between the human mind and the immense complexity of the world.

Apart from such psychological guessing, a more theoretical explanation will be suggested, to wit Leibniz’s mistaken treating of the reasoning (supposed by him as liable to mechanization) and the perception (seen by him as entirely non-mechanical) as mutually exclusive mental activities. Thus, he might have held the mechanistic view in the former, and the opposite in the latter issue.

In what follows, the prospective conclusion is anticipated with a typographical device to distinguish, where necessary, between ‘two Leibnizs’; let one of them be referred to as Leibniz_{e}, for engineer, and the other one as Leibniz_{m}, for metaphysician.

1. Leibniz_{e} attitude to AI. At the very start, it should be noted that Leibniz_{e} — as a forerunner of strong AI — did by no means anticipate the limitative results of Gödel [1931] and Turing [1936-37]. Sharing the epistemological attitude of his century, he was even more optimistic than the early Hilbert School as for the possibility of solving any problem properly posed.

Leibniz’s Idea of Automated Reasoning Compared with Modern AI

While Hilbert’s contention was concerned with mathematics alone, Leibniz_{e} believed that all scientific and philosophical problems can be definitely solved in a foreseeble time. In this respect, he was confident like Descartes. However, while Descartes atrributed the power of reasoning to the mind alone, and discounted linguistic devices, Leibniz_{e} extended that power to the mind-imitating machines equipped with a suitable symbolic system.

Thus that claim concerning AI shared by Leibniz_{e} with Turing rests on a stronger epistemological assumption than that of Turing. Turing, owing to his own [1936-37] and Gödel’s [1931] mathematical results, was aware of the existence of problems undecidable for any machine; that is to say, undecidable with the use of a purely formalistic scheme in proving, in which solely the physical form of symbols is referred to. However, he opposed those who used these results to argue for the minds’s superiority to the reasoning machine. In his [1950, Sec. 4] essay, he wrote as follows.

The short answer to this argument [that there is a disability of machines to which the human intellect is not subject] is that although it is established that there are limitations to the powers of any particular machine, it has only been stated, without any sort of proof, that no such limitations apply to the human intellect.

When Turing speaks of the lack of any proof, he means a mathematical proof. However, there may be other reasons to believe in a greater ability of the human intellect, for instance, those adduced by Penrose [1989, p. 111 f].

It seems to me that it is a clear consequence of the Gödel argument that the concept of mathematical truth cannot be encapsulated in any formalistic scheme. Mathematical truth is something that goes beyond mere formalism. This is perhaps clear even without Gödel’s theorem. For how are we to decide what axioms or rules of procedure to adopt in any case when trying to set up a formal system? Our guide in deciding on the rules to adopt must always be our intuitive understanding of what is ‘self-evidently true’, given the ‘meanings’ of the symbols of the system. How are we to decide which formal systems are sensible ones to adopt — in accordance, that is, with our intuitive feelings about ‘self-evidence’ and ‘meaning’ — and which are not? The notion of self-consistency is certainly not adequate for this. One can have many selfconsistent systems which are not ‘sensible’ in this sense, where the axioms and rules of procedure have meanings that we would reject as false, or perhaps no meaning at all. ‘Self-evidence’ and ‘meaning’ are concepts which would still be needed, even without Gödel’s theorem.

However, without Gödel theorem it might have been possible to imagine that the intuitive notions of ‘self-evidence’ and ‘meaning’ could have been employed just one and for all, merely to set up the formal system in the first place, and thereafter dispensed with as part of clear mathematical argument for determining truth. Then, in accordance with a formalist view, these ‘vague’ intutive notions would have roles to play as part of the mathematician’s preliminary thinking, as a guide toward finding the appropriate formal argument; but they would play no part in the actual demonstration of mathematical truth. Gödel’s theorem shows that this point of view is not really tenable in a fundamental philosophy of mathematics. The notion of mathematical truth goes beyond the whole concept of formalism.

This quotation is to testify how an intensely practicing mathematician like Penrose (a mathematical physicist) may see the human intellect’s superiority to a resoning machine. Such a testimony is no decisive argument but it should counterbalance Turing’s belief, not being supported by a decisive proof either, that the human mind equals a sufficiently involved machine. One can rewrite Turing’s words as cited above, just omitting ‘no’, and so his own positions will be equalled with that of his opponents: it has only been stated, without any sort of [mathematical] proof, that such limitations apply to the human intellect. It does not seem possible to decide which side is here obliged by the onus probandi rule. However, Turing’s opponents, as represented by Penrose, may have arguments which are not mathematical but take advantage of mathematicians’ experiences.

Now, what about Leibniz_{e}? Let the following biographical event renders his attitude toward what we nowadays call strong AI (cf. Ross [1984, p. 12]. After he constructed his mechanical calculator in 1670, he was so proud of his invention (applauded, indeed, by most brilliant minds in Paris and London) that he thought of commemorating it with a medal bearing the motto Superior to Man. To understand this emphasis, one should recall that at Leibniz’s time “even educated people rarely understood multiplication, let alone division (Pepys had to learn his multiplication tables when already a senior administrator)”. In this respect, Leibniz’s calculator, which surpassed Pascal’s machine (1641) as it was capable of multiplying and dividing, was actually superior to quite a number of people.

However, as will be reported later (Sec. 2), Leibniz_{e} believed that his arithmetical machine is just the beginning of a development that should result in logical machines to match humans in the ability of reasoning. And that, in principle, there are in science and philosophy no unsolvable problems, either for men of for logical machines, but in practice (he presumably thought) the machines should act better (as carefully equipped for their cognitive tasks).

Thus, though on different premisses, Leibniz_{e} shared with Turing and other strong AI proponents the belief in the prospective likeness of the reasoning power with humans and machines.

Once having been so identified, Leibniz_{e}’s point that the human reasoner would not surpass mechanical reasoning devices should be attentively examined, and then compared with Leibniz_{m}’s view. The latter is to the effect that each organic body is a kind of divine machine, or natural automaton, which infinitely surpasses all artificial automata because of its infinite complexity (see Monadology, 64). Hence the human mind should surpass artificial machines in dealing with complexity of problems to be addressed.^{2}

2. On ‘filum cogitationis’ — an algorithmic method of reasoning. The algorithmic procedure of problem-solving consists in mechanically following a set of fixed instructions which describe transformations of characters (ie, symbols) treated as physical objects. Hence any introduction of algorithm has to be preceded by, so to speak, physicalization of language. This idea of algorithm is clearly stated by Leibniz_{e} in the following passage.^{3}

Filum autem meditandi voco quandam sensibilem et velut mechanicam mentis directionem, quam stupidissimus quisque agnoscat. [...] Scriptura enim et meditatio pari passu ibunt, vel ut rectius dicam, scriptura erit meditandi filum. — GP vii, 14; Briefwechsel, I, 102, to Oldenburg. See Couturat [1901], p. 91, fn. 2; p. 96, fn. 4. “What I call a thread of thought is a certain sensory and machine-like guidance to the mind to be practicable even for most stupid ones. For, the following of a text and the thinking will proceed in step, that is, a writen text will be a thread for thought.”

That even most stupid beings can profit from this method, means that no intelligence is needed to algorithmically solve problems; just mechanical rules, concerning sensible and palpable properties, should be followed, like commands followed by a computer carrying out a program. Such palpability is due to the sensible qualities of characters which constitute the language used in reasonings. Owing to a suitable correspondence between notions and characters, as postulated by Leibniz, and owing to the mechanical characters-processing (‘caeca cogitatio’), one safely arrives at the solution to be found.

Here are other statements of the same programme.

2 This remarkable split does not seem to have attracted due attention in Leibniz literature. When discussing it in my talks addressed to some audiences of Leibniz scholars, I had enjoyed their kind interest, but – except for some Breger’s and Schnelle’s publications (cf. References) – the problem of Leibniz’s relation to AI does not enjoy a treatment it deserves.

3 This point becomes more conspicuous when seen against the contrastive background of the Cartesian Method; see Marciszewski [1994a, Chap. 3]; another Leibniz’s metaphor to render algorithmic procedures is that of caeca cogitatio – the blind thinking (see ibid. p. 61, 178). A clear and insightful discussion of the concept of algorithm as related to the decision problem and modern computers is found in Gandy [1988] and Davies [1988].

Ad inventionem ac demonstrationem veritatum opus est analysi cogitationum, quae quia respondet analysi characterum..., hinc analysin cogitationum possimus sensibilem reddere, et velut quodam filo mechanico dirigere; quia analysis characterum quoddam sensibile est. — Analysis linguarum, 11 Sept. 1678. (C, 351) “What is required for finding and proving truths, it is an analysis of thoughts. Since it corresponds to analysis of characters [...], the analysis of thoughts can be physicalised through characters, and proceed as if guided by a mechanical thread.”

Erit enim in promptu velut Mechanicum meditandi filum, cujus ope idea quaelibet in alias, ex quibus componitur, facillime resolvi possit; imo charactere alicujus conceptus attente considerato, statim conceptus simpliciores, in quos resolvitur, menti occurrent: ... resolutio conceptus resolutioni characteris ad amussim respondet. — GM iv, 461; Briefwechsel, I, 380, to Tschirnhaus, May 1678. See Couturat [1901], p. 91, fn. 4. “There will be on hand something like a mechanical device to assist thinking, such that with its help any idea could be most easily resolved in its constituents; to wit, with careful examining a character denoting a concept, at sight the simpler concepts in which that one resolves will appear to the mind: the resolution of concepts exactly corresponds to the resolution of respective symbols.”

In other texts there appear the notions of calculus and of machine. They should be also considered in order to compare the whole Leibniz’s programme with the nowadays concepts of algorithm and of formalized and mechanized reasoning; formalization, i.e. the rendering of a reasoning in symbols to be processed by an algorithmic calculus, forms a prerequisite for mechanization. Here are the statements in question.

Nihil enim aliud est Calculus, quam operatio per characteres, quae non solum in quantitate, sed et in omni alia ratiocinatione locum habet. — GM iv, 462, Briefwechsel, I, 381, to Tschirnhaus, May 1678. See Couturat [1901], p. 96, fn. 2. “The calculus is nothing else but operating with characters, what occurs not only in computing quantities but also in any other reasoning.”

This Leibniz_{e}’s insight, not without Thomas Hobbes’ influence, does anticipate the modern notion of logical calculus as dealing with characters but not those which refer to numbers or quantities. He is aware of the novelty of these ideas, as in another place he remarks:

[...] calculus ratiocinator, seu artificium facile et infallibiter ratiocinandi. Res hactenus ignorata. — GP vii, B, II, 8. See Couturat [1901], p. 96, fn. 5. the reasoner calculus, that is, a device for reasoning in an easy and infallible way — the thing unknown so far.

This kind of statements includes the famous “Calculemus” which occurs in the following context:

Quando orientur controversiae, non magis disputatione opus erit inter duos philosophos, quam inter duos Computistas. Sufficiet enim calamos in manus sumere sedereque ad abacos, et sibi mutuo dicere: Calculemus! — GP vii,

200. See Couturat [1901], p. 88, fn. 3.

“When arise controversies, no more a dispute will be necessary among two philosophers than among two calculators. For it will be enough to take pencils and abacuses in hands, and say to each other: let us compute!”

When the notion of machine gets added to those of sensible characters and of calculus, Leibniz’s theory of mechanical problem-solving becomes surprisingly close to the modern idea of logical computing, even in such details as printouts of inferences produced by a computer operated by a suitable software (at present called a prover):

Ut Veritas quasi picta, velut Machinae ope in charta expressa, deprehendatur. [...] Illud Criterion [...] quod velut mechanica ratione fixam et visibilem reddit veritatem. — GP vii, 10, Briefwechsel I, 145, to Oldenburg, 28 Dec. 1675. See Couturat [1901], p. 99, fn. 2; p. 100, fn. 3. “[A device should enable that] the truth like in a picture, as if by a machine printed on a chart, be perceived. [This would be] that criterion which will produce the truth as if established in a mechanical way and made clearly visible.”

In such insights and images, Leibniz_{e} might have been inspired by his successful construction of arithmetical machine. Once reasoning is conceived as computing, a reasoning machine can be seen as a computing machine. However, to obtain a deeper insight into Leibniz’s views, we should start not from the concept of a machine but rather from the concept of a formal language. It is the latter in which we are to look for the core of Leibniz’s programme. For, once we obtain a language defined in a purely formal manner, i.e., with respect to physical forms of expressions (and their combinations) alone, we can arithmetize it through assigning numbers to expressions, and to syntactic constructions, and then take advantage of a mathematical machine.

This is why a formalization of inferences has to be prior to their mechanization. Hence, a modern counterpart of the Leibnizian filum cogitationis is the Hilbert programme of formalizing the language of mathematics, the programme stated to ensure decidability of mathematical problems. It should have resulted in an algorithm to check whether a reasoning in question is logically correct. To commit such a task to the care of machines is the next step whose success would be granted provided both a suitable formalization and an advanced data-processing technology.

2. Leibniz_{e} compared with Hilbert. It was Hilbert who at the Second International Congress of Mathematics held in Paris in 1900, expressed an extreme optimism like that of Leibniz — a faith in the mathematician’s ability to solve any problem he might set for himself. This Leibniz-Hilbert analogy is crucial for the point of this essay; Leibniz in his methodological programme is as close as possible to Hilbert’s formalism and finitism while his philosophical views on mind and matter are as remote as possible from finitistic conceptions.

Let us examine the analogy in question. With Hilbert and Bernays [1934-39], the first step was to formalize the language of mathematics. That is to say, an artificial symbolic language and rules of building well-formed formulas ought to be fixed. Furthermore, axioms as well as formal rules of inference, i.e., the rules referring only to the physical form of formulas (not to their meaning), should be stated. Both the set of primitive expressions, being building blocks of formulas, and the set of rules, are finite, and every proof is to be performed in a finite number of steps.

Leibniz_{e}’s stress on the sensibility and palpability of characters used in reasoning resembles the formalistic point of Hilbert, and his belief expressed in the famous Calculemus is to the effect that every demonstration can be performed in a finite number of steps. As for differences, these are as follows. With Leibniz_{e}, (i) axioms are certain real definitions in the form of equality, (ii) the sole inference rule is that of definitional replacement; thus, (iii) the proof reduces to an analysis of defined concepts which terminates in some semantic primitives.^{4} With Hilbert, both the form of axioms and the form of inference rules is much more differentiated, in accordance with the modern methodology of deduction. Moreover, (iv) Leibniz_{e} is even more optimistic than Hilbert, for the method postulated is by him regarded as universal, i.e., applicable to the whole knowledge, not only to mathematics.

However, these variations do not affect the analogy crucial for the present discussion, namely that consisting in formalism and finitism, as far as a theory of proof is concerned. The postulate of finitism, for its practicability, requires a more precise statement than that found in Hilbert himself; however, for the present comparison it is enough to conceive it as demanding possibility to mechanically solve each problem in a finite number of steps, which amounts to the existence of a suitable algorithm.^{5}

4 The problem of the creative role of real definitions as axioms in a reasoning is discussed in my [1993] and [1995b] essays.

5 A thorough examination of Hilbert’s approach is found in Murawski [1994] while my paper [1994b] involves a discussion on how formalization of inferences is related to their mechanization.

3. Leibniz_{m}’s infinitistic theory of matter. One may object a lack of coherence between the finitistic filum cogitationis methodology, as reported above, and Leibniz_{m}’s infinitistic approach in his metaphysical conceptions of matter and mind. The relation between these two points in the Leibnizian thought deserves a closer examination.

The first expression of the infinitistic point of view is found in the dissertation De Arte Combinatoria, 1666, where, advised by Erhard Weigel, he appended “Demonstratio existentiae Dei ad mathematicam certitudinem exacta”. The last axiom in this demonstration reads:

Cujuscumque corporis infinitae sunt partes, seu, ut vulgo loquuntur, Continuum est divisibile in infinitum. “Every body has infinitely many parts, or, as it is commonly said, the continuum is infinitely divisible.”

This statement constitutes the very core of the demonstration: were it not so that every body whatever has an infinite number of parts, then it would not be necessary to resort to the infinite power, hence to that of God, which in Definition 3 is meant as an original capacity (potentia principalis) to move the infinite. Note that Leibniz_{m}, unlike Aquinas and other masters of Gottesbeweise, does not claim the necessity of closing the chain with the First Mover, but the necessity of infinite power to be possessed by the Mover; and such a power is required because of the infinite difficulty of the task to move the infinite set of pieces of matter.

This crucial statement will be confusing, unless the activity of moving is conceived as an intellectual operation (again, a point alien to other authors trying to prove God’s existence). In fact, each of us is able to physically move a piece of matter, say a chair, even if the matter which it consists of is divisible in infinitum; that we exercise such a power is independent of whether the question of infinite divisibility of matter proves answered in the negative, as with atomists, or in the affirmative, as with Leibniz_{m} and, possibly, modern physics.

Let the following quotations support the point that Leibniz, in a way, anticipated some recent guesses as to the actual infinite divisibility of matter. “So we know that particles thought to be ‘elementary’ twenty years ago are, in fact, made up of smaller particles. May these, as we go to still higher energies, in turn be found to be made from still smalller particles? This is certainly possible.” (Hawking [1992, p. 66].) “For modern particle physicists, [...] every new accelerator, with its increase in energy and speed, extends science’s field of view to tinier particles.” (Gleick [1991, p. 115].)

According to Ulam [1976, Ch. 15], the most interesting question in physics is whether there exists actual infinity of ever tinier structures. He suggests to consider that we deal with a strange structure having infinitely many levels, each level possessing its specific nature. This is – he continues – not only a philosophical riddle but also a fascinating vision in physics. When mentioning quarks, he comments that we may have reached the point in which we should consider an infinite sequence of structures.

Ulam’s picture of the physical world is astonishingly close to that of Leibniz_{m}. It becomes even closer if we take into account the limitations of physical research as implied by Gleick’s account. One cannot increase the energies applied in laboratories ad infinitum. Hence, for this reason too, there must be a limit of structural complexity beyond which no human mind can penetrate. The lack of knowledge of those so deep levels of complexity puts up an impassable barier to human technological power to control processes at those levels. Other bariers have to arouse from the finiteness of human memory and of the time given the mind for its operations; these limitations affect both humans and computers.^{6}

Hence, if ‘moving’ (in the mentioned Demonstratio) means controlling, like a software moves a computer, the more involved the object to be controlled, the greater intellectual power is required. Thus, the increasing human capacity to change the world, and in this sense to move things, is proportional to the advances in knowing ever more minute structures of matter, be it genetic code, be it atomic structure. Hence, in the case of an infinite structural complexity, an infinite intellectual power is required to handle it, i.e., to influence a course of events according to an intended plan.

Thus – let us emphasize this once more – any finite mind, when inquiring into more and more involved structures, has to meet a limit of its cognitive capabilities. The more forceful is a mind, the more distant is that limit, but somewhere it must exist; the infinite mind alone is free of any such limitations. This implies a failure of the belief that for every problem challenging the human mind there is a suitable algorithm, i.e. filum cogitationis, to solve the problem mechanically in a finite number of steps. This is a consequence of the three Leibniz’s tenets, viz. that (i) controlling the material world requires an intellectual power to the extent relevant to the degree of complexity, (ii) the material world possesses infinite computational complexity, and (iii) the human mind’s power is not infinite.

The question of coherence between the above points and the programme for logic (to make logic capable of mechanical solving any problem whatever) cannot be settled by assigning each view to a different stage of Leibniz’s

6 That incommensurability between the human mind and the unboundness of universe, as seen by Leibniz_{m}, is extensively discussed by Drozdek [1997] (this book). His thought-provoking analysis should be continued by taking into account the set-theoretical distinction between dense ordering and continuous ordering; it was alien to Leibniz_{m} himself, but when rendering his term continuum in modern concepts, we are to consider which concept corresponds to his intentions.

development. The juvenile insight regarding the infinite complexity of bodies persisted through all the changes of his views, and is found in the final phase, that of Monadology; also his insistence on formalization of reasoning is constant.

As for the concept of matter, in 1686 he wrote that “every body, however small, has parts which are actually infinite, and in every particle there is a world of innumerable creatures”,^{7} and that “there is no body so small that it is not actually subdivided”.^{8} A similar statement in a text of 1689 reads “there is no portion of matter so small that there does not exist in it a world of creatures, infinite in number”.^{9} In Leibniz’s article “Système noveau [...]” published in Journal des Savants, 1695, we read that “everything in matter is but a collection or accumulation of parts ad infinitum”.^{10} In Monadology of 1714 it is stated in item 65 what follows.

Et l’Auteur de la Nature a pû practiquer cet artifice divin et infiniment merveilleux, parce que chaque portion de la matière n’est pas seulement divisible à l’infini comme les anciens ont reconnû, mais encor sous-divisée actuellement sans fin, chaque partie en parties, dont chacune a quelque mouvement propre: autrement il seroit impossible, que chaque portion de la matière pût exprimer tout l’univers. “The Author of nature was enabled to practise this divine and infinitely marvellous artifice, because each portion of matter is not only infinitely divisible, as the ancients recognised, but is also subdivided without limit, each part into further parts, of which each one has some motion of its own: otherwise it would be impossible for each portion of matter to express the whole universe.” — (GP vi, 607-23; P. [1973], p. 190.)

4. On scale-dependent limitations of research. The texts quoted above testify Leibniz_{m}’s persistency in his infinitistic theory of matter. One of them, the juvenile Demonstratio existentiae Dei, hints at the connexion between a theory of matter and a theory of mind. According to Demonstratio, the infinite mind alone is capable of managing the infinitely complex universe. It is in order to consider later texts to express the same idea.

A text on Metaphysical consequences of the principle of reason (ca. 1712) attributes the full knowledge of matter to the omniscient, hence infinite, mind alone.

7 GP vii, 309-18. See P. [1973], p. 82.

8 C 518-23. See P. [1973], p. 91.

9 E. Bodeman, Die Leibniz-Handschriften der Königlichen Bibliothek zu Hannover, Hannover 1889, 115-17. See P. [1973], p. 108.

10 GP iv, 477-87. See P.[1973], p. 16.

“Each corpuscle is acted on by all the bodies in the universe, and is variously affected by them, in such a way that the omniscient being knows, in each particle of the universe, everything which happens in the entire universe. This could not happen unless matter were everywhere divisible, and indeed actually divided ad infinitum.” (P. [1973], p. 176)

The latest testimony of the connexion between Leibniz’s philosophy of matter and his theory of mind is found in the correspondence with Clarke, 1715-16. Let it be discussed against the contrastive background of scientific atomism as represented by the physicist Herman von Helmholtz (1821– 1894). He shared the view close to the philosophy of ancient atomism that the whole material world can be simply explained by interactions of material particles.^{11}

We come to the conclusion that the task of science consists in reducing natural phenomena to invariable forces of attraction and repulsion whose intensity entirely depends on the distance. The decidability of this problem constitutes the condition of complete comprehensibility of Nature. [...] The task of science will be completed when all the physical phenomena get reduced to those simple forces, and when a proof is provided that it is the sole possible solution.

Thus, atomistic philosophy of matter implies finitistic epistemology. This relation is discussed by Leibniz – in order to oppose the finitistic view – in the following passage of the Correspondence with Clarke.

“On this [i.e., atomistic] theory a limit is set to our researches; reflexion is fixed and as it were pinned down; we suppose ourselves to have found the first elements — a non plus ultra. We should like nature to go no further; we should like it to be finite, like our mind; but this is to ignore the greatness and the majesty of the Author of things. The last corpuscle is actually subdivided ad infinitum and contains a world of new created things, which the universe would lack if this corpuscle were an atom, that is a body all of a piece and not subdivided.” (GP vii, 352 ff; P. [1973], p. 220)

In Helmholtz’s finitistic paradigm no additional assumptions are necessary to justify the belief in solvability of any physicists’ problem, provided a sufficient advancement of the research in question. (Even if the number of atoms is infinite, in each particular problem one deals with a finite number of them.) Is it possible to reach such conclusion within Leibniz_{m}’s infinitistic paradigm?

The answer depends on some additional assumptions. Let us imagine that besides his infinitistic contention Leibniz_{m} endorsed what should be called an “explanation by downward resolution”, downward explanation, for short, and should be defined as follows: for each system, a downward explanation of its structure and functions takes into account structures

11 Reported after Einstein and Infeld [1947, Ch. 1].

and functions of its constituents: first, of immediate constituents, then (if necessary) of those forming the immediate constituents, and so on. (While upward explanation would refer to the structure and functions of that whole whose the system in question is a constituent). In such a reductionist way, we explain activities of organic cells by activities of their molecules, and those, in turn, by activities of atoms, and those by motions of elementary particles, and so on.

In the infinitistic framework of Leibniz_{m}, the chain of levels of complexity is infinite. This implies that solely the infinite mind, in the sense of actual infinity, can deal with all of them. But what about dealing with each of them? This would require less from the mind, only potential infinity. Suppose, the human mind has potentially infinite ability of development. Then for each problem there may come a time in which the mind’s capability will match its complexity.

Then, though a new problem may require going still deeper “down”, again it would be a finite number of steps. For each new problem one could devise a special algorithm, in accordance with the Calculemus postulate. The more involved a problem, the more steps must be done to solve it, but always there exists its solution in a finite number of steps.

Are there any reasons to suppose that Leibniz might have cherished such an optimistic perspective? If so, then Leibniz_{m} and Leibniz_{c} might be reconciled through atributing the human mind a potentially infinite power (the actual infinity being reserved for the divine mind). No explicit statements of such a kind are found in his writings, but this conjecture does not seem to be inconsistent with the rest of his views; as he believed in the eternal existence of minds, he might have hoped that the eternal life involves a potentially infinite intellectual development.

However, when we return to more earthly affairs, and consider the development of science in a finite perspective, such eschatological speculations have to be put aside. The historical development of science reveals limitations of our research, up to the unsolvability of some problems which result from a change of scale; let them be called scale-dependent limitations.

The most famous of them is Heisenberg’s uncertainty principle, another one is the increase of observation abilities only with the increase of energies to be adopted (note, we cannot increase energies infinitely). Another scale limitation, most relevant to the present issue, is due to the fact of memory size limitations in a computer; analogous limitations must be supposed to exist in human brains. An intuitive reasoning requires much less brain memory than a formalized, or algorithmized, reasoning.

Hence some problems which are solvable in an intuitive way, without a verbalization, may prove unsolvable in a verbalized form (while verbalization is necessary to formalize a reasoning, and this, in turn, is necessary for its mechanization).

When holding a finistic opinion like that of Helmholtz, one may a priori hope to evade scale-dependent limitations. However, Leibniz_{m}’s infinitistic conception of matter, when linked with assuming the human mind’s finiteness (cf. point (iii), Sec. 3), should have resulted in a limitative theorem — to the effect that there occur unsolvable problems. They occur because of reaching, when going sufficiently long along an infinite path of resolution, such an increase on the scale of complexity which dramatically changes the relationship between the subject of research and the finite mind’s capabilities.

However, Leibniz_{c} did not draw such consequences from what was being held by Leibniz_{m}. The explanation of this riddle in its entirety deos not seem possible without a very thorough and extensive study on this subject. But the problem becomes more manageable when restricted to the issue of relations between reasoning and what Leibniz_{m} called perception.

Let us try this way.

5. An overlooked relation between perception and reasoning. When there appears inconsistency with a great thinker, one may try to explain this by resorting to Bergson’s idea that the philosopher’s insights exceed any verbal means of their adequate expression, hence the apparent cleavage (this is the strategy adopted in my [1996b, p. 245 ff] paper).

However, when following this advice, a historian or a follower should conjecture what deficiencies in the thinker’s means of expression – likely, may be, to be remedied with a newer conceptual equipment – are responsible for the fact in question. The conjecture to be here offered for discusion runs as follows.

There is in Monadology a direct attack on the claim that the functioning of the mind can be explained in terms of mechanism (which looks as if addressed to Leibniz_{e}). It is found in item 17.

On est obligé d’ailleurs de confesser que la perception et ce qui en depend est inexplicable par des raisons mecaniques, c’est à dire, par les figures et par les mouvements. Et feignant qu’il ait une Machine, dont la structure fasse penser, sentir, avoir perception; on pourra la concevoir aggrandie en conservant les mêmes proportions, en sorte qu’on y puisse entrer, comme dans un moulin. Et cela posé, on ne trouvera en la visitant au-dedans, que des pieces qui se poussent les unes les autres, et jamais de quoi expliquer une perception. Ainsi c’est dans la substance simple, et non dans le composé ou dans la machine qu’il la faut chercher. “We are moreover obliged to confess that perception and that which depends on it cannot be explained mechanically, that is to say by figures and motions. Suppose that there were a machine so constructed as to produce thought, feeling, and perception, we could imagine it increased in size while retaining the same proportions, so that one could enter as one might a mill. On going inside we should only see the parts impinging upon one another; we should not see anything which would explain a perception. The explanation of perception must therefore be sought in a simple substance, and not in a compound or in a machine.” (P. [1973], p. 181.)

Note, there is not necessarily an inconsistency in supposing both that the system in question is a machine (as put down in the second sentence) and that its functioning cannot be explained mechanically. For, as we know from the item 64 of Monadology, there are two kinds of machines, and the system endowed with perception is a divine machine, not being a machine in the sense of a human artifact.

Unfortunately, Leibniz_{m} yields no argument why we should not observe anything which would mechanically explain a perception; he simply states that. Neither mentions reasonings as activities of such ‘non-mechanical machine’, though he lists thought, feeling and perception. Should, for instance, reasoning fall under what he calls thought?

Suppose, there is possible the following agreement between Leibniz_{m} and Leibniz_{c}: the former accepts the latter’s claim that reasonings can be explained mechanically, as in an artificial logical machine, while the latter acknowledges the irreducibility of perception to mechanical moves. This solution assumes that no perception is involved in reasoning, since its involvement would deprive reasoning of mechanical character.

Now suppose that, after a time, Leibniz_{m} finds a new evidence, namely to the effect that there are mathematical proofs necessarily involving perception which could not be adequately verbalized. Had Leibniz_{m} lived not earlier than Georg Cantor, he could have produced a nice example of perception-involving argument; to wit, the diagonal slash procedure to prove that the set of real numbers is not countable (see Cantor [1890//91], Kertèsz [1983], Penrose [1989]).

This argument requires both an evidence supplied with one’s eyes as well as bold imagination to extend the perceived picture towards infinity. Is not either of them an indubitable instance of perception? No verbal statement seems necessary to render the course of that reasoning which results in the firm conviction that there are more real numbers than natural numbers.

Thus we reach a double conclusion — a historical conjecture and a theoretical suggestion. The former is to the effect that Leibniz would have reasonably limited his AI project concerning a logical machine, had he become aware of the role of perception in reasonings, eg. those of the diagonal kind.

The theoretical suggestion – independent of a possible course of Leibniz’s thought, if confronted with modern mathematical proofs – is to the effect that one should check Leibiniz_{e}’s project as being close to the modern strong AI project. This is to be done so that one picks up those mathematical demonstrations which evidently involve perceptions, as once upon a time tried by Immanuel Kant, and then launches an attack on the problem of their automatizing.^{12} Does the attack succeed, this should shed light both on the modern issue of automatization of reasoning and Leibniz_{e}’s engineering dream.

References

— Albertazzi and Poli [1991]. Albertazzi, L. and Poli, R. (eds.), Topics in Philosophy and Artificial Intelligence. Mitteleuropäisches Kulturinstitut, Bozen 1991.

— Beth [1959]. Beth, Evert B., The Foundations of Mathematics. A Study in the Philosophy of Science., North-Holland, Amsterdam 1959.

— Beth [1970]. Beth, Evert W., Aspects of Modern Logic, Reidel, Dordrecht 1970.

— Breger [1988]. Breger, H.,Das Postulat der Explizierbarkeit in der Debatte um die künstliche Intelligenz in: Leibniz. Tradition und Aktualität. V. Internationaler Leibniz-Kongreß. Vorträge. Hannover 14.-19. November 1988. Leibniz-Gesellschaft, Hannover 1988.

— Breger [1989]. Breger, H., Maschine und Seele als Paradigmen der Naturphilosophie bei Leibniz in: von Weizsäcker and Rudolph (eds.) [1989].

— Bodeman [1889]. Bodeman, E., Die Leibniz-Handschriften der Königlichen Bibliothek zu Hannover, Hannover 1889, 115-17.

— Boden [1990]. Boden, Margaret A. (ed.), The Philosophy of Artificial Intelligence, Oxford University Press, Oxford 1990.

— Couturat [1901]. Couturat, Louis, La logique de Leibniz, Alcan, Paris 1901.

12 This Kant’s problem is extensively discussed by Evert Wilem Beth at several places, e.g. in [1959, Sec. 20] and 1970, Ch. 4.

— Cantor [1890//91]. Cantor, Georg, Über eine elementare Frage der Mannigfaltigkeitslehre, “Jahresber. Dtsch. Math. Ver.” 1, 1890//91, 75-78 and 278-281.

— Davis [1988]. Davis, Martin, Mathematical Logic and the Origin of Modern Computers in: Herken (ed.) [1988], 149-174.

— Drozdek [1997]. Drozdek, Adam, Leibniz: Struggles with Infinity, this book.

— Einstein and Infeld [1947]. Einstein, Albert and Infeld, Leopold, Evolution of Physics, Cambridge University Press, Cambridge (Mass.) 1947.

— Fodor [1976]. Fodor, Jerry A., The Language of Thought, The Harvester Press, Hassocks (Sussex, England) 1976.

— Gams [1995]. Gams, Matjaz̆, Strong vs. Weak AI. “Informatica: An International Journal of Computing and Informatics”, vol. 19, no. 4, November, 479-493.

— Gandy [1988]. Gandy, R., The Confluence of Ideas in 1936 in: Herken (ed.) [1988], 55-111.

— Gleick [1991]. Gleick, J., Chaos. Making a New Science, Macdonald, London 1991.

— Gödel [1930]. Gödel, Kurt, Die Vollständigkeit der Axiome des logischen Funktionenkalküls, “Monatshefte für Mathematik und Physik”, 37, 349-360.

— Gödel [1931]. Gödel, Kurt, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I, “Monatshefte für Mathematik und Physik”, 38, 173-198.

— Hawking [1992]. Hawking, S. W., A Brief History of Time, Bantam Press, London 1992.

— Herken (ed.) [1998]. Herken R. (ed.), The Universal Turing Machine. A Half-Century Survey. Oxford University Press, Oxford 1988.

— Hilbert and Bernays [1934-39]. Hilbert, D. und Bernays, P., Die Grundlagen der Mathematik, Springer, Berlin, 1st vol. 1934, 2nd vol. 1939.

— Kertèsz [1983]. Kertèsz, Andor, Georg Cantor 1845-1918. Schöpfer der Mengenlehre, “Acta Historica Leopoldina”, 15, Halle 1983.

— Leibniz [1982]. Leibniz, G. W., Vernunftprinzipien der Natur und der Gnade. Monadologie. Französisch-Deutsch. Meiner, Hamburg.

— Marciszewski [see the end of this list].

— McCulloch and Pitts [1943]. McCulloch, W. S. and W. Pitts, A Logical Calculus of the Ideas Immanent in Nervous Activity, “B. Math. Biophy.” 5, 1943, 115-133. Reprinted in Boden [1990].

— Murawski [1994]. Murawski, Roman, Hilbert’s Program: Incompleteness Theorems vs. Partial Realizations in: J. Woleński (ed.) Philosophical Logic in Poland, Kluwer, Dordrecht etc. 1994 (Synthese Library vol. 228).

— Newell [1980]. Nevell, A., Physical Symbol Systems, “Cognitive Science” 4, 135-83.

— P. [1973]. Parkinson, G. H. R. (ed.), Gottfried Wilhelm Leibniz: Philosophical Writings, Everyman’s Library, London 1973.

— Penrose [1988]. Penrose, R., On the Physics and Mathematics of Thought in: Herken (ed.) [1988], 491-522.

— Penrose [1989]. Penrose, R., The Emperor’s New Mind. Concerning Computers, Minds, and The Laws of Physics, Oxford University Press, Oxford - New York - Melbourne 1989.

— Ross [1984]. Ross, G. MacDonald, Leibniz, Oxford University Press, Oxford 1984.

— Schnelle [1988]. Schnelle H., Turing Naturalized: von Neumann’s unfinished Project in: Herken (ed.) [1988], 539-559.

— Schnelle [1991]. Schnelle, H., From Leibniz to Artificial Intelligence, in: Albertazzi and Poli (eds.) [1991], 61-75.

— Turing [1936-7]. Turing, A. M., On Computable Numbers, with an Application to the Entscheidungsproblem. “P. Lond. Math. Soc.” (2), vol. 42 (1936-37), 230-265; a correction, ibid. vol. 43 (1937), 544-546.

— Turing [1950]. Turing, A. M., Computing Machinery and Intelligence, “Mind”, vol. 59, no. 2236 (Oct.), 433-460. Reprinted in Boden [1990].

— Ulam [1976]. Ulam, S. M., Adventures of a Mathematician, University of California Press, Berkeley (USA) 1976.

— von Neumann [1951, lecture held in 1948]. von Neumann, J., The General and Logical Theory of Automata in: Collected Works, vol. V, ed. A. H. Taub, vol. 5, Pergamon Press, New York 1963.

— von Neumann [1958]. von Neumann, J., The Computer and the Brain. Yale Univ. Press, New Haven 1958.

— von Weizsäcker and Rudolph (eds.) [1989]. von Weizsäcker, C. F. and E. Rudolph (eds.) Zeit und Logik bei Leibniz. Studien zu Problemen der Naturphilosophie, Mathematik, Logik und Metaphysik. Klett-Cotta, Stuttgart 1989.

— Marciszewski [1993]. Arguments Founded on Creative Definitions in: E. C. W. Krabbe et al. (eds.) Empirical Logic and Public Debate. Essays in Honour of Else M. Barth, Rodopi, Amsterdam 1993.

— Marciszewski [1993//94]. Why Leibniz Should not Have Believed in filum cogitationis, “Studies in Logic, Grammar and Rhetoric” 12//13, Bia/lystok 1994, 5-16. The same text appeared in: Leibniz und Europa. VI. Internationaler Leibniz-Kongreß. Votrträge, I. Teil, Leibniz Gesellschaft, Hannover 1994, 457-464.

— Marciszewski [1994a]. Logic from a Rhetorical Point of View, Walter de Gruyter, Berlin – New York 1994.

— Marciszewski [1994b]. A Ja/skowski-Style System of Computer-Assisted Reasoning in: J. Wole/nski (ed.) Philosophical Logic in Poland, Kluwer, Dordrecht etc. 1994 (Synthese Library vol. 228).

— Marciszewski, Witold and Murawski, Roman [1995a]. Mechanization of Reasoning in a Historical Perspective, Rodopi, Amsterdam - Altanta GA 1995.

— Marciszewski [1995b]. Real Definitions and Creativity in: Vito Sinisi and Jan Wole/nski (eds.), The Heritage of Kazimierz Ajdukiewicz, Rodopi, Amsterdam – Atlanta GA, 1995.

— Marciszewski [1996a]. Leibniz’s Two Legacies. Their Implications for Knowledge Engineering, “Knowledge Organization” (Frankfurt), 23 (1996) No.2, 77-82.

— Marciszewski [1996b]. Hätte Leibniz von Neumanns logischen Physikalismus geteilt?, “Beiträge zur Geschichte der Sprachwissenschaft” (Münster), 6 (1996), 245-259.