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Witold Marciszewski holds the chair of , Methodology and of Witold Marciszewski Science at Warsaw University, Poland. Warsaw University, Poland Mcmber ofLcibniz Gescllschaft.Presi­ dentofFoundationforInfOlmatics,Logie and Mathematics, Poland. Editor of the electronic joumal Mathesis Universalis Leibniz'sTw o Legacies. for "Logic andIntelligence, Mechanized Dcduction, Lcibnizian Perspective", as­ Their Implications fo r sociated with LogBank - a database for logic, informatics, etc. Authorofscveral books relevant to the

Marciszcwski, W.: Leibniz' two legacies and their implica­ knowledge, and (2) his implicit questioning of the same tions regarding lm owledge engineering. possibility because ofthe role attributed by him to percep­ Know!.Org. 23(1996)No.2, 1'.77-83, 37 refs. tion as characteristic of organic life. Knowledge engineering anticipated by Leibniz in such projects The former makes him closer to Alan Turing as a as ideal language for science reasoning automata, library pioneer of hard AI, the latter - to John von Neumann as organization, etc. owes to him philosophical presuppositions as to the scope of possible automation. The paper deals with an one who acknowledged peculiarities of organisms, seeing ambiguity in his position. His attitude as an engineer implies their enormous advantages over electronic devices. hard AI (as represented by A. Turing) while his metaphysical These views, though opposing each other, have a point involve an insuperable physical between in common, to wit either makes a stand against Dcscartes' organisms and artificial machines, which is highly relevant to conceptions of and knowledge; thus Descartes efficiency ofinformation processing (this physicalism claiming provides us with a remarkable contrastive background to the importofhardware, accords with J. vonNeumann's insights). better perceive Leibniz's two approaches3• The paper's sections: 1. Leibniz VS. Descartes in views on knowledge. 2. On physicalism and anti physicalism in logic. 3. To express Descartes' position in a most concise way, Turing's claim as to the insignificance of hardware. 4. Von let us put it as fo llows: 'the mind does not belong to the Neumann's claim as to the significance of hardware. 5. Why same world to which does ' (cp Ryle [1949]). Thus Leibniz would not have accepted logical physicalism. 6. Why Descartes has created the of a physics-inde­ Leibniz would have accepted logical physicalism. (Author) pendent ; the article "the" is to hint at the enormous impact of that theory, to the extent of its a commonsense approach (extremities of be� 1. Leibniz vs Descartes in Views on Knowledge haviouris111 may be partly explained as a revolutionary Gottfried Wilhelm Leibniz (1646-1716), who antici­ reaction to that paradigm). Leibniz's point results from pated so enormous areas of modern knowledge, is also a denying that denial, so that it reads 'the mind 'does' forerunner of what we nowadays call 'knowledge engi­ belong' (etc). Thus Leibniz paves the way to what nowa­ neering' - by his creations of library systems and his days starts to be called 'physics of mind' (cp Penrose proj ects of collective research, the latter fruitfully mate­ [1988]). rialized with the etablishing of learned societies and Now, there arehvo theoreti cally possible concretisations ministries of science according to some of his projectsl. of this general point: either (1) one reduces to We need this term to cover the issues of both correct matter (as, eg, in hard AI), or (2) one acknowledges their 'producing' and efficient 'organizing and managing ' distinctness and interaction (as, eg, Popper and Eccles human knowledge. The former is traditionally handled by [1997]; as to hard and weak AI, cp Gams [1995]). Leibniz and logic, including methodology of sci­ in his philosophy never endorsed (1) but in his practical ences, the latter - by a cluster of new specialized knowledge-engineering projects he came close to it, while disciplines. However, these two fields are not unrelated to in the main stream of his philosophy it was point (2) each other. To use an example to suggest the argument, which he firmly held. In this sense we can speak of let it be recalled that the procedures offonnalizing proofs, Leibniz's two legacies. His approaching point (1) was though belonging to the sphere of mathematical produc­ connected with the of his 'ars combinatoria' as a tion (and so handled by mathematical logic) turned out to method ofproblem solving which was combina­ be an indispensable tool for databases, in particular tory and finitist, hence feasible for mechanical devices. mathematical ones, which belong to the domain ofknow 1- In the sequel, the fairly 'materialistic' point (as that of edge organization and managementz. strong AI) is designated by the term 'anti-physicalism' Let us start from a comparison of Leibniz views to and the opposite one by 'physicalism'; this terminology those of Rene Descartes (1596-1650). Why should we may seem rather odd, but is justifiedas fo llows. The view start in this way? The answer is as fo llows. There are two that physical devices (mechanical, electronic, etc) in views of Leibniz to be discussed in this essay, which (technical complications notwithstanding) can oppose each other, namely (1) his explicit in the do the same job as organisms and do, involves the possibility of automation of the processes of producing irrelevance of a physical kind of hardware: it is not matter

Know!. Org. 23(1996)No.2 77 Witold Marciszewski: Leibniz's Two Legacies

https://doi.org/10.5771/0943-7444-1996-2-77 Generiert durch IP '170.106.35.234', am 27.09.2021, 09:57:11. Das Erstellen und Weitergeben von Kopien dieses PDFs ist nicht zulässig. · but software what . Thus the stress put on the 2. On Physicalism and Alltiphysicalism ill Logic import of what constitutes a physical component deserves Physicalism holds human and acts to be to be denoted as 'physicalism', while its opposite is called determined by physical laws (Webster [1971]). Logical 'antiphysicalism' . Physicalism, LP for short, holds reasoning processes to be Before we enter into the discusion of both approaches, determined by laws deriving from physical properties of it is in order to hint at their implications for knowledge the brain, hence from some hardware properties. engineering. Knowledge is produced by , In the heroic of logical people used hence a step is needed towards a theory of intelligence as to employ the term 'physicalism' in a different sense; that data-processing faculty. Let us assume that in producing stmy, though, seems to be half-forgotten, so one can give knowledge three kinds of data processing are involved this word a new , as suggested in Marciszewski and three skills respectively, to wit 'reasoning' (includ­ and Murawski [1995]. An alternativesuggestion is due to ing computing), 'abstraction ', and 'ordering'. Reason­ Schnelle [1988) who uses the phrase 'naturalization of ing with computing is the unique member of this triad logic'. However, it seems desirable to have a term related which so far, to some extent, has been successfully to the phrase 'physics of thought' (see below). Moreover, mechanized, lhat is, made feasible for machines, esp. the use of the adjective 'natural' in contexts like 'natural electronic ones; hence it plays a special role in the present logic' has been already established for what Gentzen discussion (cp Marciszewski and Murawski [1995]). called 'das natiirliche Schliessen '. As to reasoning, therefore, we can already sec its role It was the famous physicist Roger Penrose [1988) who in knowledge organization and engineering, for instance was bold enough to claim into ' the mathematics that of inferential in expert systems and olldphysics of thought'. His can be combined in a databases. As for the other skills, the question is not fertile way with those ofJohn von Neumann [1958) which settled yet in an empirical way, hence a support should be prove crucial fo r the story in question. expected from philosophy. Abstraction is a -mat­ However, when associating physics with logic and a ter of AI research but at a most primitive stage, namely theory of mind, one has to regard the strong hold over that of recognition (from that to, eg, abstracting philosophers got by the Cartesian paradigm concerning transfinite cardinals is a rather long way). Should we, for the mind-matter . With respect to that paradigm, the future, envisage its fu ll mastering by electronic de­ any phrase like "the physics of thought" is even worse vices, as antiphysicalism does claim? If so, then it is worth than a philosophical heresy; it is felt as a category­ while to devote and money to such a promising mistake, like saying that numbers happen to be warm, or research. Ifnot, then it is wiser to spare resources for more that some thoughts are yellow. The term 'categ01Y­ feasible and payable AI projects. mistake ' is due to Ryle [1949]. In the same book the The same dilemma appears with regard to the skill of Cartesian doctrine is rendered as fo llows. "Human bodies ordering which includes creation of structures, as math­ [ ...) are subject to the mechanical laws which govern all ematical, syntactic, musical, technical, political ones, etc other bodies in space. [ ... J Minds are not in space, nor are (if we endorse such interpretation of Georg Cantor's well­ their operations subject to mechanical laws." (p. l1). ordering theorem)4. When the mechanical laws (like those stated by Newton) The ordering issue turns to be even more involved than are identified with the totality of physical laws, the mind­ the abstraction issue, since any non-trivial ordering pre­ body problem is doomed to be "solved" either in the supposes acts of assessing certain values. In the face of an Cartesian way or in the behaviouristic way (endorsed by enormous multitude of elements from which relevant Ryle). However, modern physics offers a more sophisti­ ones are to be selected for the structure in question one has cated approach, and that seems to accord with Leibniz's to be able to judgc which ones are duly important, insights. relevant, etc. This, in turn,presupposes a system of values Had Leibniz more influence on modern minds, than or goals. Such a system is inborn to any living entity, as Descartes seems to have even in our times, then the idea these have such goals (imparting values to means) as self­ of a physics of thought would be less shocking. For preservation and reproduction (to mention the most primi­ Leibniz this idea would be rooted in the ofthe pre­ tive ones). It is hard to imagine how an electronic device established harmony between of the monads could share such attitudes; however if there are people and the motions of the bodies. As he puts it, there is "the who can imagine such a thing, let them do their best concord and the physical union of the and the body, (anyway, were the present author a VIP in a ministry of which exists without the one able to change the laws science, he would never grant a financial support to of the other"'. research based such an antiphysicalist philosophy). 011 If so, then the laws of thought must exactly mirror the These are only a few examples of connexions between physical laws offunctioning ofthe entity in question, and the theme of this essay and the issues of producing and vice versa. Hence, since electronic automata are SUbjected organizing knowledge. However, let this suffice to en­ to different physical laws than organic automata, ie courage those being interested in these issues, as well as monads, the laws governing their intellectual processes those who like inquiring into philosophical and logical must be different as well. This is a physicalistic thesis on presumptions of AI research inspired by Leibniz's ideas.

78 Knowl. Org. 23(l996)No.2 Witold Marciszewski: Leibniz's Two Legacies

https://doi.org/10.5771/0943-7444-1996-2-77 Generiert durch IP '170.106.35.234', am 27.09.2021, 09:57:11. Das Erstellen und Weitergeben von Kopien dieses PDFs ist nicht zulässig. the relevance of hardware to intellectual performances, ie, all ofthem are ' universal' . A discrete-state machine is inherent in mature writings of Leibniz. On the other one that in a deterministic way passes step by step from a hand, the younger Leibniz's belief in the possibility of definitestate to another state, each step being determined constructing an artificial reasoning automaton to entirely by an appropriate rule. In other words, each state is a replace human reasoners implies the irrelevance of hard­ function of the previous state and an impulse. Imagine, ware in this important respect. eg, a wheel which clicks round through 1200 once a These opposing views may complement each other in second, but may be stopped by a lever operated from an attempt to express a live fundamental surpass­ outside; a lamp is to light in one of the positions of the ing either formulation. It seems a great task for Leibniz wheel. Let the machine states, ie, three possible positions

of the wheel, be referred to as s ' 82, S3 input signals as io scholarship to inquire into relations holding between J and t (for 'transition') is to denote that two-place thosc polcs ofLeibniz thought. Thc prcsent paper docs not i1 ; aim at such a remote target. Instead, it tries to clarify the function which assigns a to each pair Sk' ik' tenets oflogical physicalism and logical anti-physicalism 1(81 ,;0) =82 I(S2,io) =s3 t(S3, io) = Sl and, farthermore, to present reasons for either point as seen by Leibniz. Thus it should be treated as a prelimary t(SI, ir) = SI t(s2,i[) = s, t(S3 , i1) = S3 study to pave the way to the more ambitious task of interepreting the alleged discrepancy. The input signal i consists in stopping the wheel and In a natural way, the main body of this study should t thus preserving the currentstate, while input the signal io consist offour parts, two ofthem providing paradigmatic means the lack of such a move, and so allowing the wheel statements of antiphysicalism and physicalism, the fa nner to reach the next from among its three possible internal represented by Alan Turing, the latter by John von states. Let another function assign each internal state an Neumann. Then there follow two items regardingLeibniz: external one which consists either in lighting or in non­ one concernedwith his supposed anti- and the other with lighting thc lamp. his pro-physicalist attitude. Turing's [1936-7] result is to the cffect that any procedure which can be computed at all, ie, any procedure 3. Turing's Claim as to the Insignificance of Hardware for which there is an algorithm, can be computed by his (1) Is a human brain a universal Turing machine? machine callcd, therefore, univcrsal. As Turing [1950] (2) Is the material that constitutes a thinking device, argued, a physical stuff from which such a machine is esp. a brain, of any consequence? made, ie, its hardware component, is irrelevant to its Turing's answer to the questions stated above, as found pCiformances in any respect, also with regard to methods in his [1950] article, is as fo llows. A human brain is rcally of reasoning. In this sense, his claim opposes logical a kind of computer. From the so-called universal Turing physicalism. machine it differs in that it may involve a random element, ie have instructions like that: " throw the die (the 4. Von Neumann's Claim regarding the Significance of throwing may have the counterpart in an electronic Hardware process) and put the resulting numbcr into store n (say, For the sake of convenience, let us repeat the questions 1000)" . Moreover, unlike the universal machine, it has posed in the preceding section. only a finite store (memory). (l) Is a human brain a universal Turing machine? To explain that the hardware to be used is irrelevant, (2) Is the material that constitutes a thinking device, Turing takes advantage ofthe that Charles Babbage's esp. a brain, of any consequence? Analytical Engine was a real prototype of modern elec­ Whilc Turing [1950] answers YES to (I) - with the tronic computers although it was a mechanical device, proviso that a brain may involve a random element, and using wheels and cards (Boden [1990, 46]; Babbage's NO to (2), Von Neumann [1958] answers YES to (2), ideas, going back to 1834, are discussedby Gandy [1988]). which implies NO to (I). (Cp. Schnelle [1988], Penrose Here is Turing's [1950] comment. "Since Babbagc's [1988]). Von Neumann concludes his essay as follows: machine was not electrical, and since all digital comput­ "Thus logic and mathematics in the central nervous ers are in a sense equivalent, we see that this use of system, when viewed as languages, must structurally be electricity cannot be of theoretical importance. [ ...J In the essentially different from those languages to which our nervous system chemical phenomena are at least as common refers" (ie those commonly used by important as electrical. In certain computers the storage logicians and mathematicians). This puts limitations to system is mainly acoustic. The feature of using electricity the project of creating , unless a is thus seen to be only a very superficial similarity. Ifwe human creator proves to be able to imitate the wish to find such similarities we should look rather fo r of the human brain and from the mathematical analogies of function." process of evolution (a definitionof AIis found in Boden That all digital computers arc equivalent fo llows from (ed.) [1990], Schnelle [1988], Sterelny [1991], Gams the fact that they can mimic any discrete-state machine, [1995]).

Know!. Org. 23(1996)No.2 79 Witold Marciszcwski: Leibniz's Two Legacies https://doi.org/10.5771/0943-7444-1996-2-77 Generiert durch IP '170.106.35.234', am 27.09.2021, 09:57:11. Das Erstellen und Weitergeben von Kopien dieses PDFs ist nicht zulässig. Von Neumann's point docs not imply any postulate of 5. Why Leibniz would NOT have accepted logical symbolic reconstruction of those neural systems that physicalism would yield an alternative logic or mathematics (a differ­ Leibniz held it possible to build a logical machine ent set of theorems, or different meanings of operators). matching humans in the ability of reasoning and surpass­ What is at variance is a different -processing ing them as to its infallibility: 'lit en"are ne passim us 'it technology' to produce and theorems - quidam si velimus, et ut Veritas quasi picta, velut Ma­ when compared with that of fo rmalized-systems, Turing chine ope in charta expressa deprehendatur.' (letter to machines and digital computers. Technology involves Oldenburg, Oct. 28, 1675, quoted by Couturat [1901]). hardware, ie a physical component, as well as software; In his philosophy there were premises to judge that hence von Neumann's point can be called physicalist. Is programme impossible, but the "Zeitgeist" led him to the it right, then AI requires a human-like hardware, contrary opposite. It was the time of extreme optimism regarding to the claim involved in Turing's project. potentialities ofthe human mind (eg, Descartes was ready According to von Neumann, the hardware difference to prove all philosophical in one chat). It was only between a neural device and a digital computer consists in needed to findproper ways of improving the actual human the fo rmer's (1) being partly analog (eg, chemical) and mind; in some programs, as that of Leibniz, those ways only partly digital; (2) using a recording system that is not involved an ideal language combined with a universal digital but statistic, which meant)that the sense of a signal calculus. Once having such a system, one could feed it to depends on its intensity rendered as frequency ofoscilla­ a machine as well. tions. Though at his time nobody heard of a Turing machine, Here is an example which combines some recent the logical idea of formalized reasoning, as algorithmic neurological findings (Fischbach [1992], Crick and Koch computation (claimed by Hobbes), was in vogue owing to [1992]) with a logician's reflexion. In visual awareness the schoolmen, followed by Leibniz. A fo rmalized rea­ a significant role is played by 40-cycle-per-second oscil­ soning requires just a sheet of paper (Turing's tape), a lations in fi ring rate which synchronize the firing of pencil (,calamus'), and an eraser. The steps could be so neurons responding to different parts of a perceptual arranged that a single word was either written or erased scene, and so the whole , eg, one's face emerges. in each step. 'Nihil eni11l aliLid est calculus, quam opel'atio There are specialised cells responsible fo r reassembling a per characteres, quae in omni ratiocinatione locum face picture from scattered components (a parallel process­ habet.' (letter to Tschirnhaus, May 1678, see Couturat ing). Such integration is accompanied by abstraction as [1901]). the resulting picture corresponds to faces with similar The technological assumption required to justify features rather than to one face alone. Leibniz's project ofa fu lly successful reasoning machine To find a logical point, let LISfancy the way by which runs as fo llows: whatever can be thought by the mind can the human mind must have made from perceiving, say also be recorded both on a sheet of paper and in an aptly (instead of faces), the SUll, the moon and round tree devised mechanism, as cogs of the arithmetical machine trunks, to the abstract of a circle (which, in turn, were aptto record data and operations (cp. Breger [1988]). may have suggested the technological idea of a wheel). When discussing such a programme, one should keep in The process starts from non verbalized, even not apper­ mind that still at the beginning of the 20th century (eg, ceived (in Leibniz's sense) percepts being unconscious Hilbert's 1900 programme) nobody was able to guess the counterparts of statements like " the sun is round". In the results concerning our cognitive limitations, as long course of information processing, such true state­ Heisenberg'S principle and the undecidability or incom­ ments result in true Euclid theorems on the circle; hence pleteness theorems (initiated by Godel [1930], [193 1]; cpo it is a '-preserving ' process, characteristic of reason­ Church [1936], Davis [1988], Gandy [1988]). ing. Those theorems speak against the possibility of an Thus, should be defined broader, including algorithmic solution of some mathematical problems. intellectual percepts of mathematical and other abstract Another argument came from the research on the nervous objecls. This can be seen, eg, in Euclid's proofs, where the system, guided by comparisons with digital computers. It perception of an object, both concrete and typical, leads to proved that an enormous number of operations must be general proposilions (the famous Locke-Kant problem performed at the unconscious level, while their success (cp. Beth [1970], Beth and Piaget [1966], reported by depends on properties of the organic hardware involved. Marciszewski [1994]). The logical step in question is due Thus they are capable neither ofbeing verbally recorded, to applying quantifiers, a fact that shows a possible to be later translated into a piece of software, nor ofbeing mutual, dependence of perception and reasoning. Since performed by a digital machine. perceiving is due to the statistical (not the digital) The fact last mentioned as we11 as G6del's limitative of brain signals, the dependence confirmsvon Neumann's results may shed light on each other; their optimistic contention that such a logical process requires a piece of component is to effect that the human mind can do more hardware (hence a physical entity) different from that than any home-made machine, while the pessimistic one fo und in a digital computer. - that artificial machines, because of their less advanced

80 Know!. Org. 23(1996)No.2 Witold Marciszewski: Leibniz's Two Legacies https://doi.org/10.5771/0943-7444-1996-2-77 Generiert durch IP '170.106.35.234', am 27.09.2021, 09:57:11. Das Erstellen und Weitergeben von Kopien dieses PDFs ist nicht zulässig. hardware, in some cases fa il to strengthen human abili­ general quantifier elimination conflates with modus ties. AJI that might have been anticipated by Leibniz, were ponens; thus, so to speak, a 'macro-rule' replaces a set of he more sensitive to consequences of his own metaphys­ single rules. ics, and less eager in following his time's slogans. Obviously, the systems compared are identical as to the set of theorems and the meanings of logical constants

6. Why Leibnizwould have accepted logical physicalism (hence no alternative logic is at stake), but are different technologically, ie as to the mechanism of producing That Leibniz would not accept logical physicalism is conclusions, depending on the hardware involved. The easier to defend than the answer in the affirmative. connexion betwecn the quantifiers and the perception Premisses for the fanner were stated by Leibniz explicitly, (requiring an organic hardware) as well as the macro-rule while those for the affirmativc statement may be guessed technology may form a basis for "the logical language as being implicit in his concepts of perception and of truly used by the central nervous system" (von Neumann organic machincs (cp Breger [1989], SchneJle [1991D. [1958, 82]. The example of Mizar MSE suggests a way of For the same reason, though, the affirmative answer is imitating organic reasoning with the macro-rules strat­ deeper rooted in Leibniz's thought. egy, but the entanglement of reasoning with perception, Leibniz failed to see the connexions between percep­ characteristic oforganic reasoners, is hardly imitatable by tion and reasoning - those exemplified above. Had he computers. noticed them, he would have acknowledged the essential Had Lcibniz had our present logical knowledge with its difference in the "technology" of reasoning between limitative theorems, accompanied by suitable biological natural and artificial machines. As to perception, he premises, he would not have expected the full-scale voiced its non-mechanical nature in the fo llowing way: mechanization of reasoning. Instead, he would have 'pel'cepNon alld that which depends it canllot be 011 welcomed such limitations as supporting his belief in the explained mechanically, that is to say, by figlfres and range of physical differences between natural and artifi­ motions.' (Monadology, item 17). cial hardware - the belief that ' each organic body is a Did Leibniz admit processes ofreasoning, unlike those kind of a divine machine, or a natural automaton, which of perception, to be of mechanical nature? This is likely infinitely SllllJasses all art(flcial automata. ' (Monadology, if we consider his fascination with Hobbes' idea that item 64). reasoning is like computing. In the latter there does not exist any direct link with perception. In reasoning it does, Notes but that vital fact was not likely to be discovered until 1 These creations are so commonly known that there is no need to modern quantification logic came into , esp. in dwell upon them except for, possibly, Leibniz's role in establishing a computerized form of infercntial logic mainly due to state offices, like ministries, to manage science in the countly in Gentzen [1934-35],. question. In fa ct, his project of the l'russian Academy in Berlin For, it is the rules of manipulating quantifiers (and like made it more similar to a state office thanto a learned society in the operators, as that of description) that makes us aware of style of the Royal Society of London. Even greater was this similarity in the case of the Russian Academy, what complied with the involved relations between the concrete (as given in the autocratic mode of I1lling of Peter the Great. (l owe this to a perception) and the general. The data-proccssing done by spoken communication, viz. the address of Vladimir S. Kirsanov, neural "face cells" (as Fischbach [1992] calls them) the President of the Russian Academy of Sciences, held at the which results in perceiving many faces ofthe same class, opening of thc Lcibniz-Symposion in Leipzig, April 1996.) forms a generalization, rendcred by the rule of introduc­ 2 Cf. eg, the 'QED Manifesto' in the electronic journalj\tfa thesis ing thc general quantifier. Universalis whose Internet address is as fo llows: http:// The rule of introducing the existenlial quantifier de­ www.pip.com.pl 3 The fact that Descartes' position is so strikingly one-sided, fines another kind of reasoning in which a perception disregarding complexities ofmind-malter relations, makes it even yields the premiss in question. Usually, sllch a premiss more useful for the present purposes. The narrower a view, the remains unverbalized, hence not manageable by a digital greater may prove its cognitive value, provided its limitations computer (unless one becomes able to feed it with non­ being due to a fitting idealization,even a countelfactual one. In our verbal representations oftheobjects perceived, and estab­ century a nice example ofsuch a strategy is fo und in the programmc lish logical rules to process such representations). for science furthered by logical empiricism. An even more impres­ sive paradigm (firmly opposing both empiricism and logicism) The rule of concretization (ie eliminating the general which preserves its vitality for fo ur centuries, is that produced by quantifier) is of special consequence for the present Descartes. Any reflection on knowledge, mind and logic has to discussion as it can exemplify von Neumann's claim involve references to his so illuminatively clear (even if false) regarding the difference between the textbook logic and ideas. the logic of our brain. The example is found in 4 More on this issue can be found in my paper 'Polish notation, Marciszewski [1994, 145-9] where the reasoning of an well-ordering and praxeology' for the conference on the work of apc is reconstructed in terms of a computerizcd system, Jan Lukasiewicz, Dublin, July 1996. Its electronic version is to be termed Mizar MSE, of quantification logic. The system found in the database associated with the electronic journal accepts an orthodox "textbook formalization" as well as Mathesis Un iversa/is. The path is as fo llows: hltp:llwww.pip.c0111.pIlLogBanklindex.html another one, closer to actual reasonings, in which the

Know!. Org. 23(1996)No.2 81 Witold Marciszewski: Leibniz's Two Legacies

https://doi.org/10.5771/0943-7444-1996-2-77 Generiert durch IP '170.106.35.234', am 27.09.2021, 09:57:11. Das Erstellen und Weitergeben von Kopien dieses PDFs ist nicht zulässig. At the LogBank home page accessed in this way findthe Section 37( 1930)p.349-60. lI Academic Meetings, subscetion on the Dublin Conference and G6de1, K. [1931] ber fo rmal unentscheidbare Satze derPriJlcipia select the paper quoted abovc. MatiTematica und verwandter Systeme, I. Monatshefte f. 5 Prillciples de /a nature et de fa gracefimdes en raison, item 3. Mathematik lind Physik, 38(1931)1'.173-98. This statement is taken from the English translation, Leibniz Berken R. (Ed.) [1998] The Universal Turing Machine. A Half­ [1973]. To render all the nuances ofthis important text, it is worth Century Survey. Oxford: Oxford University Press while to quote it in the French original and in a suggestive German Leibniz, G. W. [1973] Philosophical Writings. Ed. by G. B. R. translation. "Ainsi il y a harmonie parfaite entre les perceptions de Parkinson. Everyman's Library, London, etc. (includes la Monade et les mouvements des corps, preetablie d'abord entre Monadology). Ie systeme des causes effieienteset celuy des causes finales, ct e' cst Leibniz, G. W. [1982] Vernunftprinzipien der Natur llnd der en cela que consiste l'accord ct l'union physique de l'arne el du Gnadc. Monadologie. Franz6sisch-Deutsch. Hamburg: Meiner corps, sans que J'un puisse changer les loix de l'autre". Here is the Verlag. Germantext: "DaherbestehteinevollkommeneHarmoniezwisehen Marciszewski, W. [1994] Logic fi·oma Rhetorical Point ofVicw. den Perzeptionen der Monade und den Bcwegungen der Korper, BerlinlNew York: Walter de Gruyter. Reihe: Grundlagen del" die von Anbeginn an zwischen dem System dcr Wirlmrsaehen und Kom111unikation und Kognilion, Ed.: R. Posner and G. Meggle. dem del" Zweekursachen prastabilisiert ist; und eben darin besteht Marciszewski, W. [1994aJ A laskowski-stylesystem of compuler­ die Ubereinstimmung und die natiirliche Vereinigung von Secle assisted reasoning. In: Wolenski (Ed.) [1994]. lind Karper, ohue dan cines die Gesetze des anderen Zll andern Mareiszewski, W., Murawski, R. [1995] Mechanization of Rea­ vermochte". See Leibniz [1982], p.6-. soning in a Historical Perspective. Amsterdam/Atlanta: Rodopi. Series: Poznan Studies in the Philosophy of the Sciences and the Humanities, Ed. by L. Nowak. References Pcnrose, R. [1988] On thc physics and mathematics ofthought. In: Albertazzi, L. and R. Poli (Eds.) [1991] Topics in Philosophy and Rerken (Ed.) [1988] p. 491-522. Artificial Intelligence. MiHeleuropaisehes Kulturinstitut, Bozcn. Penrose, R. [1989J The Emperor's New Mind. Conccrning Com­ Beth, E. W. [1970] Aspects of ModernLogic. Dordrecht: Rcidcl. puters, Minds, and The Laws of Physics. Oxford - Ncw York - Beth, E. W., Piaget, 1. [1966] Mathematical Epistemology and Melbourne: Oxford University Press. 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82 Know!. Org. 23(1996)No.2 Witold Marciszewski: Lcibniz's Two Legacies

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