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Mathesis Universalis     No.5 - Winter 1998
When using any part of this text - by Witold Marciszewski - refer, please, to the original URL.

in the 350th Anniversary of His Birth?

Introduction to this Issue

There is no reason that every philosopher read every philosophical classic. Leibniz, though, should be read by quite a many, at least those involved in the intellectual foundations of information society. It was he who saw the universe as an immense system of information machines.

The dates as that celebrated in 1996 - the 350th Anniversary of Leibniz's birth - provide a special opportunity to reflect on his topicality. This issue (though delayed for technical reasons) is to hint at some Leibniz' ideas which retained their vitality to our days.

This issue contains contributions which appeared in the volume On Leibniz's Philosophical Legacy - in the 350th Anniversary of His Birth, edited by Halina 安i璚zkowska in the series "Studies in Logic, Grammar and Rhetoric", no 14, this series being published since 1980 by the Chair of Logic, Informatics and Philosophy of Science" at the University of Bia造stok (previously University of Warsaw, Bia造stok Branch)

The first three contributions deal directly with Leibnizian issues, and the remaining are variously related, even if indirectly, to the main subject.

Halina 安i璚zkowska in the essay Language as the Mirror of the Mind offers a substantial motive to read Leibniz in our days. In AI and cognitive science, the vivid debate between symbolism and connectionism corresponds to a significant problem with Leibniz.

Leibniz proves a symbolist, namely the one who postulated a universal system of symbols (Characteristica Universalis) to precisely mirror and enhance a system of thoughts. Thereby, Leibniz would have endorsed the recent notion of LOT (Language Of Thought), to which the behavioristic approach was so hostile. However, with Leibniz it is not clear whether every natural language should adequately mirror a system of thought or just a specially elaborated perfect language would match that system. Ms. 安i璚zkowska's contribution consists in discovering and discussing two kinds of Leibniz's explicit statements, opposing each other in that matter.

There is yet a more fundamental split in Leibniz's views on language, as discussed by Witold Marciszewski in his Leibniz's Idea of Automated Reasoning Compared with Modern AI. If we take Hilbert's programme for comparison, then Leibniz appears both as an eager forerunner of that programme and its eager opponent.

He represented a Hilbert-like approach when postulated algorithms to solve any problem whatever -- if duly formalized, that is, stated in a manner as precise as should have been enabled by his Characteristica. On the other hand, in Monadology, he claimed infinite complexity of organic machines constituting the universe, and that should have produced problems to be hardly solvable in finite sequences of steps.

Adam Drozdek's (Duquesne University, USA) paper Leibniz: Struggles with Infinity addresses the question of reconciling Leibnizian finitism with his vision of infinity in nature.

Drozdek stresses the enormous role of infinity in Leibniz's outlook and research, and thoroughly examines approaches to continuum as found in his numerous texts. The very term `continuum' is taken in its Leibnizian sense, proving vague enough when compared with its counterpart in modern set theory. The discussion offered by Drozdek encourages to attack the problem again, in the light of modern distinctions between infinities, while at its present stage it brings thought-provoking insights into some intricacies of Leibniz's thought.

In the same year, there is a historical reason to commemorate Descartes as well, to wit the 400th anniversary of his birth. There are also reasons to discuss his approach to science and to the universe in the context of commemorating Leibniz. In some points, understanding Leibniz does profit from understanding Descartes.

A reason of special import is inquired by Jerzy Kopania in the essay Descartes' Great Thesis on Nature. The author - a historian of philosophy with a rich logical background - avails himself of basic logical concepts to reconstruct Descartes' view on the structure of scientific theories, i.e., those concerning the material universe.

That view results in what is by the author called the Great Thesis on Nature. It is to the effect that " rational inquiry into nature cannot lead beyond it", and is substantiated by the thesis of material homogeneity of nature, which means that material effects should be explained by material causes alone; and this, in turn, follows from the idea of extension as exhausting the essence of matter (hence no other factors, eg. non-material forms, as claimed by the Schoolmen, are needed to explain physical phenomena).

To fully understand this great thesis, the thesis complementary to it has to be considered - as stated at the end of Kopania's essay - to wit: rational inquiry into the mind leads beyond it, to the transcendent mind as the cause of that of ours.

All that provides us with an excellent contrastive background to grasp the great thesis of Leibniz. Let it put in a nutshell, using the modern notion of code, or software. The thesis runs as follows: the inquiry into nature reveals that beyond extension there is a software, and that requires a mind as its author; and since matter is infinite as dividing into ever deeper and tinier structures, the corresponding software requires the infinite mind.

This is the very essence of Leibnizianism, namely -- let us repeat -- the combining of both software and structural infinity as attributes of matter, both denied by Descartes (his infinity of matter was purely geometrical, one could cut a body in an arbitrary way). Needless to say how close are these ideas to modern science in which some behaviour of matter has to be explained, eg, with recourse to genetic code, and the structural infinity of ever tinier particle structures is seriously considered by scientists.

Such insights into Leibniz are available just through those insights into Descartes which we owe to Kopania's penetrating examination. This is why his essay so nicely fits into this issue.

Now, let us think about Leibniz's dream of reasoning machines. He believed in practicability of such a project, encouraged by the success of his calculating machine as well as theoretical considerations concerning the closest similarity between computing and reasoning. This dream is being materialized owing to computer programs called provers. Ms. Anna Zalewska's paper A Criterion of Decidability of some Algorithmic Formulas is based on a work to result in a prover which she constructed for Salwicki's algorithmic logic (included in a broader category, called logic of programs).

The construction of a prover required the adjusting of algoritmic logic to constraints of automated proving. For thus modified system, the author states a criterion of decidability which makes it possible to automatically prove validity of algorithmic logic formulas.

Algorithmic logic appears again in two communications completing each other: The Norms from the Point of View of a Certain Logic of Programs by Anna Zalewska, and Norms and Programs by Andrzej Malec. The former is more concerned with logical foundations, the latter with legal applications. The authors take advantage of the fact that both a legal norm and a computer program transforms an existing situation into a postulated situation. Such formal analogies make it likely that each member of the pair would profit from a joint development.

This idea is only sketched in the said texts, but even in a sketchy form it proves its belonging to the Leibnizian legacy. As for algorithmic logic, its relation to the great Leibniz's project of an automatic prover is mentioned above. As to legal issues, their approaching from a logical point was Leibniz's concern as well (as shown in his De casibus perplexibus in lege). Though he was far from algorithmic approach to the law, the first step in this direction was made in his precising the language of that discipline.

If we look for most significant points of Leibniz philosophical creed, then -- besides the great thesis on the ubiquity and infinity of software -- we encounter his radically deterministic approach. At the same time, we observe how much attention he payed to the notions of space and time.

The last two items in the volume are concerned with logical connexions between determinism and conceptions of time. These are: Some Remarks about Intuitionistic Tense Logic by Dariusz Surowik, and Omniscience, Omnipotence and Related Notions by Kazimierz Trz瘰icki. Their role for understanding Leibniz is a bit similar to that played by the study of Descartes as considered above, namely that of a contrastive background.

Both authors develop a version of indeterminism which goes back to Jan ㄆkasiewicz. However, in spite of the fact that ㄆkasiewicz himself provided a logic to precisely state indeterministic insights, namely his multi-valued logic, our authors prefer other logical devices: those of tense logic (created after ㄆkasiewicz). Surowik combines them with intuitionistic logic (thus resorting to multi-valuedness but not in ㄆkasiewicz's style). Trz瘰icki develops some insights concerning the notion of freewill; these are based on thorough historical erudition, and have a formal logic of tenses in their background.

Let me sum up with a more general comment on the approaches found in this volume. They may resemble what is called "Whig interpretation of history". This notion was used by Volker Peckhaus (University of Erlangen), a thorough historian of logic, in a review of him concerning W. Marciszewski's and R. Murawski's Mechanization of Reasoning in a Historical Perspective (Amsterdam 1995, Rodopi). Peckhaus does not share this approach, and he has good reasons for that. This is why a comment on this volume's intentions will be in order.

When applied to the history of ideas, Volker's expression is like a metaphor, since literally it refers to political history. First it was used by H.~Butterfield in his The Whig Interpretation of History (London 1931). However, it is a fitting metaphor if, for example, one refers to Jan ㄆkasiewicz's programme for history of logic, followed by quite a number of logic historians.

In his book, Butterfield examined critically the tendency of historians to see the past as the story of the conflict between progressives and reactionaries, in which the progressives, or Whigs, win and bring about the modern world. For ㄆkasiewicz, for instance, Stoic logic was more `progressive' than Aristotelian logic since in our times the former has proved more general and more fundamental.

In this volume, to take a most recent example, J. Kopania presents Descartes as a `progressive' in methodology of natural science, contrasting his attitude with that of the Schoolmen, while the present author in his comment to Kopania's contribution suggests that it is Leibniz who proves more `progressive' (in some respect, at least).

To hint at a rational core of the "Whiggish interpretation", let me first observe that it proves more reasonable in the history of science than in the political history. For the former is a cumulative process in which previous achievements contribute to later ones. Neverthelesss, a caution is needed.

We should cautiously distinguish between a historical reconstruction of the past and what may be called a diachronic comparative research. In the volume commented, it is the study by 安i璚zkowska} which is closest to the former (though not without a modern perspective) while Marciszewski's approach exemplifies the latter.

In the latter one does not claim that, for instance, Leibniz's projects belonged to a causal chain to result in modern computer science. Instead, one compares two systems of ideas, distant in time, to recognize their logical relations. Once having done so, one can ask whether logical relations have influenced the actual progress or have not. The mere fact of logical connexions does not yield any historical answer, it just may assist a better understanding of the concepts to be used in a genuine historical research.

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