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\def\ParzystyHead{\line{\HeaderFont\vphantom{/S}Witold Marciszewski\hfill}}

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\lasta Witold Marciszewski\\

%The University in Bia/lystok
%Department of Logic, Methodology and Philosophy of Science

\t LEIBNIZ'S IDEA OF AUTOMATED REASONING\\
\lastt COMPARED WITH MODERN AI\f{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.}\\


Gottfried Wilhelm Leibniz (1646-1716) is duly merited as the first who
anticipated Artificial Intelligence. If so, let us ask: which~AI? That
called {\it strong} which --- following Turing [1950], Newell [1980],
etc. --- answers in the affirmative the famous Turing's question {\it
`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?'.

\looseness=1
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 --
{\q WM}). ``Cognitive explanation requires not only causally interrelated
mental states, but also mental states whose {\it causal} relations
respect the {\it 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
laws but hardly related to semantic rules. Such a contrastive background
helps to appreciate the case of {\it 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 {\it 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 `{\it 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.


\rsub 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 G\"odel [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.

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 G\"odel'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 {\it 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.

\cytat
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.
\\


\noindent 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].

\cytat
It seems to me that it is a clear consequence of the G\"odel 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 G\"odel'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 self-consistent 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 G\"odel's theorem.\hfill\break
\indent \kern5mm However, without G\"odel 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 {\it preliminary} thinking, as a guide
toward finding the appropriate formal argument; but they would play no
part in the actual demonstration of mathematical truth. G\"odel'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.
\\

\vskip1mm

\noindent 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 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
{\it 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
{\it 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 {\it each organic body is a kind of divine machine, or
natural automaton, which infinitely surpasses all artificial automata}
because of its infinite complexity (see {\it Monadology}, 64). Hence the
human mind should surpass artificial machines in dealing with complexity
of problems to be
addressed.\f{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.}

\rsub 2. On `filum cogitationis' --- an algorithmic method of reason\-ing.\\
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.\f{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
{\it 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].}

\cytat
\indent\kern5mm {\sl 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; {\it Briefwechsel}, I,
102, to Oldenburg. See Couturat [1901], p.~91, fn.~2; p.~96,
fn.~4.\hfill\break
``What I call {\it 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.''
\\



\noindent 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.

\cytat
\indent\kern5mm {\sl Ad inventionem ac demonstrationem veritatum opus est analysi
co\-gi\-ta\-tio\-num, quae quia respondet analysi characterum..., hinc analysin
cogitationum possimus sensibilem reddere, et velut quodam filo mechanico
dirigere; quia analysis characterum quoddam sensibile est.}
--- {\it Analysis linguarum}, 11 Sept. 1678. (C, 351)\hfill\break
``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.''\hfill\break
\indent\kern5mm {\sl 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; {\it
Briefwechsel}, I, 380, to Tschirnhaus, May 1678. See
Couturat [1901], p.~91, fn.~4.\hfill\break
``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 cons\-ti\-tu-\break
ents; 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 re\-so\-lu-\break
tion of concepts exactly corresponds
to the resolution of respective symbols.''
\\


\noindent In other texts there appear the notions
of {\it calculus} and of {\it 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.


\cytat
\indent\kern5mm {\sl Nihil enim aliud est {\it Calculus}, quam operatio per
characteres, quae non solum in quantitate, sed et in omni alia
ratiocinatione locum habet.} --- GM iv, 462, {\it Briefwechsel}, I, 381,
to Tschirnhaus, May 1678. See Couturat [1901], p.~96, fn.~2.\hfill\break
``The calculus is nothing else but operating with characters, what
occurs not only in computing quantities but also in any other reasoning.''
\\


\noindent 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:

\cytat
\indent\kern5mm [...] {\sl calculus ratiocinator, seu artificium facile et
infallibiter ratiocinandi. Res hactenus ignorata.} --- GP~vii, B, II, 8.
See Couturat [1901], p.~96, fn.~5.\hfill\break
the reasoner calculus, that is, a device for reasoning in an easy and
infallible way --- the thing unknown so far.
\\


\noindent This kind of statements includes the famous ``Calculemus'' which occurs
in the following context:

\cytat
\indent\kern5mm {\sl 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.\hfill\break
``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: {\it let us
compute!}\/''
\\


\noindent When the notion of {\it machine} gets added to those of {\it sensible
characters} and of {\it 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):

\cytat
\indent\kern5mm {\sl Ut Veritas quasi
picta, velut Machinae ope in charta expressa,
deprehendatur.} [...] {\sl Illud Criterion} [...] {\sl quod velut
mechanica ratione fixam et visibilem reddit veritatem.} ---
GP vii, 10, {\it Briefwechsel} I, 145, to Oldenburg, 28 Dec.\ 1675. See
Couturat [1901], p.~99, fn.~2; p.~100, fn.~3.\hfill\break
``[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.''
\\


\noindent 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 {\it 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.


\rsub 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.

\looseness=1
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 {\it 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.\f{The problem of the
creative role of real definitions as axioms in
a~reasoning is discussed in my [1993] and [1995b]
essays.}
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.\f{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.}

\pu%%%%%%%%%%%%%%%%

\rsub 3. Leibniz$_m$'s infinitistic theory of matter.\\
One may object a lack of coherence between the finitistic {\it 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 {\it 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:

\cytat
\indent\kern5mm {\sl Cujuscumque corporis
infinitae sunt partes, seu, ut vulgo
loquuntur, Continuum est divisibile in infinitum.}\hfill\break
``Every body has infinitely many parts, or, as it is commonly said, the
continuum is infinitely divisible.''
\\


\noindent 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 ({\it potentia
principalis}) to move the infinite. Note that Leibniz$_m$, unlike Aquinas and
other masters of {\it 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.

\cytat
Let the following quotations support the point that Leibniz, in
a~way, anticipated some recent guesses as to the actual infinite
divisibility of matter.\hfill\break
\indent\kern5mm ``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].)\hfill\break
\indent\kern5mm 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.
\\


\noindent 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.\f{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 {\it continuum} in modern concepts, we are to consider
which concept corresponds to his intentions.}

Hence, if `moving' (in the mentioned {\it 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. {\it 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
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 {\it 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'',\f{GP vii, 309-18. See P.\ [1973], p. 82.}
and that ``there is no body so small that it is not actually
subdivided''.\f{C 518-23. See P. [1973], p.~91.}
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''.\f{E.~Bodeman, {\it Die Leibniz-Handschriften der K\"oniglichen
Bibliothek zu Hannover}, Hannover 1889, 115-17. See~P. [1973], p.~108.}
In Leibniz's article ``Syst\`eme noveau [...]'' published in {\it Journal
des Savants}, 1695, we read that ``everything in matter is but
a~collection or accumulation of parts
{\it ad infinitum}''.\f{GP iv, 477-87. See P.[1973], p.~16.}
In {\it Monadology} of 1714 it is stated in item 65 what follows.


\cytat
\indent\kern5mm {\sl Et l'Auteur de la
Nature a p\^u practiquer cet artifice divin et
infiniment merveilleux, parce que chaque portion de la mati\`ere n'est pas
seulement divisible \`a l'infini comme les anciens ont reconn\^u, mais
encor sous-divis\'ee actuellement sans fin, chaque partie en parties, dont
chacune a quelque mouvement propre: autrement il seroit impossible, que
chaque portion de la mati\`ere p\^ut exprimer tout
l'univers.}\hfill\break
``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.)
\\



\rsub 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 {\it Demonstratio existentiae
Dei}, hints at the connexion between a theory of matter and a theory of
mind. According to {\it 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 {\it Metaphysical consequences of the principle of reason}
(ca.~1712) attributes the full knowledge of matter to the omniscient,
hence infinite, mind alone.

\cytat
``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 {\it ad infinitum}.'' (P.~[1973], p.~176)
\\


\noindent 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.\f{Reported after Einstein and Infeld [1947, Ch.\ 1].}


\cytat
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.
\\


\noindent 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 {\it Correspondence
with Clarke}.


\cytat
``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 {\it 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 {\it 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)
\\


\noindent 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'', {\it downward
explanation}, for short, and should be defined as follows: {\it for each
system, a downward explanation of its structure and functions takes into
account structures 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 {\it all} of them. But what
about dealing with {\it 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 {\it 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
{\it 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 {\it 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.

\rsub 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 {\it 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.

\vskip1mm

\cytat
\indent\kern5mm {\sl On est oblig\'e d'ailleurs de confesser que la} perception
{\sl et ce qui en depend est} inexplicable par des raisons mecaniques,
{\sl c'est \`a
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 m\^emes proportions, en
sorte qu'on y puisse entrer, comme dans un moulin. Et cela pos\'e, 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 compos\'e ou dans la machine qu'il la
faut chercher.}\hfill\break
``We are moreover obliged to confess that {\it perception} and that which
depends on it {\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.)
\\


\noindent 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 {\it 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], Kert\`esz [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.% \f{Unfortunately, I did not succeed -- before the dead line for this
% paper -- to ask more competent collegaues, esp.\ those engaged in the QED
% project, if the diagonal argument has been formalized in any of the most
% advanced systems of automated reasoning. This remains an intriguing task
% for a further investigation. Anyway, I happened to win a minor experience.
% Once I asked a leading expert in a leading project (within the QED group) to
% automatize the proof of the first theorem in Euclid's Book One. The
% theorem is known as one which needs an ad hoc premiss (not included in the
% axioms) based on a visual perception. The experts, which succeeded in
% gathering a big system of automatized proofs in several branches of
% mathematics, did not succeed in that demonstration. This is no argument
% for its impossibility, as new attempts can be made, but seems to suggest
% that {\it there is} a problem. For QED project, see the newsletter
% {\it Mathesis Universalis} (http:////www.pip.com.pl//MathUniversalis//),
% Nos.~1, 2 and~3.}

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.\f{This Kant's problem is extensively discussed by Evert Wilem Beth at
several places, e.g. in [1959, Sec.~20] and 1970, Ch.~4.}
Does the attack succeed, this should shed light both on the modern issue of
automatization of reasoning and Leibniz$_e$'s engineering dream.

\pu

\r References\\

\parindent0pt
\parskip3pt plus1pt

--- Albertazzi and Poli [1991]. Albertazzi, L.\ and Poli, R. (eds.), {\it
Topics in Philosophy and Artificial Intelligence}. Mitteleurop\"aisches
Kulturinstitut, Bozen 1991.

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

--- Beth [1970]. Beth, Evert W., {\it Aspects of Modern Logic}, Reidel,
Dordrecht 1970.

--- Breger [1988]. Breger, H.,{\it Das Postulat der Explizierbarkeit in der
Debatte um die k\"unstliche Intelligenz} in: {\it Leibniz. Tradition und
Aktualit\"at. V. Internationaler Leibniz-Kongre{\bbbit}. Vortr\"age. Hannover
14.-19. November 1988}. Leibniz-Gesell\-schaft, Hannover 1988.


--- Breger [1989]. Breger, H., {\it Maschine und Seele als Paradigmen der
Naturphilosophie bei Leibniz} in: von Weizs\"acker and Rudolph
(eds.) [1989].

--- Bodeman [1889]. Bodeman, E., {\it Die Leibniz-Handschriften der
K\"oniglichen Bibliothek zu Hannover}, Hannover 1889, 115-17.

--- Boden [1990]. Boden, Margaret A.\ (ed.), {\it The Philosophy of
Artificial Intelligence}, Oxford University Press, Oxford 1990.

--- Couturat [1901]. Couturat, Louis, {\it La logique de Leibniz}, Alcan,
Paris 1901.

--- Cantor [1890//91]. Cantor, Georg, {\it \"Uber eine elementare Frage der
Mannigfaltigkeitslehre}, ``Jahresber.\ Dtsch.\ Math.\ Ver.'' 1,
1890//91, 75-78 and 278-281.

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

--- Drozdek [1997]. Drozdek, Adam, {\it Leibniz: Struggles with Infinity},
this book.

--- Einstein and Infeld [1947]. Einstein, Albert and Infeld, Leopold,
{\it Evolution of Physics}, Cambridge University Press, Cambridge (Mass.)
1947.

--- Fodor [1976]. Fodor, Jerry A., {\it The Language of Thought}, The
Harvester Press, Hassocks (Sussex, England) 1976.

--- Gams [1995]. Gams, Matja\u{z}, {\it Strong vs.\ Weak AI}.
``Informatica: An International Journal of Computing and Informatics'', vol.\
19, no.\ 4, November, 479-493.

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

--- Gleick [1991]. Gleick, J., {\it Chaos. Making a New Science},
Macdonald, London 1991.

--- G\"odel [1930]. G\"odel, Kurt, {\it Die Vollst\"andigkeit der Axiome des
logischen Funktionenkalk\"uls}, ``Monatshefte f\"ur Mathematik und
Physik'', 37, 349-360.

--- G\"odel [1931]. G\"odel, Kurt, {\it \"Uber formal unentscheidbare
S\"atze der {\it Principia Mathematica} und verwandter Systeme}, I,
``Monatshefte f\"ur Mathematik und Physik'', 38, 173-198.

--- Hawking [1992]. Hawking, S.\ W., {\it A Brief History of Time},
Bantam Press, London 1992.

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

%%%\eject%%%%%%%%%%%%%%%%

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

--- Kert\`esz [1983]. Kert\`esz, Andor, {\it Georg Cantor 1845-1918.
Sch\"opfer der Mengenlehre}, ``Acta Historica Leopoldina'', 15, Halle 1983.

--- Leibniz [1982]. Leibniz, G.\ W., {\it Vernunftprinzipien der Natur und
der Gnade. Monadologie. Fran\-z\"o\-sisch-Deutsch}. Meiner, Hamburg.

--- Marciszewski [see the end of this list].

\eject


{

\parskip2,5pt plus1pt



--- McCulloch and Pitts [1943]. McCulloch, W.\ S.\ and W.\ Pitts,
{\it 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, {\it Hilbert's Program: Incompleteness
Theorems vs.\ Partial Realizations} in: J.~Wole\'nski (ed.) {\it
Philosophical Logic in Poland}, Kluwer, Dordrecht etc. 1994 (Synthese
Library vol. 228).

--- Newell [1980]. Nevell, A., {\it Physical Symbol Systems}, ``Cognitive
Science'' 4, 135-83.

--- P.\ [1973]. Parkinson, G.\ H.\ R.\ (ed.), {\it Gottfried Wilhelm Leibniz:
Philosophical Writings}, Everyman's Library, London 1973.

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

--- Penrose [1989]. Penrose, R., {\it 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, {\it Leibniz}, Oxford University
Press, Oxford 1984.

--- Schnelle [1988]. Schnelle H., {\it Turing Naturalized: von Neumann's
unfinished Project} in: Herken (ed.) [1988], 539-559.

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

--- Turing [1936-7]. Turing, A.\ M., {\it 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., {\it Computing Machinery and Intelligence},
``Mind'', vol.\ 59, no.\ 2236 (Oct.), 433-460. Reprinted in Boden [1990].

--- Ulam [1976]. Ulam, S.\ M., {\it Adventures of a Mathematician},
University of California Press, Berkeley (USA) 1976.

%%\eject

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

--- von Neumann [1958]. von Neumann, J., {\it The Computer and the Brain}.
Yale Univ.\ Press, New Haven 1958.

--- von Weizs\"acker and Rudolph (eds.) [1989]. von Weizs\"acker, C.\ F.\
and E.\ Rudolph (eds.) {\it Zeit und Logik bei Leibniz. Studien zu
Problemen der Naturphilosophie, Mathematik, Logik und Metaphysik}.
Klett-Cotta, Stuttgart 1989.

}

\eject


\hrule\vskip1pt\hrule

\medskip


\font\mss=p1ss10
\font\mssit=p1ssi10

\parskip0pt

\def\\ #1\par #2\par{{
--- #1\par
\smallskip
{\baselineskip11pt\mss
\noindent #2\par}
\bigskip
}}

{\mss\baselineskip11pt
The references to this author's contributions are listed apart
for their additional function: they should round off the text with minute
notes to hint at some preparatory steps leading to the point of this
essay.\par}


\bigskip

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

A study of the role of non-verbal abstraction processes in
concept-formation, being an important kind of what Leibniz calls
perceptions, and contributing to reasonings in a way comparable to that
of axiomatic definitions (cp.~[1994a]).



\\ Marciszewski [1993//94]. {\it Why Leibniz Should not Have Believed in
{\it filum cogitationis}}, ``Studies in Logic, Grammar and Rhetoric''
12//13, Bia/lystok 1994, 5-16. The same text appeared in: {\it Leibniz
und Europa. VI.\ Internationaler Leibniz-Kongre{\bbbit}. Votrtr\"age, I.\
Teil}, Leibniz Gesellschaft, Hannover 1994, 457-464.

It forms a considerable part of Sections 2-4 of the present essay.



\\ Marciszewski [1994a].
{\it Logic from a Rhetorical Point of View}, Walter de Gruyter, Berlin --
New York 1994.

The book contributes to the problems of the present essay through
extensive discussion of formalization procedures as contrasted with
non-verbal reasonings, while axiomatization procedures are used as
a device to study non-verbal processes of concept-formation.



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

The paper comments on the proof-checker Mizar MSE, being a preliminary
part of the system Mizar.
Special attention is paid to how that computerized
system of reasoning renders obviousness of inferences (belonging to the
Leibnizian domain of perception).



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

The book tells the history of logic as leading to mechanization of
reasoning --- through (i) the medieval nominalism merited for the ideas of
logical form and of extensionality, (ii) the ideas of formalized language and
logical calculus due to Leibniz, Lambert et al., (iii) the success of those
ideas with algebraization of logic by Boole et al., (iv) the modern methods of
formalization and mechanization of reasoning.



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

The problem, handled also in [1993], is here related to Ajdukiewicz's
theory of real and nominal definitions.



\\ Marciszewski [1996a].
{\it Leibniz's Two Legacies. Their Implications for Knowledge Engineering},
``Knowledge Organization'' (Frankfurt), 23 (1996) No.2, 77-82.

This text (invited by KO Editors to commemorate Descartes' and
Leibniz's anniversaries in 1996) presents Leibniz's views on knowledge
against the contrastive background of those of Descartes, and then relates
them to some Turing's [1950] and von Neumann's [1951], [1958] ideas.



\\ Marciszewski [1996b]. {\it H\"atte Leibniz von Neumanns logischen
Physikalismus geteilt?}, ``Beitr\"age zur Geschichte der
Sprachwissenschaft'' (M\"unster), 6 (1996), 245-259.

The text identical with the author's lecture raed at a session of {\mssit
Leibniz Gesellschaft} in Hannover (1995, June). In a part, it is a German
version of [1996a].