Concept for a Dynamic Logic According to Leibniz and Gödel

Lecture at the X. International Leibniz Congress, Hannover 18-23 July 2016, revised version. (The original version is available in: Wenchao Li (Ed.) „Für unser Glück oder das Glück Anderer”, Vorträge des X. Internationalen Leibniz-Kongresses, Hildesheim 2016, Vol. IV, p. 221-233)

 

 

Abstract: The first beginnings of a dynamic logic go back to Aristotle. For him the continuum and the time cannot be composed of dimensionless points, but they consist of moving, smallest units in the transition from possibility (dynamis) to reality (energeia). How can they be calculated? Leibniz understood them as differentials dx and dy, for which he developed his own calculus (the differential calculus), which transforms at the limit into the calculus of ordinary numbers. In his Monadology he makes a similar distinction between the ordinary perceptions and the small perceptions of which they are composed, but which are indistinguishable for us. Their relationship is the model of a dynamic logic that leads from the semi-finished and still developing elements of a space of possibility to the fixed elements of reality. There must be affinities between the semi-finished elements of the space of possibility (Leibniz speaks of appetitus), from which their dynamics evolve and through which they combine to form aggregates that step into reality. If, as a result of the multitude of all possible connections, a selection is made that leads from possibility to reality, the selection for Leibniz follows the principle of harmony. Gödel has adopted these ideas point by point in his own work, when he speaks of continuum, incompleteness, impredicativity and fecundity. This is the foundation of the mathematical program of a dynamic logic.

Keywords: monadology, small perceptions, differential calculus, incompleteness theorems, affinity, continuum, impredicativity, harmony principle

1 Introduction

Although Leibniz was the favourite philosopher for Gödel - as well as for Einstein -, whose works he studied throughout his life, there are hardly any statements in which he refers to Leibniz in a coherent way. But in the various works of Gödel there is a red thread that goes back to Leibniz. It is his own decision not to look for new formal systems or a universal language, which could also refer to Leibniz, but to base logic on a pre-logical space of statements that are vague, semi-finished and within the realm of possibility, before they result in protocol sentences and scientifically verified statements that are represented by the means of formal logic. This logic is no longer a static system of relationships and conclusions of statements about rigid objects and their symbolic representations, but a logical development model oriented to natural growth processes. At the same time, this approach gives a new perspective on the paradoxes of logic and mathematics, if their special place and status is taken into account during the development process.

The model for this approach are the small perceptions that Leibniz introduced in his Monadology. Leibniz distinguishes in the Monadology between the »small perceptions«, »of which we are not consciously aware« (»perceptions, dont on ne s’ aperçoit pas« in the original French text) and the perception in the usual sense ^[1]. The small perceptions are the ground from where the perceptions come into consciousness according to comparable rules, just as the real emerges from the possible. In another analogy, the differentials introduced by Leibniz can be understood as the ground, where real numbers are embedded and from where they emerge. Leibniz and Gödel have developed with the differential calculus and the incompleteness theorems first ideas for a logic of the space of possibility and its transition into reality.

This thesis is to be carried out at four steps:

– The Possibility Space as ground for Reality

– The "appetitus" as elementary interrelation between the elements in the space of possibility

– The connection or the continuum as an overall property of the space of possibility

– And finally, harmony as a principle through which reality emerges from the diversity of possibilities. In Gödel's drafts, it corresponds to the fertility principle (fruitfulness), a kind of meta-principle for mathematical axiomatics.

2 Ground

The ground is not a causality chain like 'from A follows B, from B follows C and so on' as in Kant or formal logic according to Frege, where each element is the cause for its successor, but a space of possibility from which the real emerges. This can vividly be the soil, which contains a variety of possibilities, the furrow of the field provided with seeds or seedlings or in another picture an embryo, which develops protected in the womb. Mathematically speaking, this can be the whole of differentials dx, dy, ..., from which the real numbers arise, and based on this the transition of Germs from topological spaces into their physical applications as demonstrated using Geometric Calculus developed in the 20th century. – In a broad philosophical sense it may be the "causa" in the double meaning of ground and matter (case). The subject matter itself is the ground from which something is understood and seen in the horizon of all its alternative possibilities.

The ground contains nothing impossible. In this respect, the tertium non datur applies in a trivial way: Something cannot be possible and impossible at the same time. But it will become apparent that for the ground the postulations of metaphysics and the determinations of reflection (identity, difference, contradiction and sufficient reason) are only valid in a restricted way. The ground consists of unfinished elements that differentiate themselves from each other in their further development, contradict each other or, conversely, can grow together into higher organic units. In the realm of the possible, two germs (options) can coexist, even if both contradict each other in their further development ^[2]. If one of them becomes real, the other cannot become real. In this respect, the tertium non datur is violated in the space of possibility and is basically only valid to a limited extent.

Gödel's incompleteness theorems can be read backwards. In an incomplete system there may be elements that cannot be clearly separated due to their incompleteness. They carry an exactly definable degree of indistinguishability. There are many approaches and examples for this: (a) When algebra operates with letters instead of numbers, it is deliberately left undefined which numbers are to be used ^[3]. (b) Within the natural numbers there is an indeterminable bound, from which the large numbers elude the view, "what Gödel called the 'big jump'" ^[4]. This thought is beyond the imagination of the mind. Everyone knows the formal rule, how to imagine and form the following number ›n + 1‹ for an arbitrarily large number n, and yet Gödel's intuition is at the same time plausible that somewhere in the whole as on the horizon the numbers become indistinguishable. Every computer scientist knows this, if the address space for representable numbers is finite or when a finite number has been found with the Bremermann-limit, beyond which it is no longer possible to calculate and decode data. (c) Since it can be shown that the universe of all sets is not a set, there must be among the sets largest sets which cannot be exceeded. For Gödel, exceeding them is a "second jump" ^[5]. Gödel was convinced that a maximum principle could be found for this limit, which he compared with the gravitational law of physics, by means of which the open questions of set theory can be solved ^[6]. The sets beyond this limit are indistinguishable. (d) The best known and most accurate example are the differentials dx and dy as introduced by Leibniz. These are vanishing quantities whose location can only be approximated in limit processes and which can nevertheless be calculated using the differential calculus developed by Leibniz. Today they are described with ε-spheres. The elements within an ε-sphere cannot be distinguished. It can only be defined: If an arbitrary element ε₀ is selected from an ε-sphere, then there exists a smaller ε-sphere e', in which this element is not contained.

The smallest units of the ground cannot be individualized. There are no atoms. They are indistinguishable within defined limits. It is impossible to assign simple predicates to them; both the elements and their properties are only given artificial names with an uncertain reference set. If an element is selected, this can only be arbitrary, and it is pure chance which element was hit from the reference set. (According to Wang, even Gödel had difficulties with this approach developed by Paul Cohen. "In Cohen's proofs one makes generality or impossibility statements about what one does not know. Nobody can understand this." ^[7]).

In this way the paradox of chance and sufficient reason is solved. Chance (or probability) does not violate the principle of sufficient reason if it applies within the ground in which the unfinished elements are not yet fully distinguishable.

Summarizing: In principle, the usual meanings of the determinations of reflection (identity, difference, contradiction and reason) apply only to a limited extent. The classical postulations of metaphysics are not yet fully valid. (i) For unfinished elements A, identity A = A can only be postulated approximately. (ii) If two elements A and B are indistinguishable, the postulation of difference is violated. (iii) There may be elements A and B which are both possible but cannot be realized simultaneously. (iv) And the postulation 'Everything has a sufficient reason' is only temporarily valid in the space of possibility in motion: It can turn out that an element A can only develop further and attain continuance, if it grows together with another element B, so that formally in the further development B proves to be a cause of element A.

If, as usual, the real objects are described as zero order, the signs referring to them as first order, the metalanguage about these signs as second order and so on, then the virtual, unfinished objects of the possibility space can be called the first negative order, the totality of all possible names for possible objects as second negative order and so forth ^[8].

The idea of the possibility space reaches very far. It contains not only the diversity of all possibilities, but also the traces of the past and the future:

"In every substance there are traces of all that it has encountered and of all that it will still encounter. But the infinite multiplicity of perceptions makes it impossible to distinguish them from each other." ^[9]

A theological meaning is connected with the trace: The idea of tzimtzum, the God who withdraws from the world and leaves only a trace, goes back to Isaac Luria (1534-1572). Modern science takes the place, which god has left, and their triumphal march is in pessimistic interpretations seen as the "European nihilism" (Nietzsche) or the "abandonment by being" (Heidegger). But this can also be seen optimistically: God does not show Himself in the real things, which are frozen in their finality, as it is believed by a vulgar materialism, but in the variety of possibilities and the freedom and liveliness given to man with them.

3 appetitus

The different possibilities are not simply located equally and externally next to each other according to the model of a power set (the set of all subsets of a given set). The set of the next higher cardinality does not simply result as the totality of all possible combinations from the elements of a given set. For Gödel only an upper limit for the next higher cardinality can be estimated with the number of all possible combinations. Instead, a selection has to be made from the totality of all combinatorial possibilities. Since it is a selection from a mathematical and not from a natural given set, only a formal procedure can be found for the selection, a rule on how such a selection is to be made.

The simplest approach is the axiom of choice established by Zermelo in 1904, without which no axiomatization of set theory would have been possible: a single, arbitrary element is selected from each set (selection rule). Additional conditions can be set step by step for the selection: If it is an ordered set, the smallest element should be selected. If the size of all elements is known, the statistical average value should be used. Further, it may be required that the element to be selected is representative of the set due to its particular nature.

While these selection procedures are applied from the outside to a given set of possibilities, Leibniz assumed that the elements have an internal dynamic and, thanks to their properties, combine themselves into certain groups (appetitus). There must be an inner affinity between them. As Kant wrote in the first edition of the Critique of Pure Reason: "The ground of the possibility of the association of the manifold, so far as it lies in the object, is named the affinity of the manifold." ^[10] Hegel went on with that and, starting from the example of chemically reacting substances, described it as elective affinities (Wahlverwandtschaften, see Hegel in the Science of Logic on the real measure).

But for Hegel there are also other examples of elective affinities: tones seek each other and certain tones sound harmonious or less harmonious together. In a further abstraction, I consider it conceivable that there are certain numbers within the natural numbers that fit better or worse together and form higher orders within the mere successor relation. Primary number research deals with questions of this kind, and it is one of the most astonishing results of recent mathematics when the mathematician Manjul Bhargava (*1974) together with Jonathan Hanke was able to prove that for questions such as 'Can any natural number be represented as the sum of 4 square numbers' "a test with exactly 29 numbers is sufficient to answer the question. [...] Here are the 29 numbers: 1, 2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 34, 35, 37, 42, 58, 93, 110, 145, 203, 290" ^[11] . For reasons that are unclear to me, these numbers form a reference set for inductive regularity. - However, while Kant and Hegel refer to physical elements with affinity or elective affinity, Leibniz understands appetitus as a characteristic of the monads, which must not be confused with the physical world.

Whatever the relations between the elements of the space of possibility are formulated, mathematically they are relations. The totality of all relations is the space of all possible relations. The relations considered here are themselves in the space of possibility, i.e. there can also be relations that contradict each other. Only one of them can be realized.

Gödel's proof idea of the incompleteness theorem states that objects and relations in this space cannot be sharply separated from each other. In the first step, following traditional logic, each object can be described and defined by all its relationships to other objects and each relationship by all the objects to which it applies. Furthermore, following Gödel, every statement, operation and Turing machine can be represented as a Gödel number and thus as an element by the procedure of Gödelization. Each number can be both an object and a relation. If with Leibniz the relations are understood as affections (appetitus), then each element can show both what something is in its incompleteness in the space of possibility, as well as its inclination (relation) to others.

With this approach, Gödel was able to show that the liar paradox can be formulated in sufficiently powerful formal systems, and later also the unsolvable halting problem of theoretical informatics. By this idea of proof, Gödel demonstrated the fundamental incompleteness of formal systems.

4.Cohesion

However, it is not sufficient to ask for the relations (inclinations) between the elements of a set, but both the elements and their relations must be arranged in such a way that they form a continuum. The best-known example, which is decisive for Gödel, is interval nesting for the definition of real numbers. With each interval an elementary relation is given between all elements of the interval, which consists in the common membership to the same interval. A real number is self-referential defined as the intersection of all intervals to which it belongs.

At this point at the latest, it becomes clear how a dynamic logic can go back to Aristotle, who introduced the concept of continuum (synecheia) in his physics using the example of the antinomies of Zenon ^[12] .

As an example for the small perceptions Leibniz had mentioned the sound of the sea. If you hear the sound of the sea, you will basically hear the sounds of the individual drops of water below the threshold, the movement of which makes up the sound of the sea. However, it is not enough to imagine that a whole sea is formed from a countable number of water drops. Rather, two things must be added: From the point of view of the sea, the individual water drops are indistinguishable, and from the point of view of the individual water drops, there must be cohesion so that they join together to form the sea. This cohesion is more than just an enumeration. It can be described by the specific properties of the aggregate state of the liquid water.

The fluidity of water is an example of a cohesion that arises in the continuum of the sea. The water drops and the sea are not two states of the same substrate, they are not only two different orders of the same material or two relations of different range, as for example the order of the atoms lies within the order of the molecules or the molecules within the organism, but with the cohesion a property is given, which applies only to the respective totality and not to the isolated elements. Cohesion is an example of Impredicativity in the sense of Poincaré, Russell and Carnap: Something has cohesion if it belongs to a totality that has cohesion.

Gödel has explicitly opposed Russell's ban on self-referential definitions (vicious circle). Instead, he has given the self-referential definition a very specific role: With it one can define what distinguishes a whole from its parts. Self-referential definitions are a property that applies only to the whole and not to the isolated parts. For him, this characteristic is a separate, independent principle with a scope of validity to be precisely defined: it applies like the violation of the tertium non datur and the paradox of chance and reason only within the space of possibility. Self-referential definitions such as cohesion are only valid in a dynamic logic that describes something that is still in development and whose success is endangered. Only when looking at the finished, realized objects self-references can be excluded. Both the tertium non datur and Russell's ban apply there. (Note: Cantor had already transformed a paradox into a definition in a similar way. It is a paradox already investigated by Leibniz that there are as many square numbers as natural numbers, although the square numbers are clearly a subset of the natural numbers. Cantor turned this around and made it the definition of infinite quantities: An infinite set is defined by the property that it contains subsets which have the same number of elements as the total set.)

The cohesion meant here can be seen when traces and writings are examined. If you follow a track, you are looking for characteristics that show how you can conclude from one element to the next along a track. It is assumed that there is a consistent continuity within the track. And anyone who wants to recognize a script by archaeologically discovered signs looks for the inner cohesion of the given signs, which are supposed to make sense.

5 Harmony, Fertility (Fruitfulness)

The terms used so far can be compared with the three basic terms of classical physics: The soil provides the mass from which everything emerges and is weighted. The relations are the forces that exist between the elements within the amorphous mass. Continuity is energy when understood as the ability to sustain a process continuously from beginning to end and to draw on and use all necessary resources. - Music can be taken as a further analogy: The pitch class is the soil from which the music emerges. Intervals and chords are the relations of tones, melodies show the inner continuity of the musical process. (Addition: During the Leibniz Congress the piece Unnatural Nature for clarinet solo written by Xie Xin for a competition on the occasion of the Leibniz Memorial Year was performed, in which this thought became very audible ^[13] .)

The harmony remains to be added. Ground, appetitus and cohesion are still value-free. They can remain in a static state comparable to absolute heat death. Only with harmony and fertility does life emerge. By harmony is meant not only that something is beautiful, but that all principles mentioned so far are joined into a higher unity.

Gödel strangely demanded this principle not for the elements of a set, but for the creative process. In his contribution on Russell, he acknowledges Leibniz's concern. He understands his Characteristica universalis explicitly not as an "utopian project", but as a "calculus of reasoning to a large extent". Leibniz had expected that with the help of some scientists this would be a matter of perhaps 5 years and of great benefit ("fecundity"). ^[14]

In a similar way, Gödel expects a special "fruitfulness" from new axioms of set theory. Until the end of his life he searched for axioms with which set theory could be justified in a new way and the continuum hypothesis could be answered (see the three manuscripts of 1970 ^[15] . The new axioms are only successful if they can be used fruitfully.

"Success here means fruitfulness in consequences, in particular in 'verifiable' consequences, i.e., consequences demonstrable without the new axiom, whose proofs with the help of the new axiom, however, are considerably simpler and easier to discover, and make it possible to condense into one proof many different proofs." ^[16]

In the closer sense, the continuum hypothesis for Gödel is the question of whether and how cardinal numbers can be used for calculations. Is it possible to find a formula in which one cardinal number is derived from the other? Such a formula would be of similar importance as the formula ›F = m · a‹ (force equals mass times acceleration), with which the basic concepts of physics are linked. He assumed that for the power of the continuum the formula can be established:

    2^ℵ₀ = ℵ₂

In contrast to the usual continuum hypothesis, he sets ℵ₂ and not ℵ₁ for the cardinality of the continuum, since he is convinced that the cardinality of the continuum includes not only the totality of all subsets of natural numbers, but also the continuity of their relations with each other.

Without going into further detail, the question here is by which axioms it is possible to move from one cardinality to the next and a formula like this can be established. It is not enough to find the fruitfulness of an axiom in terms of its consequences, but also to set an axiom of fruitfulness for the set itself, thanks to which this set is able to produce the subsequent set of greater cardinality.

The fertility principle is basically identical with the harmony principle of Leibniz. Gödel has congenially hit upon what Leibniz meant and perhaps even put it more clearly. If formally the power set is compared with the totality of all possible worlds, then another principle is necessary to understand the real world from this totality. For Leibniz, this is the harmony principle, and it is Gödel's - albeit not yet fully formulated - idea that this can be generalized for set theory, where the set of higher cardinality stands for the reality that emerges from the power set of a set of lower cardinality.

At this point, a reflection intertwines with its self-reflections one last time in three ways: The highest principle of possibility space allows the transition to reality. This principle, in turn, can only be found if the set of principles is self-reflectively subject to its own harmony (fertility). And these principles are to be gained by continuing the thoughts that were not yet fully elaborated by Leibniz and Gödel, and only showing their own value in them when they can in turn be further elaborated in the future.

6 Epilogue January 2020

This article was written in 2016 for the International Leibniz Congress in Hannover on the occasion of the 300th anniversary of Leibniz's year of death. Since then the focus of the work has shifted: I am less interested in a further investigation of the history of ideas from Leibniz to Gödel now, than in working out the idea of a dynamic logic. This is a move away from classical logic. I see approaches above all in the work of Gotthard Günther (1900-1984) and George Spencer-Brown (1923-2016) ^[17] . Both asked in different ways for a logic that no longer thinks exclusively linear and transitive, but cyclic and intransitive. Günther goes beyond the use of positive predicate calculus and designs the concept of negative languages (Negativsprachen), with which it my be conceivable to elaborate and formalize the negative order of possibility spaces. Spencer-Brown emphasizes the fundamental importance of medium and form and develops a new approach to logic from the relationship between the two. The medium can be understood as a new view of the space of possibility. I would like to add that both approaches lead to a new understanding of the imaginary numbers, which can be used to work out a dynamic logic. First results are available in articles on imaginary numbers and on Spencer-Brown's Laws of Form ^[18] .

Appendices

Plato and Aristotle

In Timaeus Plato describes chaos with its germs (traces of form, ichne ton eidon) from which something can be created. This is reminiscent of Aristotle, and Heidegger believes that Aristotle developed concepts such as dynamis and genos independently and that Plato reacts to them in his late writings ^[19] . But there is a contrast between the two: While for Aristotle nature strives out of itself to something (entelecheia), for Plato it requires the master builder (demiurge), who forms the still chaotic nature from the outside ^[20].

The place with Plato reads:

"At first, they were all without reason and measure. But when the world began to get into order, fire and water and earth and air had only certain faint traces of themselves, and were altogether such as everything might be expected to be in the absence of God ; this, I say, was their nature at that time, and God fashioned them by form and number." ^[21]

Gödel's Critique of Hegel's Beginning of the Science of Logic

Gödel studied Hegel intensively, but only a few of his comments on Hegel are known. In some respects Hegel even went further than Leibniz for him ^[22], but he criticized him for lack of clarity, a lack of meaningful predication and the inability to develop complex concepts from simple concepts ^[23]. Apart from these formal and methodological questions, he criticized how Hegel begins the Science of Logic.

For Gödel, the beginning is the transition from the possible to the real and not becoming as a synthesis of being and nothing. The possible also has existence for him. Insofar he agrees with Hegel to begin with being. But he thinks it is wrong to speak of nothingness instead of the possible like Hegel. The possible is more than nothing. According to his impression, Hegel, contrary to the own inner logic of his thinking, implicitly introduced the concept of becoming far too early with the concept of time.

"9.4.10 Independently of Hegel's primitive terms [which begin with being, nothing and becoming], the process is not in time nor an analogy with history. It is right to begin with being, because we have to have something to talk about. But becoming should not come immediately after being and nothing: this is taking time too seriously. It is very clear that possibility is the synthesis between being and nothing. It is an an essential an natural definition of possibility to take it as the synthesis of being and nothing. - Possibility is a weakened form of being." ^[24]

In contrast, Gödel understands his philosophy as a monadology in the sense of Leibniz. The monads are located in an area of the possible before space and time, and with them one can explain what appears in space and time. Hegel takes the opposite approach. He begins with reality and introduces possibility as the result of a reflection that recognizes the possibility contained in reality ^[25].

In contrast to Russell, Wittgenstein or Popper, however, Gödel shares Hegel's "dealing with contradictions. It is simply a systematic way of obtaining new concepts" ^[24].

Gödel's ontological proof of God

"In 1972 Gödel told me that his study of Leibniz had no influence on his own work except in the case of his ontological proof." ^[26]

Gödel of course was aware of Kant's argument that the ontological proof of God fails to accept existence as a predicate like all other predicates. I can describe a coin with as many predicates as I want (what metal it is made of, how big, old, heavy, what color it is, and so on), but it does not follow from this whether it is in my possession (whether it exists for me).

As far as I understand Gödel's proof of God, Kant's criticism does not apply to him, because Kant equates predicates and categories. But the ontological proof of God does not conclude from categories of the existing (size, quality, relation and so on) on God's existence, but from value characteristics. If there is something good, and if everything can be evaluated according to how good it is, then there must be a nature of good by which the good can be measured. The proof of existence concludes from the existence of something good to the existence of a nature of good. (In my opinion, Gödel is very close to Hegel with this thought. Hegel understands "the predicates good, bad, true, beautiful, right etc." as judgments of the concept ^[27] and develops from them in several steps the objectivity underlying them, with which he wants to refute Kant's criticism of God's proofs ^[28]).

Gödel speaks of values as "positive" and their negation as "negative". Central to his proof is an axiom that something can only have a value if this value corresponds to its nature ("because it follows from the nature of the property" ^[29]).

Humanly speaking, something has no value if, contrary to its own nature, it only seems to be good, can only present itself as good.

It may be left open whether this can be recognized as evidence or whether it is a statement of faith on its part. But it seems clear to me that an axiom of this kind is necessary in order to find the group of axioms of set theory sought by Gödel and their fertility. So it seems to me no coincidence that in the same year 1970 Gödel wrote both his final proof of God and the manuscripts for the continuum hypothesis.

Sigles

CW = Kurt Gödel: Collected Works, 5 Bd., Oxford 1986, 1990, 1995, 2003, 2003

HW: = Georg Wilhelm Friedrich Hegel: Werke in 20 Bänden. Auf der Grundlage der Werke von 1832-1845 neu ediert. Red. E. Moldenhauer und K. M. Michel. Frankfurt/M. 1969-1971

KrV (A and B) = Immanuel Kant: Kritik der reinen Vernunft, 1st and 2nd Edition, Riga 1781 and 1787

Wang = Hao Wang: A logical Journey, from Gödel to Philosophy, Cambride, Massachusetts 1996

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Notes

2016, 2020