Leibniz 2: Symbols of Synthesis in Intuitions


Fred Williams not titled [Lysterfield landscape] 1966-1967

While testing the hypothesis I establish with a high degree of probability that impersonality was intuited at its origin by governance actors who had not necessarily followed a scientific process of explaining or defining the concept. Is this possible? Wittgenstein thought it was, through repeated encounters with a tacit rule. Leibniz’s method of hypothesising does not exclude the possibility that these actors can have conceptualised impersonality symbolically, as a discovery. To me it looks like a simple deduction.

Leibniz: “Since the individual concept of each person includes once and for all everything which can ever happen to him, one sees in it a priori proofs or reasons for the truths of each event and why one has happened rather than another … And we are now maintaining that everything that happens to some person is already contained virtually in his nature or concept, just as the properties of the circle are contained in its definition. Thus the difficulty still subsists. To answer it squarely, I say that there are two kinds of connection or sequence. One is absolutely necessary, for its contrary implies a contradiction, and this deductive connection occurs in eternal truths like those of geometry. The other is necessary only ex hypothesi, and by accident, so to speak, and this connection is contingent in itself when its contrary implies no contradiction.”

Leibniz: “It is most desirable to demonstrate truths which are relatively simple, but which the prejudiced opinions of men keep them from admitting, by means of simpler ones.”

Leibniz invented the infinitesimal calculus concurrently with Isaac Newton. He also invented the binary numbering system. Leibniz was a philosopher of science, and the calculus and dyadic notation were part of his lifelong project of systematising the sciences as a whole. Much of his writing was an effort to distinguish between universal truths, possible truths, and probable truths as understood within a “general science” that lays foundations for methods of discovery that employ symbols modelled on the signs of algebraic combinations. The methods of hypotheses are to be found in Leibniz’s unfinished project for a universal science. 

Leibniz was interested in symbols that depict composites, compounds, and combinations. All of science endeavours to create symbolic ways of resolving composites into single concepts. It is a goal of a workable hypothesis to reduce the central concept close to a primary truth or essence as though the concept were a symbol that represents the sum of its parts. Leibniz used the word ‘primitive’ to describe a concept whose meaning is so pared down that it requires no symbol because it cannot be reduced further, so that it represents itself symbolically, and can be intuited. 

Leibniz:  “Once a hypothesis or a manner of generation is found, one has a real definition from which others can also be derived, and from them those can be selected which best satisfy the other conditions, when a method of actually producing the thing is sought. Those real definitions are most perfect, furthermore, which are common to all the hypotheses or methods of generation and which involve the proximate cause of a thing, and from which the possibility of the thing is immediately apparent without presupposing any experiment or the demonstration of any further possibilities. In other words, those real definitions are most perfect which resolve the thing into simple primitive notions understood in themselves. Such knowledge I usually call adequate or intuitive, for, if there were any inconsistency, it would appear here at once, since no further resolution can take place.”

In order to demonstrate the probability of a hypothesis I begin with solid intelligible concepts. Like the concepts of ideal types these are a product of analysis and synthesis. Leibniz’s emphasis on the utility of simple, easily understood hypotheses is apparent in his identification of a three-step process for combining facts and ideas — analysis-synthesis-analysis — with a definite emphasis on the importance of the synthesis. 

He offers the wonderful analogy of a shop full of objects of knowledge, the compound result of many analyses, “the great wealth of observations already available to our century”. With all this knowledge we can set to work on remedies for all of the social ills. But these wares are not listed or arranged in order. While browsing through them, we waste time on trivia. In our impatience we put analysis ahead of synthesis. 

The priority, instead, should be synthesis, putting the objects in order, grouping them by their qualities, by their similarities and dissimilarities, by their familial connections, by the sequences of their discovery, and by their heuristic progressions. Abstract general formulas for truth-finding — including ideal types — are manufactured by combining data. 

The shop of knowledge is chock-full of composite-compound concepts. What we ideally aim for are synthesised primary-primitive concepts that can be represented as symbols in the hypothesis. Concepts become simple only when the composites are resolved into primitives. Symbols help us reduce complex arguments to elementary reasoning or calculation. They abbreviate the complexity of phenomena, in order that knowledge can be easily presented and remembered. 

The best hypothesis will pare down concepts until they are close to being understood in themselves, represented in primitive terms. Afterwards comes a return to analysis in order to solve a particular problem “just as if neither we nor others had discovered anything before”. Analysis is needed “for discovering the means when the thing to be discovered or the proposed end is given”. But, Leibniz emphasised, original synthesis is the only “work of permanent value”. 

Leibniz:  “I have often observed that of the great geniuses of discovery, some are more inclined to analysis, others to the art of combinations. Combination or synthesis is the better means for discovering the use or application of something, as for example, given the magnetic needle, to think of its application in the compass. Analysis, on the contrary, is best suited for discovering the means when the thing to be discovered or the proposed end is given. Analysis is rarely pure, however, for usually, when we search for the means, we come upon contrivances which have already been discovered by others.”

Leibniz:  “Synthesis is achieved when we begin from principles and run through truths in good order, thus discovering certain progressions and setting up tables, or sometimes general formulas, in which the answers to emerging questions can later be discovered. Analysis goes back to the principles in order to solve the given problems only … It is more important to establish syntheses”. 

Synthesis requires creation of representational symbols and reasoning by which the meaning of the sign is defined, so that the represented reality can be verified. The process is circular: analysis-synthesis-analysis. Synthesis creates new knowledge. New knowledge is formulated as a testable hypotheses. Only then, having identified the end, we return to the task of discovering the means. Synthesis and hypothesis enable social science to properly select objects of analysis. The role of the sign or symbol of such synthesis is important, as it is in language, numbers, and code.

Of relevance to my hypothesis is Leibniz’s argument that synthesised knowledge can be intuited, even though intuitive knowledge is rare:

Leibniz:  “When everything which enters into a definition or distinct knowledge is known distinctly, down to the primitive concepts, I call such knowledge adequate. And when my mind grasps all the primitive ingredients of a concept at once and distinctly, it possesses an intuitive knowledge. This is very rare, since for the most part human knowledge is merely confused … It is therefore only when our knowledge of confused concepts is clear, and our knowledge of distinct concepts is intuitive, that we see their whole ideas.”

Although intuitive knowledge is gained over the long run by going through the process of clearing up conceptual confusions, Leibniz offers support for the supplementary idea, which is that adequate primitive knowledge — such as of a repeatedly encountered basic rule of governance, and including jurisprudence— is intuited even though one has not oneself undertaken the synthesis. It is at least logically possible to regard this form of discovery as equivalent to the hypothesis.

Leibniz: “The conjectural method a priori proceeds by hypotheses, assuming certain causes, perhaps, without proof, and showing that the things which now happen would follow from these assumptions. A hypothesis of this kind is like the key to a cryptograph, and the simpler it is, and the greater the number of events that can be explained by it, the more probable it is.”

Leibniz argued that some things are so primitive they can only be known intuitively:

Leibniz:  “When every ingredient that enters into a distinct concept is itself known distinctly, or when analysis is carried through to the end, knowledge is adequate … Yet for the most part, especially in a longer analysis, we do not intuit the entire nature of the subject matter at once but make use of signs instead of things, though we usually omit the explanation of these signs in any actually present thought for the sake of brevity, knowing or believing that we have the power to do it …  Such thinking I usually call blind or symbolic; we use it in algebra and in arithmetic, and indeed almost everywhere. When a concept is very complex, we certainly cannot think simultaneously of all the concepts which compose it. But when this is possible, or at least insofar as it is possible, I call the knowledge intuitive. There is no other knowledge than intuitive of a distinct primitive concept.”

I go further than Leibniz in saying that the intuition of a primitive, primary feature of the system environment is a ‘scientific’ mental faculty potentially possessed by anyone engaged in processes of cognition and problem-solving during active experiments with progressively more complex forms of governance. I do not refer in such cases to rational laws or explicit regulations, but rather to an instinct or intuition about what is required for the functioning and maintenance of complex systems. My hypothesis is an adaption of Leibniz’s arguments and fits with inferences drawn about governance code from Wittgenstein’s theories of rule-following

Michael G. Heller  ©2021

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