Leibniz 3: Binaries in Contradiction and Extrapolation

Fred Williams 'Lysterfield Landscape' 1966

If I hypothesise that a binary code is a rule born of a contradiction, what are the methods by which an hypothesis of the origins of the binary code can be constructed? In Leibniz’s method of universal science there is to be found the requirement for a contradiction or opposition to be built in to every hypothesis. Logically, the symbol of the contradiction in the hypothesis could take a binary form. 

Objective judgement of the methodological quality of the hypothesis is based on detection of a contradiction if the hypothesis is to encompass simplicity, understandability, multiplicity of applications, and minimal assumptions.

Leibniz:  “As long as we have only a nominal definition, we cannot be sure of the consequences drawn from it, for if it concealed some contradiction or impossibility, we could draw conflicting conclusions.”

Leibniz:  “Demonstration is reasoning by which some proposition is made certain. This is achieved whenever it is shown that the proposition necessarily follows from certain suppositions (which are assumed to be certain). By necessarily I mean in such a way that its contrary implies a contradiction.”

Leibniz:  “It is obvious that all necessary propositions, or propositions which have eternal truth, are virtual identities and can be demonstrated or reduced to primary truths by ideas or definitions alone, that is, by the analysis of terms, so that it is made clear that their opposite implies a contradiction and conflicts with some identity or primary truth.”

The initial measure of whether an idea is true is that the concept is possible, whereas if the idea contains a contradiction it is almost certain that the idea is false. The only way that the central assertion made in the hypothesis can demonstrate that it is not a contradiction is by including its opposite in order to allow verification. 

Single concepts are not believable until they display general validity by illustrating their intrinsic opposites. If a hypothesis is to be believed it cannot conceal the contradiction. The opposite in the concept is marked by its sign. The symbol is of utility and the claim is demonstrably truer if its opposite is within the sign, revealing the contradiction that would otherwise make it a false claim. 

The explanatory power of a hypothesis with regard to multiple phenomena is increased by inclusion of circumstances that would conflict with the hypothesis, and the greater the number of circumstances that are explained by a hypothesis, the more probable and truthful it is likely to be. When the hypothesis contains a contradiction without itself expressing a contradiction this has further implications in the social sciences. The fact that there is an opposition or a contradiction suggests a choice. If I hypothesise that ‘good governance is impersonal governance’, or that ‘effective and sustainable governance is impersonal governance’, I am implying a choice between personal and impersonal.

The rule or the code draws attention not only to the fact of there being a contradiction but also to the choice that is made in pursuit of truth. In this sense the contradiction is a code. The rule to which the code refers is a choice, the existence of the contradiction is the claim made in the hypothesis, and the code is the contradiction. Here we have two opposing forms of action expressed in unison so that the choice of going one way or the other will be instantly perceived. If an impersonal action is to have cognitive or prescriptive value it has to be seen as a contradiction. Impersonal has no meaning in and of itself. It can only be understood by referring to its opposite. 

While fulfilling the binary requirement of hypothetical reasoning (that it should contain its opposite) I also reveal the first feature of ‘code’ in governance, namely the instruction to reject one thing, and knowing why we reject it, because the opposite of the particular form is the universal form. 

I have a binary code expressed in a binary hypothesis. That is perfectly natural and logical. When the binary code represents a synthesis of two composites, it is compound coding.

It follows that a successful binary formulation would be the sign of a successful reduction of composite-compound knowledge into primitive concepts through the process of synthesis. As we saw earlier, the art of synthesis is like the art of the ideal type — defining, resolving, reducing, connecting, and combining phenomena. Each side of the formulation is a unity of observations and possibilities. In social science the attempt to explain the evolution of society and its governance shows — in the demonstration — why the thing was necessary and why its opposite was denied.

Leibniz:  “If we call something necessary, we deny the possibility of its opposite. It therefore suffices to demonstrate the necessary connections between things and their consequences in this way: by deducing them from a clear and distinct intuition (that is, from a definition when this intuition is expressed in words), through a continuous series of definitions which imply them; that is, through a demonstration.”

Leibniz’s writings can offer further justifications the utilisation of binary concepts in a social science context. Almost the only non-mathematical use Leibniz made of his discovery of binary numbering or dyadic notation was the following ex nihilo theological observation:

Leibniz:  “Everyone agrees that the decimal progression is arbitrary, so that sometimes other numbering systems are employed. This fact caused me to think of the dyadic … which is the simplest and most natural. I decided at the outset that it would have no more than two characters: 0 and 1 … Some people have admired in it the surprising analogy between the origin of all numbers out of 1 and 0 and the origin of all things from God and Nothing.”

If God was the origin of all things, so ex nihilo, one emerges from nothing. Leibniz’s own instruction that “a concept from which nothing can be removed must be simple and primitive” encouraged him to conceive God and nothingness as the only perfectly primitive concepts. 

His other-worldly God hypothesis was a product of its times. Today it is indemonstrable and unscientific. Social science has no conception of anything prior to the origin of species and of the group and society. Institutions of governance are the product of social evolution, not of divine creation. 

Yet analogies of the origin of all numbers out of 1 and 0, or the origin of 1 from 0, or of something out of nothing, are potentially productive for extrapolating to other contexts. The early modern rejection of divinely-ordained rulership and moral order was a factor in justifications that propelled the transition from personal to impersonal order. During this period of history there really was no earthly precedent for depersonalisation and separation of powers. The primitive concept in the personal-impersonal hypothesis thus does possess some ‘ex nihilo’ justification. 

In his lifelong project to discover the foundations of a general science or ‘characteristica universalis’ Leibniz hoped to turn complex arguments about political rule and universal jurisprudence into simple calculations. Until the end of his life he believed it possible to do.

Leibniz:  “Indeed, justice follows certain rules of equality and of proportion which are no less founded in the immutable nature of things, and in the divine areas, than principles of arithmetic and geometry”. 

The implication is that some of the common disputes in law can be decided automatically, applying mathematical criteria with impartiality. Leibniz invented a famous calculating machine, and he is often referred to in writings on the history of computing and artificial intelligence. Although his ‘characteristica universalis’ did not succeed in reducing universal jurisprudence to numerical calculations it can be credibly claimed that Leibniz foretold the potential for robot judges with artificial intelligence. 

Leibniz came close to formulating a personal-impersonal hypothesis with a binary distinction. 

Leibniz:  “Friendship is particular and charity is universal.”

Leibniz:  “Charity is nothing else than a general friendship which extends to all, but with distinction, for it must be regulated by justice according to the degrees of perfection which can be found or introduced in objects.”

Since there is no efficient or legitimate functional role for particularism in governance interactions, ‘general friendship’ is not a bad way of describing equal justice for all. The contradiction between discretion and general rules is one that Leibniz did recognise. It is not so fanciful to imagine circumstances when there occurs a ‘universalistic’ ex nihilio transition from one to the other. Ex nihilo could be a useful device to include in the toolbox of social science where a binary theory of causation is called for. 

Leibniz: “To set up a hypothesis or to explain the method of production is merely to demonstrate the possibility of a thing, and this is useful even though the thing in question often has not been generated in that way.”

I have identified two ways in which a formula of binaries can be adapted for use in a Leibniz-standard hypothesis. Symbolic representation of complex reality presented in a binary reduction of the contradiction is consistent with the Leibniz’s method for systematising knowledge. And a binary concept of origins may depict societal transitions when new things replace old things, where a contradiction, in a combination, has guided the system pressure to choose.

Michael G. Heller  ©2021