Ambiguity

An imaginary dialogue between Irenotheus (i.e. the German logician, mathematician, and philosopher Gottfried Leibniz) and Theophilus (the character in Leibniz’s New Essays on Human Understanding who is thought to portray the philosopher’s own views). The Essays were finished by Leibniz in 1704, but not published until 1765.

Replica (1923) of Leibniz Calculating Machine. The original dates to c.1700 and is located in the Landesbibliothek Hannover, Germany, Inv.-Nr. 56956. Photo: Deutsches Museum.

Theophilus: It is quite an astonishing mechanism that you’ve built, this ‘step-reckoner’. With this, nobody will ever need to worry about arithmetic again. And so simple! Just cylinders and rotors, one has but to turn the lever…

Irenotheus: I am glad that you see its power. But actually this is just the first and simplest step towards my actual goal.

T: Which is? Square roots and powers?

I: Perhaps... I was thinking, rather, of taking this beyond mathematics. Most of the disputes that happen in this world cannot yet be settled by arithmetical operations. But I see no impossibility with doing so in principle.

T: I cannot imagine how this could work. Even you acknowledge the mere fact that these things are beyond arithmetics…

I: But they don’t have to be. Our common words stand for worldly objects in the way that numbers stand for forms — certainly there is much still to discover in the science of both, but a good language can save our treatment of objects like it saved our treatment of mathematical forms.

T: So you suggest that we should abandon the language we learnt as children and adopt, so to speak, a better one?

I: Quite right — though of course, we only need do so when there is disagreement. However, I do not mean that we should simply make yet another language. Instead, this time we should reconstruct the alphabet of human thought — concentrating not on its pragmatic, social use, but on a complete sampling of our most fundamental concepts, using a symbol or number for each. I shall call this vocabulary ‘universal characteristic’.

T: But if we were to have a symbol for each concept of our mind, no library could possibly fit the dictionary of this language! After all, even in our imprecise language, most of our concepts are complex enough to need a few words to represent them — even for the concept of ‘universal characteristic’ you need two words…

I: Quite the opposite! By saying that I want an ‘alphabet of thought’ I expect that this language will be much shorter, just like the list of the twenty-six symbols of the alphabet is shorter than the list of meanings carried by the combinations of sounds they depict. The majority of concepts, even those that in natural languages are expressed by a single word, are not basic and can be considered as predicate concepts (i.e., concepts which serve to specify or modify other simpler concepts) — often several times.

T: I see, although this already assumes that the possibility of attaching a predicate to a basic concept is already contained in the essence of that very concept. And so this universal characteristic would have to deconstruct each everyday concept down to the most fundamental basic concept, separating it from all its predicate concepts?

I: Indeed. The whole reason why I say that this would settle disputes, like my step reckoner settles arithmetic, is that the very same formal rules by which we construct complex concepts from basic ones will also allow us to decide whether arguments are valid. One will simply be able to tell if it is valid by seeing if the conclusion is a predicate, or compound of predicates, which is embedded in the basic concepts that build the premisses. If some suggested predication never appears in the analysis of the basic concept, just like the number nine never appears in the series of the powers of two, we reject it. These rules are completely independent from the content of specific concepts, just like the general rules of mathematics. Once spelt out, these rules will constitute the ‘calculus for reasoning’, a ‘calculus ratiocinator’.

Frontispiece from Gottfried Leibniz, Dissertatio de arte combinatoria [1666] (Leipzig, 1690). Image: Deutsche Fotothek Mathematik & Kombinatorik. Licensed under Creative Commons BY-SA 3.0.

T: I am starting to see what the idea is, and indeed it would be marvellous to solve every dispute with a “let’s calculate!” — maybe even by using a machine like yours. But I confess that I think your plan is impossible, not just because of the years it would take great minds to write up both your universal characteristic and your calculus, but, fundamentally, because there is ambiguity in our compound concepts which one can never dissolve.

I: But dissolving ambiguity is all it does!

T: Those wheels would be spinning for eternity, with no result. To explain how we reduce the innumerable concepts we possess to an alphabet you appealed to the idea that simple concepts contain more complex ones about them.

I: Precisely — everything we can say about something is already embedded in our concept of it. We err, and thus disagree, when we misplace the relationship between a predicate and its subject: when we think that a predicate that is embedded in its subject, is not – or, indeed, think that a predicate that is not embedded in its subject, is.

T: I have no quarrel with this. But I doubt we actually can perform this analysis in all cases. Surely you will accept that of our concepts, some are necessary, and some are contingent — neither necessarily true nor necessarily false.

I: Of course. There are necessary truths, which would remain true in any possible world, and then there are other, contingent truths, that in a less perfect world would not be so.

T: I think your goal can be pursued as far as necessary truths are concerned — such as the truths of mathematics which your step reckoner splendidly computes. But this is because there is always a finite number of concepts that lead from our question to the primary mathematical truth upon which its answer hinges. There is always a finite number of times we have to turn the handle of your machine to get the answer. Similarly, for predicating around necessary truths, we can, in a finite number of steps, work our way back to the basic concepts and verify the validity as described before.

I: But certainly there will be a path to a necessary truth for all concepts, since they all originate from God.

T: Yes and no, I’m afraid. This path may well exist for God, who in one stroke of mind can contemplate the whole series from the basic concept to the contingent truth in question. So it is not impossible. But consider that if it truly were possible for us, we would have no reason to distinguish necessary from contingent truths. Contingency for a concept just is the fact that, when we analyse it, we go through reasons and reasons, ever deeper into the predicates of some unknown notion, indefinitely — never ending.

I: Does this not amount to saying that the causal order of contingent objects is infinite, and will never lead back to God?

T: It is important to distinguish the analysis of concepts and the causal order of objects. Of course, this distinction might seem unnecessary in mathematics. But for everything else, our concepts might be confused, ever changing, be built upon each other in the most unruly way: in a word, ambiguous. Even if the causal order of the objects to which they refer is finite, we might not find the way to mirror it, and further, would have no way to know that we have found it, even if we did. Contingent concepts spring from habits, conventions, prejudice and so on — nothing guarantees that their analysis will not include vicious loops and similar failures.

I: I see your point — the universal characteristic is by its very nature a collection of concepts, not of objects. The order is precisely that of predicates and concepts, not of causes. We must focus on the analysis of our predicates, not on what they represent. For necessary truths, such as those of mathematics, the analysis is always possible, but for the contingent ones, those I wanted to expand my calculus to include, this is uncertain.

T: Each of us can only contemplate a little portion of the indefinitely long analysis of each contingent concept. To make matters worse, it is clear that since each of us has a different perspective on the cosmos, no two people will have exactly the same portion of that analysis in mind. For example, history may change the perspective one has on a concept: think of how different our scientific view of ‘motion’ is from that of the Aristotelians who talked about it not long ago.

I: Perhaps then those wheels really would be spinning for eternity — it then seems that this ambiguity is necessarily embedded into the very contingency of contingent concepts, and any calculus would only help with questions which only involve necessary truths.

Amedeo Robiolio is a PhD Candidate in the Department of Philosophy at King’s College London.

3 August 2021.