What is an axiom?

1. In De Corpore, Hobbes writes: ‘A primary proposition is one in which the predicate is a name which explains the subject by means of a number of names.’ The example immediately given is the primary proposition: ‘A human is a rational, animated body’, which progressively reveals multiple determinations or aspects of the single given subject, in this case a human. This, Hobbes tells us, is what it is to be a definition, and the sense of such claims constitute truths that are indemonstrable in principle. He goes on: ‘There are some who add certain other propositions, which they call “primary” or “principles”, namely axioms or common notions. However, they are not really principles, because they can be proved (even if their self-evidence means that they do not need to be proved).’ (Hobbes, De Corpore, §3.9) And yet it is not clear how an axiom or common notion would be proven, since they are neither propositions nor definitions.

2. Axiom is one of the categories of thought which, like ‘definition’ and ‘proof’, is so ubiquitous and silent that it is almost impossible to thematize directly. As in any effort to define definition or to demonstrate the nature of proof, the attempt at thinking the axiom is bound to an unavoidable circularity. We say that any movement of thought is conditioned by and presupposes the operation of an axiom. As such, to define or determine what the nature of axiom is in itself will circularly involve the use of axioms in general, and will rely on the deployment of at least one axiom in particular.

3. Axiom names a rule or protocol by which we pass from one thought to another and consider this passage legitimate. An axiom may either be explicitly stated or recognized, or implicitly operative without being an object of conscious reflection. In its first valence, where axiom names the movement of a thought, it is universally necessary as a generic precondition for any thinking; in its second valence, where axiom names the process of sanctioning such a passage or transition as legitimate, its validity or authority in providing such a sanction cannot be demonstrated. We would like to know that our axioms are well-grounded, that they are correct, adequate, or valid such that we are justified in relying on and deploying them. But precisely this is what cannot be demonstrated or proven, since any demonstration or proof relies on at least one axiom, as does any given conception of adequacy and justification.

4. Axioms are not however indifferent; concretely distinct lines of thought follow from different axiomatic systems. We call the usage or deployment of a particular axiom its being posited. As they do not admit of demonstrative justification, they demand a different kind of evaluation. Axioms need to be tested. Once made the object of explicit philosophical reflection, axioms open onto the domain of hypothetical reason. The question is no longer: what is the best axiom? Or, what are the necessary features of a legitimate axiom? But rather: what this axiom let me do? What does an axiom or combination of axioms make possible? Axiomatics is the space in which the unassailable necessity of thought converges with the existential imperative to experiment. It shrouds the clarity and distinctness of a priori demonstrative knowledge in the mist of hypothesis, but the absolute knowledge thereby dissipated is compensated for by the open circuit of conceptual creation and the joys of exploring each new space in thought.

Place and Expression

This is a response I gave to a paper entitled ‘Leibniz on Place’ by Jen Nguyen at DePaul’s philosophy graduate conference yesterday, 2.11.2017.

Nguyen’s paper raises a fascinating series of issues in the context of Leibniz’s metaphysics, which to my mind come down to this central question: where is a place? In some ways, as she notes, Leibniz’s view on place is commonsensical enough: a place will turn out to be a point of view on the world; this is its intrinsic determination, as opposed to the purely extrinsic or formal definition, according to which a place is a set of coordinates in abstract three-dimensional space. And if this squares with common sense, it’s due to a strange sort of phenomenological intuition, rather like how, to borrow from Wilfrid Sellars’ terminology, we sense an incompleteness in the ‘scientific image’ of the world and supplement it with the ‘manifest image’. We want to say: no, this place, this room, is not defined by the amount of space, the quantitative distance between the walls, the placement of the windows, the positions of these tables and chairs; rather it is a matter of the way we perceive it, what it opens up for us, how we are determined within it, a question of perspective and orientation, affect and delimitation: it is how we express it.

But if things get off to a sensible enough start, they also get weird very quickly. Leibniz never leaves a thought half developed; he will always follow it to the end, however unintuitive and bizarre. For, as Nguyen argues, the central question here becomes: what are the metaphysical foundations of place? And to answer this, it won’t be enough to jettison the definition of place as a local configuration of abstract space for being merely extrinsic. We need an intrinsic, genetic and concrete definition. Leibniz always begins so innocently; we start by making a true claim. ‘You are here today.’ True. Now we ask: what is the basis, the ground for the truth of this claim? And Leibniz responds: it is in the nature of the things themselves. And this because the alternative verges on the incomprehensible: we would claim that it is true that you are here, but somehow this truth would have nothing to do with your nature or the nature of this place; that seems absurd. But if Leibniz says that true predication is based in the nature of the subject, then place becomes a particularly odd focal point through which to, if you like, test his metaphysics: for if I say, truly, that ‘you are here’, then ‘here’ becomes one of your true attributes. So that ‘crossing the Rubicon’ is a true attribute of Caesar; but, moreover, the Rubicon does not exist at all outside true attributions like this, in which it constitutes a real site for a subject’s becoming.

In fact, all attributes are evental, they involve durations and passages. This is why the standard critique of the ‘metaphysics of substance’ misses Leibniz and, incidentally, Spinoza. For Spinoza and Leibniz converge here, by way of inverted philosophical movements. According to Spinoza, there is an absolutely infinite single substance; according to Leibniz, there is an absolute infinity of distinct substances. Could they be more apparently opposed to one another? Spinoza says: the characteristics of an individual mode are all extrinsic, there are no essential qualities (those all belong to substance); Leibniz says: every individual substance’s characteristics are intrinsic, there are no accidental qualities. But something odd happens here; where there is no line any longer separating an individual’s essential from accidental, or intrinsic from extrinsic properties – a dialectic commences that implicates the whole world in each of its parts and vice versa, where monism and monadology ceaselessly pass over into one another by means of the category of expression; and Nguyen’s emphasis on the significance of this category, expression, is an aspect of her paper I appreciate greatly. For Spinoza, there is only one substance, but it only exists as expressed in its modes; for Leibniz, there is only the world, but it is made up exclusively of the monads that express it. For both, individuals are expressive of the whole, and their attributes and characteristics are all relational.

What distinguishes monads—what individuates one—is nothing other than its singular difference, its unique point of view on one and the same world. But each monad in fact expresses the whole world, everything at once. In what, then, does this difference consist? As with most things in Leibniz, it is a matter of ratio. If I perceive or express the entire world as a whole, it’s obviously not the case that I perceive or express everything clearly. In fact, the vast majority of what I perceive remains totally obscure to me. One of Leibniz’s great lessons is that these obscure perceptions are not nothing; those of my perceptions that are unconscious or obscure are still real differences that make a difference. Imagine the contrary: that the only differences that make a difference are those that are clearly expressed in consciousness, as if the world ceased to exist when I don’t look at it. Leibniz’s innovation is to suggest that when we don’t express part of the world clearly, we still always do so obscurely. Nguyen cites Leibniz in the Discourse: every substance “expresses, however confusedly, everything that happens in the universe, past, present, and future.” (Discourse on Metaphysics, §9) My unique difference, then, my singularity or individuation, is the particular ratio of clarity and obscurity with which I express the world as a whole. And what I express or perceive clearly is a matter of my body, my bodily relations of proximity and distance, the intensity and distinctness of what strikes me in these embodied situations.

So let’s go back to the river Rubicon. It is the locus or site of a crossing, an evental attribute which is essential to Caesar. It is a place, but not indifferently or abstractly, as a piece of three-dimensional space (as in the model of Newton’s space as absolute container). Rather, if it is a place, it is because it is where an event transpires, in the strange perfect future anterior mood of all truths understood by God: Caesar will always have crossed the Rubicon. So where is it? In some way, since the claim is analytically true, we have to say that it’s not just the event of crossing it but the river Rubicon itself that is virtually contained in the complete concept of Caesar. But not just Caesar: as a real place where events transpire, it is contained no less in the complete concepts of all those who ever will have crossed it, bathed in it, drank from it, even heard about it or just reflected on it. To follow Nguyen’s example from Moby Dick, even falling silent in the cathedral is an event, a manner in which the place of the cathedral expresses itself in the conduct of the wayward sailors. But since each monad expresses the whole world, we have to go with Leibniz to the limit: the Rubicon is part of the complete concept of every single person who ever will have been part of the world; it’s just that, for the most part, its mode of inclusion in all these minds is very obscure and minimal, a difference that makes almost no difference, where the difference made is unassignably small or imperceptible.

We can thus say that a place just is the spatiotemporal locus of a becoming or an event, on which each monad has a unique perspective or point of view, a unique distribution of expressive clarity and obscurity. It is a part of the world in which all singularities become what they are. A place exists only in the perception of a monad, or as a virtual part of a complete concept, but it is also more than this, since no single concept or mind expresses it in the full richness of its real articulation; when the world just is the totality of monads that are points of view on it, each finite place becomes an infinity mirror, a site of infinite expression. And for each place the question is then transformed: how can we express it more clearly, how do we diminish the proportional obscurity of our understanding, the passivity of our bodies and minds? How do we become active—not in general, not indifferently, but concretely and determinately active in each particular place? How do we open ourselves to the texture and rhythm of a place? Or, under what conditions does a place enter into its own becoming, which is expressed in and through us? These are the kinds of questions that Nguyen’s paper raises, and I am grateful for the opportunity to think them together with you that she has provided us here today.