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.