Explanations come in discrete units, logically minimum quanta. It is logically impossible for the situation to be otherwise. We can reason about continua of various different kinds (the “continuum” of the Real numbers being a prominent example, although it is to be noted that within that branch of formal logic known as “model theory,” there are examples of continua that are “more continuous” than even the Real numbers.) But we cannot reason “in” a continuum. Our ideas may have vague boundaries, but they are still unitary quanta, or at least collections of such quanta. Our concepts are even more sharply defined. We assemble these units into larger structures that become arguments (in the good, philosophical sense) and, ideally, explanations. But a continuum gives us nothing to work with. Like trying to nail mercury to the wall, every time we attempt to grasp it, it slips around and away in out grasp, so that either we (1) end up speaking about the continuum itself as a whole, at which point the continuum qua whole has become our quantum, (2) we isolate individual points on the continuum, and these become our quanta as we extrapolate connections amongst them, (3) or, alternatively, we end up spouting nothing but nonsense.
I’ve touched on this subject before. But rather than making coy suggestions in the final paragraph as a rhetorical flourish, I think it time I spoke to the subject more directly. As is often the case, I’ll barely be able to gloss the topic in this post. But, of course, the whole purpose of a blog post is to provide a small quantum of ideas that might lead interested readers off in interesting directions.
To return briefly to questions I opened above about the continuum: the analogy I used above was talking about the continuum or being in the continuum. To be “in” the continuum, in the sense of not reducing it to explicit relations between logically determinable quanta, would be (in effect) to be inside a (logical?) space without any form of reference. This is because any such reference would itself be the kind of minimal unit, a quantum of explanation, to and from which the whole rest of the space could be understood. But the hypothesis is that no such quantum exists or is available; there is only the continuum – and that continuum is not taken from the “outside,” as its own sort of quantum, but engaged from the “inside,” purely in itself as a continuum, lacking all unitary quanta which might anchor the logical possibilities of explanation. So, lacking the kinds of reference that quanta of explanations alone could provide, one could not possibly determine if one was moving or still; if one was in one place, all places, or no place at all.
Perhaps this is an important, even essential, characteristic of mystical experiences? Most writers on the subject insist that their words and descriptions are inherently false, shoe-horning qualitatively different forms of experiences into the fragmented and discontinuous structures of language and concepts. Insofar, we must then agree with William James, not only on the phenomenological grounds he examines in his Varieties of Religious Experience, but on logical grounds of explanation itself, that the mystic’s experience carries no probative weight for anyone other than the mystic. This is not to deny the phenomenological fact of the mystic’s experience. But explanation is, by necessity, logical and therefore public. The idea of “private explanation” is sheer nonsense on stilts, an oxymoron that is really just an example of barnyard noises, emotional outbursts without rational content, regardless of their superficial grammatical structure.
Now, I just said that explanations are public. This claim itself was introduced without explanation, but that is typically how things work: one makes an assertion and then works backwards from it to its logical underpinnings. In this instance, in order to qualify as an explanation, a claim or an assertion must connect, via legitimate modes of inquiry (i.e., “logically”) to the rest of the world so as to qualify an an objective truth – or, at least, as something that has a chance of being an objective truth. But the objective world – that world about which we make objective claims – is “out there,” and publicly shared with and by all of us. The subjective world is real, but to the extent that it is genuinely subjective, it is also limited to just and only the particular subject’s experience. The objective world does not exhaust all that there is, but it does exhaust all that can be explained within the bounds of logically justifiable inquiry.
Now, as stated so far, the ideas of the “subjective” and the “objective” represent ideal poles on a spectrum, with humanly possible experience spread out somewhere in between these (frankly inaccessible) ideals. Mystics are unhappy with their descriptions, and yet their descriptions carry weight. Because regardless of how “utterly subjective” those experiences seemed, they weren’t entirely so; the fact that they were experienced by humans who could frame those experiences (even if only badly) in language proves they weren’t completely buried in an unqunatizable continuum.
A logical quantum is not to be mistaken for a physical “thing.” For one thing, qua quantum, the logical unit need not possess any metrical characteristics at all. By this I mean to say that such a quantum is neither large nor small, since these considerations are themselves scarcely the primitive units of explanation. Thus, measure concepts like “large” or “small,” “greater than” or “lesser than,” make little or no appearance in studies such as topology, affine geometry, mereology and its synthetic cousin mereotopology. Alfred North Whitehead built a great deal of his Natural Philosophy, and vital elements from his Metaphysics, out of just such primitive logical quanta. And it is worth emphasizing in this regard that what Whitehead was developing in these and other works was not a theory of “stuff;” he is not talking about what the bits and pieces of reality are, in and of themselves; he is discussing the logical foundations of explanation itself.
Indeed, are we required to raise the question of ontology (“what there is”) when we talk about the logical quanta that are structurally necessary in constructing an explanation? Can we talk about the structure of explanation without reducing that talk to some discourse about how this “stuff” makes things happen with that “stuff”? The answer is “yes,” and the example we have to hand of why the answer is “yes” is Whitehead himself. It is a common misreading of Process and Reality, to view Whitehead’s great metaphysical work as an exercise in talking about the “stuff” of which reality is made. But a careful reading will reveal that, in fact, Whitehead is talking about the structures and relational characters which are the organizing features of the world, regardless of which sorts of “stuff” our particular inquiry might be focusing on. (My uses of the terms “ontology” and “metaphysics” in the preceding are somewhat non-standard, but not at all unreasonable.)
Notice that the above does not deny the ontological reality of a/the continuum, only that the continuum – qua continuum – serves no cognitively legitimate role in explanation. There are, of course, other functions – understanding, experience, and so on – which are every bit as much a part of human reality, and intensely more intimate a part of human reality, as/than “cognitively legitimate explanation.” But there are limits to what we can do with these other functions; limits, as well, to what we might want to do with them. Thus, a collection of theorems (more or less independently developed by Löwenheim and Skolem) demonstrate that one does not need a continuous theory to model a continuum. A model, when it is a good model, is the structural framework of explanation.
So, here’s the thing: what if there is a core of principles (and here I hearken back explicitly to my oft repeated quadrivium of “logic, principles, evidence, and facts”) which capture the central, operative concepts of any broadly conceived attempt at explanation? These principles would be just below the level of “logic,” and hence would qualify – in the above structural/relational sense – as metaphysical in character. Such a metaphysics would then characterize the quantum of explanation.