Logic and Space

Logic and Space

There is a gleam in the eye of this metaphysician, or a glimmer of a gleam. Might logic be derivable from space? Then we would have a metaphysics based solely on space (assuming time can be regarded as a dimension of space).[1] By “logic” I don’t just mean the logical systems studied in university philosophy departments and elsewhere—propositional and predicate calculus (or even modal logic, tense logic, and the like). I mean the entire realm of what we call logical necessity (aka metaphysical necessity): anything consisting of necessary structure or necessary relations. So, I mean to be including, especially, geometry: and not just the academic subject but the actual form of space—its structure. Logic as it is typically studied today is really the logic of quantifiers and truth-functional connectives; I am talking about the logic of space (as we can speak of the logic of time, i.e., tense logic, or the logic of obligation, i.e., deontic logic)—necessary a prioritruths of space. So, my question is whether logic in this broad sense can be derived from the nature of space. Is space the ultimate cradle of necessity? More ambitiously: is space everything?

What are the basic formal properties of space? They are not difficult to discern: finite parts, infinite extension, inclusion, adjacency, impalpability, aggregation. Space is made up of regions that aggregate into adjacent spaces and form inclusive spaces; it is infinite as to extent and fine structure. We can view space as what is studied in geometry in the manner of Euclid: necessary or logical truths about spatial figures. This logical study is one of the earliest forms of logic as a human intellectual discipline—axioms, theorems, universal generalizations. The logic of reasoning is another form of logic in the broad sense, as is arithmetic (the logic of numbers), as is metaphysics and philosophy generally (the logic of reality as a whole). The current proposal, then, is that the logic of space is prior to all other forms of logic. How does this work? As follows: space is a logical structure; it consists of parts necessarily linked—fixed, unchangeable, discoverable a priori. You can infer that other regions exist from the existence of a given region, because one region entails others—and so on ad infinitum. Nothing can be (wholly) in one place and also in another; nothing can both be in a place and not be in it; everything is either in a place or not in that place. We speak naturally of logical space; possible worlds are laid out in logical space; propositions are conceived quasi-spatially. We think of necessity spatially—as existing everywhere, of one thing being included in another, of one thing following from another (as if tracking it through space). We implicitly recognize the affinity. Moreover, both space and logic are abstract not concrete, a matter of the intellect. When the child begins to grasp the structure of space, particularly its extent, this is mirrored in his grasp of the infinite power of logic—its ability to generate infinities. Every proposition entails and is entailed by an infinity of propositions, as every part of space entails an infinity of other parts of space. Logic, like space, is vast: there is the endless successor relation in mathematics and the endless adjacency relation in space. The necessities of space are hard necessities, like the necessities of logic. Logic is implicit in space; we don’t first have a logically neutral space to which we add necessities. Logic is already present in space. Space has a logical (necessary) nature—parts standing in logical relations to each other. Since material objects are themselves products of space, any logical features they possess derive from those of space. This isn’t to say that logic is reducible to physics; rather, physics (the theory of material objects) is up to its neck in logic—in natural necessities. Space and logic are intertwined; thus, space has the resources from which what we call logic can be derived. Predicate calculus is an offshoot of space, de re and originally. Logic and space are coevals. Space is pregnant with logic. Therefore, we don’t need to supplement it with anything extraneous in order for logical necessity to exist. We might say: the world is the totality of spatial facts. All reality is preceded by and based upon an antecedent spatial reality (compare “all ideas derive from antecedent impressions”). This doctrine deserves to be called spatialism. It is intended as a metaphysical or ontological doctrine not an epistemological one, but there are obvious epistemological consequences of it; we may therefore speak of epistemological spatialism. It contrasts with both materialism and idealism, though it is closer to materialism (material objects are not fundamental in spatialism). All you need is space. Pan-spatialism. To be is to be spatial. Logical space is a by-product of physical space. Cue the gleam.

How do we get from spatialism about logic to standard metaphysical necessities and analytic necessities? Quite easily: the former concern necessities of material objects (progeny of space) and the latter arise from meaning and language that depend on human beings and their brains (minds). For example, the necessity of “bachelors are unmarried males” has its origins in necessities of space; indeed, we speak of inclusion in both cases (the meaning of “bachelor” contains the meaning of “male”). Meaning takes its rise ultimately from space, like everything else, and its necessities reflect the necessities of space—the most basic necessities of all. Space is woven into everything, because space is the foundation of everything. No space, no nothing; with space everything. All is vacuum, as a pre-Socratic might put it. Because space is not pure nothingness; space is an existent thing with real potential—a living thing, we might say. It has a nature, a real essence, a mode of being. It preceded the big bang, cosmologically, and underwent a transformation (this is the cosmology of metaphysical spatialism). It may not have looked much like the space we see around us every day (it had no matter in it to start with—no particles, no solid objects). But we can designate it now without knowing anything much about it then (“that stuff”). It had the seeds of everything built into it—matter, mind, and logic. The religious-minded might well identify this amazing stuff with God; the secular-minded will see just a mysterious natural substance of unknown origin. The important point is that it has the potential to generate all that we see today, according to the spatialist creed. Can that creed be verified? Probably not. Can it be falsified? Not in any obvious way. But it can be a gleam in the metaphysician’s eye, or the glimmer of a gleam. On purely aesthetic grounds, it scores highly.[2]

[1] See my “Space, Time, and Logic”. The present paper goes for an even more exiguous foundation. The view might be called “non-reductive spatialism”.

[2] Like all paradigm shifts, this one is hard to take in, to comprehend. The intuition behind it, Kantian in spirit, is that space is the form of everything intelligible (and unintelligible): material objects, minds, and logical necessities. In the beginning was the place. What other kind of being could there be? Even the abstract partakes of a spatial flavor—numbers, propositions, platonic forms. This is the form of all intentionality, and all reality. Not matter, not mind: these are but aspects of the spatial. If it helps, think of it in evolutionary terms: all life has its origins in a single life-form, modified and yet preserved. Well, space was the original life-form of the cosmos, modified yet preserved. Logic itself is an expression of this basic reality. The Force, as invoked in Star Wars, is really space in its most primitive and elusive form—what lies behind everything, the source of all potency (and poetry). It is what mysticism is gesturing at, however ineptly. Consciousness itself is an expression of space (the real essence thereof). The metaphysical spatialist is simply trying to put the pieces together into one big picture, a unifying reality. If he speaks somewhat in riddles and paradoxes, we must forgive him; he is only trying to make sense of it all. Atomism once sounded speculative and bizarre but is now a commonplace; maybe the same will one day be true of spatialism. Sober truth can arrive mysteriously clothed.