Infinite Metaphysics





Infinite Metaphysics


“To see a World in a Grain of Sand

And a Heaven in a wild Flower,

Hold Infinity in the palm of your hand

And eternity in an hour.” (William Blake)


As Spinoza noted, we must not let metaphysics be shaped by the human perspective; we must maintain a resolutely objective stance. The human perspective favors the finite, mainly because of the limits of our senses—they are not geared to the infinite. We can conceive the infinite but we can’t perceive it. Thus we tend to think of the infinite as the finite stretched, extended or continued, instead of viewing the finite as the infinite condensed or truncated. We are skewed towards the finite, to the point of finding the infinite baffling. But that has nothing to do with ontological issues about the infinite—with its position in nature. Viewed objectively, nature has the infinite written into it at every turn; it hums with infinity. Space and time are infinite, both in their extension and their continuous structure. Matter may not be infinite in extension but it too has an infinitely divisible mathematical structure. Mathematics is par excellence the home of the infinite. Minds too are infinite in scope: consider the infinite potential of language and thought with their combinatory powers. Less obviously, meaning itself is infinite, as rules in general are, because it applies to indefinitely many cases. God is infinite (if there is such a being). Laws of nature are infinite, given that they extend over arbitrarily many natural events. The world is really the totality of infinities. We don’t tend to acknowledge the ubiquity of the infinite, owing to our perceptual and cognitive biases, but attention can be drawn to it: reality is steeped in the infinite, up to its neck in it.

            Blake is thus right that the infinite is written into the finite: there is always a deep structure of infinity in the most finite of things. Take that grain of sand: it is both part of an infinite spatial and temporal manifold and also infinitely divisible spatially. It does not stand apart from the infinite but incorporates the infinite into its being. Likewise we hold the infinite in the palm of our hand whenever we pick up an object, since every object is both a constituent of something infinite (space, the universe) and also a container of the infinite. What we call the finite is the infinite as it presents itself to us—a reflection or token of the infinite. Without the infinite the finite is nothing, something far less than a grain of sand. Pan-infinitism is true. When Berkeley speaks of “finite spirits” he perpetrates a falsehood: our minds may not be up to the level of God’s mind but there is plenty of infinity in the human mind both as to content and architecture (whatever may be said of animal minds). Even to speak of grains of sand as finite objects, by contrast with the infinity of the universe, is misleading, because grains are imbued with infinity too. It is not that infinity is cordoned off at the margins of things; it is woven into things.

            We tend to mystify the infinite, regarding it as somehow supernatural or awe-inspiring. We speak reverently of it, as of the divine. But really it is as natural and humdrum as anything else in nature. Nature simply is an infinite object. Only a misguided empiricism could blind us to this fact—a desire to reduce the real to the perceptible. Ontology cannot be dictated by epistemology. Even if infinity is beyond our cognitive capacities in some way, that implies nothing compromising as to its robust reality. In fact, it lies at the foundation of things. We might say that infinity is the most basic law of nature—what nature most fundamentally is. We are skewed away from it, but it is indifferent to that limitation. To a more discerning eye the universes swells with infinity, stretching out in every direction. Even if we had no conception of it at all, living in a totally finite conceptual world, it would still form the scaffolding of reality. To be infinite is to be to no more ontologically suspect than to be finite. Indeed, one can imagine a metaphysical system that regards the finite as merely the appearance of the infinite (Spinoza’s system is like this).  The infinite is a little bit like consciousness: mysterious from our human perspective but really a natural phenomenon like any other. After all, it is merely a question of size. The infinite is an ingredient of reality in good standing.

            We can raise ontological questions about infinity. How many kinds of infinity are there? Are some types of infinity abstract and some concrete? Did the big bang create infinity or did the universe already contain it? Does it play any explanatory role in the universe? Is there any possibility of defending a projective view of infinity? How is infinity related to necessity? Is there something inherently paradoxical about infinity?  Could there be a completely finite world? Are the things that are actually infinite necessarily infinite? Is all infinity really of one basic kind? Is the property of being infinite a unitary property (as opposed, say, to a family resemblance property)? Can infinity be destroyed? Can it be caused? Why does it exist? That is, we can undertake traditional metaphysics with respect to infinity. We don’t have to leave it to the mathematicians or be cowed by its majesty.         

            Blake’s lines raise an interesting question about perception. I said that we don’t perceive infinity, but Blake speaks of seeing the world in a grain of sand and holding infinity in one’s palm. What kind of seeing and holding might these be? It might just be metaphor, but I think the poet intends more. A natural suggestion is that Blake is speaking of a kind of seeing-as: we can see the grain as a world and feel something in the palm as infinite. But this strikes me as stretching the concept of seeing-as, because there is no experience of such things in the act of seeing a grain or holding a ball (say). Nor does Blake use the locution “seeing as”. What would it be like to see a grain as a world?  [1] What Blake says is that we might see a world in a grain of sand (or perhaps feel infinity in a ball). Presumably he means something like this: we can use our imagination to supplement what we are seeing or feeling. Given that the infinite is present in the finite object, we can strive to find it there. We can bring our understanding to bear on our senses in such a way that it is as if we are seeing or feeling the infinite—what we might call “seeing as-if”. Blake is stressing the proximity of the observable world to the world we are apt to think transcends it: these are not two realms cut off from each other, but closely intertwined. The infinite is woven into the fabric of observable reality not remote from it. It ought to be perceptible—even though it is not (literally). The grain of sand is an emblem of the infinite, a manifestation of it; the distinction between finite and infinite is not an ontological distinction. A finite segment of space, say, is not opposed to the infinity of space; it is a part of infinite space and it contains infinity within its boundaries (in the form of infinitesimal points). Similarly for an hour of time: finite but enmeshed in the infinite.

            It is an interesting fact that the question of the reality of the infinite is neutral among standard metaphysical systems: idealism, materialism, dualism, pluralism. All these systems are consistent with the full reality of the infinite. The infinite does not favor mind over matter or matter over mind; the world could be material and infinite, or mental and infinite, or both and infinite. Infinity has no ontological preferences. But a dualism of the finite and the infinite is quite misguided: the two are much too closely connected for that to be plausible. Each presupposes the other; they are not independent. A double-aspect theory suggests itself.



  [1] It is not as if the grain is ambiguous between a unit of sand and an entire world, like the duck-rabbit drawing, presenting one aspect and then the other; nor is there any sensory impression of a world when looking at a sand grain.

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