at least equally difficult to resolve. Could the universe be at once infinite and plenary? If it were entirely full of bodies, there would then be no space for motion, and it would become a frozen Parmenidean One. Yet if it were not chock-full, we would need more than microvoids internal to bodies to allow for motion. Perhaps, after all (as the Atomists held), there are empty “intervals” between bodies. But how can we determine just how big such intervals would need to be in order to make motion possible? There seems to be no way of giving a generally satisfactory answer to this last question. Maybe because of this difficulty, the very idea of interval (diastema) was expanded by the Neoplatonists to become extension, ultimately the “spatial extension” posited by Philoponus. Yet this latter idea, especially under the guise of “cosmical extension,” returns us to the deeply perplexing issue of whether the cosmos itself has a place. A place for the cosmos may be asserted—as it is by Simplicius—but then we must ask: a place where? Is its place a place in the universe at large, that is, in a space that exceeds the world-place itself? And is such a space finite or infinite?
By this circuitous route, we return once again to Archytas, who is reported to have posed the following conundrum.
If I came to be at the edge, for example at the heaven of the fixed stars, could I stretch my hand or my staff outside, or not? That I should not stretch it out would be absurd (atopos), but if I do stretch it out, what is outside will be either body or place. . . . If it is always something different into which the staff is stretched, it will clearly be something infinite.122
Alexander of Aphrodisias claimed that this thought experiment comes to naught, since what is outside the cosmos is nothing at all, not even a void.
He will not stretch out his hand; he will be prevented, but prevented not as they say by some obstacle bordering the universe (to pan) on the outside, but rather by there being nothing (to meden einai). For how can anyone stretch something, but stretch it into nothing? How can the thing come to be in what does not even exist?123
Simplicius insists similarly that Archytas’s conundrum is question-begging: “In imagination it assumes in advance what it seeks to prove, that there is something, whether empty or solid, outside the universe.”124
Despite these telling objections, Archytas’s provocative puzzle kept arising in ancient and medieval debates, and it still haunts contemporary cosmological thinking. For it will always occur to the cosmologically curious to ask, what lies beyond the last boundary of the known world? If there is some thing there, then I can (at least in principle) get to this thing and even reach beyond it. If there is no thing, then there might be, not nothing (as Alexander assumes), but empty space. This observation indicates that Archytas’s exclusive alternative of “body or place” needs to be supplemented. If place is always bounded—as it is for Archytas and Aristotle alike—then it is not what we encounter when we stretch out our hand or staff beyond the final frontier of the cosmos. What such extracosmic stretching gets us into is something else, and its increasingly unrefusable name is space. This word (or its equivalent in other languages: spatium, Raum, espace, etc.) is required if we are to designate a domain that, itself unbounded, affords sufficient room for motion of all kinds, including the modest motion of a hand or staff as it reaches out tentatively beyond the world’s outer limit.
But space thus regarded is precisely what “infinite space” means—at least minimally. Infinite space is space for (motion) and space without (bounds). In its twofold character, such space brings together two of the most ancient terms in Greek philosophy, attributable to Plato and Anaximander, respectively: “room” (chōra) and “the boundless” (to apeiron). Their conjunction, which is conceptual as well as historical, suggests that if the cosmos indeed has a place, it is a place in space: space at once endlessly voluminous and boundaryless. Moreover, the world not only has a place, it is in place: it is in the very place of infinite space, occupying particular stations in the regions that make up the spatial universe. Just as Archytas’s conundrum drives us to the idea of infinite space from the known fact of the cosmos, so this same space preserves a place—indeed, innumerably many places—for the world from whose edge we are asked to stretch out our hand or staff, or (in Lucretius’s version) throw a long javelin. The Archytian axiom abides, but only as applicable to a much larger domain than Archytas himself envisioned. To be is still to be in place, but a place that is part of an unending space.
5
The Ascent of Infinite Space
Medieval and Renaissance Speculations
God, however, is infused into the world He makes, which is placed wherever He makes it.
—Thomas Bradwardine, De causa Dei contra Pelagium
Physical objects are not in space, but these objects are spatially extended. In this way the concept of “empty space” loses its meaning.
—Albert Einstein, Relativity, the Special and the General Theory, A Popular Exposition
I
From Archytas’s challenging conundrum we can derive a more momentous question: not whether an outstretched hand or staff can reach out into something (or nothing) but whether the whole world (i.e., the physical cosmos as one entity) can move. And if the world moves, in what, into what, does it move? These questions vexed philosophers and theologians of the Middle Ages—construing this period as the entire era stretching between A.D. 600 (a date that marks the demise of Hellenistic and Neoplatonic philosophy) and A.D. 1500 (when the Renaissance was fully alive in Italy). Whichever way you answer such questions, the stakes are high. For if the world cannot move—if it is bound forever to occupy the same place, that place being coextensive with the outermost sphere, as Aristotle and Aristotelians assumed—then a surrounding space that exceeds the place of the cosmos, were such space to exist, would be idle. But if the world does move (i.e., laterally by displacement, rather than spinning in place like a top), then there must be an encompassing space in and through which to move, a space that extends beyond the discernible heavens. Once more, the issue is that of place versus space, only now on the grandest scale. Theologically considered (and everything in the Middle Ages was eventually, if not always immediately, so considered), this issue amounts to whether God has the power to create and occupy space sufficient to surpass the place of the cosmos—in short, space unbounded by any particular cosmic constraints and thus ultimately infinite in extent.
One form this discussion took was whether God could create something possessing infinite magnitude. Aristotle, predictably, denied any such ability, since for him there was only a finite amount of matter in the universe to begin with and this could not be increased; he could entertain the idea of the indefinitely small (though only in potentia), but the infinitely large was out of the question.1 Far from taking this restriction as problematic, Aristotle regarded it as a sign of the perfection of the universe: its very delimitation in size, like the confinement of the places within it, was a matter for admiration. (Of course, for Aristotle the two delimitations are closely related, given that place is quantitatively determined on his own analysis: questions of place are matters of magnitude, and vice versa.) But Aristotle’s espousal of this double finitude left a particularly puzzling question: Does the outermost sphere (which, as encompassing all lesser spheres, provides a place for them) itself have a place! Or is it an unplaced placer, not entirely unlike the Unmoved Mover posited at its periphery? Aristotle himself hinted at—and his Hellenistic commentator Themistius developed in the fourth century A.D.—the idea that the moving parts of this super-sphere have places, for these parts change place as they move in a perfectly circular fashion. But what of the final sphere itself? Does it have its own proper place? Aristotle was inclined to think not: “The heavens,” he maintained, “are not, as a whole, somewhere or in some place.”2 Is this to say that the heavens are nowhere? Averroës (ca. 1126—ca. 1198) gave an ingenious analysis of this paradoxical situation. According to “the Commentator,” the outermost sphere has a place, not in relation to anything more encompassing (there is not anything more encompassing than this sphere), but in relation to the earth as the fixed center of all the celestial spheres. The earth is the immobile body at the center that provides place to the otherwise unplaced outer sphere. Roger Bacon (ca. 1220-1292), building on Averroes, distinguished between “place per se”—this is