Imagine a rocket taking us away from the earth, a thousand miles away in the first half minute, another thousand in the next quarter, another in the next eighth, and the next sixteenth and so on, so that by the end of the minute every finite distance from the earth will have been surpassed: “At the end of the minute we find ourselves an infinite distance from the earth,” according to JosÃ© Benardete (1964: Infinity: An Essay in Metaphysics, Oxford: Clarendon Press, p. 149). Accelerating beyond the speed of light is unrealistic, but not logically impossible: Benardete’s reasoning, about the nature of space, is not bad reasoning. So it is interesting that space as we naturally conceive of it does, as follows, need some such speed limit.
The meaning of ‘space’ comes, in the first instance, from our experiences of such spaces as those inside rooms, and those of the surrounding landscapes, all parts of an apparently boundless space: We can easily imagine going further and further in any direction, from any conceivable place in space. And if we try to conceive of space as having a boundary, then we naturally wonder what is on the other side of that boundary; the thought of a boundary to space is essentially the same as the thought of an impenetrable object occupying the space beyond that boundary. And it is similarly easy to imagine an object going faster and faster. But where does that get us?
Looking back towards the earth, from spatial infinity, we look through space that must have been traversed instantaneously at the end of that minute, because this space cannot be in that endless sequence of thousands of miles (if it was there, then it would be further away than any finite distance). Now, that sequence is a continuous stretch of space, with the earth at one end and some sort of endlessness facing us, and how could that endlessness possibly connect with the rest of the continuous stretch of space between us and earth? Clearly there can be no dividing line, because this sequence of elements of constant length cannot tend to one, and no final thousand miles was traversed. So, we cannot be an infinite distance from the earth.
The infinite speed that we would have ended up with is relatively reasonable, because over those thousands of miles we would have covered an infinite distance in a finite time, so that our average speed was already in that sense infinite. But it is certainly strange that averaging over speeds that were so very far from infinite should have resulted in one that was infinite. And in reality there is a light speed limit in space; and note that the necessity for some such speed limit would be compatible with similar spaces having higher speed limits. So, is such a speed limit indicated, or is there a better resolution?
Many believe that the rocket would instead vanish at the end of the minute. Vanishing at spatial infinity is not like vanishing into thin air, it is more like disappearing into the distance, cohering with intuitions about the unimportance of distant things. And if we centre coordinate axes on the earth, then the rocket will indeed have gone beyond their finite measures by the end of the minute, so that were we to think of space as all and only what is measured by such axes (as we may learn to do in mathematics) then it would indeed have vanished. But, if we centred coordinate axes on the rocket then it would instead be the earth that vanished; and it is of course absurd that the earth should vanish because it fired a rocket into outer space.
It might be objected that the rocket, not the earth, is moving away, and so the rocket, not the earth, should vanish. The spaces most familiar to us (rooms, roads, fields) do contain things that tend to be at rest (relative to the walls, the buildings, the ground) unless forced to move, so such spaces do seem to be stationary. Such was the ancient view of space; but, those spaces are parts of the surface of the earth, which spins and revolves around the Sun, relative to the Sun, and in that more modern view our familiar spaces are moving relative to the Sun. Space is neither stationary nor moving; rather, it is a space (an absence) in which objects move relative to each other.
Reference frames moving with constant speeds are privileged, so it might be objected that it is the rocket, not the earth, which is accelerating away. But in fact the earth is constantly accelerating around the Sun, which is similarly accelerating, while the rocket could have constant speeds almost all of the time. Furthermore, consider two identical objects moving apart in an otherwise empty space: Would they both vanish? But then, what if one of them had instead stopped and turned and followed the other at finite distances? Would both still vanish? That would only make sense if space was stationary. Would neither vanish? That would mean that the vanishing of one of them could depend on the motion of the other, after all.
So our problem cannot be solved by having Benardete’s rocket vanish (or teleport) at spatial infinity. We are therefore left with a paradox: Space must, but also cannot, contain infinite distances, if objects can go at any speed; there is this informal logical need for a physical speed limit. In view of the elementary nature of the above, I am tempted to speculate that Einstein, given only nineteenth century physics, may have noticed such a need. But in any case, the space in which we evolved does seem to be such that the objects within it cannot accelerate beyond the speed of light. So it seems reasonable to conclude that when we think about logically possible objects we should assume some such speed limit, in the interests of coherence.

## Monday, August 03, 2015

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