The previous argument we entered into, wherein we proved the existence of a finite and indivisible lower bound to space, had the helpful side effect of proving that space is composed of discrete units and is therefore not a continuous whole. However, this argument was not accepted by all, particularly those with some knowledge of calculus. Indeed, very often in discussions on this issue one encounters resistance precisely along these lines. Since the mathematical conception of space underlies the notion of space that many of us have come to accept, we naturally resist the notion that space does not completely correspond to its mathematical map, which has no finite lower bound. However, actual space cannot actually correspond to the mathematical representation in light of which we define the concept of space. Let us quickly revisit our initial argument to show why.
Our argument consisted (in brief) of pointing out that any real partition of space must contain some amount of space (this is clear from the definition). Since this is the case, there must necessarily be a smallest unit of space. We proved this by examining the contrary, namely, the belief that there is no smallest unit of space and that space is infinitely partitioned. We showed that, if this was so, there must be an infinite number of partitions in any given finite section of space, each containing some actual amount of space (else they would not be divisions of space at all), and therefore leading to the conclusion that there was an infinite amount of space in a finite amount of space. Other than being inherently contradictory, this argument would also make motion impossible, which is absurd, since things do move. Therefore, the contrary—that space is composed of finitely small units—must be true.
Distinguished interlocutors pointed out that any of these finite segments, even the smallest possible unit that was proved to exist, could be further divided into another segment, thereby proving that the smallest possible unit was not the smallest possible unit—showing an apparent contradiction in our position. The appearance of contradiction arises because we insist on our ability to further subdivide the smallest possible unit that the preceding argument firmly established. But by what right can we actually do this? It is certainly true that, given any real number, one can always find a smaller one through simple division. This also shows that the 3-dimensional real “linear” space that we intuitively think of (constructed by extending the real number line in all three dimensions) can always be divided further, since it is composed of real numbers. However, at this point the mathematical model breaks down in relation to actual space, since we have already demonstrated that there must be a smallest actually existing unit of space. If one takes this smallest unit and mathematically divides it into halves, one has not created a smaller unit of space, but has created a unit of a unit. This construct, while mathematically reasonable, and physically existent insofar as it is a division of a unit, is not in itself an actual unit of space.
To explain this (perhaps abstract) point by reference to a concrete example—suppose one takes a glass of water and sets to dividing it in halves. One will first have two half-glasses, then four quarter-glasses, and so on. Each new division we create will be a division of the water, since there is still water in each new division. However, when we finally have divided our glass of water into millions of separate glasses, each containing only one molecule each, what now can we do? Our mathematical method of procedure has let us divide the water by two, each time producing new divisions. Now, however, if we were to slice up the molecule of water even further, we would find that we no longer had any water at all, but simply a scattering of oxygen and hydrogen atoms. This is because water cannot be actually divided into any smaller piece than one water molecule. One could draw a water molecule on a piece of paper and proceed to slice it up in a million mathematically distinct segments—in fact, given infinite time and infinite paper, one could even infinitely divide it. But these divisions, while being no doubt impressive, would in no way exist.
The point of what we have gone through is to establish firmly that space is composed of a finitely-bounded quantity of discrete units. This conclusion comes analytically from our usual concept of space. Additionally, if space is itself composed of discrete units, then nothing in space can itself be continuous. Therefore, we must assert, despite their usefulness, that continuous quantities do not exist in any actual sense. That is, assuming that what we usually refer to as “space” is itself actual and not simply different than what we think it to be but essentially different than what we think it to be…