early draft of essay on aristotle, dedekind and cantor

[wraith estuary]

In Aristotle's Physics VI, the continuous is defined as a thing whose extremities are 'together', meaning located in the same place. Indivisibles cannot have extremities, for then they would be able to be divided into parts, meaning they would not be indivisibles. Because things that are continuous are united in their extremities as one, continuous things cannot be divided into indivisibles and, in fact, are infinitely divisible.

Aristotle continues on to show that motion is contuinuous, ie, infinitely divisible. Assuming that motion and, thus, time, can be divided into indivisibles, a motion AB over a distance YZ takes up an indivisible, discrete unit of time; but considering a faster motion CD over the same distance YZ, we find that CD must divide the motion AB into smaller units, contradicting the supposition that motion and, consequently, time, can be reduced to indivisibles. This process can be repeated indefinitely, suggesting that motion and, thus, time, is continuous.

Next, Aristotle argues that the present is indivisible. First, the present is the boundary between the past and the future, and the boundary belongs to both past and future. If the present was not indivisible, then the present would be divisible into parts, and would thus contain both parts of the past and future, making it not present. Thus, the present is indivisible. Contunuing, Aristotle claims that there is no motion in the present: for if there was, the present would be divisible, as per the appeal to the continuity of motion.

A thing that is changing has a definite end of change, but cannot be said to have begun changing at any point. This is because if, on the contrary, there was a beginning of change, then the moment immediately preceding it would be simultaneous with that moment, which is impossible because of the contunity of time.

Aristotle writes:

"But there are two senses of the expression 'the primary when in whcih something has changed.' One the one hand it may mean the primary when containing the completion of the process of change-- the moment when it is correct to say 'it has changed': on the other hand it may mean the primary when containing the beginning of the process of change. Now the primary when that has reference to the end of the change is something really existent: for a change may really be completed, and there is such a thing as an end of change, which we have in fact shown to be indivisible because it is a limit. But that which has reference to the beginning is not existent at all: for there is no such thing as a beginning of a process of change, and the time occupied by the change does not contain any primary in which the change began."

Aristotle is, conceptually, anticipating Dedekind here: dividing the rationals into two cuts, the lower of which being unbounded above, expresses the idea of a change with no beginning, tending to a limit: a real number, which is an Aristotelian indivisible, insofar as it expresses its status as a number. The Aristotelian end of change becomes Dedekind's limit point.