One of Zeno’s famous paradoxes is his challenge to the mathematicians view that any finite line segment can be divided by a point. If so, the resulting lines can be subdivided. Zeno’s question is then “If lines can be divided and subdivided, what would the size of the lines be after fully dividing the line?” The absurdity does not lie in the question, but in any answer a mathematician could give. If a mathematician says there are lines with sizes as an outcome, then the sum of the infinite number of lines would make the original line infinite (we said originally that it was finite), and so a contradiction. Otherwise if the result is points, or lines of zero length, the sum of the lengths of points of course would be zero (but we said the line was of non-zero length), and so another contradiction. There are many other possible answers to the question. The reason the question is so famous is that all the answers so far have been unsatisfactory or absurd. I can only guess at why this question has been so mistreated over the 2500 years since it was asked (for example, authors often put any absurdity with Zeno himself, when the questions were intended to show absurdity in his opponents. Authors also accuse Zeno of all sorts of foolish intentions for his questions, such as that motion is impossible). Zeno’s intention, however, is not in question: he was a student of Parmenides and was simply making arguments to defend his teacher’s doctrine that there was only One thing in the world. Regardless, Zeno’s question has outlasted any answer.
Adolf Grunbaum is eminent in the position that Zeno’s paradoxes are refuted by modern mathematics. In his essay “Modern Science and Refutation of the Paradoxes of Zeno”, he began by making exactly the mistaken claim that “Zeno attempted to demonstrate the impossibility of motion” (p. 165 Zeno’s Paradoxes) Grunbaum goes on to introduce briefly the common notion among mathematicians, introduced by Cantor in the 19th century, that there are different kinds of infinities and so we are faced with choosing a particular kind of infinity for the result of the infinite subdivisions of a line. He argues that the kind of infinity of points on a line is “super-denumerable” and cannot be added the way Zeno proposed. Normal addition is reserved for the familiar denumerable infinity that proceeds like the natural numbers (1, 2, 3, etc). However, the divisions Zeno proposed begins as denumerable, (one division or point, then another, etc). And the number of divisions is the same as the number of points. Grunbaum, like Cantor before him, argues that the result of this process, while denumerable at each finite stage, results in something that is not denumerable but “super-denumerable”, refuting inductive logic. As we divide we are working with line-segments of some size, and in the limiting case what results are not segments of zero size (which could be summed), but something with no concept of size whatever. This is exactly the argument that induction is false- that even though we can make repeatable observations of an object on a controlled experiment, as soon as we stop looking the object transforms into something entirely different. Before, I defined numbers as a kind of difference, what we are summing (or subtracting) includes how it is summed, so points that are “super-denumerable” are not the same as other points because summation is different for “super-denumerable” points.
But there is something deeper going on here than this mathematical play of words. A line segment can be defined by its endpoints. Indeed, the endpoints are all that is needed to make a formula for a straight line. The “stuff” between the endpoints, using Aristotle’s terminology, would be the substance. Now we have come to the question “What is a line segment?” Aristotle would say that first and foremost it is its form (or formula), which is to say, its endpoints. Aristotle added to this of course, including in any “what” also its cause or “why” (Aristotle’s Metaphysics). Such expansive thinking has long gone out of use, but Grunbaum would have us believe that the substance of a line-segment is a thing so different from its form as to be completely incommensurable- that Aristotle’s conception of being (which was a marriage of form and substance) must be utterly divided, leaving us with much deeper problems than what we had before with Zeno’s question. Do we abandon form (endpoints) in favor of substance, since the “stuff” of the line-segment would ultimately be a collection of Grunbaum’s “super-denumerable points.” In that case, what would addition or any other mathematical concept be, since we must compare endpoints to measure, and use the result to add, without these formal concepts there is little left of mathematics at all.
Zeno’s subdivisions could be placed in an increasing order as in a sequence used in the axiom of completeness- the axiom that “distinguishes the real numbers” (Abbot, Understanding Analysis). The problem becomes that ordered in this way the limiting “super-denumerable point” is unlike the other points in the sequence, which are endpoints of line-segments. A “super-denumerable point”, as Grunbaum states, has the property that “no point is immediately adjacent to any other.” (p. 169 Zeno’s Paradoxes) In other words, there are only endpoints/there are no zero-length segments. Perhaps Grunbaum is claiming that Zeno’s infinite process of division is not plausible, since to fully subdivide we would need all the points between any two points on the original line to be there as finished divisions. But that would reject the axiom of completeness (and consequently the real number system), where infinite divisions of this kind happen, eventually creating a zero-length segment (a least upper bound to an infinite increasing sequence is an adjacent point, if it were not adjacent to the sequence of points, it would not be a least upper bound). If Zeno’s division is somehow plausible, but somehow without creating adjacent points, what follows is pure nonsense: the resulting points are not “zero-length segments” because zero-length segments require adjacent endpoints in a way that the segment is of zero length. Thus, the ideas of a “zero-length segment” and a “super-denumerable point”, according to Grunbaum’s line of reasoning, must be totally different things. The result of Zeno’s subdividing is then something neither with size, nor of zero size, or perhaps it is both, but it doesn’t matter anymore, we will just call it a “super-denumerable point”, or “linear Cantorean continuum of points” (Grunbaum p. 169 Zeno’s Paradoxes).
At this point an appropriate question is: “What purpose does the mystical belief in super-denumerable points have?” The mystical desire to control oneself has an obvious purpose of freedom and control, but the super-denumerable is not defensible on logical grounds. It would behoove proponents of the view to explain why we should get worked up about it. My speculation is that thinkers become frustrated with how little they end up knowing as a result of thinking, and so the motivation is for teachers of math to feel we know a lot, and be able to say what we know to students. Support for belief in super-denumerable points is that knowledge justifies itself, whether it makes sense or not. Another defense is that this type of math is part of academic culture. The reason math is so important to research is another embarrassment: that math makes sense while other cultures do not. Now, the culture of number is freed from sense and can expand and take on an inclusive attitude to other views.
Aristotle’s substance, at least one of his definitions of it, was a subject that could not be predicated. Another definition was form or essence, and form and substance are deeply connected. In searching for substance, predicates divide the subject. “The human is a man” (or “Woman”) is a division of human. Metaphor, in this view, is destructive to a search for substance. It expands words: “Humans are stars” makes a mess of things, only adding possible predicates. Unless you search for the intersection of “human” and “star”, in which case you are dividing both. Regardless, seeking knowledge of substance is a process of division- so too with a process of division of line segments. And what is the result of this infinite division, this search for substance in the excised world of pure form that is mathematics? It is merely the division itself – the point – which is what we started with when we were looking for substance. What is our point? Or is it changed into a super-denumerable point? Or some other kind of point or division of a line? What is this scalpel? Have I used it violently in searching for it?