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 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 begin 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.” 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. The super-denumerable point is not the result of the division process — it’s a different kind of thing asserted to exist at the place where the process would have to end. The inductive evidence (each division produces a smaller segment) is simply declared irrelevant at the limit. Grunbaum doesn’t show that Zeno’s question has an answer. He introduces a category that sits outside the framework in which Zeno’s question is meaningful. Zeno asked what you get at the end of infinite division. Grunbaum says: at the limit you get something to which the concepts of size and summation don’t apply. But that is exactly what Zeno was pointing at — that the process leads somewhere the mathematician’s tools can’t follow. Naming that place doesn’t illuminate it. The question isn’t defeated. It’s excluded. And Zeno, being a good questioner, would simply ask the next question: what is this thing with no concept of size, and how does it compose a line we can measure?

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.

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?

OP: https://questionsarepower.org/2015/02/13/zeno-grunbaum-and-order/