On the Continuum and Indivisibles
Εἰ δ’ ἐστὶ συνεχὲς καὶ ἁπτόμενον καὶ ἐφεξῆς, ὡς διώρισται πρότερον, συνεχῆ μὲν ὧν τὰ ἔσχατα ἕν, ἁπτόμενα δ’ ὧν ἅμα, ἐφεξῆς δ’ ὧν μηδὲν μεταξὺ συγγενές, ἀδύνατον ἐξ ἀδιαιρέτων εἶναί τι συνεχές, οἷον γραμμὴν ἐκ στιγμῶν, εἴπερ ἡ γραμμὴ μὲν συνεχές, ἡ στιγμὴ δὲ ἀδιαίρετον. Οὔτε γὰρ ἓν τὰ ἔσχατα τῶν στιγμῶν (οὐ γάρ ἐστι τὸ μὲν ἔσχατον τὸ δ’ ἄλλο τι μόριον τοῦ ἀδιαιρέτου), οὔθ’ ἅμα τὰ ἔσχατα (οὐ γάρ ἐστιν ἔσχατον τοῦ ἀμεροῦς οὐδέν· ἕτερον γὰρ τὸ ἔσχατον καὶ οὗ ἔσχατον).
–Aristotle, Physics 6.1
There is a concept which is absolutely intrinsic to all of geometry and mathematics. This particular concept is utilized by every single High School student that has ever graphed a line, and yet this concept is so incredibly difficult to understand that most people cannot wrap their heads around it. I’m talking about the concept of the continuum. Basically, the idea is that geometric geometrical objects are composed of a continuous group of indivisibles, objects which literally have no size, but which cannot be considered “nothing.” Despite the fact that these individual objects have no size, they form together into groups which, as a whole, can be measured in length or height or breadth. In mathematics, objects such as lines, planes, volumes, and all other sorts of space are considered to be continua, continuous and contiguous collections of these indivisibles into a unified whole. Because these infinitesimals have no size, themselves, even finite spaces contain an infinite number of these points.
Nearly every mathematician on the planet subscribes to this point of view. However, this was not always the case. Only a little more than 100 years ago, this view was considered extremely controversial and was only held by a fringe minority of scholars. Four centuries before that, this concept was nearly unthinkable. Though it has become, without question, the prevailing view of mathematicians, even today there remain a tiny handful of scholars who object to the use of the infinitesimal, the infinite, the individible, and the continuum in modern math. One such person is Dr. Norman Wildberger, an educator and mathematician for whom I have the utmost respect.
Still, I disagree with Dr. Wildberger’s philosophy on this particular issue.
There are a number of videos which Dr. Wildberger has published discussing his views on infinities, infinitesimals, and continua (particularly in his Math Foundations series). However, I chose the above video because it is the most recent one which he has published on that subject, and because it covers most of his reasoning within a single video. His debate opponent, Dr. James Franklin, provides some good responses to Dr. Wildberger’s claims, but I think Dr. Franklin missed some key opportunities and was not so clear in his answers as I would have preferred. As such, I wanted to give my responses to Dr. Wildberger’s objections, here.
His argument is framed entirely around three basic questions:
- What is a ‘set?’
- Is there a ‘set of all natural numbers ?’
- What is the theory of ‘subsets of ?’
Starting at the beginning, Dr. Wildberger draws a definition for ‘set’ from his university’s first-year Calculus textbook: “a well-defined collection of distinct objects.” Dr. Wilberger then goes on to complain that this fairly brief definition can introduce some ambiguities, and that this definition is not clear enough to differentiate those things which are sets from those which are not. As an example, he gives Russell’s Paradox: let be defined as the set of all sets which do not contain themselves. Is this an actual set? How can we differentiate things which are not sets from things which are sets?
Personally, I feel that it is more than a bit disingenuous of Dr. Wildberger to pretend like an introductory definition aimed at students fairly represents the past century’s worth of scholarship on this issue. The base definition that is presented in the video is a part of what is often termed “naive set theory.” Logicians and mathematicians who desired to remove paradoxes from mathematics– including Bertrand Russell, for whom Wildberger’s example paradox was named– understood that categorizing something as a set based solely upon the manner in which its elements are defined was weak, and that better options needed to be found. Dr. Wilberger quotes a number of mathematicians as having “vigorously opposed Cantor,” but completely ignores the fact that Russell and Zermelo and Fraenkel and numerous other 20th Century set-theorists (as well as nearly every mathematician currently in the field) would certainly agree with these detractors that such a definition is inadequate in describing sets.
It is at this point that Dr. Wildberger asserts that mathematicians “retreated into a cocoon of axiomatics,” deliberately phrasing his objection in a manner which makes it sound like they were backpedaling rather than progressing in the field. He goes on to decry the fact that set theorists began to define the behavior of sets axiomatically. He claims that axiomatic set theory leaves the notion of a “set” undefined, and simply postulates things which we want to be true about those sets. This is not quite true– a “set” is certainly not simply an amorphous, entirely undefined notion on axiomatic set theory. However, I think I understand Dr. Wildberger’s real objection, here. Axioms describe the behavior of sets without being proven from the definitions of sets. It is generally preferable, in mathematics, to keep the number of initial axioms at an absolute minimum, and to instead prove theorems which describe behavior.
Unfortunately for Dr. Wildberger, the minimum number of axioms necessary to mathematics is significantly non-zero. That is to say, it is literally impossible to formulate mathematics without using axioms. For anyone that might disagree with this, I’d challenge you to prove something as simple as 1=1 without resorting to any axioms. Given this, it seems that Dr. Wildberger’s objection is not so much that mathematics is axiomatic, but rather that it is too axiomatic for his preference.
How, then, does Dr. Wildberger propose that we limit our axioms to his tastes? It seems that he wants to tie mathematics solely to that which can be actually expressed and calculated in the real world. As an example, Dr. Wildberger discusses a number which I would probably write as . If you are unfamiliar with Knuth Up-Arrow notation, I’ve written about it once before, but Dr. Wildberger writes the number in a bit more recognizable notation:
He then states that this number is special, due to its fairly low information complexity, compared to most of the numbers between 1 and z. The vast majority of the numbers between 1 and z would be impossible to express in our universe. There is literally not enough space and matter in the observable universe to express such numbers. He then jumps from this fairly true statement to the conclusion that the very “idea of their existence is, in fact, problematic.” To support this idea, Dr. Wildberger asks the audience whether or not z has a prime factorization. He asserts that it does not have a prime factorization, and therefore feels very secure in his conclusion.
However, Dr. Wildberger’s view, here, seems to be far more problematic than the one against which he contends. For example, the ancient Greeks would not have even been able to express the number z, despite its low complexity in modern notation. Does that mean that z didn’t exist for Archimedes, even though it exists for us? The ancient Greeks could have expressed the number 1,299,709, but they most certainly could not have calculated its prime factorization– does that mean that Euclid or Diophantus would have said that this number has no prime factorization?
Ironically, Dr. Wildberger’s objections are an attempt to axiomatize the Argument from Ignorance fallacy. He is saying that, because we are ignorant of a means for expressing some numbers or performing some calculations, then those numbers or calculations cannot exist. This is, of course, preposterous. I can very simply prove that Dr. Wildberger’s z certainly does have a prime factorization, even if I am not aware of precisely what those prime factors are. In fact, the ancient Greeks were able to prove this 2500 years ago, despite not even being able to express z. Our inability to express or calculate a particular number says nothing at all about that number’s ontology, and it is more than absurd to claim that such ignorance overrides the mathematical induction which proves that such a number does exist.
Moving on to #3 in his list, now, Dr. Wildberger focuses in on the Axiom of Choice, which is the truly controversial axiom of set theory. Unfortunately, he doesn’t really tell us why he finds this axiom to be objectionable. He simply parallels the Axiom of Choice with Algorithmic methodology, and makes vague assertions about the preferences of computer scientists. The axiom of choice, itself, isn’t very controversial. It simply says that, given a quantity of non-empty sets, it is possible to define a new set which has at least one element in common with each of the other sets. The controversial part came when mathematicians, starting with Zermelo, began extending this axiom to infinite sets. Those who opposed the idea of actual infinites, including Wittgenstein (as Dr. Wildberger quoted), opposed the utilization of the Axiom of Choice with infinite sets. Such people view infinity as a sequence which moves from one element to the next. Modern set theorists, on the other hand, view infinity as every bit as much a completed quantity as is 1, 2, or 3.
Here’s an example. Let’s say I define a set of numbers k such that for integers where . We could fairly easily use this algorithm to determine the first few numbers of our sequence: , , , , , , et cetera. According to modern set theory, all possible coexist simultaneously, so we could talk about even if we haven’t yet calculated the value of that number. According to Dr. Wildberger’s approach, cannot exist until after you’ve calculated , so any attempt at discussing the properties of prior to this calculation would be incoherent. On the set theoretical approach, I would be absolutely justified in saying that is an even number, even without calculating it or any of the which lead up to it. On Dr. Wildberger’s method, a person could not make any claims about what the properties of are until they have actually calculated that number.
Now, I’ll bring this conversation back full-circle (pun completely intended) in order to discuss the implications of Dr. Wildberger’s view on indivisibles and the continuum. If Dr. Wildberger’s view is taken to its logical ends, then there can be no such thing as lines or circles. The points which we suppose to lie along the line do not exist unless we first calculate them exactly, which means that the line itself does not exist until we’ve calculated all of its points, exactly. However, it is impossible to calculate all of the points on a line, therefore that line cannot actually exist. All of geometry, and all of the fundamentals of mathematics for the past 2500 years, would implode under Dr. Wildberger’s view. Restricting the concept of number to only those values which can be pragmatically calculated with exactness is an unnecessary and self-defeating goal. Rather than restoring rigour, precision, and a deductive nature to the study of mathematics, all this would really do would be to eliminate all of the advancement which math has had since we moved beyond basic arithmetic of small numbers.
I began this post with a quote from Book VI of Aristotle’s Physics, in which the philosopher argues that “nothing which is continuous can be composed of indivisibles.” I have no doubt that Dr. Wildberger would find himself in complete agreement with Aristotle, here. Unfortunately for the both of them, the modern world finds Aristotle’s view of continua and indivisibles to be just as primitive and unsophisticated as is that eminent philosopher’s view that the Sun orbits around the Earth, or that heavier objects fall faster than lighter objects. In fact, all three of these concepts began to see their refutation at just about the same time. During the 1600’s, mathematicians like Galileo, Cavalieri, Toricelli, Newton, and Leibniz started to overturn the domineering view of Aristotle on this particular issue, and asserted that continua really could be composed of indivisibles. Two centuries after that, Cantor laid the foundations for viewing infinites as completed sets, and his work would be extended by men like Zermelo, Fraenkel, and Von Neumann in the 20th Century to the point where Aristotelian mathematics had finally been completely abandoned. Despite Dr. Wildberger’s protestations, this is not a bad thing. This is not something which has turned mathematics into nonsense. The modern understanding of continua, indivisibles, and infinites has allowed mathematics and physics to advance more in the last 100 years than it had in the previous 2500 years.
Even a finite continuum is composed of an infinite number of indivisibles.