Category Archives: Foundations of Mathematics

Finite versus “infinite”

There are several approaches to the modern theory of “real numbers”. Unfortunately, none of them makes complete sense. One hundred years ago, there was vigorous discussion about the ambiguities with them and Cantor’s theory of “infinite sets”. As time went by, the debate subsided but the difficulties didn’t really go away.

A largely unquestioning uniformity has settled on the discipline, with most students now only dimly aware of the logical problems with “uncomputable numbers”, “non-measurable functions”, the “Axiom of choice”, “hierarchies of cardinals and ordinals”, and various anomalies and paradoxes that supposedly arise in topology, set theory and measure theory.

Some of the stumbling blocks have been described at length in my Math Foundations series of YouTube videos. In this blog we concentrate on the problems with real numbers and arithmetic with them.

The basic division in mathematics is between the discrete and the continuous. Discrete mathematics studies locally finite collections and patterns, and relies on counting, beginning with the natural numbers 1,2,3,⋯ and then extending to the integers, including 0, as well as -1,-2,-3,⋯, and to rational numbers, or fractions, of the form a/b with a and b integers, subject to the condition that b non-zero and with a particular notion of equality.

Continuous mathematics studies the “continuum” and functions on it, and relies on measurement, which these days involves also “irrational numbers” like √2,√5 and π that the ancients wrestled with, as well as more modern “irrational numbers” such as e and γ arising from integrals and infinite series. But what do these words and objects actually precisely mean and refer to? We should not presume that just because we use a common term or notion familiar from everyday life, that its mathematical meaning has been properly established.

Up to a hundred years ago, the notion of the “continuum” seemed intuitively straightforward, but difficult to pin down precisely. It appeared that we could rely on our intuition of space, following the philosopher I. Kant’s view that somehow humans had an innate understanding of three-dimensional geometry. However with the advent of modern physics, and in particular relativity and quantum mechanics, the true nature of the “continuum” grew increasingly murky: if time is relative and perhaps finite in extent, and space has an inherent graininess which renders it certainly not infinitely divisible, then what exactly are we modelling with our notion of the “infinite number line”?

While engineers and scientists work primarily with finite decimal numbers in an approximate sense, “real numbers” as infinite decimals are idealized objects which attempt to extend the explicit finite but approximate numbers of engineers into a domain where infinite processes can be ostensibly be exactly evaluated. To make this magic work, mathematicians invoke a notion of “equivalence classes of Cauchy sequences of rational numbers”, or as “Dedekind cuts”.

Each view has different difficulties, but always there is the crucial problem of discussing infinite objects without sufficient regard to how to specify them. I have discussed the serious logical difficulties at length around video 80-105 in the Math Foundations series.

For example the video Inconvenient truths with sqrt(2) has generated a lot of discussion. However not everyone approves of casting doubt on the orthodoxy: the video has more than 1000 likes, but also 316 dislikes. I doubt if I am saying anything in this video which is actually incorrect though — you can judge for yourselves.

Let’s return to the safe side of things. A finite sequence such as s = 1,5,9 may be described in quite different ways, for example as the “increasing sequence of possible last digits in an odd integer square”, or as the “sequence of numbers less than 10 which are congruent to 1 modulo 4”, or as the “sequence of digits occurring in the 246-th prime after removing repetitions”. But ultimately there is only one way to specify such a sequence s completely and unambiguously: by explicitly listing all its elements.

When we make the jump to “infinite sequences”, such as a sequence somehow implied by the expression “m = 3,5,7,⋯” the situation changes dramatically. It is never possible to explicitly list “all the elements” of an infinite sequence. Instead we are forced to rely on a rule generating the sequence to specify it. In this case perhaps: m is the list of all odd numbers starting with 3, or perhaps: m is the list of all odd primes. Without such a rule, a definition like “m = 3,5,7,⋯” is really rather meaningless.

We can say the words “infinite sequence”, but what are we actually explicitly talking about??

To a computer scientist, an “infinite sequence” is modelled by a computer program, churning out number after number perhaps onto a hard drive, or in former years onto a long tape. At any given point in time, there are only finite many outputs. As long as you keep supplying more memory, or tape, and electricity the process in principle never stops, but in practice will run short of resources and either grind forwards ever more and more slowly (the next output will take two years, hang on just a while!) or just come to a grinding halt when power or memory is inevitably exhausted.

So in this case the sequence is not to be identified by the “completed output tape”, which is a figment of our imagination, but rather by the computer program that generates it, which is concrete and completely specifiable. However here we come to an essential difficulty with “infinite processes”: the program that generates a given “infinite sequence”: is always far from unique. There is no escape from this inescapable fact, and it colours all meaningful aspects of dealing with “infinity”.

A finite set such as {2 4 6 8} can also be described in many ways, but ultimately it too is only properly specified by showing all its elements. In this case order is not important, so that for example the elements might be scattered over a page. Finite sets whose elements cannot be explicitly shown have not been specified, though we might agree that they have been described.

An example might be: let S be the set of all odd perfect numbers less than 10^{100000}. [A perfect number, like 6 and 28, is the sum of those of its divisors less than itself, i.e. 6=1+2+3 and 28=1+2+4+7+14.] Such a description of S does not deserve to be called a specification of the set, at least not with our current understanding of perfect numbers, which doesn’t even allow us to determine if S is empty or not.

With sets the dichotomy between finite and infinite is much more severe than for sequences, because we do not allow a steady exhibition of the elements through time. It is impossible to exhibit all of the elements of an “infinite set” at once, so the notion is an ideal one that more properly belongs to philosophy—it can only be approximated within mathematics.

The notion of a “completed infinite set” is strongly contrary to classical thinking; since we can’t actually collect together more than a finite number of elements as a completed totality, why pretend that we can? It is the same reason that “Leprechaun heaven” or “hierarchies of angels” are not generally recognized as proper scientific entities. Infinite sets, angels and Leprechauns may very well exist in our universe, but this is a philosophical or religious inquiry, not a mathematical or scientific one.

The idea of “infinity” as an unattainable ideal that can only be approached by an endless sequence of better and better finite approximations is both humble and ancient, and one I would strongly advocate to those wishing to understand mathematics more deeply. This is the position that Archimedes, Newton, Euler and Gauss would have taken, and it is a view that ought to be seriously reconsidered. I believe it is also closer to the view of modern giants such as H. Poincare and H. Weyl, both of whom were skeptical about our uses of “infinity”.

Why is any of this important? The “real numbers” are where Cantor’s “hierarchies of infinities” begins, and much of modern set theory rests, so this is an issue with widespread consequences, even within algebra and combinatorics. Secondly the “real numbers” are the arena where calculus and analysis is developed, so difficulties with their essential arithmetic foundation lead to weakness in the calculus curriculum, confusion with aspects of measure theory, functional analysis and other advanced subjects, and are obstacles in our attempts to understand physics.

In my opinion, it is possible, perhaps even likely, that we need to understand mathematics in the right way before we will be able to unlock the deeper secrets of the universe.

By reorganizing our subject to be more careful and logical, and by removing dubious axiomatic assumptions and unnecessary philosophizing about “real numbers” and “infinite sets” we make it easier for young people to learn, appreciate and contribute.

This also strengthens the relationship between mathematics and computing.

It is time to acknowledge the orthodoxy that silently frames our discipline. We need to learn from our colleagues in physics and computer science, and begin the slow, challenging but important and ultimately rewarding task of restructuring mathematics properly.

Let alpha be a real number

PM (Pure Mathematician): Let alpha be a real number.

NJ (Me): What does that mean?

PM: Surely you are joking. What do you mean by such a question? Everyone uses this phrase all the time, probably you also.

NJ: I used to, but now I am not so sure anymore what it means. In fact I suspect it is nonsense. So I am asking you to clarify its meaning for me.

PM: No problem, then. It means that we are considering a real number, whose name is alpha. For example alpha = 438.0457897416622849… .

NJ: Is that a real number, or just a few decimal digits followed by three dots?

PM: It is a real number.

NJ: So a real number is a bunch of decimal digits followed by three dots.

PM: I think you know full well what a real number is, Norman. You are playing devil’s advocate. Officially a real number is an equivalence class of Cauchy sequences of rational numbers. The above decimal representation was just a shorthand.

NJ: So the real number alpha you informally described above is actually the following: {{32/141,13/55234,-444123/9857,…},{-62666626/43,49985424243/2,7874/3347,…},{4234/555,7/3,-424/55,…},…}?

PM: Well obviously that equivalence class of Cauchy sequences you started writing here is just a random collection of lists of rational numbers you have dreamed up. It has nothing to do with the real number alpha I am considering.

But now that I think about it for a minute, I suppose you are exploiting the fact that Cauchy sequences of rationals can be arbitrarily altered in a finite number of places without changing their limits, so you could argue that yes, my real number does look like that, although naturally alpha has a lot more information.

NJ: An infinite amount of more information?

PM: If you like.

NJ: What if I don’t like?

PM: Look, there is no use you quibbling about definitions. Modern pure mathematicians need real numbers for all sorts of things, not just for analysis, but also modern geometry, algebra, topology, you name it. Real numbers are not going away, no matter what kind of spurious objections you come up with. So why don’t you spend your time more fruitfully, and write some papers?

NJ: Have you heard of Wittgenstein’s objections to the infinite shenanigans of modern pure mathematics?

PM: No, but I think I am about to.

NJ: Wittgenstein claimed that modern pure mathematicians were trying to have their cake and eat it too, when it came to specifying infinite processes, by bouncing around between believing that infinite sequences could be described by algorithms, or they could be defined by choice. Algorithms are the stuff of computers and programming, while choice is the stuff of oracles and slimy intergalactic super-octopi. Which camp are you in? Is your real number alpha given by some finite code or by the infinite musings of a god-like creature?

PM: I think you are trying to ensnare me. You want me to say that I am thinking about decimal digits given by a program, but then you are going to say that that repudiates the Axiom of Choice. I know your strategy, you know! Don’t think you are the first to try to weaken our resolve or the faith in the Axioms. Mathematics has to start somewhere, after all.

NJ: And your answer is?

PM: Sorry, my laundry is done now, and then I have to finish my latest paper on Dohomological Q-theory over twisted holographic pseudo-morphoids. Cheers!

NJ: Cheers. Don’t forget to take alpha with you.


The Banach-Tarski paradox: is it nonsense?

How can you tell when your theory has overstepped the bounds of reasonableness? How about when you start telling people your “facts” and their faces register with incredulity and disbelief? That is the response of most reasonable people when they hear about the “Banach-Tarski paradox”.

From Wikipedia:

The Banach–Tarski paradox states that a ball in the ordinary Euclidean space can be doubled using only the operations of partitioning into subsets, replacing a set with a congruent set, and reassembly.

The “theorem” is commonly phrased in terms of two solid balls, one twice the radius of the other, in which case it asserts that we can subdivide the smaller ball into a small number (usually 5) of disjoint subsets, perform rigid motions (combinations of translations and rotations) to these sets, and obtain a partition of the larger ball. Or a couple of balls the same size as the original. It is to be emphasized that these are cut and paste congruences! This was first stated by S. Banach and A. Tarski in 1924, building on earlier work of Vitali and Hausdorff.

Doubling_of_a_sphere,_as_per_the_Banach-Tarski_Theorem (1)

This “theorem” contradicts common sense. In real life we know that it is not easy to get something from nothing. We cannot take one dollar, subtly rearrange it in some clever fashion, and end up with two dollars. It doesn’t work.

That is why most ordinary people, when they hear about this kind of result, are at first disbelieving, and then, when told that the “proof” involves “free groups of rotations” and the “Axiom of Choice”, and that the resulting sets are in fact impossible to write down explicitly, just shake their heads. Those pure mathematicians: boy they are smart, but what arcane things they get up to!

This theorem is highly dubious. It really ought to be taken with a grain of salt, or at least generate some controversy. This kind of logical legerdemain probably should not go unchallenged for decades.

The logical flaws involved in the usual argument are actually quite numerous. First there are confusions about what “free groups” are and how we specify them. The definition of a finite group and the definition of an “infinite group” are vastly different kettles of fish. An underlying theory of infinite sets is assumed, but as usual a coherent theory of such infinite sets is missing.

Then there is a claim that free groups can be found inside the group of rotations of three dimensional space. This usually involves some discussion involving real numbers and irrational rotations. All the usual difficulties with real numbers that students of my YouTube series MathFoundations will be familiar with immediately bear down.

And then finally there is an appeal to the Axiom of Choice, from the ZFC axiomfest, which claims that one can make an infinite number of independent choices. But this contradicts the Law of (Logical) Honesty that I put forward several days ago. I remind you that this was the idea:

Don’t pretend that you can do something that you can’t.

You cannot make an infinite number of independent choices. Cannot. Impossible. Never could. Never will be able to. No amount of practice will help. Whistling while you do it won’t make it happen. You cannot make an infinite number of independent choices.

So we ought not to pretend that we can; that is what the Law of (Logical) Honesty asserts. We can’t just say: and now let’s suppose that we can make an infinite number of independent choices. That is just an empty phrase if we cannot support it in ways that people can observe and validate.

The actual “sets” involved in the case of transforming a ball of radius 1 to a ball of radius 2 are not sets that one can write down in any meaningful way. They exist only in a kind of no-mans land of speculative thinking, entirely dependent on these set-theoretic assumptions that pin them up. Ask for a concrete example, and explicit specifications, and you only get smiles and shrugs.

And so the Banach-Tarski nonsense has no practical application. There is no corresponding finite version that helps us do anything useful, at least none that I know of. It is something like a modern mathematical fairy tale.

Shouldn’t we be discussing this kind of thing more vigorously, here in pure mathematics?



The Alexander Horned Sphere: is it nonsense?

Modern topology is full of contentious issues, but no-one seems to pay any notice. There are many weird, even absurd, “constructions” and “arguments” which really ought to generate vigorous debate. People should have differences of opinions. Alternatives ought to be floated. The logical structure of the entire enterprise ought to be called into question.

But not in these days of conformity and meekness, amongst pure mathematicians anyway. Students are indoctrinated, not by force of logic, clarity of examples and the compelling force of rigorous computations, but by being browbeaten into thinking that if they confess to “not understanding”, then they are tacitly admitting failure. Why don’t you understand? Don’t you have what it takes to be a professional pure mathematician?

Let’s have a historically interesting example: the so-called “Alexander Horned Sphere”. This is supposedly an example of a “topological space” which is “homeomorphic”… actually do you think I could get away with not putting everything in quotes here? Pretty well everything that I am now going to be talking about ought to be in quotes, okay?

Right, so as I was saying, the Alexander Horned sphere is supposedly a topological space which is homeomorphic to a two-dimensional sphere. It was first constructed (big quotation marks missing on this one!) by J. W. Alexander in 1924, who was interested in the question about whether it was possible for the complement of a simply-connected surface to not be simply connected.

Simply-connected means that any loop in the space can be continuously contracted to a point. The two-dimensional sphere is simply connected, but the one-dimensional sphere (a circle) is not. Alexander’s weird construction gives a surface which is topologically a two-sphere, but its complement is like the complement of a torus: if we take a loop around the main body of the sphere, then we cannot contract it to a point. And why not? Because there is a nested sequence, an infinitely nested sequence of entanglements that our contracting loop can’t get around.


This image was made by Ryan Dahl, Creative Commons license.

Here is a way of imagining what is (kind of) going on. Put your two arms in front of you, so that your hands are close. Now with both hands, make a near circle with thumb and index finger, almost touching, but not quite, and link these two almost loops. Now imagine each of your fingers/thumbs as being like a little arm, with two new appendage finger/thumb pair growing from the end of each, also almost enclosing each other. And keep doing this, as the diagram suggests better than I can explain.

At any finite stage, none of the little almost loops is quite closed, so we could still untangle a string that was looped around say one of your arms, just by sliding it off your arm, past the finger and thumb, around the other arms finger and thumbs, and also navigating around all the little fingers and thumbs that you have grown, something like Swamp Thing.

Yes…but Alexander said “Let’s go to infinity!” And most of the topologists chorused” Yes, let’s go to infinity!” And most of their students dutifully repeated: “Yes, let’s go to infinity, … I guess!” And lo… there was the Alexander Horned Sphere!

But of course, it doesn’t really make sense, does it? Because it blatantly contravenes a core Law of Logic, in fact the one we enunciated two days ago, called the Law of (Logical) Honesty:

Don’t pretend that you can do something that you can’t.

The construction doesn’t work because it requires us to grow, or create, or construct, an infinite number of pairs of littler and littler fingers, and you just can’t do that!! All that we can logically contemplate is a finite version, as shown actually in the above diagram. And for any finite version, the supposed property that Alexander thought he constructed disintegrates.

The Alexander Horned Sphere: but one example of the questionable constructs that abound in modern pure mathematics.