Category Archives: Foundations of Mathematics

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.