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.