The History and Philosphy of Non-Orientability

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August Ferdinand Moebius: The Original Topological Tyrant

Ferdi, as we shall henceforth call him in affection, was an only child born in a town in Saxony whose name is unpronounceable in our language, or almost. Saxony went on to become part of Germany, but the name of Ferdi’s home town is still very hard to say. Try it yourself: “Schulpforta.” That’s a good approximation. “Schulpforta, Saxony.” It has a certain ring to it.

Ferdi stayed at home in the unpronounceable town with his mother (who was a descendant of that original theological tyrant, Martin Luther) until he was sent to the College of Schulpforta (shew!) in 1803. After graduation, he started studies at the University of Ferdi Leipzig.

Like so many families, not only Saxonian ones, Ferdi’s mom and dad wanted him to be a lawyer. “Nope,” said Ferdi. “It is the numbers that numb me!” He took up the studies of mathematics, astronomy, and physics.

In 1813, Ferdi trucked it over to Gottingen, a city that is a little easier to say. There, he studied with Gauss, who was director of the Observatory in Gottingen but also, well, the greatest mathematician of the day. Ferdi’s dissertation was called The Occultation of Fixed Stars. (This has nothing to do with witches.)

Ferdi dodged the army, and was appointed to an “Extraordinary Professorship” in Leipzig. (Funny how those things worked back then...). Unfortunately, Ferdi was not a very entertaining or scintillating lecturer, and had to advertise (really) and offer his courses free of charge.

To make a long story short, Ferdi then, between the years of 1816 and 1827:

got passed over for the chair of the mathematics department at Leipzig.
moved sullenly to the University of Jena, where they gave him the full professorship he deserved.
held the title of “Observer” at the “Observatory” in Leipzig
got hitched
passed up the chance to review Grassmann’s Die lineale Ausdehnundslehre, ein neuer Zweig der Mathematik, which contained lots of findings similar to Ferdi’s own. (Graussman went on to submit the paper for a prize, and won.)
wrote secondary works on astronomy on the subjects of the “occulations of the planets, principles of astronomy, and celestial mechanics.”
published Der barycentrische Calkul, a classic on analytical geometry, Uber eine besondere Art von Umkehrung der Reihen, a paper which introduced the Moebius function, and Lehrbuch der Statik, a “geometric treatment of statics.”

It was only after old Ferdi’s death in 1868 that the reason for his current acclaim was unearthed. Ferdi had been working to win a contest on polyhedra theory given by the Academie des Sciences, and in a memoir post-humously presented to that scientific society, discussed the now-famous Moebius strip.

So it goes.

Philosophy: Klein Bottles and Kant

Immanuel Kant was a German philosopher in the eighteenth century, a man so intent on engaging in and reforming the current debates of his day, that he demanded there be a Copernican Revolution in philosophy. Just as the astronomer Copernicus proclaimed that the earth (and planets) revolved around the sun, Kant proclaimed the same thing in the language of philosophy: “Our understandings of what the outside world looks like,” Kant seemed to tell us, “depend on the ‘position’ and ‘movement’ of our own thoughts!”

In Kant’s field in that day, there were two schools battling to claim they had discovered the best way of understanding this outside world. The Rationalists held reason sacred, claiming our human knowledge unquestionable. The Empiricists disagreed: “Knowledge comes from experience, not from ourselves,” they said.

Kant believed neither school of thought, but set out on his own. He thought the key to resolving the whole problem might be to change the way we look at it: instead of asking how we can “bring ourselves” to understand the world, to ask instead: “How does the world come to be understood by us?” He began answering this question in his most well-known work: A Critique of Pure Reason.

Immanuel Kant Once we have achived some sort of knowledge, Kant says, we can ask how this becomes possible. He separates the judgements we make about our world into two classes. A priori judgements are universal; they are based on internal human reason and not in experience. A posteriori judgments are more limited; they come from very specific interactions we have with the world.

Here we come to Kant’s relevance to our study of mathematics. Kant believed that forms of mathematics like arithmetic and geometry comprise sorts of a priori judgements. We know that 2 and 2 make four: this is grounded in our reason. Facts like these apply to many experiences we have in the world, but do not come from these experiences. They come from us, Kant says.

We understand the spatial and mathematical world we live in by these a priori judgements, says Kant. This special framework of judgements, rooted in our human intuition, is what helps us relate the facts we know about things (like 2 + 2 = 4) to the things themselves (like a square). Kant called space and time “pure forms of sensible intuition” in his Critique. They are absolute, and derived from our very minds.

Kant had made a solid case, up until now. “But what about the things we can’t perceive?” you may ask. “How do they even exist under this system Kant has set up?”

And it is a very good question.

If you will recall our original discussion of orientability, you may remember that a right handed cut-out may “flip over” into 3-space to cover a congruent left-handed handprint. Our three-dimensional minds, equipped with their fixed a priori judgements about space and time, can handle this fine. But what about the three-dimensional equivalent of a right handprint? We cannot merely “flip this over” in the world we know to make it “cover” a 3D “left” handprint. In fact, what do “left” and “right” really mean if something has no other-handed counterpart?

All these problems are solved when we let go of our a priori judgements about time and space. This is what angered the followers of Kant so. They needed static intuition-based judgements to base any description of the world in-- yet the fourth dimension frequently evades this set of judgements.

(As you and I know, the addition of the fourth dimension into our thinking can produce its own kind of “Copernican Revolution” in our minds.)

For homework, answer the following questions:

Describe the advent of the Copernican discovery that the Earth was not the center of the universe. What social and religious consequences did this have? Do you think a world exposed to fourth dimensional geometry might mirror Copernicus' shaken world?
Differentiate betweein a priori and a posteriori judgements. (There are no strictly right or wrong answers!) What kind do you think the knowledge that a triangle has three sides is? How about knowledge of, say, the Battle of the Bulge in World War II?

This section written by EEC.

The picture of Moebius come from St. Andrew's MacTutor project. Kant's picture comes from The Proceedings of the Friesian School, Fourth Series.

For further reading, look at the following books:

Banchoff, Thomas F. Beyond the Third Dimension. New York: Scientific American Library, 1996.

Fauvel, John, et al., eds. Moebius and His Band. Oxford: Oxford University Press, 1993.

On the web, you can look at the following sites:

MacTutor's biography of Moebius.

An introduction to Kant from the Proceedings of the Friesian School, Fourth Series.

For more information on sources and other ideas for further reading, see the bibliography.

Created 981125 by jacr. Updated 981201 by jacr. URL is ./history.htm