The Series

As the first couple of posts on this blog, I want to explore a couple of ideas that have been bouncing around in my head about how mathematics changes over time. I figured I would try a 3-part series since there are about that many ideas strung together in my mind. I want to focus on group theory because it’s relatively simple to describe to a layperson, and because I recently became aware of a foundational group theory article from the 19th century which makes some fascinating rhetorical moves. (It was written by Arthur Cayley, whose name may be familiar to some readers.)

Part I of the series (i.e., this post) will give an extremely mild introduction to group theory. I’ll also describe some of how higher mathematics is taught, and contrast it with the way in which mathematicians actually think.

In Part II, I will discuss that paper from the 19th century, noting in particular how Cayley sometimes wants mathematical symbols to refer to something “real” beyond the symbols, and sometimes he doesn’t. He does this in a very specific pattern that I suspect is central to many instances of mathematical progress.

Finally, in Part III, I will describe how the rhetorical/conceptual structure of Cayley’s paper relates to its logical structure. I’ll also take the opportunity to express my thoughts on the role of logic in mathematics–is it a foundation? is it a toolbox? or is it the precise reason why mathematics is like the fine arts?

Part I

Groups

Group theory studies “groups”. No matter who you are, you have encountered many groups. I will describe four of them. The first two are easy to describe:

whole numbers

moves in the “shell game”

Whole numbers combine together under addition. Moves in the shell game combine together by doing one after another. This combination rule is what makes a group a group. (Of course, the combination rule has to have “regular behaviour” in ways I will not go into.) And the reason why this matters is because you can use the same standard toolkit of symbols (+,-,=) with any group, with completely equivalent meaning.

Here are two more groups which require a little more explaining:

12-hour clock faces

skateboard flip tricks

When you subtract 2 hours from 1 o’clock, you do not get -1 o’clock; you get 11 o’clock. It turns out that this combination rule is “regular” enough for clocks to count as a group!

Finally, while I am extremely ignorant of skateboarding, I do have an example based on an observation of the way skateboarders talk to each other. I once overheard a group of skateboarders comparing different tricks involving jumping in the air, and having your skateboard rotate underneath you. The tricks they were discussing were characterized by the number of rotations your skateboard performs, on which axis. It is possible to add together two rotations on separate axes, producing a more complex rotation, and a cooler trick. In the words of the skateboarders, one trick “plus” another trick was a third trick, showing that had grasped something essential: that rotations in 3D space form a group.

As far as group theory is concerned, the “elements” being combined together do not really matter. Two groups are called “isomorphic” if they have the same basic structure. For instance, consider musical intervals under addition. (I mean here intervals where “an octave” is treated as equivalent to “unison”.) Because there are 12 tones in the chromatic scale, and because the addition of intervals works the same way as addition of hours, the 12-hour clock face group and the musical-interval group are isomorphic. In a certain sense, they are the exact same group.

This “isomorphism” is just one kind of relationship that can hold between groups. Group theorists are often more interested in the web of relationships between groups rather than individual groups themselves. Cayley, the guy whose paper I will discuss in my next post, actually proved that the shell game groups (called “symmetric groups”) are a kind of reservoir of structure, containing within themselves the qualities of every other finite group. You could say that the symmetric groups are more fundamental in group theory than numbers are!

Groups can arise in a wide variety of forms, and they form a web of relationships. This is mirrored by the way in which the concept of a group appears. That is, there are a variety of interrelated ways to define groups. But because not everyone is familiar with the way definitions occur in higher mathematics, I will take a brief digression.

Definition, Theorem, Proof?

At higher levels of mathematics, a particular style of exposition has become the norm. It is sometimes referred to “Definition, Theorem, Proof” because texts cycle through these three subgenres of mathematical writing as they proceed. This style effective, but it lacks context when there are not enough prose sections accompanying the math. Where did this style of writing come from? It has ancient roots in Euclid’s Geometry, but there are modern influences as well. In the nineteenth century, mathematicians and philosophers were thrilled by innovations in logic. Some of them dreamed that imprecise human language could be replaced by logic, and that this would resolve all sorts of problems, from pseudoscience to fascism. The mathematical dimension of this project was the belief that mathematics comprises all of the trivially-true logical propositions. This project fell to pieces for various reasons, including some disturbing truths about the nature of logic that were discovered in the 20th century. More on this in Part III.

I suspect that some of the present norm is influenced as much by the turn towards logic in mathematics over the last 200 years as by the reference to ancient Greek geometry. In any case, I’ve included an image from Thomas Hungerford’s Algebra, a contemporary textbook. This sample includes two prose paragraphs, interspersed into a cycle of “Definition, Theorem, Proof”. A math student learning from this book would typically copy out the definitions and theorems and work through the proof to ensure their understanding. If you are a non-mathematician, I suggest you focus on how, in the definition portion, a “group” is slowly built up from simpler structures. Hungerford’s text works hard to display its logical structure.

It is easy to carry on from the form of exposition of a subject and to suppose that this mirrors the way practitioners think about it. This text sample would suggest that mathematicians think about groups in terms of a very specific definition, in terms of a single hierarchy from general “nonempty sets” down to specialized “abelian groups”. But I claim that mathematicians’ thinking about math is similar to all other thinking: ideas exist in a dynamic web, and what seems like a single concept inevitably separates into many perspectives. (And this perspectival splitting improves the concept!)

Multiple articulations

I want to share with you an example of an accomplished mathematician giving a window onto how they think about group theory. The conceptual structure of this example is starkly different from the structure of Hungerford’s textbook. The backstory of how this example came to be is interesting, so I will share it.

The mathematician Bill Thurston published his fascinating article “On Proof and Progress in Mathematics” because he disagreed so strongly with a paper about the sociology of mathematics. He criticized the sociologists’ choice to “project[] the sociology of mathematics onto a one-dimensional scale (speculation versus rigor)”, and claimed that this caused them to misrepresent his work. Thurston’s account of how he thinks about mathematics prompted Terence Tao, another mathematician, to express the plurality of ways he thinks about group theory. Should we trust the sociology of mathematics produced by mathematicians, or the sociology produced by sociologists? I am biased, but I find Thurston and Tao’s expressions quite convincing.

Tao gives the following list of ways to think about groups on his blog. If you are not a mathematician, I suggest skimming this list and picking out the names of each “way”. I will do a little bit to categorize these “ways” below.

the notion of a group \(G\) can be thought of in a number of (closely related) ways, such as the following: (0) Motivating examples: A group is an abstraction of the operations of addition/subtraction or multiplication/division in arithmetic or linear algebra, or of composition/inversion of transformations. (1) Universal algebraic: A group is a set \(G\) with an identity element \(e\), a unary inverse operation \(\cdot^{-1}: G \rightarrow G\), and a binary multiplication operation \(\cdot: G \times G \rightarrow G\) obeying the relations (or axioms) \(e \cdot x = x \cdot e = x\), \(x \cdot x^{-1} = x^{-1} \cdot x = e\), \((x \cdot y) \cdot z = x \cdot (y \cdot z)\) for all \(x,y,z \in G\). (2) Symmetric: A group is all the ways in which one can transform a space \(V\) to itself while preserving some object or structure \(O\) on this space. (3) Representation theoretic: A group is identifiable with a collection of transformations on a space \(V\) which is closed under composition and inverse, and contains the identity transformation. (4) Presentation theoretic: A group can be generated by a collection of generators subject to some number of relations. (5) Topological: A group is the fundamental group \(\pi_1(X)\) of a connected topological space \(X\). [This says that a group expresses the possible ways to travel in loops through a space.] (6) Dynamic: A group represents the passage of time (or of some other variable(s) of motion or action) on a (reversible) dynamical system. (7) Category theoretic: A group is a category with one object, in which all morphisms have inverses. (8) Quantum: A group is the classical limit \(q \rightarrow 0\) of a quantum group. … [(9)] Sheaf theoretic: A group is identifiable with a (set-valued) sheaf on the category of simplicial complexes such that the morphisms associated to collapses of \(d\)-simplices are bijective for \(d > 1\) (and merely surjective for \(d \leq 1\)).

Some of these ways of thinking about groups are identifiable as (sketches of) definitions, for instance (1)-(4), (7) and (9). Way (5) may appear to be somewhat tautological (“a group is the fundamental group of…”) but because of the way a “fundamental group” is constructed, this is really saying that groups express something already group-like about topological spaces. In this sense, ways (5) and (6) show groups in terms of things whose structure they can express, rather than trying to describe groups directly.

Way (0) simply gives examples of groups and says that groups in general are “abstractions” of these. I.e., these central examples serve as paradigms of groups, but without serving as “definitions”.

I cannot pretend to know what way (8) is talking about, but it seems to be saying that the structure of a “quantum group” approximates that of a group more and more closely as the parameter \(q\) approaches zero. In other words, if we consider a quantum group for \(q = 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4},...\) we will get an infinite sequence of things that are not groups, but which get closer and closer to becoming a group. In a like way, perhaps way (8) is not a definition of a group but specifies an infinite sequence of definitions which get closer and closer to defining groups.

Tao’s articulation of these interrelated ways of viewing groups much more closely resembles what I claim thinking, including mathematical thinking, always looks like. It raises, for me, the question of why groups are not presented this way in textbooks. I am sure some of it has to do with the pedagogical constraints. It is hard enough to begin to reason about groups using just one definition, and the simplicity certainly makes things easier. But perhaps the web-nature of mathematical thinking would be better transferred on to students if it was shown more explicitly through the style of mathematical exposition.

Conclusion

Hopefully you have a taste of what a group is if you’re a non-mathematician, and perhaps if you are a mathematician you have had the opportunity to think a bit more deeply about the way you think about math. Next week for Part II, I will prepare a post discussing the early group theory paper I mentioned. I hope to share a fascinating rhetorical pattern that I’ve noticed before in mathematics, but have never heard described.