The art of definition
Definitions are basic objects in mathematics. Even so, I’ve never seen the art of definition explicitly taught, and I have rarely seen the need for a definition explicitly discussed.
Have you ever noticed how damn hard it is to make a good definition and yet how utterly useful a good definition can be?
The basic definitions inform the research of any field, and a good definition will lead to better theorems than a bad one. If you get them right, if you really nail down the definition, then everything works out much more cleanly than otherwise.
So for example, it doesn’t make sense to work in algebraic geometry without the concepts of affine and projective space, and varieties, and schemes. They are to algebraic geometry like circles and triangles are to elementary geometry. You define your objects, then you see how they act and how they interact.
I saw first hand how a good definition improves clarity of thought back in grad school. I was lucky enough to talk to John Tate (my mathematical hero) about my thesis, and after listening to me go on for some time with a simple object but complicated proofs, he suggested that I add an extra sentence to my basic object, an assumption with a fixed structure.
This gave me a bit more explaining to do up front – but even there added intuition – and greatly simplified the statement and proofs of my theorems. It also improved my talks about my thesis. I could now go in and spend some time motivating the definition, and then state the resulting theorem very cleanly once people were convinced.
Another example from my husband’s grad seminar this semester: he’s starting out with the concept of triangulated categories coming from Verdier’s thesis. One mysterious part of the definition involves the so-called “octahedral axiom,” which mathematicians have been grappling with ever since it was invented. As far as Johan tells it, people struggle with why it’s necessary but not that it’s necessary, or at least something very much like it. What’s amazing is that Verdier managed to get it right when he was so young.
Why? Because definition building is naturally iterative, and it can take years to get it right. It’s not an obvious process. I have no doubt that many arguments were once fought over whether the most basic definitions, although I’m no historian. There’s a whole evolutionary struggle that I can imagine could take place as well – people could make the wrong definition, and the community would not be able to prove good stuff about that, so it would eventually give way to stronger, more robust definitions. Better to start out carefully.
Going back to that. I think it’s strange that the building up of definitions is not explicitly taught. I think it’s a result of the way math is taught as if it’s already known, so the mystery of how people came up with the theorems is almost hidden, never mind the original objects and questions about them. For that matter, it’s not often discussed why we care whether a given theorem is important, just whether it’s true. Somehow the “importance” conversations happen in quiet voices over wine at the seminar dinners.
Personally, I got just as much out of Tate’s help with my thesis as anything else about my thesis. The crystalline focus that he helped me achieve with the correct choice of the “basic object of study” has made me want to do that every single time I embark on a project, in data science or elsewhere.
mathbabe 
I think it’s great that you’re focusing on the importance of definition in mathematics. One of the things that bugs me greatly about the use of mathematics in academic articles in finance and economics is that they don’t understand that you have to define your terms very precisely or you’re not doing mathematics, rather something more like debating how many angels can stand on the head of a pin. I was gratified to get a lot of web repeats (like at http://ineteconomics.org/blog/inet/economics-not-math) for an article I wrote a few months ago in which I wrote “Economics pretends to be mathematics, but it is not mathematics. There is a major difference. No mathematician uses a term in a formula, or a statement of a theorem, unless that term has first been defined with excruciating precision. Economists may think they’ve defined their terms, but they should try reading some real mathematics to see what a precise definition truly is.”
LikeLike
Learning to reason from a definition was one of the key things for Pure Maths, I found. It was such a powerful thing to be able to do. Clearly, forming a definition is iterative, just like coming up with a good proof is. I didn’t realise for quite a while that mostly, when we are shown proofs, we’re not shown the first proof of a theorem – we’re usually shown a proof that arose later, with better motivation, better notation, whatever, that’s arisen in the interim. It’s nice to hear that this is also happening with definitions.
LikeLiked by 1 person
In topology there has in the last few years been a refinement of the definition of a triangulated category. There is a relatively simple and extremely motivatible object called a stable infinity category. Every stable infinity category has an invariant underlying homotopy category which can be proven to be triangulated, with the octahedral axiom falling out as a theorem. This explains the necessity of the octahedral axiom via a better definition :).
LikeLike
very nice. another perspective on stable infinity cats: it has long been known/rumored that the derived category has some annoying/undesirable behavior because cones are not functorial or some such. I think the stable infinity category records all possible choices of the cones of a morphism. so this way the stable infinity category is a cleans up some problems with an older definition.
as far as the octahedral axiom, I learned from Arend that one way to view it is just the triangulated category analogue of the fact that in an abelian category- say A, B, C are modules -that (C/A)/(B/A) is isomorphic to C/B.
LikeLike
Great post, especially at a time when high school geometry is still focusing on presenting the subject as a deductive system (one in which a minimum number of propositions suffice for the deduction of the rest, and a minimum number of terms suffice for the definition of the others). Most H.S. geometry books have a section on forming definitions. This is the first time most students are required to develop and reason, from good definitions.
It is all down hill from there. Instead of building a foundation that can last a lifetime only lip service (about 10% of exercises) is paid to writing definitions, the rest of the student time is spent in application. Since writing definitions is not on any state tests, it will be either not be empathized or ignored after September.
The only classes I took that taught definition writing in any mathematical sense were the symbolic logic classes in philosophy.
LikeLike
I share with my students (pre-calc and calculus) that the definition of pi is the ratio of circumference to diameter. (And from that comes what they would call the formula for circumference.) And that the area ‘formula’ can be proved.
I once asked my calc II students to find a proof of the Pythagorean theorem, and they all brought in examples. Yikes! That’s when I realized they generally had no idea what a proof was.
I am teaching linear algebra this semester, and am working with the students on understanding the importance of the definitions.
LikeLike
The downside is that definitions are always just a work-in-progress, always just an approximation, always just a heuristic.
Isn’t this what Gödel’s incompleteness theorems show?
LikeLike
I was lucky enough to talk to John Tate too, albeit at a much less advanced level mathematically. In one of the largest math departments around, by sheer luck his office happened to be immediately across the hall from ours in Austin.
In 2010 after being out of touch for 18 years, I looked up and had coffee with one of my UT officemates (the Catholic-raised sexist, for anyone who read my comment from yesterday). During our conversation he recalled a memory of a routine observance he and our other officemate made of a humorous (to them) Tate trait, namely, a clockwork-like impatience with clueless undergrads. They would sense the volcano about to erupt, look at each other and their watches, and correctly predict within seconds when Professor Tate was going to shoo the hapless student out of his office.
I was always a Tate fan and didn’t recall either observing the phenomenon myself or them ever telling me about it, but the anecdote could be plausible. However for the record, Professor Tate kindly allowed me to chat with him in his office a few times without ever once cutting things short. (For a few months he was at the top of my list of potential thesis advisers. Wish I were a better mathematician – that proximity to greatness was like a lounge act getting to hang with the Beatles!)
LikeLike
When i discovered I liked math, I decided to start at the beginning. Who else has read all 13 elements of Euclid? What surprised me is that it starts with a bunch of definitions of the basic concepts which are not rigorous at all. E.g. “A point is that which has no part”, “a line is breadthless length” etc.. which appeal to intuition way more than is necessary. I understood you cannot write a dictionary from the void, yet you could probably appeal to simpler concepts that that of “part”, “breadth” and “length” to define points and lines, right?
Much later, I found a satisfactory answer in Hartshorne’s beautiful yellow book – no, not that one – Geometry: Euclid and beyond. It defines geometry in a very algebraic way, having a set called “points” and another one called “lines” with a relation between them (namely, you can say what it means for a point to be on a line). The statement of the axioms of (Euclidean) geometry are then virtually identical and the consideration of whether the parallel axiom really is one becomes very natural and straightforward (it is very easy to exhibit alternate geometries).
Incidentally, the book is a wonderful read that spends a lot of time defining concept, making the proof part much simpler and clearer. I recommend it to anyone who likes algebra. Anecdotally, I put this book down, picked up Hendrik Lenstra’s book on Galois theories and decided I was going to do a PhD in Algebraic Geometry. So I guess I recommend both books.
Happy reading.
LikeLike
In classical logical traditions the concepts of definition and determination are closely related — and they are even more so if you view the overarching concept of constraint from from an information-theoretic point of view, as C.S. Peirce did beginning in the 1860s. So we have much discussion of these intertwined concepts in pursuing and applying the insights of Peirce’s theories of information and inquiry. Here’s a recent thread that leads on to a mass of notes I’ve been gathering —
• Definition and Determination
LikeLike
An excellent discussion of the logical character of various kinds of definitions can be found at the Stanford Encyclopedia of Philosophy: http://plato.stanford.edu/entries/definitions/.
Re Euclid, I would consider “point” and “line” as primitive terms in his geometry, that is, as undefined, and view his explications as informal ways of making the reader familiar with the concepts as he is going to use them.This is essential in axiomatizing anything. As it is impossible within a given theory to define or “prove” everything you want to say (Goedel’s result is not relevant here), some things need to be assumed without proof, if only within that context.
Even then, some statements may be ambiguous. A famous example is Newton’s F = ma. Is it a definition or a substantive axiom with a truth-value? On the right-hand side is a primitive term, m, and a defined term a. But what is the status of F? A lot of ink has been used discussing this and no real consensus has been reached.
Some contemporary economists can be terribly slipshod in their treatment of data and setting out their theoretical propositions logically. One example is Krugman. His treatment of data and their graphs can be shockingly awful. This is irrespective of his macroeconomic theoretical perspective.
LikeLike
Grabbing a tangent on the Newton line. I was under the impression that it is m := a/F i.e. a definition of a not directly observable property of objects he called ‘inertial mass’ in terms of directly observable quantities like acceleration and force.
LikeLiked by 1 person