Usage of set theory in undergraduate studies

Question

I would like to ask my colleagues their thought on good practices concerning set theorical framework in undergraduate studies. For example, have there been any attempt to use another mathematical formalism, such as ETCS? (for research issues, see this question).

While most, if not all, of our mathematics are thought, done, written using set theory, our younger students seem to struggle with these concepts. Some can well put a $4\times 6$ matrix in row reduced echelon form but plainly do not understand the meaning of a question like "If $A,B$ are two square matrices of size~$n$, prove that $\ker(AB)$ contains $\ker(B)$." The difficulties with $\varepsilon,\delta$ definition of limits may be of a similar nature.

In fact, one may argue that all set theoretical concepts presently are more or less eliminated from the lower levels of mathematical education. One may even argue that it should be so. I remember that each one of the first years of middle school (from 6th grade on, the French and US systems coincide here!) taught me one new definition in set theory; sets and mappings at the age of 11, then equivalence relations, then sets of equivalence classes (to define vectors)... And a few years later, students are taught quotient groups like $\mathbf Z/n\mathbf Z$ as sets of equivalence classes, a definition which they of course take litteraly.

While Set theory is very useful to formalize things, at least once you're used to it, it is true that it allows stupid questions, requires abuses of notations (so that one does not distinguish between the $1$ of $\mathbf Z$ with the $1$ of $\mathbf R$, not forgetting thoses of $\mathbf Q$ and $\mathbf C$). In some sense, modern mathematicians, especially algebraists, speak sets but think categories. This may be related with the fact that the precise definition of the axioms of set theory (ZFC, say) are not so well known among mathematicians, and even not really taught (for example, no mention of the replacement axiom in my own mathematical education). In contrast, a more recent book like Terence Tao's Analysis begins with a precise exposition of these axioms, up to this replacement axiom.

I can't really make my mind between one attitude and the opposite. So what do you think?

Answer 1

The underlying formalism, whatever it is, should be introduced very carefully.

The important thing is to teach concepts and methods. Formalism should be used whenever it is helpful for the student, but it should never be used on the grounds that "this is what mathematics really is" (it is not!) or "it is more precise this way" (but completely obfuscating!), or some such.

The examples you listed have very little to do with set theory. The $\epsilon\delta$ definitions are not hard because of set theory but because it is hard for humans to understand the difference between $\forall \exists$ and $\exists \forall$. The general idea of a map is not bound to set theory either, and neither are equivalence relations or quotients. All of these can be done in type theory, for example. In fact, if you open a random textbook it will read like type theory, not set theory.

If there is one things we do want to pick up from various formalisms, it is that we should not use broken notation. We should teach properly the difference between free and bound variables (something many mathematicians cannot get a handle on because they were taught 17th century syntax), that by itself will clean up a lot of confusion. We should always, always distinguish a function $f$ from its value $f(x)$ at $x$. We should never confuse an expressions $x^2 +1$ with a function $x \mapsto x^2 + 1$, or think that polynomials are functions. We should never say that one variable depends on another. And so on.

I taught freshmen logic and set theory. The first time around I naively explained what a formal proof was. They all learnt how to produce formal proofs, but had no idea what they were for. The next time around I taught logic and sets informally, and made the mistake of teaching logic first, and then sets "axiomatically". As it turns out "pure logic" is too pure, we had nothing to speak about. The third time around I "covered" logic in two lectures and went on to teaching "sets". I introduced things as we went along, and it was mostly about how to read and write proofs, how to transcribe a statement from natural language to a formula and back, how to deal with unions, intersections, subsets, quotients, direct and inverse images, etc. Mostly things which they are supposed to learn by osmosis in other courses. I don't think I got very far, though, and I am still not sure what the point of the course is, other than to hit students with very abstract stuff early on.

Answer 2

Students must surely get used to the language of set theory, but are in difficulties because of a lack of a training in grammmar.

I found it necessary in an analysis course to go through a proof of a particular statement of the form $ A \subseteq B$ by asking: "What is the first line of the proof? and getting them to see it has to be: "Let $a \in A$." and then lower down the board ask for the last line, i.e. "Thus $a \in B$." After doing this on a number of occasions they get the idea.

I have also used the teaching method of "reverse chaining" (see also "backward chaining" in wikipedia) to teach proof structure. You write out a proof say that the limit of a product is the product of the limits, for which there is little chance they could do from scratch; then you blank out bits, so there are still lots of clues, and the exercise is to fill in the bits, using the clues from the rest of the proof. This can be analogous to one of us writing out a sketch proof and then trying to filling in the details. This is also very easy to mark!

See also wikipedia on "errorless learning". This, like reverse chaining, is based on the idea that you learn from success, which is standard in animal training, and for children, and is also true for grown up humans! (Surprise, surprise! But actually this was explained to me by an excellent psychologist in the 1960s on observing me teaching my handicapped child a simple sorting task, and saying: "You are very good and clear about saying "No" and less clear about saying "Yes". You should do it exactly the other way round." I thought: "That is a good lesson for a mathematician!" Also she suggested making the cards bigger and clearer!)

Another problem is that the language we use today in mathematics has a certain artificiality. When we say $2 + 2 = 4$ we don't mean the left had side is the same as the right hand side, but that the operation of adding gives the right hand side. So what ever language you use, a student has to get used to it, and used to expressing things in that language. John Baez gave the example that the picture

$$ \matrix{|| & ||| \cr ||& |||}$$

is much clearer that the expression $2 \times (2 +3)= 2 \times 2 + 2 \times 3$, which uses all sorts of conventions.

The above is a bit rambling but I hope has some useful points!

Answer 3

The original question mentions beginning students who are defeated by such statements as $\ker(AB) \supseteq \ker(B)$, and I suppose that it is this student population that the original poster ACL is concerned with, rather than more advanced students.

Such students need a very little bit of set language (not axiomatic set theory). And they need a very little bit of formal logic: logical connectives and quantified sentences. Maybe 10 pages altogether of what used to be chapter 0 in every more or less advanced math text. And they ought to be shown how to translate between set operations and logical connectives, e.g. AND $\leftrightarrow \cap$.

(Actually, it seems to me that I had an entire career in mathematics without really needing much more about foundations than what I just mentioned, and Zorn's lemma.)

I think the students to whom ACL refers almost surely will not know the definition of linear independence when they finish their linear algebra course because they do not understand how to use logical connectives and quantifiers.

It baffles me we don't teach these things explicitly and repeatedly, early and often, since one cannot actually do any mathematics, even at the level of a first linear algebra course, without this much "grammar", as Ronnie Brown calls it.

Answer 4

Historically, mathematics did not begin with set theory, and most working mathematicians today do not really care about foundations. Furthermore, it is known that not much axiomatic strength is actually needed for undergraduate math:

The body of research in reverse mathematics has established that weak subsystems of second-order arithmetic suffice to formalize almost all undergraduate-level mathematics. http://en.wikipedia.org/wiki/Reverse_mathematics

Some math is eternal. Low-level number theory, algebra, geometry, and analysis all have historical roots going much deeper than set theory or category theory or even formal logic itself. Let's not kid ourselves: As much as we may be devoted to them, Set Theory and Category Theory are ideologies that may not withstand the test of time. Ordinary mathematics can be formalized within them, but this is not the only way to go. The major players in the early 20th century French school of mathematics (Lebesgue, Borel, Baire, Poincare) all had severe reservations about set theory up to the ends of their lives. We should not deprive our students of essence of the beautiful eternal mathematics by forcing them to view everything through the lens of a modern ideology. Foundational studies can come later, after people have a good grasp on the unquestionable reality of basic mathematics.

Answer 5

As a set theorist, I feel some obligation to offer an answer here. First, the difficulties students may have in proving set theoretic containments like the one you mention above or in constructing $\epsilon$-$\delta$ proofs is not a matter of them struggling with set theory but rather of them struggling with something new: constructing a proof. In the case of $\epsilon$-$\delta$ proofs, a large part of this difficulty is in understanding quantifiers and how they work (for instance "for all ... exists ..." is not the same as "exists ... for all ...". This is because math is not a trivial subject to learn and some difficulty is required as the students minds stretch and grow. Surely there is no way around this.

That really has nothing to do with set theory so far. In spite of a common misconception, set theorists do no actually care how it is that ordered pairs are defined. Or how exactly one codes the notion of a function. Set theory is not in competition with category theory, in spite of what category theory thinks. A very good analogy is provided by computer science: set theory is machine language (or maybe better: a low level language like C) and category theory is object oriented programming. While object oriented programming may provide a useful way of thinking about how to write a program, there still needs to be a machine language for the computer to run on. Moreover, there are occasionally things for which it is just better (or even necessary) to code in a low level language.

Set theory provides an exact standard by which to discuss questions like "is there a subset of the real line which is uncountable but not of cardinality $|\mathbb{R}|$" (Hilbert's First Problem) or, maybe better, "is there an almost free, non free group?" (Whitehead's problem) or "If $h$ is a homomorphism a commutative Banach algebra into $C[0,1]$, is $h$ continuous?" (the negation being Kaplanski's conjecture). With the exception of the first question, these were asked, to my knowledge at least, without any thought that there was a foundational issue involved. Surely these are questions which could reasonably be asked regardless of how one sets up their foundations. To my knowledge category theory has never resolved these questions; set theory has in as satisfactory a manner possible (or at least until we adopt a more complete set of axioms). Now, one can argue at length about whether such questions are asked in poor taste or whether we should allow them to be asked at all. Readers interested in the question of "why care about set theory" should take a look at this (which might have been titled "why care about the uncountable").

Answer 6

There are some kids with that wonderful attitude of asking "why?" about anything. The existence of a few syntactically simple axioms beyond which you cannot ask why anymore, as opposed to the discouraging "turtles all way down" approach, can be comforting to such minds, I think.

In these cases, I feel that the existence, and even the variety, of foundations should be mentioned, and made intriguing, early, which is a very delicate task.

The problem, of course, is that probably a child isn't generally prepared to face the rigors of formality, and the complexity of what there is between axioms and doing $2+2$.

This too often leads to never mentioning foundational issues, not even at college: I did physics at university and the most foundational stuff I was ever exposed to were the Cantor set, Dedekind cuts and $\epsilon / \delta$ definitions.

I think that in early education, the operational approach, as opposed to theoretical definitions, can be more appropriate, partly due to the fact that elementary school is expected by the adults to teach children how to perform numerical calculations. Later, however, when the confidence with the object one is manipulating all the time, at least the glimpse of foundations should be given: even only so that those interested can dig it. I sorely regret that didn't happen to me when I was younger, for example.

It is interesting that the opposite approach (heavy set theory from an early stage) was experimented in the past, and with little success: New Math.

Answer 7

My comments to the main question seemed like they merit a full answer.

If we are talking about introductory level courses then the approach should be naive. It is often all the set theory needed from the working mathematician, and many of my undergrad teachers didn't even know what are the axioms of ZFC (a truth I'd learned during my masters).

It is important to add some logic into the mix, what is a structure and what is an isomorphism of structures. I can give from my own limited experience:

I did my undergrad (and masters) in Ben-Gurion where a course has been tailored not from books, but by collecting pieces of set theory and logic together. I have some disagreements on its current structure, but the idea is that we teach set theory naively from "ZF+The real numbers are atoms" because that's how most people would see mathematics when they approach it naively. I, for example, disagree with the insistence of not mentioning the axiom of choice. In the set theory part we explain what are sets, their basic properties, we teach about induction over $\mathbb N$ and a bit of the theory of partial orders. We also teach the basics of cardinality (as much as you can squeeze from no-choice environment anyway).

We also teach very basic propositional calculus and predicate calculus. Nothing fancy, and we don't talk about proofs and soundness or anything much. We discuss isomorphisms and definability if time permits (e.g. this year) but not always we have this privilege (e.g. last year).

As an intro course I think it is fine, it doesn't go too deep into axioms and what are proofs and so on. Students still don't understand what they need all that for, but the majority of the students are computer science students (the course, however, was designed over 20 years ago when the computer science department was a subset of the math department), and they just want to learn programming and so.

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For advanced undergrad courses it is perfectly reasonable to present the full axioms of ZFC and discuss advanced topics like forcing, large cardinals, infinitary combinatorics, and maybe even [very basic] inner model theory.

This, coupled with a course about logic and basic model theory, should give an undergrad an excellent grasp of the basics of set theory.

On the other hand, it is reasonable to suggest a course about categorical foundations in which ETCS and other structural set theories are presented and an algebraic approach to set theory is taken. I don't know what sort of perquisites should be for such course, though.

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But to repeat what I said at first, for freshmen students (or as a firs course in set theory) the course should be presented in a naive approach, relying on the fact that we all understand what it is to put three files into a folder in the computer, or three books in our bag. Such course should present the basic structure of sets and some basic logic.

If a full year is given, it might be wise to make it into two parts: the first is very naive and basic sets manipulations and basic logic, with the added value of very basic combinatorial results from discrete mathematics course. The second part should focus on slightly more advanced topics such as basic order theory (partial orders and such), cardinals and cardinality, some applications of the axiom of choice, and some more advanced logic (from definability to elementary equivalence, depending on your taste and time limits).