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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?

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ETCS was originally defined precisely for pedagogical purposes, and the textbook Sets for mathematics is written to that end. –  David Roberts Jan 7 '13 at 9:34
Set theory should be taught naively to students. It should be guided by the axioms which you believe describe properties of sets (do they have power sets? and so on) but it should be done naively. Most mathematicians require only this from set theory, and it's fine. It should not obstruct a student from studying classical set theory (read: ZFC) or algebraic set theory if they choose so later in their career, and it should not burden the student with additional axioms and so. –  Asaf Karagila Jan 7 '13 at 9:47
Asaf, why teach such a beautiful subject only naively? You wouldn't insist that algebraic topology or statistics be taught only naively, would you? (Although it would be possible to do so.) Naive set theory is merely for introducing the subject, for those who know nothing. But students can go far beyond this. The actual subject of set theory is filled with fascinating questions that undergraduates can understand: transfinite recursion, different cardinalities, large cardinals, Souslin trees, infinite combinatorics. Most universities offer such kind of courses in set theory. –  Joel David Hamkins Jan 7 '13 at 10:34
Joel, I am not insisting that this is going to be a naive course and never mention this topic again. I'm not even saying to exclude an advance course in set theory from the list of required courses. But I do think that at the very basic level, a naive approach is sufficient and this approach can take you very far in mathematics as much as your set theory needs are concerned. I think that freshmen will not grasp the delicacy behind large cardinals. I know that I was well into my grad studies when I finally understood why they don't form an inherent inconsistency. Similarly with forcing. [cont] –  Asaf Karagila Jan 7 '13 at 10:41
[...] But the question seemed to ask, at least in my view, whether introductory courses should follow ZFC or ETCS (or variants thereof). To this I answered in my previous comment. An introductory course should be in naive set theory, it should mention the axioms and the philosophy the naive set theory is modeled after, but it should not interact with it too often. In advanced undergrad courses it would be fine to talk about either ZFC or ECTS. Both have their merits, although ECTS would probably have a category theory perquisite, which sounds quite grad-level to me already. –  Asaf Karagila Jan 7 '13 at 10:43

7 Answers 7

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.

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+1, especially for "We should always, always distinguish a function $f$ from its value $f(x)$ at $x$". But I don't get your next point: If we define (say) $y$ := $x^2+1$, then what is wrong with saying that (the variable) $y$ depends on (the variable) $x$? –  John Bentin Jan 7 '13 at 13:49
Hmm, I sense another blog post. –  Andrej Bauer Jan 7 '13 at 18:47
Though I'm philosophically sympathetic to these concerns about notation, there is a risk that such an attitude will provoke the physicists to insist more loudly on teaching their own "math for physics" classes rather than letting the mathematicians teach mathematics. –  Timothy Chow Jan 7 '13 at 21:55
There is nothing wrong with slight abuse of notation if one is aware of it. But it is very much wrong when students are never taught the non-abusive version, so they do not even get a chance. As far as physicists are concerned, they are entirely right. The sort of analysis we teach their physics students is detached from what they use later on. We should teach infinitesimal analysis to physics students, that is what they need. –  Andrej Bauer Jan 8 '13 at 7:17
In reply to Andrej: I liked doing some Euclidean geometry for first year students, since one could then show that the value of a proof was to convince that something surprising was true! We don't want to convince students that detailed formal proofs are about writing boring arguments for boring conclusions. Compare that with Miguel's theorem (see wikpedia) where the proof uses a theorem on cyclic quadrilaterals, and its converse. –  Ronnie Brown Jan 8 '13 at 14:35

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!

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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.

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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.

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Equally speaking, the relative freedom of speech and from slavery, as well the wonders of the technological era, all those are quite new and will certainly fail to withstand the test of time. Should we all go live about our lives in a cabin somewhere in the woods? :-) –  Asaf Karagila Jan 7 '13 at 17:17
No Asaf, this is not a good analogy. First these are political/moral issues, not science, and I absolutely disagree that freedom may fail to withstand the test of time, but we can debate that elsewhere. Technologies change, but these are empirically demonstrable things unlike set theory, and new technologies build on old. Furthermore I am not saying you can't teach foundations, only that you should do that after they know a good deal of "real" mathematics. If a new revolution in mathematics comes, people should be equipped to deal with it by knowing the basic truths that must be preserved. –  Monroe Eskew Jan 7 '13 at 17:23
Oh you are reading too much into my analogy. Let me try once more (with feeling), rigor is a new invention in mathematics. It actually dates roughly as set theory. Should we start by being non-rigorous? May we assume that all functions are continuous and infinitely differentiable almost everywhere like the good people of the late 18-th and early 19-th people assumed? Mathematics moves forward, and while we cannot ditch the past completely, there is so much to teach an undergrad that starting with "real mathematics that will survive the collapse of the current foundation" seems strange to me. –  Asaf Karagila Jan 7 '13 at 20:11
@Asaf, Weierstrass was not a set theorist, and his pathological functions had no inspiration in set theory, in fact Cantor was his student. Rigor in analysis is its own thing, and owes little to set theory. To equate set theory with rigor itself is nothing more than ideology. Find your local complex analysis expert and ask when was the last time they used anything more than a small fragment of ZC, or even went beyond what can be formalized in 2nd order arithmetic. I am just saying that the current foundations may collapse, but more basic things will live on. –  Monroe Eskew Jan 8 '13 at 4:21
@Marco: I go further and say that particular theories like ZFC are NOT necessary for rigor, or creativity. I do not deny that they are "exciting," but I say they are somewhat removed from the absolute math for which they serve as foundation. Now, to address the "subjectivity," if you know anything about relative consistency results, or even just some history of math, it is clear that basic math is much more eternal than current foundations, which may actually die. Finally, I DO NOT DISMISS THE IMPORTANCE OF FOUNDATIONAL STUDIES. You totally misunderstand me. –  Monroe Eskew Jan 8 '13 at 4:27

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").

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"In spite of a common misconception" -- I can't think of anyone who has that misconception. I think everyone agrees that everyone agrees that [sic] the choice of coding is irrelevant in practice. Also, your computer science analogy suggests that you do not believe categorical foundations can function autonomously, that it has to run on top of a (presumably membership-based) set-theoretic substrate ("there still needs to be a machine language..."). Finally, "category theory has never resolved" - category theory never set itself the task of resolving e.g. CH, since it had already been resolved. –  Todd Trimble Jan 9 '13 at 15:24
Would you consider ETCS a machine language (in your analogy) like ZFC? –  Toby Bartels Jan 9 '13 at 15:24
Dear Justin. Thank you for your answer. Three comments - which apply to all of the other answers. 1) I understand that writing a proof is difficult. It's difficult for all of us, in fact, even if those proof are no more hard for the grown-ups we've become. However, my impression is that building mathematical objects as sets makes the thing more complicated for the young students. As a colleague likes to recall, it was asked as an exercise in France to compute the cardinality of the set $\mathfrak P(\mathfrak P(\mathfrak P(\mathfrak P(\mathfrak P(\emptyset)))))$. I'm not sure it helps... –  ACL Jan 9 '13 at 15:54
.. understanding something at proofs. 2) About coding: Not that I don't love my living country, but on the other side of the Atlantic Ocean, things are not so clear. I know a lot of people for example who really protest at the idea that a quotient group as simple as $\mathbf Z/n\mathbf Z$ would not be equal to the set of equivalent classes mod $n$. –  ACL Jan 9 '13 at 15:55
"Set theory is not in competition with category theory, in spite of what category theory thinks." -- Fully agreed that set theory is not in competition with category theory, but "in spite of what category theory thinks"?? I don't think I've ever heard a category theorist say that set theory is in competition with category theory. On the contrary, I have heard category theorists say explicitly that there's no such competition. –  Tom Leinster Jan 10 '13 at 0:16

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.

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.

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).

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I cannot agree with the suggestion that forcing is an appropriate topic for an advanced undergraduate course, especially if category theory is not considered to be at the advanced undergraduate level. For one thing the main definitions and results of forcing are much more subtle. (For example, I get irritated when people handwave over things like the use of countable transitive models or generic filters, or when they say that $\Vdash$ is definable without clarifying that it is $\Vdash \phi$ that is definable for each $\phi$, etc.) –  Zhen Lin Jan 7 '13 at 13:29
Zhen Lin, in my undergraduate studies I was required to take a course which gave a short exposition to categories, but there was no course offering forcing (at all, not even for grad level). On the other hand in my current university there is a course which teaches forcing to undergraduates. I don't think they skip on categories in their undergrad, but the point is that some universities do it like this and others like that. I will also point out that categories were taught in a very handwavy way to me. The course that actually discussed categories was offered when I was in my masters, though. –  Asaf Karagila Jan 7 '13 at 14:35
Zhen Lin, after giving it some thought I removed that line. –  Asaf Karagila Jan 7 '13 at 21:22
I'm jealous, Asaf! I never saw category theory formally in undergrad, and only had the chance to take a course in logic leading to Gödel's incompleteness theorems, no set theory at all. –  David Roberts Jan 7 '13 at 23:19
David, the course only touched categories (and also discussed modules, and all sort of products). None of the guys understood it back then. Only a year later when I took a course in representation theory I understood the definition of a tensor product (in a huge coincidence, just two hours before the rep. theory professor gave the definition). As for set theory, the courses I took in logic and set theory were painful. We proved that you can only read a wff in a unique way, and it took us two classes. The exam consisted mostly of a complete proof of the completeness theorem. [...] –  Asaf Karagila Jan 8 '13 at 0:59

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.

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I have first-hand experience in teaching 2+2 from foundations, to an 11-year-old. The trick is to explain sets as "boxes" that contain "other boxes". (You can even use actual boxes!) As soon as the child can conceptualize a box that contains "all the finite boxes" (omega), she can come up with "infinity plus one" on her own. It involves a lot of careful wording and repetition, but it's totally doable, without scary-looking notation. –  Neil Toronto Jan 9 '13 at 4:14

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