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?
Best Answer
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.