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Toby has just commented on an answer of mine to an old question on MO about teaching derivatives (link is to my answer). A particular phrase in my answer seems to have caused a bit of … interest. I said:
Many students will just want the rules. But if the students refuse to learn, that’s their problem. My job is to provide them with an environment in which they can learn. Of course, I should ensure that what they are trying to learn is within their grasp, but they have to choose to grasp it. So I’m not going to give them a full exposition on the deep issues involving the ZF axioms if all I want is for them to have a vague idea of a “set” and a “function”, but I am going to ensure that what I say is true (or at the least is clearly flagged as a convenient lie).
Toby’s comment was:
But the question is not whether the students refuse to learn but whether the environment that we are providing is good. Explaining an intuitive concept with a complicated definition is not conducive to that good environment.
I’m not sure whether or not the other comments are relevant.
This is a discussion I’d love to have. But the comment threads at MO are not well-designed for having such discussions. So since Toby is often here, I’m taking this opportunity to transfer it over here. This may not work, we’ll see.
So first, I’d like to ask Toby to expand a bit on his comment as I’m not clear as to the point being made. I could read that comment in such a way that it agrees completely with my quoted paragraph. But I may be misreading it.
My comment wasn’t a direct response to your answer but to your immediately preceding comment, which in turn was a response to a comment by JBL, which was a response to the part of your answer that you highlighted above. (In particular, by ‘the question’ I meant the question implicitly raised by JPL rather than one implicity raised by you or by the M.O questioner.) On the other hand, there is a larger context which I was also thinking about in my comment, which was the discussion of how best to introduce the concept of the derivative in an introductory calculus class. (So anybody reading this on the nForum should read at least Andrew’s answer at M.O and all of its comments, and probably also the original question and some of the responses, as comments and answers, to that.)
I teach at a community college to students who, almost universally, take my classes because they are required for a non-mathematical programme. Many (but by no means all) of them come to the class thoroughly frustrated with school ‘math’. To make things worse, I’m faced with a large college-mandated curriculum to cover in too short a time for some of these students; I haven’t yet figured out how to fit in any actual mathematics. But I want to teach them something that will actually be useful to them. This does not include (a) precise definitions, but it does include (b) hand-wavy explanations of what things mean, as well as both (c) rules for calculation and (d) experience setting up word problems. Of these, my students generally like (b&c) but not (a&d). With (d), I have to disagree with them for their own good, but if they don’t like it when my attempt at (b) becomes too much like (a), then I blame myself.
Much of the problem with that particular M.O question, of course, is that people are teaching classes to many different types of students and talking as if we’re all teaching the same class. In your classes, you may well be doing the best that you possibly could. But you seem to be arguing (in the entire paragraph quoted above) that if one covers material within one’s students’ grasp and they do not choose to grasp it, then they alone are to blame. Ignoring the possibility that one is explaining things poorly (which in your case I doubt), just because something is within their grasp and nominally part of the syllabus doesn’t mean that it’s a useful thing to cover. Their boredom may be appropriate.
There’s also the problem that the calculus curriculum, in particular, really is silly. I’ve been pushing Bridging the Vector Calculus Gap all over M.O today, so I may as well mention it again here. Their paper on freshman calculus is most relevant. If one abandons a rigorous definition of limit, as my college-mandated text does, starting with that becomes even more pointless.
If by (b) you mean to eschew giving the precise definition of a derivative or a limit, yet still try to get students to understand something about how the derivative is defined in order that it have the desired meaning, rather than being just a a list of rules for how to differentiate all the functions they are familiar with, then that seems to me in substantially the same spirit as Andrew’s answer, but adjusted appropriately for the level of students.
But when you do (b), you get to use whatever definition you like, without regard for whether that definition could be explained to the students in the slightly more rigorous of theoretical class down the hall. Consider these two approaches to teaching calculus:
Introduce limits with a hand wave based on $\epsilon$–$\delta$ analysis. Define derivatives in terms of limits, work out a few that way, then give the rules and use those from then on. Introduce Leibniz notation for derivatives, but warn students that differentials have no meaning by themselves. Define integrals as limits of Riemann sums (and fudge it, since even this is too hard) and state the fundamental theorem without proof. Suddenly introduce differentials again to do integration by substitution or by parts, calculating them by a subtle variation on how you calculated derivatives before.
Introduce differentials with a hand wave based on nonstandard analysis (or differential topology, or SDG, or anything else with a rigorous notion of infinitesimal). Give the rules (with at least one more hand wave), and define derivatives as ratios of differentials. Introduce the prime notation for derivatives of functions. Introduce limits for their own sake (hand wave) later on. Introduce integrals with another wave of the hand and state the fundamental theorem without proof. Like the other integration techniques, integration by substitution or by parts is simply the application of an old idea to a new situation.
There is more hand waving in (2) than in (1), and it would take more work to fill in the gaps in (2). For some purposes, (2) would therefore be a bad idea. But for other purposes, it’s a good idea. In particular, it removes the problems that annoyed the OP on M.O, of the form ‘Differentiate $f(x) = 2x^2 + 3x - 2$ by determining $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}h$.’ (This is quoted from half a page of similar problems in a calculus text that I’m supposed to use.) Students in an applied course who find those problems pointless are not refusing to learn; they’re correct.
I would love to try (2), but I’ve never had that much freedom in designing my own calculus course yet. I admit that am somewhat apprehensive about giving students any latitude to think that “infinitesimals” and “infinite quantities” are even sometimes valid, since they are often quite willing to treat “infinity” as a number even when they are explicitly told not to. Although I know of course that infinity and infinitesimals can be made precise as “numbers” in plenty of ways, I am unsure of my ability to convey to beginning calculus students how and why it’s valid to use infinity in some ways, but not in other ways. (A physicist might say “you have to learn through experience when it’s okay and when it gives problems”, but as a mathematician I don’t feel comfortable with that.) At least saying “the symbol $\infty$ does not represent a number, but rather a limit process” gives a consistent thing to fall back on that answers all questions in the same way; even without a precise definition of limit, the idea of limit is straightforward; more straightforward than the transfer principle of NSA or the intuitionistic logic of SDG.
When you teach (2), what do you use for a textbook?
Where I am teaching, my flexibility is this: There is a required book, which the students buy before they ever talk to me, and a required set of topics to cover, but the detailed syllabus is only suggested. If I use another syllabus, then I must tell them, which I do, and they pay no attention. As long as nobody complains and I give my students a grade, then I can do what I want.
I am starting to teach an Applied Calculus course, and (since we’re not being rigorous anyway), I’m trying (2). It has a web page. In the back of my mind, I assume that all variable quantities are real-valued smooth partial functions with open domain on a fixed but anonymous $1$-dimensional smooth manifold, interpreting the differential operator in the sense of calculus on manifolds, so that I know what’s true, but I never say any of those words.
I don’t agree that the idea of limit is any more straightforward than the idea of infinitesimal change, although it turns out to be easier to make precise (at least in classical mathematics).
I didn’t say I thought that the idea of limit was more straightforward than the idea of infinitesimal change! I said I thought it was more straightforward than the transfer principle or intuitionistic logic. What I was trying to say is that the mere idea of infinitesimal change, though no harder (and arguably easier) than the idea of limit, seems to me to be easier to “go wrong” with, in addition to being harder to make precise.
OK, yes, it is easier to go wrong. For purposes of this class, if we stick with smooth variables[*], then we will be all right, but this means that we can apply only smooth operations to them, and thus no taking roots of zero. I should draw their attention to the fact that even $\mathrm{d}(\root{3}u)$ is undefined where $u = 0$ (as suggested by the Power Rule, which gives division by zero) and remark that calculus as we are studying it therefore doesn’t apply there (although a more sophisticated analysis can).
[*] The term ‘smooth variable’ should be interpreted in much the same way as ‘random variable’, which Lawvere would argue is in accordance with the original intended meaning of ‘variable’. In class, I often say ‘smoothly varying quantity’, although this becomes simply ‘variable’ on second reference.
[…] the mere idea of infinitesimal change, […] being harder to make precise.
Maybe that’s more a matter of being brought up with one or the other. As you know, Anders Kock in his textbooks is meaning to demonstrate that the notion of infinitesimal is easier to teach and use in an elementary fashion.
Urs 10: for some purposes nilpotent infinitesimals suffice. But in general for calculus, one has often the case when working with first order infnitesimals that in computation with first order infinitesimals gives the result in whichthe first order contributions cancels and the second order remnants matter and so on. So, while for geometry many things can be done systematically with nilpotent infinitesimals, the non-nilpotent ones like in nonstandard analysis are needed; also the calculations there are not just algebraic manipulations, but many other theorems appear using infinitesimals (and infinitely large ones!) where sometimes transfer principle is really useful. So I think that SDG is really not a full replacement for the methods of nonstandard analysis, though for some purposes it may be. (I know that in Moerdijk-Reyes there are some nonnilpotent infinitesimals as well, but it is not full story yet, and I personally do not understand if there are also various sizes of infinitely large there).
My impression that it’s easier to go wrong with infinitesimals is based on my experience teaching calculus and the sorts of mistakes that students make in trying to use “infinity.” It makes me sad, because I would love to teach with infinitesimals, so I don’t think that I am just being biased by my upbringing.
Also there is integral calculus to consider, about which nilpotent infinitesimals seem to have very little to say.
Also there is integral calculus to consider, about which nilpotent infinitesimals seem to have very little to say.
Actually, integral calculus comes in naturally in the $\infty$-categorical version of contexts for nilpotent infnitesimals:
For $X$ a manifold , let $T X = X^{(\Delta^\bullet_{inf})}$ be its infinitesimal singular simplicial complex (which is not an interal hom, therefore the parantheses in the exponent). A morphism $X^{(\Delta^\bullet_{inf})} \to \mathbf{B}^n \mathbb{R}$ is canonically identified with a closed smooth $n$-form on $X$. The homotopy classes of such morphisms, relative boundary, are naturally identified with the integral of these forms over $X$.
I used to like to call this phenomenon “integration without integration”. There is detailed discussion of this at Lie integration (where it serves to show that the formal Lie integration of the $L_\infty$-algebra $b^{n-1} \mathbb{R}$ is $\mathbf{B}^n \mathbb{R}$) and at infinity-Chern-Simons theory (schreiber) (where it serves to give a intrinsic $\infty$-topos theoretic way to speak about $n$-volume holonomy of a circle $n$-bundle with connection).
Notice that I am not claiming to participate in the discussion about how to teach calculus. I am just thinking about the general question of wich mathematical notions of infinitesimal are natural. But actually the proof of the “integration without integration”-statement above is a pretty simple exercise in Stokes’ lemma. So if prepared with care, it might actually be usable forteaching. I’d have to think about that.
Something which just occurred to me now is this: in SDG can we talk about non-analytic functions? Doesn’t the microlinear property/axiom ensure we have Taylor expansions? (perhaps this is where my reasoning has gone wrong) Or at the very least, we can’t restrict a function $R\to R$ non-analytic at 0 along $D_\infty\to R$ and then recover the original function, and this seems to me to be counter to the general feel of SDG. As I said, I may have this completely wrong…
On the teaching side, I applaud what Toby is doing; I wish I could do something similar if I was teaching. We have approx. 900 first year students studying the main mathematics course at that level (mostly engineers!), and multiple lecturers teaching the course in parallel sessions, so it is out of the question :-(
@Urs #13: So if I ask you to integrate $x^2$ from $0$ to $5$, you can do it using Lie algebras in SDG? Can you explain why it gives you the area under a curve because it’s adding up lots of little rectangles?
@David #14: My understanding is that the microlinear property ensures we have Taylor expansions, but doesn’t guarantee that these expansion converge to the function for any non-infinitesimal input.
@Mike - ah, that makes sense.
So if I ask you to integrate $x^2$ from 0 to 5, you can do it using Lie algebras in SDG?
So my claim is that in any $\infty$-SDG context there is naturally an incarnation of the following classical fact and its ingredients:
the classical fact is this: on the set $\Omega^1(D^1)$ of smooth 1-forms on the standard interval , consider the equivalence relation
$\Omega^1_{cl}(D^2) \stackrel{\to}{\to} \Omega^1(D^1)$induced by the two hemisphere boundary embeddings $D^1 \hookrightarrow D^2$. The quotient of this is naturally an $\mathbb{R}$-torsor with 0, hence is $\mathbb{R}$ itself. The image of any 1-form $f d x$ in the quotient is $\int_0^1 f(x) d x$.
$\Omega^1_{cl}(D^2) \stackrel{\to}{\to} \Omega^1(D^1) \stackrel{\int_0^1 }{\to} \mathbb{R}$The good thing about this (for me, at least) is that this can be rephrased very abstractly in any suitable $\infty$-SDG context (which I would call a "cohesive $\infty$-topos with infinitesimal cohesion").
But what it does not give is what you ask for here:
Can you explain why it gives you the area under a curve because it’s adding up lots of little rectangles?
No, I can’t see how to get the algorithm called the Riemann integral internally this way. The perspective here is different. I believe the way to think about what I say exists nicely is this: integration of functions is a special case of the general concept of push-forward in cohomology (here: in de Rham cohomology). All things cohomological exist naturally in $\infty$-toposes. Synthetic de Rham cohomology exists in suitable SDG-like $\infty$-toposes. Combined this gives a notion of integration of forms, hence of functions, in SDG-like $\infty$-toposes.
So, as I said, I don’t claim that this has any relevance for teaching calculus. It is just to point out that in contexts with nilpotent infinitesimals, there is a nice way to speak of integration.
My infinitesimals are 1-forms; whether these are nilpotent depends on which multiplication you use.
@Toby: how does the argument go that, say, the derivative of $x^2$ is $2x$, with infinitesimals that are 1-forms?
@Mike from the notes of Toby’s that I’ve looked at, you form $d(x^2) = 2x dx$, and only worry about extracting the $2x$ later. But I haven’t put in the thought as to how this links with the high-end stuff so much.
So if I understand well then Toby is talking about formal differential calculus, not really about infinitesimals in the sense of either SDG or NSA. While it is true that the fact $d x \wedge d x = 0$ is closely related to the nilpotency of infinitesimals in SDG, I would not quite call a differential form an “infinitesimal”, strictly speaking. It is rather a function on a space with infinitesimal extension. That difference will not matter for the purposes of Toby’s in teaching, but I think it does matter in the context of some of the discussion we had here.
Here is a nice abstract way to do $d (x^2) = 2 x d x$ with infinitesimals and functions on infinitesimal spaces (not meant as being relevant for teaching this stuff, just for its own sake, sorry if this hijacks this thread here):
Go to the $\infty$-topos $\mathbf{H}$ of synthetic differential infinity-groupoids. In there is canonically for each $X$ the $\infty$-groupoid $\mathbf{\Pi}_{inf}(X)$ of infinitesimal paths in $X$ (the de Rham space of $X$). Morphisms $\mathbf{\Pi}_{inf}(X) \to \mathbf{B}^n \mathbb{R}$ are infinitesimal flat parallel transport over $n$-dimensional infinitesimal paths with values in $\mathbb{R}$. These are canonically identified with closed $n$-forms. The functor $\mathbf{\Pi}_{inf}$ has a right adjoint $\mathbf{\flat}_{inf}$. There is a canonical morphism $\mathbf{B}^n \mathbb{R} \to \mathbf{\flat}_{inf} \mathbf{B}^{n+1} \mathbb{R}$. This is the (higher analog) of the Maurer-Cartan form on the $(n+1)$-group $\mathbf{B}^n \mathbb{R}$. In particular for $n = 0$ this is the ordinary Maurer-Cartan form $\theta = d x$ on $\mathbb{R}$.
Now let $f : X \to \mathbb{R}$ be a function. For instance $X = \mathbb{R}$ and $f(x) = x^2$. Then form the composite
$d f = f^* \theta : X \stackrel{f}{\to} \mathbb{R} \stackrel{\theta}{\to} \mathbf{\flat}_{inf}\mathbf{B}\mathbb{R}$or equivalently its adjunct
$d f : \mathbf{\Pi}_{inf} X \to \mathbf{B} \mathbb{R} \,.$This is canonically identified with a closed 1-form, namely with the familiar $d f$. For instance $d f = 2 x d x$ for $f(x) = x^2$.
(I should say that a detailed write-up of the technical details behind this story I have so far for the context of smooth infinity-groupoids only, need to do it for the synthetic-differential infinity-groupoids, too. But once one establishes the “de Rham theorem for $\infty$-stacks” as indicated in that last entry, the discussion is analogous. )
@David: Yes, but what I meant was, the usual argument goes something like $(x+dx)^2 = x^2 + 2x dx + (dx)^2$, and we ignore the $(dx)^2$ so that the change in $x$ is $2x dx$. In NSA, we ignore the $(dx)^2$ because it’s a higher order infinitesimal; in SDG we ignore it because it equals zero. Maybe with 1-forms we can ignore it because $dx \wedge dx = 0$, but what I don’t see is what “$x+dx$” even means, if $x$ is a number and $dx$ a 1-form.
A 1-form is not an infintiesimal, but a function on infinitesimals.
Precisely, let $D$ be the infinitesimal interval, then a 1-form on the line $X = \mathbb{R}$ is a function on $X^{(\Delta^1_{inf})} \simeq X \times D$ that vanishes when restricted along the inclusion $X \to X^{(\Delta^1_{inf})}$.
So if you start with a function $f : X \to \mathbb{R}$ such as $f : x \mapsto x^2$ you can form another function $\tilde f$ by precomposing with $+ : X \times D \to X$. With $x$ a (generalized) point in $X$ and $\epsilon$ a (generalized) point in $D$, this yields the new function
$\tilde f : (x, \epsilon) \mapsto f(x + \epsilon) = f(x) + \epsilon f'(x) + (\mathcal{O}(\epsilon^2 ) = 0) \;\;\;\;\; (1) \,.$We can also precompose $f$ with the projection $X \times D \to X$ to get another function that we should still call $f$. Then the difference
$\tilde f - f : X \times D \to X$has the property that it vanishes when restricted along $X \to X \times D$, so is a 1-form. Indeed, this is the function that sends
$(\tilde f - f ) : (x, \epsilon) \mapsto \epsilon f'(x) \,.$Dually the function algeba of $X \times D$ is
$Hom(X \times D, \mathbb{R}) \simeq C^\infty(X) \otimes \mathbb{R}[\epsilon]/(\epsilon^2)$where I am again using the letter $\epsilon$, but now for something different, for the generator of $C^\infty(D)$.
In this notation we may identify
$\tilde f - f = \epsilon f' \;\;\;\; (2)$So the $\epsilon$ here is a function on the infinitesimal space $D$ and this is what can be written $d x$.
But in this step there is a slight abuse of notation, and that’s the one appearing in the above discussion: the first $\epsilon$ in (1) is a generalized point of $D$, as such it could be added to the generalized point $x \in X = \mathbb{R}$ under the inlcusion $D \hookrightarrow X$. But then in (2) I used the same letter for a function on $D$. In the first sense it is an infinitesimal, in the second it is a function on an infinitesimal space.
I have started typing at differentiation a section optimistically titled
In algebraic geometry people often talk about the duality between
the infinitesimals (of all orders: note the filtration there and the filtration of infinitesimal neighborhoods) and
the regular differential operators (of finite order, by the definition, with the order filtration).
I know a bit about it, but do not know how to canonically treat this issue so I never wrote a sensible exposition of it in $n$Lab. I do not know if it is treated in standard exposition of SDG.
One of the hand-waves in method #2 in comment #4 above (the parenthetical “with at least one more hand wave”) is for the product rule $\mathrm{d}(u v) = v \,\mathrm{d}u + u \,\mathrm{d}v$. For this you do need to say (and I did say) something like “and we ignore this doubly infinitesimal square here”. Once you have the product rule, $\mathrm{d}(x^2) = 2x \,\mathrm{d}x$ is an easy special case.
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