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I am, by nature, a timid, centrist soul. I listen to Coldplay, drink lattes, and always begin calculus with a unit on limits. I trot out the Squeeze Theorem and the Intermediate Value Theorem and the definition of continuity like a diligent textbook follower.

But writing a book about calculus has radicalized me.

I’ve now fallen in with those hooligans and scofflaws who denigrate limits. Not the mathematical concept, of course; I just question the idea that, before meeting derivatives and integrals, the student must undergo a thorough, decontextualized study of the local behavior of abstract functions.

Where did the limit framework come from? In short, Augustin-Louis Cauchy and sundry 19th-century pals. Shaken by the tricky questions of Fourier analysis, which cast the whole concept of “convergence” into doubt, they sought to rebuild calculus from the ground up, at a level of technicality and arithmetical precision that no prior generation had felt any need for.

But calculus students haven’t seen any Fourier analysis! They don’t feel the intellectual need that Cauchy felt. The IVT to them is a piece of blinding obviousness, dressed up in ornate technicality.

Next time I teach calculus, I intend to join the rebels by diving right into differentiation, circling back to notions of convergence and continuity only when they arise naturally in context (e.g., discussing what “nondifferentiable” behavior might mean). If it was good enough for the Bernoullis, it’s good enough for me.

Go for it, man! But any time someone tells me that the IVT is “obvious,” suggest that they look at the function f(x)=X^2-2 on the RATIONAL numbers between 0 and 2. So there is something deep going on with real numbers. But yes, for an intro calculus course, “there is something deep going on with real numbers” may be all that needs to be said. Good luck with the book!

Complaining about the rational numbers excluding roots and then jumping to real numbers is a fallacy (Equivocation or motte and bailey fallacy). You can get to the algebraic numbers like this, but not reals.

I don’t think anyone was “Complaining about the rational numbers excluding roots and then jumping to real numbers”. Looking at how a function on the rationals fails to abide by the intermediate value theorem is part of what motivates the extension of the rationals in various ways. For polynomials failing to pass through intermediate values, it does indeed suffice to extend only as far as the algebraics; however, to build the full edifice on which differentiation stands, one needs more than that. A function proportional to its own derivative (i.e. an exponential) needs more than the analytics, even if only to say what the constant of proportionality is (2^x maps rationals to rationals just fine, but you can’t differentiate it without ln(2), which isn’t algebraic). The reals are designed to suffice to deal with all cases, quite possibly including many that no-one ever has any practical use for; what matters is that you don’t need more than the reals to make differentiation work systematically. If you can find a dense sub-set that suffices for all your needs, that’s great: but the assurance that at least some extension of the rationals does suffice comes from the fact that (by design) the reals do suffice (albeit they may be a bit over the top; after all, almost all reals are non-exhibitable).

Sorry: 2^x maps rationals to analytics, of course !

In the UK, when calculus is taught, limits are only given passing mentions when the definition of the derivative is introduced, or when the teacher is deriving the chain or product rules. Otherwise, limits (and especially the formal epsilon-delta definition!) are completely ignored and left until university.

Putting off the off-putting subject of limits is, if I may steal a phrase, blindingly obvious! Things like this, where students are told to trust the process and believe that someday it will all come clear why they had to accept those strange and obvious subjects *on faith*, is EXACTLY what puts off so many students at early stages. I bought into and was able to get to the circle-around stage where some of those arcane things did finally make sense, but I know plenty of folks who were long gone before then. As you make it very clear, a lot of those things simply don’t need more than a passing mention, if that, until the more solid foundations have been laid. Thank you for articulating it and for joining the ranks of the radicals who want to teach math in a way that makes sense, vs a way that makes some 19th century mathematician feel good.

To an “engineer”, math is a tool. Engineers are Babylonians. Engineers are Babylonians. Teach me a ever expanding collection of tricks and techniques to solve a wider set of problems. I don’t really care why they work, just that they work. Engineers will ask “when will I use this in the ‘real world'”? The theoretical underpinnings for calculus are unnecessary.

To a mathematician, math is an art from. Engineers are Greeks. From minimal foundation, where will logic take us. Show me the austere beauty of pure reason. That many of mathematical constructions become useful is irrelevant.

As it turns out Calculus is useful, and must be learned by engineers and mathematicians alike. The pure mathematician sees starting with the formal definition of limits as akin to beginning the day with clean underwear. It doesn’t matter if other people will never see your underpants, you know whether they are clean or dirty. The engineer might be going commando (maybe not). But, long as he isn’t exposing himself is that a problem?

As you have to teach both, and neither, I would guess that there are more engineers in high-school AP class.

If you gloss over limits, you should at least give a hand-waive to the mathematicians that you are skipping certain foundations, and at some point they will need to back-track to shore this up. And, If you go deep into the limits, you might tell the engineers that they can sleep through this part.

Yeah, yeah, I get your point, and I’m somewhat sympathetic to it. For years people were using Calculus quite happily without worrying about limits. Yet there were always those (including, famously, George Berkeley) who thought: This Calculus stuff may work, but there’s something about it that doesn’t quite add up.

And I’ve always been a little unsatisfied with books and articles that try to define Calculus as something like “the mathematics of change and motion.” That’s what Calculus is USED FOR, but it’s not what Calculus fundamentally IS. You can do Calculus without ever talking about anything moving or changing. As William Dunham put it in his (highly recommended) book A_Calculus_Gallery while discussing Weierstrass’s delta-epsilon limit definition: “Here nothing is in motion, and time is irrelevant,” which means we don’t have to “consider concepts of time and space before talking of limits.”

My own high school Calculus teacher took a highly theoretical approach. We spent a lot of time on limits, and proving things with deltas and epsilons, before we ever got to derivatives. I would never teach a Calculus class that way, but I have mixed feelings about being taught that way myself, and I do think it was pretty good preparation for becoming a math major in college. I wouldn’t expect or demand such rigor from a beginning Calculus class, but I think there’s value in at least giving them a glimpse of what rigor looks like. And I certainly agree that “before meeting derivatives and integrals, the student [needn’t] undergo a thorough, decontextualized study of the local behavior of abstract functions.” But I’m pretty sure that, the next time I teach Calculus, I’m going to stick with the approach where I introduce limits (without delving deeply into theory or proofs or pathological counterexamples) before going on to derivatives.

Why? Two reasons. One: I want to try to present things in a way that makes as much sense as possible. And I think I can explain derivatives, and what they are and why they work the way they do, better to someone who has the concept of a limit under their belt.

And two: Limits can be mind-blowing fun. “You know how you can’t divide by zero? Well, limits give you a way around that restriction.” “Here’s a function. When we plug in 3, it’s undefined, but when we look at the graph, it looks like f(3) ought to be 10. What’s going on?” And, infinity. Calculus gives you a way of talking about things “going to infinity” or “getting infinitely small,” in a way that isn’t just hand-wavey gosh-wow that’s really big, but where you can actually be clear about exactly what you’re talking about and back up your claims with logical proof.

Yeah, there’s something to be said against limits, but probably not a whole briefcase-ful.

“If it was good enough for the Bernoullis, it’s good enough for me.”
Your argument is akin to someone saying that Van Gogh and Riemann didn’t have vaccines so I will not vaccinate myself.
Bernoulli’s lived around 1700-1800. Naive set theory (Cantor) was created somewhere before 1900 and given a more rigorous standing in the twentieth century. The Bernoullis didn’t have set theory. Are you going to teach calculus without sets? With infinitesimals?

The reason limits didn’t have a rigorous standing was because they were a mean to an end (derivatives). By exploring and giving proper standing to limits we learnt that continuity is where mathematical analysis is rooted, derivatives are just a branch inside it. At some point in becomes necessary to redefine even the incremental ratio definition and use more abstract approaches.

Mathematical analysis professors still teach today in a way that forsakes the underlying sets. There is something that becomes less evident with this approach.

If you teach mathematics you should teach rigor. If you do not teach rigor then you are just forcing your students to remember by heart stupid useless formulas.

Hey, thanks for commenting! Looks like we disagree on ALL OF THE THINGS. I tend to adopt a conciliatory tone in such cases but the flu is diminishing my self-regulation capacities and your comment was unusually pugnacious & spunky so I SHALL RESPOND IN KIND.

For one, I think your reading of math history is anachronistic and Bourbaki-inflected (cf: Viktor Blasjo and Michael Barany). Gut check: What would Newton have said about vaccines? “Yes please, I wish not to die.” What would he have said about deltas and epsilons? “What the heck is this garbage for?” You’d need to show him Fourier analysis before he’d want the epsilons, and OUR STUDENTS DON’T KNOW FOURIER ANALYSIS.

Another gut check: the IB curriculum does more or less what I describe, without apparent calamity. In fact, it’s generally better than the AP Calculus curriculum (or most first-year calculus courses at US colleges).

Last: I super-110%-disagree with your final word on pedagogy! The opposite of “rote” is not “rigorous”! As I see it, the cycle of learning math is: (1) Play with ideas (e.g. what happens when I add a tiny increment to a square’s side length?); (2) Streamline and make rigorous those ideas (its area grows by 2 times the side length times the little increment); (3) Now they can be applied with rote automaticity (the derivative of x^2 is 2x).

You’re skipping Step #1, which drives student learning. Then you’re calling Step #2 the opposite of Step #3, when in fact it leads directly *into* Step #3!

“pugnacious” such a latin term it’s almost strange to hear it in english!
I didn’t mean to be aggressive, but I think that your last sentence “If it was good enough for the Bernoullis, it’s good enough for me.” really got to me as someone saying “the old ways are always better” and since it’s something you hear often these days I couldn’t believe I read it here. I already met a guy who wanted to read the euclidean way to do geometry, so I know these people exist even in math.

I think I understood next to nothing of what you wrote in your answer, so I need you to answer to some questions, if you want this conversation to reach somewhere.

“For one, I think your reading of math history is anachronistic and Bourbaki-inflected (cf: Viktor Blasjo and Michael Barany).”

What does it mean? What of what I said was Bourbaki inflected and what books/arcticles/sentences/something of Blasjo and Barany I need to read to see this?
I thought I just stated dates of when things were defined. The reason I brought Cantor in the discussion was just to add fuel to the fire, since even set theory wasn’t invented at the time of the Bernoullis. I called it “Naive” because I thought that Cantor didn’t give foundation to set theory before Russel paradox was even presented. Am I wrong? How Bourbaki come into this?

“Gut check: What would Newton have said about vaccines? “Yes please, I wish not to die.” What would he have said about deltas and epsilons? “What the heck is this garbage for?” You’d need to show him Fourier analysis before he’d want the epsilons, and OUR STUDENTS DON’T KNOW FOURIER ANALYSIS.”

I do not want to comment the sentence about vaccines since right now in the first world people are dying because they do not want to vaccinate but still refuse to accept reality.
I do not understand the other comment. I always thought that before epsilon-delta people used infinitesimals, without even defining them properly: “something not zero and positive but smaller that all other positive numbers.”. As Edward Welbourne Synthetic Differential geometry gives us a way to do it correctly, and even nonstandard analysis defined infinitesimals properly, something not exaclty 18th century-like. Since infinitesimals were something very ill defined I would not think that either Newton or Leibniz would offend epsilons and deltas.

“Another gut check: the IB curriculum does more or less what I describe, without apparent calamity. In fact, it’s generally better than the AP Calculus curriculum (or most first-year calculus courses at US colleges).”

Here I need some references because I do not know what those are. Still I don’t understand how you would define continuity and differentiability bypassing limits. All standard notion of continuity pass through the definition of topology, so since I know 8 definition of topological space I know 8 definitions of continuous function. There is one way to define a continuous function without passing through the topological one, that is defining a continuous function to be a Dedekind cut, still I do not think that this approach would be somewhere pedagogically acceptable.

“Last: I super-110%-disagree with your final word on pedagogy! The opposite of “rote” is not “rigorous”!”

I do not even understand from where this came from.

“As I see it, the cycle of learning math is: (1) Play with ideas (e.g. what happens when I add a tiny increment to a square’s side length?); (2) Streamline and make rigorous those ideas (its area grows by 2 times the side length times the little increment); (3) Now they can be applied with rote automaticity (the derivative of x^2 is 2x).”

I think it is more accurate to say “the cycle of teaching math”. Since I will do (1), you will do (1), but your students without any outside involvement would never do (1). Presenting the right example to motivate a definition or the need for it is always a good thing though. I still have an example stuck in my head as to why topological boundary is an incorrect notion to do analysis.

“You’re skipping Step #1, which drives student learning. Then you’re calling Step #2 the opposite of Step #3, when in fact it leads directly *into* Step #3!”

Why are you saying I am skipping it? When I said you can’t make examples to motivate the theory?
The last sentence I do not understand at all. To me you said:
(1) Ideas,
(2) Rigour,
(3) Exercise.

I said that if you abandon epsilons and deltas you are abandoning rigour. I do not see how this sentence brought you to what you wrote.

P.S. Cauchy used epsilons and deltas in his proofs but did not defined them. Because of this he “proved” that pointwise limits of continuous functions is continuous.

Thanks for keeping this conversation going – I totally see where several things I said were cryptic and unhelpful. Let me try to elaborate more usefully:

1) The basic historical narrative – which I always heard as a math student and which I think you are also telling – is something like this. “Newton & Leibniz developed calculus, but it was a rough and unproven calculus. It worked, but no one could fully explain why. It took several centuries for calculus to find firm foundations, in the form of delta-epsilonic rigor.”

Barany and Blasjo have persuaded me that this narrative is basically ahistorical and false. There was no failure of rigor in early calculus; indeed, all of their work could have been verified by Archimedes-style arguments from the method of exhaustion, but Newton and Leibniz viewed that as a tedious and unnecessary chore. Instead, although “Euclidean proof” is an ancient idea, “rigor” is a distinctly 19th-century one. It was a response to a new intellectual need.

As I understand it, Bourbaki (and others at the time) were the first to tell the history the way we now tend to hear it – with Berkeley’s early philosophical criticism foregrounded, and a sense of “something missing” from calculus until it was reframed as the kind of mathematics that Bourbaki preferred.

B) After that, Viktor Blasjo’s site “Intellectual Mathematics” is a trove of great material on the history of calculus and analysis, esp. his infinitesimal calculus course and his history of math reader.

2) That’s what I can say about history; onto pedagogy. First, I should clarify: My original post is about teaching calculus, not analysis! Obviously limits should be taught – in analysis. It’s calculus where they have little pedagogical use.

As I said, the practice I’m suggesting is far from uncommon.

One example: the IB is the “international baccalaureate,” a highly regarded international curriculum. Its higher-level math course is far more challenging and interesting than AP (“advanced placement”) calculus, which is the standard course in the US. The former has virtually no discussion of limits; the latter has an extensive one, spanning perhaps 20% of the course, to no obvious benefit.

Another: although it is standard practice at research universities in the US to teach delta-epsilon limits in an introductory calculus class, every British colleague I’ve spoken with finds this absurd. They wait until analysis to introduce the concept.

3) Here is the statement of yours that prompted my response: “If you teach mathematics you should teach rigor. If you do not teach rigor then you are just forcing your students to remember by heart stupid useless formulas.”

I interpreted this as a claim that “rigorous” and “rote” are opposites. Indeed, that’s what you say – in the absence of rigor, all math education devolves into rote memorization of formulas.

This view is so opposed to my own – so bizarre and so wrong to me – that I don’t know what to make of it. Of course rigor plays an important role in mathematics education, but that role is not to prevent mathematics from feeling “stupid,” “useless,” or memorization-driven! Indeed, take a poll of students, and i suspect many will associate their experiences of rigor with precisely those qualities.

Also: do you really mean this to apply to primary and secondary math education? Or when you say “math education” do you mean “the education of advanced undergraduates”?

(FOREWORD: I hope everything is clear, and even if it is not I hope that I never seem hostile or badmannered. I am not sure if you will respond in any way, or even if you will read it at all. I want to say that even in our disagreement I like your blog and what you write and I have been following you from the post “why do we pay pure mathematicians”. Maybe if the conversation continues it would be better to change it to an email conversation and tone down the number of questions answered by a single post.)

As always wordpress doesn’t consider conversations longer than 4 comments, so I will tag this post instead of the other.

I think there is some colossal misunderstanding in the usage of some words that poisons our conversation. What does the word “rigour” mean to you?

Most definitions in the online dictionaries don’t really appeal to me. The best one I found says “The quality of being extremely thorough and careful.”, and I think that I will not find a better one.

What is the opposite of being rigourous? It is being sloppy. Ironically enough mathematics is permeated with sloppiness.

After reading Barany’s paper I failed to see why you would link it in here. In there he says that Cauchy was the first to want to have a written proof of the IVT, although in his book he wrote that as a tautology and in the appendix gave a more correct proof. It even seems that he does not use even the continuity hypothesis, but just says that the thesis must follow obviously. His definition of continuity is
“continuous functions are those functions for which the difference f(x+α)−f(x) is infinitesimally small when α is” and is not used in his proof. Question: how can the proof be true if the only hypothesis is not used explicitly? Simple: it is not true, since when f is not continuous the IVT thesis fails, thus showing the sloppiness.

After reading the first chapter I was almost dumbfounded to not find even one definition.

Let us continue with the sloppy math and look at the definition of continuity of Cauchy presented by Barany. To be able to understand it we need a meaning for “function”, for “difference and sum” and for “infinitesimally small”.
Cauchy died in 1857, before set theory.

I don’t know how they thought about math things before set theory but from what I read they just didn’t care.
The “definition” of a function as a rule which gives an output from an input is really too much of an undefined concept. What would it mean? You have a universe of concepts/objects in which you move/live and you can take some of these concepts and manipulate them into something different inside this universe. Pretty vague if you ask me.

With set theory we gain sets. A function is just a well defined set. We like to have a well defined domain and codomain before trying to write functions (even though more often than not it happens otherwise.).

Let us say that the reals were well defined and as such difference and sum were easily defined concepts.
What is the infinitesimally small? Since I sum it to a real number is it a real number? Is it on the real line? Is it something else? Is it somewhere else? What is it?
This is sloppiness.

Classic analysis says that infinitesimals must be real numbers and gives you epsilontic formalism, an infinitesimal is just a sequence converging to zero.
Nonstandard analysis, born in the 1900, says infinitesimals are not on the number line, and need to be added to the reals. You must be able to sum, invert, multiply and order them. You get a bigger set of reals which I now know are called hyperreals.
A third way is to introduce a quantity “e” such that “e^2=0” and so “f(x+e)=f(x)+e f'(x)”, but I’m not 100% about this one.

Since the beginning I never said that the epsilon was the only way, but a way is needed nonetheless. If you choose to forego epsilons than what are you going to choose?
The dx in the calculus book of Blasjo is something that isn’t defined, from that moment begins the sloppiness.

Moving forward: I do not distinguish between calculus and analysis. I didn’t know those were separate concepts and I don’t know where to draw the line. My best guess is that calculus is the introduction to mathematical analysis, that is one-variable and multi-variable calculus. Still this guess doesn’t sit right with me.

In my first post I asked how would you define continuity without epsilons and deltas, and I am still curious about it, Blasjo doesn’t define it, but to be fair he doesn’t define the derivative and without the definition he cannot prove the properties of the derivative although he says they are evident.

On your last point on rigour.
Your interpretation of my words about rigour and useless formulas is just a little bit off. Mathematics is about rigour and memorization, if they were opposites such thing couldn’t be true. If I remove rigour from math I would get not rigorous math, that is sloppy math, which I still need to memorize. Sloppy means that some things are not explained thoroughly and well pinned to the ground. This sloppiness can leave some question unanswered and a sour taste in your mouth.

When I say that abandoning epsilon and delta you abandon rigour I mean if you just introduce sloppily some dx and call it infinitesimally small. If you choose one of the other two stances mentioned above you are still being rigorous. If you say that in high school or before you don’t need to delve so deeply in technicalities is your choice but you must admit that you are loosing something.

Since the topic of this post was the IVT and you wrote “it takes years to pin down”, what is your definition of continuity? Note that this is a different question to the one asking how would you define continuity without epsilontic analysis.

I am not sure at what you mean for primary/secondary education since I studied limits only in my last year of high school. To me teaching to high school students would mean to show how interconnected is all the knowledge given to them so that even if something is forgotten it can always be recovered. Still I know that some things must be edulcorated for high school students before giving it to them, but I would always prefer not to cut corners.

In many ways, the important thing about it isn’t the theorem itself: it’s the exercise of clarifying what our intuitions about continuity are and encoding them in a rigorous way that makes it possible to even have a rigorous statement of what the theorem asserts (much less prove it). That the theorem is easy to prove once one has built this edifice is just the verification that we’ve built the right edifice.

However, I would agree with Ben that this isn’t the right place to start when first introducing students to calculus. I was educated in the UK where, as edderiofer says, school (i.e. up to age 18) calculus only takes a cursory look at limits, trusting that the situations where we’d need them would be ones in which it was easy enough to see whether there was a limit or not, without needing the more stringent rigour that I was later taught at university. That intuitive notion of limits was enough to give us a taste of things we could make sense of – product rule, chain rule and so on, plus the exponential function – without having to slog through a mountain of epsilons and deltas. Those who went on to specialise in science and engineering go the useful parts of calculus without the tedious book-keeping; and those of us who went on to mathematics had a solid sense of what those epsilons and deltas were buying us by the time we had to face them.

In any case, conceptually it’s possible to deal with derivatives without those epsilons and deltas (look up “Synthetic Differential Geometry” by Anders Koch, some time – warning: heavy use of category theory) so I look on them as what computer programmers refer to as “implementation details”: the high-level description of what tools you get and what they can do for you is worth understanding in its own terms, without being distracted by how those tools are implemented. If the implementation is done right, you don’t need to know about it, you only need to understand the high-level abstraction that it implements.

I think that in synthetic differential geometry the problem would be to be able to speak about not differentiable functions! The idea of changing the logic underneath from classical to intuitionistic is a lot scarier than using epsilon and deltas. Still you made me remember that people really brought back pre1900 math with topos theory and nonstandard analysis. It’s funny to think that the nonsense of old is now placed on solid ground.

You build a sequence which is explicitly Cauchy by dichotomy. The images converge to zero since if it did not it would mean that in a neighborhood of it the function doesn’t change sign which is absurd.

What it is you need to know besides the things required to understand the statement?

There is the proof by connectivity and the one by compact sets, which are maybe less basic since you need some knowledge.
Even though:
f^{-1}(0,+∞) and f^{-1}(-∞,0) are disjoint open sets in the domain if the function didn’t have zeroes the domain couldn’t be an interval.

The “things required to understand the statement” are exactly the defining properties of the reals that make the epsilon/delta method work; these are the properties that formalise out intuitions about continuity and the reals having “enough” values for those intuitions to work. These are equally the intuitions that prevent a union of disjoint open sets from being an interval, of course. (Contrast the rationals, in which {x: x.x 2} are both open and their union is all of the rationals.)

Back when I semi taught myself calculus in the local community college library during high school (it wasn’t offered at school), I skipped over the limits chapters. Didn’t make much sense to me at the time.

But I have a bolder challenge for you: why not dive into numerical methods early on? The analytic solutions are just a collection of special cases. With numerical methods you can take on some real world problems in the first semester.

And I think that the basic numerical methods provide much more understanding that the exercises provided in typical calculus books.

Numerical integration of differential equations is to calculus what manipulatives are to learning arithmetic.

After seeing numerical methods blow up in pathological cases, deeper understanding of limits and the like become interesting.

“How our understanding of calculus has evolved over more than three centuries, how this has shaped the way it is taught in the classroom, and why calculus pedagogy needs to change

He contends instead that the historical order—which follows first integration as accumulation, then differentiation as ratios of change, series as sequences of partial sums, and finally limits as they arise from the algebra of inequalities—makes more sense in the classroom environment.”

Go for it, man! But any time someone tells me that the IVT is “obvious,” suggest that they look at the function f(x)=X^2-2 on the RATIONAL numbers between 0 and 2. So there is something deep going on with real numbers. But yes, for an intro calculus course, “there is something deep going on with real numbers” may be all that needs to be said. Good luck with the book!

Complaining about the rational numbers excluding roots and then jumping to real numbers is a fallacy (Equivocation or motte and bailey fallacy). You can get to the algebraic numbers like this, but not reals.

I don’t think anyone was “Complaining about the rational numbers excluding roots and then jumping to real numbers”. Looking at how a function on the rationals fails to abide by the intermediate value theorem is part of what motivates the extension of the rationals in various ways. For polynomials failing to pass through intermediate values, it does indeed suffice to extend only as far as the algebraics; however, to build the full edifice on which differentiation stands, one needs more than that. A function proportional to its own derivative (i.e. an exponential) needs more than the analytics, even if only to say what the constant of proportionality is (2^x maps rationals to rationals just fine, but you can’t differentiate it without ln(2), which isn’t algebraic). The reals are designed to suffice to deal with all cases, quite possibly including many that no-one ever has any practical use for; what matters is that you don’t need more than the reals to make differentiation work systematically. If you can find a dense sub-set that suffices for all your needs, that’s great: but the assurance that at least some extension of the rationals does suffice comes from the fact that (by design) the reals do suffice (albeit they may be a bit over the top; after all, almost all reals are non-exhibitable).

Sorry: 2^x maps rationals to analytics, of course !

In the UK, when calculus is taught, limits are only given passing mentions when the definition of the derivative is introduced, or when the teacher is deriving the chain or product rules. Otherwise, limits (and especially the formal epsilon-delta definition!) are completely ignored and left until university.

Putting off the off-putting subject of limits is, if I may steal a phrase, blindingly obvious! Things like this, where students are told to trust the process and believe that someday it will all come clear why they had to accept those strange and obvious subjects *on faith*, is EXACTLY what puts off so many students at early stages. I bought into and was able to get to the circle-around stage where some of those arcane things did finally make sense, but I know plenty of folks who were long gone before then. As you make it very clear, a lot of those things simply don’t need more than a passing mention, if that, until the more solid foundations have been laid. Thank you for articulating it and for joining the ranks of the radicals who want to teach math in a way that makes sense, vs a way that makes some 19th century mathematician feel good.

Are you teaching Mathematicians or Engineers?

To an “engineer”, math is a tool. Engineers are Babylonians. Engineers are Babylonians. Teach me a ever expanding collection of tricks and techniques to solve a wider set of problems. I don’t really care why they work, just that they work. Engineers will ask “when will I use this in the ‘real world'”? The theoretical underpinnings for calculus are unnecessary.

To a mathematician, math is an art from. Engineers are Greeks. From minimal foundation, where will logic take us. Show me the austere beauty of pure reason. That many of mathematical constructions become useful is irrelevant.

As it turns out Calculus is useful, and must be learned by engineers and mathematicians alike. The pure mathematician sees starting with the formal definition of limits as akin to beginning the day with clean underwear. It doesn’t matter if other people will never see your underpants, you know whether they are clean or dirty. The engineer might be going commando (maybe not). But, long as he isn’t exposing himself is that a problem?

As you have to teach both, and neither, I would guess that there are more engineers in high-school AP class.

If you gloss over limits, you should at least give a hand-waive to the mathematicians that you are skipping certain foundations, and at some point they will need to back-track to shore this up. And, If you go deep into the limits, you might tell the engineers that they can sleep through this part.

Heresy! Limits are what Calculus is all about!

Yeah, yeah, I get your point, and I’m somewhat sympathetic to it. For years people were using Calculus quite happily without worrying about limits. Yet there were always those (including, famously, George Berkeley) who thought: This Calculus stuff may work, but there’s something about it that doesn’t quite add up.

And I’ve always been a little unsatisfied with books and articles that try to define Calculus as something like “the mathematics of change and motion.” That’s what Calculus is USED FOR, but it’s not what Calculus fundamentally IS. You can do Calculus without ever talking about anything moving or changing. As William Dunham put it in his (highly recommended) book A_Calculus_Gallery while discussing Weierstrass’s delta-epsilon limit definition: “Here nothing is in motion, and time is irrelevant,” which means we don’t have to “consider concepts of time and space before talking of limits.”

My own high school Calculus teacher took a highly theoretical approach. We spent a lot of time on limits, and proving things with deltas and epsilons, before we ever got to derivatives. I would never teach a Calculus class that way, but I have mixed feelings about being taught that way myself, and I do think it was pretty good preparation for becoming a math major in college. I wouldn’t expect or demand such rigor from a beginning Calculus class, but I think there’s value in at least giving them a glimpse of what rigor looks like. And I certainly agree that “before meeting derivatives and integrals, the student [needn’t] undergo a thorough, decontextualized study of the local behavior of abstract functions.” But I’m pretty sure that, the next time I teach Calculus, I’m going to stick with the approach where I introduce limits (without delving deeply into theory or proofs or pathological counterexamples) before going on to derivatives.

Why? Two reasons. One: I want to try to present things in a way that makes as much sense as possible. And I think I can explain derivatives, and what they are and why they work the way they do, better to someone who has the concept of a limit under their belt.

And two: Limits can be mind-blowing fun. “You know how you can’t divide by zero? Well, limits give you a way around that restriction.” “Here’s a function. When we plug in 3, it’s undefined, but when we look at the graph, it looks like f(3) ought to be 10. What’s going on?” And, infinity. Calculus gives you a way of talking about things “going to infinity” or “getting infinitely small,” in a way that isn’t just hand-wavey gosh-wow that’s really big, but where you can actually be clear about exactly what you’re talking about and back up your claims with logical proof.

Yeah, there’s something to be said against limits, but probably not a whole briefcase-ful.

“If it was good enough for the Bernoullis, it’s good enough for me.”

Your argument is akin to someone saying that Van Gogh and Riemann didn’t have vaccines so I will not vaccinate myself.

Bernoulli’s lived around 1700-1800. Naive set theory (Cantor) was created somewhere before 1900 and given a more rigorous standing in the twentieth century. The Bernoullis didn’t have set theory. Are you going to teach calculus without sets? With infinitesimals?

The reason limits didn’t have a rigorous standing was because they were a mean to an end (derivatives). By exploring and giving proper standing to limits we learnt that continuity is where mathematical analysis is rooted, derivatives are just a branch inside it. At some point in becomes necessary to redefine even the incremental ratio definition and use more abstract approaches.

Mathematical analysis professors still teach today in a way that forsakes the underlying sets. There is something that becomes less evident with this approach.

If you teach mathematics you should teach rigor. If you do not teach rigor then you are just forcing your students to remember by heart stupid useless formulas.

Hey, thanks for commenting! Looks like we disagree on ALL OF THE THINGS. I tend to adopt a conciliatory tone in such cases but the flu is diminishing my self-regulation capacities and your comment was unusually pugnacious & spunky so I SHALL RESPOND IN KIND.

For one, I think your reading of math history is anachronistic and Bourbaki-inflected (cf: Viktor Blasjo and Michael Barany). Gut check: What would Newton have said about vaccines? “Yes please, I wish not to die.” What would he have said about deltas and epsilons? “What the heck is this garbage for?” You’d need to show him Fourier analysis before he’d want the epsilons, and OUR STUDENTS DON’T KNOW FOURIER ANALYSIS.

Another gut check: the IB curriculum does more or less what I describe, without apparent calamity. In fact, it’s generally better than the AP Calculus curriculum (or most first-year calculus courses at US colleges).

Last: I super-110%-disagree with your final word on pedagogy! The opposite of “rote” is not “rigorous”! As I see it, the cycle of learning math is: (1) Play with ideas (e.g. what happens when I add a tiny increment to a square’s side length?); (2) Streamline and make rigorous those ideas (its area grows by 2 times the side length times the little increment); (3) Now they can be applied with rote automaticity (the derivative of x^2 is 2x).

You’re skipping Step #1, which drives student learning. Then you’re calling Step #2 the opposite of Step #3, when in fact it leads directly *into* Step #3!

“pugnacious” such a latin term it’s almost strange to hear it in english!

I didn’t mean to be aggressive, but I think that your last sentence “If it was good enough for the Bernoullis, it’s good enough for me.” really got to me as someone saying “the old ways are always better” and since it’s something you hear often these days I couldn’t believe I read it here. I already met a guy who wanted to read the euclidean way to do geometry, so I know these people exist even in math.

I think I understood next to nothing of what you wrote in your answer, so I need you to answer to some questions, if you want this conversation to reach somewhere.

“For one, I think your reading of math history is anachronistic and Bourbaki-inflected (cf: Viktor Blasjo and Michael Barany).”

What does it mean? What of what I said was Bourbaki inflected and what books/arcticles/sentences/something of Blasjo and Barany I need to read to see this?

I thought I just stated dates of when things were defined. The reason I brought Cantor in the discussion was just to add fuel to the fire, since even set theory wasn’t invented at the time of the Bernoullis. I called it “Naive” because I thought that Cantor didn’t give foundation to set theory before Russel paradox was even presented. Am I wrong? How Bourbaki come into this?

“Gut check: What would Newton have said about vaccines? “Yes please, I wish not to die.” What would he have said about deltas and epsilons? “What the heck is this garbage for?” You’d need to show him Fourier analysis before he’d want the epsilons, and OUR STUDENTS DON’T KNOW FOURIER ANALYSIS.”

I do not want to comment the sentence about vaccines since right now in the first world people are dying because they do not want to vaccinate but still refuse to accept reality.

I do not understand the other comment. I always thought that before epsilon-delta people used infinitesimals, without even defining them properly: “something not zero and positive but smaller that all other positive numbers.”. As Edward Welbourne Synthetic Differential geometry gives us a way to do it correctly, and even nonstandard analysis defined infinitesimals properly, something not exaclty 18th century-like. Since infinitesimals were something very ill defined I would not think that either Newton or Leibniz would offend epsilons and deltas.

“Another gut check: the IB curriculum does more or less what I describe, without apparent calamity. In fact, it’s generally better than the AP Calculus curriculum (or most first-year calculus courses at US colleges).”

Here I need some references because I do not know what those are. Still I don’t understand how you would define continuity and differentiability bypassing limits. All standard notion of continuity pass through the definition of topology, so since I know 8 definition of topological space I know 8 definitions of continuous function. There is one way to define a continuous function without passing through the topological one, that is defining a continuous function to be a Dedekind cut, still I do not think that this approach would be somewhere pedagogically acceptable.

“Last: I super-110%-disagree with your final word on pedagogy! The opposite of “rote” is not “rigorous”!”

I do not even understand from where this came from.

“As I see it, the cycle of learning math is: (1) Play with ideas (e.g. what happens when I add a tiny increment to a square’s side length?); (2) Streamline and make rigorous those ideas (its area grows by 2 times the side length times the little increment); (3) Now they can be applied with rote automaticity (the derivative of x^2 is 2x).”

I think it is more accurate to say “the cycle of teaching math”. Since I will do (1), you will do (1), but your students without any outside involvement would never do (1). Presenting the right example to motivate a definition or the need for it is always a good thing though. I still have an example stuck in my head as to why topological boundary is an incorrect notion to do analysis.

“You’re skipping Step #1, which drives student learning. Then you’re calling Step #2 the opposite of Step #3, when in fact it leads directly *into* Step #3!”

Why are you saying I am skipping it? When I said you can’t make examples to motivate the theory?

The last sentence I do not understand at all. To me you said:

(1) Ideas,

(2) Rigour,

(3) Exercise.

I said that if you abandon epsilons and deltas you are abandoning rigour. I do not see how this sentence brought you to what you wrote.

P.S. Cauchy used epsilons and deltas in his proofs but did not defined them. Because of this he “proved” that pointwise limits of continuous functions is continuous.

Thanks for keeping this conversation going – I totally see where several things I said were cryptic and unhelpful. Let me try to elaborate more usefully:

1) The basic historical narrative – which I always heard as a math student and which I think you are also telling – is something like this. “Newton & Leibniz developed calculus, but it was a rough and unproven calculus. It worked, but no one could fully explain why. It took several centuries for calculus to find firm foundations, in the form of delta-epsilonic rigor.”

Barany and Blasjo have persuaded me that this narrative is basically ahistorical and false. There was no failure of rigor in early calculus; indeed, all of their work could have been verified by Archimedes-style arguments from the method of exhaustion, but Newton and Leibniz viewed that as a tedious and unnecessary chore. Instead, although “Euclidean proof” is an ancient idea, “rigor” is a distinctly 19th-century one. It was a response to a new intellectual need.

As I understand it, Bourbaki (and others at the time) were the first to tell the history the way we now tend to hear it – with Berkeley’s early philosophical criticism foregrounded, and a sense of “something missing” from calculus until it was reframed as the kind of mathematics that Bourbaki preferred.

For sources on this:

A) Math historian Michael Barany’s paper on the Intermediate Value Theorem is perhaps the place to start: https://www.ams.org/notices/201310/rnoti-p1334.pdf.

B) After that, Viktor Blasjo’s site “Intellectual Mathematics” is a trove of great material on the history of calculus and analysis, esp. his infinitesimal calculus course and his history of math reader.

2) That’s what I can say about history; onto pedagogy. First, I should clarify: My original post is about teaching calculus, not analysis! Obviously limits should be taught – in analysis. It’s calculus where they have little pedagogical use.

As I said, the practice I’m suggesting is far from uncommon.

One example: the IB is the “international baccalaureate,” a highly regarded international curriculum. Its higher-level math course is far more challenging and interesting than AP (“advanced placement”) calculus, which is the standard course in the US. The former has virtually no discussion of limits; the latter has an extensive one, spanning perhaps 20% of the course, to no obvious benefit.

Another: although it is standard practice at research universities in the US to teach delta-epsilon limits in an introductory calculus class, every British colleague I’ve spoken with finds this absurd. They wait until analysis to introduce the concept.

3) Here is the statement of yours that prompted my response: “If you teach mathematics you should teach rigor. If you do not teach rigor then you are just forcing your students to remember by heart stupid useless formulas.”

I interpreted this as a claim that “rigorous” and “rote” are opposites. Indeed, that’s what you say – in the absence of rigor, all math education devolves into rote memorization of formulas.

This view is so opposed to my own – so bizarre and so wrong to me – that I don’t know what to make of it. Of course rigor plays an important role in mathematics education, but that role is not to prevent mathematics from feeling “stupid,” “useless,” or memorization-driven! Indeed, take a poll of students, and i suspect many will associate their experiences of rigor with precisely those qualities.

Also: do you really mean this to apply to primary and secondary math education? Or when you say “math education” do you mean “the education of advanced undergraduates”?

(FOREWORD: I hope everything is clear, and even if it is not I hope that I never seem hostile or badmannered. I am not sure if you will respond in any way, or even if you will read it at all. I want to say that even in our disagreement I like your blog and what you write and I have been following you from the post “why do we pay pure mathematicians”. Maybe if the conversation continues it would be better to change it to an email conversation and tone down the number of questions answered by a single post.)

As always wordpress doesn’t consider conversations longer than 4 comments, so I will tag this post instead of the other.

I think there is some colossal misunderstanding in the usage of some words that poisons our conversation. What does the word “rigour” mean to you?

Most definitions in the online dictionaries don’t really appeal to me. The best one I found says “The quality of being extremely thorough and careful.”, and I think that I will not find a better one.

What is the opposite of being rigourous? It is being sloppy. Ironically enough mathematics is permeated with sloppiness.

I took some time to think about what you wrote and about the things you linked. I’ve read Michael Barany’s https://www.ams.org/notices/201310/rnoti-p1334.pdf and the first chapter of Viktor Blasjo’s calculus book http://intellectualmathematics.com/calculus/ .

After reading Barany’s paper I failed to see why you would link it in here. In there he says that Cauchy was the first to want to have a written proof of the IVT, although in his book he wrote that as a tautology and in the appendix gave a more correct proof. It even seems that he does not use even the continuity hypothesis, but just says that the thesis must follow obviously. His definition of continuity is

“continuous functions are those functions for which the difference f(x+α)−f(x) is infinitesimally small when α is” and is not used in his proof. Question: how can the proof be true if the only hypothesis is not used explicitly? Simple: it is not true, since when f is not continuous the IVT thesis fails, thus showing the sloppiness.

After reading the first chapter I was almost dumbfounded to not find even one definition.

Let us continue with the sloppy math and look at the definition of continuity of Cauchy presented by Barany. To be able to understand it we need a meaning for “function”, for “difference and sum” and for “infinitesimally small”.

Cauchy died in 1857, before set theory.

I don’t know how they thought about math things before set theory but from what I read they just didn’t care.

The “definition” of a function as a rule which gives an output from an input is really too much of an undefined concept. What would it mean? You have a universe of concepts/objects in which you move/live and you can take some of these concepts and manipulate them into something different inside this universe. Pretty vague if you ask me.

With set theory we gain sets. A function is just a well defined set. We like to have a well defined domain and codomain before trying to write functions (even though more often than not it happens otherwise.).

Let us say that the reals were well defined and as such difference and sum were easily defined concepts.

What is the infinitesimally small? Since I sum it to a real number is it a real number? Is it on the real line? Is it something else? Is it somewhere else? What is it?

This is sloppiness.

Classic analysis says that infinitesimals must be real numbers and gives you epsilontic formalism, an infinitesimal is just a sequence converging to zero.

Nonstandard analysis, born in the 1900, says infinitesimals are not on the number line, and need to be added to the reals. You must be able to sum, invert, multiply and order them. You get a bigger set of reals which I now know are called hyperreals.

A third way is to introduce a quantity “e” such that “e^2=0” and so “f(x+e)=f(x)+e f'(x)”, but I’m not 100% about this one.

Since the beginning I never said that the epsilon was the only way, but a way is needed nonetheless. If you choose to forego epsilons than what are you going to choose?

The dx in the calculus book of Blasjo is something that isn’t defined, from that moment begins the sloppiness.

Moving forward: I do not distinguish between calculus and analysis. I didn’t know those were separate concepts and I don’t know where to draw the line. My best guess is that calculus is the introduction to mathematical analysis, that is one-variable and multi-variable calculus. Still this guess doesn’t sit right with me.

In my first post I asked how would you define continuity without epsilons and deltas, and I am still curious about it, Blasjo doesn’t define it, but to be fair he doesn’t define the derivative and without the definition he cannot prove the properties of the derivative although he says they are evident.

On your last point on rigour.

Your interpretation of my words about rigour and useless formulas is just a little bit off. Mathematics is about rigour and memorization, if they were opposites such thing couldn’t be true. If I remove rigour from math I would get not rigorous math, that is sloppy math, which I still need to memorize. Sloppy means that some things are not explained thoroughly and well pinned to the ground. This sloppiness can leave some question unanswered and a sour taste in your mouth.

When I say that abandoning epsilon and delta you abandon rigour I mean if you just introduce sloppily some dx and call it infinitesimally small. If you choose one of the other two stances mentioned above you are still being rigorous. If you say that in high school or before you don’t need to delve so deeply in technicalities is your choice but you must admit that you are loosing something.

Since the topic of this post was the IVT and you wrote “it takes years to pin down”, what is your definition of continuity? Note that this is a different question to the one asking how would you define continuity without epsilontic analysis.

I am not sure at what you mean for primary/secondary education since I studied limits only in my last year of high school. To me teaching to high school students would mean to show how interconnected is all the knowledge given to them so that even if something is forgotten it can always be recovered. Still I know that some things must be edulcorated for high school students before giving it to them, but I would always prefer not to cut corners.

BTW, the intermediate value theorem is proved in a line or two.

Not from scratch it isn’t.

In many ways, the important thing about it isn’t the theorem itself: it’s the exercise of clarifying what our intuitions about continuity are and encoding them in a rigorous way that makes it possible to even have a rigorous statement of what the theorem asserts (much less prove it). That the theorem is easy to prove once one has built this edifice is just the verification that we’ve built the right edifice.

However, I would agree with Ben that this isn’t the right place to start when first introducing students to calculus. I was educated in the UK where, as edderiofer says, school (i.e. up to age 18) calculus only takes a cursory look at limits, trusting that the situations where we’d need them would be ones in which it was easy enough to see whether there was a limit or not, without needing the more stringent rigour that I was later taught at university. That intuitive notion of limits was enough to give us a taste of things we could make sense of – product rule, chain rule and so on, plus the exponential function – without having to slog through a mountain of epsilons and deltas. Those who went on to specialise in science and engineering go the useful parts of calculus without the tedious book-keeping; and those of us who went on to mathematics had a solid sense of what those epsilons and deltas were buying us by the time we had to face them.

In any case, conceptually it’s possible to deal with derivatives without those epsilons and deltas (look up “Synthetic Differential Geometry” by Anders Koch, some time – warning: heavy use of category theory) so I look on them as what computer programmers refer to as “implementation details”: the high-level description of what tools you get and what they can do for you is worth understanding in its own terms, without being distracted by how those tools are implemented. If the implementation is done right, you don’t need to know about it, you only need to understand the high-level abstraction that it implements.

I think that in synthetic differential geometry the problem would be to be able to speak about not differentiable functions! The idea of changing the logic underneath from classical to intuitionistic is a lot scarier than using epsilon and deltas. Still you made me remember that people really brought back pre1900 math with topos theory and nonstandard analysis. It’s funny to think that the nonsense of old is now placed on solid ground.

You build a sequence which is explicitly Cauchy by dichotomy. The images converge to zero since if it did not it would mean that in a neighborhood of it the function doesn’t change sign which is absurd.

What it is you need to know besides the things required to understand the statement?

There is the proof by connectivity and the one by compact sets, which are maybe less basic since you need some knowledge.

Even though:

f^{-1}(0,+∞) and f^{-1}(-∞,0) are disjoint open sets in the domain if the function didn’t have zeroes the domain couldn’t be an interval.

The “things required to understand the statement” are exactly the defining properties of the reals that make the epsilon/delta method work; these are the properties that formalise out intuitions about continuity and the reals having “enough” values for those intuitions to work. These are equally the intuitions that prevent a union of disjoint open sets from being an interval, of course. (Contrast the rationals, in which {x: x.x 2} are both open and their union is all of the rationals.)

Back when I semi taught myself calculus in the local community college library during high school (it wasn’t offered at school), I skipped over the limits chapters. Didn’t make much sense to me at the time.

But I have a bolder challenge for you: why not dive into numerical methods early on? The analytic solutions are just a collection of special cases. With numerical methods you can take on some real world problems in the first semester.

And I think that the basic numerical methods provide much more understanding that the exercises provided in typical calculus books.

Numerical integration of differential equations is to calculus what manipulatives are to learning arithmetic.

After seeing numerical methods blow up in pathological cases, deeper understanding of limits and the like become interesting.

Yes, join the Rebel Alliance!

We’ve been underground (or at least oppressed by the Empire) for at least a century. But there is so much evidence that this works for students.

See http://launchings.blogspot.com/2014/06/beyond-limit-i.html

and forthcoming

Calculus Reordered

A History of the Big Ideas

David M. Bressoud

https://press.princeton.edu/titles/13397.html

“How our understanding of calculus has evolved over more than three centuries, how this has shaped the way it is taught in the classroom, and why calculus pedagogy needs to change

He contends instead that the historical order—which follows first integration as accumulation, then differentiation as ratios of change, series as sequences of partial sums, and finally limits as they arise from the algebra of inequalities—makes more sense in the classroom environment.”

Wait, did I miss something? What is this calculus book?

Announcement forthcoming!