# The History of Calculus / Honor Your Dissatisfaction

I was just rereading an email exchange with a friend (actually the O of this post), and found that I had summarized the history of calculus from the 17th to 20th centuries, up through and including Abraham Robinson’s invention of nonstandard analysis, in the form of a short play! I’m sharing it with you.

Mainly this is for fun, but it’s also part of my ongoing campaign promoting the value of honoring your dissatisfaction. The dialectic between honoring our impulse to invent ideas to understand the world better and honoring our dissatisfaction with these ideas is where mathematics comes from.

Here’s the play!

# The History of Calculus, in 4 Extremely Short Acts

Featuring a lot of oversimplification and a certain amount of harmless cursing

Act I

Late 17th century

Leibniz, Newton: Look everybody, we can calculate instantaeous speed!

Everybody: How??

Leibniz: well, you consider the distance traveled during an infinitesimal interval of time, and you divide distance/time.

Everybody: Leibniz, what do you mean, “infinitesimal”? Like, a millisecond?

Leibniz: No, way smaller than that.

Everybody: A nanosecond?

Leibniz: Nah, dude, you’re missing the point. Smaller than any finite amount.

Everybody: So, zero time?

Leibniz: No, bigger than that.

Some people: Oh, cool! Look we can use this idea to accurately calculate planetary motion and stuff!

Other people: WTF are you talking about Leibniz? That makes no effing sense.

Act II

18th century

Bernoullis, Euler, Lagrange, Laplace, and everybody else: Whee, look at everything we can calculate with Newton and Leibniz’s crazy infinitesimals! This is awesome!

Bishop George Berkeley: But nobody answered the question of WTF they are even talking about. “What are these [infinitesimals]? May we not call them the ghosts of departed quantities?”

Lagrange: Hold on, let me try to rebuild this theory from scratch, I will make no mention of spooky infinitesimals, and will do the whole thing using the algebra of power series.

Everybody: Cool, good luck with that.

Act III

19th century

Cauchy: Lagrange, homie, it’s not gonna work. $e^{-1/x^2}$ doesn’t match its power series at zero.

Lagrange: Sh*t.

Everybody: I think we don’t actually understand this as well as we thought we did.

Ghost of departed Bishop Berkeley: OMG I HAVE BEEN TRYING TO TELL YOU THIS.

Cauchy: How about we forget the whole “infinitesimal” thing and just say that the average speeds are approaching a certain limit to whatever desired degree of accuracy. As long as we can identify the limit and prove that it gets as close as we want it to, we can call that limit the “instantaneous speed” without ever trying to divide some spooky infinitesimals by each other.

Everybody: Awesome.

Weierstrass: I have an even better idea. Let’s formalize Cauchy’s thinking into some tight symbols and quantifiers. “Let us say that the limit of a function $f(x)$ at $c$ is a number $L$ if for every $\varepsilon > 0$ there exists a $\delta > 0$ such that whenever $0 <|x-c|<\delta$, it follows that $|f(x)-L|<\varepsilon$…”

All the mathematicians: AWESOME. Down with spooky infinitesimals! Calculus can be built soundly on the firm footing of “for any $\varepsilon>0$ there exists a $\delta>0$ such that…” and you never have to talk about any spooky sh*t!

All the mathematicians, in private: … but thinking about infinitesimals sure streamlines some of these calculations…

[Meanwhile all the physicists and engineers miss this whole episode and continue blithely using infinitesimals.]

Act IV

20th century

Scene i

Mathematicians: Infinitesimals are satanic voodoo!

Mathematicians: Whatever dude, don’t you know about Weierstrass and $\varepsilon$ and $\delta$?

Physicists and engineers: Um, no, and I don’t care either! What’s the point when everything already works fine?

Mathematicians, in public: No, dude, there are all these tricky convergence issues and you will F*CK UP EVERYTHING IF YOU’RE NOT CAREFUL!

Mathematicians, in private: … but those infinitesimals are indispensible as a heuristic guide…

Scene ii

Abraham Robinson: Um, whatever happened to infinitesimals?

Mathematicians: I mean we rejected them as satanic voodoo because nobody was ever able to tell us WTF THEY ARE.

Robinson: I have a proposal. How about we consider them to be [fancy-*ss definition based on formal logic and other fancy sh*t]. Would you say that constitutes an answer to “wtf they are?”

Mathematicians: … why, yes!

Some mathematicians: omg awesome I can now RESPECTABLY use infinitesimals in calculations, I don’t have to hide anymore!

Other mathematicians: Whatever, I have no need to do the work to master this fancy sh*t. It doesn’t do anything good ole’ Weierstrass $\varepsilon$ and $\delta$ couldn’t do.

Physicists and engineers: wow, you guys are way over-concerned with the little stuff. Literally.

# End

(Long-time readers of this blog will recognize the bit of dialogue with Leibniz from something I shared long ago.)

The point is that the whole episode is driven by uncertainty about what is even being discussed. The early developers of calculus shared the conviction that there was something there when they talked about “infinitesimals”, but none of them (not even Euler) gave a definition that was satisfying to everybody at the time (let alone to a modern audience). But this encounter, between the intuition that there’s something there and the insistence of the world to honor its dissatisfaction until a really satisfying account was given, was a generative encounter, resulting in several hundred years’ worth of powerful math progress.

# Pershan’s Essay on Cognitive Load Theory

Just a note to point you to Michael Pershan’s motherf*cking gorgeous essay on the history of cognitive load theory, centered on its trailblazer, John Sweller.

I’m serious.

I tend to think of Sweller as, like, “that *sshole who thinks he can prove that it’s bad for learning if you think hard.”

On the other hand, any thoughtful teacher with any experience has seen students get overwhelmed by the demands of a problem and lose the forest for the trees, so you know that he’s talking about a real thing.

Michael has just tied it together for me, tracing how Sweller’s point of view was born and evolved, what imperatives it comes from, other researchers who take cognitive load theory in related and different directions, where their imperatives come from, and how Sweller’s relationship to these other directions has evolved as well. I have more empathy for him now, a better sense of his stance, and a better sense of why I see things so differently.

Probably the biggest surprise for me was seeing the connection between Sweller’s point of view on learning, and the imperatives he is beholden to as a scientist. I get so annoyed at the limited scope of his theory of learning, but apparently he defends this choice of scope on the grounds that it supports the scientific rigor of the work. I understand why he sees it that way.

The remaining confusion I have is why the Sweller of Michael’s account, ultimately so clear on the limited scope of his work (“not a theory of everything”) and the methodological reasons for this limited scope, nonetheless seems to feel so empowered to use it to talk about what is happening in schools and colleges. (See this for an example.) Relatedly, I’m having trouble reconciling this careful scientific-methodology-motivated scope limitation with Sweller’s stated goal (as quoted by Michael) to support the creation of new instructional techniques. The problem I’m having is this:

Is his real interest in supporting the work of the classroom or isn’t it?

If it is, well, then this squares both with the fact that he says it is, and that he’s so willing to jump into debates about instructional design as it is implemented in real classrooms. But it doesn’t square with rigorously limiting the scope of his theory, entirely avoiding conversations about obviously-relevant factors like motivation and productive difficulty, which he says he’s doing for reasons of scientific rigor, as in this quote:

Here is a brief history of germane cognitive load. The concept was introduced into CLT to indicate that we can devise instructional procedures that increase cognitive load by increasing what students learn. The problem was that the research literature immediately filled up with articles introducing new instructional procedures that worked and so were claimed to be due to germane cognitive load. That meant that all experimental results could be explained by CLT rendering the theory unfalsifiable. The simple solution that I use now is to never explain a result as being due to factors unrelated to working memory.

On the other hand, if his interest is purely in science, in mapping The Truth about the small part of the learning picture he’s chosen to focus on, then why does he claim he’s doing it all for the sake of instruction, and why does he feel he has something to say about the way instructional paradigms are playing out inside live classrooms?

Michael, help me out?