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. 
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 
All the mathematicians: AWESOME. Down with spooky infinitesimals! Calculus can be built soundly on the firm footing of “for any 
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!
Physicists and engineers: What are you talking about, what about CALCULUS?
Mathematicians: Whatever dude, don’t you know about Weierstrass and 
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
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.
So. Honor your dissatisfaction.
Research in Practice 
Ben, you make me so happy! This is perfect. I want to share with my students.
But I don’t get the part about e to the -x^2/2 not matching its power series at zero. If you take the power series for e^x, and substitute -x^2/2 in for x, you get a series that gives 1 when x is 0. Can you tell me more about the problem? (Or point me to what I should read…)
Hey Sue – sure – ok 3 things:
(1) The exponent is -1/x^2, not -x^2/2. If you sub in, you get a power series with x in the denominator, so if you let x=0 it’s undefined. (Of course this is technically true of the formula for the function itself. But the limit at x=0 is zero, so if you define f(0)=0 you get a continuous and smooth function. In fact it heads extremely rapidly to zero as x goes to zero from either side, because of the exponential; in fact how rapidly it heads to zero causes its key feature, see point (2).)
(2) The real point is what happens if you use the formula for Taylor expansion in terms of the derivatives. The key feature of the function mentioned in (1) is that at the origin, all the derivatives are zero. So the Taylor expansion is zero. But yet the function is not zero. So the function cannot be recovered from its Taylor series. Lagrange’s idea had been to base calculus on identifying a function with its Taylor series and then taking the derivative as the coefficient of the linear term in the Taylor series, but this example shows that the Taylor series is somehow not rich enough to capture the function.
(3) Thank you for honoring your dissatisfaction! (Folks, this is what it looks like.)
Hee hee. I just assumed my favorite e to a weird power function, and thought you’d written what came into my head when I was reading. Funny how that works.
Too tired now to play with your cool problem child, but I think I will. I am guessing this is a common example in real analysis?
Ha ha oh I totally get that. (And yes, it’s a standard example. Actually
or 
does the same thing, the squared wasn’t essential.)