From The 5 Elements of Effective Thinking:
Calculus may hold a world’s record for how far an idea can be pushed. Leibniz published the first article on calculus in 1684, an essay that was a mere 6 pages long. Newton and Leibniz would surely be astounded to learn that today’s introductory calculus textbook contains over 1,300 pages. A calculus textbook introduces two fundamental ideas, and the remaining 1,294 pages consists of examples, variations, and applications—all arising from following the consequences of just two fundamental idea.
12 thoughts on “Pushing an idea”
The theory of evolution may be comprable. A simple idea that bears on almost everything we observe or do.
Thanks. I’ll share this with my students.
Looking forward to your review.
What are the two ideas? I thought it was just one idea — that you can rigorously define ‘change’ (or ‘motion’) at a single point in time…
Derivative and integral
You might try to reduce calculus down to just one idea, but I’d say that’s the limit. *rimshot*
A. Webb “swoosh”. An integral is just the inverse of a derivative. So I’d say just one idea. But a lot of fields can be reduced like that biology “a self regulating reproducing and consuming being is alive”, physics pretty much done with the 3 Newton laws for mainstream things, and then expaned for Quantum and gravitiational systems but still ultimately you have a Lagrangian you are minimizing just the math is different.
I suppose it’s fair to say that derivative and integral were two different ideas that turned out to be the same thing. The ideas existed independently long before it was realized that they are inverse functions — sort of like the concepts of “the morning star” and “the evening star” were around for a long time before anyone realized they were the same celestial body.
And what’s wrong with that? What’s wrong with biding together in the same book, let’s say, Lagrange multipliers and partial derivatives?? What’s wrong about writing 1300 pages about a universe that spans from just two little features that can be described in 6 pages???
I mean you can essentially explain what “War and Peace” is about in little under 10 min, right?
jcborras: I didn’t interpret the quote as negative. A modern calculus book needs to be 1300 pages. Maybe it should be longer. I took the quote to say that Leibniz’ six-page article was filled with potential that has taken a great deal of time and space to develop and have proved to be extraordinarily useful.
Another example that comes to mind is Maxwell’s four equations. They capture all there is to know about electricity and magnetism, in a sense, but they take years of study to unpack.
If the two ideas in question are ‘derivative’ and ‘integral’ then the largeness of a book on the implications seems sorta vacuously unsurprising. Consider the idea of ‘inverse of a function’. A sizable book like _Applied Cryptography_ can be written just to survey a small subset of the implications. (Roughly: that subset of implications involving Alice trying to choose functions which will be easy for Bob to invert but hard for Carl to invert.) Similarly one might be able to spend a lifetime typing text without reaching an obvious stopping point when describing examples, variations, and implications of ‘algorithm’ or ‘prime number’ or ‘random’ or ‘finite’.
Taking the comments here forward, isnt the basic concept of limits common to both derivatives and integrals and hence, be considered fundamental in calculus?
Just thinking aloud.