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Comment by btilly | Hacker News Reader

Calculus can be rigorously done with limits, continuous functions, little-o notation or infinitesmals. The key word is "or", use one approach and don't jump around. (After you understand multiple you can see how they connect and then jump around.)

Of those approaches infinitesmals look the simplest but require the most complex mathematical machinery to really do right. Yes, you can construct a non-standard model of the real number system with a transfer principle that lets you transfer between proofs done in the one to the other which lets you formulate calculus using infinitesmals. But using a construction using the axiom of choice to prove that the derivative of x^2 is 2x is kind of like using a sledgehammer to push a tack into a cork board.

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Great point. I think one of the issues with calculus is that it can be studied at so many levels. Most high school classes give lip service to limits (somehow they're important), and then get into symbol manipulation (move the exponent down, subtract 1, etc.).

I think the most important thing in the beginning is to learn to think conceptually about calculus, about how to break a shape into a simpler model and solve it (not just manipulate models that are given to you). Later on (in analysis, which I've never formally taken :) ) the details about the transfer principle, epsilon-delta proofs, etc. can emerge, but realistically only a very, very small fraction of students need that. That fraction can be increased if they are able to use calculus regularly (with infinitesimals, say) and then wonder the details of how it all works.

I like to ask: Would it be such a bad thing if everyone understood calculus as Newton did, at a baseline level? Many people taking calculus leave with no understanding whatsoever, but the same isn't said for geometry or even algebra (you can make references to parallel lines, equations and feel that people will 'get it', but mention a derivative or limit to someone 10 years after calc class and you'll get a blank stare). I advocate having rough mental models that are successively refined over time (vs. presenting the most rigorous version first).

History note for you. Newton understood calculus through the idea of fluxions. It is often claimed that an insistence on teaching calculus with Newton's fluxions instead of Leibniz' infinitesmals held back research in England for a long time. (Another random note. Bishop Berkeley's famous comment about "ghosts of departed quantities" was a criticism of fluxions, not infinitesmals. In fact he had positive things to say about infinitesmals.) So yes, I do think it would be bad for students to understand calculus like Newton did. :-)

That said, to my eyes the primary problem with how we teach calculus is that we introduce too many big concepts at the same time. The key picture that people should come away with is the tangent line. But when you say that the derivative is a function defined at each point as the slope of the tangent line at that point, most people experience brain freeze.

Instead I would like to see students get to the point where they can calculate tangent lines. Then stop. Work out lots of tangent lines for lots of functions. Introduce them to how the tangent line can be used to estimate error bars. (Typical word problem, if a rock take 5 +- 0.5s to fall off a cliff, how tall is the cliff?) And then only after the picture and mechanics of the tangent line is well established should we move on to defining the derivative.

This approach, if done right, would be rigorous but informal. It should avoid the problem of causing brain freeze by presenting too many layers of generality at once. And it would really cement the key image behind half of calculus.

Ah, thank you for the clarification -- yes, Newton loved his fluxions, didn't he? In future examples I'll s/Newton/Leibniz/ :).

I think the general point is true about the brain freeze and too many big topics -- it's way too easy to just throw a definition out there, say "this represents slope which is the rate of change" and move on to the next idea.

There is also this:

http://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis

It uses intuitionistic logic, so no axiom of choice.

No axiom of choice, but most working mathematicians reject constructivism.

Plus there are some pretty big differences. Do you really want to explain to students why you don't consider f(x) = 1 if x >= 1 and 0 otherwise to be a valid definition of a function? And (in some ways more importantly) why most of the rest of the world does? Furthermore the intermediate value theorem is pretty useful and I wouldn't want to lose it.

Just mentioning another approach. I don't really understand it myself, but I kind of find it intriguing because of its relation to computability aspects.

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