I recently stumbled over an article in the BetterExplained blog (Learning Calculus: Overcoming our artificial need for precision). The blog itself contains a lot of interesting articles. However, the blog has a tendency to bash „school mathematics“ in favor of „insight“. The subtitle of the blog „learn right, not rote“ says it all.
Now, I have been blogging here against school mathematics as taught today too. So why am I complaining? As I think about it, I come to the conclusion that both critiques go into directly opposite directions. Effectively, I want more theory in form of logic and thinking, and the blog wants less in favor of „useful“ mathematical thinking.
But let us concentrate on the blog entry in question: „Overcoming our artificial need for precision“. If you read the article, you know that it makes a lot of valid points. For instance, we really do not need many decimal digits of anything from an engineering point of view. In fact, in engineering it is considered an error to compute a result with more digits then the underlying measurements provide. There is that famous joke saying that 85.784% of all statistics are unnecessarily accurate.
This is certainly a valid insight! But does it mean less mathematics? I doubt. In fact, there are several heavy mathematical theories treating inaccuracy. One is taught in numerical math and is called „propagation of errors“. Then there is „interval arithmetic“. And finally, of course, there is statistics treating errors as statistical errors. So we see that we need complete math theories to understand inaccuracy and its consequences. No wonder, this is out-bracketed from first year math.
We simplify our world as if it was exact, not, because we neglect the inaccuracy, but because we cannot treat it on an elementary level. I think everyone, the teachers and the students, know that this kind of exactness is only a simplification. I add and admit that it would be time well spend to teach how to handle measurement errors. But this is a teaching subject on its own.
To summarize, I claim that correct mathematical thinking (my aim in teaching) is necessary even if we give up our „artificial“ need for precision. On the contrary, it is precise mathematical thinking which solves the problems with uncertain measurements. I tried to convince myself that this is indeed what the blog entry tries to say. But frankly I think the normal reader will understand quite the contrary. He will understand that precise mathematical reasoning is rubbish because the objects derived from this thinking do not exist in real life.
Let me also mention constructive mathematics. This is a now abandoned branch of mathematics, which neglects non-constructive mathematical objects. E.g., the intermediate value theorem sounds like this: „A continuous function negative on one end and positive on the other end of an interval can be made arbitrarily small in the interval“. Note, that it does not claim the existence of a zero! However, this way of thinking becomes very complicated and thus is not used in mathematical teaching or practice. I am a big friend of constructive methods. But admittedly, it is much easier to think of the constructed target as a mathematical object.
Finally, I do not understand the sentence at the end of the article: „Again, my goal is to understand the ideas behind Calculus, not simply rework the mechanics of its proofs.“ I would phrase like this: „My goal is to understand the ideas behind proofs, not to rework the mechanics of Calculus.“
Hi Rene, thanks for the post! I enjoy having discussions on these topics. I’ll try to get to your points in order.
Actually, I’m not opposed to theory at all, I enjoy it quite a bit :). I do find that theory is often taught as just that (a set of results to memorize) without much discussion about the insight that led to it. For example, there is the Pythagorean Theorem that most people learn, but very rarely is it shown as an example of relationship between areas in general (for example, you can see it almost as a conservation law of area).
That’s a great point about the potential for misinterpretation. I did mean what you hoped: our thinking should be precise, even if our measurements are not. I do want that idea to come through: our numbers are accurate within their own scale, but there are classes of numbers (infinitesimals) that we can’t interact with directly. But clear, logical thinking about how these numbers would behave (in limits, etc.) can lead to new insights about Calculus.
For the last sentence, I meant it as you say — I want to really understand the ideas behind the proofs, not go through the symbol manipulation of the proofs themselves.
Thanks again for the response!
Speaking of precision, if I am doing numerical searches, lets say trying to find zeta(3) in terms of common constants, it would be nice if EMT had more precision than 16 significant digits of precision. Any chance you will be changing precision to something greater than 16?
For the zeta function you can use Maxima via
For special purpose, you should use special programs.