Old-School Slashdotter Discovers and Solves Longstanding Flaw In Basic Calculus (mindmatters.ai)

Longtime Slashdot reader johnnyb (Jonathan Bartlett) shares the findings of a new study he, along with co-author Asatur Zh. Khurshudyan, published this week in the journal DCDIS-A: Recently a longstanding flaw in elementary calculus was found and corrected. The "second derivative" has a notation that has confused many students. It turns out that part of the confusion is because the notation is wrong. Note -- I am the subject of the article. Mind Matters provides the technical details: "[T]he second derivative of y with respect to x has traditionally had the notation 'd2 y/dx 2.' While this notation is expressed as a fraction, the problem is that it doesn't actually work as a fraction. The problem is well-known but it has been generally assumed that there is no way to express the second derivative in fraction form. It has been thought that differentials (the fundamental 'dy' and 'dx' that calculus works with) were not actual values and therefore they aren't actually in ratio with each other. Because of these underlying assumptions, the fact that you could not treat the second derivative as a fraction was not thought to be an anomaly. However, it turns out that, with minor modifications to the notation, the terms of the second derivative (and higher derivatives) can indeed be manipulated as an algebraic fraction. The revised notation for the second derivative is '(d 2 y/dx 2) - (dy/dx)(d 2 x/dx 2).'"

The report adds that while mathematicians haven't been getting wrong answers, "correcting the notation enables mathematicians to work with fewer special-case formulas and also to develop a more intuitive understanding of the nature of differentials."

Comments

Congrats

[yodleboy]

Figured I'd better say congratulations before the inevitable flood of people shitting on your contribution to math gets up to speed.

Re:Congrats

[johnnyb]

Thanks! I appreciate it. Given that this was my first peer-reviewed mathematics paper, I had no idea how long the process was. I submitted the paper over a *year* ago. The necessary changes were minor. But the actual time it took to go through the process was excruciating. I'm happy to finally be on the other side :)

Re:Summary's accuracy seems questionable

[johnnyb]

It's a bit of both. Some of the facts of the matter were known, but it was assumed that this was just "the way it was". That is, no one considered it an open problem. For instance, we view the inability to divide by zero just a fact of mathematics, not a flaw. Likewise, this was not known to be a flaw, it was just assumed that this was the way things worked.

If you need to point to a definitive flaw, it was in our understanding of how it was supposed to work - the relationship between our understanding and the notation. Once *that* flaw was discovered, the actual notation just spilled right out. That is, the flaw was that people were *not* treating dy/dx *sufficiently* as a fraction, due to 19th century preferences against infinitesimals. Once you realize that dy/dx really is a fraction, and has to be treated accordingly, everything automatically works.

It's almost humorous because there was no real advanced work to do. Literally everything needed is available in intro calculus. The problem was (a) the mathematics community had a habit of *not* treating dy/dx as a fraction, and (b) new students who didn't know better were simply taught *what* to do, not *why* to do it, and continued to repeat the mistake for over a century.

Re:What about partial derivatives?

[johnnyb]

I've actually got a second paper on partial derivatives just about ready to go. It was originally part of this paper, but it got a little long, and I wanted to rethink and clarify a few concepts. Anyway, partial differentials have the same notational problem *plus* one more. The problem is that there are several partial differentials which all go by the same name. Once you name them properly (i.e., give them each a distinct name) the problems go away.