Some of you know that I published a Calculus book last year. My goal in this was mostly to counter-act the dry, boring, and difficult-to-understand textbooks that dominate the field. However, when it came to the second derivative, I realized that not only is the notation unintuitive, there is literally no explanation for it in any textbook I could find.
… the notation itself is problematic. Although it is written as a fraction, the numerator and denominator cannot be separated without causing math errors. This problem is somewhat more widely known, and has a workaround for it, known as Faa di Bruno’s formula.
My goal was to present a reason for the notation to my readers/students, so that they could more intuitively grasp the purpose of the notation. So, I decided that since no one else was providing an explanation, I would try to derive the notation myself. More.
The story we ran on the topic at Mind Matters has gone viral via Slashdot, with five thousand views since yesterday afternoon. (A paper about calculus?) Figures, Bartlett must have a point about the problem.
Paper: Bartlett, Jonathan
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See also: Walter Bradley Center Fellow Discovers Longstanding Flaw in Elementary Calculus Jonathan Bartlett: The flaw doesn’t lead directly to wrong answers but it does create confusion. The lead author, Jonathan Bartlett, noted that the likely source of the bad notation was a philosophical issue. Because no one wanted to give differentials that same ontological status as other numbers, everyone presumed that the notational problems were simply the result of this fact, and no one pursued it further. (Mind Matters)
Is Standard Calculus Notation Wrong? Bartlett: The reason why this was not noticed before, I believe, is because, since the 1800s, mathematicians have not wanted to believe that infinitesimals are valid entities. Therefore, they were not concerned when the second derivative did not operate as a fraction – it didn’t need to, because it indeed wasn’t a fraction. Infinities and infinitesimals are the non-materialistic aspects of mathematics, just as teleology, purpose, and desire are the non-materialistic aspects of biology.