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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."
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."
Figured I'd better say congratulations before the inevitable flood of people shitting on your contribution to math gets up to speed.
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.
Engineering and the Ultimate
I have a "math issue" that has stumped most of my professors and online math forums. Linear regression typically uses the "least squares" algorithm. However, the power of 2 seems arbitrary to me, and possibly over-emphasizes outliers.
One professor at first said that the power of 2 makes the "best fit" in an objective sense, but later admitted that he doesn't really know, and couldn't find an answer before the end of the semester.
While it is true that the power of 2 may simplify the computation process*, that doesn't necessarily means it produces a better result in terms of line or curve fitting. Now that we have computers to do the number crunching, perhaps it's time to embrace arbitrary or different powers (superscripts).
(Disclaimer, I'm not a math expert.)
* In other words, power-of-2 produces the simplest known algorithm. But my question revolves around best data fit, not computational resources nor algorithm or formula brevity. Note that when using other powers, one may have to add an absolute value function because power-of-2 automatically provides the equivalent. I actually did a simulation that tested different powers; "blurring" known datasets and seeing which power best matched the original. I couldn't find any significant difference, but probably didn't try enough samples. I tested with fractional powers also, such as 1.5, 2.5, etc.
Table-ized A.I.
because the old way ways "you can treat the derivative as a fraction,
Except the second derivative notion isn't a fraction. It's a way of writing "the second derivative of Y with respect to X" in a short form. Not all '/' create "fractions". Unless, of course, you want to argue that I'm putting a lot of "< divided by quote>" fractions in my /. postings.
The error is not in the notation, it's in the inability to overload the / operator when dealing with more complex and abstract mathematical concepts. It's like not being able to differentiate between "e as a variable" and "e as a constant". Do you ever think the the mass of an object times the speed of light squared is equal to ~2.718? Einstein says so, it must be true.
Except that, in the first derivative, it *is* used as a fraction. Otherwise you couldn't reformulate your equation for integration (i.e., you have to multiply both sides by dx, which is treating it as a fraction). So, to say that in one case, it is a fraction, but this next case it isn't, but still written as a fraction, even though it *could* be written as a fraction, but we just decided not to, seems strange, at least to me.
Engineering and the Ultimate
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.
Engineering and the Ultimate