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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."
There's no "flaw" in calculus. They've proposed a notation which if one used it would allow a broader range of formal manipulations to be valid. This is interesting but it isn't groundbreaking.
Figured I'd better say congratulations before the inevitable flood of people shitting on your contribution to math gets up to speed.
The messed up notation by Newton is not used and instead the much saner stuff from Riemann is used. Newton was smart, but a hack and a crank. And he tried to suppress Riemann notation. Mathematics would probably have done better without Newton.
Surely you mean Leibniz (1646-1716), not Riemann (1826-1866).
Escher was the first MC and Giger invented the HR department.
It took me a few minutes to get to the nub of the matter.
If you're mentally reading the notation d^2 y / dx^2 as the second derivative of y divided by dx squared, you're doing it wrong.
Because what this notion really intends to mean is d(d(y)/dx)/dx, which as the paper points out is a different order of operation.
A more compact notation less misleading than the traditional d^2 y / dx^2 might be (d/dx)^2 dy, which expands via two repeated function applications to d(d(y)/dx)/dx, with the underlying operations now in the right order.
Calculus was never my best thing, so I might be all wet, but it seems to make sense.
I never liked the dx/dy notation much, regarding it more as a cryptic code than anything conceptually helpful (when its not cryptic, it's not helpful, because that's the common case you already know).
With the right lambda notation (riffing on what I proposed above) the fundamental operator nature of d() could be correctly expressed, even if you don't want into these algebraic manipulations, which mostly strike me as far too detailed and tedious.
This is my thought as well. Interestingly, I developed this while writing a book (Calculus from the Ground Up) to use for my homeschool co-op calculus classes. I was trying to find a good way to explain the notation, and I literally had 20 calculus books that I read through trying to find a good explanation for the standard notation in any of them. None of them even attempted an explanation, just "this is the way it is, but don't treat it as a fraction." So, I tried to deduce the notation myself. That's when I realized that it was not just limited, it was actually wrong. So I wrote the paper and finished the book (it's Appendix B in the book).
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.
The messed up notation by Newton is not used and instead the much saner stuff from Riemann is used.
The advantage of the Newtonian notation is that it is a lot faster and easier for, unsurprisingly, basic Newtonian mechanics where you only really differentiate with respect to time. This is why it is used extensively in this area of physics. Leibnitz's (not Riemann's!) notation is a lot more versatile which is not surprising: Leibnitz was a mathematician who was interested in the abstract concept whereas Newton was a physicist who only developed calculus so he could describe mechanics and so did not really need a broader, more flexible notation.
It is actually quite a common that fundamental physics can find itself ahead of maths. For example String theory today is really a joint venture between maths and physics since they are having to develop the maths needed to describe the physical models they work on.
Finally, Newton was neither a hack or a crank but he was a somewhat evil genius. He could be quite nasty and viscous, sometimes in extremely petty ways. For example he discredited Leibnitz and he fell out with Robert Hooke and had all contemporary portraits of him destroyed which so angered a modern artist that she spent the time an effort painting multiple portraits of Hooke from contemporary descriptions so that, today, there are more portraits of Hooke than Newton!
dy/dx doesn't represent instantaneous rate of change. That would be nonsense. The d in dx and dy means "small difference that will eventually go to zero". This is why dy/dx is a fraction. It represents the limit of a small change in y divided by small change in x, as the changes go to zero. This is why we teach students about the limit definition of the derivative as being what the derivative really is. As far dy and dx being tricks of notation, they're really not. They really are small changes. There's no instantaneous rate of change. dy and dx are always finite real numbers. They never actually become zero. dy/dx is the ratio that is approached as they get smaller and smaller.
As far as this guy's new version of the second derivative, I call bullshit. I seriously doubt that this is correct. And the notation d^2y/dx^2 actually makes sense when you think about. It's really just d(dy/dx)/dx, that is, a small change in dy/dx divided by a small change in x, where dy/dx is a small change in y divided by a small change in x. Writing it in the other way is just a good way of doing it. If you draw out what this means graphically, is becomes clear that it's really a small change between two consecutive small changes in y divided by two small changes in x, that is d(dy)/dx^2, hence d^2y/dx^2.
This guy's new version, on the other hand, doesn't make sense at all. I mean, how do you get that from taking the derivative of the first derivative. Let's take a pretty standard function: x=1/2*t^2+2*t+12. x'=t+2; x''=1, whereas his version would be x''=1-t, which doesn't make any sense, unless he has completely redefined everything. I mean, d^2y/dx^2 would have to be something like 2t+5 and d^2x/dx^2 would have to be something like 2, and then we get x''=2t+5-(t+2)*2. I didn't read the paper so I don't know what it would actually be, but there's no doubt that x''=1, so if his method is to make any sense at all it would have to give the same results in the end. I just don't see how it could.
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