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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.
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
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!
It's not arbitrary. There's actually a good reason for minimizing (y-yobs)^2, assuming that your observations have a Gaussian distribution. The resulting estimators provide a maximum likelihood estimator of the parameters of the distribution, if and only if it really was Gaussian. Thus, of course, if it isn't Gaussian (outliers of various sorts, et.c), the x^2 may not be the best bet. There is an entire field of 'robust estimators' of quantities, which are more resistant to outliers than least squares. There are also cases in which the underlying distribution is pathologically different from Gaussian; it could be Lorentzian (Cauchy), in which case it is so completely unlike a Gaussian, it doesn't even have a defined standard deviation (it is infinite). There are weighted methods which can fix this too.
So, in short, least squares is the right answer (in the sense that it yields results which provable have the maximum likelihood describing the data at hand) if you have a perfect Gaussian variate; otherwise, it may well not be.