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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."
I appreciate the new form is technically more accurate but the expansion is pretty large compared to the original form... I wonder if the extra length doesn't wash out the understandability gains you get out of the original form.
"There is more worth loving than we have strength to love." - Brian Jay Stanley
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 d, dx, dy, etc. are not things to be generally operated on.
Writing the second derivative as d^2 / dx^2, or worse, d^2y / dx^2 is doubly absurd. (I'm using the ^ to denote supersripting, not exponentiation.)
d represents the instantaneous rate of change (which itself is a flawed concept - a rate of change cannot be instantaneous as a rate depends on the passage of time), dx represents that instantaneous rate of change of x. d/dx represents the instantaneous rate of some value, possibly some value dependent on x, with respect to the instantaneous rate of change of x. dy/dx, dv/dt, etc. are all the same deal. That rate of change of some variables with respect to other variables.
What is that instantaneous rate of change? The slope of a line (plane, or whatever if you've got more free variables) tangent to your function at a given point, presuming such a thing exists.
How do you determine that tangent line? You take the target point and some point h past it ((f(x) vs f(x+h)) (or before it!) and determine what the line does when you consider h approaching 0. You make sure you can define that shit from both ends and both ends agree. If that works out, have a limit, you've got a derivative, and baby, you've got the fundamental theorem of calculus goin'.
Whoever tried to slap that shit together as a fraction or take shortcuts and try to manipulate those symbols in a way that looks sort of like algebraic manipulation is a clown. Trying to fix that is going to be an uphill battle, but using more of the busted notation isn't really the solution.
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.
"d/dx" is an operator on functions that has about a half dozen tidier, less-confusing alternate notations, while "dy/dx" is a limit of a ratio of nonzero numbers that is misleadingly written as a fraction because people in the 18th century weren't as bothered about the whole 'dividing by zero' thing.
The fact that algebraically treating dy/dx as a fraction works in any situation at all is a minor stroke of luck that honestly should be concealed, since thinking that way already hurts the progress of a huge number of basic calculus students. It's analogous to the fact that undefined algebraic manipulations with zero/infinity work often enough to make them tempting to use. In professional hands "wrong" calculations like this can sometimes point the way to the right ones, but for students it's just a terrible idea all around to even bring it up.
Call differentiation "quark" instead. The new form for d2y/dx2 could be called a double quark or "fred" for short or f for really short. For really rigorous treatment call it f(x). In the UKoGBnNI it shall be known as noddy on Tuesdays unless the year is 2022.
Nah. He's right. The current notation is bullshit. It looks like standard algebra and people try to manipulate the symbols as such.
I mean, you could shorten d/dx to 1/x, right?
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.
Thanks for your opinion on the history of Calculus, person who is not even aware of the most basic of facts and provides no real rationale.
Slashdot: providing anti-social weirdos a soapbox, since 1997.
god that's settled. Now we can figure out that P=NP problem that nobody can give a coherent answer on why its even a thing.
Only the State obtains its revenue by coercion. - Murray Rothbard
Unfortunately we can't go back through history and give every single human who ever lived privilege and time to see what they're capable of.
So we can only assess those who have made prominent contributions. There are many others who had privilege and time and still didn't contribute what Newton did, so it's still fair to call him a genius.
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.
Leibniz' notation is normally treated as a "suggestive kind", never to be understood literally. The origin of notation d^2/dx^2 goes from applying d/dx to d/dx, but d/dx only means "a derivative w.r.t. x" and nothing else. Sometime taking this notation literally and doing manipulations as if it were the regular fractions work (and that's b.t.w. is attributed to the early discoveries of many differentiation and integration rules), but it doesn't work most of the time. Any decent book on Calculus should point out that fact. Working with fractions helps to discover some rules, yes, but it's never rigorous, it's more like discovering something in a heuristic way, but then you still need a rigorous proof and that involves going back to basic definition of limits, not arguing in terms of "infinitesimals" (yes, I'm aware of Robinson's "non-standard calculus", but IMHO it's not a mainstream approach. Cheers.
but Lagrange's notation > Leibniz's notation anyhow.
If you want to do wizardry by manipulating the notation itself, then by all means use '(d 2 y/dx 2) - (dy/dx)(d 2 x/dx 2)'.
Kudos to johnnyb
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!
Old-School Slashdotter Discovers and Solves Longstanding Flaw In Basic Calculus
Can occasionally be heard yelling at younger mathematicians: "Get off my lambda"
It must have been something you assimilated. . . .
I appreciate the new form is technically more accurate
It's only technically more accurate if you read the standard form as a fraction. If you actually read the standard form as intended - a notation indicating the second derivative of y with respect to x - the standard notational form is just as accurate as is the Newtonian notation of dots to denote derivatives with respect to time.
I mean the Riemann Integral. But you are correct, differentiation came first.
Most ACs are not even worth the keystrokes to insult them. Be generically insulted by this and ignored otherwise.
He could not have. He was dead at the time all information became available. But you seem to be stupid, so you are probably incapable of understanding causality.
Most ACs are not even worth the keystrokes to insult them. Be generically insulted by this and ignored otherwise.
It does not work for almost all cases. (Mathematical "almost all".) And it does not warn you of it. It is basically not mathematics, but clever shifting around of symbols with pitfalls which works purely by accident. Not a good thing.
Most ACs are not even worth the keystrokes to insult them. Be generically insulted by this and ignored otherwise.
Nah. He's right. The current notation is bullshit. It looks like standard algebra and people try to manipulate the symbols as such.
I mean, you could shorten d/dx to 1/x, right?
The current notation is mathematically correct. It express that fact that the second derivative is the incremental limit of the first derivative.
The new notation for d^2/dx^2 loses the corect mathematical meaning and gains nothing of value.
d^2/dx^2 (f) (x) = lim (h->0) (d/dx (f)(x+h) - d/dx (f) (x))/h
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.
Riemann lived 2 centuries after Newton. And your conclusions aren't correct, they aren't even wrong!!!
"Those that start by burning books, will end by burning men."
The current notation is bullshit. It looks like standard algebra and people try to manipulate the symbols as such.
And that's why they call it "learning calculus" instead of "coming up with calculus all on your own". Not understanding how something works or what the terms mean can lead to horrible results.
Very cool! I think the paper will help me understand more deeply problems with the notation I've fought with many times!
However, I'm a bit disappointed that the notion of partial vs. full derivative wasn't raised, which I think is very relevant to the question...
Exactly! If gweiher here had more time on his hands, he would have written supercalculus by now. But no, his solemn duty to try to top dead people on Slashdot takes up his otherwise suuuuper valuable time.
Congratulations dude!
That would make you a jerk.
Finally the first commenter who knows what dy/dx really is. It is the limit of an expression containing a fraction, so not a "real" fraction in algebraic sense.
In the form of a frog meme?
"Eve of Destruction", it's not just for old hippies anymore...
Often in maths, a mere change of notation, analogous equation in another field, or just looking at things in a slightly different way will open up whole new areas of maths.
Fermat's Last Theorem took forever to prove and the proof relies on translating the problem to a completely unrelated area of maths, solving it there, and then translating the results back.
And if you do things like use polar coordinates, etc. some areas of maths burst open with good sense and nice equations.
Something as simple as a notation change can work wonders. But this is just for convenience of amateurs who don't understand what a derivative actually is and does. It's like saying "Don't use the word multiplication for vectors, because it's not the same as for scalars". We know. Anyone handling it knows. Anyone dumb enough to confuse the notations is going to find out very quickly that nothing works. Sure, it might help if you've literally never done those kinds of equations before, but likely then you'll not be making any ground-breaking mathematical discoveries any time soon.
Things don't tend to survive hundreds of years for no reason, especially when they are one pen-stroke away from being changed, and have themselves gone through several notational iterations in their time.
I got through a degree in maths without thinking "Well, this notation is stupid", including three years of advanced calculus.
If you don't understand the notation, that's the very least of your worries as regards actually doing any calculus.
Will revolutionize calculus teaching
There is a strong indicator: Leibnitz did the same thing independently at the same time. If you factor in that the scientific community was pretty small back then, that means the prerequisites were all there, the question had been asked and it just took somebody to put it together. That makes the results a "good" scientific result, but not a "genius level" one. And I am not talking about his contributions to physics, I am talking about his contributions to calculus, see the original story.
As to Riemann, I confused him with Leibnitz.
Most ACs are not even worth the keystrokes to insult them. Be generically insulted by this and ignored otherwise.
And alternatively, I really know what I am talking about and regard ACs as basically scum. However, I cannot see how you come to "pretentious", unless you are an authoritarian follower that things people that have a name may not be criticized. Is that it? Newton was so great, nobody is allowed to criticize him? That would be a pretty bad stance. As for "self-centered", were to you see any evidence for that?
So I got confused on a name in the history of mathematics. But this is /. and I did not look it up to make sure. It is a data point, not something that requires insight.
BTW, I do not "like" or "dislike" Newton. That is you projecting. I am of the opinion that the formalism he created for calculus is pretty bad and he did a huge disservice to a lot of people by creating and pushing it and trying to suppress better alternatives. This is something I am qualified to judge. Decidedly a negative contribution to calculus overall.
I am also wondering why you post as an AC. /. has pseudonymity. Use it. If you are scared of the NSA, they can already identify anything you post, AC or not.
Most ACs are not even worth the keystrokes to insult them. Be generically insulted by this and ignored otherwise.
How does this differ from the Grassmann algebra of differential forms? https://en.wikipedia.org/wiki/...