Walter Bradley Center Fellow Discovers Longstanding Flaw in Elementary Calculus(mindmatters.ai)
mindmatters.ai
Walter Bradley Center Fellow Discovers Longstanding Flaw in Elementary Calculus
https://mindmatters.ai/2019/04/walter-bradley-center-fellow-discovers-longstanding-flaw-in-elementary-calculus/
8 comments
The article's title is misleading. The mathematician has suggested an improvement or clarification to the notation used to describe a second derivative which makes it easier to manipulate formulaically.
The article says:
> The flaw they discovered is one of notation.
I don't think the title is misleading given the clarity of the article. Like most titles regarding technical subjects, it needs to be carefully qualified to be understood.
But was there a flaw of calculus notation?
Bartlett introduces changes to Leibniz's notation that:
1. Allows for direct algebraic manipulation of differential relationships, for both simple and complex equations, and which extends very nicely to multivariable cases.
2. Makes understanding the relationships between higher order derivatives clearer.
3. Clearly highlights that differentials and derivatives are the two steps of differentiation.
4. Removes misunderstandings about Leibnitz's notation where the notation looks like a ratio, an interpretation that works for the first derivative, but confusingly and inconveniently does not for high order derivatives.
5. Results in equations with far fewer subscripts and superscripts. Surely a Fields Medal is in order? :)
6. Removes a now unnecessary concept from differential algebra (i.e. the Leibniz non-algebraic non-ratio).
That last one simplifies the foundations of calculus, not just notation.
Given that, it is a fix to calculus, not just its notation.
> The flaw they discovered is one of notation.
I don't think the title is misleading given the clarity of the article. Like most titles regarding technical subjects, it needs to be carefully qualified to be understood.
But was there a flaw of calculus notation?
Bartlett introduces changes to Leibniz's notation that:
1. Allows for direct algebraic manipulation of differential relationships, for both simple and complex equations, and which extends very nicely to multivariable cases.
2. Makes understanding the relationships between higher order derivatives clearer.
3. Clearly highlights that differentials and derivatives are the two steps of differentiation.
4. Removes misunderstandings about Leibnitz's notation where the notation looks like a ratio, an interpretation that works for the first derivative, but confusingly and inconveniently does not for high order derivatives.
5. Results in equations with far fewer subscripts and superscripts. Surely a Fields Medal is in order? :)
6. Removes a now unnecessary concept from differential algebra (i.e. the Leibniz non-algebraic non-ratio).
That last one simplifies the foundations of calculus, not just notation.
Given that, it is a fix to calculus, not just its notation.
Well, I guess that's debatable. However, the way I see it, the old version was presented as a fraction, but couldn't be used as a fraction. The new version is also presented as a fraction and can be used as a fraction.
Perhaps it is due to my old-school nature, but if I wrote a fraction in school that didn't work as a fraction, that would be a mistake, would it not? Especially if it was possible to write it correctly? I couldn't go to my professor and just say, "well, actually, while it looks like a fraction, it was actually just a piece of notation that needed future clarification."
Perhaps it is due to my old-school nature, but if I wrote a fraction in school that didn't work as a fraction, that would be a mistake, would it not? Especially if it was possible to write it correctly? I couldn't go to my professor and just say, "well, actually, while it looks like a fraction, it was actually just a piece of notation that needed future clarification."
[deleted]
> Well, I guess that's debatable.
What part of it is debatable? The article clearly states that it is just a notational improvement: there was really no flaw in calculus, it was more a wart in notation. This change would not result in different answers unless the original calculations were done incorrectly.
I'd regard the current notation more like an irregular verb, or something that is spelled differently from how it is pronounced: it's inconvenient to deal with, but everyone adept at calculus knew about it and how to deal with it. This article is just describing a notational improvement that would make it easier to get right.
What part of it is debatable? The article clearly states that it is just a notational improvement: there was really no flaw in calculus, it was more a wart in notation. This change would not result in different answers unless the original calculations were done incorrectly.
I'd regard the current notation more like an irregular verb, or something that is spelled differently from how it is pronounced: it's inconvenient to deal with, but everyone adept at calculus knew about it and how to deal with it. This article is just describing a notational improvement that would make it easier to get right.
[deleted]
The fact that there was a well-known kludge to get around the problems inherent in the notation doesn't make the notation right. As the paper pointed out, the new notation didn't drop magically from the sky, it is literally the application of the quotient rule to the derivative. The reason that no one noticed this notation before, is that they didn't think to apply the quotient rule to the quotient dy/dx.
If I had a problem, and my solution involved forgetting to apply the quotient rule to a prominent quotient, it would still be wrong, even if I also came out with a set of kludges that allowed me to get right answers in the most common cases.
If I had a problem, and my solution involved forgetting to apply the quotient rule to a prominent quotient, it would still be wrong, even if I also came out with a set of kludges that allowed me to get right answers in the most common cases.
It seems similar to Roman numerals vs Arabic numerals. We could stick with Roman numerals, but mathematics would be pretty much at a standstill if we had to use such an unwieldy system.
A major improvement in notation is not insignificant.
A major improvement in notation is not insignificant.