Further to

Some of you know that I published a Calculus book last year. My goal in this was mostly to counter-act the dry, boring, and difficult-to-understand textbooks that dominate the field. However, when it came to the second derivative, I realized that not only is the notation unintuitive, there is literally no explanation for it in any textbook I could find.

… the notation itself is problematic. Although it is written as a fraction, the numerator and denominator cannot be separated without causing math errors. This problem is somewhat more widely known, and has a workaround for it, known as Faa di Bruno’s formula.

My goal was to present a reason for the notation to my readers/students, so that they could more intuitively grasp the purpose of the notation. So, I decided that since no one else was providing an explanation, I would try to derive the notation myself. More.

The story we ran on the topic at Mind Matters has gone viral via Slashdot, with five thousand views since yesterday afternoon. (A paper about *calculus?*) Figures, Bartlett must have a point about the problem.

Paper: Bartlett, Jonathan *(open access)*,

Follow UD News at Twitter!

*See also:* Walter Bradley Center Fellow Discovers Longstanding Flaw in Elementary Calculus Jonathan Bartlett: The flaw doesn’t lead directly to wrong answers but it does create confusion. The lead author, Jonathan Bartlett, noted that the likely source of the bad notation was a philosophical issue. Because no one wanted to give differentials that same ontological status as other numbers, everyone presumed that the notational problems were simply the result of this fact, and no one pursued it further. *(Mind Matters)*

and

Is Standard Calculus Notation Wrong? Bartlett: The reason why this was not noticed before, I believe, is because, since the 1800s, mathematicians have not wanted to believe that infinitesimals are valid entities. Therefore, they were not concerned when the second derivative did not operate as a fraction – it didn’t need to, because it indeed wasn’t a fraction. Infinities and infinitesimals are the non-materialistic aspects of mathematics, just as teleology, purpose, and desire are the non-materialistic aspects of biology. *(Uncommon Descent)*

Bravo and congratulations to Jonathan Bartlett.

News, that’s a notable achievement and it’s good they are being noticed. Kudos to our authors and their colleagues. KF

PS: Somehow, I don’t see News Anchors holding interviews on the merits of adjusting notations for higher order derivatives!

No disrespect to Mr. Bartlett (I am not an expert on calculus) but 5000 hits is not “viral” by any measure. As a bare minimum 1,000,000 hits are required. 5 -10 million by most accounts. So, even by the most generous measure, his article is 0.5% viral.

BB

Are you incapable of praising someone for their accomplishments? If you want to criticize Bartlett’s paper, fair enough. But if all you have is childish criticism of what constitutes “viral” you are no better than ET.

Hi Ed- Thank you for your puerile bearing false witness. It seems you are no better than a belligerent infant.

ET

See BB, do you really wasn’t to be lumped in with this sort of behavior?

EG

Absolutely not. ET is not someone I would like to be associated with. Even indirectly. That is why, if you have not noticed, I never respond to his comments. Life is too short to attempt to have a discussion with someone like that.

LoL! “Ed George” and “Brother Brian” are different socks from the same person. Talk about a sad and deluded loser.

That said, Brother Brian has “responded” to my comments if you count spewing bald assertions as a response. It never says anything that cannot be easily swatted.

But yes, Ed Brother Brian George isn’t any better than I am. And its jealousy is duly noted- or is that envy?

BB

On this we agree. But my earlier comment about you making a negative statement still stands. Is the number of hits required to say something has gone viral relevant to this OP?

Hmmm, Ed. I don’t think BB is being hard on Johnny: he is criticizing, validly in my opinion, News’s headline. Johnny’s paper has stimulated an interesting discussion here, but as I said on the other thread, it is difficult to know what the source of the interest at Mind Matters is because Mind Matters doesn’t have comments. I looked at the comments thread at Slashdot, and it was a typical mishmash as happens on comments threads, with some support of johnny and some saying his point wasn’t useful and some pretty obviously not understanding calculus.

Johnny obviously knows calculus, has written a textbook whose approach is in ways similar to mine (so I approve! 🙂 ), and has offered a novel suggestion which may or may not wind up having an impact. I think he deserves some recognition for those things.

But, to repeat, New’s headline was not very accurate. (I’ll note that inaccurate headlines in all venues is probably more than norm than not.) I just don’t think BB should be derided for pointing that out, especially as he said nothing derogatory about Johnny or his paper at all.

I feel like I am being tag-teamed by Sybil….

Sybil? Who’s she?

Yes it is if those 5,000 hits came in one hour. And no, 1,000,000 hits is not the benchmark:

Going Viral: Three Definitions of Viral Videohazel- “Sybil” was a lady diagnosed with multiple personalities. It was also the name of a book that was required reading in psych classes.

Ah, that Sybil! Multiple-sock conspiracy disorder!

Hazel, yes I am possibly being too harsh on BB. I realize that his comment was about the title, not about Johnny’s article. I think he said as much. I just don’t see the point of mocking the title of an OP. News isn’t the first, and won’t be the last, to invoke hyperbole in the headlines of her pieces.

Yes, Ed, I agree, as I said, that inaccurate and often hyperbolic headlines are the norm many places today. One would wear oneself out if one felt like pointing them out all the time. I have several pet peeves: inaccurate headlines are one, photoshopped nature pictures that pretend to be real is another, as is “less” vs “fewer”. We all have to pick our battles. 🙂

Hazel

“Conspiracy” being the operative word. This is not the first time that ET has accused me of being a sock, even after explaining the nature of my on-site name. He/she obviously has some unhealthy obsession over this person. This other person must be devilishly handsome and very intelligent. 🙂

Hazel@17, I have the same pet peeve over “less” vs “fewer”. My pet peeves also include “affect” vs “effect”.

Folks, if something seems to be a fraction, but isn’t (it’s inherently the limit of a bound ratio, i.e. the trigonometric tangent of the angle of elevation or depression of the tangent-line to a curve at relevant point) — it specifies a slope function; then the symbol dy/dx appears to be treated as a fraction in certain operations [solve the “variable separable” differential eqn), that is in fact a clear point for confusions. Yes, work-arounds exist, but that is already an implicit admission. The alternative fluxion and D-operator notations have a point. Then, the second order is even more involved: the original function’s slope function has its slope function evaluated. In the mix, what is an infinitesimal lurks, and how this came to be re-legitimised scarce fifty years past, after having been the gateway to Calculus then deprecated. Has somebody written the Coffee Table book yet? Would our JB want to do the honours if not? KF

Hold on to your hats, it’s going viral at Hacker News now. link

F/N: I should note on why one may want higher order derivatives. The dynamics may have the equivalent of a displacement, driven by accumulation of velocity, but velocity itself is driven by acceleration (an effect of force), and that too is variable — “jerk” is discussed and there are onward terms. Often, second to fourth or perhaps fifth order will be good enough and dynamics are captured in differential equations. Yes, equations whose terms are derivatives. The old analogue computers used chains of integrating elements to set up and solve the equations physically, e.g. ball and disk mechanical integrators accumulating rotation. The general trick is to set up the highest order derivative on the LHS and drive it from a sum of lower order units on the RHS, cascaded as a chain and fed back, with appropriate modifications and injected signals. This was also done electronically using operational amplifiers, which are capable of performing a wide array of operations physically. KF

Ed George:

LoL! Except for the fact that only Brother Brian has explained the nature of its on-site name.

Whoopsie… 😛

ET

Except for the fact that I discussed the nature of my on-site name with Barry a few months ago.

Sure you did…

Brother Brian at 3: 5000 hits over a day or so is “viral” by the standards of newbie Mind Matters. Compare it with the showing for O’Leary’s piece on new research as to whether cats can recognize their names, not with Justin Bieber’s fan club. Thanks.

PS: RAWR!!!

Hazel at 10: “Viral” may have a definition somewhere but it is commonly used to mean that the stats spiked the way viruses multiply, with the anticipated impact on people providing the service.

If you (or others participating in this discussion) write about or teach calculus, it would be good to hear more of your thoughts on Dr. Bartlett’s approach.

Hi News. Actually, I have taught calculus and I’ve written a bunch about his paper on the other thread. I think I made it clear that I’ve offered positive support for johnny’s effort, even though I am skeptical about some aspects of his results.

And, I understand that viral may be relative, and if that maybe if Mind Matters or Slashdot normally gets few views, 5000 in a day would be viral in respective to those sites. Generally I consider viral to mean spreading out among difference sources, spread by different people, so in that sense, I wouldn’t say having a discussion on just one site would be viral.

But this isn’t a big deal: as I said above, I have a pet peeve about inaccurate, misleading, or exaggerated headlines, which are quite common, and the word viral is very commonly used without any specific definition.

I suppose 5,000 hits is viral for THIS site.

News, I think he is asking some important questions and has raised some serious questions. Certainly, the rehabilitation of infinitesimals 50 years ago now will have repercussions. KF

F/N: I see this thread has over 5,000 hits, so let me welcome new visitors and also let me draw attention to the main discussion here at UD: https://uncommondescent.com/intelligent-design/is-standard-calculus-notation-wrong/ KF

PS: For those looking for an in-brief, UD contributor JB has worked with colleagues to author a paper — here: http://online.watsci.org/abstr.....-pdf/4.pdf — on an issue connected to the notation commonly used for the second derivative in Calculus, after having authored a book on introductory Calculus. He has suggested that the usual notation dy/dx is there because there is a connexion to fractions and we should seek to find out how this can be fleshed out.

Note:

dy/dx is

lim h –> 0

of

[f(x + h) – f(x)]/[(x + h) – x]

Treating dy/dx as a quotient, one may use the standard result for derivative of a quotient. When one does so, one sees:

d/dx[ dy/dx] =

d^2y/dx^2– [dy/dx]*[d^2x/dx^2]If x is TRULY independent, the second term reduces to 0 [as dx/dx is 1 thence its rate of change or slope is 0], but if there is an underlying parameter such as time that is the true independent variable (notice, ontological and causal considerations now lurk) that does not necessarily hold.

An interesting exchange of ideas emerges, tied to this.

Where, underlying, one needs to appreciate infinitesimals and transfinite hyperreals, perhaps this may help: https://uncommondescent.com/mathematics/logic-first-principles-17-pondering-the-hyperreals-r-with-prof-carol-thomas-including-infinitesimals/

Yes, things have got a lot more complicated with the old familiar number line.