[math-fun] The Bernoulli Manifesto
Back in 2004, Peter Luschny wrote an open letter to Donald E. Knuth about his conviction that the proper definition of the Bernoulli number B(1) should be 1/2, not -1/2. Professor Knuth declined to embrace the idea. http://www.luschny.de/math/zeta/OpenLetter.pdf Peter has now put up a detailed response to Knuth, calling it The Bernoulli Manifesto. http://luschny.de/math/zeta/The-Bernoulli-Manifesto.html
I have responded << Viewed on an iMac with Safari browser under OS 10.8.4 , eight of these line drawings display as unreadable thumbnail size; from "The real Bernoulli function" to "Euler's summation formula, as written by Euler" . At a quick glance you make very detailed and coherent argument. I have considerable sympathy, having recently expended much effort in documenting a similarly entrenched dichotomy regarding the definition of binomial coefficients --- wherein it is noteworthy that both Mathematica and Maple find themselves again on the wrong side of the fence --- see https://www.dropbox.com/s/anykne0pd55ehjg/binomial.pdf I hope you will not remain so offended by Knuth's response to your initial post, which was frankly far less convincing. These questions are genuinely difficult to resolve; and no professional is going to casually discard his entrenched conventions without a good deal of persuasion. WFL >> On 9/15/13, Hans Havermann <gladhobo@teksavvy.com> wrote:
Back in 2004, Peter Luschny wrote an open letter to Donald E. Knuth about his conviction that the proper definition of the Bernoulli number B(1) should be 1/2, not -1/2. Professor Knuth declined to embrace the idea.
http://www.luschny.de/math/zeta/OpenLetter.pdf
Peter has now put up a detailed response to Knuth, calling it The Bernoulli Manifesto.
http://luschny.de/math/zeta/The-Bernoulli-Manifesto.html
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Minuscule point: It seems to me that in Knuth's response to Luschny, he writes cos(pi s) where he means to write cos(2pi s). --Dan
On 15/09/2013 17:05, Hans Havermann wrote:
Back in 2004, Peter Luschny wrote an open letter to Donald E. Knuth about his conviction that the proper definition of the Bernoulli number B(1) should be 1/2, not -1/2. Professor Knuth declined to embrace the idea. ... Peter has now put up a detailed response to Knuth, calling it The Bernoulli Manifesto.
As Knuth says (and Luschny agrees), there are lots of little issues of this kind. Perhaps what is called for is a widely agreed upon set of New Conventions, with a website and (if anyone will publish it) a manifesto in a widely read journal, so that mathematicians wanting to break with the mistakes of the past can just begin their papers with something like In this paper I use the so-called New Conventions of mathematical notation documented at www.newconventions.org and in [1]; in particular, note that the definitions used here for the Bernoulli numbers $B_k$ and the gamma function $\Gamma(z)$ may be unexpected. (I don't actually think it makes sense to change the definition of the gamma function -- better just to abandon it in favour of the factorial function -- but never mind that. It's just an example.) For this to have any chance of success, the new conventions would be appealing enough *collectively* to persuade a reasonable number of mathematicians to adopt them wholesale. There would probably need to be a preliminary round of opinion-canvassing. All of which would doubtless seem like a total waste of time to many mathematicians. But if the end result were half an hour less of wasted head-scratching for each of 20k undergraduates once a year (let's assume, contrary to fact, that by the time a mathematician is properly trained all these poor conventions cease to cause any extra work or annoyance) then it would be worth a substantial one-off effort. I doubt it'll ever happen. Herding cats, etc. -- g
It would cause much less confusion if a new name is associated with the new definition, e.g. a "new-Bernoulli" number. -- Gene
________________________________ From: Gareth McCaughan <gareth.mccaughan@pobox.com> To: math-fun@mailman.xmission.com Sent: Sunday, September 15, 2013 1:50 PM Subject: Re: [math-fun] The Bernoulli Manifesto
On 15/09/2013 17:05, Hans Havermann wrote:
Back in 2004, Peter Luschny wrote an open letter to Donald E. Knuth about his conviction that the proper definition of the Bernoulli number B(1) should be 1/2, not -1/2. Professor Knuth declined to embrace the idea. ... Peter has now put up a detailed response to Knuth, calling it The Bernoulli Manifesto.
As Knuth says (and Luschny agrees), there are lots of little issues of this kind. Perhaps what is called for is a widely agreed upon set of New Conventions, with a website and (if anyone will publish it) a manifesto in a widely read journal, so that mathematicians wanting to break with the mistakes of the past can just begin their papers with something like
In this paper I use the so-called New Conventions of mathematical notation documented at www.newconventions.org and in [1]; in particular, note that the definitions used here for the Bernoulli numbers $B_k$ and the gamma function $\Gamma(z)$ may be unexpected.
(I don't actually think it makes sense to change the definition of the gamma function -- better just to abandon it in favour of the factorial function -- but never mind that. It's just an example.)
For this to have any chance of success, the new conventions would be appealing enough *collectively* to persuade a reasonable number of mathematicians to adopt them wholesale. There would probably need to be a preliminary round of opinion-canvassing.
All of which would doubtless seem like a total waste of time to many mathematicians. But if the end result were half an hour less of wasted head-scratching for each of 20k undergraduates once a year (let's assume, contrary to fact, that by the time a mathematician is properly trained all these poor conventions cease to cause any extra work or annoyance) then it would be worth a substantial one-off effort.
I doubt it'll ever happen. Herding cats, etc.
-- g
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How about "Bernewli" numbers? Sent from my iPhone On Sep 15, 2013, at 16:57, Eugene Salamin <gene_salamin@yahoo.com> wrote:
It would cause much less confusion if a new name is associated with the new definition, e.g. a "new-Bernoulli" number.
-- Gene
________________________________ From: Gareth McCaughan <gareth.mccaughan@pobox.com> To: math-fun@mailman.xmission.com Sent: Sunday, September 15, 2013 1:50 PM Subject: Re: [math-fun] The Bernoulli Manifesto
On 15/09/2013 17:05, Hans Havermann wrote:
Back in 2004, Peter Luschny wrote an open letter to Donald E. Knuth about his conviction that the proper definition of the Bernoulli number B(1) should be 1/2, not -1/2. Professor Knuth declined to embrace the idea. ... Peter has now put up a detailed response to Knuth, calling it The Bernoulli Manifesto.
As Knuth says (and Luschny agrees), there are lots of little issues of this kind. Perhaps what is called for is a widely agreed upon set of New Conventions, with a website and (if anyone will publish it) a manifesto in a widely read journal, so that mathematicians wanting to break with the mistakes of the past can just begin their papers with something like
In this paper I use the so-called New Conventions of mathematical notation documented at www.newconventions.org and in [1]; in particular, note that the definitions used here for the Bernoulli numbers $B_k$ and the gamma function $\Gamma(z)$ may be unexpected.
(I don't actually think it makes sense to change the definition of the gamma function -- better just to abandon it in favour of the factorial function -- but never mind that. It's just an example.)
For this to have any chance of success, the new conventions would be appealing enough *collectively* to persuade a reasonable number of mathematicians to adopt them wholesale. There would probably need to be a preliminary round of opinion-canvassing.
All of which would doubtless seem like a total waste of time to many mathematicians. But if the end result were half an hour less of wasted head-scratching for each of 20k undergraduates once a year (let's assume, contrary to fact, that by the time a mathematician is properly trained all these poor conventions cease to cause any extra work or annoyance) then it would be worth a substantial one-off effort.
I doubt it'll ever happen. Herding cats, etc.
-- g
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Wikipedia calls the original def. "First" Bernoulli numbers and the version with B(1) = 1/2 "Second" Bernoulli numbers. --Dan On 2013-09-15, at 1:57 PM, Eugene Salamin wrote:
It would cause much less confusion if a new name is associated with the new definition, e.g. a "new-Bernoulli" number.
-- Gene
________________________________ From: Gareth McCaughan <gareth.mccaughan@pobox.com> To: math-fun@mailman.xmission.com Sent: Sunday, September 15, 2013 1:50 PM Subject: Re: [math-fun] The Bernoulli Manifesto
On 15/09/2013 17:05, Hans Havermann wrote:
Back in 2004, Peter Luschny wrote an open letter to Donald E. Knuth about his conviction that the proper definition of the Bernoulli number B(1) should be 1/2, not -1/2. Professor Knuth declined to embrace the idea. ... Peter has now put up a detailed response to Knuth, calling it The Bernoulli Manifesto.
As Knuth says (and Luschny agrees), there are lots of little issues of this kind. Perhaps what is called for is a widely agreed upon set of New Conventions, with a website and (if anyone will publish it) a manifesto in a widely read journal, so that mathematicians wanting to break with the mistakes of the past can just begin their papers with something like
In this paper I use the so-called New Conventions of mathematical notation documented at www.newconventions.org and in [1]; in particular, note that the definitions used here for the Bernoulli numbers $B_k$ and the gamma function $\Gamma(z)$ may be unexpected.
(I don't actually think it makes sense to change the definition of the gamma function -- better just to abandon it in favour of the factorial function -- but never mind that. It's just an example.)
For this to have any chance of success, the new conventions would be appealing enough *collectively* to persuade a reasonable number of mathematicians to adopt them wholesale. There would probably need to be a preliminary round of opinion-canvassing.
All of which would doubtless seem like a total waste of time to many mathematicians. But if the end result were half an hour less of wasted head-scratching for each of 20k undergraduates once a year (let's assume, contrary to fact, that by the time a mathematician is properly trained all these poor conventions cease to cause any extra work or annoyance) then it would be worth a substantial one-off effort.
I doubt it'll ever happen. Herding cats, etc.
-- g
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="Dan Asimov" <dasimov@earthlink.net> Wikipedia calls the original def. "First" Bernoulli numbers and the version with B(1) = 1/2 "Second" Bernoulli numbers.
Oh my! Surely these should be "Zeroth" Bernoulli numbers and "First" Bernoulli numbers respectively? Wikipedia aside, the heat at the core of this dispute seems to be over nothing more than the assignment of the title "THE Bernoulli numbers". Since Lushny presented this as a contest, perhaps B^(1) really ought to be crowned as "THE WORLD CHAMPION Bernoulli numbers"? It's a superpower war over control of the tiny notational duchy consisting of the letter "B"! This is contentious only because of the scarcity of single letters (hence the trend in computing to eschew these for anything other than trivial temporary indices, and to use longer more descriptive identifiers for things with any significant persistence or scope). Maybe we should base mathematical notation using Egyptian hieroglyphs or Chinese ideograms instead of alphabetic characters--then maybe these kinds of arguments over notation wouldn't arise? While I personally found Lushny's reasons to prefer B^(1) persuasive, I found it telling that throughout it all he could keep the discussion quite lucid merely by using the notations B^(0) and B^(1). But if you really want to reduce the clutter in your public expositions why not simply say something like: "Given the generating function for the Bernoulli polynomials B^(x)(z) = <RHS eq. (1)>, we see that when x=1 we have B^(1)(z) = <RHS eq. (3)>. Below we will write the coefficients B_n(1) simply as B_n. [* footnote: Beware! Some Authors prefer to write B_n for B_n(0) instead]" ? This seems a small tax to pay while Ye Olde Misguyded Stylinge withers away. --MLB
participants (7)
-
Dan Asimov -
Eugene Salamin -
Fred Lunnon -
Gareth McCaughan -
Hans Havermann -
Marc LeBrun -
Victor S. Miller