Talk:Bernhard Riemann

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Perhaps the Greatest Mathematician

  • Georg Friedrich Bernhard Riemann (1826-1866) was perhaps the greatest mathematician in history.
  • Leonhard Euler (April 15, 1707–September 18, 1783) was a devout Christian (Calvinist) who became the greatest mathematician of the eighteenth century and perhaps the most productive of all time.
  • Carl Friedrich Gauss (1777–1855) was a German mathematician considered to be one of the greatest of all time, sometimes called the "the prince of mathematicians."
  • Georg Cantor (1845-1918) was a Russian-German mathematician who is considered one of the greatest ever because he created the field of Set Theory.
  • Augustin-Louis Cauchy (1789-1857) was a devout Catholic, and an extraordinary French mathematician, considered to be one of the top twenty of all time.
  • Pierre Simon Laplace (b. Beaumont-en-Auge, March 23, 1749 - d. Paris, March 5, 1827) was a French astronomer and mathematician. Laplace is considered one of the greatest scientists of all time.
  • John von Neumann (1903-1957) was perhaps the most brilliant man of the 20th century, based on his remarkable achievements and impressions by other smart people who knew him.
  • Amalie[sic!] Noether (1882-1935) was a German mathematician who contributed to abstract algebra and theoretical physics. Albert Einstein described her as the most important woman in the history of mathematics.

I prefer "Riemann was one of the most influential mathematicians in history" over "Riemann was perhaps the greatest mathematician in history": while Riemann certainly was one of the greatest mathematicians of all time, many would argue that Gauss or Euler deserve to be called the greatest. --AugustO (talk) 08:25, 4 July 2018 (EDT)

Georg Friedrich Bernhard Riemann (1826-1866) is considered the greatest modern mathematician, That is as meaningful as Mozart is considered the greatest classical composer: it is just a starting point for an endless debate. --AugustO (talk) 06:15, 23 July 2018 (EDT)

It's not a close question. Riemann was the greatest modern mathematician. Deniers of Christ don't like that because Riemann was a devout Christian, but here we give credit where it is due, without liberal denial.--Andy Schlafly (talk) 12:05, 23 July 2018 (EDT)
Does there exist any source for this statement - other than your intuition? I failed to come up with surveys which put Riemann on the very top of the greatest modern mathematicians. --AugustO (talk) 15:24, 23 July 2018 (EDT)
Perhaps you're looking too much at liberal sites, where Christians are routinely marginalized. Denying Christ means denying or downplaying those who accomplished greatness as motivated by their belief in Christ. Or do you deny the logical truth of that observation?--Andy Schlafly (talk) 00:55, 24 July 2018 (EDT)

Andy, you are the only one I know who is obsessed with the religion or denomination of mathematicians. I don't deny that Riemann was great mathematician. Nevertheless you will have difficulties to find a mathematician who would call him "the greatest mathematician in history", or even "the greatest modern mathematician" - whether they know about his faith or not. Do you have any Christian sites or non-liberal rankings which put him on the very top? I'd be interested to see them.

But I'm afraid that you base your statement solely on one of your personal insights which are seldom shared by anyone else... --AugustO (talk) 07:07, 24 July 2018 (EDT)

You don't address my point: "Denying Christ means denying or downplaying those who accomplished greatness as motivated by their belief in Christ." You also don't engage in any objective analysis of Riemann's greatness, such as the fact that he is the author of the greatest unsolved problem in math. Moreover, I find your repeated edits to the entry to be silly, rather than a genuine attempt at a compromise. I've locked the page accordingly.--Andy Schlafly (talk) 15:32, 24 July 2018 (EDT)

Andy, I just saw your edit adding the Wikipedia reference to this article -- would you please make it a permalink (since Wikipedia constantly changes and what it will say five years from now could be entirely different), since I can't edit the article myself? --1990'sguy (talk) 23:55, 24 July 2018 (EDT)

  • You don't address my point Sorry, I give it a try...
  • "Denying Christ means denying or downplaying those who accomplished greatness as motivated by their belief in Christ." Riemann is judged to be one of the greatest mathematicians by those who know his religious beliefs and those who don't (those seem to make up the overwhelming majority). I have yet to see comments which disparage Riemann's religious beliefs - other than those of Isaac Newton, who is sometimes regarded as a religious nutjob.
  • You also don't engage in any objective analysis of Riemann's greatness, such as the fact that he is the author of the greatest unsolved problem in math. Was Fermat the greatest mathematician until 1994 when Andrew Wiles solved his eponymous problem? This was certainly the greatest unsolved problem back then!
  • Moreover, I find your repeated edits to the entry to be silly, rather than a genuine attempt at a compromise. My repeated edit high-lighted the fact that you are the only one I know who claims that Riemann is the greatest modern mathematician - a statement which seems silly to me in its absoluteness.
I think that most mathematicians would agree that it is very difficult to rank the greatest ones: a linear order does not suit the task. Is Gauss > Euler or Euler > Gauss? Both equations are true in some regards, but not in others. Most will agree that Newton > Leibniz, but nontheless, Leibniz's notation was more successful and mathematicians profited greatly from it.
In summa: your quest to define the "greatest modern mathematician" is a fool's errand. --AugustO (talk) 10:20, 25 July 2018 (EDT)

For example, even in the entry on Wikipedia ...

Andy, how do you get these numbers? Omitting textbooks and ancient manuscripts, there are - according to my count - 99 different works mentioned in the list:

author number of works
Leonhard Euler 6
Bernhard Riemann 5
Alexander Grothendieck 3
Carl Friedrich Gauss 3
Joseph Louis Lagrange 3
Peter Gustav Lejeune Dirichlet 3
Jean-Pierre Serre 3

Hilbert, Gödel, Neumann, Newton and a few others are listed with two works. While I could probably tell you which fields these mathematicians worked in, I would be hard-pressed to state their religious beliefs.

Andy, could you please link a webpage or cite a book which states that Bernhard Riemann is the greatest modern mathematician? You should change the introduction to the true statement "Bernhard Riemann is considered one of the greatest modern mathematicians". I imagine a high-school pupil who cites your insight in a homework and will then be berated as he cannot give any proper sources. At this moment, Conservapedia is the only place in the internet which declares Riemann to be the greatest modern mathematician. Such a bold statement needs a little bit more proof than a misread list from wikipedia.

--AugustO (talk) 06:51, 25 July 2018 (EDT)

"... Bernhard Riemann was the greatest modern mathematician."[1] Conservative (talk) 11:43, 25 July 2018 (EDT)
Leonhard Euler was Christian mathematician who wrote a Christian apologetics work. "If Gauss is the Prince, Euler is the King. Living from 1707 to 1783, he is regarded as the greatest mathematician to have ever walked this planet."[2]Conservative (talk) 11:49, 25 July 2018 (EDT)

The unnamed blogger of "SHOCKAWENOW" states that "The above cited article and the the videos below clearly show that Bernhard Riemann was the greatest modern mathematician." Unfortunately, he or she seems to engage in misdirections:

  • The article makes the case that Riemann was an exceptional mathematician, certainly one of the greatest. But its author Steve Bishop never claims that Riemann is THE greatest modern mathematician.
  • The videos call him "a giant of mathematics" and stress his general importance. But neither claims that he was THE greatest (modern) mathematician.

As the blogger of SHOCKAWENOW created this entry today, I assume that he or she is aware of this discussion and just wants to troll us with his or her lies. --AugustO (talk) 13:21, 25 July 2018 (EDT)

As can be seen HERE, the article has been updated.Conservative (talk) 13:42, 25 July 2018 (EDT)
Another obvious misdirection or lie: he or she gives a list of great mathematicians including Riemann, but claims that this list shows that Riemann was the greatest, and not one of the greatest. The website which SHOCKAWENOW misquotes ranks Riemann under the TOP 5 (as can be seen [here]), below Newton, Archimedes, Gauss, and Euler. This shows that Riemann was indeed one of the greatest mathematicians ever, Gauss and Riemann belong certainly to the greatest modern mathematicians. --AugustO (talk) 14:27, 25 July 2018 (EDT)

Even based on August's table, Riemann is still clearly the greatest modern mathematician. His only rival for that title -- the likewise devout Christian Leonhard Euler -- did not live in modern times.--Andy Schlafly (talk) 13:54, 25 July 2018 (EDT)

Why have no mathematician yet claimed that Riemann was THE greatest modern mathematician based on this list? Because its really hard to compare their diverse works! --AugustO (talk) 15:17, 25 July 2018 (EDT)
God is the greatest mathematician of all time. And His book of Numbers is among the most published books in all of history.
And Christians have access to inspiration by God Almighty.Conservative (talk) 14:29, 25 July 2018 (EDT)
And Christians should not misdirect to make a point. --AugustO (talk) 14:34, 25 July 2018 (EDT)
In all seriousness, I might be mistaken, but it seems as if the two top contenders for the greatest mathematicians of the period between 1800 and 2000 are David Hilbert and Bernhard Riemann.
Why was Riemann a less/greater mathematician than Hilbert? Conservative (talk) 14:39, 25 July 2018 (EDT)
I suppose one should look at the working period: Gauss published since 1799, Hilbert in the last decades of the 19th century. Hilbert (a Calvinist who turned agnostic), Gauss (a Lutheran Christian), and Riemann certainly some of the greatest mathematicians of all time, and I cannot judge which one of them was the greatest... --AugustO (talk) 15:17, 25 July 2018 (EDT)
When the greatest is a devout Christian, then liberal denial requires denying that there is a greatest. But surely there is the greatest in certain fields, by objective measures. Do you deny that there was a greatest basketball player, a greatest playwright, and a greatest boxer too?--Andy Schlafly (talk) 18:27, 25 July 2018 (EDT)
I doubt that many people hold to this notion that there is a greatest person in any but a very small set of fields of endeavor. Certainly not in sports (other than a few very simple measures), play writing, musical composition, or mathematics.
And I doubt that not holding this notion ("denial", as you call it) has anything to do with whether the top 3 contenders (Euler, Gauss, and Riemann, in this discussion) are Christian. And this "denial" (I call it common sense) has nothing to do with liberalism. I looked through your liberal denial page, and none of its 87 points addresses the issue of believing there is a greatest in any field.
SamHB (talk) 19:35, 27 July 2018 (EDT)

Unrelated to this dispute on whether Riemann is the greatest modern mathematician, would you replace the Wikipedia link with a permalink? I can't edit the page since it's protected, and it's good practice to use permalinks when citing wikis. --1990'sguy (talk) 18:37, 25 July 2018 (EDT)

I just unlocked it. I welcome your improvement on the link. Thanks!--Andy Schlafly (talk) 18:43, 25 July 2018 (EDT)
Done! --1990'sguy (talk) 18:44, 25 July 2018 (EDT)
It appears as if Gauss was greater than Hilbert as a mathematician. The problem with comparing Riemann with Gauss is that Gauss lived much longer. Gauss lived twice as long as Riemann. Who knows what Riemann would have accomplished if he lived as long as Gauss. Conservative (talk) 22:51, 25 July 2018 (EDT)
Gauss was greater than Hilbert in math. But even biased Wikipedia tacitly concedes, as shown above, that Riemann was a greater mathematician than Gauss.--Andy Schlafly (talk) 01:10, 26 July 2018 (EDT)

@Conservative: the mendacious troll at SHOCKAWENOW has deleted the entry on Riemann, leaving Conservapedia the only place in the internet which claims that Riemann is the greatest (modern) mathematician. Perhaps they remembered the belatedly the Eights Commandment. --AugustO (talk) 03:38, 26 July 2018 (EDT)

@Aschlafly: "Do you deny that there was a greatest basketball player, a greatest playwright, and a greatest boxer too?" Saying that XYZ is the greatest basketball player ever seems to be a good way to start a fight in a pub. There are endless debates on the internet whether Tyson could have beaten Ali. As for the greatest playwright: 100 million of Russians will agree that it is Anton Chekhov,

Has it ever occurred to you to ask some (Christian) mathematicians about their view on the absolute greatness of Riemann? Given your aversion to complex analysis, do you even understand Riemann's conjecture? The sureness of your claim seems to be rooted in ignorance, making it a prime example of the Dunning-Kruger-effect. --AugustO (talk) 03:47, 26 July 2018 (EDT)

AugustO, perhaps the blogger at Shockawenow changed his/her mind. For example, both Riemann and Gauss were giants in terms of modern mathematicians.Conservative (talk) 09:42, 26 July 2018 (EDT)
August, do you deny that Shakespeare -- a devout Christian -- was the greatest English playwright? I'm just wondering how far your denial extends.--Andy Schlafly (talk) 14:19, 26 July 2018 (EDT)

Quite amusing: you are shifting the goalposts and crying denial! Shakespeare is presented as the greatest English playwright in all English classes all over the world. Is he the greatest playwright ever? What about Aeschylus? They are quite hard to compare, aren't they? Can you even rank them? Same for Gauss and Riemann. --AugustO (talk) 17:07, 26 July 2018 (EDT)

So you won't directly answer the obvious question as to whether Shakespeare was the greatest English playwright as a devout Christian. Liberal denial is a stubborn thing. It is ultimately rooted in the denial of Christ. Once Christ is denied, then achievements of those inspired by Christ must be denied or downplayed too.--Andy Schlafly (talk) 17:12, 26 July 2018 (EDT)
Nonsense! Non-acceptance of a religion does not require non-acceptance of the achievements of that religion's adherents. I do not accept the tenets of the Roman Catholic Church, or of Judaism or Islam, but I recognize the achievements of the many scholars and experts in those religions. SamHB (talk) 19:35, 27 July 2018 (EDT)
I belief that Shakespeare is the greatest English playwright. Why? I havn't read or seen all of his works, I don't know that many other English playwrights - and I certainly enjoyed "The Importance of Being Earnest" more than "Much Ado about Nothing". But I see the his influence still at work today, and - as I said above - there seems to be a general consensus that he deserves indeed the top spot.
How much do you know about Riemann's work in detail? Or Gauss's? Can you make the informed decision for yourself who of those is the greater one? Or do you judge them only by who was more devout than the other?
--AugustO (talk) 18:11, 26 July 2018 (EDT)

How dare you suggest that AugustO and I are "deniers of Christ"?

In one of Andy's many recent edits making outlandish claims about the absolute primacy of Riemann, he gives an edit comment of "Liberal denial of Christ is not going to extend here to denial of the greatness of those who were devout Christians, as Riemann was." I am NOT a denier of Christ. And, from AugustO's extensive writings about the Greek text of the Gospels, I doubt that he is a denier.

Those outlandish claims seem to be rooted in the notion that one's mathematical prowess can be deduced from their religion. But Gauss and Euler, the two other mathematicians in my edit, were also devout Christians. In fact, virtually the entire European intelligentsia before the 20th century were. To use religious considerations in arguing various mathematicians' achievements is just silly. It makes Conservapedia look like a place where mathematical/intellectual topics are not taken seriously.

Insistence that there must logically be a "greatest basketball player, a greatest playwright, and a greatest boxer" is a real slippery slope, and I recommend that this notion not be used in arguing Riemann's greatness. While the other two endeavors are admittedly competitive, arguing that there must be a "greatest playwright" is silly. Many people consider Shakespeare to be their favorite, and many people (including myself and, apparently, Cons) list Mozart as their favorite composer. But it's silly to elevate this to some kind of objective "best" status. And particularly silly to bring in religion as a determining factor.

There are a number of specific achievements by Riemann that the article doesn't cover, such as what it is that makes the "Riemann integral" the accepted definition of integral, when integration had actually been known hundreds of years earlier. Similarly, the article by Euler fails to mention the polyhedral formula, a formula known to any sharp junior high school mathematics student, right up there with the e-to-the-pi-i thing. We could make significant contributions (the preceding two were just off the top of my head) to Conservapdia's mathematics presentation if we would stop getting caught up in who's the better Christian and whether that makes them a better mathematician. I could make such improvements myself if I hadn't gotten burned out on CP math material several years ago. (Ironically, over the Riemann integral article.)

@AugustO: Please don't make mainspace edits that explicitly and publicly annoy Andy. Putting Andy's name ("is considered by Andrew Schlafly") into the mainspace article is what caused the recent locking, and you should have known better. You know full well how to engage Andy, and you are very good at it.

@Cons: Congratulations for doing serious research and making serious edits to a serious topic.

@everyone: What's this "SHOCKAWENOW" garbage? Some anonymous internet troll? Perhaps living in his parents' basement and making Youtube videos? The internet is full of them. Especially those that like to troll CP for entertainment. Ignore him. His antics should be utterly beneath the notice of people trying to make serious pages about mathematicians. Let's not have any further talk about this loser.

SamHB (talk) 13:21, 26 July 2018 (EDT)

Gauss very strongly believed in the existence of God. But he was either a "general monotheist"/deist or at least a nominal Lutheran.[3][4]Conservative (talk) 14:00, 26 July 2018 (EDT)
Sam, you make the same fallacy that is common in liberal style on this topic: implying that the high intellectual achievers were Christian only if they lived before the 20th century, but since the beginning of the 20th century atheism rules intellectual achievement today.
Utter nonsense! I implied nothing of the kind! You got the first clause correctly (high achievers were mostly Christian if they lived before 1900), but your claim that I therefore implied that high achievers were mostly non-Christian after 1900 is totally wrong. I implied nothing of the kind. I don't know what your fallacy is called (if X is true before 1900, then it must be false after 1900), but it is very obviously wrong. The only reason I set a cutoff of 1900 is that I didn't want to get into a debate over the prevalence of Christianity among the intelligentsia before 1900 vs. after. What's going on after 1900 is not germane to the discussion of Euler, Gauss, and Riemann.
It is singularly unhelpful to tell me that I was falling into some kind of fallacy here, and seeming to suggest that this would be elucidated by the liberal style article. I glanced at the 76 points in that article, and at a few of the 74 other articles referenced in the template at the bottom, and none of that material relates to this topic. I fact, I wonder who you think your target audience is for those articles, what you are trying to convince those people of, and whether you are succeeding at it.
SamHB (talk) 19:35, 27 July 2018 (EDT)
In fact, deniers of Christ have been common for 2000 years, no more so today than before. John von Neumann, considered the smartest intellectual of the 20th century, converted to Catholicism.--Andy Schlafly (talk) 15:29, 26 July 2018 (EDT)
And, as another example, Lise Meitner converted from Judaism to Christianity later in her life, after the war. SamHB (talk) 19:35, 27 July 2018 (EDT)
And we thought you were supposed to be gone until August 28, SamHB? Not only did you prove Andy's point regarding liberal style in your edit, you also showed poor form in that same edit by interrupting the flow of Andy's comment by inserting part of your edit into the middle of his. Congratulations on showing just what a liberal's word is really worth. Northwest (talk) 21:23, 27 July 2018 (EDT)

I think we can put together a decent article

Andy: I saw your reference involving Hetu Chakra. While I don't think he is much of a mathematician, and Quora is not a place to find high-quality information, I think we (you, me, and AugustO) can reach a consensus on how to present Riemann's stature in the mathematical pantheon. Please be patient. I need to do more research on this. (I looked for the E. T. Bell book in my personal library and was surprised to find that it isn't there, so I'm going to have to borrow a copy.) In any case, my "sabbatical" against substantive contributions goes until the end of the month, at which time I wil put some work into this, as well as a thing about relativity that's in my queue. I'm only responding to emergencies at present.

Incidentally, the stuff about Riemannian geometry in the first section of the Quora page (the Hetu Chakra article was the third) is complete rubbish. We shouldn't be pushing Quora as a source of reliable information.

A few things:

  • The handling of the subject matter leaves much to be desired. The stuff about Riemann vs. Lebesgue integration can be greatly improved while still making it accessible to lay people. We can do better than just say that the Lebesgue integral is used in "some mathematical fields". We can say why. It has to do with measure theory and abstract set theory and uncountable sets. The audience will then see why it isn't relevant in day-to-day work. Incidentally, this is the same reason that the Banach-Tarski paradox does not apply in the real world. It's very abstract set theory.
  • We can also explain Riemannian geometry better. And it isn't just elliptical. And the article doesn't say what elliptical geometry means in any case. Riemannian geometry is way more general than elliptical or hyperbolic. We can't get the readers to actually understand it, but we can put it into context more clearly. The "common notions" of elliptical or hyperbolic geometry existed since the 1820's and before. What Riemann did was generalize it to arbitrary manifolds of arbitrary dimension. And, by the way, this paved the way for general relativity.
  • We really need to separate one's religious devoutness from one's mathematical prowess. While his greatness may have been divinely inspired, it's a slippery slope to say that explicitly. And it would lead to ranking Riemann, Gauss, Euler, and others based on their religious devoutness. We would have to put stuff into the Gauss article about whether he was more or less devout than Riemann, and consequently whether he was a greater mathematician. Many people have written about the prowess of these and other mathematicians; we should let those writings speak for themselves.
  • We should take out the "Wikipedia popularity poll" stuff. That struck me as a case of you grasping at straws to counter AugustO. We don't need that. We can write an article that is well-sourced.

SamHB (talk) 14:51, 28 July 2018 (EDT)

Sam, you make some good suggestions above, but there is nothing wrong with both citations to support Riemann's greatness. That is the kind of information that is useful to visitors, and is what encyclopedias should provide. I also disagree with your separation-of-accomplishment-from-religion approach. Riemann was inspired by his Christianity, and that should not be denied or downplayed.--Andy Schlafly (talk) 18:32, 28 July 2018 (EDT)

Right. There are a number of people that have commented on the greatness of various scientists, mathematicians, and so on. Opinions of their relative greatness are all over the map, not surprisingly, and we've seen it right here. We can mention several of them, including the quote from Hetu Chakra. My internet research indicates that he (Hetu Chakra) isn't really an intellectual giant, but we can include quotes from people who aren't intellectual giants. Cold facts are good, but a little drama doesn't hurt either. His "in a fairer world ..." comment is nice, and it should be included, along with various other quotes, some of which do not place Riemann as supreme. And it should be right out there in the article, not in a footnote.

As far as inspiration from Christianity, the best source is the person's own statements. No one knows where a person derives his inspirations from better than the person himself. Do you have a first-person source for this? Could you find one? It would be really good if we had first-person sources for him, as well as Gauss, Euler, Einstein, etc. The article hints at something in his collected works. Can we track that down? There is a subtle but important difference between his Christianity being important to him and it being the source of his inspiration.

From reading the godandmath article, I see that I'd completely forgotten about the Cauchy-Riemann equations. So simple, and yet so important. Very easy to include in this article.

SamHB (talk) 21:01, 28 July 2018 (EDT)

I took a computer class and I was walking across the college in frustration over an assignment the professor gave us. I silently cried out to God in frustration and immediately the solution came to me. I solved the task in the least amount of computer code necessary too solve the task and only one person tied me in the class.
Another time, I had an illness that the medical establishment has no cures for. I was in front of a library and a flash of inspiration came to me. It appeared to be divine inspiration that came out of the blue and beyond my knowledge base. I asked the librarian to do a search in a medical database using the search parameter I gave her. Within 60 days using the information I found, I not only cured my condition, but I became about 70% healthier than the general population when it comes to this aspect of health.
Don't discount the power of divine inspiration.Conservative (talk) 21:48, 28 July 2018 (EDT)
I don't. The point is that those words came out of your own mouth. Well, keyboard. If I were writing an article about you, I would definitely include this, as your personal statement of where you get inspiration. SamHB (talk) 23:19, 28 July 2018 (EDT)
A reference to the Cauchy-Riemann equations would be a great addition, Sam. So would personal statements by Riemann about his faith.--Andy Schlafly (talk) 22:32, 28 July 2018 (EDT)
I created the Cauchy-Riemann equations page, but I think someone with more expertise should check what I've written and correct/improve.
@Conservative What was the computing problem and what language did you use to solve it? I do a little (and I do mean small) bit of programming in my spare time so am just curious. - FredericBernard (talk) 09:59, 29 July 2018 (EDT)
Wow!!!! You new page is a tremendous start. Will try to improve, and will also incorporate into the Bernhard Riemann entry here.--Andy Schlafly (talk) 10:54, 29 July 2018 (EDT)

Thunder stolen, fair and square!

Congratulations FredericBernard! That's an excellent article. I'd been planning to write such a thing, but you beat me to it.

While contemplating this, I remembered something I had written explaining just what it is that makes complex analytic functions so special—it has to do with the "there exists a delta" clause allowing the delta in any direction in the complex plane. That is what makes all the difference! I looked around, and that explanation was not in the usual places where it should be, such as the Complex_analysis page. Then I found it, in a sandbox page. I had moved stuff to the sandbox after the late TK deleted a lot of my stuff (and blocked me) in a fit of rage over the (ironically!) then-existing Riemann Integral page. So I'm going to start moving some of this material back (and deleting much of it as redundant), and making the world safe for the Cauchy-Riemann equations. I've already requested the restoration of complex analytic function. A lot of work to do.

SamHB (talk) 21:23, 29 July 2018 (EDT)

Manifest of Ignorance

The whole article is riddled with errors, and the misconceptions can also be seen at the article on Infinity:

  • Riemann did not write four great paper, even less so "Four Great Papers": he had published eleven papers during his lifetime, and four papers published posthumously - under those his two Habilitationsschriften. One of those is Uber die Hypothesen, welche der Geometrie zu Grunde liegen, which is somehow enumerated as one of his "Four Great Papers". Though this Habilitatinsschrift was not published during his lifetime, he read it out in the presence of Gauss and the faculty at Göttingen, and it had an instant impact.
  • "Riemann advanced the understanding of trigonometric series, which led to profound work to determine when a function is integrable. " Riemann's second Habilitationsschrift was titled Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe: in this lecture, he introduced his version of the integral, and determined when a function is integrable.
  • Riemann did not use infinitesimals. The sentence "Bernhard Riemann used the concept of infinitesimal to develop a rigorous definition of the integral, now known as the Riemann Integral" is patently absurd and will result in ridicule by anybody with a little knowledge of the history of calculus.

The editor who wrote these sentences obviously has not read any of Riemann's original work, and bases his assumptions on cursory internet searches. It is astounding that nevertheless he feels qualifiede to rule that "Riemann is the greatest modern mathematician": mathematicians all over the world think highly of Riemann, he is one of the very best. But I have not found anyone remotely competent who shares this insight.

--AugustO (talk) 04:59, 2 August 2018 (EDT)

"Profound work to determine when a function is integrable"? Trigonometric series?

That comment doesn't make a lot of sense. Do you have a reference for it?

Aside from issues of going to infinity, a function over a compact set is (Riemann) integrable if its set of discontinuities has Lebesgue measure zero. That's pretty well known and easy to see. I don't know whether Riemann was the first person to see that. (It might have predated the formal definition of Lebesgue measure, but, for measure zero, simpler formulations, available at the time, will do.) But that characterization of integrability has nothing to do with trigonometric functions. No trigonometric functions have weird sets of discontinuities. They are all well-behaved, and understood at the (junior) high school level.

Now a function can also fail to be integrable due to being too big, that is, the integral diverges. But that is fairly easy to characterize. And, if series must be used, that is fairly straightforward. For example, the tangent ("tan") function goes to infinity at pi/2. That doesn't necessarily mean the integral diverges—lots of interesting problems involve tricky cases of functions diverging but their integrals not diverging. But the trig functions are pretty straightforward. The integral of the tangent function is the secant squared, which goes to infinity at pi/2, so the integral on any open set containing pi/2 diverges.

I don't think Riemann had anything to do with that. His work was at a much more sophisticated level. Do you have any references for your claim?

SamHB (talk) 22:47, 5 August 2018 (EDT)