44

Cited by "Imre Lakatos and the Guises of Reason" John David Kadvany, 2001:

It is remarkable that the nineteenth century was a time of error for mathematics: not trivial oversights or amateur conclusions, but fundamental mistakes in the understanding of mathematical concepts and the formulation of mathematical proofs. These mistakes were not restricted to unknown mathematicians but occurred in the works of great mathematicians such as Joseph Fourier, Augustin Cauchy, and Denis Poisson.

I'm highly interested what are the mathematical mistakes that Fourier did, especially if there are any at the fields that relate to series and transform named after him?

J. M. ain't a mathematician
  • 71,951
  • 6
  • 191
  • 335
Poliakoff
  • 557
  • 4
  • 9
  • Fourier was wrong, [Riemann was wrong](http://www.claymath.org/sites/default/files/ezeta.pdf) and arguably [we're wrong too](http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf). – Count Iblis Jun 24 '16 at 18:17
  • 8
    Makes me think of the following quotation: "Only Dirichlet, Not I, not Cauchy, not Gauss, knows what a perfectly rigourous proof is, but we learn it only from him. When Gauss says he has proved something, I think it is very likely; when Cauchy says it, it is a fifty-fifty bet; when Dirichlet says it, it is certain; I prefer not to go into these delicate matters." – Andres Mejia Jun 25 '16 at 03:13
  • 2
    @AndresMejia who is that quoting? – snulty Jun 25 '16 at 19:14
  • 2
    @snulty Jacobi [see here](https://books.google.com.ua/books?id=uTFNAAAAQBAJ&pg=PA63&lpg=PA63) – Poliakoff Jun 25 '16 at 19:21
  • Jacobi! The translation is from a popular math overflow question on famous mathematical quotations. – Andres Mejia Jun 25 '16 at 19:23

4 Answers4

62

Anyone back then working on anything that had anything to do with calculus or related topics could hardly avoid making mistakes, since there simply was no logically coherent formulation of the basic definitions at that time. Trying to prove something about continuous functions without a definition of continuity is going to lead to problems.

Fourier in particular is famous for stating that any periodic function is equal to the sum of its Fourier series. This is nonsense (see the comment below). But it's one of the all-time great errors. Trying to make sense of this, to see what could actually be proved in this direction, was one motivation for the development of modern rigorous analysis. In fact sorting this out was part of the motivation for at least three major developments that spring to mind:

  • People like Cauchy, Weierstrass et al invent epsilons and deltas. Now we can actually state and prove things about calculus rigorously.

  • But the theory of Fourier series, although it now made sense logically, still didn't work as well as we'd like; Lebesgue and others invent the Lebesgue integral and the theory of Fourier series gets a big boost.

  • Cantor was actually led to set theory, in particular transfinite numbers, in the course of investigations into Fourier series! (When you're studying sets of uniqueness for trig series the notion of the "derived set" $E'$ of $E$ comes up; this is the set of limit points of $E$. Then one can consider $E''$, etc; this leads naturally to a study of $E^\alpha$ for infinite ordinals $\alpha$.)

(The first two items above are hugely well known. For more on the third, regarding Cantor, set theory and Fourier series, you might look here or here. Will R suggests you look here; I haven't seen that, internet too slow for YouTube, but a lecture by Walter Rudin on the topic is certain to be great.)


Comment I had no idea that the assertion that there exists a (continuous) function with a divergent Fourier series would be controversial. Writing down an explicit example is not easy; any continuous function that Fourier ever encountered does have a convergent Fourier series.

But proving the existence is very simple, from the right point of view. Say $s_n(f)$ is the $n$-th partial sum of the Fourier series for $f$ and $D_n$ is the Dirichlet kernel, so that $$s_n(f)(0)=\frac1{2\pi}\int_0^{2\pi}f(t)D_n(t)\,dt.$$The norm of $s_n(f)(0)$ as a linear functional on $C(\Bbb T)$ is the same as the norm of $D_n$ regarded as a complex measure, which is in turn equal to $\|D_n\|_1$. It's easy to see that $\|D_n\|_1\ge c\log n$. So the Uniform Boundedness Principle, aka the Banach-Steinhaus Theorem, shows that there exists $f\in C(\Bbb T)$ such that $s_n(f)$ is unbounded.

David C. Ullrich
  • 84,497
  • 5
  • 61
  • 132
  • 1
    Also useful: [*Trigonometric series and theories of integration*](http://www.jstor.org/stable/2690547) by Gluchoff (1994) **AND** [*Connections in mathematical analysis: the case of Fourier series*](https://www.math.ucdavis.edu/~saito/courses/121/gonzalez.pdf) by González-Velasco (1992) **AND** [*Vibrating strings, Fourier series, and some aspects of mathematical analysis*](http://www.jstor.org/stable/24525245) by Larsen (1974) **AND** [*The influence of Fourier's series upon the development of mathematics*](https://archive.org/details/jstor-1640940) by Van Vleck (1914). – Dave L. Renfro Jun 24 '16 at 16:29
  • 1
    Bressoud's *A Radical Approach to Real Analysis* was particularly illuminating for me when this stuff was discussed there. – J. M. ain't a mathematician Jun 24 '16 at 16:36
  • 1
    Cauchy introduced epsilon-delta proofs early in the 19th century, not at the end (Cauchy didn't state the definition in that fashion, but his proofs followed this pattern). Bolzano first stated the definition itself in terms of epsilons and deltas in 1817, though Weierstrass later cleaned it up. – Paul Sinclair Jun 24 '16 at 17:52
  • @PaulSinclair Oh. Thanks. – David C. Ullrich Jun 24 '16 at 17:57
  • Also concerning the third (and perhaps more accessible to a slightly more general audience) is [this](https://www.youtube.com/watch?v=hBcWRZMP6xs&list=WL&index=14) lecture by Walter Rudin. If you already know the stuff then it's nothing new, but it's an interesting story for those who don't. – Will R Jun 24 '16 at 19:05
  • 5
    _"[..] any periodic function is equal to the sum of its Fourier series. This is nonsense."_ - Wait, it is!? I remember learning that to be true in Calc 3. If it's false, what use is a Fourier series!? – BlueRaja - Danny Pflughoeft Jun 24 '16 at 19:06
  • 2
    @BlueRaja-DannyPflughoeft Same here. What exactly is incorrect about that statement? Is the problem simply that it's not as precise as stating "for any periodic function, there exists a Fourier series that converges to the original function"? Is it that Fourier's definition of "periodic function" somehow permits exceptions to the rule, such as functions that are undefined over some continuous space? (I'm not sure what Fourier's definition was; I do know that the definition was rather hotly contested at the time, and that he was rather central to the controversy.) – Kyle Strand Jun 24 '16 at 20:45
  • Neither of those possibilities strikes me as "nonsense," though. – Kyle Strand Jun 24 '16 at 20:45
  • 10
    @BlueRaja-DannyPflughoeft I assumed you were joking. It is simply not true that every periodic function is equal to its Fourier series. Not even every continuous function. You need hypotheses somewhat stronger than continuity. You say this contradicts what you learned in Calc 3. I hope it doesn't actually say in the book that any periodic function equals its Fourier series; I hope the instructor never said so. _But_ in a calculus course a person could easily get that idea, because the emphasis there is not on this sort of detail, the emphasis would be on how to use the things. Which is ok... – David C. Ullrich Jun 24 '16 at 20:56
  • @KyleStrand: well, literally it isn't nonsense. It makes sense and is false. But colloquially of course the word "nonsense" is used to mean "something easily seen to be false". I suppose "easily" in this example has to be taken as, "easily, by someone with an understanding of real analysis". If real analysis had existed at the time then no doubt Fourier would have understood it and wouldn't have dreamt of making such a statement, because he'd have know more about functions that don't behave "nicely". – Steve Jessop Jun 25 '16 at 13:50
  • 3
    @DavidC.Ullrich Could you please provide a few examples of function that is not equal to the sum of its Fourier series? – Poliakoff Jun 25 '16 at 17:27
  • @Polyakoff I added some explanation to the bottom of the answer. – David C. Ullrich Jun 25 '16 at 19:08
  • What's the dirichlet kernel can I ask? – snulty Jun 25 '16 at 19:21
  • 1
    @snulty $D_n(t)=\sum_{k=-n}^ne^{ikt}=\sin((n+\frac12)t)/\sin(\frac12 t)$. – David C. Ullrich Jun 25 '16 at 19:53
  • 3
    @LOLOLOLOLOLOLOLOLOLOLOLOLLOLOL Well of course. I acknowledged most of what you said above. Nobody's said Fourier was dumb. Quibbling over the word "mistake" is silly - the OP had read about Fourier making mistakes and asked for an example - what I said is an example of what the quote was referring to, regardless of whether you want to call it a "mistake". But I do believe that he _was_ willing to accept a trig series as the definition of a "function" - if so then yes, he _would_ have recognized the example, given an appropriate presentation. Your idea that I can't write one down is funny. – David C. Ullrich Jun 27 '16 at 15:01
  • 2
    David, just because "tro" is missing from the beginning of their username, it doesn't mean s/he's not a troll. – Vincenzo Oliva Jun 27 '16 at 19:15
  • @VincenzoOliva Heh. Thanks. Of course I recognized his or her trollitude. He or she is a pathetically inept troll, making a few reasonable points. I figured I could totally destroy his or her day by pointing out I agreed with the _spirit_ of much of what (s)he said. – David C. Ullrich Jun 27 '16 at 22:39
  • I do think that's an honourable attitude (the upvote to the comment, which adds to that to the answer, is mine). :) Just wanted to emphasize that your answer is excellent and you should not worry about nuisances (and in a way, I'm sure you don't). – Vincenzo Oliva Jun 27 '16 at 23:36
  • So is the consensus that Fourier wasn't wrong, he was just using a different definition of "function" than we do? – DanielV Jun 28 '16 at 00:25
  • @DanielV : Yes, that's the correct History. Read the quote I posted 2 posts down. – Disintegrating By Parts Jul 18 '16 at 02:32
20

Presumably the reference is to:

referring to Fourier's example of convergent series of continuous functions which tends to a Cauchy discontinuous function, into Fourier's Mémoire sur la Propagation de la Chaleur (1812).

But we are not speaking of "mistakes" as calculation errors or things like that; what Lakatos is discussing are "exceptions" to some general theorem where the proof neglects some condition necessary for the general validity of the proof.

Mauro ALLEGRANZA
  • 89,210
  • 6
  • 59
  • 133
7

Fourier was right about his conjecture concerning expanding functions in a Fourier series, when you consider what the "functions" were at the time. It's fun to pounce on Fourier, and I noticed a lot of people have drunk that Kool-Aid. However, the facts don't support what people typically claim. Set Theory didn't exist, and general functions were not conceived. The general functions at the time of Fourier were piecewise arcs, and Fourier did demonstrate the convergence to the mean of the left and right limits for such functions. It is false that Dirichlet gave the first proof of this fact. In fact, Dirichlet's proof was almost identical to that given by Fourier, and Fourier gave the "Dirichlet kernel." It is very possible that Dirichlet took his proof from Fourier's manuscript that had been denied publication. It is true that Fourier also gave several wrong demonstrations, but the Dirichlet kernel proof should really be called the Fourier kernel.

Imagine doing what Fourier did in a time when the following had not yet been defined: (1) The Riemann integral (2) a Real Number (3) Set Theory and general functions (4) Completion of a space and Convergence of a Cauchy Sequence (5) Functional Analysis (6) Inner Product Space (7) The Cauchy-Schwarz inequality. It's important to keep Historical perspective, and to keep in mind that a large part of Analysis came out of trying to resolve Fourier's claims.

Quoting from the well-regarded 1926 Introduction to the Theory of Fourier's Series and Integrals by H. S. Carslaw,

Debunking False Claims

Disintegrating By Parts
  • 79,842
  • 5
  • 49
  • 126
  • Thank you: I was not aware that the "Dirichlet kernel" had appeared in Fourier's unpublished/denied-publication paper(s). – paul garrett Jun 27 '16 at 19:22
  • @paulgarrett : Fourier's original treatise on heat conduction was blocked from publication until he had gained enough prominence to force its publication in original form (except for a few minor corrections.) His treatise was submitted in 1811 and published in 1822, which was still before Dirichlet's 1829 paper. That's what makes it even more interesting. I'm not sure why the false narrative remains so prevalent, especially after being debunked in Carslaw's text, which was well-kown and well-regarded in his time ~1925. And you'd think they would have listened to Darboux. – Disintegrating By Parts Jun 27 '16 at 20:30
  • Good answer. The only thing I'd dispute is the idea that anyone here has said anything that contradicts anything you said. The OP had read about mistakes Fourier made and asked for examples; answering the question is not pouncing on Fourier. – David C. Ullrich Jun 27 '16 at 23:19
  • @DavidC.Ullrich : I didn't say anything inaccurate. The evidence of how widespread the error is how nobody credits Fourier with the Dirichlet kernel or the proof of convergence using this kernel. – Disintegrating By Parts Jun 27 '16 at 23:30
  • I read "a lot of people" as "a lot of people here" (literally; on a second reading I was surprised to see the word "here" wasn't there). If that's not what you meant sorry. Curiously, the first sentence of that quote from Carslaw does seem inaccurate; it seems to be referring to a false theorem! Presumably he clarifies this on the next page. (What does it say on the next page? My recollection is that Dirichlet proved all this assuming bounded variation; did Fourier also mention bounded variation?) – David C. Ullrich Jun 27 '16 at 23:46
  • 1
    @DavidC.Ullrich : Piecewise smooth is what Dirichlet and Fourier would have assumed. The bounded variation development was due to C. Jordan, who introduced the concept in his 1881 paper on the subject of Fourier series. Riemann introduced his integral to study the Fourier Series, and Lebesgue also stated that as his purpose for proposing a new integral as well. It look a long time to get to that point. BTW, even one in this thread would not be a lot of people. – Disintegrating By Parts Jun 27 '16 at 23:55
5

In addition to other good answers, it might be worth noting that if we do not insist that functions be "pointwise", and do not insist that convergence of partial sums of Fourier series be pointwise, then there's no difficulty in making a very broad interpretation (not Fourier's original) that "everything is represented by its [Fourier] series" completely legitimate. A starting point is the (originally disturbing-to-me) fact that while Fourier series of continuous functions do converge in $L^2$, there are problems with pointwise convergence. But no problems with $L^2$ convergence even for $L^2$ functions. The simplicity of the Hilbert-space aspects is exploited in ($L^2$-) Sobolev spaces (initiated c. 1906-7 by Beppo Levi and G. Frobenius in the guise of "energy norms", and then systematically by Sobolev in the 1930s)... which leads to the assertion that every distribution on the circle has a Fourier series converging to it in a suitable Sobolev space. Termwise differentiation is always justified (if interpreted distributionally). And so on.

E.g., the objection that $\sum_{n\in\mathbb Z} 1\cdot e^{2\pi inz}$ does not converge pointwise is essentially irrelevant to the provable fact that it does converge to the periodic Dirac $\delta$ in the Sobolev space $H^{-1/2-\epsilon}$ for every $\epsilon>0$, where $H^s$ is the completion of smooth functions on the circle with respect to the $H^s$ norm defined at first on test functions by $|f|^2_s=\sum_{n\in\mathbb Z} |\widehat{f}(n)|^2\cdot (1+n^2)^s$.

My interpretation of such possibilities is that, in many applications (both to physical sciences and to more esoteric mathematical situations) the reason Fourier series work so well is that, despite our inherited penchant for worrying about pointwise behavior, it's not pointwise behavior that matters very much, but, rather, various averaged versions.

paul garrett
  • 46,394
  • 4
  • 79
  • 149
  • 1
    Indeed. I'd add that even if we're concerned with pointwise behavior, the failure of pointwise convergence really doesn't matter, because whatever we wish we could do with pointwise convergence we can do just as well with pointwise _summability_. Hence it's always seemed to me that the main reason for worrying about pointwise convergence is to give people something to do. Hoping Lennart Carleson is too old to figure out how that internet thing works... – David C. Ullrich Jun 27 '16 at 23:29
  • @DavidC.Ullrich, heh... but, as you know, ... and as I explain to my PhD students when they're nonplussed by my slightly snarky remarks about various "famous problems"... "fame" is not always because of the utility of the putative result, but often as much simply that an easy-to-ask question has been unanswered for a long time, so will score many status points if answered. As we know, "status" and "advancing collective human understanding" are not identical. :) And then there're the never-quite-definitively-answerable, if pointless questions that generate endless publishable papers... :) – paul garrett Jun 30 '16 at 13:18