Timeline for Has incorrect notation ever led to a mistaken proof?
Current License: CC BY-SA 4.0
44 events
| when toggle format | what | by | license | comment | |
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| Nov 18, 2022 at 1:00 | answer | added | Pace Nielsen | timeline score: 2 | |
| Nov 16, 2022 at 0:26 | comment | added | Joel David Hamkins | OK, I'll try to post this within the next few days. | |
| Nov 15, 2022 at 23:53 | comment | added | Mike Shulman | @JoelDavidHamkins In that case, yes, I would like to see an answer explaining it! | |
| Nov 15, 2022 at 2:33 | comment | added | Joel David Hamkins | Yes, certainly, although it was primarily a philosophical project in mathematical foundations. His system was inconsistent, but consistent fragments have been saved. The neologicists celebrate "Frege's theorem." | |
| Nov 15, 2022 at 1:57 | comment | added | Mike Shulman | @JoelDavidHamkins Would you say that the underlying mathematics that Frege was doing was precise and correct from a modern perspective? | |
| Nov 14, 2022 at 16:40 | answer | added | Pace Nielsen | timeline score: 8 | |
| Nov 14, 2022 at 15:40 | comment | added | Joel David Hamkins | Frege's notation is arguably an instance. He doesn't state general comprehension explicitly, but it is built into his notation in a way that causes inconsistency. I can post an answer explaining this if you like. | |
| Dec 20, 2021 at 21:43 | answer | added | user44143 | timeline score: 12 | |
| Oct 5, 2019 at 18:59 | comment | added | Toby Bartels | @LSpice : This is late, but since I just replied to a new comment of yours farther down: The simplest argument that $\mathrm d^2y/\mathrm dx^2$ can't really be a ratio is that $\mathrm d^2y/\mathrm dy^2=0$, so $\mathrm d^2y=0$, so $\mathrm d^2y/\mathrm dx^2=0$. This is all about the numerator, not the denominator; that's why, as Mike said, there's no way to consistently get the second derivative by dividing the second differential, no matter what you propose to divide it by. | |
| Oct 30, 2018 at 23:10 | comment | added | Mike Shulman | @TimothyChow In the abstract, I might agree with you that these complicated systems should be called something more substantial than "notation". But the point is that there are people who call them just "notation", and I'm looking for data to support or refute the argument that they should be taken more seriously than being considered "just notation". | |
| Oct 30, 2018 at 22:05 | comment | added | Mike Shulman | @TimothyChow I suppose we could argue about terminology until doomsday. I would only point out that most people who make use of string diagrams use them on paper without involving any computers. | |
| Oct 30, 2018 at 15:57 | comment | added | Timothy Chow | @MikeShulman : I would be inclined to classify those examples as "computer programming" even if they are not fully implemented as such. If this is what you're interested in, then I think you should be asking not about "notation" but about "symbolic calculus." | |
| Oct 30, 2018 at 3:18 | comment | added | Mike Shulman | @TimothyChow Your faith in "hand-crafted notation" is greater than mine. Especially when the notation is highly complicated and the proof of its correctness is highly nontrivial, as for string diagrams and type theory. | |
| Oct 30, 2018 at 2:21 | comment | added | Timothy Chow | @MikeShulman : Your clarification of what you're looking for comes perilously close to a proof that no example can exist. If some notation has been systematically developed but does not accurately reflect the underlying mathematics, then surely this reflects incomplete understanding of the mathematics on the part of the developers of the notation, and you would categorize it as a non-example ("of fluxion type"). Unless maybe the math is perfectly well understood but a "bug" slips into the notation? This could happen in computer programming but it's hard to imagine with hand-crafted notation. | |
| Aug 11, 2018 at 23:01 | comment | added | Alexander Woo | @darijgrinberg: Related issue: plethystic notation | |
| Aug 11, 2018 at 21:29 | comment | added | Joel David Hamkins | I bet there must be some good examples involving the $o(f(n))$ and $O(g(n))$ notation. | |
| Aug 10, 2018 at 20:01 | comment | added | LSpice | @MikeShulman, I knew (in a distant way) about iterated tangent bundles, but didn't know that they provided a framework in which $\mathrm dx^2$ was an honest square. Thanks! | |
| Aug 10, 2018 at 18:08 | comment | added | Mike Shulman | @LSpice For instance, if you take the differential of $dy = f'(x) dx$ using the product rule naively, you get $d(dy) = d^2y = (f''(x) dx)dx + f'(x) d(dx) = f''(x) (dx)^2 + f'(x) d^2x$. There is an $f''(x)$ in there, and $(dx)^2$ really is a square, but there's an extra term so that you can't just divide by it to get the $f''(x)$ out. Moreover, the extra term is exactly what makes the chain rule argument come out right instead of wrong. (This sort of "second differential" can be made precise using, among other approaches, iterated tangent bundles.) | |
| Aug 10, 2018 at 18:06 | comment | added | Mike Shulman | @LSpice this is very tangential to the question, but there is a systematic meaning of differentials according to which it is valid to treat dx as a square, whereas as far as I can tell there is nothing (whether or not it is a square) that you can divide a "second differential" $d^2y$ by to get the second derivative. | |
| Aug 10, 2018 at 17:56 | history | edited | Mike Shulman | CC BY-SA 4.0 |
clarified further what I'm looking for
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| Aug 10, 2018 at 17:24 | comment | added | Robert Furber | @Number Yes they would, at least to me. I was introduced to umbral calculus by Rota's version of it, so I've not seen the classical literature. | |
| Aug 10, 2018 at 12:06 | comment | added | LSpice | Probably one really does just have to give up on the idea of regarding $\frac{\mathrm d^2y}{\mathrm dx^2}$ as a fraction, but I think that what your argument really shows is that one can't regard $\mathrm dx^2$ as a square. (The partial-derivative argument given by @StevenLandsburg is the one that convinces me that any attempt to treat derivatives as fractions is going to go wrong.) | |
| Aug 10, 2018 at 8:32 | answer | added | Michael Bächtold | timeline score: 43 | |
| Aug 10, 2018 at 2:15 | comment | added | Nate Eldredge | I'm a little shy about going into details, but one that has bitten me is that a Lie group with a (non-bi-invariant) Riemannian metric has two different "exponential maps", both denoted $\exp$... | |
| Aug 10, 2018 at 1:55 | comment | added | Bill Dubuque | Would errors in early umbral calculus be of interest? | |
| Aug 10, 2018 at 1:43 | answer | added | Timothy Chow | timeline score: 31 | |
| Aug 9, 2018 at 23:33 | comment | added | Gerry Myerson | As Gauss said, in regard to Wilson's Theorem, "In our opinion, truths of this kind should be drawn from notions rather than from notations." | |
| Aug 9, 2018 at 22:48 | comment | added | darij grinberg | @AndrejBauer and MikeShulman: This will need a question, not an answer... I hope to get to it soon (next week?). | |
| Aug 9, 2018 at 21:30 | comment | added | Mike Shulman | Yes, @darijgrinberg that would be useful, especially if it included enough explanation for a non-combinatorialist about what a raising operator is to understand the issue. | |
| Aug 9, 2018 at 21:05 | comment | added | Andrej Bauer | It would be useful if @darijgrinberg collected his string of comments into an answer, even if the answer does not exactly address the question. | |
| Aug 9, 2018 at 18:52 | comment | added | darij grinberg | ... in Macdonald polynomial theory. Macdonald himself leaves the task of making the concept rigorous to the reader in his book (like he does to so many other things). I'm not sure to what extent this is an instance of what this question was asking for: it could well be that the proofs aren't as much wrong as merely presented without some necessary context. | |
| Aug 9, 2018 at 18:49 | comment | added | darij grinberg | ... necessarily be any clearer than the original sources, seeing that this is a matter of confusion rather than a specific question. Adriano Garsia has his own interpretation of raising operators (A. M. Garsia, Raising operators and Young's rule), who suggests that the operators should act on tableaux rather than on symmetric functions (see the sentences after equation 3.3); I'm not sure to what extent his suggestions can be used as a replacement for the uses of raising operators ... | |
| Aug 9, 2018 at 18:49 | comment | added | darij grinberg | @WillSawin: It's tricky. For an example of a paper heavily using raising operators, see arxiv.org/abs/1008.3094v2 . It tries to build them on a rigorous foundation (§2.1--2.2), by letting them act on Laurent polynomials instead of them acting on symmetric functions; but it soon slips back into pretending that they act on symmetric functions themselves (e.g., the first computation in §2.4 relies on associativity of that "action"). I have long thought about asking here on MO if there is a good way of making sense of these operators; but I'm afraid that the answers will not ... | |
| Aug 9, 2018 at 18:33 | comment | added | Will Sawin | @darijgrinberg Can you explain more about this example for non-combinatorialists? | |
| Aug 9, 2018 at 18:26 | answer | added | Per Alexandersson | timeline score: 11 | |
| Aug 9, 2018 at 18:01 | answer | added | Joel David Hamkins | timeline score: 86 | |
| Aug 9, 2018 at 17:59 | comment | added | darij grinberg | Often it's a matter of controversy -- e.g., I don't believe any argument that uses symmetric-functions raising operators (and I know I'm not alone with that), but several authors use them without worrying. | |
| Aug 9, 2018 at 17:56 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Aug 9, 2018 at 17:53 | comment | added | darij grinberg | I think the only reason why we combinatorialists have (almost) never gotten wrong results out of this is that we tend to check our conjectures on the computer before proving them... As for incomplete proofs, however, combinatorics seems positively teeming with them. | |
| Aug 9, 2018 at 17:50 | comment | added | Mateusz Kwaśnicki | In his 2000 article Nick Laskin apparently got the notation $\nabla^\alpha f$ for the fractional derivative wrong and played with it as if it was a local operator. This error still persists among some physicists, for example, it is still there in Laskin's 2018 book; see here for further links. Not sure if this qualifies, so I leave this as a comment. | |
| Aug 9, 2018 at 17:48 | answer | added | Steven Landsburg | timeline score: 51 | |
| Aug 9, 2018 at 17:27 | comment | added | GH from MO | I don't know but we lost a Mars probe due to mixing up newtons with pounds: articles.latimes.com/1999/oct/01/news/mn-17288 | |
| Aug 9, 2018 at 17:26 | comment | added | Steven Landsburg | There must have been cases where someone defined $y$ as an implicit function of $x$ via an equation $f(x,y)=0$ and then wrote something like $dy/dx=(df/dx)/(df/dy)$ (which "follows" from treating apparent fractions as fractions, but of course gets the sign wrong). Whether this has ever made it past a blackboard into a preprint is less certain. | |
| Aug 9, 2018 at 17:16 | history | asked | Mike Shulman | CC BY-SA 4.0 |