text:
Most interesting mathematics mistake?
Some mistakes in mathematics made by extremely smart and famous people can eventually lead to interesting developments and theorems, e.g. Poincaré's 3d sphere characterization or the search to prove that Euclid's parallel axiom is really necessary unnecessary.
But I also think there are less famous mistakes worth hearing about. So, here's a question:
What's the most interesting mathematics mistake that you know of?
EDIT: There is a similar question which has been closed as a duplicate to this one, but which also garnered some new answers. It can be found here:
Failures that lead eventually to new mathematics
soft-question big-list gm.general-mathematics
Ilya Nikokoshev
$\begingroup$ Closed: big-list questions don't need to keep cycling back to the front page, after some point. $\endgroup$ – Scott Morrison♦ Mar 7 '10 at 6:41
$\begingroup$ doesn't "cycling back to the front page" could also mean that it is still of interest? e.g. this one has been just been edited and therefore got to the front page again. Therefore it gets closed??? I don't get the logic behind that... $\endgroup$ – vonjd Mar 12 '10 at 18:28
$\begingroup$ Well, cycling well-viewed topics back to the front comes at the cost of pushing newer questions out of immediate visibility faster, so I understand the motivation. On the other hand, as the site grows, we get new perspectives on old questions which, and as vonjd points out, are apparently still of interest. We shouldn't close things just because the site old-timers are tired of seeing them. This discussion is probably on meta somewhere already.... $\endgroup$ – Cam McLeman Mar 12 '10 at 18:40
$\begingroup$ I agree with Cam - and in this case additionally: the big-list-tag means it is a big list and it can only become a big-list because many people make it a big list - so to close big-lists because they became big-lists is kind of absurd. Perhaps the underlying mechanism of bringing things to the front page should be changed in the software then. Just closing it is no solution $\endgroup$ – vonjd Mar 12 '10 at 18:46
$\begingroup$ A discussion thread was started on meta partly inspired by this: tea.mathoverflow.net/discussion/284/…. $\endgroup$ – Jonas Meyer Mar 13 '10 at 0:19
C.N. Little listing the Perko pair as different knots in 1885 (10161 and 10162). The mistake was found almost a century later, in 1974, by Ken Perko, a NY lawyer (!)
For almost a century, when everyone thought they were different knots, people tried their best to find knot invariants to distinguish them, but of course they failed. But the effort was a major motivation to research covering linkage etc., and was surely tremendously fruitful for knot theory.
Update (2013):
This morning I received a letter from Ken Perko himself, revealing the true history of the Perko pair, which is so much more interesting! Perko writes:
The duplicate knot in tables compiled by Tait-Little [3], Conway [1], and Rolfsen-Bailey-Roth [4], is not just a bookkeeping error. It is a counterexample to an 1899 "Theorem" of C.N. Little (Yale PhD, 1885), accepted as true by P.G. Tait [3], and incorporated by Dehn and Heegaard in their important survey article on "Analysis situs" in the German Encyclopedia of Mathematics [2].
Little's `Theorem' was that any two reduced diagrams of the same knot possess the same writhe (number of overcrossings minus number of undercrossings). The Perko pair have different writhes, and so Little's "Theorem", if true, would prove them to be distinct!
Perko continues:
Yet still, after 40 years, learned scholars do not speak of Little's false theorem, describing instead its decapitated remnants as a Tait Conjecture- and indeed, one subsequently proved correct by Kauffman, Murasugi, and Thistlethwaite.
I had no idea! Perko concludes (boldface is my own):
I think they are missing a valuable point. History instructs by reminding the reader not merely of past triumphs, but of terrible mistakes as well.
And the final nail in the coffin is that the image above isn't of the Perko pair!!! It's the `Weisstein pair' $10_{161}$ and mirror $10_{163}$, described by Perko as "those magenta colored, almost matching non-twins that add beauty and confusion to the Perko Pair page of Wolfram Web's Math World website. In a way, it's an honor to have my name attached to such a well-crafted likeness of a couple of Bhuddist prayer wheels, but it certainly must be treated with the caution that its color suggests by anyone seriously interested in mathematics."
The real Perko pair is this:
You can read more about this fascinating story at Richard Elwes's blog.
Well, I'll be jiggered! The most interesting mathematics mistake that I know turns out to be more interesting than I had ever imagined!
1. J.H. Conway, An enumeration of knots and links, and some of their algebraic properties, Proc. Conf. Oxford, 1967, p. 329-358 (Pergamon Press, 1970).
2. M. Dehn and P. Heegaard, Enzyk. der Math. Wiss. III AB 3 (1907), p. 212: "Die algebraische Zahl der Ueberkreuzungen ist fuer die reduzierte Form jedes Knotens bestimmt."
3. C.N. Little, Non-alternating +/- knots, Trans. Roy. Soc. Edinburgh 39 (1900), page 774 and plate III. This paper describes itself at p. 771 as "Communicated by Prof. Tait."
4. D. Rolfsen, Knots and links (Publish or Perish, 1976).
Daniel Moskovich
$\begingroup$ Little (with Tait and Kirkman) compiled his tables combinatorially. He drew all possible 4-valent graphs with some number of vertices (in this case 10), and resolved 4-valent vertices into crossings in all possible ways. He ended up with 210 knots. Then he worked BY HAND to eliminate doubles, by making physical models with string. He failed to bring these two knots to the same position, and concluded that they must be different. It took almost 100 years to find the ambient isotopy which shows that there are the same knot, but the quest to show they are different was fruitful. $\endgroup$ – Daniel Moskovich Dec 16 '09 at 7:22
$\begingroup$ +1 for the update that shows that there was a mistake wrapped in a mistake hidden in a mistake. :) $\endgroup$ – Michael Nov 7 '13 at 21:45
$\begingroup$ Ken Perko attempted to make another edit, by adding the following to the citation of Conway's paper: CONWAY WAS NOT MISLED BY THIS FALSE THEOREM OF C.N.LITTLE. HE FOUND THREE COUNTEREXAMPLES AMONG HIS 11-CROSSING NON-ALTERNATING KNOTS AND CORRECTLY WEEDED OUT THE DUPLICATE KNOT TYPES. Cf. Hoste-Thistlethwaite-Weeks, The first 1,701,936 knots, Math. Intelligencer 20 (1998) FOOTNOTE 8 and Jablan-Radovic-Saxdanovic, Adequacy of link fanilies, Publictiones de L'Institute Mathematique, Nouvelle Serie, Tome 88(102) (2010), 21-52. $\endgroup$ – S. Carnahan♦ Nov 23 '13 at 5:07
$\begingroup$ (Perko's comment continues...) FOR CONWAY, KNOT THEORY WAS A HIGH SCHOOL HOBBY AND HIS CHECKING AND EXTENSION OF THE NINETEENTH CENTURY TABLES "AN AFTERNOON'S WORK." He just didn't look very closely at the 10-crossing knots. $\endgroup$ – S. Carnahan♦ Nov 23 '13 at 5:08
$\begingroup$ Image for the real Perko pair is gone :( $\endgroup$ – მამუკა ჯიბლაძე Sep 29 '15 at 17:09
An error of Lebesgue. 1905 or so. Take a Borel set in the plane, project it onto a line, the result is a Borel set. Obvious: the projection of an open set is open, and the Borel sets in the plane are the least family containing the open sets, closed under countable unions and countable decreasing intersections.
But wrong. Projection doesn't commute with countable decreasing intersection.
Studying this error lead Suslin to begin the line of study now called "descriptive set theory", 1917 or so.
Gerald Edgar
$\begingroup$ Projection doesn't even commute with finite intersection. $\endgroup$ – domotorp Nov 2 '14 at 12:33
$\begingroup$ To generate Borel sets, it is enough to use countable decreasing intersections. Will correct. $\endgroup$ – Gerald Edgar Nov 2 '14 at 12:57
$\begingroup$ The story was recounted by Kazimierz Kuratowski in his ``A half century of Polish mathematics." Of course neither Suslin nor his advisor Lusin were Polish, but Waclaw Sierpinski, a Polish mathematician who was interned in Russia at that time, witnessed the conversation between Suslin and Lusin in which the student communicated the discovery to the professor. $\endgroup$ – Margaret Friedland Apr 10 '15 at 17:55
All of the (in retrospect) misguided attempts to prove Euclid's Parallel Postulate, which eventually lead Gauss to develop hyperbolic geometry.
Alex Basson
$\begingroup$ (and/or Lobachevsky, and/or Bolyai) This gets my vote as one of the most fruitful mistakes, and one of the longest perpetuated. $\endgroup$ – Aaron Mazel-Gee Oct 17 '09 at 18:46
Kempe's "proof" of the four-color theorem, which didn't prove the four-color theorem, but did:
Prove the five-color theorem
Somehow manage to go unnoticed for a dozen years
Lay the foundations for major tools in structural graph theory, and despite being fundamentally flawed, serve as the starting point for the eventual successful proof(s) of 4CT.
Harrison Brown
$\begingroup$ The news of that "proof" was even given in an Iranian public newspaper at the time when a few Iranian (could) read anything but religious texts! $\endgroup$ – Amir Asghari Dec 7 '17 at 22:40
A story I heard in grad school:
Once upon a time, a set theorist was writing a paper on inner models, and in it he wrote, "... and we will call such models nice." When he got his manuscript back from the typist (this was back in the pre-LaTeX days of technical typists), he saw that his handwriting had been misread, and the line came out as: "... and we will call such models mice." The name stuck, and to this day if you browse almost any recent volume of the Journal of Symbolic Logic, you will find set theory articles on "mice."
John Goodrick
$\begingroup$ I've heard a version of this story too, but I've also heard that Jensen denied that this was the origin of "mice". I never asked Jensen himself about it, so I don't know what to believe. $\endgroup$ – Andreas Blass Oct 9 '11 at 23:55
$\begingroup$ You know what will be a great paper title? "Of mice and men" $\endgroup$ – Aleks Vlasev Oct 10 '11 at 6:50
$\begingroup$ Note that this story is not true. $\endgroup$ – Danu Nov 8 '15 at 8:52
Maybe it's not true, but there's the story of the "Grothendieck prime":
One striking characteristic of Grothendieck's mode of thinking is that it seemed to rely so little on examples. This can be seen in the legend of the so-called "Grothendieck prime". In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. "You mean an actual number?" Grothendieck asked. The other person replies, yes, an actual prime number. Grothendieck suggested, "All right, take 57."
But Grothendieck must have known that 57 is not prime, right? Absolutely not, said David Mumford of Brown University. "He doesn't think concretely."
from here: http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf
Kevin H. Lin
$\begingroup$ But does this qualify as an interesting mistake? $\endgroup$ – Todd Trimble♦ Nov 7 '13 at 14:17
$\begingroup$ @Todd, yes, in the sense that it is a fun mistake. $\endgroup$ – Joël Nov 7 '13 at 14:49
$\begingroup$ @Joël I agree that it's an amusing mistake, but I was reading the question more in terms of mistakes that led to interesting developments (and I think the highest voted answers went with that reading). $\endgroup$ – Todd Trimble♦ Nov 7 '13 at 15:39
$\begingroup$ I was told this story by my lecturer in a graduate algebra course back in 2015, except the lecturer ended with the punchline "All right, take 59". There was a dead silence for about three seconds until we all realized what had happened and everyone started laughing. $\endgroup$ – Improve Mar 28 '16 at 2:47
$\begingroup$ Grothendieck was not the first eminent mathematician to give 57 as an example of a prime. Hermann Weyl (American Mathematical Monthly 1951, p.532) mentioned Goldbach's conjecture about "primes of the smallest possible difference 2, like 57 and 59." $\endgroup$ – John Stillwell Oct 29 '16 at 18:47
It was "proved" in 1961 that the first right derived functor, $\lim^1_{\leftarrow}$ of the inverse limit functor is zero on Mittag-Leffler systems.
However, recently a counter-example was found by Neeman and Deligne: http://www.springerlink.com/content/aeem2yx884nnufxn/
the L
$\begingroup$ wait... really? this is serious, i use that a lot... dammnit! $\endgroup$ – Sean Tilson Mar 31 '10 at 6:35
$\begingroup$ Apparently all you need is that the abelian category have a set of generators, so for example it is true in the category of abelian groups. Also, my advisor, and i believe some others, use it differently then it is stated on wikipedia and in the paper... i think $\endgroup$ – Sean Tilson Apr 6 '10 at 18:15
$\begingroup$ Cf. mathoverflow.net/a/35864/27465 and jlms.oxfordjournals.org/content/73/1/65. A sufficient extra structure in an abelian category for this to hold is: Grothendieck's axioms AB3, AB4* and having a set of generators. In particular, it is true in module categories (and even categories of "almost modules"). $\endgroup$ – Torsten Schoeneberg Jan 19 '14 at 0:28
An insignificant mistake, but amusing nonetheless: in Cayley's famous 1854 paper where he defines the concept of an abstract group, as an illustration he proves that there are three groups of order 6 (up to isomorphism). This is because he does not realize that the groups $Z_2\times Z_3$ and $Z_6$ are isomorphic.
This is found on page 51 of A. Cayley, Desiderata and suggestions: No. 1. The theory of groups, American J. Math. 1 (1878), 50-52. An interesting related paper is G. A. Miller, Contradictions in the literature of group theory, American Math. Monthly 29 (1922), 319-328.
Richard Stanley
$\begingroup$ Very nice one and very much in spirit of computer bugs! $\endgroup$ – Ilya Nikokoshev Dec 24 '09 at 18:51
$\begingroup$ Iǘe had a couple of students who are now going to be proud! :P $\endgroup$ – Mariano Suárez-Álvarez Dec 25 '09 at 23:09
$\begingroup$ Can you provide a reference? I just checked his paper "On the theory of groups, as depending on the symbolic equation θ^n = 1" (1854) which Wikipedia gives as the first definition of an abstract group. He says, "And we have thus two, and only two, essentially distinct forms of a group of six", and then gives their Cayley tables. The paper I was reading is pdfserve.informaworld.com/26005_751318052_910849049.pdf linked from informaworld.com/smpp/content~content=a910849049&db=all $\endgroup$ – aorq Feb 3 '10 at 1:05
$\begingroup$ It took me a while to track down the correct reference. It is page 51 of A. Cayley, Desiderta and suggestions: No. 1. The theory of groups, American J. Math. 1 (1878), 50-52. An interesting related paper is G. A. Miller, Contradictions in the literature of group theory, American Math. Monthly 29 (1922), 319-328. $\endgroup$ – Richard Stanley Mar 1 '10 at 15:51
From wikipedia (http://en.wikipedia.org/wiki/Uniform_convergence), about uniform convergence:
"Augustin Louis Cauchy in 1821 published a faulty proof of the false statement that the pointwise limit of a sequence of continuous functions is always continuous. Joseph Fourier and Niels Henrik Abel found counter examples in the context of Fourier series. Dirichlet then analyzed Cauchy's proof and found the mistake: the notion of pointwise convergence had to be replaced by uniform convergence."
$\begingroup$ I have always loved that way Abel wrote this (in a footnote): «it appears to me that this theorem suffers exceptions»... $\endgroup$ – Mariano Suárez-Álvarez Dec 15 '09 at 23:49
$\begingroup$ Some (e.g. A. Robinson) say that this is a mis-interpretation of the situation. When Cauchy says the sequence converges at all points this includes infinitesimals and such things not recognized as real numbers nowadays. Abel's counterexample $\sum (1/n) \sin(nx)$ in fact does not converge at certain points $x$ infinitely close to $0$. We can hardly fault Cauchy if he did not use the notion of real number from Dedekind and Cantor, since that would not come until 50 years later. $\endgroup$ – Gerald Edgar Dec 16 '09 at 16:40
Frege's proposed axioms in Die Grundgesetze der Arithmetik.
Frege was trying to derive the concept of "number" from more basic concepts, and he tried to axiomatize higher-order logic (essentially, a kind of set theory), but his intuitive-seeming axioms were logically inconsistent. Russell first found the inconsistency, which we now call Russell's Paradox.
In fairness to Frege, he was suspicious of his flawed axiom, before Russell wrote to him about his paradox. In the introduction he writes:
"If we find everything in order, then we have accurate knowledge of the grounds upon which each individual theorem is based. A dispute can arise, so far as I can see, only with regard to my Basic Law concerning courses-of-values (V), which logicians perhaps have not yet expressly enunciated, and yet is what people have in mind, for example, where they speak of the extensions of concepts."
Anish Gupta
$\begingroup$ +1, although I don't see this as a "mistake" in the sense of the other examples. He didn't do what he intended, but he wasn't wrong at any point. An inconsistent formal system is a perfectly fine mathematical setup, just one that people are mostly uninterested in. $\endgroup$ – Nikolaj-K Feb 23 '15 at 18:17
Poincare defined the fundamental group and the homology groups and proved that $H_1$ was $\pi _1$ abelianized. So the question came up whether there were other groups $\pi_n$ whose abelianization would give the $H_n$. Cech defined the higher $\pi_n$ as a proposed answer and submitted a paper on this. But Alexandroff and Hopf got the paper, proved that the higher $\pi_n$ were abelian and thus not the solution, and they persuaded Cech to withdraw the paper. Nevertheless a short note appeared and the $\pi_n$ started to be studied anyway...
Taken from http://www.intlpress.com/hha/v1/n1/a1/ ,page 17
Peter Arndt
I believe Kummer's failed attempt at a proof of Fermat's last theorem led to the discovery of ideals.
GMRA
$\begingroup$ I'm told that Kummer actually didn't care about Fermat's last theorem; it just happened that the techniques he developed were applicable. $\endgroup$ – Qiaochu Yuan Oct 17 '09 at 20:54
$\begingroup$ It was actually Lame who came up with that bad proof. $\endgroup$ – Ben Webster♦ Oct 18 '09 at 1:13
$\begingroup$ Oh, ok my mistake. $\endgroup$ – GMRA Oct 18 '09 at 14:27
$\begingroup$ Harold Edwards wrote a wonderful account of this history in his paper "The background of Kummer's proof of Fermat's last theorem for regular primes". It doesn't seem to be available online, but the mathsci net review is: ams.org/mathscinet-getitem?mr=57:12066a $\endgroup$ – Ben Linowitz Jan 6 '10 at 2:19
$\begingroup$ I don't know whether it is appropriate to say "discovery" of ideals. Maybe "recognition of the importance/relevance of ideals"? $\endgroup$ – Kevin H. Lin Apr 5 '10 at 6:14
Pontryagin made a famous mistake while computing the stable homotopy groups of spheres (specifically, π2) which led to the discovery of the Kervaire invariant. I won't spoil what the mistake was: watch this video of Mike Hopkins' talk (second video on the page), starting about 7 minutes in.
$\begingroup$ A mistake in a close field (I can't do another answer): before Milnor, everybody thought it obvious that two topological n-spheres could have different structures as differentiable manifolds. I'm pretty sure Milnor himself thought he made a mistake when some invariant turned out to be different for two topological spheres. For details, see the third volume of his collected papers. $\endgroup$ – Ilya Grigoriev Mar 13 '10 at 8:02
$\begingroup$ @IlyaGrigoriev: Do you mean "before Milnor, everybody thought it obvious that two topological n-spheres could NOT have different structures as differentiable manifolds"? $\endgroup$ – Jim Conant Sep 29 '15 at 13:30
$\begingroup$ @JimConant: Yes, of course. $\endgroup$ – Ilya Grigoriev Oct 1 '15 at 23:09
$\begingroup$ I am glad this mistake had some good consequence. i had only heard that it delayed Bott's proof of his periodicity theorem, since it seemed to contradict that result. $\endgroup$ – roy smith Feb 7 '16 at 0:13
$\begingroup$ @roysmith: I think you're mixing a couple of stories. The computation of $\pi_2$ was sorted out by 1950, probably much earlier, but certainly before Bott had really started down the path to periodicity. See Pontrjagin's 1950 paper, Homotopy classification of the mappings of an (n+2)-dimensional sphere on an n-dimensional one, or G. Whitehead (Ann. of Math. 52 (1950)). Eckmann's review of the former clearly indicates the role of the Arf invariant. There were early homotopy theory calculations that seemed to contradict Bott periodicity; Bott tells the story in various autobiographical pieces. $\endgroup$ – Danny Ruberman Mar 26 '18 at 17:02
Supposedly Stefan Bergman attended a course on orthogonal functions while an undergraduate, and misunderstood what he was hearing, believing that the functions were supposed to be analytic. This led him to the Bergman kernel and Hilbert spaces of analytic functions, which has developed into a whole field of study at the junction of complex analysis and operator theory, and also with important ramifications in the more geometric parts of SCV. If the story is true, this was certainly an extremely fruitful mistake!
engelbrekt
$\begingroup$ Paul Cohen told a version this story in his Complex Analysis class at Stanford when I was a graduate student. $\endgroup$ – Dan Ramras Nov 7 '13 at 22:02
Not just a great mistake, but also a great documentation of a mistake: Stallings's How not to prove the Poincare Conjecture. (I think this paper is my answer to every community-wiki question.)
HJRW
$\begingroup$ Whitehead's similar mistake is very interesting, too, as it lead him to the construction of contractible 3-manifolds that aren't balls. $\endgroup$ – Ryan Budney Dec 15 '09 at 21:22
$\begingroup$ There is also a paper by Cartier with a similar title: "Comment l'hypothèse de Riemann ne fut pas prouvée." $\endgroup$ – Joël Nov 7 '13 at 14:50
$\begingroup$ Stallings's page went down. The paper is at web.archive.org/web/20140630032821/http://math.berkeley.edu/… $\endgroup$ – Samuel Lelièvre Feb 22 '16 at 15:41
Poincaré's discovery of homoclinic points grew out of a extremely serious mistake he made in his original submission for a prize essay contest sponsored by Acta Mathematica in 1888. His original 200 page manuscript, on the restricted three-body problem, was evaluated by Weierstrass, Mittag-Leffler, and Phragmén, who had great difficulty following his arguments. Poincaré responded with a dozen further explanations, totaling 100 pages. After many further exchanges, the editors finally decided to accept the manuscript (this was, after all, Poincaré, and he must know what he's doing) and awarded him the prize.
But around the time of publication, Phragmén was still puzzled by some points and Mittag-Leffler wrote to Poincaré. They received back a telegram from Poincaré asking that publication be stopped immediately! Poincaré realized that his belief that the stable and unstable manifolds could not intersect transversally was wrong, and that such intersection points, which he later called homoclinic points, immediately forced very complicated dynamically behavior, invalidating much of his work. He wrote to Mittag-Leffler:
"I have written this morning to Mr. Phragmén to tell him about an error which I have committed and he has undoubtedly informed you of my letter. But the consequences of this error are more serious than I first thought. It is not true that the asymptotic surfaces are closed, at least not in the sense that I meant before. What is true, is that if one considers the two parts of that surface (which I yesterday still believed coincided with each other) they intersect along infinitely many asymptotic trajectories and furthermore their distance is an infinitesimal of higher order than $\mu^p$ however big p is.
I don't conceal from you the trouble this discovery gives me."
Mittag-Leffler immediately halted the presses and recalled all copies of this issue he could get, destroying them all (except for a few, one of which remains in the library of the Mittag-Leffler Institute). They asked Poincare to pay for the suppression of this issue, which he did.
Poincare then wrote a new essay, incorporating many of the added notes from the original, and this was the version that Acta Mathematica published (with no mention of the earlier one). Eventually Poincaré used this as the basis of his three volume classic Les méthodes nouvelles de la mécanique céleste.
A riveting account of this story is contained in Poincaré's discovery of homoclinic points by K. G. Anderson, Archive History of Exact Sciences, 48(2) (1994), 133–147.
Douglas Lind
$\begingroup$ And, apparently, Poincaré had to spend (all or most of) his prize money to pay for the destruction and reprinting of that Acta issue. Would be good to get an authoritative source for this. $\endgroup$ – David Roberts Feb 6 '16 at 12:57
$\begingroup$ The article by K. G. Anderson cited above does give authoritative account of these issues. It's not clear how much of Poincare'd prize was spent on the recall however. $\endgroup$ – Douglas Lind Feb 8 '16 at 6:11
Steiner's count 7776 of the number of the number of plane conics tangent to 5 general plane conics certainly deserves a mention here. He gave this answer in 1848, and it wasn't fixed until 1864, when Chasles pointed out the error and came up with the correct value of 3264. You can regard this as the first recognition of needing appropriate compactifications in order to do valid calculations in enumerative geometry.
Gary Kennedy
$\begingroup$ I must add to this that this story is even linked with modern developments, in the following direction: if the 5 conics are all real, how many among the 3264 conics tangents to all 5 of them are real? In 1997, Ronga, Tognoli and Vust found an example where all 3264 tangents conics are real. In 2005, Welschinger proved that if the 5 conics are ellipses, no two nested one inside the other, then at least 32 of the 3264 tangent conics must be real. This is related to a lot of deep modern tools. $\endgroup$ – Benoît Kloeckner Apr 11 '15 at 8:41
$\begingroup$ Eisenbud and Harris consider this example so important that they named a book after the number 3264: D. Eisenbud, J. Harris: 3264 and all that. Cambridge University Press (2016). In a nutshell, what is is to be appreciated in this story is that mathematicians a hundred years later succeeded in completing a formal framework which guarantees that "number of plane conics tangent to 5 general plane conics" is well-posed. (The deeply problematic word to make sense of in the latter is "general".) $\endgroup$ – Peter Heinig Sep 23 '17 at 8:43
Hilbert's program, whose development was induced by on assumptions shattered by Gödel.
Thomas Riepe
$\begingroup$ Just stumbled over this answer, and would like to point out that this sentence 'does not parse', as they say. I will not touch it, out of respect to any writing which is not technically-wrong, yet it might be good if you reformulated it. The 'shattered' is a bit extreme, by the way. And in case you don't have umlauts ready, here is an ö to copy and paste. @Thomas Riepe $\endgroup$ – Peter Heinig Sep 23 '17 at 8:48
$\begingroup$ I think "induced by" could be replaced by "founded" and then it would read sensibly. $\endgroup$ – Todd Trimble♦ Sep 23 '17 at 17:06
Goodrick's "story from Grad school" is incorrect. According to Ronald Jensen, the set theorist in question, he felt that the concept was important enough that it deserved a name which had not already been used elsewhere in mathematics. And 'mice' was it. (Also, note that 'mice' is a noun, and 'nice' is an adjective --- it would not make sense.)
$\begingroup$ But the urban legend is so funny... $\endgroup$ – Ilya Nikokoshev Oct 19 '09 at 20:02
$\begingroup$ I have heard 3 versions of the origin of the name. They all originated with Jensen, and were told at a rate of one per decade. Last I checked, he actually does not seem to remember the reason for the name. $\endgroup$ – Andrés E. Caicedo Oct 26 '10 at 4:52
Perhaps not under this heading but I enjoy reading in Marshall Hall Group Theory book:
"Let p be any old prime."
John mac
$\begingroup$ I can't find this in my copy of Hall's book :-( Google led me to a suggestion that it was on "p419" but the only mention of primes on p419 of my copy is in the middle of Theorem 20.9.13 where it's an odd prime. I have the second printing (published 1976). Maybe this is the problem? $\endgroup$ – eric Sep 11 '15 at 12:24
$\begingroup$ @eric the mistake is indeed "old" instead if "odd" in the first edition! $\endgroup$ – Andrea Dec 30 '15 at 14:43
Bringing in sort of tragic flavor to this question, - the following came to my mind:
He was not a very careful person as a mathematician. He made a lot of mistakes. But he made mistakes in a good direction. I tried to imitate him. But I've realized that it's very difficult to make good mistakes.
(Shimura on Taniyama, seen it in the "BBC Horizon Season 1996 Episode 2 - Fermat's Last Theorem" available on Youtube)
მამუკა ჯიბლაძე
$\begingroup$ This quotation can be found in print (with slightly different wording and punctuation) in Shimura's obituary of Taniyama in the Bull. London Math. Soc., March 1989, p. 190. $\endgroup$ – John Stillwell Mar 26 '18 at 23:19
$\begingroup$ Shimura wrote: "Though he was by no means the sloppy type, he was gifted with the special capability of making many mistakes, mostly in the right direction. I envied him for this, and tried in vain to imitate him, but found it quite difficult to make good mistakes." $\endgroup$ – John Stillwell Mar 26 '18 at 23:27
In chapter 3 of What Is Mathematics, Really? (pages 43-45), Prof. Hersh writes:
How is it possible that mistakes occur in mathematics?
René Descartes's Method was so clear, he said, a mistake could only happen by inadvertence. Yet, ... his Géométrie contains conceptual mistakes about three-dimensional space.
Henri Poincaré said it was strange that mistakes happen in mathematics, since mathematics is just sound reasoning, such as anyone in his right mind follows. His explanation was memory lapse—there are only so many things we can keep in mind at once.
Wittgenstein said that mathematics could be characterized as the subject where it's possible to make mistakes. (Actually, it's not just possible, it's inevitable.) The very notion of a mistake presupposes that there is right and wrong independent of what we think, which is what makes mathematics mathematics. We mathematicians make mistakes, even important ones, even in famous papers that have been around for years.
Philip J. Davis displays an imposing collection of errors, with some famous names. His article shows that mistakes aren't uncommon. It shows that mathematical knowledge is fallible, like other knowledge.
Some mistakes come from keeping old assumptions in a new context.
Infinite dimensionl space is just like finite dimensional space—except for one or two properties, which are entirely different.
Riemann stated and used what he called "Dirichlet's principle" incorrectly [when trying to prove his mapping theorem].
Julius König and David Hilbert each thought he had proven the continuum hypothesis. (Decades later, it was proved undecidable by Kurt Gödel and Paul Cohen.)
Sometimes mathematicians try to give a complete classification of an object of interest. It's a mistake to claim a complete classification while leaving out several cases. That's what happened, first to Descartes, then to Newton, in their attempts to classify cubic curves (Boyer). [cf. this annotation by Peter Shor.]
Is a gap in a proof a mistake? Newton found the speed of a falling stone by dividing 0/0. Berkeley called him to account for bad algebra, but admitted Newton had the right answer... Mistake or not?
"The mistakes of a great mathematician are worth more than the correctness of a mediocrity." I've heard those words more than once. Explicating this thought would tell something about the nature of mathematics. For most academic philosopher of mathematics, this remark has nothing to do with mathematics or the philosophy of mathematics. Mathematics for them is indubitable—rigorous deductions from premises. If you made a mistake, your deduction wasn't rigorous, By definition, then, it wasn't mathematics!
So the brilliant, fruitful mistakes of Newton, Euler, and Riemann, weren't mathematics, and needn't be considered by the philosopher of mathematics.
Riemann's incorrect statement of Dirichlet's principle was corrected, implemented, and flowered into the calculus of variations. On the other hand, thousands of correct theorems are published every week. Most lead nowhere.
A famous oversight of Euclid and his students (don't call it a mistake) was neglecting the relation of "between-ness" of points on a line. This relation was used implicitly by Euclid in 300 B.C. It was recognized explicitly by Moritz Pasch over 2,000 years later, in 1882. For two millennia, mathematicians and philosophers accepted reasoning that they later rejected.
Can we be sure that we, unlike our predecessors, are not overlooking big gaps? We can't. Our mathematics can't be certain.
The reference to the said article by Philip J. Davis is:
Fidelity in mathematical discourse: Is one and one really two? Amer. Math. Monthly 79 (1972), 252–263.
From that article, I find particularly amusing the following couple of paragraphs from page 262:
There is a book entitled Erreurs de Mathématiciens, published by Maurice Lecat in 1935 in Brussels. This book contains more than 130 pages of errors committed by mathematicians of the first and second rank from antiquity to about 1900.There are parallel columns listing the mathematician, the place where his error occurs, the man who discovers the error, and the place where the error is discussed. For example, J. J. Sylvester committed an error in "On the Relation between the Minor Determinant of Linearly Equivalent Quadratic Factors", Philos. Mag., (1851) pp. 295-305. This error was corrected by H. E. Baker in the Collected Papers of Sylvester, Vol. I, pp. 647-650.
A mathematical error of international significance may occur every twenty years or so. By this I mean the conjunction a mathematician of great reputation and a problem of great notoriety. Such a conjunction occurred around 1945 when H. Rademacher thought he had solved the Riemann Hypothesis. There was a report in Time magazine.
José Hdz. Stgo.
$\begingroup$ IIRC, one reason this is interesting is that this got people looking at calculus of variations in more detail/anxiety than had been done previously (i.e. in arguments where one assumed a minimizing function existed, and then used properties of said function, it wasn't always clear that a minimizer existed) $\endgroup$ – Yemon Choi Dec 16 '09 at 6:22
$\begingroup$ Re Poincaré's explanation: IMHO, it's frequently not just memory lapse, but just plain laziness. Whenever you see "it is obvious that", take a moment to see if it's really so obvious. I find this a useful rule of thumb when refereeing papers. $\endgroup$ – Todd Trimble♦ Sep 29 '15 at 11:52
I don't know if this is really a mistake: Fermat's "missing proof" for Fermat's last theorem.
shenghao
I find this one (it is not in the same vein as the ones that have been posted here so far, this is not a pure math mistake) to be interesting and instructive to students: patriot missile failure due to poor understanding of binary decimals
J M
Petrovskiĭ-Landis solution to the second part of Hilbert 16th problem. They "proved" the existence of a bound for the number of limit cycles of planar polynomial vector fields of fixed degree. Ilyashenko pointed out the mistake.
The problem remains wide open but the basic idea of Petrovskiĭ-Landis ( complexification of real differential equations ) lead to the study of holomorphic foliations.
Jorge Vitório Pereira
Samuel I. Krieger made many attempts at significant contributions to the field of mathematics, unfortunately some of his efforts did not pan out.
In 1934, he claimed that the 72-digit composite number 231,584,178,474,632,390,847,141,970,017,375,815,706,593,969,331,281,128,078,915,826,259,279,871 was the largest known prime number.
He also attempted to show that the number 2^256(2^257-1) was perfect, implying that 2^257-1 is a prime number. 2^257-1 is actually a composite number: its smallest prime factor is 535,006,138,814,359.
Finally, he claimed to have a counter example to Fermat's Last Theorem x^n + y^n = z^n using the numbers x = 1324, y = 731 and z = 1961 with an undisclosed n. A reporter supposedly called Krieger to ask how the left and the right hand side could be equal, when the left hand side could only end in a 4 or a 6 plus 1, and the right hand side could only end in 1.
Wlog
$\begingroup$ A reporter who knew about modulo 10 arithmetics, now that's quite a story! $\endgroup$ – Ilya Nikokoshev Oct 22 '09 at 7:43
$\begingroup$ I believe his 1935 claim for a much larger prime number is also incorrect: news.google.com/… sagecell.sagemath.org/… $\endgroup$ – Christopher D. Long Oct 27 '14 at 22:56
$\begingroup$ There is a little bit more about Krieger at outofthenormmaths.wordpress.com/2011/11/09/blitz-krieger Amazon.com claims to have a book, Krieger's Mathematical Formulae, that he published in 1934, but the page also says, Out of Print--Limited Availability. $\endgroup$ – Gerry Myerson Dec 6 '16 at 22:50
Then there's always the Martian Climate Orbiter Newtons vs Pounds of thrust embarrassment.
For surfaces of constant mean curvature, it is alleged that Hopf thought that all compact CMC surfaces in $\mathbb{R}^3$ were round spheres. CMC surfaces are what you get if you have a soap film bounding a fixed volume, so after a childhood full of blowing bubbles this is a pretty reasonable thing to think. And it even happens to be mostly true: Hopf proved that immersed CMC spheres are round, and Alexandrov proved with a nice reflection argument that embedded CMC surfaces of any genus must actually be round spheres.
But a bit later, Wente discovered a collection of CMC tori. Ivan Sterling has some nice pictures of these on his website, as does MSRI. There are many very pretty connections between these surfaces and algebraic geometry, so to me they sort of mark the start of the modern "integrable systems" era of CMC research.
I should probably add that nobody actually seems sure if Hopf believed that compact CMC surfaces are spheres, but it makes a good creation story for the subfield!
Matt Noonan
Lakatos' work "Proof and refutation" contains many examples of mistakes concerning the development of Euler's polyhedron formula, along with an extensive treatment of what mistakes are and how they can crucially contribute to the development of mathematics.
M Carl
Euler conjectured that there were no pairs of orthogonal Latin squares for orders $n \equiv 2 (\text{mod}~ 4)$. Nearly two hundred years later, this was proved false for every $n \equiv 2 (\text{mod}~ 4)$ except $ 2 $ and $ 6 $. Here's the link to Euler's paper. Regardless, Euler's work certainly helped spur research into Latin squares.
Douglas S. Stones
$\begingroup$ It seems odd to call a conjecture a mistake. $\endgroup$ – Jim Conant Sep 29 '15 at 13:23
protected by François G. Dorais♦ Nov 7 '13 at 13:17
Not the answer you're looking for? Browse other questions tagged soft-question big-list gm.general-mathematics or ask your own question.
Which math paper maximizes the ratio (importance)/(length)?
Widely accepted mathematical results that were later shown to be wrong?
What recent discoveries have amateur mathematicians made?
What are some famous rejections of correct mathematics?
Theorems demoted back to conjectures
If d/dx is an operator, on what does it operate?
The error in Petrovski and Landis' proof of the 16th Hilbert problem
Projection of Borel set from $R^2$ to $R^1$
Why are smooth numbers called "smooth"?


Okay, so this one already came out as of the making of this list... but it was definitely my most anticipated game of this year, or even of the past five years. And it basically lived up to my expectations, which is awesome. Still working on 100%-ing it!
I didn't really know what to think of this game when I first saw the announcement trailer, but the more I read about it the more excited I get. Fallout 4 was pretty disappointing to me, and I've stayed far away from FO76 (as cool as some of the locations and creatures in that game look), but if The Outer Worlds is as polished at release as it looks in promo material right now, I think it'll be amazing. I don't think I'm a big fan of the snarky humor vibe I'm getting from it, but it looks visually cool, and most importantly they seem to be putting a big emphasis on roleplaying and player choice. Those alone make me super hyped for TOW.
We have zero details on this thing, yet just the prospect of a new, full-fledged Animal Crossing makes me so happy. I don't know what new mechanics or features they could implement, but I'd probably still buy it even if it was just an HD remake of New Leaf, which it won't be. Never have I wanted so badly to be in debt.
Yet another game we have almost zero details on. But I loved the campaign in Titanfall 2, and I've wanted a current-gen, single-player Star Wars experience for so long now. Plus, I think the post-ROTS era is a potentially really interesting setting, focusing on the rise of the Empire and the survival of the last few Jedi. It's a period of time we haven't seen much from - Rogue One and Rebels both focused on the time right before A New Hope, and Solo gives us some idea of what it was like, but that movie starts a whole six years after ROTS. (It's probably been covered a bit in the novels and comics, but I'm less familiar with those.) I'd love to see Kanan Jarrus and/or Ahsoka Tano show up at some point, but new characters could be cool as well.
I've been wanting a story-focused, combat-less immersive sim for as long as I've known what the genre is, and The Occupation looks to be an excellent fulfillment of that wish. Plus I love the art style, and I love journalism stories, so I am very excited for this one.
The description of this game is an "open-world, post-apocalyptic Kung-Fu fable RPG," and I don't think I've loved anything more based solely on the pitch. Plus, the main character they've been showing off looks like if Rocket Raccoon was a samurai. If this game is good, it'll be amazing.
This is a late add, since it was just announced in February, but I am SO EXCITED. I never had a WiiU, so I didn't get the chance to participate in the community of the first game, but now we get a new one on Switch!
Fire Emblem Awakening is probably in my top ten games ever, so I can't wait to see what the series is like on Switch.
I'm not much of a horror-shooter guy, but the world and art style here looks so fascinating. It's like an SCP Foundation story set in an alternate Soviet reality. My hope is that it's got some puzzle solving and/or immersive sim elements, because I usually don't find straight-up shooters that interesting. But I might have to play this for the story and worldbuilding alone.
I still haven't played ABZU, but I've heard nothing but great things about it, and this game already looks more up my alley - an open-world action game, you've got a bird friend, the art style is gorgeous and the animation looks amazing, and I'm really curious as to what the story is. I'm all in.
I love open-world games and I adore this art style/aesthetic, so I'm excited to see where this one goes.
I could never get into Alan Wake, but the trailer for this game looks so rad. Fingers crossed!
Doom 2016 was a lot of heckin' fun. This one's got a hookshot.
Ori and the Blind Forest was 90% amazing and beautiful, 10% a bit too hard for me. I mean, I finished it, it was just frustrating sometimes. But I can't wait for this sequel to come out. The first one was among the most beautiful games I think I've ever played.
We don't know much about this game yet, but the premise sounds cool, and I really loved the last two Wolfenstein games. Time will tell if I like the co-op gameplay.


text:
It's not easy being in Michael A Grammar. Faced with deportation, hard drive meltdowns and plucking bass players from life drawing classes, the band have somehow managed to battle against the odds and record their amazing self-titled debut released 29th September on Melodic Records.
Combining the bands two E.Ps to date with four new tracks, the bands sonic development is perfectly demonstrated with a collage of sounds ranging from funky post-punk grooves, interstellar guitar lines down into propulsive motorik jams influenced by the likes of Broadcast (from which their name derives), Slowdive and a wired-up version of The Beta Band
If Michael A Grammar's previous two E.Ps (Random Vision and Vitamin Easy) cemented how devastatingly accomplished this young band has become in the last 18 months, then their debut album 'Michael A Grammar' (made up of those first two EPs and 4 brand new tracks) is set to confirm that good things come in threes. It's the Brighton four-piece's triumphant debut album and it brilliantly captures their sonic evolution and live prowess from conception through to the present day, and gives us a heady insight into their future.
"Had we set out to record everything straight away, we'd probably have drifted into an infinite tundra of sub-satisfactory jingles and drowned early on," explains guitarist and singer Joel Sayers. "This record marks the voyage of change that has been underway since our beginning, documented in the way we know best."
Whilst most young bands simply start up, make a record, and then tour it out, Michael A Grammar boast a freshness contrary to having already experienced their fair share of ups and downs. Following the release of their first Vitamin Easy EP when just 20 years old (four transcendent, psychedelic tracks of Slowdive-style sparkle that surfaced from sessions in a Victorian coach house whilst attempting to recreate lost songs from a crashed hard drive) the band's original bassist Daniel Ondieki was facing deportation back to his native Kenya.
"Unfortunately Daniel had to leave due to visa issues with "The Combine", though it could've been worse had Nigel Farage been in the hot-seat with a stick up his arse," says Joel. "Thankfully we found Clémentine with her clothes off at a life drawing class and invited her into the family. We're now waiting on the fifth member…"
Michael A Grammar's evolving line-up was unveiled on their next installment, the Random Vision EP. Recorded in the basement of their Nottingham student house by Joel and fellow founding member Frankie Mockett (both guitar/vocals). They were joined by John Davies (drums) and French bassist Clémentine Blue, whose driving riffs enhanced the group's widescreen and stoned out jams towards a more buoyant 'not give a fuck' blistering live show. This EP, recalling Boards of Canada's mind-ensconcing atmospherics, The Beta Band chowing down on ritalin or Broadcast (whose song 'Michael A Grammar' lends the band its name) propelled them forwards. A feature on Mary Ann Hobbs' XFM podcast ensued, as did a session for BBC 6music's Marc Riley, live appearances at The Great Escape and Camden Crawl followed shortly after an invite by British Sea Power to be main support on their sold out tour, an invite to play Green Man and a European autumn headline tour are next on the horizon.
Marking Michael A Grammar's biggest progression to date, the album's anthemic final third (which they've named collectively as the suitably intergalactic 'Lunar Sea') kicks off with 'Mondays' – it skilfully sets the tone with petroleum powered melodies fired from layered Brit Pop guitars, devastating distortion and woozy vocal effects. 'Nature's Child' explodes into a ambush of axe-wielding noise and 'You Make Me' is a patchwork quilt of manipulated psychedelic organ textures and varying sources of inspiration;
"The words are a mix of notes I'd scribbled down in various places," Joel reveals. "Some come from listening to Souls of Mischief; another line comes from a poem in the film Patton." Shifting from the DIY charm of a friend's old office in Portslade, Brighton, to a fully-equipped studio for the first time, the tracks were recorded live with external producer, Julian Tardo.
"Each part of the album is an indicator of our present," they say. "Obviously as things change, so too does the music and the way it's recorded. As a result we feel more confident within ourselves which has inspired us to bring it back down to earth and record in the manner we did." All live recordings, single takes and a ferocious attitude gives these tracks a contagious energy.
Everything has been leading towards this moment; Michael Grammar is the story of four young friends progression personally and sonically into adulthoodthrough their past, present, and if you listen to its entirely improvised hidden ending, maybe a glimpse into their near future
Tracklisting / /
1. 'Upside Down'
2. 'All Night Afloat'
3. 'Light of a Darkness'
4. 'King & Barnes'
5. 'The Day I Come Alive'
6. 'Suzanna'
7. 'Upstairs Downstairs'
8. 'The Way You Move'
9. 'Mondays'
10. 'You Make Me'
11. 'Nature's Child"
12. 'Don't Wake Me'

redpajama_set_name: RedPajamaCommonCrawl


During the decision support lots of data gets analysed to help the decisionmaker understand her situation. This situational awareness is the deciding factor that is needed for making optimal decisions. For the automatic procedure artificial intelligences are used, which are designed as software modules for different applications such as medical engineering or robotics.
Information is being visualised and edited for easier processing.
Patterns in large data sets are recognized, which allows inferences to be drawn.
Workflows are analyzed and enhanced in realtime.
The systems advise physicians like an expert for medical problem solving would.
Unusual situations are recognized and analysed.


Spontaneous alpha-band oscillations represent ongoing fluctuations in neuronal excitability, with periods of strong oscillations indicating neuronal inhibition (Watson et al., 2018). Visual detection studies have indicated that heightened neuronal excitability (weak alpha-band oscillations) biases subjects to report seeing a target stimulus even when no stimulus had been presented (Limbach and Corballis, 2016; Iemi et al., 2017). In their eNeuro publication, Iemi and Busch (2018) focus on the nature of this bias: is it due to a deliberate preference to report this perception (decision bias), or is it due to a genuine subjective, albeit incorrect, perception of seeing something (perceptual bias)?
They used variations of a two-interval forced choice task, in which subjects were asked to report in which of two intervals a critical stimulus had appeared. The rationale was based on two assumptions. First, bias should facilitate performance on trials in which excitability in the to-be-reported interval happens to exceed excitability in the other interval. Second, while a decision bias (i.e., "I report the interval in which stronger excitability was sensed") should have that effect irrespective of what the decision is about, a perceptual bias (i.e., "this rather looks like something, as opposed to nothing") should affect performance only in decisions about stimulus presence versus stimulus absence.
Indeed, Iemi and Busch (2018) found that weak EEG alpha oscillations (strong excitability) before the stimulus-present interval and strong alpha oscillations (weak excitability) before the stimulus-absent interval biased subjects to correctly detect the stimulus in the appropriate interval. No effect of ongoing oscillations was found when the decision was not about stimulus presence versus absence, but instead was about stimulus orientation. This finding implies that fluctuations of neuronal excitability due to ongoing neuronal oscillations in the alpha band amplify sensory signals as well as internal noise, thereby amplifying the subjective impression of seeing something, even when nothing had been presented.
Conceptually, this perceptual bias can be compared to the effect of the ISO setting (i.e., the sensitivity of a camera to light) (Fig. 1). With low ISO (weak excitability), the camera sensor would not pick up faint objects in the dark; however, low ISO is accompanied by a low noise level. Thus, the photographer might miss an object, but would not mistake the noise for something real. With high ISO (strong excitability), the sensor can pick up even faint objects, but the noise level is also high. Thus, high ISO might "bias" the photographer to see real objects, as well as illusory objects that actually reflect noise.
Effect of perceptual bias on perceptual decisions. Left, An image taken at low ISO corresponding to what a subject would see in a brain state of weak excitability. No objects are detected, whether real, but faint (yellow box) or illusory (blue box). Right, An image taken at high ISO corresponding to what a subject would see in a brain state of strong excitability. In this case, even faint objects can be detected (yellow box), but the increased noise level can result in false detection of illusory objects (blue box). (Image courtesy of Niko Busch.).
These results bring up an exciting follow-up question: do these ongoing fluctuations of excitability and perceptual bias, caused by fluctuations in the strength of neuronal oscillations, affect performance in real-world detection tasks such as radiology or airport security screenings?
Iemi L, Busch NA (2018) Moment-to-moment fluctuations in neuronal excitability bias subjective perception rather than strategic decision-making. eNeuro 3:ENEURO.0430-17.2018.
Iemi L, Chaumon M, Crouzet SM, Busch NA (2017) Spontaneous neural oscillations bias perception by modulating baseline excitability. J Neurosci 37:807–819.
Limbach K, Corballis PM (2016) Prestimulus alpha power influences response criterion in a detection task. Psychophysiology 53:1154–1164.
Watson BO, Mingxin D, Buzsáki G (2018) Temporal coupling of field potentials and 699 action potentials in the neocortex. Eur J Neurosci. Advance online publication. Retrieved May 19, 2018. doi:10.1111/ejn.13807.


The PlayStation Portable, officially abbreviated PSP, is a handheld game console. In addition to games, have you ever thought of watching movie on your PSP? You might know that PSP can watch video clips, but how to put video to PSP? Well, let's take a brief look at what video formats your PSP support.
All movie files you can view on your PSP come with an .MP4 extension. If you want to watch DVD movie on your PSP, you're going to need DVD ripper software to rip the DVD movie to the video format that your PSP support. MacX DVD Ripper Pro is just the program you need to solve this problem on Mac. This simple yet powerful free DVD to PSP converter utility enables you to convert DVD to MP4 and other PSP accepted formats with a few mouse clicks.
Follows the instruction below, and you can convert DVD movies to PSP on Mac in 3 steps.
Make sure you have the DVD to PSP converter installed.
Run this Free DVD ripper software on Mac, then insert your DVD on the DVD-ROM and click "DVD Disc" button to load your DVD or add DVD folder by clicking "DVD Path".
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When the settings are done, click the "RUN" button to start conversion. The converting speed depends on the file size of DVD movies and your CPU. New movie DVD releases are all supported.
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How to use Mac the Ripper to rip DVD to iPad, iPhone iPod on Mac? MacTheRipper User guide article. You will learn from the tutorial how to rip DVD to VOB files and convert VOB files to video that compatible with iPad, iPhone, iPod etc.


text:
Home » Spider Control
Habitat: Common household pest in the fall when they are looking for a warm place to overwinter. They are commonly found around doors, windows, house plants, basements, garages, and in almost all terrestrial habitats. They do not spin a web but roam at night to hunt for food. Wolf spiders are often confused with the brown recluse, but they lack the unmistakable violin-shaped marking behind the head. The wolf spider is shy and seeks to run away when disturbed.
Description: Wolf spiders range from about 1/2 inch to 2 inches in length, hairy, and are typically brown to gray in color with various markings or lines. Wolf spider mothers carry their large egg sacs around with them. When the young spiderlings hatch they climb onto their mother's back and ride around until partially grown. Wolf spiders are not poisonous, though as with all spiders, bites may cause reactions in certain individuals.
Also known as the brown recluse.
The name "violin spider" describes a characteristic marking on this animal: there is a violin-shaped patch on the broad, almost heart-shaped cephalothorax (the "head" as opposed to the abdomen). The overall color is usually a grayish-yellow-brown, the oblong abdomen covered with gray hairs. The legs are darker than the body and are long and slim. Females are larger then males. The webs are small, irregular and untidy. These spiders are usually seen walking or running around, not in a web.
Size: Length: females to 1/4 inch, not counting legs; including legs in a typical pose, they are about 1 inch long.
Habitat and conservation: The name "recluse" describes this spider well. It commonly hides in little-used drawers, closets, attics and basement areas, and behind baseboards and furniture. They cannot climb smooth surfaces and are often found trapped in bathtubs and sinks. Their habit of hiding among packed-away garments and towels creates a common situation for human bites, when a person dons clothing from storage without inspecting or shaking it out first. In nature, they live under and in crevices of rocks.
Foods: Dietarily, brown recluses are little different from other spiders. They eat small insects and other spiders. As a running spider (not a spider that uses webs to catch prey), recluses chase down prey like a wolf. The purpose of their venom is to subdue their prey. In fact, brown recluses cannot easily bite humans unless they are pressed against our skin (as when they are suddenly trapped between a garment and our bodies, or if they are exploring our bed sheets and we roll on top of them).
Jumping Spider, common name for any of a group of hunting spiders that can leap 10 to 40 times their body length. With over 4000 described species, they comprise the largest family of spiders. Jumping spiders are particularly diverse in tropical regions, but occur in habitats ranging from rain forests to above the timberline on Mount Everest in the Himalayas. More than 300 species have been described in the United States.
Jumping spiders are usually less than 2 cm (less than 0.8 in) in length with females generally larger than males. They are among the most ornate of spiders; many species are brightly colored and strikingly patterned, with stout bodies, short legs, and a very large pair of eyes on the front of the face. The jumping spider has four pairs of eyes, with the large principal eyes giving it sharper vision than any other animal of similar size. It can identify prey, predators, and mates from up to 30 cm (up to 12 in) away.
The jumping spider is an active predator, usually hunting during daylight. It will stalk to within a few body lengths of the prey, crouch, crawl slowly forward, and then lift its front legs and pounce. It accomplishes its spectacular jumps by means of muscular contractions in the body that force body fluids into the legs, causing the legs to extend rapidly. Most jumping spiders feed on insects, while others feed primarily on web-building spiders.
The male's front pair of legs are colored and have distinctive bands of hair. In many species the male performs complex courtship displays in which he bobs his body and waves his front legs in a highly specific manner. After mating, the female lays her eggs in a silk-lined shelter under stones or bark, or on the surface of plants. The female will often guard the eggs and newly hatched young.
The hobo spider is a member of the spider family Agelenidae, a common group that has many species throughout California and the United States. Agelenid spiders can have very dense populations in certain habitats. The members of this family construct a snare referred to as a funnel web, which is a trampolinelike, horizontal web constricting back into a funnel or hole. The web is typically found in a crack between bricks or under wood, stones, or vegetation. The spider waits in the mouth of the funnel for prey to fall onto the horizontal surface, and then it rushes out, grabs the prey, and takes it back to its funnel to consume. If you go outside on a dewy morning, you can often see many of these funnel webs.
The hobo spider shares traits with many of its relatives in the Agelenidae family, including coloration and web-building characteristics. It is a brown spider about 1/4 to 5/8 inch in body length and lives in a funnel web. There are dozens of similar looking spider species in California that build funnel webs, including members of the genera Agelenopsis (2 species), Calilena (13 species), Hololena (21 species), Novalena (4 species), and Rualena (8 species). There is even a unique wolf spider genus, Sosippus (1 species), in California that, unlike its free-hunting relatives, builds a funnel web. Therefore, if you see a funnel web in California, there are many other spiders you should suspect before even considering the hobo spider as a possibility. None of these species causes necrotic wounds or serious injury to humans.
To distinguish funnel-weaving spider species, the arachnid's reproductive structures must be examined, a task that requires the skills of a qualified arachnologist. Each spider species has a distinctive "lock and key" design of the male and female reproductive organs. Through evolution, the physical features of the males and females have become unique for each species and hence are used by arachnologists for species differentiation. Until someone has examined reproductive features on dozens of hobo spiders and also related species, it is unlikely that a hobo spider would be correctly identified with the naked eye. Therefore, virtually any "hobo spider" identification by a non-arachnologist in California should be suspect.
If you have access to a magnifying device (hand lens, microscope, etc.), there is an easy way to determine if the spider you have IS NOT a hobo spider. First, the spider must be associated with a funnel web; otherwise it could easily be one of the hundreds of other non-agelenid spider species in California. All funnel-weaving spiders have eight eyes arranged in two rows. If you look at the spider head on, however, most of the agelenid spiders in California have their two rows of eyes curved so strongly that it appears that their eyes are actually in three horizontal rows with four eyes in the middle row with two eyes above and two eyes below this row. This is known as the 2-4-2 eye pattern. Exceptions are the spiders of the genus Tegenaria, which have eyes that are in the more common pattern of two rows of four. This is also the most common eye pattern for spiders in general, and unless you remove the spider from a funnel web, it will probably not be a funnel-weaving spider.
Nonetheless, it is not uncommon to find a funnel-weaving spider with two straight rows of eyes in California. There are two species of Tegenaria in the state, T. pagana and T. domestica. Neither species is native to the United States. Their coloration is similar enough to that of the hobo spider, T. agrestis, that anyone except a spider expert might confuse them with the hobo spider. Tegenaria pagana is found in coastal California regions to slightly inland. Tegenaria domestica, however, is an extremely common spider, about 1/4 to 1/2 inch in body length, found throughout California often inside homes, as is evident by the name "domestica." In fact, T. domestica is found throughout the world, having been carried by commerce. Neither of these spiders is considered poisonous to humans even though they are closely related to the hobo spider.
Harvestman
Harvestman, or Daddy Longlegs, a small eight-legged animal related to the spiders. It lives in temperate and tropical regions. There are over 3,200 species, 200 of which are native to the United States. Unlike a spider, a harvestman has long, stiltlike legs and a segmented abdomen, and lacks silk glands. The legs each have seven joints, making them highly flexible and allowing the animal to run rapidly over leaves and grass.
Harvestmen feed on small insects, decayed animal matter, and the juices of fruits and vegetables. A pair of glands in the carapace secrete an unpleasant-smelling substance that deters predators. The female has an ovipositor, a specialized tubular organ that is extended from the abdomen to deposit eggs in damp soil.
The longbodied cellar spider is the most common of the Pholcidae in the United States. It has extremely long legs and is found in many types of buildings throughout the year. Although these spiders can develop large populations in protected locations, they are not known to be in any way harmful. This spider is commonly found in cellars, warehouses, garages, caves, and other dark, quiet, protected spots.
The females are from 7 to 8 millimeters and the males 6 millimeters in length. The carapace is very wide. The abdomen is about three times as long as it is wide and cylindrical in shape. The eyes are arranged in two triads of larger, light-colored eyes on the top of the cephalothorax and a pair of dark, small eyes on the front. The color of the carapace and abdomen is a pale tan or yellow with a gray mark in the center of the carapace.
The forelegs of the adult female are about 50 millimeters in length. The webs are an irregular mesh of threads.
Life History/Behavior
Both male and female spiders can be found in heated structures anytime of the year. It takes approximately one year for the spiders to mature and they can live for at least another two.
The eggs, which are about 1 millimeter in diameter, are deposited in clusters of 25 to 60 and wrapped in a thin layer of silk. The sac is held by the female in her chelicerae as she hangs inverted in her web.
An unusual behavior is that the spiders will rapidly vibrate in a circular fashion in the web if threatened, making it difficult to see them.
Description/Identification
Female black widow spiders are about .5 inch long, to about 1.5 inches long when the legs are spread. Males are about half the female's size, with smaller bodies and longer legs. Both females and males have shiny, globular abdomens and are mostly black, and sometimes brown.
Females usually have a reddish hourglass shape on the underside of their abdomens. In some species the females have a series of red spots and two crosswise bars on the underbelly.
Male black widows frequently have yellow and red bands and spots over their backs, as do both sexes of black widows in their immature stages.
Newly hatched spiderlings are predominately white or yellowish-white, gradually acquiring more black and varying amounts of red and white with each molt.
Juveniles of both sexes resemble the male and are harmless to humans.
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