Georg Cantor Quotes

My beautiful proof lies all in ruins.

The essence of mathematics is in its freedom.

No one shall expel us from the paradise which Cantor has created for us. {Expressing the importance of Georg Cantor's set theory in the development of mathematics.}

In Mathematics the art of proposing a question must be held of higher value than solving it.

My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? I have studied it…I have followed its roots, so to speak, to the first infallible cause of all created things. —GEORG CANTOR

Before the work of Georg Cantor in the nineteenth century, the study of the infinite was as much theology as science; now, we understand Cantor’s theory of multiple infinities, each one infinitely larger than the last, well enough to teach it to first-year math majors. (To be fair, it does kind of blow their minds.)

A small but typical example of how ‘philosophy’ sends out new shoots is to be found in the case of Georg Cantor, a nineteenth-century German mathematician. His research on the subject of infinity was at first written off by his scientific colleagues as mere ‘philosophy’ because it seemed so bizarre, abstract and pointless. Now it is taught in schools under the name of set-theory.

In der Mathematik ist die Kunst Fragen zu stellen wertvoller als Probleme zu lösen.

Je le voie, mais je ne le crois pas.

Engineers had not framework for understanding Mandelbrot's description, but mathematicians did. In effect, Mandelbrot was duplicating an abstract construction known as the Cantor set, after the nineteenth-century mathematician Georg Cantor. To make a Cantor set, you start with the interval of numbers from zero to one, represented by a line segment. Then you remove the middle third. That leaves two segments, and you remove the middle third of each (from one-ninth to two-ninths and from seven-ninths to eight-ninths). That leaves four segments, and you remove the middle third of each- and so on to infinity. What remains? A strange "dust" of points, arranged in clusters, infinitely many yet infinitely sparse. Mandelbrot was thinking of transmission errors as a Cantor set arranged in time.

what wonderful power there is in the real numbers, since one is in a position to determine uniquely, with a single coordinate, the elements of an n-dimensional continuous space.

In particular, in introducing new numbers, mathematics is only obliged to give definitions of them, by which such a definiteness and, circumstances permitting, such a relation to the older numbers are conferred upon them that in given cases they can definitely be distinguished from one another. As soon as a number satisfies all these conditions, it can and must be regarded as existent and real in mathematics. Here I perceive the reason why one has to regard the rational, irrational, and complex numbers as being just thoroughly existent as the finite positive integers.

The idea of considering the infinitely large not only in the form of the unlimitedly increasing magnitude and in the closely related form of convergent infinite series...but to also fix it mathematically by numbers in the definite form of the completed infinite was logically forced upon me, almost against my will since it was contrary to traditions which I had come to cherish in the course of many years of scientific effort and investigations.

The truth, though is that Cantor's work and its context are so totally interesting and beautiful that there's no need for breathless Prometheusizing of the poor guy's life. The real irony is that the view of (Infinity) as some forbidden zone or road to insanity-which view was very old and powerful and haunted math for 2000+ years- is precisely what Cantor's own work overturned. Saying that (Infinity) drove Cantor mad is sort of like mourning St. George's loss to the dragon: it's not only wrong but insulting.

The transfinite numbers themselves are in a certain sense new irrationals, and in fact I think the best way to define the finite irrational numbers is entirely similar; I might even say in principle it is the same as my method for introducing transfinite numbers. One can absolutely assert: the transfinite numbers stand or fall with the finite irrational numbers; they are alike in their most intrinsic nature; for the former like these latter are definite, delineated forms or modifications of the actual infinite.

One other thing to keep in mind, though, is that Cantor's transfinite math will end up totally undercutting Aristotelian objections like the above (b) to Dedekind's proof, since Cantor's theory will constitute direct evidence that actually-infinite sets can be understood and manipulated, truly handled by the human intellect, just as velocity and acceleration are handled by calculus. So one thing to appreciate up front is that, however abstract infinite systems are, after Cantor they are most definitely not abstract in the nonreal/unreal way that unicorns are.

De Forest came up with the idea of imprinting the sound directly onto the film. That meant that no matter what happened with the film, sound and image would always be perfectly aligned. Failing to find backers in America, he moved to Berlin in the early 1920s and there developed a system that he called Phonofilm. De Forest made his first Phonofilm movie in 1921 and by 1923 he was back in America giving public demonstrations. He filmed Calvin Coolidge making a speech, Eddie Cantor singing, George Bernard Shaw pontificating, and DeWolf Hopper reciting “Casey at the Bat.” By any measure, these were the first talking pictures. However, no Hollywood studio would invest in them. The sound quality still wasn’t ideal, and the recording system couldn’t quite cope with multiple voices and movement of a type necessary for any meaningful dramatic presentation.

If talking pictures could be said to have a father, it was Lee De Forest, a brilliant but erratic inventor of electrical devices of all types. (He had 216 patents.) In 1907, while searching for ways to boost telephone signals, De Forest invented something called the thermionic triode detector. De Forest’s patent described it as “a System for Amplifying Feeble Electric Currents” and it would play a pivotal role in the development of broadcast radio and much else involving the delivery of sound, but the real developments would come from others. De Forest, unfortunately, was forever distracted by business problems. Several companies he founded went bankrupt, twice he was swindled by his backers, and constantly he was in court fighting over money or patents. For these reasons, he didn’t follow through on his invention. Meanwhile, other hopeful inventors demonstrated various sound-and-image systems—Cinematophone, Cameraphone, Synchroscope—but in every case the only really original thing about them was their name. All produced sounds that were faint or muddy, or required impossibly perfect timing on the part of the projectionist. Getting a projector and sound system to run in perfect tandem was basically impossible. Moving pictures were filmed with hand-cranked cameras, which introduced a slight variability in speed that no sound system could adjust to. Projectionists also commonly repaired damaged film by cutting out a few frames and resplicing what remained, which clearly would throw out any recording. Even perfect film sometimes skipped or momentarily stuttered in the projector. All these things confounded synchronization. De Forest came up with the idea of imprinting the sound directly onto the film. That meant that no matter what happened with the film, sound and image would always be perfectly aligned. Failing to find backers in America, he moved to Berlin in the early 1920s and there developed a system that he called Phonofilm. De Forest made his first Phonofilm movie in 1921 and by 1923 he was back in America giving public demonstrations. He filmed Calvin Coolidge making a speech, Eddie Cantor singing, George Bernard Shaw pontificating, and DeWolf Hopper reciting “Casey at the Bat.” By any measure, these were the first talking pictures. However, no Hollywood studio would invest in them. The sound quality still wasn’t ideal, and the recording system couldn’t quite cope with multiple voices and movement of a type necessary for any meaningful dramatic presentation. One invention De Forest couldn’t make use of was his own triode detector tube, because the patents now resided with Western Electric, a subsidiary of AT&T. Western Electric had been using the triode to develop public address systems for conveying speeches to large crowds or announcements to fans at baseball stadiums and the like. But in the 1920s it occurred to some forgotten engineer at the company that the triode detector could be used to project sound in theaters as well. The upshot was that in 1925 Warner Bros. bought the system from Western Electric and dubbed it Vitaphone. By the time of The Jazz Singer, it had already featured in theatrical presentations several times. Indeed, the Roxy on its opening night in March 1927 played a Vitaphone feature of songs from Carmen sung by Giovanni Martinelli. “His voice burst from the screen with splendid synchronization with the movements of his lips,” marveled the critic Mordaunt Hall in the Times. “It rang through the great theatre as if he had himself been on the stage.

In order for A to apply to computations generally, we shall need a way of coding all the different computations C(n) so that A can use this coding for its action. All the possible different computations C can in fact be listed, say as C0, C1, C2, C3, C4, C5,..., and we can refer to Cq as the qth computation. When such a computation is applied to a particular number n, we shall write C0(n), C1(n), C2(n), C3(n), C4(n), C5(n),.... We can take this ordering as being given, say, as some kind of numerical ordering of computer programs. (To be explicit, we could, if desired, take this ordering as being provided by the Turing-machine numbering given in ENM, so that then the computation Cq(n) is the action of the qth Turing machine Tq acting on n.) One technical thing that is important here is that this listing is computable, i.e. there is a single computation Cx that gives us Cq when it is presented with q, or, more precisely, the computation Cx acts on the pair of numbers q, n (i.e. q followed by n) to give Cq(n). The procedure A can now be thought of as a particular computation that, when presented with the pair of numbers q,n, tries to ascertain that the computation Cq(n) will never ultimately halt. Thus, when the computation A terminates, we shall have a demonstration that Cq(n) does not halt. Although, as stated earlier, we are shortly going to try to imagine that A might be a formalization of all the procedures that are available to human mathematicians for validly deciding that computations never will halt, it is not at all necessary for us to think of A in this way just now. A is just any sound set of computational rules for ascertaining that some computations Cq(n) do not ever halt. Being dependent upon the two numbers q and n, the computation that A performs can be written A(q,n), and we have: (H) If A(q,n) stops, then Cq(n) does not stop. Now let us consider the particular statements (H) for which q is put equal to n. This may seem an odd thing to do, but it is perfectly legitimate. (This is the first step in the powerful 'diagonal slash', a procedure discovered by the highly original and influential nineteenth-century Danish/Russian/German mathematician Georg Cantor, central to the arguments of both Godel and Turing.) With q equal to n, we now have: (I) If A(n,n) stops, then Cn(n) does not stop. We now notice that A(n,n) depends upon just one number n, not two, so it must be one of the computations C0,C1,C2,C3,...(as applied to n), since this was supposed to be a listing of all the computations that can be performed on a single natural number n. Let us suppose that it is in fact Ck, so we have: (J) A(n,n) = Ck(n) Now examine the particular value n=k. (This is the second part of Cantor's diagonal slash!) We have, from (J), (K) A(k,k) = Ck(k) and, from (I), with n=k: (L) If A(k,k) stops, then Ck(k) does not stop. Substituting (K) in (L), we find: (M) If Ck(k) stops, then Ck(k) does not stop. From this, we must deduce that the computation Ck(k) does not in fact stop. (For if it did then it does not, according to (M)! But A(k,k) cannot stop either, since by (K), it is the same as Ck(k). Thus, our procedure A is incapable of ascertaining that this particular computation Ck(k) does not stop even though it does not. Moreover, if we know that A is sound, then we know that Ck(k) does not stop. Thus, we know something that A is unable to ascertain. It follows that A cannot encapsulate our understanding.

Modern mathematics contains much more than that, of course. It includes set theory, for example, created by Georg Cantor in 1874, and “foundations,” which another George, the Englishman George Boole, split off from classical logic in 1854, and in which the logical underpinnings of all mathematical ideas are studied. The traditional categories have also been enlarged to include big new topics—geometry to include topology, algebra to take in game theory, and so on. Even before the early nineteenth century there was considerable seepage from one area into another. Trigonometry, for example, (the word was first used in 1595) contains elements of both geometry and algebra. Descartes had in fact arithmetized and algebraized a large part of geometry in the seventeenth century, though pure-geometric demonstrations in the style of Euclid were still popular—and still are— for their clarity, elegance, and ingenuity.

Das 'Wesen' der Mathematik liegt 'gerade' in ihrer 'Freiheit

Pythagoras, should he want to continue on for a degree in modern-day mathematics, would have to learn to abide far more counterintuitive results than numbers that cannot be written as ratios between whole numbers. From the square root of −1, to Georg Cantor’s revelation of infinite domains infinitely more infinite than other infinite domains, to Kurt Gödel’s incompleteness theorems, mathematics has constantly displaced the borders between the conceivable and the inconceivable, and Pythagoras would be in for some long hours of awesome mind-blowing.

A false conclusion once arrived at and widely accepted is not easily dislodged and the less it is understood the more tenaciously it is held.

Infinity earned its rightful place as a legitimate mathematical concept towards the end of the nineteenth century. The two big players behind this were German mathematicians Georg Cantor and his champion David Hilbert. No longer just accepting infinity as a general notion but actually investigating it rigorously did not go down well. Contemporary mathematicians described Cantor as a 'corrupter of youth'. This had a detrimental effect on Cantor, who already suffered from depression. Thankfully, Hilbert saw the power of what Cantor had done. He described Cantor's work as 'the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity', and famously said, 'no one shall expel us from the Paradise that Cantor has created'.

Cantor began a practice, long associated with Vallee, of introducing new talent via radio. Gracie Allen made her first radio appearance with Cantor: Burns and Allen would occasionally be mentioned, only half-jokingly, as a Cantor “discovery,” but George Burns had his own grim version of that affair (see BURNS AND ALLEN). A more legitimate discovery was Harry Einstein. Cantor was in Boston in 1934 when he happened to hear, on a local radio station, a man doing a funny Greek dialect. Einstein was then the advertising director of Boston’s Kane Furniture Company. He had been dabbling radio for years and had created a character named Nick Parkyakakas, a comedy candidate for mayor who could be heard on WNAC Mondays and Fridays at 10:30. Cantor thought it the funniest Greek impersonation he had ever heard: by wire, he offered Einstein a slot on NBC, and the following Sunday Parkyakakas played to the nation for the first time.

In a speech at the 1939 New York World’s Fair, he attacked by name some of the nation’s most prominent advocates of right-wing politics. He was most vocal about Father Charles Coughlin, the “radio priest” whose pulpit of the air was seen by some as a major dispenser of racial disharmony and anti-Semitism. Cantor also denounced George Sylvester Viereck, a German-American poet, frequent contributor to Coughlin’s Social Justice magazine, and admitted admirer of Hitler and Mussolini.

Os homens chamarão você de corvo. Ele, de "Vossa Graça", Cantores elogiarão cada coisinha que ele fizer, enquanto os maiores do seus feitos passarão despercebidos. Diga-me que nada disso o perturba.