Statistics. Evolution. Montyhall. Pi. Chaos. The romantic's favorite mathematician. The reticent and relentlessly abstract logician Kurt Gödel might seem an unlikely candidate for popular appreciation. But that's what Rebecca Goldstein aims for in her new book Incompleteness, an account of Gödel's most famous theorem, which was announced 75 years ago this October. Goldstein calls Gödel's incompleteness theorem "the third leg, together with Heisenberg's uncertainty principle and Einstein's relativity, of that tripod of theoretical cataclysms that have been felt to force disturbances deep down in the foundations of the 'exact sciences.' " What is this great theorem?
Mathematicians, like other scientists, strive for simplicity; we want to boil messy phenomena down to some short list of first principles called axioms, akin to basic physical laws, from which everything we see can be derived. Then Gödel kicked the whole thing over. Gödel's incompleteness theorem says: P is not provable using the given axioms. What is it about Gödel's theorem that so captures the imagination? A circle with the center everywhere. Posted by Alexandre Borovik in Uncategorized. Trackback A collection of quotes: Hermes Trismegistus, “thrice-great Hermes” “God is an infinite sphere, the center of which is everywhere, the circumference nowhere.”
Book of the 24 Philosophers. Alain of Lille “God is an intelligible sphere, whose center is everywhere, and whose circumference is nowhere.” Pascal: “The whole visible world is only an imperceptible atom in the ample bosom of nature. Also apparently, “Let him contemplate all nature in its awful and finished magnificence; let him observe that splendid luminary, set forth as an eternal lamp to enlighten the universe; let him view the earth as a mere speck within the vast circuit described by that luminary; let him think with amazement, that this vast circuit itself is only a minute point , compared with that formed by the revolutions of the stars…All that we see in of the creation, is but an almost imperceptible streak in the vast expanse of the universe.
Like this: Like Loading... Detexify symbol classifier. Want a Mac app? Lucky you. The Mac app is finally stable enough. See how it works on Vimeo. Download the latest version here. Restriction: In addition to the LaTeX command the unlicensed version will copy a reminder to purchase a license to the clipboard when you select a symbol. You can purchase a license here: Buy Detexify for Mac If you need help contact mail@danielkirs.ch. What is this? Anyone who works with LaTeX knows how time-consuming it can be to find a symbol in symbols-a4.pdf that you just can't memorize. How do I use it? Just draw the symbol you are looking for into the square area above and look what happens! My symbol isn't found! The symbol may not be trained enough or it is not yet in the list of supported symbols. I like this. You could spare some time training Detexify. The backend server is running on Digital Ocean (referral link) so you can also reduce my hosting costs by using that referral link.
Why should I donate? Hosting of detexify costs some money. No. Yes. Mysterious number 6174. March 2006 Anyone can uncover the mystery The number 6174 is a really mysterious number. At first glance, it might not seem so obvious. But as we are about to see, anyone who can subtract can uncover the mystery that makes 6174 so special. Kaprekar's operation In 1949 the mathematician D. It is a simple operation, but Kaprekar discovered it led to a surprising result. When we reach 6174 the operation repeats itself, returning 6174 every time. We reached 6174 again! A very mysterious number... When we started with 2005 the process reached 6174 in seven steps, and for 1789 in three steps.
Only 6174? The digits of any four digit number can be arranged into a maximum number by putting the digits in descending order, and a minimum number by putting them in ascending order. 9 ≥ a ≥ b ≥ c ≥ d ≥ 0 and a, b, c, d are not all the same digit, the maximum number is abcd and the minimum is dcba. which gives the relations for those numbers where a>b>c>d. For three digit numbers the same phenomenon occurs. And. Erdos article. By Charles Krauthammer Washington Post Writers Group WASHINGTON - One of the most extraordinary minds of our time has "left. " "Left" is the word Paul Erdos, a prodigiously gifted and productive mathematician, used for "died. " "Died" is the word he used to signify "stopped doing math.
" It wasn't just his vocabulary that was eccentric. He had no home, no family, no possessions, no address. Erdos traveled with two suitcases, each half-full. He seemed sentenced to a life of solitariness from birth, on the day of which his two sisters, age 3 and 5, died of scarlet fever, leaving him an only child, doted upon and kept at home by a fretful mother. But in reality he did: hundreds of scientific collaborators and 1,500 mathematical papers produced with them. Mathematicians tend to bloom early and die early. Erdos didn't. Erdos was unusual in yet one other respect.
Not so Erdos. That sociability sets him apart from other mathematical geniuses. Erdos didn't just share his genius. Intro to group theory. Representing complex numbers as 2×2 matrices. Solving Fermat: Andrew Wiles. Posted 11.01.00 NOVA Andrew Wiles devoted much of his career to proving Fermat's Last Theorem, a challenge that perplexed the best minds in mathematics for 300 years. In 1993, he made front-page headlines when he announced a proof of the problem, but this was not the end of the story; an error in his calculation jeopardized his life's work. In this interview, Wiles recounts how he came to terms with the mistake, and eventually went on to achieve his life's ambition.
Anyone who thinks that mathematics doesn't involve passion and emotion should hear directly from Andrew Wiles. Enlarge Photo credit: © WGBH Educational Foundation A childhood dream NOVA: Many great scientific discoveries are the result of obsession, but in your case that obsession has held you since you were a child. ANDREW WILES: I grew up in Cambridge in England, and my love of mathematics dates from those early childhood days. Who was Fermat and what was his Last Theorem? x2 + y2 = z2 and x3 + y3 = z3 He did more than that. Yes. Cr.yp.to. Tagged union. Mathematically, tagged unions correspond to disjoint or discriminated unions, usually written using +. Given an element of a disjoint union A + B, it is possible to determine whether it came from A or B.
If an element lies in both, there will be two effectively distinct copies of the value in A + B, one from A and one from B. In type theory, a tagged union is called a sum type. Notations vary, but usually the sum type comes with two introduction forms and has type have type under the assumptions respectively, then the term . An enumerated type can be seen as a degenerate case: a tagged union of unit types. A tagged union can be seen as the simplest kind of self-describing data format. Advantages and disadvantages[edit] The primary advantage of a tagged union over an untagged union is that all accesses are safe, and the compiler can even check that all cases are handled.
The main disadvantage of tagged unions is that the tag occupies space. Examples[edit] which corresponds to this tree: World's shortest explanation of Gödel's theorem. World's shortest explanation of Gödel's theorem A while back I started writing up an article titled "World's shortest explanation of Gödel's theorem". But I didn't finish it, and later I encountered Raymond Smullyan's version, which is much shorter anyway. So here, shamelessly stolen from Smullyan, is the World's shortest explanation of Gödel's theorem.
We have some sort of machine that prints out statements in some sort of language. It needn't be a statement-printing machine exactly; it could be some sort of technique for taking statements and deciding if they are true. But let's think of it as a machine that prints out statements. In particular, some of the statements that the machine might (or might not) print look like these: For example, NPR*FOO means that the machine will never print FOOFOO. Now, let's consider the statement NPR*NPR*.
Either the machine prints NPR*NPR*, or it never prints NPR*NPR*. If the machine prints NPR*NPR*, it has printed a false statement. Hope this helps! The Philosopher's Game. This file is a transcription of a 1563 translation by William Fulke (or Fulwood -- the sources disagree) of Boissiere's 1554/56 description of Rythmomachy. It is entry 15542a in the Short Title Catalog of Pollard and Redgrave, and on Reel 806 of the corresponding microfilm collection. Annotation will occur occasionally throughout; they will appear in square brackets and italics, [like this]. Spelling will be erratic; I'm transcribing quickly, so I will often be modernizing the spelling, but will leave original spelling whenever I consider there to be doubt about the meaning.
I also will sometimes modernize the punctuation and paragraph breaks in the interests of readability. This is not intended to serve as a definitive critical edition, merely a working copy, good enough to understand the game. My thanks to Peter Mebben, who pointed me in the direction of this source, and provided the first draft of the dedication. [Page] The Lord Robert Duddedlye. For why? Of diverse kyndes of playinge. Rithmomachy. Rithmomachy (or Rithmomachia, also Arithmomachia, Rythmomachy, Rhythmomachy, or sundry other variants; sometimes known as The Philosophers' Game) is a highly complex, early European mathematical board game.
The earliest known description of it dates from the eleventh century. A literal translation of the name is "The Battle of the Numbers". The game is much like chess, except most methods of capture depend on the numbers inscribed on each piece. It has been argued that between the twelfth and sixteenth centuries, "rithmomachia served as a practical exemplar for teaching the contemplative values of Boethian mathematical philosophy, which emphasized the natural harmony and perfection of number and proportion. History[edit] Very little, if anything, is known about the origin of the game. The name, which appears in a variety of forms, points to a Greek origin, the more so because Greek was little known at the time when the game first appeared in literature. Hand-made Rithmomachy set R. This week in mathematical findings. December 5, 2009 John Baez A while back, my friend Dan Christensen drew a picture of all the roots of all the polynomials of degree at most 5 with integer coefficients ranging from -4 to 4: 1) Dan Christensen, Plots of roots of polynomials with integer coefficients, 2) John Baez, The beauty of roots, Click on the picture for bigger view.
You can see lots of fascinating patterns here, like how the roots of polynomials with integer coefficients tend to avoid integers and roots of unity - except when they land right on these points! Now you see beautiful feathers surrounding the blank area around the point 1 on the real axis, a hexagonal star around exp(i π / 6), a strange red curve from this point to 1, smaller stars around other points, and more.... People should study this sort of thing! But based on the above picture, there seem to be a lot of interesting conjectures to make about this set as d → ∞ for fixed n. 4) Eric W. And. 100 Incredible Open Lectures for Math Geeks. While many math geeks out there may have been teased for their love of numbers, it’s math that makes the world go round, defining everything from the economy to how the universe itself operates. You can indulge your love of mathematics in these great lectures and lecture series, which are a great diversion for those diligently working toward traditional or online master’s degree programs in mathematics.
Some are meant to review the basics and others will keep you on the cutting edge of what renowned researchers are doing in the field, but all will help you expand your knowledge and spend a few hours enjoying a topic you love. Basic Math These lectures cover some pretty basic mathematical issues that can be a great review or help younger math lovers get a handle on a subject. Such lectures are an excellent resource for students who are completing online degrees in applied mathematics with the intention of entering careers as mathematics educators in middle schools and high schools. Algebra. Surprises in Mathematics and Theory.
Gil Kalai is one of the great combinatorialists in the world, who has proved terrific results in many aspects of mathematics: from geometry, to set systems, to voting systems, to quantum computation, and on. He also writes one of the most interesting blogs in all of mathematics—Combinatorics and more; I strongly recommend it to you. Today I want to talk about surprises in mathematics. I claim that that there are often surprises in our field—if there were none it would be pretty boring. The field is exciting precisely because there are surprises, guesses that are wrong, proofs that eventually fall apart, and in general enough entropy to make the field exciting. The geometer Karol Borsuk asked in 1932: Can every convex body in be cut into pieces of smaller diameter? The shock, I think more than a surprise, came when Jeff Kahn and Kalai proved in 1993 that for large enough the answer was not .
Large enough, we need at least pieces. Pieces are enough. Is a fixed constant. Is really different from. Are Impossibility Proofs Possible? Alan Turing is of course famous for his machine based notion of what computable means—what we now call, in his honor, Turing Machines. His model is the foundation on which all modern complexity theory rests; it is hard to imagine what theory would be like without his beautiful model.
His proof that the Halting Problem is impossible for his machines is compelling precisely because his model of computation is so clearly “right.” Turing did many other things: his code-breaking work helped win the war; his work on AI is still used today; his ideas on game playing, biology, and the Riemann Hypothesis were ground breaking. Today I plan to talk about the role of negative results in science, which are results that say “you cannot do that.” I claim that there are very few meaningful negative results; even Turing’s result must be viewed with care.
You may know that Turing took his own life after being treated inhumanely by his own government. He signed the form, but he answered the question “No.” Real numbers as infinite decimals. One of the early objectives of almost any university mathematics course is to teach people to stop thinking of the real numbers as infinite decimals and to regard them instead as elements of the unique complete ordered field, which can be shown to exist by means of Dedekind cuts, or Cauchy sequences of rationals.
I would like to argue here that there is nothing wrong with thinking of them as infinite decimals: indeed, many of the traditional arguments of analysis become more intuitive when one does, even if they are less neat. Neatness is of course a great advantage, and I do not wish to suggest that universities should change the way they teach the real numbers. However, it is good to see how the conventional treatment is connected to, and grows out of, more `naive' ideas.
I shall illustrate this later with a discussion of the square root of two and the intermediate value theorem. Constructing the real numbers as infinite decimals. That defines the set I am constructing. 1. 2. Introduction to Matrix Algebra. Discrete Mathematics. Square root. Introducing the Surreal Numbers (Edited rerun) Mathematics: The only true universal language.
MathBin.net - Math pastebin for equation rendering. Pronunciation Guide for Mathematics. The Methodology of Mathematics. Robert Recorde invents the equals sign. On the Importance of Mathematics. James Tauber : Poincaré Project.