User marko amnell - MathOverflow

Did Pogorzelski claim to have a proof of Goldbach's Conjecture?

2010-09-10T15:33:12Z

In 1977, Henry Pogorzelski published what some believed was a claimed proof of Goldbach's Conjecture in Crelle's Journal (292, 1977, 1-12). His argument has not been accepted as a proof of Goldbach's Conjecture, but as far as I know it has not been shown that his argument is incorrect.

Pogorzelski's argument is said to depend on the "Consistency Hypothesis," the "Extended Wittgenstein Thesis," and "Church's Thesis." Pogorzelski has a Ph.D. in mathematics (his advisor was Raymond Smullyan).

Daniel Shanks says in Solved and Unsolved Problems in Number Theory (fourth edition, 1993) that: "It seems unlikely that (most) number-theorists will accept this as a proof [of Goldbach's Conjecture] but perhaps we should wait for the dust to settle before we attempt a final assessment." (page 222)

Did Pogorzelski claim to present a proof of Goldbach's Conjecture? If so, and this claimed proof has not been disproven after 33 years, I am curious why this would be the case, given that Shanks considers it important enough to mention in his book.

Answer by Marko Amnell for Your favorite surprising connections in Mathematics

2010-02-17T00:26:42Z

Paul Vojta's discovery of the unexpected parallels between value distribution theory (Nevanlinna theory) in complex analysis and Diophantine approximation in number theory. See, e.g., Vojta's paper "Recent Work on Nevanlinna Theory and Diophantine Approximation". Serge Lang and William Cherry discuss the matter in their book Topics in Nevanlinna Theory.

Answer by Marko Amnell for Looking for book with good general overview of math and its various branches

2010-01-11T12:45:46Z

Saunders MacLane, Mathematics: Form and Function. Very good overview of undergraduate mathematics, showing interconnections between different areas. As might be expected from one of the inventors of category theory, MacLane defends categories as a foundation for mathematics.

Answer by Marko Amnell for Cryptomorphisms

2010-02-19T23:51:51Z

Theodore Hailperin found a finite set of axioms for Quine's NF set theory. This finite axiomatization consists of a short list of particular instances of the NF axiom scheme of "stratefied comprehension." The advantage of Hailperin's alternate axiomatization is that it eliminates the necessity of referring to the concept of type in the definition of NF. See Hailperin's article "A set of axioms for logic" [Journal of Symbolic Logic, Volume 9, Issue 1 (1944), pp. 1-19].

Answer by Marko Amnell for Open source mathematical software.

2010-03-22T20:19:36Z

Michael Rubinstein's lcalc is a fast and easy-to-use program for calculating values of L-functions including the Riemann zeta function. It can be downloaded from:

http://pmmac03.math.uwaterloo.ca/~mrubinst/L_function_public/L.html

Answer by Marko Amnell for Which came first: the Fibonacci Numbers or the Golden Ratio?

2010-07-01T12:02:42Z

The book A Mathematical History of the Golden Number by Roger Herz-Fischler is an exhaustive study of nearly all references to the golden ratio, from the earliest times, and is available as a free e-book. As has been pointed out by others, the golden ratio is older than the Fibonacci numbers. On page 53, Herz-Fischler notes that a pentagram appears as "a pot mark on a jar" dating from 3100 BC in Egypt.

Answer by Marko Amnell for What is the history of the name "Chinese remainder theorem"?

2010-01-22T16:41:20Z

The book A History of Mathematics: An Introduction by Victor J. Katz says:

"...probably the most famous mathematical technique coming from China is the technique long known as the Chinese remainder theorem. This result was so named after a description of some congruence problems appeared in one of the first reports in the West on Chinese mathematics, articles by Alexander Wylie published in 1852 in the North China Herald, which were soon translated into both German and French and republished in European journals..." (page 222)

This seems to suggest that the name "Chinese Remainder Theorem" was introduced soon after Wylie's 1852 article.

But the book Historical Perspectives on East Asian Science, Technology, and Medicine, edited by Alan Kam-leung Chan, Gregory K. Clancey and Hui-Chieh Loy says:

"A. Wylie introduced the solution of Sun Zi's remainder problem (i.e. "Wu Bu Zhi Shu") and Da-Yan Shu to the West in 1852, and L. Matthiessen pointed out the identity of Qin Jiushao's solution with the rule given by C. F. Gauss in his Disquisitiones Arithmeticae in 1874. Since then it has been called the Chinese Remainder Theorem in Western books on the history of mathematics."

This is ambiguous, as it is not clear whether the author means that the name "Chinese Remainder Theorem" came into use after 1852 or after 1874. But the phrasing does suggest that the name came into use before 1929.

In 1881, Matthiessen published the following article:

L. Matthiessen. "Le problème des restes dans l'ouvrage chinois Swang-King de Sum-Tzi et dans l'ouvrage Ta Sen Lei Schu de Yihhing." Comptes rendus de l'Académie de Paris, 92 :291-294, 1881.

But does the name "Chinese Remainder Theorem" ("le théorème chinois des restes") appear in this article?

Answer by Marko Amnell for Experimental Mathematics

2010-02-09T23:55:41Z

I'm surprised that no one has mentioned the Mandelbrot set yet, arguably the most famous new mathematical object of the last 30 years, at least among the general public. Benoit Mandelbrot discovered it in 1979 as a result of computer experiments. He says that when he first saw it he was so surprised by its appearance that he thought it must be the result of a computer malfunction. In his book Fractals and Chaos, Mandelbrot argues that his discovery of the Mandelbrot set contributed to the revival of experimental mathematics and led to a general change in the attitude of mathematicians to experiments in mathematics. On page 25 he writes:

"The culture of mathematics during the 1960s and 1970s

Within that culture the Mandelbrot set could not have been discovered. Hence its discovery marked a historical departure. Today -- but not yesterday -- only a minority among mathematicians would agree with the opinion due to someone who did not discover that set, that the study of M reflects "a rather infantile and somewhat dull mathematical sensibility" (Brooks 1989)."

Answer by Marko Amnell for (Preferably rare) Audio/Video recordings of famous mathematicians?

2011-03-16T15:54:01Z

Bertrand Russell, co-author (with Alfred North Whitehead) of Principia Mathematica, interviewed on BBC television in 1959 (three parts):

http://www.youtube.com/watch?v=OziPcicgmbw

http://www.youtube.com/watch?v=TedtMmUq8ig

http://www.youtube.com/watch?v=L7I9pgqiLo0

Answer by Marko Amnell for A non-technical account of the Birch—Swinnerton-Dyer Conjecture

2011-03-12T12:33:12Z

There is a short non-technical description of the Birch and Swinnerton-Dyer Conjecture in Keith Devlin's book The Millennium Problems. See Chapter 6, pages 189-211. Devlin's exposition is meant for a broad audience and may be at the level you are looking for. He tries hard to illustrate the problem and starts by comparing the Conjecture to the old Greek problem of finding sides that are rational numbers for a triangle with an area that is a positive whole number. He then provides elementary descriptions of the group of rational points of an elliptic curve, the rank of the group, reduction mod p and the Hasse-Weil L-function L(E,s).

Answer by Marko Amnell for Books you would like to read (if somebody would just write them...)

2011-02-02T08:57:37Z

I would like to read a comprehensive, step-by-step introduction to the Langlands Programme written for non-experts. An Introduction to the Langlands Program (edited by Joseph Bernstein and Stephen Gelbart) is good, but it is a collection of articles, not a textbook or monograph. Stephen Gelbart's "An Elementary Introduction to the Langlands Program" (Bulletin of the AMS, Vol. 10, No. 2, 1984, pp. 177-219) has the right approach, but while quite long, is not a book-length treatment. David Nadler's excellent new article "The Geometric Nature of the Fundamental Lemma" is another example of the sort of expository approach I would like to see in a full-length book about the Langlands Programme.

Answer by Marko Amnell for Gosper's Mathematics

2010-10-05T16:08:36Z

Some of Gosper's results are studied, and proved, in the article "Pages from the Computer Files of R. William Gosper," by Mourad E. H. Ismail, Yu Takeuchi and Ruiming Zhang (Proceedings of the American Mathematical Society, Volume 119, Number 3, November 1993). Gosper's work is also discussed in Wolfram Koepf's book Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. See also Wikipedia's list of hypergeometric identities.

Answer by Marko Amnell for Gossip about Grothendieck and distributive lattices

2010-09-19T10:44:18Z

Gian-Carlo Rota made a related comment in his article "The Many Lives of Lattice Theory" (Notices of the AMS, Volume 44, Number 11, December 1997, p. 1442):

"Dedekind outlined the program of studying the ideals of a commutative ring by lattice-theoretic methods, but the relevance of lattice theory in commutative algebra was not appreciated by algebraists until the sixties, when Grothendieck demanded that the prime ideals of a ring should be granted equal rights with maximal ideals. Those mathematicians who knew some lattice theory watched with amazement as the algebraic geometers of the Grothendieck school clumsily reinvented the rudiments of lattice theory in their own language. To this day lattice theory has not made much of a dent in the sect of algebraic geometers; if ever it does, it will contribute new insights. One elementary instance: the Chinese remainder theorem. Necessary and sufficient conditions on a commutative ring are known that insure the validity of the Chinese remainder theorem. There is, however, one necessary and sufficient condition that places the theorem in proper perspective. It states that the Chinese remainder theorem holds in a commutative ring if and only if the lattice of ideals of the ring is distributive."

I think this sheds some light on the question "What did Rota think would happen?" as Ben Webster interpreted the question.

Answer by Marko Amnell for Widely accepted mathematical results that were later shown wrong?

2010-08-19T22:47:41Z

In 1803, Gian Francesco Malfatti proposed a solution to the problem of how to cut out three circular columns of marble of maximal area from a triangular piece of stone. Malfatti's solution was three circles that are tangent to each other and to the sides of the triangle (known as Malfatti circles). His solution was believed to be correct until 1930, when it was shown that Malfatti circles are not always the best solution. Then, in 1967, Goldberg showed that Malfatti circles are never the optimal solution. Finally, in 1992, Zalgaller and Los' gave a complete solution to the problem.

http://en.wikipedia.org/wiki/Malfatti_circles

Answer by Marko Amnell for What are your favorite puzzles/toys for introducing new mathematical concepts to students?

2010-08-15T00:44:17Z

The Tangle is a plastic manipulative toy that can be used to introduce students to knot theory. This is what the Tangle looks like:

Colin Adams has published a book entitled Why Knot: An Introduction to the Mathematical Theory of Knots with Tangle.

The publisher's blurb says: "Each copy of Why Knot? is packaged with a plastic manipulative called the Tangle®. Adams uses the Tangle because 'you can open it up, tie it in a knot and then close it up again.' The Tangle is the ultimate tool for knot theory because knots are defined in mathematics as being closed on a loop. Readers use the Tangle to complete the experiments throughout the brief volume."

The Tangle that is included with the book is much longer than the one shown in the photograph above, so it can be bent to create fairly complicated knots.

Answer by Marko Amnell for What are some famous rejections of correct mathematics?

2010-02-04T04:35:56Z

Gauss essentially invented the Fast Fourier Transform in 1805, but the importance of his work was not understood for a century.

"A 1965 paper by John Tukey and John Cooley [2] is generally credited as the starting point for modern usage of the FFT. However, a paper by Gauss published posthumously in 1866 [3] (and dated to 1805) contains indisputable use of the splitting technique that forms the basis of modern FFT algorithms.

"Gauss was interested in the problem of computing accurate asteroid orbits from observations of their positions. His paper contains 12 data points on the position of the asteroid Pallas, through which he wished to interpolate a trigonometric polynomial with 12 coefficients. Instead of solving the resulting 12-by-12 system of linear equations by hand, Gauss looked for a shortcut. He discovered how to separate the equations into three subproblems that were much easier to solve, and then how to recombine the solutions to obtain the desired result. The solution is equivalent to estimating the DFT of the data with an FFT algorithm."

http://www.mathworks.com/access/helpdesk/help/techdoc/math/brentm1-1.html

"Recent studies of the history of the fast Fourier transform (FFT) algorithm, going back to Gauss[1], provide an example of exactly the opposite situation. After having been published and used over a period of 150 years without being regarded as having any particular importance, the FFT was re-discovered, developed extensively, and applied on electronic computers in 1965, creating a revolutionary change in the scale and types of problems amenable to digital processes."

http://www.springerlink.com/content/g5762llr5knw8505/

Answer by Marko Amnell for On the series 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ...

2009-11-08T18:04:47Z

Also, it is possible to prove that for any numbers a and b for which gcd(a,b)=1, the sum of all the 1/p's for p prime that satisfy p = a (mod b) diverges.

Comment by Marko Amnell

2010-10-25T04:24:38Z

There is a new book "The Prime Number Theorem" by G.J.O Jameson, which the author says is intended precisely to fill the gap of a book devoted entirely to introducing and proving the PNT. amazon.com/…

Comment by Marko Amnell

2010-09-14T15:56:03Z

deepdyve.com/lp/de-gruyter/… If you do not have an account, click the button "Rent for free."

Comment by Marko Amnell

2010-09-10T19:05:19Z

Thanks. That is very informative, but it raises the further question of why people such as Daniel Shanks believed that Pogorzelski had claimed to have a proof of Goldbach's Conjecture.

Comment by Marko Amnell

2010-09-10T19:01:36Z

I have changed the title and rewritten the question. I do not want to change it too much, however, or the comments will not make much sense.

Comment by Marko Amnell

2010-09-10T16:08:18Z

@Cam: Good point. I added the information to the question.

Comment by Marko Amnell

2010-09-10T16:03:57Z

I replaced the word "false" with "incorrect" when describing the purported proof.

Comment by Marko Amnell

2010-09-10T15:55:36Z

Pogorzelski published his purported proof in Crelle's Journal (292, 1977, 1-12) and has a Ph.D. in mathematics (his advisor was Raymond Smullyan). If his claimed proof has not been disproven after 33 years, I am curious why this would be the case, given that Shanks considers it important enough to mention in his book.

Comment by Marko Amnell

2010-08-26T07:50:16Z

This is not a disagreement with what Carl Mummert says but it is worth remembering that when Zermelo first proposed his axioms for set theory, there was considerable scepticism that they really would avoid contradictions. People like Bertrand Russell, Philip Jourdain and Henri Poincaré criticised his axioms. Russell wrote that "I suspect that his axioms will not really avoid contradictions, i.e., I suspect new contradictions could be manufactured specially designed to be consistent with his axioms." [quoted on p. 91 of Ebbinghaus's biography of Zermelo tinyurl.com/2fskff7 ]

Comment by Marko Amnell

2010-08-15T18:10:58Z

Thanks for the correction. The paper by Brooks and Matelski is available online. The picture of the Mandelbrot set is on the second last page. math.harvard.edu/archive/118r_spring_05/docs/…

Comment by Marko Amnell

2010-08-15T17:05:54Z

(continued): which suggests that prehistoric cultures may have had some familiarity with the golden ratio. In any case, the question was just about which came first, the golden ratio or the Fibonacci numbers. The golden ratio was definitely understood in the ancient world. The first unequivocal mention of it appears to be by Euclid in The Elements. There is evidence that the Fibonacci numbers were understood in ancient India, with Wikipedia citing a date as early as 200 B.C. That is 100 years after Euclid, but close enough that one could claim the question is not settled.

Comment by Marko Amnell

2010-08-15T16:52:12Z

The section I quoted is at the bottom of page 53; the link I gave should direct you to that page. You have to scroll down the page to see it. You are right that the mere use of a pentagram does not prove that the ancient Egyptians were aware of the golden ratio and its significance. But if you read further in that section, entitled "Examples Before Pythagoras (before c. -550)," the author lists many examples from various times and geographical locations, that show at least some understanding of the golden ratio. I think the earliest evidence he cites is from 4,500 B.C. in Palestine (page 57),

Comment by Marko Amnell

2010-07-11T12:18:24Z

The Scandinavian is probably Sören Illman. Well, Finland is actually not part of Scandinavia, but Illman did clarify the work of Gleason, Montgomery and Zippin. The 1998 Kluwer book is probably "Parametric Lie Group Actions On Global Generalised Solutions Of Nonlinear Pdes: Including A Solution To Hilbert's Fifth Problem" by Elemer E. Rosinger. tinyurl.com/2wapl44 I believe the author is a contributor to MathOverflow so hopefully he will see your question and answer it.

Comment by Marko Amnell

2010-03-17T20:01:12Z

From Wikipedia: "In a manuscript of De revolutionibus, Copernicus wrote, 'It is likely that ... Philolaus perceived the mobility of the earth, which also some say was the opinion of Aristarchus of Samos,' but later struck out the passage and omitted it from the published book." en.wikipedia.org/wiki/…

Comment by Marko Amnell

2010-03-17T18:02:02Z

@Ilya: Sorry, I did not see your comment until after I had posted my comment. To answer your question: Yes, it is certain that Copernicus was aware of Aristarchus's priority because the original draft of his 1543 book has survived and it included a passage that refers to Aristarchus, which Copernicus later crossed out so as not to compromise the originality of his theory.

Comment by Marko Amnell

2010-03-17T16:42:41Z

It is not quite true that before Copernicus and Kepler everyone placed the Earth at the centre of the system. The first person to place the Sun at the centre was Aristarchus of Samos, 1,800 years earlier. His system was rejected in favour of the geocentric model of Ptolemy and Aristotle, and then revived much later by Copernicus and Kepler.