Excerpt from GENE515
This partly answers a question asked of me when I was 21 years old, by my doctoral thesis advisor Oliver G. Selfridge. When I say "Man" I am echoing and older text, and not excluding Woman.

Excerpt from GENE515

What is Man, that he may know Number? What is Number that it may be known by Man?

As we are mathematicians, we are in the image of our creator, The Mathematician, who has other attributes beyond our comprehension, and is Transfinite.

He freely gives us this world, and the cosmos beyond, and the flora and fauna over which to be stewards, and our fellow human beings to love, which is in the image of His love, which is transfinite.

We have free will, and for those of use who choose to be mathematicians, he gives us the integers as toys, in which is His book coded.

We play with those toys, some of us in solitude, some of us playing together. And when we put aside childish things, behold, we still have the gift of Number, and they are more than first we knew.

Eureka!, and Aha!, and knowing what Mozart meant when he said that he did not write music, but it was already there and he plucked it from thin air as it blew past. And what Ramanujan said was given him by a Goddess, And what Gauss could see as a child, and Riemann in the looking glass of Primes, and Galois by candlelight in the brief hours before his fatal duel.

Euclid, alone, has looked on beauty bare. But we mathematicians today are not alone, far from it, cradled in the same Web woven of Number, binary and octal and hex, decimal and alphanumeric, vector and raster, and more in cables, trunks, and as wifi in the very air about us.

By knowing Number more deeply, we more deeply know ourselves, and our Creator.

Every word begins and ends with the empty word; the empty word begins and ends with itself.

(225 comments) - (Post a new comment)

Loving Euclid: Euclid alone has looked on beauty bare? magicdragon2 2006-03-04 07:14 am UTC (link)

"At the age of eleven, I began Euclid, with my brother as my tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world. From that moment until I was thirty-eight, mathematics was my chief interest and my chief source of happiness." -- Bertrand Russell, (1872-1970) The Autobiography of Bertrand Russell, (3 vols.) Allen & Unwin: London, 1967-1969. (Reply to this)

Andre Weil: Mathematical state of lucid exaltation (1) magicdragon2 2006-03-04 07:25 am UTC (link)

"Every mathematician worthy of the name has experienced ... the state of lucid exaltation in which one thought succeeds another as if miraculously... this feeling may last for hours at a time, even for days. Once you have experienced it, you are eager to repeat it but unable to do it at will, unless perhaps by dogged work..." Andre Weil, (6 May 1906 - 6 August 1998) The Apprenticeship of a Mathematician. Andre Weil in Wikipedia. Andre Weil was one of the great mathematicians of the 20th century, whether measured by his research work, its influence on future work, exposition or breadth. He is known for his foundational work in number theory and algebraic geometry. He was a founding member, and de facto the early leader, of the influential Bourbaki group. The philosopher Simone Weil was his sister. Born in Paris to Alsatian parents who fled the annexation of Alsace-Lorraine to Germany, he studied in Paris, Rome and Gottingen and received his doctorate in 1928. He spent two academic years at Aligarh Muslim University from 1930. Sanskrit literature was a life-long interest of his. He had a one-year position in Marseilles, and then spent six years in Strasbourg. He married Eveline in 1937. A conscientious objector by conviction, and a Jew, Weil was in Finland when World War II broke out; he had been travelling in Scandinavia since April 1939. Eveline returned to France, but he did not. A famous anecdote was confirmed in his autobiography: after having been arrested under suspicion of espionage in Finland, when the USSR attacked on 30 November 1939, he was saved from being shot only by the intervention of Rolf Nevanlinna. He returned to France via Sweden and the United Kingdom, and was detained at Le Havre in January 1940. He was charged with failure to report for duty, and was imprisoned in Le Havre and then Rouen. It was there in the military prison in Bonne-Nouvelle, a district of Rouen, from February to May, that he did the work that made his reputation. He was sent to trial on May 3, 1940. Sentenced to five years, he asked to be sent to a military unit instead, and joined a regiment in Cherbourg. After the fall of France, he met up with his family in Marseilles, where he arrived by sea. He then went to Clermont-Ferrand, where he managed to join Eveline, who had been in the German-occupied region. In January 1941 they left by sea from Marseilles, and sailed to New York. During the war, Weil went to the United States where he was supported by the Rockefeller Foundation and Guggenheim Foundation. He was at the Universidade de Sao Paulo for two years from 1945, where he spent much time with Oscar Zariski. He taught at the University of Chicago from 1947 to 1958 before settling at the Institute for Advanced Study in Princeton. (Reply to this)

Andre Weil: Mathematical state of lucid exaltation (2) magicdragon2 2006-03-04 07:26 am UTC (link)

He made substantial contributions in many areas, the most important being profound connections between algebraic geometry and number theory. This began in his doctoral work leading to the Mordell-Weil theorem (1928, and shortly applied in Siegel's theorem on integral points). Mordell's theorem had an ad hoc proof; Weil began the separation of the infinite descent argument into two types of structural approach, by means of height functions for sizing rational points, and by means of Galois cohomology, not to be named as that for two more decades. Both aspects have steadily developed into substantial theories. Among his major accomplishments were the 1940 proof, while in prison, of the Riemann hypothesis for local zeta-functions, and his subsequent laying of proper foundations for algebraic geometry to support that result (from 1942 to 1946, most intensively). By modern standards his claim to have a proof had a very easy ride, but wartime conditions were one factor, and the fact that the German experts made little or no comment another. The so-called Weil conjectures were hugely influential from around 1950; they were later proved by Bernard Dwork, Alexander Grothendieck, Michael Artin, and Pierre Deligne, who completed the most difficult step in 1973. He had introduced the adele ring in the late 1930s, following Claude Chevalley's lead with the ideles, and given a proof of the Riemann-Roch theorem with them (a version appeared in his Basic Number Theory in 1967). His 'matrix divisor' (vector bundle avant le jour) Riemann-Roch theorem from 1938 was a very early anticipation of later ideas such as moduli spaces of bundles. The Weil conjecture on Tamagawa numbers proved resistant for many years. Eventually the adelic approach became basic in automorphic representation theory. He picked up another credited Weil conjecture, around 1970, which later under pressure from Serge Lang became known as the Shimura-Taniyama conjecture based on the presentation of the basic ideas at the 1955 Nikko conference. His attitude towards conjectures struck many in the field as oblique; he wrote that one should not dignify a guess as a conjecture lightly, and in the Shimura-Taniyama case the evidence was only there after extensive computational work. Other significant results were on Pontryagin duality and differential geometry. He introduced the concept of uniform space in general topology. His work on sheaf theory hardly appears in his published papers, but correspondence with Henri Cartan in the late 1940s proved most influential. His discovery that the so-called Weil representation, previously introduced in quantum mechanics by Irving Segal and Shale, gave a proper framework for understanding the classical theory of quadratic forms, and was also a beginning of a substantial development connecting representation theory and theta-functions. His books, unusually for mathematics, had an important influence on research. (In one major case possibly negative: Alexander Grothendieck is supposed to have complained of the 'aridity' of Weil's Foundations of Algebraic Geometry. This is a good joke, if unintentional.) Through Bourbaki's writings and seminars, Weil's ideas can also be traced in the mainstream of post-war mathematics. More trivially, he invented the notation [O with a slash through it] for the empty set (q.v.). (Reply to this)

Andre Weil: Mathematical state of lucid exaltation (3) magicdragon2 2006-03-04 07:27 am UTC (link)

Andre Weil is not be confused with Hermann Weyl, who helped Weil receive a Guggenheim fellowship in 1944; or with Andrew Wiles, another famous mathematician who, like Weil, has done important work in elliptic curves. Pronunciation: "Weil" is vay, while "Weyl" is vile, and "Wiles" is just wiles. Books: Arithmetique et geometrie sur les varietes algebriques (1935) Sur les espaces a structure uniforme et sur la topologie generale (1937) L'integration dans les groupes topologiques et ses applications (1940) Foundations of Algebraic Geometry (1946) Sur les courbes algebriques et les varietes qui s'en deduisent (1948) Varietes abeliennes et courbes algebriques (1948) Introduction a l'etude des varietes kahleriennes (1958) Discontinuous subgroups of classical groups (1958) Chicago lecture notes Basic Number Theory (1967) Dirichlet Series and Automorphic Forms, Lezioni Fermiane (1971) Lecture Notes in Mathematics, vol. 189, Essais historiques sur la theorie des nombres (1975) Elliptic Functions According to Eisenstein and Kronecker (1976) Oeuvres Scientifiques, Collected Works, three volumes (1979) Number Theory for Beginners (1979) with Maxwell Rosenlicht Adeles and Algebraic Groups (1982) Number Theory: An Approach Through History From Hammurapi to Legendre (1984) Souvenirs d'Apprentissage (1991) as The Apprenticeship of a Mathematician (1992) See also Weil cohomology Weil conjecture disambiguation page Weil conjectures Weil conjecture on Tamagawa numbers Weil distribution Weil divisor Siegel-Weil formula Weil group, Weil-Deligne group scheme Weil-Chatelet group Chern-Weil homomorphism Chern-Weil theory Hasse-Weil L-function Weil pairing Weil reciprocity law Weil representation Borel-Weil theorem De Rham-Weil theorem Mordell-Weil theorem. External links John J. O'Connor and Edmund F. Robertson. Andre Weil at the MacTutor archive. (Reply to this)

Sylvia Townsend Warner: math swoon, ecstasy, orgy magicdragon2 2006-03-04 07:31 am UTC (link)

"For twenty pages perhaps, he read slowly, carefully, dutifully, with pauses for self-examination and working out examples. Then, just as it was working up and the pauses should have been more scrupulous than ever, a kind of swoon and ecstasy would fall on him, and he read ravening on, sitting up till dawn to finish the book, as though it were a novel. After that his passion was stayed; the book went back to the Library and he was done with mathematics till the next bout. Not much remained with him after these orgies, but something remained: a sensation in the mind, a worshiping acknowledgment of something isolated and unassailable, or a remembered mental joy at the rightness of thoughts coming together to a conclusion, accurate thoughts, thoughts in just intonation, coming together like unaccompanied voices coming to a close." -- Sylvia Townsend Warner (1893-1978), Mr. Fortune's Maggot, 1927. (Reply to this) (Thread)

	the joy of suddenly discovering a hitherto unknown truth magicdragon2 2006-03-04 10:09 am UTC (link)
	"The joy of suddenly learning a former secret and the joy of suddenly discovering a hitherto unknown truth are the same to me -- both have the flash of enlightenment, the almost incredibly enhanced vision, and the ecstasy and euphoria of released tension." -- Paul R. Halmos, I Want to be a Mathematician, Washington: MAA Spectrum, 1985. (Reply to this) (Parent)

	Darwin wrote: "Mathematics... like a new sense" magicdragon2 2006-03-04 07:36 am UTC (link)
	Mathematics seems to endow one with something like a new sense. -- Charles Darwin, in N. Rose (ed.), Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988. (Reply to this)

J.J. Sylvester: higher and higher states of conscious magicdragon2 2006-03-04 07:43 am UTC (link)

"... there is no study in the world which brings into more harmonious action all the faculties of the mind than [Mathematics], ... or, like this, seems to raise them, by successive steps of initiation, to higher and higher states of conscious intellectual being...." -- J. J. Sylvester, (3 September 1814 London - 15 March 1897 Oxford) Presidential Address to British Association, 1869 [for full citation, see below]. James Joseph Sylvester in Wikipedia James Joseph Sylvester was an English mathematician. Sylvester was born "James Joseph" but adopted the surname "Sylvester" when his older brother did so. His brother was emigrating to the United States, a country which at that time required all immigrants to have a given name, a middle name, and a surname. Sylvester began his study of mathematics at St John's College, Cambridge in 1831 and was ranked second in Cambridge's famous mathematical examinations, the tripos. Yet he did not obtain a degree, because graduates at that time were required to state their acceptance of the Thirty-Nine Articles of the Church of England, and Sylvester declined to do so because he had been raised Jewish. In 1838 Sylvester became professor of natural philosophy at University College London UCL. In 1841, he was awarded a BA and an MA by Trinity College, Dublin. In the same year he moved to the United States to become a professor at the University of Virginia but soon returned to England. On his return to England he studied law, alongside fellow British lawyer/mathematician Arthur Cayley, with whom he made significant contributions to matrix theory while working as an actuary. One of his private pupils was Florence Nightingale. He did not obtain a position teaching university mathematics until 1855, when he was appointed professor of mathematics at the Royal Military Academy at Woolwich, from which he retired in 1869, because the compulsory retirement age was 55. In 1877 Sylvester again crossed the Atlantic Ocean to become the inaugural professor of mathematics at the new Johns Hopkins University in Baltimore. In 1878 he founded the American Journal of Mathematics, the first American mathematical journal. In 1883, he returned to England to take up the Savilian Professor of Geometry at Oxford University. He held this chair until his death, although in 1892 the University appointed a deputy professor to the same chair. Sylvester invented a great number of mathematical terms such as the totient function φ(n). His collected scientific work fills four volumes. In 1880, the Royal Society of London awarded Sylvester the Copley Medal, its highest award for scientific achievement; in 1901, it instituted the Sylvester Medal in his memory, to encourage mathematical research. Sylvester House, an undergraduate dormitory at Johns Hopkins, is named in his honor. Bibliography Primary: 1904-10. Collected Mathematical Papers in 4 vols. Edited by H. F. Baker. New York. 1839. "On rational derivation from equations of coexistence, that is to say, a new and extended theory of elimination, Part I," Philos. Mag. 15: 428-435. 1857. "On the partition of numbers," Quart. J. Math. I: 141-152. 1869. "Presidential address to Section A of the British Association" in Ewald, William B., ed., 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford Uni. Press: 511-22. 1897. "Outlines of seven lectures on the partition of numbers," Proc. Lond. Math. Soc. 28: 33-96. Secondary: Franklin, Address Commemorative of Sylvester, (Baltimore, 1897) See also: Chebyshev-Sylvester constant Coin problem Sylvester's identity Sylvester's sequence Sylvester matrix (resultant matrix) Sylvester's theorem Sylvester-Gallai theorem Sylvester's law of inertia External links: MacTutor archive: J. J. Sylvester (Reply to this)

	Thomas Mann, mathematics versus lusts of the flesh magicdragon2 2006-03-04 07:49 am UTC (link)
	"I tell them that if they will occupy themselves with the study of mathematics they will find in it the best remedy against the lusts of the flesh." -- Thomas Mann, (6 June 1875-12 Aug 1955) The Magic Mountain. 1927. (Reply to this)

	William Leybourn: Pleasure of "Mathematicks" magicdragon2 2006-03-04 07:53 am UTC (link)
	"But leaving those of the Body, I shall proceed to such Recreation as adorn the Mind; of which those of the Mathematicks are inferior to none." -- William Leybourn, (1626-1700) Pleasure with Profit, 1694. William Leybourn (1626-1719) (alias Oliver Wallingby): The Compleat Surveyor (Reply to this)

	Kant: "pure reason... without the aid of experience" magicdragon2 2006-03-04 08:18 am UTC (link)
	"The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience." -- Emmanuel Kant, (22 Apr 1724-12 Feb 1804) The Mathematical Intelligencer, v. 13, no. 1, Winter 1991. (Reply to this) (Thread)

	Sherlock: When you have eliminated the impossible magicdragon2 2006-03-04 09:25 am UTC (link)
	"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." -- Sherlock Holmes, The Sign of the Four (Reply to this) (Parent)

Math: system of the world magicdragon2 2006-03-04 08:21 am UTC (link)

"It is true that Fourier had the opinion that the principal aim of mathematics was public utility and explanation of natural phenomena; but a philosopher like him should have known that the sole end of science is the honor of the human mind, and that under this title a question about numbers is worth as much as a question about the system of the world." Carl Jacobi, in N. Rose, Mathematical Maxims and Minims, Raleigh, NC: Rome Press Inc., 1988. The System of the World can refer to several things: The System of the World (novel), a 2005 book by Neal Stephenson; The third book of Isaac Newton's Philosophiae Naturalis Principia Mathematica (Reply to this)

Mathematician washes his hands of applications magicdragon2 2006-03-04 08:24 am UTC (link)

"If he is consistent a man of the mathematical school washes his hands of applications. To someone who wants them he would say that the ideal system runs parallel to the usual theory: 'If this is what you want, try it: it is not my business to justify application of the system; that can only be done by philosophizing; I am a mathematician.'" -- J. E. Littlewood, (9 June 1885-6 Sep 1977) A Mathematician's Miscellany, Methuen Co. Ltd., 1953. (Reply to this) (Thread)

	Hadamard: Practical application ... by not looking for it magicdragon2 2006-03-04 09:01 am UTC (link)
	"Practical application is found by not looking for it, and one can say that the whole progress of civilization rests on that principle." -- Jacques Hadamard, in H. Eves, Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972. (Reply to this) (Parent)

Pure mathematics, may it never be of any use to anyone - magicdragon2, 2006-03-04 09:07 am UTC

Hardy: Pure mathematics = more useful than applied - magicdragon2, 2006-03-04 09:58 am UTC

nothing mysterious in applicability of mathematics - magicdragon2, 2006-03-04 09:37 am UTC

Weaving and geometry in [only] two or three dimensions magicdragon2 2006-03-04 08:56 am UTC (link)

"Mathematicians have long since regarded it as demeaning to work on problems related to elementary geometry in two or three dimensions, in spite of the fact that it it precisely this sort of mathematics which is of practical value." -- Branko Grunbaum (1926 - ), [University of Washington] and G. C. Shephard, Handbook of Applicable Mathematics, Vol V: Combinatorics and Geometry, Wiley, Chichester, 1985. "UW mathematician Branko Grunbaum has taken his inspiration from decorative patterns used in arts and crafts to advance basic theories of geometry. Since the early 1980s, Grunbaum and colleague G. C. Shephard of the University of East Anglia, Norwich, England, have pioneered new ways of analyzing the intricate patterns found in tilings and textiles and have elaborated what they call a 'theory of patterns.'" "The results of this work are far-ranging. They have important ramifications for divisions of mathematics including group theory, combinatorics, geometry and topology; they may benefit other technical fields such as crystallography and engineering; and they may find use in the fields of design, art, and anthropology." "Grunbaum and Shephard observe that the beginnings of geometry were, in ancient times, stimulated by practical problems of building, surveying, and decorating. 'From the beginnings of civilization, peoples of every culture have manufactured and used objects decorated with repeating geometric patterns. This is still true today and everywhere patterns of many kinds can be seen all around us,' they write. 'Moreover, repeating patterns frequently arise naturally in many areas of science and engineering, and their investigation has proved a useful tool in such diverse areas as crystallography and the ethnological classification of primitive artifacts.'" "What is surprising, they note, is that there had been relatively little work on these patterns from a theoretical standpoint. The practice of weaving is a typical example. 'Weaving is one of the oldest activities of mankind and so it is hardly surprising that there exists a vast literature on the subject. But this literature is almost entirely concerned with the practical aspects of weaving; any treatment of the theoretical problem of designing fabrics with prescribed mathematical properties is conspicuously absent,' note the researchers." "In 1980, Grunbaum and Shephard published 'Satins and Twills: An Introduction to the Geometry of Fabrics,' which revealed subtle problems in combinatorics and geometry. Traversing uncharted territory, they had to establish new concepts with an entirely new vocabulary to describe textile patterns." "Much the same situation was encountered with repetitive ornamentation. Previous studies had been restricted to analyzing the patterns with the well-established tool of symmetry groups. But Grunbaum found them to be 'very coarse tools for the characterization and description of repeating patterns.' Much finer classifications were developed by the team; the concepts are elaborated in Grunbaum and Shephard's detailed text, Tilings and Patterns." "The researchers were able to elucidate 'certain mysterious aspects' of interlace patterns frequently found in Islamic and Moorish art. They verified that, despite the complexity of the designs, most of the interlaces are formed by strands of a small number of shapes, often just a single shape stretching over many repeats of the design. A plausible explanation is that the early artisans used stencils to draw the patterns; for practical reasons, the stencils were made as small as possible for a given pattern, and they may have consisted of just one translational repeat unit." "Other work by Grunbaum contributed to the mathematical understanding of aperiodic tilings, in which the constituent units repeat but the pattern lacks symmetry over the long range. This work subsequently became of interest to solid matter physics because of the discovery of actual substances with this type of aperiodic symmetry, called quasicrystals; and it was applied to an analysis of decorations of ancient Peruvian fabrics that are not covered by the usual symmetry groups." (Reply to this) (Thread)

mathematician compared to a designer of garments magicdragon2 2006-03-04 09:05 am UTC (link)

"The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity for clothing such creatures, but this was long ago; to this day a shape will occasionally appear which will fit into the garment as if the garment had been made for it. Then there is no end of surprise and delight." -- Dantzig George Dantzig (1914-2005) or David van Dantzig (1900-1959) ? (Reply to this) (Parent)(Thread)

Dantzig: paradox of reality of the number concept - magicdragon2, 2006-03-04 09:11 am UTC

Winston S. Churchill on Mathematics magicdragon2 2006-03-04 09:09 am UTC (link)

"Some of my cousins who had the great advantage of University education used to tease me with arguments to prove that nothing has any existence except what we think of it. ... These amusing mental acrobatics are all right to play with.They are perfectly harmless and perfectly useless. ... I always rested on the following argument... We look up to the sky and see the sun. Our eyes are dazzled and our senses record the fact. So here is this great sun standing apparently on no better foundation than our physical senses. But happily there is a method, apart altogether from our physical senses, of testing the reality of the sun. It is by mathematics. By means of prolonged processes of mathematics, entirely separate from the senses, astronomers are able to calculate when an eclipse will occur. They predict by pure reason that a black spot will pass across the sun on a certain day. You go and look, and your sense of sight immediately tells you that their calculations are vindicated. So here you have the evidence of the senses reinforced by the entirely separate evidence of a vast independent process of mathematical reasoning. We have taken what is called in military map-making a cross bearing.' ... When my metaphysical friends tell me that the data on which the astronomers made their calculations, were necessarily obtained originally through the evidence of the senses, I say, 'no.' They might, in theory at any rate, be obtained by automatic calculating-machines set in motion by the light falling upon them without admixture of the human senses at any stage. When it is persisted that we should have to be told about the calculations and use our ears for that purpose, I reply that the mathematical process has a reality and virtue in itself, and that once discovered it constitutes a new and independent factor. I am also at this point accustomed to reaffirm with emphasis my conviction that the sun is real, and also that it is hot--in fact hot as Hell, and that if the metaphysicians doubt it they should go there and see." Winston S. Churchill, My Early Life, Fontana, London, 1972, pp 123-124. (Reply to this) (Thread)

Winston S. Churchill: Mathematics - that I saw it all magicdragon2 2006-03-04 10:08 am UTC (link)

"I had a feeling once about Mathematics - that I saw it all. Depth beyond depth was revealed to me - the Byss and Abyss. I saw - as one might see the transit of Venus or even the Lord Mayor's Show - a quantity passing through infinity and changing its sign from plus to minus. I saw exactly why it happened and why the tergiversation was inevitable but it was after dinner and I let it go." -- [Sir] Winston Spencer Churchill (1874-1965) in H. Eves, Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988. (Reply to this) (Parent)

	"... what Gauss could see as a child..." magicdragon2 2006-03-04 09:30 am UTC (link)
	"If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries." -- Karl Friedrich Gauss (1777-1855) In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956. p. 326. (Reply to this)

Blake and Pope on Mathematics magicdragon2 2006-03-04 09:34 am UTC (link)

"God forbid that Truth should be confined to Mathematical Demonstration!" -- William Blake, Notes on Reynold's Discourses, c. 1808. "What is now proved was once only imagin'd." -- William Blake, The Marriage of Heaven and Hell, 1790-3. "See skulking Truth to her old cavern fled, Mountains of Casuistry heap'd o'er her head! Philosophy, that lean'd on Heav'n before, Shrinks to her second cause, and is no more. Physic of Metaphysic begs defence, And Metaphysic calls for aid on Sense! See Mystery to Mathematics fly!" -- Alexander Pope, (1688-1744) In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956. (Reply to this)

generalization by abstraction magicdragon2 2006-03-04 09:44 am UTC (link)

"We lay down a fundamental principle of generalization by abstraction: 'The existence of analogies between central features of various theories implies the existence of a general theory which underlies the particular theories and unifies them with respect to those central features....'" -- E. H. Moore, (1862 - 1932) in H. Eves, Mathematical Circles Revisited, Boston: Prindle, Weber and Schmidt, 1971. (Reply to this) (Thread)

One should always generalize magicdragon2 2006-03-04 09:49 am UTC (link)

"Talk with M. Hermite. He never evokes a concrete image, yet you soon perceive that the more abstract entities are to him like living creatures." -- Jules Henri Poincare, (1854-1912) In G. Simmons, Calculus Gems, New York: McGraw Hill Inc., 1992. "We think in generalities, but we live in details." -- Alfred North Whitehead, W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966. "One should always generalize." (Man muss immer generalisieren) -- Carl Jacobi, In P. Davis and R. Hersh The Mathematical Experience, Boston: Birkhauser, 1981. (Reply to this) (Parent)

Whitehead, Bell, on abstraction - magicdragon2, 2006-03-04 09:57 am UTC

Math via Hard Work, Midnight Oil, yet disregard for rigor magicdragon2 2006-03-04 10:11 am UTC (link)

"Mathematical discoveries, small or great are never born of spontaneous generation They always presuppose a soil seeded with preliminary knowledge and well prepared by labour, both conscious and subconscious." -- Jules Henri Poincare, (1854-1912) "All great theorems were discovered after midnight." -- Adrian Mathesis, in H. Eves, Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988. "It was, no doubt, partially because of his very disregard for rigor that he was able to take conceptual steps which were inherently impossible to men of more critical genius." -- Rudoph E. Langer, [about Fourier] In P. Davis and R. Hersh The Mathematical Experience, Boston: Birkhauser, 1981. (Reply to this)

	Mathematics of Gone With the Wind magicdragon2 2006-03-04 10:19 am UTC (link)
	"...She knew only that if she did or said thus-and-so, men would unerringly respond with the complimentary thus-and-so. It was like a mathematical formula and no more difficult, for mathematics was the one subject that had come easy to Scarlett in her schooldays." -- Margaret Mitchell, Gone With the Wind. (Reply to this)

	Agatha Christie: the Mystery of Mathematics magicdragon2 2006-03-04 10:22 am UTC (link)
	"I continued to do arithmetic with my father, passing proudly through fractions to decimals. I eventually arrived at the point where so many cows ate so much grass, and tanks filled with water in so many hours I found it quite enthralling." -- Agatha Christie, An Autobiography. (Reply to this)

	angel of topology, devil of abstract algebra magicdragon2 2006-03-10 05:27 pm UTC (link)
	I am reminded by the lovely blog Ars Mathematica that: Hermann Weyl (a leading mathematicians of the twentieth century, and a man who early combined general relativity with the laws of electromagnetism) once said "In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics." (Reply to this)

Alain Badiou: 'mathematics is ontology.' magicdragon2 2006-03-12 08:37 pm UTC (link)

Alain Badiou Creates a Buzz With Views of Philosophy's Relevance to Other Disciplines By RICHARD BYRNE The Chronicle of Higher Education "... To reach that analysis of creation, however, requires the reader to navigate contemporary mathematics. Much of the alleged inaccessibility of Mr. Badiou's work is rooted in his reliance on set theory to discuss ontology, the branch of philosophy that deals with existence. "Indeed, Being and Event makes the striking claim that 'mathematics is ontology.' And chunks of the book are studded with equations and theorems that may frighten off the scholar who fled to the humanities to escape mathematics. "'It's a phobia,' grinned Mr. Badiou when Mr. Critchley brought up the topic of some scholars' resistance to the mathematical concepts that he employs. 'My goal is to change a phobia into love,' he said. And though the clusters of equations in Being and Event look complicated, Mr. Badiou's reliance on them is explained with little difficulty." (Reply to this)

Answer to everything: 42 [The Tao begot one...] magicdragon2 2006-03-27 07:56 pm UTC (link)

============== Laotse Tao te king Translated by Gia Fu Feng ============== Forty-two The Tao begot one. One begot two. Two begot three. And three begot the ten thousand things. The ten thousand things carry yin and embrace yang. They achieve harmony by combining these forces. Men hate to be "orphaned," "widowed," or "worthless," But this is how kings and lords describe themselves. For one gains by losing And loses by gaining. What others teach, I also teach; that is: "A violent man will die a violent death!" This will be the essence of my teaching. (Reply to this)

Life, the Universe and Zeta [asymptotically] magicdragon2 2006-03-27 08:01 pm UTC (link)

42 *IS* The answer to Life, the Universe and Zeta If you integrate the nth power of the absolute value of the Riemann zeta function on the the critical line between heights -T and T and divide by 2T, you will get a sort of nth moment on average. Random matrix theory predicts the growth of this function to be asymptotic to a "geometric factor" (coming from an integral over the unitary group) times the n^2 power of the logarithm of T. It turned out that the random matrix theory prediction is off by an "arithmetic" factor, so that the correct asymptotics is a(n)g(n) (log T)^(n^2) where g(n) is the geometric factor from above and a(n) is a rational number. The article Prime Numbers Get Hitched is about the prediction a(3)=42. (Reply to this)

	cross-ratio: an invention of the devil? magicdragon2 2006-04-07 03:54 am UTC (link)
	"You might say it was a triumph of algebra to invent this quantity [cross-ratio] that turns out to be so valuable and could not be imagined geometrically. Or if you are a geometer at heart, you may say it is an invention of the devil and hate it all your life." - Robin Hartshorne, Geometry: Euclid and Beyond, p. 341. (Reply to this)

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