- How to become heartless man, his heart like a juice or nectar is hidden in a corner. The way hidden fire in stone, in the same way the human heart - no matter how brutal it may be, excellent and gentle expressions are hidden.
- Who are intolerant love, which does not seem like the moods of others, who do not hesitate to implantation seeming to blur, she is hysterical, not love.
- Man is upset or the circumstances or pre-conditionings. Situations which fall to sacrifice humans can survive only under those circumstances.
- Thieves not only to avoid punishment, he also wants to avoid humiliation. Not so much afraid of the punishment of the offense.
- Education is needed to make life a success, not of degree. Our degree - our stewardship, our humility, simplicity of our lives. If it was not degree, was not awakened when our soul is wasted paper degree.
Friday 31 May 2013
Munshi Premchand Quotes
A better way to make iPSCs?
Embryonic stem cells can develop into most cell types and have much therapeutic potential. However, its method of extraction from human embryos remains highly controversial.
That is what made Shinya Yamanaka’s invention of induced pluripotent stem cells (iPSC) a Nobel-winning achievement.
iPSCs are body (somatic) cells which have been reprogrammed to function like embryonic stem cells, thereby sidestepping the controversial use of killing the embryos while harvesting the stem cells. This is done by introducing four regulatory factors (pieces of DNA) into the cells.
“Each of the ‘Yamanaka’ factors, when expressed in cells will activate or repress target genes. Expressed together they can, at low- frequency, convert cells to a relatively stable state of gene-expression similar to that seen in embryonic stem cells,” Dr. K. VijayRaghavan pointed out in an email to this correspondent. Dr. VijayRaghavan is the Secretary, Department of Biotechnology, and former Director of Bangalore-based NCBS. He was not involved in the work.
However, the efficiency of iPSC production is traditionally quite low. Only about 0.01 per cent of the cells successfully become iPSC.
Duanqing Pei and team at the Guangzhou Institutes of Biomedicine and Health, China, may have found a way to improve this performance. The results were published on May 27 in Nature Cell Biology. They suspected that the four factors may counteract each other when introduced together, making iPSC production less efficient.
So the team used the same four Yamanaka factors — Oct4, Klf4, c-Myc and Sox2 — but experimented with the sequence, timing and combination of their introduction.
They found that an OK+M+S (Oct4+Kllf4 followed by c-Myc and then Sox2) combination achieved the highest reprogramming efficiency among all the other combinations they tried. The sequential introduction showed five times more efficiency than the simultaneous introduction protocol. This better performance was recorded both in mouse and human differentiated cells.
“We can generate iPSCs with less money and time and higher efficiency,” said Dr. Pei in an email to this correspondent.
More importantly, such experiments shed light on the individual roles of the Yamanaka factors in reprogramming — something that according to Dr. Pei is still not very clear at molecular or mechanistic level.
What is known is that the Yamanaka factors turn on/off target genes. In this way, they reprogramme the DNA of the somatic cell such that it attains pluripotency — the ability to develop into all cell types that make up the body, just like embryonic stem cells. The research shows that it is not enough that these factors just function, but it is also crucial when and how much they do.
“The correct balance of on/off genes that switch a cell to an iPS fate is achieved only by chance in very few cells. The sequential induction of the Yamanaka factors seems to increase this chance,” elaborated Dr. VijayRaghavan.
Further, the team confirmed a 2010 discovery that a process called mesenchymal-to-epithelial-transition (MET) is involved in the transformation of a mouse fibroblast cell into an iPSC. The current work showed that the cells were actually undergoing a reverse process —early EMT followed by MET — before finally attaining pluripotency.
The Yamanaka factors were shown to play roles in the EMT-MET processes. “Now, we know that each of the factors appear to perform a specific task to convert a fibroblast to iPSC, a division of labour,” said Dr. Pei.
Understanding how induced pluripotent cells are made will allow us to program human cells in a particular direction. As Dr. VijayRaghavan emphasised, “This is of fundamental importance in understanding how humans develop and in our understanding and treatment of disease and in regeneration.”
Why do ants favour feeding on sugary products?
In an ant colony, food is gathered by the foragers who constitute about 10 per cent of the colony members. The worker ants usually transfer the food or other fluids among the colony members through mouth-to-mouth (trophallaxis). Ants feed on a variety of food items, not just limiting themselves to the sugary products.
They can feed on pieces of solid food, seeds or grains, nectar, sugar, candy, honeydew, dead insects, etc. However, their food preferences vary with the nutritional demands of the colony members.
For example, the foragers might collect the protein-rich food for the queen, when she is actively laying the eggs. Similarly, protein-rich foods are preferred when the colony has a high larval population.
When the colony has no larvae, the foragers preferably gather the sugar-rich foods, because adult colonies consuming protein-rich food have extremely high mortality due to protein toxicity.
Even if they find a food which is rich in protein but poor in sugars, they extract the sugars and eject the protein as a waste. Hence, it is quite normal to see the foragers attracted towards sugary products (carbohydrates), unless there is a demand for protein by the egg-laying queen or the larvae.
How do flowers transform into fruits in fruit bearing plants like tomato etc. while this is not the case in other flowering plants like Jasmine etc.?
In general, flowers transform into fruits in a process called, ‘fertilization’. Although the fertilization process is mostly similar in all the plants, the presence of reproductive organs in flowers and the pollination process may greatly vary among the plant species.
For example, the male and female reproductive organs are present in the same flower of a tomato plant; whereas separate male and female flowers are present in the same pumpkin plant. But, the palmyrah has separate male and female trees. Hence, for a simple understanding, let us consider tomato plant as an example in this answer.
The female reproductive organ has an ovary. The ovary contains ovules (egg), which develop into seeds upon fertilization. On top of the ovary is a long tube, called style. The tip of the style is called stigma, which is usually sticky or hairy in nature.
The male part of the flower also has a long filament on which the anther is present. The anther produces the yellow, dust-like pollen grains that contain the sperm cell. During the process of fertilization, the pollen grain first lands on the receptive hairy or sticky stigma.
The pollen grain then germinates and grows a pollen tube. The growing pollen tube reaches the ovary through the style tube, and penetrates the ovule through a tiny pore. After fertilization, the ovary will develop into a fruit and the ovules will develop into the seeds.
Sometimes, a barrier called ‘self-incompatibility’ interrupts the normal fertilization process. The pollen germination, pollen tube growth or ovule penetration is halted at one of its stages in plants having self-incompatibility.
Most of the jasmine species has strong self-incompatibility as well as cross-incompatibility. Hence, the chances of natural fertilization and fruit formation are very less in jasmines. However, fertilization and fruit formation successfully take place in jasmine hybridization programmes. Usually, jasmine fruits contain only one seed.
Thursday 30 May 2013
List of Greatest Mathematicians
List Revisions
The major change over the last year was to expand the List to a Top 105 for which the five new names ended up being Daniel Bernoulli, George Birkhoff, Felix Hausdorff, George Pólya and Thabit ibn Qurra. From the Top 105, five mathematical physicists (if Aristotle can be so called) are set aside, without numbers, so that the List can be treated as a Top 100 List of pure mathematicians with special importance, depth and breadth. In addition, I added twenty also-rans to create a Top 125 list.Ignoring ordering within a List I'm happy with my present Top 3, Top 4, Top 11, Top 29. (If you don't like my #30, who would you replace him with? I like my choices for #31-#40, but there's none I desperately want to promote to the Top Thirty.)
Although I'm happy with my Top 11, I'm not sure who to remove to make it a Top 10. Grothendieck has great depth, breadth and importance and might rank ahead of Leibiz and Euclid. I leave him at #11, with the excuse that he only barely meets the "born before 1930" criterion.
Criteria
To qualify, a mathematician must have historical importance, "depth" and "breadth." Historical importance (or influence) deals with the question "Did this person change the course of mathematical history?" Many lists consider only historical influence, and will end up with a very different list from mine. Ramanujan had only minor influence, but appears high on my list because of his great genius.When I say a mathematician has "Depth," I mean that his work was particularly creative, revolutionary, or difficult. Cantor and Godel have great "depth" because their work was especially creative and revolutionary. Weierstrass and Dirichlet have great "depth" becuase they were able to prove difficult theorems that had stumped others.
With "Breadth" I promote mathematicians who did excellent work in varied fields. For example, I rank Fermat higher than many would because he was a key early developer of both analytic geometry AND number theory and also did good work in geometry, probability, and even optics. Leibniz did not prove any particularly difficult theorem, but is ranked high because of his historical importance and great breadth. Due to outstanding breadth, I also rank Von Neumann and Leonardo 'Fibonacci' higher than others might. On the other hand, Jakob Steiner was an outstanding genius whom I've "demoted" somewhat due to his focus exclusively on geometry.
Of course, this is all arbitrary and fuzzy. The exact "rankings" are just my feelings ... and I often rearrange the list on whim!
Is it pretentious of me to "Rank" the Greatest Mathematicians?
I've tried to do a good job of ranking the mathematicians, but realize it's a silly conceit, and that no ranking would satisfy everyone.Some will wonder, not whether my rankings are "correct," but whether Great Mathematicians should be ranked at all, especially by someone like me, with no apparent qualifications. I ask myself this as well. But to simply list "100 Great Mathematicians" would be even less satisfying: it would seem silly to have both Archimedes and Lambert appear on the same single list. And to make, for example, two lists ("50 Greatest" and "50 Near Greats") would combine the worst of both approaches: How could one justify the distinction between the #50 slot and the #51 slot? Like it or not, the ranking is needed for the List to have any value. I've tried to justify the rankings in the mini-bios.
As a separate matter, some of my rankings are controversial, despite that I've tried to base them partly on expert opinions I've gleaned from books and Internet resources. One correspondent thought Pascal should rank above Fibonacci. (This was one of the correspondents who thought ranking at all was wrong!) Since this is probably a popular opinion, I'll comment here. Pascal was certainly a spectacular genius, but had very low actual influence. Wallis and Cavalieri, who are each shown far below Pascal in rank, were each more influential to the early development of calculus; moreover Pascal's brilliant geometry was inspired in large measure by Desargues' work. I wonder if I've placed Pascal too high, not too low. Leonardo `Fibonacci', on the other hand was a versatile and important teacher who was one of the very best number theorists before Fermat. It isn't widely acknowledged but Leonardo proved the n=4 case of FLT more than 400 years before Fermat did.
Many people view Newton, Gauss and Archimedes as an almost "Divine" Trinity, being the three greatest mathematicians ever, while others would add Euler and make this a Divine Quaternity. Euler was superlative in several ways and it is tempting to rank him above Gauss, but Gauss was the greatest theorem prover ever and handily solved several problems that had stumped Euler. I just keep changing my mind about the best order to rank the four greatest. If you still disapprove of my rankings within the Top Four, just pretend I've ranked them all as tied for #1 !
The criteria of "depth" (work that was particularly creative, revolutionary, or difficult) and "historical importance" are probably not controversial, but my insistence on "breadth" (excellent work in multiple fields) may result in rankings different from others. Since the fuzzy measures of "depth" and "breadth" apply unequally to the candidates, the final "rankings" become arbitrary. Leibniz may lack the "depth" for the Top Ten, but his breadth and importance are enormous. Abel, Weierstrass and Dirichlet probably have less importance and "breadth" than others in the Top Thirty but, as measured by skill at proving difficult theorems, they each had great "depth." Some think Lebesgue should rank higher because of his importance, but, lacking "breadth," he should perhaps be lower. Et cetera. In any event, I hope those commenting on my rankings will base their comments on my criteria, not the criteria they would have chosen!
There are two versions of the list; the content is identical but greatmm.htm is the start of a seven-page set containing the List and biographies, while mathmen.htm combines the list and bios into a single, very large, page.
Mathematicians from the Modern Era
There are many living mathematicians with extraordinary genius, who will certainly appear on a future List like this, but for several reasons it would be uncomfortable or difficult to include them on my list. Therefore I've adopted 1930 as an arbitrary cut-off for birth year. Even with this cutoff, the List includes seven people still living, or who were alive after 1985. These names are Atiyah, Chern, Grothendieck, Kolmogorov, Selberg, Serre, Weil. While most of those just named exhibited great breadth, many of the great geniuses born after 1930 specialized of necessity. This is one reason I include Conway and Milnor on an expanded list even though they were born after 1930 -- they each exhibited outstanding breadth.
About my List of Greatest Mathematicians
The mini-biographies are of different lengths. Some of the greatest (e.g. Jacobi) have biographies much shorter than less obvious candidates like Kepler. This is due in part to a desire to justify Kepler's inclusion, while Jacobi's inclusion is not in doubt.
I've tried to add a quotation to each of the mini-biographies: either something that genius said, or something some other genius said about him.
I've learned a great deal while preparing this list, not only about Renaissance and Modern mathematicians, but about ancient mathematics as well. While preparing the very brief summary of ancient mathematics I stumbled upon descriptions of Babylonian Multiplication. In the note you'll see why this came as a pleasant surprise to me. I'm not really qualified to make a list like this -- it started as a practice exercise while learning HTML tags! -- but many Websurfers were stumbling on My List of Mathematicians, so I've devoted considerable effort to making this a list I can be proud and confident about. By now, I've devoted many hours to reading biographies, and reacting to others' opinions, and by now I'm fairly satisfied with the validity of my List of Greatest Mathematicians, but I'd be happy to make it better!
Mathematical physicists
I had trouble deciding whether to include great mathematical physicists like Maxwell, Einstein, and Galileo, who would not qualify for the List if only their contributions to pure mathematics are considered. I've changed my mind back-and-forth about whether to include them, and finally compromised by adding these names (and Aristotle and Alberuni) as a special addendum to increase the List to 105. (In the following discussion references to my "List of 100" usual refer to the complete List of 105.)
Hipparchus, Cardano, Huygens, Kepler and Daniel Bernoulli are also great mathematical physicists, but I have included them in the main List of 100 because of their great historical importance in the development of mathematics.
Why Einstein might be on the List
I get many comments that Einstein doesn't belong on a List of Greatest Mathematicians, despite that I indicate my reasons in his mini-bio. (I admit that this is my List, and I may have occasionally allowed whims to influence some of the choices. I might not have included Omar al-Khayyám if it weren't for his poetry.)
But I even receive comments that Einstein wasn't even a great physicist, despite that his 1905 papers revolutionized physics, and that his 1915 General Relativity has been called the most creative physics ever. I won't comment on this fringe iconoclastic view except to show the comments of Hilbert (not known for humility) on the matter: "Every boy in the streets of Gottingen understands more about four-dimensional geometry than Einstein. Yet, in spite of that, Einstein did the work and not the mathematicians."
And, of course, all great scientists have built on others' work: Cantor and Gödel are regarded as two of the most original thinkers ever, yet Dedekind anticipated much of Cantor's work, and von Neumann's thinking inspired Gödel. Abel's theorem of quintics was first stated and partially proved by Ruffini; and so on.
Math is Beautiful
While preparing the mini-bios, I was struck by how many great mathematicians emphasized the beauty of their work. The quotations I've chosen by Boole, Cayley, Hardy, Weyl, and Banach all contain the word "beauty." If words like "poetry" or "ecstacy" are considered, Kepler, d'Alembert, Steiner, Weierstrass, and Weil can be added to that list.Betrand Russell wrote "Mathematics ... possesses not only truth, but supreme beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show."
Buckminster Fuller once said "When I am working on a problem, I never think about beauty. I think only of how to solve the problem. But when I have finished, if the solution is not beautiful, I know it is wrong."
Paul Dirac wrote "... it is more important to have beauty in one's equations that to have them fit experiment ... [because any discrepancy with experiment] may ... get cleared up with further development of the theory."
Gosta Mittag-Leffler wrote "The mathematician's best work is art, a high perfect art, as daring as the most secret dreams of imagination."
Controversies
In the mini-bios of ancient mathematicians I state some traditions that are disputed. Assertions that Hindu mathematicians knew the laws of motion may be exaggerated (though recently I found that the writings of al-Biruni confirm some ancient Hindu knowledge often ignored in the West.) Some historians do not believe Thales predicted a solar eclipse. Though the Pythagorean school had great influence, some historians believe Pythagoras himself was mythical. Some say that the compass/straight-edge construction regime often associated with Plato was introduced only later by Apollonius. Some historians believe that al-Khayyám the mathematician and al-Khayyám the poet were two different people. I am certainly not qualified to arbitrate these controversies, but I will say that my readings in social sciences have taught me to be very skeptical of skeptics. And I wonder if those who think Khayyam the poet was not the astronomer and mathematician have read poems such as:Ah, by my Computations, People say,or
Reduce the Year to better reckoning? -- Nay,
'Twas only striking from the Calendar
Unborn To-morrow and dead Yesterday.
Up from Earth's Centre through the seventh Gate
I rose, and on the Throne of Saturn sate,
And many Knots unravel'd by the Road;
But not the Knot of Human Death and Fate.
-- Omar Khayyam (trans. by Edward Fitzgerald)
Miscellania
Dates and Nationalities of Mathematicians on List
Of the Top 105 mathematicians, 13 fluorished during the Antique Classical Age, 9 during the Middle Ages, then (based on death year) 1, 8, 9, 29, 31 in respectively the 16th, 17th, 18th, 19th, 20th centuries, and 5 are still living (or lived into the 21st). Comparing the state of mathematics as Lagrange found it and as Cayley left it, the 19th-century concentration should be no surprise. Most of the 20th-century names were in the early part of that century: no matter how great the latest geniuses may be, they can't have as much importance, at least in applied math, as those that came before them.As implied by the numbers just presented, the List includes roughly three times as many mathematicians per generation from the modern era compared with the 18th century. Based on the numbers of mathematicians and the continuing high pace of advances, this actually constitutes a bias towards the past, justified by the extreme historic influence of the earlier time.
Of the 105 Great Mathematicians, there are 23 from France; 20 from Germany; 12 from the ancient Greek world; 9 from England; 7 from Switzerland; 6 from Italy; 5 from India; 3 each from Hungary, Norway, Persian domain and Russia, 2 each from China and Holland; and 1 each from Austria, Iraq, Ireland, Lithuania, Poland, Scotland ant the U.S.A. (It's not always clear which country to assign: I put Cantor in Germany, Lagrange in Italy, Einstein in Switzerland.)
I.Q. Estimates
Recently I learned that a team of psychologists led by Catherine M. Cox tried to estimate the childhood IQ's of certain famous people born between 1450 and 1850. This sounds "iffy" to me, but the reader may be interested to know that the team scored Leibniz and Pascal highest among mathematicians. (The top six on the Cox list had four non-mathematicians: Johann Wolfgang von Goethe, Hugo Grotius, Thomas Wolsey and Pietro Sarpi. Goethe ranked 1st, Leibniz 2nd.) Bygone mathematicians scoring 170 or higher in that survey include 10 others on my list (Newton, Laplace, Lagrange, Déscartes, Kepler, Huygens, Cardano, Hamilton, d'Alembert, Galileo) and 3 not on my list (Buffon, Napier, Lazare Carnot). That famous Cox list has been revised (by whom I don't know) with five names promoted above Goethe and Leibniz: Newton, Voltaire, da Vinci, David Hume and Michelangelo. Mathematicians not mentioned above who are sometimes seen on lists of great polymaths or geniuses include two women (Hypatia and Kovalevskaya), and several men (including Alhazen, Poincaré, Ludwig Wittgenstein, Einstein, and von Neumann). Wittgenstein is sometimes shown with the highest IQ ever but is missing from my list since his emphasis was logic and philosophy.Complex Numbers
The discoverer of imaginary and complex numbers would surely deserve a prime position on this List, but these seem to have been a very gradual development. That some quadratics have only "fictitious" solutions was known in ancient Greece; Cardano is often credited but his work was preceded and succeeded by his own Italian colleagues. Wallis (and perhaps Vieta) began serious use of complex numbers; Leibniz, de Moivre, Johann Bernoulli and d'Alembert developed them further; yet even Euler (who popularized the symbol i) once made elementary mistakes with imaginary numbers. It was finally Jean-Robert Argand (not quite "great" enough to appear on my list) and then Carl Gauss who integrated complex numbers into the mathematical mainstream.
Euler's Discovery: π2/6 = 1-2 + 2-2 + 3-2 + 4-2 + ...
The story of Euler's famous discovery may be interesting. The following facts were already well known to him ((3) was discovered by Isaac Newton):- If f(x) is a polynomial with zeros (roots) at a, b, etc. then f(x) = (x-a) (x-b) ...
- lim sin(x)/x = 1 as x --> 0
- sin x = x - x3/3! + x5/5! - ...
By itself, this doesn't constitute a rigorous proof (similar sleight of hand can be used to derive 1 + 1 = 3), but it is how Euler first derived his formula, and it can be made rigorous. Euler's intuition and the simplicity of his method here may serve as inspiration!
Heegner Numbers
In the bios, I refer to fields like "algebraic number theory" that I don't understand at all. I have caught a few glimpses that amazed me, however. Consider Euler's formula, that generates 40 consecutive primes for consecutive n: n2 + n + 41. Now consider Martin Gardner's famous "April Fool's integer": e√163 π = 6403203 + 744. (In this "equality" the left-side is actually smaller than the right-side integer, but by less than a trillionth of a unit. Too close to be a "coincidence", right?)163 = 4*41 - 1 is a Heegner number and the fact that it yields both Euler's prime-generation formula and the "April Fool's integer" (also called Ramanujan's constant) is not a coincidence (though the "April Fool's equation" might seem completely unrelated to prime numbers). If you doubt this, substitute another Heegner number, such as 67 = 4*17 - 1, to get 16 consecutive primes and the approximation e√67 π = 52803 + 744. It was Carl Gauss himself who discovered these "magical" Heegner numbers (though not the mysterious "almost equations") and conjectured that 163 was the largest of them. (Ramanujan's formula for π is also related to the Heegner number 163.)
An Elegant Diagram
| The names Plato, da Vinci, Kepler and Einstein appear among "other contenders" even though they didn't specialize in mathematics. These men were important because their work inspired other mathematicians. While reading about Gaspard Monge, to judge whether he qualified for the List or not, I learned that Kepler had anticipated the invention of Descriptive Geometry when he related the dodecahedron to the "Snowflake" shown at right. |  |
Prime Numbers
In about 1767, Euler proved that (2^31 - 1), was prime. This record beat the previous highest known prime by a factor of 4096 and was not surpassed for 100 years. This record-setting prime, whose hexadecimal value is 7FFFFFFF, is immortalized in the C Programming system as "INT_MAX"! (While Euler's prime has 10 decimal digits, a prime discovered in 2008 has almost 13 million digits. But perhaps the prime number Gold Medal should go to Edouard Lucas whose clever 19th-century method found a 39-digit prime that held the record for 75 years. Euler's prime, Lucas' prime and the present record-holder are all Mersenne numbers but, perhaps surprisingly, neither the prime which first beat Euler's nor the prime that first beat Lucas' record was a Mersenne.)
Musical scales
According to tradition, Pythagoras' understanding of music began when he passed a blacksmith shop and, finding it harmonious, inspected the anvils and noticed some harmonious anvils had length ratio 2 : 1 (what we now call an "octave") and others 3 : 2 (what we now call a "perfect fifth"). Pythagoras devised a musical scale based on the same seven notes as ours, with the whole-step ratio 9 : 8, but the half-steps B-C, and E-F in a (2^8 : 3^5) ratio. Five whole-steps and 2 half-steps made an "octave"; three whole-steps and 1 half-step a "fifth", and the interval between octave and fifth was thus a "perfect fourth" (4 : 3 ratio). In this system a whole-step exceeds two half-steps in the ratio 3^12 : 2^19 = 1.01364; this "error" term is called the "Pythagorean comma." (For centuries, the octaves and perfect fifths of an instrument were tuned as described, and B-sharp needed to be sharper than C by a Pythagorean comma.)Archytas considered the question of dividing the "perfect fifth" into two harmonious intervals. He had observed that a ratio is the product of arithmetic and harmonic means, suggesting that C-E-G should have the ratios 4 : 5 : 6 rather than the 4 : 5.0625 : 6 of Pythagoras' scheme. (5/4 and 6/5 are the arithmetic and harmonic means of the pair (1, 3/2).) Today the 5 : 4 and 6 : 5 ratios are called major and minor thirds. In the Renaissance, the notes C-D-E-F-G-A-B-C were given the ratios 24 : 27 : 30 : 32 : 36 : 40 : 45 : 48.
But neither scheme provides harmony in all keys; and the cross-key errors are much more severe in the Renaissance system than in Pythagoras' scheme. An equal-tempered scale was desired; it was Simon Stevin who suggested half-steps whose size was the twelfth root of 2. The "fifth" ratio is now 1.4983 -- almost "perfect," and the same in every key. Unfortunately the major and minor "thirds" were each off by almost 1%.
Huygen's 31-tone scale gives 1.4955 as the ratio for the 18-step "fifth" and 1.2506 for the 10-step "third." (A 53-tone equal-tempered scale would be even more harmonious.) But of course, an organ with 31 keys per octave instead of just 12 is rather unwieldy!
Note on Babylonian Multiplication
I was slightly amazed when I learned that Babylonians used the formula xy = ((x+y)^2 - (x-y)^2) / 4 to perform multiplications with just additions, subtractions and table lookups. I've used the same method on computers with slow multipliers and was surprised to find, on an expert programmer message-board, that hardly anyone else was familiar with it. (Manual multiplication today is done by memorizing products up to 9-times-9; the obvious Babylonian analog would have required fitting 59-times-59 terms onto tablets; this explains why they invented the approach requiring only the first 59 square numbers.) (See, for example,http://www.math.tamu.edu/~don.allen/history/babylon/babylon.html.)
Great Mathematicians' Self-Appraisals
Hardy once described himself as "for a time the 5th best mathematician in the world." My guess is he referred to the early 1920's and his four superiors included Weyl, Littlewood, Noether and perhaps Cartan or the retired Hilbert. At this time, the mathematical powers of Klein and Hadamard were waning, while von Neumann, Kolmogorov and Weil had not reached their maturity.Von Neumann described himself as "only the 3rd best mathematician of my time." Assuming he didn't consider Hilbert (who had retired when von Neumann was in his 20's) to be "of his time", one wonders whom von Neumann regarded as #1 and #2. Interestingly, the likeliest candidates (Hermann Weyl, André Weil, Kurt Gödel and Carl Siegel) were all faculty colleagues of von Neumann at Princeton's Institute of Advanced Study. (Many other great mathematicians served as faculty at I.A.S. including Einstein, Veblen, Atiyah, Deligne, Selberg, Milnor and Witten.) Despite Von Neumann's modesty, I've ranked him as "best of his time" because of his huge breadth: foundations, analysis, game theory, etc.
There were two mathematicians who surely considered themselves the greatest of their time. About themselves they wrote:
Isaac Newton: I do not know what I may appear to the world; but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.
René Déscartes: I hope that posterity will judge me kindly, not only as to the things which I have explained, but also to those which I have intentionally omitted so as to leave to others the pleasure of discovery.
(I've ranked Isaac Newton a bit higher than others might, and Déscartes a bit lower. This is partly because of the respective modesty they show in their self-appraisals above.)
Other Lists
Pickover's List
Cliff Pickover has a list of the "Ten Most Influential Mathematicians." His Top Ten include seven of my Top Ten (all except Archimedes, Grothendieck, and Lagrange), but bypass my #11-#13 to include Déscartes, Galois and Pascal. His omitting Archimedes and including Déscartes may make sense since his criterion emphasizes historical importance, but Pascal seems an odd choice: most of the names on my list were more "influential."
Compared with this and other lists, I seem to overrate Fermat, and underrate Napier. Pickover names Cardano and Napier as "runners-up." I'm doubtful about John Napier, who had great influence (he discovered 'e', developed logarithms and the use of decimal point, invented "Napier's bones", and seems to have conceived of continuous functions), but proved no great theorems.
Eells' List
W.C. Eells produced a list of the top 100 mathematicians of all time, in order (though the list contains no mathematician born after 1843 except Poincaré). His Top Ten include two names (Laplace and Cardano) missing from my Top Twenty. His Top Forty include 6 names missing from my Top Hundred: (in Eells' order) Napier, Ptolemy, Regiomontanus, Maclaurin, Tartaglia, Heron. Eells' #41-#60 include 14 names missing from my List: Chasles, Cremona, Desargues, Roberval, Boskovic, Barrow, Sturm, Stevin, De Morgan, Taylor, Briggs, Camot, Maupertius, Babbage.
On my List of Hundred there are 74 names born before 1843; 17 of these are missing from Eells' List of 100; these 17 include ten ancients (e.g. Panini, Eudoxus, Brahmagupta, Alhazen), Dedekind, Eisenstein, Kummer, Liouville, Maxwell, Chebyshev, Jordan.
Polyanin's List
Andrei D. Polyanin has a List of the Top 24 Mathematicians. It contains seventeen names from my Top 21 (all but Grothendieck, Fermat, von Neumann, Brahmagupta), five other names from my Top Hundred (d'Alembert, Fourier, Kolmogorov, Laplace, Lobachevsky), and two Russian mathematicians missing from my list (Sophie Kovalevskaya, Andrei Markov).
Ioan James' List
Ioan James has a List of the 60 Remarkable Mathematicians from Euler to Von Neumann, which restricts to mathematicians born between 1705 and 1905. (His criterion of "remarkable" isn't quite the same as "great" but let's compare the lists anyway!) My List of Hundred includes 59 mathematicians born in that period; the two lists are fairly close with 45 names in common. However there is a strong difference in the Lists' temporal biases: 25% of the names on James' List were born between 1880 and 1899.
Comparing the lists shows that mine has a bias toward the past: I'm missing only two names from James' list who were born before 1825 (Germain and Kronecker) while he is missing six of mine from the same time period (Boole, Eisenstein, Lambert, Lobachevsky, Plücker, Steiner). For the period from 1825 to 1905, James is missing six names (Jordan, Lebesgue, Littlewood, Minkowski, Peano, Siegel) from my List and I am missing 13 names from his List (Aleksandrov, Alexander, Courant, Kovalevskaya, Lefschetz, Mittag-Leffler, E.H. Moore, R.L. Moore, Smith, Takagi, Veblen, Wiener, Zariski), with nine of these born in the 1880-1899 period James emphasizes. (Combining James' and Polyanin's Lists with Eells' Top 60, there are 38 names missing from my Top 105, or 28 names missing from my Top 125.)
Cardano's List
Girolamo Cardano prepared a list of the 12 men who "excelled all others in the force of genius and invention." This list includes six from my List: Archimedes, Euclid, Apollonius, al-Khowârizmi, Archytas, Aristotle; and six other names: Ptolemy, al-Kindi, John Duns Scotus, Richard Swineshead the Calculator, Galen of Pergamum (physician), and Jabir ibn Aflah (aka Heber of Spain, astronomer whose works were translated into Latin). When Copernicus obsoleted Ptolemy's cosmology during Cardano's lifetime, Cardano removed Ptolemy and added Marcus Vitruvius Pollio (engineer) to keep the list at twelve names.
Simmons' Most Influential Scientists
John Galbraith Simmons prepared a list of the 100 Most Influential Scientists in order. His Top Twelve seem to me to be excellent choices; they are Newton, Einstein, Bohr, Darwin, Pasteur, Freud, Galileo, Lavoisier, Kepler, Copernicus, Faraday, Maxwell. Five of these names are in my List of 105. John Balchin and John H. Tiner have each prepared their own lists of 100 influential scientists; each includes Simmons' Top Twelve. Other mathematicians who appear on all three lists are Archimedes, Euclid, Huygens. Mathematicians appearing on two of the lists are Aristotle, Descartes, Euler, Gauss, Pascal, Pythagoras, and one missing from my List: Charles Babbage.
Hart's List
Michael Hart's List of the 100 Most Influential People Ever seems well reasoned. His List, which considers only historical significance and not genius, includes nine mathematicians from my List (Newton, Einstein, Euclid, Déscartes, Kepler, Euler, Aristotle, Galileo, Maxwell) and four other mathematical physicists (Copernicus, Heisenberg, Planck, Fermi). Hart's list also includes Lavoisier, Faraday, Dalton, Plato, Rutherford, Roentgen, Francis Bacon and eighty people who did no work in pure physical science (44 political, military or religious leaders, 2 explorers, 16 inventors, 9 social philosophers, 5 artists, 4 biologists).
Among scientists, Hart omits Bohr altogether (and ranks Freud and Kepler behind Aristotle, Euclid, Adam Smith, Dalton, Heisenberg, etc.) but otherwise his top scientists agree fairly well with Simmons. (In addition to the Top 12 Scientists, seven other scientists not on my List appear on all four of the lists just described -- Dalton, Fermi, Harvey, Heisenberg, Leeuwenhoek, Mendel, Rutherford.) Some names seen near the top of ther top scientist lists include Nikola Tesla, Leonardo da Vinci, Marie Curie, Thomas Edison, and one name from my List: Alan Turing.
Prizes and Medals
The Royal Society's Copley Medal is among the most prestigious of science prizes (fewer are given than Nobel's). Eleven men on my List received that prize: Atiyah, Cayley, Einstein, Gauss, Klein, Hardy, Littlewood, Plücker, Poisson, Sylvester, Weierstrass. Several other great mathematicians have also won the Copley Medal, e.g. Chandrasekhar, Chasles, Clausius, Gibbs, Lamb, Le Verrier, Penrose, Rayleigh, Stokes, Sturm, Waring. (The Copley Medal is usually awarded late in a career and never posthumously; this explains the absence of certain greats who died at a relatively young age, e.g. Riemann and Maxwell.)Prestigious prizes where preference is given to Britons include the De Morgan Prize and Sylvester Medal. From my list, Atiyah, Cantor, Cayley, Darboux, Hardy, Klein, Littlewood, Poincaré, Sylvester have won one or both of those prizes. Other winners include Besicovitch, Carleson, Lamb, Penrose, Rayleigh, Russell, J.G. Thompson, Titchmarsh, among others.
Until recently, the Wolf Prize was perhaps the most prestigious of math awards; winners from my list include Atiyah, Chern, Erdös, Kolmogorov, Selberg, Serre, Siegel, and Weil. In addition the Fields Medal was won by Atiyah, Grothendieck, Selberg, and Serre; the Abel Prize by Atiyah and Serre. Five mathematicians born before 1930 but not on the List have the distinction of winning at least two of these three most prestigious prizes: Lars Ahlfors, Lennart Carleson, Kunihiko Kodaira, Peter Lax, and John T. Tate. (There are several other major prizes given for mathematics. Mathematicians born 1930 or later who have won at least three major Prizes include Pierre Deligne, Mikhail Gromov, Lars V. Hormander, John W. Milnor, David Mumford, Stephen Smale, John G. Thompson.)
Female Mathematicians
I hope no one takes offense that 29 of the Top Thirty are males, Emma Noether being the sole woman on the list. Other great female mathematicians include Marie-Sophie Germain, Sophia Vasilyevna Kovalevskaya, Hypatia (the last librarian of Alexandria), and perhaps Theano (Pythagoras' wife). Part of the reason for the dearth of women among famous mathematicians may be discrimination; even the great Noether might have been almost overlooked except that colleagues like Hilbert and Weyl insisted that her greatness be recognized.Omissions from the List
I started with just Ten names, and gradually grew this list up to One Hundred over several years. I know I'm finally done because there's no one left that I fervently want to include. To demonstrate this, I'll "dispose" of the most obvious remaining candidates.Certain great geniuses are omitted because they lacked great historical influence, e.g Roberval, who didn't publish, Bolzano, who was censored by the Austrian government, or Clifford and Gregory, who each died young.
Regiomontanus and Ptolemy are famous mathematicians of great historic importance whom I omit. Regiomontanus' work is important primarily because he lived shortly after Gutenberg's printing press invention; his actual creative achievements in spherical trigonometry were anticipated by Nasir al-Din al-Tusi. Similarly, Ptolemy's great fame exaggerates his genius. Careful comparison of the errors in their works reveals that Ptolemy borrowed both his data and his methods from Hipparchus. Napier, Stevin, al-Kashi and perhaps Chang are strong candidates based on genius and historical importance, but none of them is credited with any non-trivial theorem. I would include members of the Maxwell/Galileo group before any of these.
I may include Thales of Miletus (he produced the first known "proof" and was very influential) and Panini (who anticipated both Boolean logic and modern mathematical grammars) but the list includes no other mathematicians prior to Pythagoras, though some names from India's ancient Vedic period are known, e.g. Lagadha, the first astronomer known to have used trigonometry, and Baudhayana, who taught the Pythagorean theorem, and Apastambha, perhaps the greatest mathematician before Archytas. Another famous Vedic mathematician who left more than fragmentary work, was Pingala (a Jainist who was famous for his work in music theory and combinatorics).
I omit some famous mathematicians like Ferdinand Lindemann and Andrew Wiles (lacking in breadth, and their "revolutionary" discoveries relied heavily on earlier work). (Some other mathematicians frequently included on "Great" lists are Babbage, Barrow, Buffon, Chasles, Clebsch, Cramer, DeMoivre, Heron, and Mittag-Leffler.)
Two correspondents have asked me to include Oliver Heaviside among the greatest mathematicians. This self-taught man was certainly an amazing genius, and should certainly appear on a list of greatest applied mathematicians, but would probably fall behind Maxwell, Galileo, and Napier on that list, and none of those appear in the Top 100 either.
Finally, I've imposed an arbitrary rule: the List includes no mathematicians born after 1930.
Proposed candidates
The list following (chronological by death year) includes the greatest mathematicians, various proposed candidates, and some, like Plato, da Vinci or Einstein, often proposed as inspirational to mathematicians.
(Names in deep purple are on my Top Thirty List, with short mini-bios.)
(Names in bold-face are on the Top Hundred, or prime contenders.)
(Names in light blue have mini-bios but are not in Top Hundred.)
|
|
|
|
|
The Greatest Mathematicians of All Time
At some point a longer list will become a List of Great Mathematicians rather than a List of Greatest Mathematicians. I've expanded the List to an even Hundred, but you may prefer to reduce it to a Top Seventy, Top Sixty, Top Fifty, Top Forty or Top Thirty list, or even Top Twenty, Top Fifteen or Top Ten List. | ||||
| ||||
|
| |||
| I've appended five additional names to the List of One Hundred Greatest Mathematicians. Maxwell, Einstein, etc. are among the greatest applied mathematicians in history, but may lack the importance as pure mathematicians to qualify for The Hundred. Nevertheless, I would certainly include them in any longer list.I think One Hundred (or 105) is a good list size, but will bring it up to 125 (not a very "round" number, but a cubical one :-), just to show some "spares." (For this extended list, I relax the birth-date rule slightly to include two greats born in the 1930's.) | ||||
Still other contenders: Ahmes Al-Kindi Apastambha Barrow Beltrami Bolyai Bolzano Bombelli Chasles Clifford Courant Cremona Copernicus Deligne deMoivre Dirac Eratosthenes Fréchet Germain Gordan Green Gregory Heron Hypatia Kronecker Landau L.daVinci Lefschetz Levi-Civita Lindemann Maclaurin Madhava Möbius Oresme Plato Ptolemy Qin Regiomontanus Roberval Russell Seki Shannon Smale Sturm Tao Tate Theaetetus J.G.Thompson Torricelli Volterra Witten Wittgenstein Zariski Zeno et cetera.
This is primarily a list of Greatest Mathematicians of the Past, but I use 1930 birth as an arbitrary cutoff, and three of the "Top 100" are still alive as I write.
Click for a discussion of certain omissions. Please send me e-mail if you believe there's a major flaw in my rankings (or an error in any of the biographies). Obviously the relative ranks of, say Fibonacci and Ramanujan, will never satisfy everyone since the reasons for their "greatness" are different. I'm sure I've overlooked great mathematicians who obviously belong on this list. Please e-mail and tell me! (Sorry if mathematician "100." displays as "00." Either my html is flawed, or Microsoft I-E doesn't like lists longer than 99.)
Biographies of the greatest mathematicians are in separate files by birth year:
Subscribe to:
Posts (Atom)