There are now two versions of the list; the content
is identical but This page
is the start of a seven-page set containing the List and
biographies, while This page
combines the list and bios into a single, very large, page.
Why Einstein is 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. In addition to Einstein, there are others who might be omitted (Hipparchus, Cardano, Kepler) or demoted (Newton, von Neumann, Hamilton, Alhazen, Huygens, Fourier) if it weren't for their scientific work outside pure mathematics.
(I also 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 note that even if Hilbert published the 1915 field equations first, those equations were just the mathematical treatment of Einstein's original thought. Anyway 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.)
Including "prime candidates" the List is finally up to Ninety-nine names. It seems best not to take it to a full 100: anyone who likes my List will doubtless have one favorite missing, so everyone can add their own missing favorite to get a full One Hundred Greatest Mathematicians! (It seems most politic not to list living mathematicians; however I've relaxed that rule to admit any mathematician born before 1930, so three living mathematicians appear on the extended list of 99.)
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!
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 Russel 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.")
In the mini-bios of ancient mathematicians I state some traditions that are disputed. Assertions that Hindu mathematicians knew the laws of motion appear to be exaggerated. 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)
Of the seventy-five mathematicians on the list, 10 fluorished during the Antique Classical Age, 7 during the Middle Ages, then (based on death year) 1, 6, 7, 22, 21 in respectively the 16th, 17th, 18th, 19th, 20th centuries, and one is still alive as I write. 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 seventy-five great mathematicians, there are 20 from France; 16 from Germany; 9 from the ancient Greek world; 7 from England; 5 from Switzerland; 4 from India; 3 from Italy; 2 each from Holland, Persia; and 1 each from China, Hungary, Iraq, Ireland, Norway, Poland, Russia. (It's not always clear which country to assign: I put Cantor in Germany, Lagrange in Italy, Einstein in Switzerland.)
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 9 others on my list (Newton, Laplace, Lagrange, Déscartes, Kepler, Huygens, Cardano, Hamilton, d'Alembert) and 4 not on my list (Galileo, 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.
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.
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):
Since 0, ±π, ± 2π, ± 3π, etc. are all zeros of the sine function, (1) suggests sin(x) = A(x) x (1 - (x/π)2) (1 - (x/2π)2) (1 - (x/3π)2) .... Although we've not proved that A(x) is a simple constant, if we assume that it is, (2) leads directly to A(x) = 1. Substitute this new polynomial for sin x into (3); equate the x3 terms on each side, and Euler's identity soon emerges!
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!
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.)
| 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. |
|
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.)
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!
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.)
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 and Kurt Gödel) 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, 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.)
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, Sofia 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.
Cliff Pickover has a list of the "Ten Most Influential Mathematicians." His Top Ten include seven of my Top Eight (all except Archimedes), but bypass my #9-#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.
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 7 names missing from my Top Seventy-five: (in Eells' order) Napier, Ptolemy, Regiomontanus, Maclaurin, Cavalieri, Tartaglia, Heron. Of Eels' #41-#70, only four names are in my Top Seventy-five; another nine names are in my Top Ninety-nine; the other 17 names include Chasles, Cremona, Roberval, Barrow, and Sturm.
On my List of Seventy-five there are 57 names born before 1843; eleven of these are missing from Eells' List of 100; these 11 include six non-Europeans (e.g. Brahmagupta and Alhazen), Eudoxus, Dedekind, Eisenstein, Hipparchus, Liouville.
Andrei D. Polyanin has a List of the Top 24 Mathematicians. It contains sixteen names from my Top 20 (all but Grothendieck, Fermat, von Neumann, Brahmagupta), five other names from my Top Seventy-five (d'Alembert, Fourier, Jacobi, Kolmogorov, Laplace), and three Russian mathematicians missing from my list (Sofia Kovalevskaya, Nikolai Lobachevsky, Andrei Markov).
Ioan James has a List of the 60 Remarkable Mathematicians from Euler to Von Neumann, which restricts to mathematicians born between 1705 and 1905. Let's divide this 200-year range into 120-year and 80-year periods. For the 120-year period from 1705 to 1825, James' List of 60 has 26 names and my List of 75 has 24 names; these lists are very similar. James is missing three names (Boole, Eisenstein, Steiner) from my List; I am missing five names (Germain, Grassman, Kummer, Chebyshev, Kronecker) from James' List. (Of these, Grassman and Chebyshev are on my expanded List of 99.)
For the 80-year period from 1825 to 1905, James' List has 34 names but my List has only 19 names. Here, James is missing two names (Einstein, Littlewood) from my List and I am missing 17 names from his List (Smith, Lie, Mittag-Leffler, Kovalevskaya, E.H. Moore, Hausdorff, Takagi, Veblen, R.L. Moore, Lefschetz, Birkhoff, Pólya, Courant, Alexander, Wiener, Aleksandrov, Zariski).
Girolamo Cardano prepared a list of the 12 men who "excelled all others in the force of genius and invention." This list includes five from my List: Archimedes, Euclid, Apollonius, al-Khowârizmi, Archytas; and seven other names: Ptolemy, Aristotle, 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.
John Galbraith Simmons prepared a list of the 100 Most Influential Scientists. It includes ten names from our list (Newton #1, Einstein #2, Kepler #9, Laplace #29, Euler #35, Huygens #40, Gauss #41, von Neumann #51, Euclid #59, Archimedes #100). Other mathematicians on the list are the Comte de Buffon (#23), Boltzmann (#24), and several quantum physicists. Simmons' Top Twelve include nine names not on our list (Bohr, Darwin, Pasteur, Freud, Galileo, Lavoisier, Copernicus, Faraday, Maxwell). Other names often seen on lists of most influential scientists include Planck (#25), Marie Curie (#26), and some names missing from Simmons' list (Aristotle, Tesla, da Vinci).
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 six mathematicians from my List (Newton, Einstein, Euclid, Déscartes, Kepler, Euler) and five mathematical physicists (Galileo, Maxwell, Heisenberg, Planck, Fermi). Hart's list also includes Aristotle, Copernicus, 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).
The Royal Society's Copley Medal is among the most prestigious of science prizes (fewer are given than Nobel's). Nine men on my List received that prize: Gauss, Cayley, Weierstrass, Klein, Einstein, Hardy, Littlewood, Poisson, Sylvester. Several other great mathematicians have also won the Copley Medal, e.g. Waring, Sturm, Le Verrier, Chasles, Plücker, Clausius, Stokes, Rayleigh, Gibbs, Lamb, Chandrasekhar, Atiyah and Penrose.
Another very prestigious prize, though preference is given to Britons, is the De Morgan Prize. Five men on my List received that prize: Cayley, Sylvester, Klein, Hardy, Littlewood. Other recipients include Rayleigh, Lamb, Russell, Titchmarsh, Atiyah and Penrose.
Of the 51 individuals on my List who died between 1690 and 2000, all but 14 were "Fellows of the Royal Society"; the exceptions are Jacob Bernoulli, Monge, Legendre, Steiner, Abel, Hamilton, Galois, Eisenstein, Dedekind, Cantor, Borel, Noether, Banach, von Neumann. Great mathematicians with that honor who are not on the List of 75 include Clairaut, Plücker, Chebyshev, Jordan, Barrow, Gregory, DeMoivre, Taylor, two other Bernoullis, Maclaurin, Cramer, Sturm, Chasles, Kummer, Clifford, Buffon, Kronecker, Lamb, Heaviside, Lie, Mittag-Leffler, Darboux, Volterra, Titchmarsh, Lebesgue, Serre, Atiyah, etc.
Eells' List contains some great early mathematicians (Ptolemy, Regiomontanus, etc.) and some 2nd- or 3rd-bests from the early modern era (Maclaurin, Cavalieri, etc.). I reject these to include some important modern mathematicians like Banach and Klein who might not even make a Top Five Living List. (With one exception, all mathematicians born after Euler who are likely "Top Three Living" candidates are listed in my Top Forty; the exception is Legendre, who might have been the 2nd-best mathematician alive for a brief period.)
Napier and Regiomontanus were noted for work with spherical trigonometry, but this field had been developed in the 13th century by Nasir ad-Din at-Tusi, and first studied even earlier.
I may include Thales of Miletus (he produced the first known "proof" and was very influential) 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 Apastamba, perhaps the greatest mathematician before Archytas. Two famous early Vedic mathematicians who left more than fragmentary work, and lived somewhat after the time of Pythagoras, are Panini (who anticipated both Boolean logic and modern mathematical grammars) and Pingala (a Jainist who was famous for his work in music theory and combinatorics).
Although I include Huygens, Einstein, Kepler, and Poisson, I omit several other great mathematical physicists whose work in pure mathematics is not important enough to qualify: Galilei, Maxwell, Heaviside, Planck, Bohr, Dirac, Heisenberg, Fermi, Pauli, Feynman.
I omit some famous mathematicians like Nikolas Lobatchewsky, Ferdinand Lindemann, and Andrew Wiles (lacking in breadth, and their "revolutionary" discoveries relied heavily on earlier work), and prolific geniuses like Paul Erdös, whose historical importance is not quite enough to qualify. I've omitted some who worked in mathematical foundations, notably Giuseppe Peano and Bertrand Russell. Finally, I've imposed an arbitrary rule that (with the exception of Alexander Grothendieck who is already regarded as one of the greatest minds ever) the List include no living mathematicians.
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 Seventy-five,
or prime contenders.)
(Names in light blue have mini-bios but are not in Top Seventy-five.)
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