Discussion about my List of Greatest Mathematicians

Controversies about the Greatest Geniuses

Since starting to develop this list, I have spent several hundreds of hours reading biographies and web articles. I have paid special attention to the opinions about geniuses by people who themselves are great geniuses. Two of the very greatest geniuses are controversial.

Galileo is an interesting case, as I discuss in his mini-bio. I became convinced he was a great genius because (a) other top geniuses like Albert Einstein praise him highly; and (b) he was the first to deduce how Archimedes tested the gold crown. (Is that true? Impressive story!)

But the amazing controversy is about Albert Einstein. I have received a largish number of e-mails from people convinced that Einstein does NOT belong on a list of greats. (Sure, he wasn't "great" at pure mathematics, and might be left off a list of great mathematicians, but that's not what concerns my correspondents. They don't think Einstein was a great physicist, or a genius of any sort. Starting from a few ignorant half-truths, they call him plagiarist. Google will find similar instances in web articles.

But everything I have read tells me that these people are 100% wrong. Einstein wasn't just a great genius, he was one of the very most creative geniuses in all of history. If I had to name The Three Most Creative Geniuses Ever I might pick Archimedes, Leonardo da Vinci and Albert Einstein.

Three incomparable geniuses lived in the 20th century: Einstein, Ramanujan and von Neumann. Eugene Wigner, himself a genius, was a close friend of von Neumann and Einstein and several other top geniuses. (And Paul Dirac was Wigner's brother-in-law.) His comments on the geniuses he knew are revealing: "[N]one if them had a mind as quick and acute as Jancsi [John] von Neumann. But Einstein's understanding was deeper even than von Neumann's. His mind was both more penetrating and more original than von Neumann's.

Why the controversy then? Albert Einstein was born into a family of secular Ashkenazi Jews. Although people I've chatted or corresponded with usually avoid mentioning this, there's no doubt that it is religious prejudice that has led to denigration of Einstein's achievement. What a sick world we live in.

Popular "Memes"

"Memes" develop, and many ordinary people decide that the memes are exaggerated. But I find some of the biggest memes ARE true. Marilyn Monroe is the sexiest actress ever? I think if you disagree you must never have watched her movies or interviews. Casablanca is the best movie ever? I pity anyone who hasn't watched this great movie.

And Albert Einstein is certainly among the very most creative geniuses who ever lived. Anyone who doesn't know this should be embarrassed.

I continue to revise the list. Recently I realized that the List of 150, which "allowed" only ten mathematicians born 1930 or later, was missing some very important names; so I added five more post-1930 mathematicians. I placed all 15 in the #106-#120 slots. Altogether, comparing today's (March 2021) List with the one 29 months earlier, I see that I've moved names around (Nash was promoted to the Top 100; Kepler was promoted to #35; etc.) but the actual names in the Top 200 have changed very little. Plato, Willard Gibbs and Karen Uhlenbeck are the only three additions to the Top 200 during those 2+ years. Alkindus, Lefschetz and Szemerédi were removed to make room for them. Mrs. Uhlenbeck becomes the 4th woman on the List of 200. Minor changes in Spring 2021 were to demote Matiyasevich, E.Moore and Roberval to make room for DeMoivre, Madhava, Heron on the Top 200. And in early 2022 I removed Shimura and Beltrami from the Top 200 to make room for Wm.V.D. Hodge and Ivan M. Vinogradov.

I realize these frequent changes make me appear indecisive, or even schizophrenic! But hey! -- this List is one of my hobbies; of course I should refine the List as I learn more and more in my readings.

In compiling this list, I've considered contributions outside mathematics. Newton contributed little to number theory, for example, but I consider him to have great breadth because of his physics, which is also his main influence. (I do give a much lower "weight" to contributions outside pure mathematics.) With different criteria, the List would appear much different. For example, here's a different ordering for the Top Twelve that might make more sense to some:

1 Leonhard Euler           4 Archimedes           7 Bernhard Riemann           10 Isaac Newton
2 Carl F. Gauss           5 Alexandre Grothendieck           8 David Hilbert           11 Henri Poincaré
3 Euclid of Alexandria           6 Joseph-Louis Lagrange           9 Pierre de Fermat           12 Évariste Galois

(The list has become fairly stable: The only major change in 2013 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 (after adding but then demoting Thabit ibn Qurra) Israel Gelfand. In 2014 I added Tarski and Frege, two of the greatest logicians ever. (Erdös and Lobachevsky were removed to make space.) Given these additions, it made sense to demote two other pure logicians on the list: Gödel and Boole.

During June and July of 2016 I made a large number of smallish changes. The Top 30 and Top 60 remained the same, though the ordering within these groups changed. There were changes in the Top 75, with Huygens and Poncelet demoted to make room for Vieta and Lie. In addition to the four major moves just mentioned, several others got demoted by four or five ranks.

September 2016 I suddenly made several overdue largish changes: Panini and Weil joined the Top 60 evicting Grassman and Monge. Also, Borel, Jacob Bernoulli and others were promoted, while Hadamard, Hipparchus, d'Alembert, etc. were demoted.

After expanding the List to 200 to include many modern mathematicians I was missing, I found that there was much more scope for on-going changes. For example, I promoted Claude Shannon to the Top 200. This may offend some -- the List is still missing deeper mathematicians -- but Shannon's several contributions to the Information Revolution give him great historical importance.

In 2018 I did some re-ordering in the Top 25, though leaving the Top 12 unchanged. I promoted Hermann Weyl and demoted Niels Abel. These changes were long overdue.

I wanted this to be a List of Great Mathematicians of the Past, but several e-mail correspondents insisted that I include modern mathematicians. So I now have a list of 200 that includes 24 living mathematicians and another ten who died in the 21st century. As you see in the following table of birthdates versus mathematician's rank, I've adopted a compromise. Birth must be before 1930 to qualify for the Top 100, and only ten of these recent guys are permitted in the Top 150. (Consider this arbitrary and stupid if you wish). Remember: If you use my List as a starting-point and come up with a List quite different from mine, I'll only be flattered!

Here are the birthyears for the Top 50, the next 50 and so on. Almost half the mathematicians born before 1000 AD who are on the List at all are shown in the Top 50. After that, ranks are more balanced except for my prejudice against incuding very recent mathematicians.
Birth period -->        Before Christ        1-999 AD        1000-1599        1600-1699        1700-1799        1800-1849        1850-1899        1900-1929        1930-1980
Greatness Cohorts
1 - 50653461394-
51 - 994263710118-
101 - 15013734610515
151 - 20021433371113
(In May 2020 I updated the post-1900 cohorts without correcting the pre-1900 cohorts -- hence the imbalance.)

Although I hadn't tried for this explicitly, as a "sanity check" I was pleased to notice a balance between the 1900-1929 cohort and the 1930-1980 cohort. Each group has precisely 14% of the Top 200, spread out (among the Fifty-sized groups) 4-8-5-11 and 0-0-15-13 respectively. If we guessed that 10 of the 15 from the post-1930 in the Top 150 actually belong in the Top 100, these would become 4-7-6-11 and 0-10-5-13. Excellent balance, except for the reluctance to put the most recent mathematicians into the Top 50.

The 17 (4+8+5) mathematicians born 1900-1929 in the Top 150 are: Grothendieck, von Neumann, Gödel, Kolmogorov; Weil, Serre, Atiyah, Selberg, Tarski, Turing, Chern, Nash; Gelfand, Erdös, Carleson, Tate. The 15 mathematicians born 1930-1980 in the Top 150 are (in birth order): Smale, Milnor, Thompson, Cohen, Langlands, Conway, Arnold, Gromov, Deligne, Shelah, Thurston, Witten, Donaldson, Gowers, Tao.

Are the Rankings "Correct"?

Since the List emphasizes the past, Grothendieck (who barely makes the 1930 birth "cutoff") does not appear in my Top Ten. Other Top Ten lists might omit Lagrange and Leibniz to make room for Grothendieck, Galois or Descartes, but Lagrange and Leibniz had extreme genius and historical importance.

There's overlap between top mathematicians and top practitioners of other areas, especially physics. I'm afraid that, almost unconsciously, I've allowed greatness at physics (or other areas) to blur in and affect my ratings. Newton belongs somewhere on the Top Ten, but he'd not get the #1 rank without his physics. Similarly, I've promoted Leibniz, von Neumann, Weyl, Pythagoras, Pascal, Archytas, Alhazen, Kepler, Grassmann, Huygens, and several others because of their work outside pure mathematics. (As an extreme case, Leonardo da Vinci appears on the extended list.) Does this affect the validity of my rankings? Perhaps -- But I always encourage readers, who might have different criteria, to use this List as a starting-point in constructing their own List.

Even an expert with very clear criteria would have trouble assigning exact ranks to mathematicians' "greatness," if such a thing even has meaning; and I am not an expert and do not have clear criteria. Nevertheless I am largely satisfied with the present rankings. I believe that the Top Ten are the only possible Top Ten! I think the Top Thirty and the Top Sixty are also reasonable.

In recent years I've made minor changes to the Top Thirty, demoting Eudoxus and Al-Khowarizmi and promoting Hamilton and Aryabhata. I'd also like to make room for Apollonius in the Top 30. (Despite his historical importance and fame, there were several Islamic mathematicians with more genius than Al-Khowarizmi, who borrowed much from the great Hindu mathematician Aryabhata.)

The only changes to the Top 60 List since early 2013 have been to replace Boole, Fourier, Poncelet and Vieta with Borel, Grassmann, Liouville and Peano. Liouville at #54 may still be too low; he was a mathematician of outstanding breadth.

A major change to the Top 90 was to include five applied mathematicians: Galileo, Einstein, Maxwell, Aristotle, Cardano. To make room for these five, I needed small demotions of Hausdorff, Artin and Pólya; a large demotion of Wallis, and a huge demotion of Boole.

Of course there will be quibbling about the precise order within these lists. Some think Ramanujan isn't historically important enough to make the Top 30, but please judge my list by the criteria I use (breadth of work and depth of genius also considered), not the criteria you'd have used.

I'm still willing to try to improve the List. :-) No List will please everybody, so please complain only if your choice for Top 75 is missing from my Top 150 (or if I have someone in the Top 75 who doesn't even belong in the Top 150). Please E-mail suggestions for the List (and, more importantly, corrections to the mini-bios) to me.


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 Gödel have great "depth" because their work was especially creative and revolutionary. Weierstrass and Dirichlet have great "depth" because 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!

Born before 1930

Some will object to my decision to ignore mathematicians born after 1930, but this decision seemed clearly right to me for several reasons:

Seven born between 1910 and 1930 appear in the Top 105. I'm quite unsure whether my "rankings" of these relatively recent mathematicians will meet the test of time. Adding more recent mathematicians would just increase my anxiety!

Anyway, my List isn't intended to be anything but a starting point. You're welcome to adopt it with additions and subtractions as you wish!

The "Maxwell Quintet"

There are five men who are among the most important mathematical scientists ever, yet may lack the importance as pure mathematicians for a list such as this -- Maxwell, Einstein, Galileo, Aristotle and Cardano. I'd definitely want to include them in any List of Greatest Mathematicans with size 100 or more, but am not sure where on the List to put them; instead I've just set them aside as an extra quintet that brings the List to 105 total.

There are several other outstanding physicists that may seem to belong in the same category, for examples: Huygens, Kepler, Alhazen. However each of these did produce work that affected the development of pure mathematics (though I've considered their importance to physics in their high rankings). The Maxwell quintet, on the other hand, lack the mathematical importance to belong on the list at all, and are mentioned solely because of their huge importance as applied mathematicians.

Other top theoretical mathematical physicists include Dirac, who is on the List of 200; and Gibbs, Boltzmann, Feynman, Heisenberg and Schrödinger who are not.

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 Legendre 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 of 100 and bios into a single, very large, page. (gmat200.htm is an even longer page with the List expanded to 200 names.)

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 eight people still living, or who were alive after 1986. These names are Atiyah, Chern, Gelfand, 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 Cardano) 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, 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.

Willard Gibbs, Ludwig Boltzmann and Richard Feynman would belong high on a List of Mathematical Physicists, but a line has to be drawn somewhere and they're omitted from the Top 200 Mathematicians.

I also include a few whose greatness was teaching or preserving ancient math, rather than doing original mathematics themselves. And again I need to draw the line. Textbook writers like Boethius or Chang, who might be important historically, are omitted from the list.

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 Kepler, Boole, Cayley, Hardy, Weyl, Dirac, Atiyah and Banach all contain the word "beauty." If words like "poetry" (or "ecstacy" -- Kepler) are considered, d'Alembert, Steiner, Weierstrass, Kovalevskaya, 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."

Gosta Mittag-Leffler wrote "The mathematician's best work is art, a high perfect art, as daring as the most secret dreams of imagination."

In his memorial address for Minkowski, Hilbert said "Our science, which we loved above everything, had brought us together. It appeared to us as a flowering garden. In this garden there are beaten paths where one may look around at leisure and enjoy oneself without effort, especially at the side of a congenial companion. But we also liked to seek out hidden trails and discovered many a novel view, beautiful to behold, so we thought, and when we pointed them out to one another our joy was perfect."

Ten Most Beautiful Theorems

The Mathematical Intelligencer conducted a survey to see which theorems its readers selected as most beautiful. The Ten Most Beautiful Theorems include no less than four by Euler, two credited to Euclid (Only 5 regular polyhedra, and the infinitude of primes), Brouwer's Fixed Point Theorem, the Hermite-Lindemann proof that pi is transcendental, the 4-color map theorem proven by Appel, Haken and Computer, and the discovery by Pythagoras' school that the square root of 2 is irrational.

This list shows the survey scores. Lagrange's Theorem and the Cayley-Hamilton Theorem scored #11 and #12; I include them and their score to show that truncating to the Top Ten was reasonable.

  1. (7.7) ei π = -1
  2. (7.5) V + F = E + 2
  3. (7.5) There is no largest prime
  4. (7.0) There are only five convex regular polyhedra
  5. (7.0) ζ(2) = π2/6
  6. (6.8) A continuous mapping from a disk to itself must have a fixed point
  7. (6.7) √2 is irrational
  8. (6.5) π is transcendental
  9. (6.2) Any planar map can be four-colored
  10. (6.0) Prime 4k+1 = a2 + b2 has exactly one solution
  11. (5.3) The order of a subgroup divides the order of the group
  12. (5.2) Any square matrix satisfies its characteristic equation


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,
  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)

First Discoverers

The simplified mini-bios ignore some first discoveries. Neither Huygens nor Leibniz was first to sum the reciprocals of the triangle numbers; it was Pietro Mengoli (1625-1686).

I don't list Lorenzo Mascheroni (1750-1800) who is credited with the amazing theorem that straight-edge is unnecessary for Platonic constructions; that theorem was actually first discovered by the forgotten Jorgen Mohr (1640-1697).

Many questions of first discovery can be tied to "threads": The biggest threads in the whole of mathematical physics are the issues of optics and celestial motion. One question where these come together is: What is the speed of light? See below.


There are several threads which flow through the stories of the greatest mathematicians. The elementary bases of science like mensuration methods, light refraction, etc. qualify. (The development and understanding of lenses by Alhazen, Galileo, Fermat, Huygens, Newton, etc. is a major thread in the early history of science . So are other issues in optics, like understanding rainbows.)

Several exercised their talents by computing π, or by trisecting the general angle, or constructing the regular heptagon. Some worked in code-breaking. Several are closely involved with the theory of celestial motions, or the theory of music.

Several of the mathematicians worked with ancient problems like Apollonius' tangent circles, or Fermat's Last Theorem. Gauss' solution of FLT-3 for complex numbers led directly to great insights into algebra. Chasles' list of the six who conceived of calculus seems useful: Archimedes, Kepler, Cavalieri, Fermat, Leibniz, Newton.

A few "Fundamental Theorems" have been worked on and extended over the centuries. In this category is the work of Gauss and Bonnet whose eponymous theorem linked geometry and topology -- a simple version had already been stated by Descartes, and several 20th-century mathematicians took it far beyond Gauss's work.

One topic of applied mathematics which shows up in the mini-bios is map projections: representing the spherical earth on planar paper. Among those who invented new map projections are Thales, Hipparchus, Ptolemy, Alberuni, Loenardo da Vinci, Lambert (several), Lagrange and Karl Gauss. (I don't mention Lagrange's map projection in his mini-bio. He improved one of Lambert's projections, one where all latitude and longitude lines are circular arcs.) Klein, Darboux, and other great mathematicians also worked with such projections, though their names may not appear on actual map-making algorithms.

The development of heliocentrism was an important thread in the development of science; I discuss it a little in the mini-bios of Ptolemy and Copernicus. Not only did the facts of celestial motion lead to Newton's Laws of Gravitation and Motion, but placing the Earth away from the center led Galileo and others to conclude that physical laws are constant throughout the universe, rather than differing between heaven and earth. (It may have been this notion that most offended the Church.)

Another interesting thread concerned the distinctions betwen constructible and non-constructible numbers. By allowing addition, then division, then square roots, then cube and higher roots, one gets integers, rationals, Euclid-constructibles, and Galois-constructibles; then come algebraic numbers; the leftover reals are called transcendentals. One of Pythagoras' disciples proved a number irrational; Omar al-Khayyám speculated that many cubic polynomial roots were not Euclid-constructible; and Fibonacci demonstrated one that wasn't. Abel and Galois achieved fame by noting algebraic numbers not Galois-constructible; and the hunt for transcendental numbers made Liouville and Hermite famous.

Determining the Speed of Light

The first calculation of the speed of light is an interesting story. Galileo and his assistant used two lanterns and a water-clock but concluded only that light's speed was at least Mach 10 (as we call it now).

However Galileo's discovery of the moons of Jupiter did lead to a measurement of light's speed! Recall that Galileo proposed that the eclipses of Jupiter's moons provided a universal clock that could be used to find longitudes. A 1675 project of Danish astronomer Ole Rømer was to pursue this by recording the exact times that Jupiter eclipsed its nearest moon, Io. (The eclipse lasts for two hours, so is far too fuzzy; the idea is to note the crisper times of immersion into, or emergence from eclipse.) The orbit of Io takes about 42.5 hours, but Rømer discovered via cumulative errors that Io's orbit appears to be 32 seconds faster when Earth is moving toward Jupiter compared with Earth moving away. This led to the very first estimate of the speed of light. Among the first to acknowledge this discovery were Rømer's contemporaries, Christiaan Huygens and Isaac Newton. Although certain crudities made Rømer's speed estimate a little off, improved measurements of the discrepancies in Io's orbit provided the most accurate estimate of the speed of light for almost another 200 years. (The calculation depended on AU -- the distance from Earth to Sun; errors in the AU estimate was by far the biggest error source in Rømer's result. Rømer had several other interesting accomplishments but his discovery of the speed of light may be most famous.)


Dates and Nationalities of Mathematicians on List

Of the Top 105 mathematicians, 12 fluorished during the Antique Classical Age, 8 during the Middle Ages, then (based on death year) 1, 9, 9, 28, 32 in respectively the 16th, 17th, 18th, 19th, 20th centuries, and 6 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.

Here is the birth-year breakdown for the Top 150/200 as of January 2022: 11/13 born BC, 6/7 born between 1 and 700, 9/13 born 700 to 1400, 1/2 in 15th century, 9/10 in 16th century, 10/13 in 17th century, 16/20 in 18th century, 29/32 1800 to 1849, 27/35 1850 to 1900, 17/29 1900 to 1929, 15/26 1930 or later.

Of the 105 Great Mathematicians, there are 23 from France; 21 from Germany; 11 from the ancient Greek world; 9 from England; 7 from Switzerland; 6 from Italy; 5 from India; 3 each from Holland, Norway, Russia, and Persian domain, 2 each from China, Hungary and Poland; and 1 each from Austria, Ireland, Lithuania, Scotland and 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.) By-gone mathematicians scoring 170 or higher in that survey include 10 others on my list (Newton, Laplace, Lagrange, Descartes, 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 or physicists 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, von Neumann, Gauss, Maxwell, Ettore Majorana, Blaise Pascal, Rudolf Clausius, Willard Gibbs, Ludwig Boltzmann, Nikola Tesla, Paul Dirac, Archimedes, Euler and Aristotle). The precocious John Stuart Mill, Thomas Young (1773-1829) `the Last Man who knew Everything', and Empedocles of Acragas are some top geniuses who didn't specialize in math, Wittgenstein is sometimes shown with the highest IQ ever but is missing from my list since his emphasis was logic and philosophy.

Hmolpedia's I.Q. Estimates

Libb Thims' Hmolpedia Project has a page with 315 historical persons that Mr. Thims estimates had an IQ of 175+. Of these, 71 appear in my list of 370. Another 15 or so from my Top 370, appear on Mr. Thim's lists for IQs 160-174.

Complex Numbers

Bombelli, called the Inventor of Complex Numbers (and who was also one of the first Europeas to treat negative numbers) might deserve a prime place on the list. But complex numbers were a gradual development. That some quadratics have only "fictitious" solutions was known in ancient Greece; Cardano worked with them before Bombelli; 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 "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 solution to the "Basel Problem" may be interesting. The following facts were already well known to him ((3) was discovered by Isaac Newton):

  1. If f(x) is a polynomial with zeros (roots) at a, b, etc. then f(x) = (x-a) (x-b) ...
  2. lim sin(x)/x = 1 as x --> 0
  3. sin x = x - x3/3! + x5/5! - ...

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, 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!

There are other proofs of Euler's discovery
    π2/6 = 1-2 + 2-2 + 3-2 + 4-2 + ...
or of the closely related
    π2/8 = 1-2 + 3-2 + 5-2 + 7-2 + ...

Here's a proof of the latter formula that seems "neat": It's a geometric proof, and since we need π in the answer, the proof involves circles!

Consider a circular lake of diameter W, with W number of lighthouses spaced equally around the circumference. You are located on the shore midway between two of the lighthouses. There is a theorem that the light intensity reaching you is independent of W.

If W is 1, the single lighthouse is opposite you, at distance 1, and one unit of light intensity reaches you. If W is 3, the lighthouse opposite you, at distance 3, provides 1/9 units of light intensity -- that's the inverse-squared-distance law. A second lighthouse forms a 30-60-90 triangle with you and the distal triangle, so its distance to you is 3/2 and its light intensity is (3/2)^(-2) = 4/9. The third lighthouse is symmetric to the second; the three light intensities 1/9 + 4/9 + 4/9 sum to 1 just as for W=1. So we've shown the theorem holds for W=1 and W=3. W=2 is also easy; what about the general case? We will show that if the theorem holds for some W it also holds for 2W, so holds whenever W is a power-of-two. (This won't complete the proof -- what about W=5? -- but is all we need to prove Euler's Identity.)

The key will be the geometric ideas depicted in the diagram at right. D is the center of a small circle; CDE a diameter of that circle, and also the radius of a circle twice as tall. ACF is an arbitrary diameter of the large circle, and B the point where that diameter intersects the small circle. Observe that CBE is a right angle (Thales' theorem) and the angle BCE is half of the angle t=BDE (Inscribed angle theorem). The point B at angle t on the small circle projects to two points, A and F, on the big circle, at angles t/2 and t/2 - π.

Now consider the triangle AEF (to reduce clutter, we've not drawn the legs (x = AE, y = FE) of this triangle). The angle AEF is right, again by Thales' theorem, so the legs satisfy x^2 + y^2 = z^2 where z is the length of the hypoteneuse, but there is another, less well-known, relationship in right triangles: x^-2 + y^-2 = h^-2, where h is the length of altitude BE.

Thus two lighthouses at A and F provide the same combined light intensity at the observer E as a single lighthouse at B. Instead of just the point B projecting to A and E on the twice-as-large circle, imagine W different points projecting to 2W new points on the large circle. And, as a consequence that the angle t becomes t/2 (see above), if the W points are spaced as desired, the 2W points will be also.

The distance between adjacent lighthouses along the circumference will be π/W, but let's multiply everything by 2/π. Now the distance between lighthouses is 2. (Starting from the observer they are at distances 1,3,5,7,.... as measured along the circumference.) With the scaling, instead of 1, the total light intensity is π^2/4. And let's extinguish half the lighthouses -- those to your left. Now the light intensity is π^2/8. Finally let W become so large that the circumference appears as a straight line. Then the total light intensity is
    π^2/8 = 1-2 + 3-2 + 5-2 + 7-2 + ...


Fermat's Little Theorem

Although Fermat did not discover it, there is a beautifully simple proof of his "Little Theorem" based on counting necklaces. A necklace can be built from p beads using a different colors in ap ways, or (ap-a) ways when we insist that at least two different colors be used. Rotating such a necklace will produce p distinct cases when p is prime; this defines equivalence classes each of size p. Therefore (ap-a) must be a multiple of p, which is what the Little Theorem states.

Fermat's Christmas Theorem

Fermat's famous Christmas is harder to prove than the "Little" Theorem but here is an elegant new proof of that theorem based on counting windmill shapes!

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 (correctly) that 163 was the largest of them. (Ramanujan's formula for π is also related to the Heegner number 163.)

An Elegant Diagram

Names like Einstein and da Vinci appear on the List even though they didn't specialize in mathematics. Even Kepler is much more famous as an astronomer. 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 only 10 decimal digits, a prime discovered in 2018 has almost 25 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.)

(An archaelogist has deciphered ancient clay tablets from Ugarit showing a song in a diatonic scale which predates Pythagoras by eight centuries. Here is that ancient song.)

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 (and at about the same time, Zhu Zaiyu of China) 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%. (Marin Mersenne suggested a constructible number for the half-step ratio, but 12 of those half-steps would make the octave ratio 2.006 instead of 2.)

I'm not sure but perhaps these are tuning values (with C set to 32 in arbitrary units). Bach's "well-temperament" is conjectured. He wrote a complex piece with 12 different major keys (and 12 minor keys), each key having its own mood. But the major keys are all equivalent with equal-temperament, almost universally used today.
Equal temperament   24.0025.4326.9428.5430.2432.0433.9435.9638.1040.3642.7645.3148.00
Bach's well-temperament     24.00     25.40     26.88     28.51     30.10     32.07     33.87     35.92     38.06     40.23     42.72     45.16     48.00  

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.)

Fibonacci weights

Ancient Mesopotamia had a monetary system based on a shekel of silver. A shekel was said to be equal in weight to 180 grains of barley. In medieval Europe, fine silver was also weighed in terms of barleycorns, one such grain being defined as about 64.9 milligrams today, but according to this article a bag of 14 weights found in the excavations at Ur imply a 46.7 milligram grain, about the weight of a grain of wheat. The weights were mostly agate stones; there was one of 3 such grains, three of 9 grains, four of 18 grains, three of 27 grains, two of 45 grains, and one of 72 grains. Ignoring the single smallest weight, and treating 9 grains as the unit. these weights are 1, 2, 3, 5, 8. A Fibonacci series was convenient for the manipulations when trying to weigh sdmall quantities, presumably of gold or silver!

Great Mathematicians' Self-Appraisals

Hardy once described himself as "for a time the 5th best mathematician in the world." What was the time, and who were the Top Four?

Hardy was doing excellent work in his 20's, but Hilbert, Poincare, Cartan, Hadamard and Cantor were established greats, and Noether, Weyl and Littlewood were hot on Hardy's heels. When Hardy was approaching his 60's, von Neumann, Weyl, Cartan, Siegel were the obvious greats with Littlewood and others also in contention. Hardy himself said he was at his peak in his early 40's, and it must be after the death of Ramanujan, that he became "fifth best in the world." At this time, the powers of Hilbert were waning and von Neumann was still very young.

The best candidates for Top Four in 1922, about when Hardy considered himself 5th-best, were Weyl, Noether, Cartan, and probably either Siegel or Littlewood.

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. The likeliest candidates are Hermann Weyl and Kurt Gödel since von Neumann had shown strong respect for their talents. Interestingly, these two, along with two other possible candidates (Carl Siegel and André Weil) were all faculty colleagues of von Neumann at Princeton's Institute of Advanced Study. (Not at AIS was Emma Noether, another of the great mathematicians of that era.) Despite Von Neumann's modesty, I've ranked him as "best of his time" because of his huge breadth: foundations, analysis, game theory, etc. (Many other great mathematicians have served as faculty at I.A.S. including Einstein, Veblen, Atiyah, Deligne, Selberg, Milnor and Witten.)

Von Neumann had huge respect for David Hilbert, and lamented that he could not compete for greatness with that man. It is possible that, despite the big gap in dates, Hilbert was one of the two that John von Neumann was thinking of when he made his remark.

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 Descartes:     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 Descartes a bit lower. This is partly because of the respective modesty they show in their self-appraisals above. By the way, Newton described his method of discovery with "I keep the subject constantly before me, and wait 'till the first dawnings open slowly, by little and little, into a full and clear light.")

To prepare my List I have frequently reviewed the comments by great mathematicians about other great mathematicians. Here's what Atle Selberg said when contrasting another mathematician's views on Weyl and Siegel:

"You see, there are two kinds of great mathematician. The first kind solves a difficult problem; it becomes thereafter easy. Such a mathematician was Hermann Weyl. The second kind solves a difficult problem, it still remains difficult; such a mathematician was Carl Ludwig Siegel."

Other Lists

Pickover's List

Cliff Pickover has a list of the "Ten Most Influential Mathematicians." His Top Ten include seven of my top nine (all except Archimedes and Lagrange), but bypass my #10-#12 to include Descartes, Galois and Pascal. His omitting Archimedes and including Descartes may make sense since his criterion emphasizes historical importance, but Pascal seems an odd choice: many mathematicians were more influential than Pascal.

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 six names missing from my Top Hundred; in Eells' order these are: Napier, Ptolemy, Regiomontanus, Maclaurin, Tartaglia, Heron. Eells' List includes 25 names missing from my Top 200; only five of these (Tartaglia, Boskovic, Barrow, Sturm, De Morgan) rank in Eells' Top 55. (The other 20 on Eells' 100 missing from my 200 are Briggs, Carnot, Maupertuis, Babbage, Pacioli, Gerbert, Cotes, Ahmes, Borda, Frisi(*), Montucla(*), Hesse, Jordanus de Nemore, Halley, Ampere, L'Hospital(*), Thomson Lord Kelvin, Boethius, Tschirnhausen(*), Zeno. The four asterisked names are missing from my List of 370.)

On my List of 100 there are about 72 names born before 1843; 18 of these are missing from Eells' List of 100, including seven from the modern era: Dedekind, Eisenstein, Kummer, Liouville, Maxwell, Chebyshev, Jordan.

Polyanin's List

Andrei D. Polyanin has a List of the Top 24 Mathematicians. It contains fourteen names from my Top 17 (all but Grothendieck, Fermat, von Neumann), three more names from my Top Twenty-Five (Abel, Jacobi, Cauchy), three more names from my Top Fifty (d'Alembert, Kolmogorov, Laplace), Fourier (my #67), and three Russian mathematicians each of whom I rank near #110 (Lobachevsky, 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, Kronecker) while he is missing four of mine from the same time period (Eisenstein, Lambert, Plücker, Steiner). For the period from 1825 to 1905, James is missing seven names (Frege, Jordan, Lebesgue, Littlewood, Peano, Siegel, Tarski) from my List of 100. My List of 200 is missing eight names from James' List (Aleksandrov, James Alexander, Richard Courant, Gosta Mittag-Leffler, Lefschetz, E.H. Moore, Robert L. Moore(*) and Teiji Takagi). another two of James' names are present on my List of 200 but missing from my Top 150: Germain, Zariski.

Mastin's List

Luke Mastin lists 110 important mathematicians. These are ones he chose to summarize the story of math's development, rather than the "greatest;" and nine of his names are not even on my List of 370; nevertheless I'll compare the lists. Mastin is missing four names (Dirichlet, Weyl, Noether, Hermite) from my Top 40, and another 12 from my 41 to 60. My List of 200 is missing 18 names from Mastin's list: Hippasus (Pythagoras' student), Democritus, Sun Tzu, Yang Hui (mentioned in Qin's mini-bio), al-Farisi, Pacioli, Ferrari, Mersenne, Goldbach, Peacock, Venn, Whitehead, Fatou, Julia, Lorenz, Julia Robinson, Wiles.

Ian Stewart's List of 17 Equations

Ian Stewart associates names with his "17 equations that changed the world." Twelve of these names appear on my List of 200: Einstein, Pythagoras, Maxwell, Napier, Gauss, Shannon, d'Alembert, Fourier, and, with two equations each, Newton and Euler.

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, W.V.D. Hodge, James Ivory, Lamb, Le Verrier, Lorentz, Penrose, Rayleigh, Salmon, Stokes, Sturm, Wm. Thomson, Waring, E.T. Whittaker, Wiles. (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.)

Newton and Atiyah both served as President of the Royal Society. Several other outstanding mathematicians served as the Society's President including William Brouncker, Christopher Wren, William Spottiswoode, George Stokes, George Airy and Lord Kelvin.

Prestigious prizes where preference is given to Britons include the De Morgan Prize and Sylvester Medal. From my list of 200, Atiyah, Cantor, Carleson, Cayley, Darboux, Hardy (both), Klein, Levi-Civita, Littlewood (both), Penrose, Poincaré, Sylvester, J.G. Thompson (both) have won one (or both) of those prizes. Among several other winners are Besicovitch (both), Coxeter, Davenport, Lamb, Mordell, Rayleigh, Russell (both), Titchmarsh (both), Whitehead. One of the top math chairs at Oxford or Cambridge has been held by 7 on my list: Wallis, Newton, Smith, Sylvester, Hardy, Dirac, Atiyah. Other Savilian or Lucasian Professors are Briggs, Barrow, Halley, Babbage, Stokes, Titchmarsh, Hawking and 23 others.

Five mathematicians on my List won Cambridge's Smith's Prize: Cayley, Maxwell, Hardy, Hodge, Littlewood and Turing. Among these, Cayley and Littlewood were also Senior Wranglers at Cambridge; Maxwell Second Wrangler, and Hardy Fourth Wrangler. Turing took the test in a year when ranks were not announced -- he might have been the highest-scoring Wrangler. Some other famous mathematicians who won Smith's Prize and were Senior or Second Wrangler include John Couch Adams, Airy, Henry Coddington, Coxeter, Eddington, Lamb, Mordell, James Stirling, Stokes, Strutt (Lord Rayleigh), Thomson (Lord Kelvin). From my List, Sylvester and Clifford were Second Wrangler.

Some other Senior Wranglers are Edward Waring, John Wilson (of Wilson's Theorem) and George Stokes.

Miss Philippa Fawcett was ineligible for Wrangler but was permitted to take the examination: she scored above the Senior Wrangler. (As an experiment, Hardy asked George Pólya to sit for the undergraduate examination when he was Visiting Professor; the results, which were not published in that year, are mysterious. Some say Pólya failed, others say that he joined Miss Fawcett in the rare feat of scoring "above the Senior Wrangler.")

The Wolf Prize, Abel Prize and Fields Medal are perhaps the most prestigious of math awards; winners from my List of 100 include Atiyah (3), Chern, Erdös, Grothendieck, Kolmogorov, Selberg (2), Serre (3), Siegel, and Weil. There are several other major prizes (e.g. the Bocher Memorial Prize) given for mathematics, or sometimes (e.g. Nobel Prize, Copley Medal) given to mathematicians. At least three dozen mathematicians not on the List of 100 have won two or more major prizes; these include 21 from my List of 200: Lars Ahlfors, Lennart Carleson, John H. Conway, Paul Cohen, Paul Dirac, Pierre Deligne (3), Simon Donaldson, Gerd Faltings, Michael Freedman, Timothy Gowers, Mikhail Gromov (3), Lars V. Hörmander, Kunihiko Kodaira, Peter Lax, Solomon Lefschetz, John W. Milnor (3), David Mumford, John Nash, Sergei Novikov, Roger Penrose, Yakov Sinai (3), Isadore Singer, Stephen Smale, Terence Tao (3), John T. Tate, John G. Thompson (3), William Thurston, Norbert Wiener, Edward Witten (3); and some names not on my list: James W. Alexander, Enrico Bombieri, Jean Bourgain, Jesse Douglas, Charles Fefferman, Maxim Kontsevich, Gregory Margulis, Shigefumi Mori, Louis Nirenberg, Daniel Quillen, Klaus Roth, Wendelin Werner, Shing-Tung Yau.

Twelve Greatest Scientists

My own "Five Greatest Scientists" List would be: Newton, Einstein, Galileo, Maxwell, and Archimedes, all of whom are also top mathematicians. To bring this up to a "Top Twelve" list, I'd add Aristotle and six non-mathematicians: Rutherford, Bohr, Faraday, Pasteur, Darwin, and Lavoisier. (For a Top Fifteen list I'll add Aryabhata, Alhazen and Kepler from my greatest mathematicians list, to represent the Scientific Revolutions in the Hindu, Islamic and Christian worlds respectively.)

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 eight from my List of 200: Archimedes, Euclid, Apollonius, al-Khowârizmi, Archytas, Aristotle, Ptolemy, al-Kindi; and four other names: John Duns Scotus, Richard Swineshead (aka John Suisset) the Calculator, Galen of Pergamum (physician), and Jabir ibn Aflah (astronomer and mathematician whose works were translated into Latin). (Sources show "Heber of Spain" as the 12th name; it is my assumption that this refers to Geber of Spain, an alternate name for Jabir ibn Aflah. There was another Geber, the very famous 8th-century scientist and chemist Jabir ibn Hayyan, and a "pseudo-Geber" named after Jabir ibn Hayyan; perhaps Cardano conflated these figures. However in another work, Cardano calls specific attention to Jabir ibn Aflah and writes that Regiomontanus borrowed his ideas from that Jabir.) 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 are Newton, Einstein, Bohr, Darwin, Pasteur, Freud, Galileo, Lavoisier, Kepler, Copernicus, Faraday, Maxwell. (This is the same set as my Top 12 Scientists if Freud and Copernicus are replaced with Archimedes and Aristotle.) Five of these names are in my List of 100. 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, Descartes, Kepler, Euler, Aristotle, Galileo, Maxwell) and four other mathematical physicists (Copernicus, Heisenberg, Planck, Fermi). Hart's list also includes Pasteur, Lavoisier, Faraday, Dalton, Plato, Rutherford, Roentgen, Francis Bacon and seventy-nine people who did no work in math or pure physical science (44 political, military or religious leaders; 2 explorers; 16 inventors; 9 social philosophers; 5 artists; 3 biologists).

Among scientists, Hart omits Bohr altogether -- though he included him in a First Edition of the List; he also 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 often seen near the top of other Top Scientist lists include Nikola Tesla, Marie Curie, Thomas Edison, and two names from my List: Leonardo da Vinci and Alan Turing.

Researchers at MIT's Pantheon project, using the statistics of on-line biographies, prepared their own list of the 100 most influential persons ever. Combining this list with Hart's yields a List of 157. This combined list includes 15 mathematicians from my List of 50; these are the nine already mentioned from Hart's List and six more: Leonardo da Vinci (ranked #6), Pythagoras (#10), Archimedes (#11), Thales (#38), Pascal (#67), Ptolemy (#80). A small sample of the non-mathematicians on Pantheon's list missing from Hart's list are Socrates (ranked #4), Mozart (#16), Dante Alighieri (#26), Cleopatra VII (#29), Herodotus (#32), Virgil (#36), Marco Polo (#40), Joan of Arc (#53), Epicurus (#56) Goethe (#62), Mahatma Gandhi (#75), Saladin (#76), Heraclitus (#84), Gilgamesh (#85), Anne Boleyn (#91), Avicenna (#97).

Female Mathematicians

I hope no one takes offense that 29 of the Top Thirty and 196 of the Top 200 are males, In addition to Emmy Noether, Sophia Vasilyevna Kovalevskaya, Marie-Sophie Germain, and Karen Uhlenbeck, other famous female mathematicians include Hypatia (the last librarian of Alexandria), and 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. (Above we mentioned Philippa Fawcett, who scored above Cambridge's Senior Wrangler.)

Omissions from the List

(In the following discussion I list some notable omissions from the List of 100. Many or most of these do appear on the List of 200.)

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 Torricelli, 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 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 named 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. It was partly to make room for him that I made an expanded list with, now, 150 names.

Some mathematicians are famous and important for their publications of others' work. There are few of these on the List of 100 (unless you count Euclid), but some appear on the Extended List of 380 candidates. Proclus Diadochus was especially important -- much of what we know about ancient Greek mathamtics is due to his writing. But I could not bring myself to mention Nicomachus of Gerasa -- his writings may have been historically important, but they contain many errors.

Finally, I've imposed an arbitrary rule: the List of 100 includes no mathematicians born after 1930. (This rule is relaxed slightly for the List of 150 and eliminated altogether for the List of 200.)

Proposed candidates

If the List of 200 still isn't long enough for you, here is a list of about 370 of some of the most important mathematicians.
    (Names in deep purple are on my Top Thirty List. Names in bold-face are candidates for the Top Two Hundred.)
    (This long list is divided into three parts, based on whether the mathematician died before 1780, died before 1962, or was living some time after 1961.)
  • Endre Szemerédi (1940-)
  • Enrico Bombieri (1940-)
  • S.R. Srinivasa Varadhan (1940-)
  • Dennis Parnell Sullivan (1941-)
  • Karen Keskulla Uhlenbeck (1942-)
  • Mikhail Gromov (1943-)
  • Pierre René Deligne (1944-)
  • Saharon Shelah (1945-)
  • Gregori A. Margulis (1946-)
  • Alain Connes (1947-)
  • Masaki Kashiwara (1947-)
  • Yuri V. Matiyasevich (1948-)
  • Lovász László (1948-)
  • Charles L. Fefferman (1949-)
  • Shing-Tung Yau (1949-)
  • Edward Witten (1951-)
  • Michael H. Freedman (1951-)
  • Shigefumi Mori (1951-)
  • Andrew Wiles (1953-)
  • Gerd Faltings (1954-)
  • Simon K. Donaldson (1957-)
  • Stephen Wolfram (1959-)
  • Richard Taylor (1962-)
  • Wm. Timothy Gowers (1963-)
  • Maxim L. Kontsevich (1964-)
  • Grigori Perelman (1966-)
  • Wendelin Werner (1968-)
  • Shinichi Mochizuki (1969-)
  • Cédric Villani (1973-)
  • Terence Chi-Shen Tao (1975-)
  • Jacob Lurie (1977-)
  • Ben Green (1977-)
  • Peter Scholze (1987-)

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