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There are mathematicians whose creativity, insight and taste have the power of driving anyone into a world of beautiful ideas, which can inspire the desire, even the need for doing mathematics, or can make one to confront some kind of problems, dedicate his life to a branch of math, or choose an specific research topic.

I think that this kind of force must not be underestimated; on the contrary, we have the duty to take advantage of it in order to improve the mathematical education of those who may come after us, using the work of those gifted mathematicians (and even their own words) to inspire them as they inspired ourselves.

So, I'm interested on knowing who (the mathematician), when (in which moment of your career), where (which specific work) and why this person had an impact on your way of looking at math. Like this, we will have an statistic about which mathematicians are more akin to appeal to our students at any moment of their development. Please, keep one mathematician for post, so that votes are really representative.

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closed as off topic by quid, Yemon Choi, Andres Caicedo, Ryan Budney, S. Carnahan Sep 5 '11 at 14:08

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It's always your advisor(s) that influence you the most, aren't they? –  Zsbán Ambrus Jul 5 '10 at 22:00
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@Zsbán: I'm not so sure about that! I have the feeling that many mathematicians are most influenced by some others, or some works, or some open problems, or even some teachers BEFORE getting to have an advisor at all! –  Jose Brox Jul 5 '10 at 23:12
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Interesting question. I noticed that all the stronger mathematicians I know (or know of) have other mathematicians that they look up to (sometimes long gone mathematicians who only communicate with us through their writings). So that the most influential may also be the most influenced (insert "shoulders of giants" Newton quote here). You would expect some self-made geniuses out there, people who feel they owe their success mostly to themselves, but I have yet to come across one. –  Thierry Zell Aug 14 '10 at 1:12
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This is a nice list, but perhaps it is long enough. I vote to close. –  quid Sep 2 '11 at 18:17
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I agree with quid, and am voting likewise –  Yemon Choi Sep 2 '11 at 20:44

62 Answers 62

Alexander Grothendieck. See, for example, his passage about opening a nut. This was very inspiring for me and was one of the key reasons that led me to abandon computer science and start studying math. I also very much like the way he uses geometric intuition in algebraic geometry, it helped me a lot and not only in algebraic geometry.

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The passage you refer to is probably the most motivating thing I've ever heard about doing mathematics...not that I can effectively implement it! –  Jon Bannon Oct 19 '10 at 18:09
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@Dr Shello: From Récoltes et Semailles, page 552: “I can illustrate the second approach with the same image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months—when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!” –  Dmitri Pavlov Apr 27 '11 at 4:17
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@Dr Shello: “A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration… the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it… yet it finally surrounds the resistant substance.” –  Dmitri Pavlov Apr 27 '11 at 4:18

Gromov.

Gromov 1982

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Gromov is, I think, one of the more creative mathematicians I know. He rarely uses much machinery or does anything technically difficult. But he often takes familiar ideas or techniques and finds surprising ways to use them. I sometimes laugh in disbelief when I read Gromov's work. –  Deane Yang Nov 14 '09 at 23:59
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Gromov is one of my heros, but I think your definition of "technically difficult" is different from mine... –  Andy Putman Nov 15 '09 at 0:41
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Well, maybe I just don't read the stuff that is technically difficult. But I love the way he can prove nontrivial theorems about, say, isometric embeddings without doing any hard PDE estimates. –  Deane Yang Nov 16 '09 at 0:16
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I think he often manages to avoid technically difficult patches by skipping over them and letting others work them out. –  Jeffrey Giansiracusa May 14 '10 at 7:00
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I once asked a very prominent symplectic geometer whether Gromov's paper on pseudo-holomorphic curves was the best place to learn the subject. The answer was: yes, but DON"T READ THE PROOFS. –  Igor Rivin Jan 17 '11 at 0:11

Thurston. When I was a graduate student, Thurston's work really inspired me to appreciate the role of imagination and visualization in geometry/topology.

A prominent mathematician once remarked to me that Thurston was the most underappreciated mathematician alive today. When I pointed out that Thurston had a Fields medal and innumerable other accolades, he replied that this was not incompatible with his thesis.

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Sir Michael Atiyah.

Besides his great technical work (his collected papers are absolutely magnificent!) especially his great interview "Beauty in Mathematics" was very inspiring to me. Another inspiring piece is his "Advice to a Young Mathematician" in the Princeton Companion to Mathematics.

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Terence Tao. He is one of many who influenced me the most. I don't have to mention how superb his blog and publications are. From his writings I found analysis of PDE as a fascinating subject and I am really happy that I found this topic not too late. It amazes me how much he produces.

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Absolutely. This is an important one for me as well. I blame Tao's clarity of exposition for turning me into a dirty analyst. It is hard work to breathe the life of geometric and physical intuition into the skeletal technical parts of harmonic analysis and PDE, but he seems so often to do just that and I find it quite inspiring, particularly since analysis is often taught so listlessly. –  Spencer Apr 7 '10 at 11:18
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Tao is one of the philosophers of mathematics and that rare oddity so many strive to be:A great researcher whose also a terrific and inspiring teacher. –  Andrew L Jul 5 '10 at 1:36
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Unfortunately, I'm too old and too off topic to be influenced by his research, but there is no doubt that he is among the leaders in changing the way mathematics are done. In that way, his influence reaches all. –  Thierry Zell Aug 14 '10 at 1:40

Serge Lang's Algebra was my first serious encounter with mathematics, the event was a very singular defining moment in my life.

Back then, I was firmly intent on becoming a poet or, at least, pursuing some kind of literary career. Like most budding poets, I loved books and I liked spending time in the library. I was very curious, I would often wander in a section and pick up a book just to see what that row was about. One day I picked up an old rebound copy of Lang's Algebra. It was dirty purplish grey and it just said Lang: Algebra in half erased white letters. I don't think I had any good reason to pick up that book, it certainly wasn't very attractive, I probably just wondered why one would write such a large tome on algebra. I sat down with the book and read the first page where he defines a monoid and proves the uniqueness of the identity element. I was fascinated. It was so beautiful. I fell in love.

I don't think I read much of Lang's book on that day, I probably only had an hour or less to spare, but I went back to the math section later and I picked up more books. The next one was Willard Van Orman Quine's Set Theory and its Logic, which is probably the worst possible way to get introduced to Set Theory but that's how I eventually became a logician instead of a poet.

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What a great story! I wonder how many more poets we could get if only they read the right books... –  Qiaochu Yuan Jan 11 '10 at 4:59
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or the wrong book :D –  Tran Chieu Minh Apr 5 '10 at 2:45
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omg I had almost the same experience, but with Hardy & Wright's Theory of Numbers! I still don't know how the book got into my high school library... –  Burhan May 30 '10 at 5:14
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Is the world in greater need of poets, or of logicians? –  Gerry Myerson Oct 20 '10 at 2:23
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Interesting, especially as Lang turned me off algebra for decades (in my opinion, it is not a bad book, just a bad first book). –  Igor Rivin Jan 17 '11 at 0:09

John Baez. "This week's finds in mathematical physics" is a great playground for young mathematicians. I was a graduate student when I first found it, and I really loved the links between various TWFs and the math they discussed. Not only does he show you the breadth of modern math, he also gives you bridges between the various areas.

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I think it is really cool that someone mentioned him on here. He is such a great guy and so interested in getting people into mathematics! I wonder if we should point this out to him, or if someone already has? –  Sean Tilson Mar 4 '10 at 1:11
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What inspired me the most is his way of talking about homology and algebraic topology as if it were a good friend instead of a punitive expedition into the twisty mazes of diagram chases and sticky glueing procedures; a terrible impression that I had gotten from taking a textbook at face value. –  Greg Graviton Oct 19 '10 at 19:13

Does Martin Gardner count, even though he is not a mathematician?

I read all of the "Mathematical Games" columns in Scientific American when I was maybe 12 or 14. And this was a non-trivial task ... I would ride my bicycle to the public library one afternoon a week to read a few more columns (the school library didn't have it). So it took maybe a year to read them all.

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Yes, I think he should definitely count! +1 –  Jose Brox Nov 16 '09 at 0:59
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Gardner's column on Catalan Numbers and Planted Plane Trees made me fall in love with mathematics all over again — after a long spell of doldrums when the tedium of my undergradgrind coursework had all but extinguished the last beamish bits of joy in it. –  Jon Awbrey Nov 17 '09 at 19:55

John Willard Milnor for his books about "Morse Theory" and the "h-cobordism theorem" (I think it is a crime that it isn't printed anymore) and for writing papers in a way that they are quite self-contained and readable.

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"Topology from the differentiable viewpoint" could be read any time, anywhere and pretty much by any one! –  Somnath Basu Oct 20 '10 at 0:15

Who: Leonhard Euler.

When: As a highschool student.

Where: On the book "Euler: the master of us all" by William Dunham.

Why: The amount of creativity and genius dispersed among the so-different works of Euler continues to amaze me just now, so it only could have a devastating effect on me 10 years ago. He not only addressed a lot of distinct topics, he layed the foundations of many branches of mathematics and solved with ease many problems that were interesting me at that moment of my life. I learned a lot from him: he really deserves the title of "master of us all".

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Yes!!! I also read this book in my senior year of high school, around the time I was applying to universities. I went into college intending to be a computer science major. However, because of Dunham's book, I decided to continue taking math courses on the side, which then of course eventually lead to a full-time interest in math. :) –  Kevin H. Lin Nov 14 '09 at 15:30
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Euler's precalculus book (something like "On the Analysis of the Infinite") is surprisingly readable. –  Noah Snyder Nov 26 '09 at 20:11
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The amount of shelf space necessary to house Euler's Opera Omnia (still being completed!) is simply staggering. –  Thierry Zell Aug 14 '10 at 1:35

Carl Friedrich Gauss.

The breadth and beauty of his work amazed me when I was a student, and it still inspires me.

He started by building on much less than what many of us take for granted: His doctoral dissertation was the Fundamental Theorem of Algebra. His work covered deep, essential results in many areas, from number theory (quadratic reciprocity, conjecture of prime number theorem) to geometry (Gaussian curvature) to statistics (least squares) to probability (Gaussian distribution). It is difficult to imagine these areas without his fundamental contributions. He also contributed to physics and astronomy.

Even though Gauss explored many areas, he took the time to revisit old results, looking for different and more satisfactory proofs.

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Indeed. Gauss' Disquisitiones Arithmeticae is arguably the most important and influential text on pure mathematics ever written. Certainly it secured an eternal place for number theory in the esteem of the mathematical community. Gauss was such a juggernaut that I find it easier to think in terms of what he didn't do than what he did: for instance, he did not anticipate Dirichlet's results on L-series and primes in arithmetic progression. Sometimes I have wondered why... –  Pete L. Clark Mar 26 '10 at 5:50
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More important and influential than Euclid's elements? –  Gerry Myerson Oct 20 '10 at 2:21
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Let us not forget the immortal quote Gauss gave us: The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length... Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated. –  DavidLHarden Apr 27 '11 at 2:32

Erdős

Sophomore year when I decided that I didn't like physics classes I just happened to be reading "The Man Who Loved Only Numbers" by Hoffman. Between this and "How to Read and Do Proofs" by Solow, I saw mathematics as something much more beautiful. This combined with reading about Erdos style of mathematics made me really attracted to research and led to my first REU experience. It was all downhill from there.

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My research has been based on fields which he has had great influence. –  Kristal Cantwell Nov 14 '09 at 19:41

Gian-Carlo Rota. I really wish I could pinpoint the moment that I came across some of his work, but I can't. And I've only just begun graduate work so it's impossible to be really honest about what sort of impact he has had on me... only time can tell.

But nonetheless, his writings are truly inspiring. It's tough to describe the wonder they have given me. Rota began as a functional analyst (PhD under Jacob Schwartz, of the Dunford & Schwartz fame) and moved over to algebraic combinatorics in the 1960s. One of his first papers that has stuck in my mind is "The Number of Partitions of a Set" in which he applies the techniques of the so-called 'umbral calculus' (which he also worked to rigorously formulate) to beautifully establish some combinatorial results. He's credited as setting the field of algebraic combinatorics on solid ground via his seminal papers On the Foundations of Combinatorial Theory. But it's not just the technical results -- his writing is just plain fun to read.

Given my personal interests, I really appreciate that although Rota did so much work in combinatorics he always seemed to lean back towards his roots in functional analysis & probability. In fact, his goal to find the true nature of classical results in analysis and probability led him to a great deal of good work in combinatorics, e.g. his work on the Rota-Baxter algebra inspired by his ambition to understand "the algebra of indefinite integration", and his work on the foundations of probability with the ambition to understand the middle ground between the discrete and the continuous. Moreso than most people he is willing to put his position out there and speak about mathematics rather than just speak mathematics. A great example of this is his book "Indiscrete Thoughts" -- definitely worth reading. You may not agree with many things he says but it's wonderful to be allowed a glimpse into the mind of a man such as Rota.

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I agree so much I'm commenting. –  Qiaochu Yuan Nov 16 '09 at 17:16
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When I was an undergraduate, Rota's paper "The Pernicious Influence of Mathematics upon Philosophy" completely changed my thinking about analytic philosophy. As a result, I did not complete a second major in philosophy that I had almost completed at that point. This wonderful paper is available here : springerlink.com/content/r29435u7722u58j2 –  Andy Putman Nov 16 '09 at 20:39
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@Qiaochu: I was wondering when you were going to show up to second this one... :) –  Harrison Brown Nov 17 '09 at 2:56
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I took Differential Equations from Rota at MIT in the early 1980s. He was a fantastic lecturer. I remember two things: he used to make fun of the people taking notes in class and would enunciate punctuation aloud. The other thing I remember is that he used to like to lecture while drinking a coke. Every day some student would buy a can of coke from the vending machines outside the lecture room and leave it on the table in front of the blackboard. The last day of the lecture EVERYONE bought him a can of coke. There were literally more than 100 cans of coke on the table that day. –  José Figueroa-O'Farrill Jul 5 '10 at 1:58
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In a probability class I took from him, the exams he set were very easy, just tests of basic computational skill. But his homeworks were fantastically difficult and creative, and he encouraged us to work together on them. For all but the very best students, this was a neccessity! The devious thing is that this didn't just teach us probability, but also how to collaborate mathematically. –  Neel Krishnaswami Oct 20 '10 at 17:51

JH Conway. He has published work in a diverse set of interesting fields. I first met his name when looking for cool computer programs to write as a kid (ie. the Game of Life) but since then his name kept appearing in mathematics that I found interesting, whether it's Monstrous Moonshine or the properties of finite state automata. He has this incredible knack for turning anything he touches into fun - whether it's knot theory, group theory, quadratic forms, or, more obviously, combinatorial games. As well as working at the frontiers of mathematics he's discovered accessible but surprising and beautiful recreational mathematics, like Conway's soldiers. All in all, an amazing guy. Once of my regrets in life is being too lazy to attend his lectures on finite simple groups when I was an undergraduate.

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Poincaré. Not so much for his mathematical writings (although what I've found in English, or struggled through in French, has been uniformly interesting [if dated, and/or, um, in a language I barely understand]) but for his thoughts on the philosophy and psychology of math. After the already-mentioned John Baez, the first thing I'll implore anyone who bothers to ask to read is "Intuition and Logic in Mathematics," fin-de-siecle thinking and all.

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It can be read at gap-system.org/~history/Extras/Poincare_Intuition.html –  Jose Brox Nov 16 '09 at 17:05
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+1 for mentioning Poincaré expository, autobiographical and epistemological essays. But his mathematical research articles are often very sketchy, and I find myself wanting to actually fully write them when I read them. –  ogerard May 14 '10 at 6:33

Gödel I was captivated by his belief of a platonic mathematical world and the belief that human can understand such a thing.

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Bernhard Riemann.

The idea of the Riemann Surface and manifolds stroke me when I was a high school student.

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G.H. Hardy. Reading "A Mathematician's Apology" in high school really changed the way I think about mathematics.

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Hopefully it changed your opinion of mathematics for the better! –  Kevin H. Lin Dec 7 '09 at 1:52

Cliched perhaps, but my fellow graduate students when I was in grad school. They're the ones that answered my questions when I got lost, shared their half-baked ideas and listened to mine, showed me just how many interesting fields of math there are and how many different perspectives people can have on the same subject, and cheered me on when things were difficult.

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Consider looking forward: Grisha Perelman, his strange history wiki G.Perelman. However the first millenium prize awarded. Even note this: Terence Tao said... "well, it's amazing"

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what he had achieved, what he had written: now is studied by several people I think, like Yau, Cao,... –  janmarqz Dec 7 '09 at 19:08

Srinivasa Ramanujan. He does not figure that much in my work right now. But studying his notebooks (via Bruce Berndt's studies) when I was a teenager taught me how to appreciate beautiful mathematics. From that moment on, I was hooked. I knew I had to be a mathematician.

As for those whose lives or personalities inspired me, or whose style of thinking influenced my methodology, too many to count...

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In terms of style of math, I'm not quite sure yet. However, I think that it is certainly true for me, and no doubt for countless others, one's advisors role is one of the most crucial influences one may have.

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Otto Forster.
He is the most brilliant expositor I have ever met. I cherish the notes I took a long time ago of courses he gave in Italy and France, in perfect Italian and French. He wrote a wonderful course on Analysis (in three volumes) which has been the reference in German Universities for 30 years, something like Rudin in the States. His book on Riemann surfaces (both compact and non compact) is a masterful blend of Algebra, Topology and Analysis, with tools ranging from cohomology of sheaves to difficult potential theory. He is a brilliant researcher and has made important contributions to complex geometry and also to algebra (Forster-Swan). Working with him was a wonderful experience and he had the generosity of letting me co-sign articles to which my contribution was negligible compared to his. I am very happy of this opportunity to express my gratitude to and admiration for this genuine scholar and real gentleman.

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Benedict Gross. I saw him lecture a few times on BSD. His enthusiasm and mastery were very inspirational. It reminded me why I want to be a professional mathematician. I had just finished my general exams the previous semester and felt tired from taking so many classes and preparing for exams. It had put a haze over the beauty of mathematics. Professor Gross made it clear again.

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@Andy. Yup,turns out,indeed they are: extension.harvard.edu/openlearning/math222/… –  Andrew L Oct 20 '10 at 4:14

Raymond Smullyan, in elementary school. His book "Alice in Puzzleland" was a childhood favorite of mine and is what and inspired a life long interest in math.

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Richard Courant. Several years before I started studying mathematics in earnest, I spent a summer working through his calculus texts. Only recently, on re-reading them, have I come to realize how much my understanding of calculus, linear algebra, and, more generally, of the unity of all mathematics and, to use Hilbert's words, the importance of "finding that special case which contains all the germs of generality," have been directly inspired by Courant's writings.

From the preface to the first German edition of his Differential and Integral Calculus:

My aim is to exhibit the close connexion between analysis and its applications and, without loss of rigour and precision, to give due credit to intuition as the source of mathematical truth. The presentation of analysis as a closed system of truths without reference to their origin and purpose has, it is true, an aesthetic charm and satisfies a deep philosophical need. But the attitude of those who consider analysis solely as an abstractly logical, introverted science is not only highly unsuitable for beginners but endangers the future of the subject; for to pursue mathematical analysis while at the same time turning one's back on its applications and on intuition is to condemn it to hopeless atrophy. To me it seems extremely important that the student should be warned from the very beginning against a smug and presumptuous purism; this is not the least of my purposes in writing this book.

Another example: while not a "linear algebra book" per se, I have yet to find a better introduction to "abstract linear algbera" than the first volume of Courant's Methods of Mathematical Physics ("Courant-Hilbert"; so named because much of the material was drawn from Hilbert's lectures and writings on the subject). His one-line explanation of "abstract finite-dimensional vector spaces" is classic: "for n > 3, geometrical visualization is no longer possible but geometrical terminology remains suitable."

Lest one be misled into thinking Courant saw "abstract" vector spaces as "$\mathbb{R}^n$ in a cheap tuxedo," he introduces function spaces in the second chapter ("series expansions of arbitrary functions"), and most of the book is about quadratic eigenvalue problems, or, as Courant saw it, "the problem of transforming a quadratic form in infinitely many variables to principal axes."

As a final example: Courant's expository What is Mathematics? is perhaps best described as an unparalleled collection of articles carefully crafted to serve as an object at which one can point and say "this is." Moreover, while written as a "popularization," its introduction to constrained extrema problems is, without question, a far, far better introduction than any textbook I've ever seen.

I should also mention Felix Klein, not only because Klein's views on "calculus reform" so clearly influenced both the style and substance of Courant's texts, but since a number of Klein's lectures have had an equally significant influence on my own perspective. For those unfamiliar with the breadth of Klein's interests, I'm tempted to say "his Erlangen lecture, least of all" (not that there's anything wrong with it).

Lest my comments be mistaken for a sort of wistful "remembrance of things past," I'd easily place Terence Tao's writings on par with Courant's, for many of the same reasons: clear and concise without being terse, straightforward yet not oversimplified, and, most importantly, animated by a sort of — je ne sais quoi — whatever it is, it seems to involve, in roughly equal proportions: mastery of one's own craft, a genuine desire to pass it on, and the considerable expository skills required to actually do so.

Finally, I can't help but mention Richard Feynman in this context, and to plug his Nobel lecture in particular. While not a mathematician per se, Feynman surely ranks among the twentieth century's best examples of a "mathematical physicist" in the finest sense of the term, not merely satisfied by a purely mathematical "interpretation" of physical phenomena, but surprised, excited, and, dare I say, delighted by the prospect! Moreover, he was equally excited about mathematics in general, see, e.g., the "algebra" chapter in the Feynman Lectures on Physics.

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Arnold Ross. He ran the summer program in Number Theory for high school students at Ohio State University, my first exposure to serious mathematics. His lectures set me on a course from which I've hardly deviated in over 40 years.

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I have to vote this up, if only because Ross's program was the inspiration for PROMYS, from which I've heavily benefited. –  Qiaochu Yuan Mar 26 '10 at 2:09

Dieudonné

The number 1 personality behind Bourbaki. Even though he was famous for taking the most extreme positions and was widely dismissed as a radical, his vision of mathematics is one that has largely been adopted by almost all mathematicians everywhere. Reading any piece of mathematical work he wrote, it his hard not to feel the respect and passion he felt for mathematics as a subject.

Dan Kan

Singlehandedly developed categorical homotopy theory into a full-fledged replacement for the homotopy theory of spaces (Kan complexes, combinatorial homotopy groups, subdivision, $Ex^\infty$, among many other things) as well as a large part of the foundations of homological algebra (Dold-Kan correspondence), category theory (adjoint functors, Kan extensions), and the modern theory of simplicial localization (with Dwyer) among numerous other achievements.

It's said that Kan's breakthrough paper on adjoint functors convinced Eilenberg and Mac Lane that pure category theory was not only a viable mathematical discipline (rather than a language), but also a deep and rich one.

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A really nice answer to my question. Thanks! –  Jose Brox Jan 28 '11 at 10:57
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@Harry Gindi: Thanks for the classification. The elaboration you give is close to things I heard before. It seems my confusion arose from the fact 'personality' has a slightly different meaning for you an me. –  quid Jan 28 '11 at 16:38

Leibniz. Not just for his mathematics (calculus, amazing insights in logic, semantics) but he was just an incredible polymath, with deep work in law, history, linguistics, chemistry, physics, metaphysics, politics, engineering, sociology, he founded 'library science', and on and on.

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Dedekind, whose championing of concepts (vs. calculation) left a longstanding impression on the way that I conceive mathematics - even long after I first started reading the masters as a student. Back then I had to grovel through the bowels of the MIT libraries but now, with many important historical works easily accessible online, there is no excuse not to read the masters.

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