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In every field there's always that person who's just years ahead of their time. For instance, Paul Morphy (born 1837) is said to have retired from chess because he found no one to match his technique that very much resembles modern chess theory.

So, who were the Paul Morphys of mathematics?

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Fermat, if one is to believe he actually did have a proof, and not just a marginal note... :) Riemann was also supposed to have some pretty advanced stuff in his papers, IIRC. – apnorton Apr 28 '14 at 14:01
No one is ahead of their time. – Asaf Karagila Apr 28 '14 at 14:35
@AsafKaragila: everybody lived for the years between birth and death, and no others. So yeah, nobody is literally ahead of their time. But I think the questioner has defined terms for this question sufficiently clearly that it's a big claim to say that no mathematician has ever been ahead of their time, in the way that Morphy is supposed to be for chess. – Steve Jessop Apr 28 '14 at 16:15
It's tempting to suggest Shinichi Mochizuki, but it's too soon to know for sure. – MJD Apr 28 '14 at 17:50
Not exactly a mathematician, but how about Babbage? I think it wasn't until computers were invented in the 20th century that people realized he'd had the key ideas in the 19th century (and had attempted to make a computer out of mechanical components). I'm not an expert on the history though. – littleO Apr 29 '14 at 2:46

25 Answers 25

up vote 70 down vote accepted

I think Asaf makes a good argument against people like Galois and Cantor. As he says, if Cantor didn't develop set theory, who would have? I think to find someone who is arguably ahead of their time, you need to find a mathematician whose work was ignored, and then reinvented in substantially similar fashion by others much later, so that you can say “look, this guy had it, but it was too soon, and then later someone else got credit for inventing the same thing.” And so I nominate the logician Charles Sanders Peirce (1839–1914). When you study the early history of logic, it sometimes seems that every other sentence is “This work was anticipated by C.S. Peirce, whose contribution was unfortunately overlooked.”

To give a very narrow and incomplete idea of Peirce's accomplishments in mathematical logic I will quote briefly from the Stanford Encyclopedia of Philosophy:

In 1870 Peirce published a long paper “Description of a Notation for the Logic of Relatives” in which he introduced for the first time in history, two years before Frege's Begriffschrift, a complete syntax for the logic of relations of arbitrary [arity]. In this paper the notion of the variable (though not under the name “variable”) was invented, and Peirce provided devices for negating, for combining relations (basically by building upon de Morgan's relative product and relative sum), and for quantifying existentially and universally. By 1883, along with his student O. H. Mitchell, Peirce had developed a full syntax for quantificational logic that was only a very little different… from the standard Russell-Whitehead syntax, which did not appear until 1910 (with no adequate citations of Peirce).

Peirce introduced the material-conditional operator into logic, developed the Sheffer stroke and dagger operators 40 years before Sheffer, and developed a full logical system based only on the stroke function. As Garret Birkhoff notes in his Lattice Theory it was in fact Peirce who invented the concept of a lattice (around 1883).

(Burch, Robert, "Charles Sanders Peirce", The Stanford Encyclopedia of Philosophy (Summer 2013 Edition), Edward N. Zalta (ed.))

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I had a hard time deciding between Peirce and Ramanujan for best answer but while Ramanujan is one of the most brilliant mathematicians to ever live, Peirce fits the criteria I was looking for better. He suggested a cardinal arithmetic for infinite numbers, years before any work by Cantor. He also showed how Boolean algebra could be done via a repeated sufficient single binary operation (logical NOR), anticipating Henry Sheffer by 33 years. He also distinguished between first-order and second-order quantification, anticipating Zermelo by about two decades – hb20007 Apr 29 '14 at 12:30
And finally, in 1886 he saw that Boolean calculations could be carried out via electrical switches, anticipating Claude Shannon, who used the same principle to found the field of information theory, by more than 50 years. In my judgement, though less influential than the bigger names, Peirce is the Morphy of mathematics. @MJD thanks for your response. – hb20007 Apr 29 '14 at 12:37
Jeezus! Did he also invent LISP? – cody Apr 30 '14 at 16:37
As far as I know, that wasn't invented until the early 1930s, by Alonzo Church. But Peirce wrote around 100,000 (!!) pages of unpublished manuscripts, many of which are still uncatalogued, so it's possible that he anticipated Church and nobody knows it yet. – MJD Apr 30 '14 at 18:33
I'm being slightly facetious. – MJD May 1 '14 at 15:55

My top vote would be for Ramanujan. With severely limited resources, he was able to formulate deep number-theoretic identities that the top mathematicians in the field at the time hadn't the imagination to conceive, let alone the slightest clue to prove. A close second would be Evariste Galois--dead at the age of 20, he had already established the foundations of what is now an entire algebraic theory named after him. The world will never know what mathematics he could have discovered had he lived.

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+1 for Galois, who was clearly ahead of his time. – lhf Apr 28 '14 at 16:16
Galois may have produced excellent work, and he knew some of the following results and not others, but to say he was "ahead of his time" is very clearly false. The connection between polynomials and permutations was hinted at by Lagrange about 40 years prior, who himself was just using the theory of symmetric polynomials, known since at least Newton. Ruffini published very similar results to Galois and discovered many simply group theoretical results to do it. Abel did the same. Cauchy used permutations in his work on polynomial equations. – Jack M Apr 28 '14 at 17:04
Ramanujan is still ahead of his time. – rookie Apr 28 '14 at 20:00
Galois, all disputes about what"ahead of their time" means aside, was certainly ahead of his age. :) – apnorton Apr 28 '14 at 23:36
+1 for Ramanujan. It's important to not forget context. Ramanujan's early work was done effectively in a bubble. Even if a lot of what he did was re-discovery, there was no way for him to know that. He had no access to peers, no access to the state-of-the-art mathematics, and as far as I know, the academic culture on the Indian subcontinent was not well-suited for rigorous mathematical development. – Emily Apr 29 '14 at 14:38

For sheer amount of time ahead, may I suggest Brahmagupta (597–668), Jayadeva (950 ~ 1000) and Bhāskara II (1114 ~ 1185), Indian mathematicians whose work in indeterminate quadratic equations and many other branches predated European attempts by more than half a millennium.

In particular, consider their work on chakravala, an elegant and powerful method to find solutions to Pell's equation $x^2 = Ny^2 + 1$. Circa 1150, Bhāskara II had a solution for the case $N=61$, while in Europe it was given as a challenge by Fermat and first solved in 1657. More than one hundred years later (in 1766), Lagrange's "general" method to solve this problem was still much more complicated and inelegant than chakravala, which for its application requires nothing but elementary arithmetic.

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I can't tell whether those Indian mathematicians were ahead of their time, or Europe was behind the times ;-) – Steve Jessop Apr 29 '14 at 8:18
@SteveJessop: Europe was far, far behind the times. IMO. – Charles Apr 29 '14 at 14:57
Also calculus, apparently: If anyone can shed some light on this book, I'd be very grateful. Translation available from Springer:… – AmadeusDrZaius Apr 29 '14 at 22:54 says Indian mathematics emerged in the Indian subcontinent[1] from 1200 BC[2] until the end of the 18th century. In the classical period of Indian mathematics (400 AD to 1600 AD), important contributions were made by scholars like Aryabhata, Brahmagupta, Mahāvīra (mathematician), Bhaskara II, Madhava of Sangamagrama and Nilakantha Somayaji. – Zerotoinfinity May 2 '14 at 17:41
The definition is "ahead of their time" so being ahead of someone else's time (i.e. European time) doesn't cut it. – PatrickT May 4 '14 at 14:50

Firstly, its worth recalling what Asaf writes in the comments:

Yes, pioneers are often considered "ahead of their time", but we forget that the reason we think of them as ahead of their time is that their ideas were thoroughly developed later because they required several decades to be processed by the mathematical community at large.

That being said, I think that everyone is bound by the dominant paradigm in which they're surrounded; yet somehow, certain key individuals are able to break with this paradigm just enough to do the ground-breaking work that they're remembered for. When this happens, their insights often seem to "come out of the blue," and it sometimes takes the wider mathematical community many years to catch up. Here's a couple of examples that I find particularly compelling.

Jean Baptiste Joseph Fourier (1768 – 1830) was one of the first people to think of a function as, well, an arbitrary function, rather than an explicitly given rule; hence the functions $f : [0,1] \rightarrow \mathbb{R}$ and $g : [0,2] \rightarrow \mathbb{R}$ given by $x \mapsto x^2$ are actually different. This was necessary to develop what we now call the Fourier transform, the then-groundbreaking (frankly I think its still groundbreaking) idea that just about every function $I \rightarrow \mathbb{R}$ can be expressed as a superposition of sine and cosine waves, where $I$ is a real interval. Even today, the Fourier transform is among the most important of tools in the arsenal of many engineers, physicists, and applied mathematicians; and the realization that functions with different domains need to be distinguished was a crucial step toward the pure mathematics of today that we know and love.

Georg Cantor (1845 – 1918) is, of course, the grandfather of modern set theory. Among other things, Cantor invented the ordinals, proved that every set could be well-ordered; and, building on Fourier's idea of a function as-an-arbitrary function, Cantor was able to formulate the concept of two sets being equipotent. Cantor realized that both $\mathbb{Z}$ and $\mathbb{Q}$ are equipotent to $\mathbb{N}$, and later, much to his shock, that $\mathbb{R}$ is not! I think he also discovered the $\aleph$ indexing of the cardinal numbers. Writing in a mathematical climate that was fiercely constructive and deeply suspicious of infinity, Cantor's work was largely ignored, and many of the top mathematicians of the day openly ridiculed his ideas. Nonetheless, he kept on working on his set theory (which he believed had theological significance) until the very end of his days, especially the continuum hypothesis. Its sucks that Cantor's work wasn't recognized sooner; once we started paying attention, pure mathematics was changed forever.

There's a lot more people I wanted to talk about (especially Godel, Samuel Eilenberg, Saunders Mac Lane, Garret Birkhoff, Alexander Grothendieck, and William Lawvere) but maybe I'll just leave it there.

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Or type theory would have been more prevalent, and we would all be doing type theory today. – Asaf Karagila Apr 28 '14 at 15:55
I don't mind, I don't know a lot about it either. But it is what Russell and Whitehead were suggesting as a foundational basis. In the absence of good logic and set theory alternative, it might have been developed faster and sooner. So when Cantor arrives some decades later, he's already drawn to this ideas and doesn't develop set theory. In that universe I didn't become a mathematician, but I did manage to take over the world and kill the evil aliens that destroyed Mt. Everest. – Asaf Karagila Apr 28 '14 at 16:12
@Asaf: but without there already being activity in set theory, Russell and Whitehead maybe don't see the need for a foundational basis because the problems they set out to solve aren't discovered yet (and Zermelo was tragically killed in crossfire with the aliens shortly after the Battle of Kathmandu). So maybe Frege was the man ahead of his time, in that he didn't get much traction for his project to formalise set theory. Thanks to a catchy paradox Russell did get traction :-) – Steve Jessop Apr 28 '14 at 16:27
@AsafKaragila I think Cantor is a good example because his ideas were so new and radical that many established mathematicians--such as Kronecker, Poincaré, Brouwer, and Weyl--rejected them. – SpamIAm Apr 28 '14 at 21:45
@littleO, thanks for your encouragement. I might write a little more when I find the time. – goblin Apr 29 '14 at 7:39

I'm surprised to see no one here has nominated Hermann Grassmann, who essentially invented the subject of Linear Algebra. His mathematical work was little appreciated in its time, and it took decades before it gained acceptance.

From Wikipedia:

"Fearnley-Sander (1979) describes Grassmann's foundation of linear algebra as follows:

' The definition of a linear space (vector space)... became widely known around 1920, when Hermann Weyl and others published formal definitions. In fact, such a definition had been given thirty years previously by Peano, who was thoroughly acquainted with Grassmann's mathematical work. Grassmann did not put down a formal definition --- the language was not available --- but there is no doubt that he had the concept. Beginning with a collection of 'units' $e_1, e_2, e_3, \dots ,$ he effectively defines the free linear space which they generate; that is to say, he considers formal linear combinations $a_1e_1 + a_2e_2 + a_3e_3 + \dots$ where the $a_j$ are real numbers, defines addition and multiplication by real numbers [in what is now the usual way] and formally proves the linear space properties for these operations. ... He then develops the theory of linear independence in a way which is astonishingly similar to the presentation one finds in modern linear algebra texts. He defines the notions of subspace, linear independence, span, dimension, join and meet of subspaces, and projections of elements onto subspaces.

...few have come closer than Hermann Grassmann to creating, single-handedly, a new subject.'

...Comprehension of Grassmann awaited the concept of vector spaces which then could express the multilinear algebra of his extension theory. A. N. Whitehead's first monograph, the Universal Algebra (1898), included the first systematic exposition in English of the theory of extension and the exterior algebra. With the rise of differential geometry the exterior algebra was applied to differential forms."

Hence, Hermann Grassmann is my nomination for a mathematician ahead of his time.

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Grassmann did this work for his PhD, but his advisor (Mobius) didn't understand or appreciate it and failed him! That pretty much ended Grassmann's mathematical career - he changed fields and went on to be a noted linguist. – Nick Alger Apr 30 '14 at 14:35
He knew that he was ahead of his time, and that people just did not notice it - a quote from his revised version of the work on "Ausdehnungslehre" (1862): But I know and feel obliged to state ... that even if this work should again remain unused for another seventeen years or even longer, will be brought forth from the dust of oblivion and when ideas now dormant will bring forth fruit.. I mainly knew him for his pioneering work in geometric algebra, and would have mentioned him if you had not already done this (+1) – Marco13 Apr 30 '14 at 16:51
This isn't quite correct. See the addendum on this MO answer. – Mike Miller Apr 30 '14 at 17:00
@NickAlger: Only your last statement is true, please correct. Mike: Thanks for linking. – Torsten Schoeneberg May 3 '14 at 15:39
@Nick Alger: sounds like what happened to Louis Bachelier with his Poincaré supervisor! – PatrickT May 4 '14 at 14:51

Archimedes, for basically discovering calculus eighteen centuries before either Newton or Leibniz. See Archimedes Palimpsest for more details. :-)

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It says on that very Wiki page (on the second line) that the method of exhaustion (which I assume you're referring to) was invented by Eudoxus (or at least known to him). – Jack M Apr 28 '14 at 22:27
@JackM: No. That's not what the Palimpsest is about. – Lucian Apr 28 '14 at 22:39
Archimedes developed the method of exhaustion by inventing heuristics which resemble the methods of integral calculus. That's why he could be considered ahead of his time. – hb20007 Apr 29 '14 at 12:06
Also, Archimedes was the first to find the tangent to a curve, other than a circle, in a method akin to differential calculus. However saying he 'discovered' calculus is an overstatement as the Fundamental Principle of Calculus was only discovered by Newton/Leibniz – hb20007 Apr 29 '14 at 12:11
In a similar vein, one could say that the Indian mathematicians of the 12th (Bhaskara II - and later the 15th centuries (Madhava - were ahead of European time. Although I've yet to read a good English translation (too expensive) of it, so I say how close it really is to 'calculus'. – AmadeusDrZaius Apr 29 '14 at 22:48

Is nobody going to even mention Blaise Pascal?!

Just check out a few of his accomplishments and see how far ahead of his time he was:

  • He made a fundamental contribution to projective geometry when he was 16(!) years old (appreciating the brilliance of Desargues's contributions at a time when analytic geometry was far more popular)
  • His correspondence with Pierre de Fermat basically invented Probability Theory
  • He independently invented one of the first mechanical calculators
  • His work with fluids generalized the results of Evangelista Torricelli

(Also note the similarity to Paul Morphy -- Pascal gave up math by the age of 30)

And that's not even mentioning his contributions to Philosophy & Theology--frankly, few people span the Dewey Decimal System like my boy BLAISE.

And there's that triangle of his...

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I have a soft spot for Pascal, however "his" triangle was already known in India and China. – PA6OTA May 1 '14 at 16:13
I'm don't know enough to either agree or disagree with him, but Devlin in The Unfinished Game seems to think that Fermat was the teacher and Pascal the student in their correspondence. – joeA May 3 '14 at 3:19
@joeA, no way, Pascal was ahead of God! – PatrickT May 4 '14 at 14:52

Diophantus of Alexandria, who invented a form of algebraic notation and much of what we now call Diophantine equations. His work was nearly ignored in his own time and rediscovered hundreds of years later by al-Khwarizmi and others.

His use of παρισὀτης (as mentioned in another answer) anticipated early versions of differential calculus 1300 years later.

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Girard Desargues more or less invented projective geometry in the 17th century, but nobody else paid significant attention to the topic until the 19th century. You might say he was both ahead of and behind his time, since he developed his geometry in a classical style (without any interpretation using numeric coordinates) at just about the time Cartesian coordinates were coming into vogue.

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Simon Stevin (1548 – 1620) had the unbelievable insight of thinking of an arbitrary number in terms of its unending decimal expansion. He was ahead of his time in the sense that the full significance of what he did was not appreciated until the 1870s when more abstract versions of the construction of the reals were given by Charles Méray and others. Unending decimals inspired Newton to introduce more general power series. The rest is history.

Another amazing contribution of Stevin was his argument immediately recognizable as a proof of the intermediate value theorem, which was not "officially" proved until the 1820s by Cauchy. See this article. Here he was ahead of his time by at least 200 years.

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I would like to add a name that nobody has mentioned: Alexandre Grothendieck.

The name comes to my mind not only for what he achieved in his active career, which was itself pathbreaking and quite visionary, but rather for the Esquisse d’un Programme, in particular for how it is described by people that worked on it. For example, this is what Leila Schneps of Université de Paris VI said about it:

“It was a wild expression of mathematical imagination, [...] I loved it. I was bowled over, and I wanted to start working on it right away. [...] Some of it doesn’t even seem to make sense at first, but then you work for two years, and you go back and look at it, and you say, ‘He knew this’.” [p.1205 - Jackson - The Life of Alexandre Grothendieck]

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Fermat, whose adequality anticipated the calculus by several decades.

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I would go for Louis Bachelier, who developed and conceptualized what we today know as Brownian motion. His thesis The Theory of Speculation applied the idea of a random walk to describe and model stock options, predating Einstein's work on physical diffusion and brownian motion which itself came before the idea of a Markov process was formalized by Andrey Markov in 1906.

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While we're talking about physics, Emmy Noether's derivation of the laws of conservation from symmetry is definitely ahead of her time in my view. And perhaps still ahead of our time. – AmadeusDrZaius Apr 29 '14 at 23:02
@AmadeusDrZaius, Louis Bachelier wrote his thesis under Henri Poincaré, the leading mathematician of his time, not in physics. By the way, Poincaré did not understand Bachelier's work and did nothing to help him. – PatrickT May 4 '14 at 14:54

What about George Boole (1815-1864) with boolean algebra (1847) that didn't have any practical use until the introduction of computers?

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Define "practical use". – Asaf Karagila Apr 29 '14 at 11:09

Though I'm not sure to what extent I agree with the premise of the question, I'd nevertheless like to mention Élie Cartan.

Much of Cartan's work was not fully appreciated until late in his life. He was the first to introduce the concepts of holonomy and spinors. He defined and classified the symmetric spaces. His technique of moving frames proved to be very successful, and I would say that his Method of Equivalence was very forward-thinking.

And really, this is such an incomplete list of his accomplishments.

(As an aside, I'm surprised to see no mention of Grothendieck at the time of this writing... or, separately, any of the developers of hyperbolic geometry.)

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Grothendieck was a mathematician of astonishing insight but he was precisely of his times. If you look at any of his contributions, they are well-motivated by existing ideas, even if they are radical abstractions of them. Also, although he was polarizing, he had a huge following throughout his career. Any rejection of his methods has to be seen as an understandable resistance to having one's foundations overturned and the prevailing values of one's subject (apparently) questioned. – Ryan Reich Apr 29 '14 at 15:08

There is a book that claims that while Babbage invented the Analytic Engine, it was actually Ada Lovelace that conceived of an abstract "Science of Operations" as she called it. She invented this well before a working computer capable of implementing her ideas was constructed, and well before Turing et al., so objectively her theory was ahead of her time.

I don't know how well developed her Science of Operations was. Does anyone know more about this? I am finding it hard to find many details.

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Archimedes was mentioned by Lucian for his palimpsest (which does show his technique for computing volumes of bodies by adding up infinitely many infinitessimals), and I agree, but I would like to say more.

I think it must also be mentioned that many believe Archimedes to be the inventor of the Antikythera device, which was very far ahead of its time, by much more than a millenium, according to the engineers who inspected it. Though not strictly a work of mathematics, it did involve complicated mathematics and does demonstrate (if one accepts Archimedes' authorship) that he was a mathematician far ahead of his time.

It is also interesting to note that both the palimpset and the device are badly decayed ancient artifacts which we had no knowledge of before discovering them (except for perhaps a vague line by someone like Pliny the Elder - I can't remember who exactly), and which completely shocked historians and mathematicians/engineers with their sophistication for the time.

Therefore one wonders what other insights or marvels Archimedes was responsible for which, if discovered, would engender similar degrees of incredulity and wonder. It is assured that we do not possess his complete works - probably by a wide margin, given the accounts of how dedicated to and driven by his work he was; of what remains outstanding, how much do you think we would find utterly outstanding?


Actually, even more needs to be said. Archimedes also presaged exponential notation with his notion of myriad when trying to figure out how many grains of sand are in the universe, and thereby developed techniques for dealing with enormous quantities, and in the estimation of some invented the odometer. Both achievements are also quite far ahead of their time. I'm sure even more can be said, but I couldn't leave out mention of these notable examples.

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Obviously every mathematician who did something extraordinary in mathematics was ahead of his time.
So , I suppose a very good example is Leibniz

He discovered Calculus which is something groundbreaking in the field of mathematics.
(and not only mathematics actually)

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I'm looking for mathematicians who were way ahead of other mathematicians of their time, maybe to the extent of their work or its implications not being understood. And I'm not sure about Leibniz discovering Calculus as Newton usually shares the credit. – hb20007 Apr 28 '14 at 14:19
RAMANUJAN and EULER .. Euler invented things like Borel transform FOurier analysis , Laplace transform and many other ideas which were studied hundred years after – Jose Garcia Apr 28 '14 at 14:57
@JoseGarcia, pretty sure Fourier invented Fourier analysis. – goblin Apr 28 '14 at 15:53
The fact that Leibnitz and Newton each developed calculus within a few years of each other means, I think, that each stands as witness that the other's work on the subject was very much of its time. If "being the first to do something" is evidence of being ahead of your time then hopefully anyone awarded a PhD in mathematics is ahead of their time, extraordinary or not :-) A more fruitful definition might be, to do something that your contemporaries consider wrong or pointless, but that some time significantly later is realised to have been important. – Steve Jessop Apr 28 '14 at 16:09
@KonstantinosGaitanas: no matter how important the "thing", I don't see how "doing something at more or less the same time that someone else did it" can be described as "being ahead of your time". Being very clever and doing something incredibly important and novel, sure. But that is not what the idiom means and neither is it how the questioner characterises it for the purpose of the question. – Steve Jessop Apr 28 '14 at 16:19

I propose Bolzano. His understanding of mathematical rigor in analysis was certainly much more advanced than that of most of his peers. For instance, he correctly criticized the rigor of even the great C F Gauss's proof of the fundamental theorem of algebra. Among other examples I could cite, he proved the Extreme Value Theorem in the 1830's using what is now called the Bolzano-Weierstrass theorem. This was rediscovered by Weierstrass in the 1860's.

Another example of his insight is his example of a continuous nowhere differentiable function in 1831. His construction was essentially fractal in nature. Not only did it take until 1872 for Weierstrass to publish his famous example of the same kind of function, but Weierstrass' example was regarded as controversial and shocking all those years later.

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All names mentioned above deserve credit! A notable mathematician was only the one ahead of his time, when his/her mathematics changed the world. I would add, for my money, John Nash.

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I think John Nash is a particularly hard sell. His two main contributions are Nash Equilibirium and the Nash Embedding Theorem. His treatment of the Nash Equilibrium seems to have been very similar to some earlier work by Neumann. I think you would also have a hard time arguing that the Nash Embedding Theorem was ahead of it's time, since he did not even give the first correct proof of the theorem. – Bill Trok May 2 '14 at 17:43
I guess you are right. I am familiar with van Neumann's book. On grounds of Neumann's earlier work, yes, I agree that Nash is hard to sell. How about on grounds of results? His equilibrium changed the face of modern economics. With these said, I guess you are right. – Mihai Oct 21 '14 at 12:21

Rufus Isaacs' work in dynamic/differential game theory was substantially more advanced than the control theory work of the time. In fact, it can be argued that most control theory folks still don't understand it.

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With the discussion of what is actually meant by "ahead of their time" in mind, and even if he is not the "father of computing" as is often claimed, I still think that Alan Turing deserves a mention for the intuition that nonlinearities may give rise to complexities and unstable vacuum solutions, and for the intuition that such phenomena (through reaction--diffusion mechanisms) are responsible for observable biological phenomena such as the patterning of animal hides. Only recently has physical, non-computational evidence of this been found.

In his famous 1952 paper The Chemical Basis of Morphogenesis he describes a reaction--diffusion model that he argues can describe patterning in animal hides. Due to limitations in computer power at the time, his numerical investigations were limited to a one-dimensional model.

Modern computer simulations in 2D do indeed produce patterns that are strikingly similar to a wealth of animal hide patterns, and also other patterns in nature such as sand dunes. Importantly, Turing had no means to compute these patterns, but understood that his model could give rise to them, via the effects of nonlinearities, which was in itself a ground breaking idea.

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You can have people 'ahead of their time'. What really happens is you have 'mainstream maths', and than a whole lot of eccentrics who are 'doing their thing', which may or may not be where maths will be in 20 or 30 years time.

There are several things one has to note here. Some of us are just not really good at sitting down and writing theorms and proofs etc. I have a degree in physics, which means you approach the world in a different way to what the mathematicians do. None the same, you can do 'real maths', but do it without having to prove the tedia involved.

The sorts of things i needed for my studies in weights and measures, and number-bases (i use base 120 natively), etc, i had to go back something like 100 years and look at what the eccentrics were doing.

So, really, people really can be 20 or 30 years ahead of the mathematics, you just got to be in the right crowd of eccentrics.

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I am going to vote for Kurt Gödel. Not for his groundbreaking results in logics, but for his early understanding of problems with computational complexity.

Specifically this was mentioned in his lost letter to John von Neumann in 1956.

Only 15 years later, Stephen Cook and Richard Karp redeveloped these ideas, and formulated them preceisly, e.g. the Karps list of 21 NP complete problems.

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My vote goes to Carl Friedrich Gauß: "Mathematical historian Eric Temple Bell said that had Gauss published all of his discoveries in a timely manner, he would have advanced mathematics by fifty years" (

See also

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