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It is said that in every field there’s that person who was 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 resembled modern chess theory.
So, who was the Paul Morphy of mathematics?
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.
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.))
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.
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.
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.
Archimedes, for basically discovering calculus eighteen centuries before either Newton or Leibniz. See Archimedes Palimpsest for more details. :-)
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:
(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...
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.
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.
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.
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]
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.
What about George Boole (1815-1864) with boolean algebra (1847) that didn't have any practical use until the introduction of computers?
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.
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.)
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?
EDIT
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.
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.
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.
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)
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.
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" (https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss#Personality)
See also
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.
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.
I am pretty sure, Clause Shannon and Alan Turing will also find a place in this list. They may not be bracketed purely as mathematicians, but as applied mathematicians, they surely deserve a place here.
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.
The problem of key distribution is one that has historically afflicted cryptography. No matter how secure a cipher is, it can be compromised if the key is not shared securely.
With businesses growing in size and having more keys to be delivered, key distribution became the major issue for post-WWII cryptographers. The Diffie-Hellman-Merkle key exchange scheme (1976) was a brilliant attempt at tackling this problem by using one-way functions in modular arithmetic.
A year later, in 1977, this idea was further developed by Rivest, Shamir and Adleman into what is known today as RSA — a system of public key crytography which uses an asymmetric key rather than a symmetric one (which had been the norm in all of history).
However, one man, Clifford Cocks, had already invented the “RSA” algorithm back in 1973.
Another mathematician, James Ellis, working at the UK's top-secret Government Communications Headquarters (GCHQ) had already invented the idea of non-secret encryption, anticipating Diffie, Hellman and Merkle by several years.
Clifford Cocks, then a recent PhD graduate from Cambridge, was among the mathematicians recruited to attempt to find the required one-way functions for public key cryptography to work. He realized that prime factorization is the answer and developed the same solution that would later become known as the RSA encryption algorithm.
Although Cocks’s idea was one of the GCHQ’s most formidable discoveries, it suffered from the problem of being ahead of its time. Neither the GCHQ nor the NSA with whom the discovery was shared were able to find a way to use the algorithm, and they simply treated it as classified information.
The work was finally declassified approximately 25 years after Cocks’s discovery.
[Source: The Code Book: The Science of Secrecy from Ancient Egypt to Quantum Cryptography]