Mathematicians ahead of their time?

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.