I am not familiar with Grothendieck's work and with algebraic geometry in general, but Turing's work was rather unique, and very much philosophical in nature[0].
I am basing the following on Turing's 1936 paper, On Computable Numbers[1], on Juliet Floyd's chapter on Turing's mathematical philosophy[2], and on Robin Gandy's 1988 paper, The Confluence of Ideas in 1936[3].
The generalization of mathematical concepts has always been an important part of mathematics. In fact, Gandy writes that this was precisely the concern with Church's claim that the lambda calculus can express all algorithms ("a vague and intuitive notion", in Church's words). It was thought that perhaps a genius could some day generalize the notion further to include lambda-definability as well as things that fall outside it. But then Turing appeared with arguments of a very different kind. For one, his construction of an automatic machine, while precise, was completely arbitrary. It was clear to everyone from his treatment that while his math concerns a specific formalism (what's known as the Turing machine ever since Church named it so), it really covers all formalisms. For another, his proof of undecidability is very different from the one normally presented as the proof for the undecidability of the halting problem (which is really due to Martin Davis). Turing's proof, Floyd notes, makes no use at all of any logical construction, and does not rely even on logical contradiction or the law of excluded middle (which are expected to hold in many "reasonable" formalisms) but on more basic philosophical principles that must hold in any and all formalisms.
Gödel called Turing's achievement a "miracle". In 1946 he said (quoted in Gandy's paper):
...with this concept one has for the first time succeeded in giving an absolute definition of an interesting epistemological notion, i.e., one not depending on the formalism chosen. In all other cases treated previously... one has been able to define them only relative to a given language, and for each individual language it is clear that the one thus obtained is not the one looked for. For the concept of computability, however,... the situation is different. By a kind of miracle it is not necessary to distinguish orders, and the diagonal procedure does not lead outside the defined notion...
[0]: Turing was a mathematical philosopher among other things, and is known for his discussions with Wittgenstein, published in Wittgenstein's Lectures on the Foundations of Mathematics.
I am basing the following on Turing's 1936 paper, On Computable Numbers[1], on Juliet Floyd's chapter on Turing's mathematical philosophy[2], and on Robin Gandy's 1988 paper, The Confluence of Ideas in 1936[3].
The generalization of mathematical concepts has always been an important part of mathematics. In fact, Gandy writes that this was precisely the concern with Church's claim that the lambda calculus can express all algorithms ("a vague and intuitive notion", in Church's words). It was thought that perhaps a genius could some day generalize the notion further to include lambda-definability as well as things that fall outside it. But then Turing appeared with arguments of a very different kind. For one, his construction of an automatic machine, while precise, was completely arbitrary. It was clear to everyone from his treatment that while his math concerns a specific formalism (what's known as the Turing machine ever since Church named it so), it really covers all formalisms. For another, his proof of undecidability is very different from the one normally presented as the proof for the undecidability of the halting problem (which is really due to Martin Davis). Turing's proof, Floyd notes, makes no use at all of any logical construction, and does not rely even on logical contradiction or the law of excluded middle (which are expected to hold in many "reasonable" formalisms) but on more basic philosophical principles that must hold in any and all formalisms.
Gödel called Turing's achievement a "miracle". In 1946 he said (quoted in Gandy's paper):
...with this concept one has for the first time succeeded in giving an absolute definition of an interesting epistemological notion, i.e., one not depending on the formalism chosen. In all other cases treated previously... one has been able to define them only relative to a given language, and for each individual language it is clear that the one thus obtained is not the one looked for. For the concept of computability, however,... the situation is different. By a kind of miracle it is not necessary to distinguish orders, and the diagonal procedure does not lead outside the defined notion...
[0]: Turing was a mathematical philosopher among other things, and is known for his discussions with Wittgenstein, published in Wittgenstein's Lectures on the Foundations of Mathematics.
[1]: https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf
[2]: https://mdetlefsen.nd.edu/assets/201037/jf.turing.pdf to appear in Philosophical Explorations of the Legacy of Alan Turing
[3]: http://dl.acm.org/citation.cfm?id=213993 published in The Universal Turing Machine A Half Century Survey, 1995