In his work, Albert Lautman thinks the genesis of some mathematical works as a dialectic that takes place between opposite notions, like between global and local. He argues that while those notions, in the natural language, are obviously linked by their opposition, the *souci* or concern that stems from it can only be studied in mathematics. In other words, the study of the crux of this opposition ends up being a mathematical study. He then suggests that the friction between such pairs of opposite notions, originally in the language, is the origin of many mathematical developments.


I would like to show that there are "lone" notions, that is notions not thought as part of a pair like one/many, intrinsic/extrinsic etc., that led through their study (or can be linked, maybe notwithstanding some historical reality of the study) to interesting mathematical developments. Hence my question :



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> What are examples of stand-alone notions (in the language) that calls (by their importance and their wide meaning) for a formalisation which can be linked to new mathematical works ?
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An example : The development of the theory of computability may be seen as an answer to the *souci* that comes from the informal concept of "computation". I can do a computation on a sheet of paper, Babbage's machine does computation, a sum is a computation... Then what unifies those ideas of computation ? Three attempts to capture the idea behind it have been made. Those are the theory of general recursive functions, Turing machines and the lambda calculus ; each of those embodies some aspect of the notion of computation, and it happens that those three definitions are equivalent. Hence, the Church-Turing thesis states, arguably, that those indeed capture the pre-conceptual notion of "computation".


This will be in a writing addressed to philosophers. So the simpler and the more fundamental, the better.



Formalisation of intuitive concepts in the language leading to mathematical progress
An example that is closely related to computation is proof. Mathematicians have been proving theorems ever since Euclid (and presumably even earlier). But it was not until the 20th century that the concept of a proof was formalized to the point where they could be studied as mathematical objects in their own right.


Though it does not have a well-known name like "Church–Turing Thesis," there is an analogous "formalization thesis" about proofs that I have mentioned [elsewhere on MathOverflow](https://mathoverflow.net/a/24919) as well as on the [Foundations of Mathematics mailing list](https://cs.nyu.edu/pipermail/fom/2010-May/014743.html):



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> Given any precise mathematical statement, one can exhibit a formal
> sentence S in the first-order language of set theory with the property
> that any mathematically acceptable proof of the original mathematical
> statement can be mimicked to produce a formal proof of S from the axioms
> of ZFC.
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