Grothendieck's Vision and Semantic Unfolding in Topology: Unification of Riemann's Geometric with Galois' Algebraic Unfolding Paradigms via Topological Covering Spaces and the Fundamental Group as a Galois Group

Points as Fiber Functors

Universal Covering of the Circle by the Helix: Simply-Connected Unfolding

Spectral Amalgamation

Homotopical Unfolding: The Fundamental Group as a Galois Group

Grothendieck's Phraseology

Grothendieck introduces the metaphor of thinking of a theorem to be proved as a nut to be opened, so as to reach "the nourishing flesh protected by the shell." He describes two approaches in order to explicate his conceptual advancement method, which is based on the second approach in the sequel. The first approach is based on the "hammer and chisel principle":

Put the cutting edge of the chisel against the shell and strike hard. If needed, begin again at many different points until the shell cracks - and you are satisfied.

He proceeds in the description of the second approach as follows:

I can illustrate the second approach with the same image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months - when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado! A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration ... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it ... yet it finally surrounds the resistant substance.

In this "rising sea" metaphora the resolution of a mathematical problem, culminated in the formulation of a theorem, is "submerged and dissolved by some more or less vast theory, going well beyond the results originally to be established." This is precisely the vision characterizing Grothendieck's perspective on the fundamental theorem of Galois theory in the context of polynomial equations. The vast theory is the topological generalization of Galois theory culminating into the conception of Poincare's fundamental group of a topological space as a Galois group and the concomitant connection with Riemann's geometric unfolding paradigm.