Manin: "Forgotten Motives: the Varieties of Scientific Experience"

[dmm]

Про многие научные темы, мне хотелось понять, что в них происходит на самом деле, без того, чтобы достигать уровня, когда умеешь решать задачи. И есть темы, которые никакая популярная литература даже и не пытается покрывать (насколько я знаю).

Вот, оказывается, что эта замечательная неформальная статья Манина в сборнике памяти Гротендика проливает свет на целый ряд понятий, которые я считал совершенно недоступными:

arxiv.org/abs/1402.2155

Comments

[dmm]

"However, the passage from the set of morphisms to the K–module of correspondences involves one more intuitive idea, and it can be most succinctly invoked by referring to physics, namely the great leap from the classical mode of description of nature to the quantum one. This leap defined the science of the XXth century. Its basic and universal step consists in the introduction of a linear span of everything that in classical physics was only a set: points of a phase space, field configurations over a domain of space–time etc. Such quantum superpositions then form linear spaces on which Hilbert–like scalar products are defined, that in turn allow one to speak about probability amplitudes, quantum observations etc."

(Italics mine)

Specifically, this sentence is the key in a variety of other contexts, which don't have anything to do with quanta (the importance of forming linear spans from all kinds of things is much wider):

"Its basic and universal step consists in the introduction of a linear span of everything that in classical physics was only a set:"

[dmm]

And there is much more; I'll be rereading this and writing more comments.