I will not return here to the ambiguity, highlighted by Jean Dieudonné, which attaches to Valéry’s use of the word “group”. From the moment one speaks of the group of all transformations of a mental object, as soon as this object is represented by a geometric continuum (for example, by a variety), one is led to consider groups of infinite dimension, like the group of all diffeomorphisms of a variety. Now, for this kind of object, we no longer have the tools (representation theory, Noether’s theorem, etc.) that permit the construction of quantitative models in physics from the symmetries of space-time (Galileo group, Lorentz group). If one holds to a strictly operationalist point of view, this is a serious situation, because no experimental control is possible anymore. (This is the case with catastrophe theory, which is an instrument of intelligibility, not of control…)
Most probably, Valéry conceived his “System” all alone; if the mind is capable of modelling a mental process on the model of a physical process, it will be able, by applying to the elements furnished by perception mathematically well-defined operations (extracted from a “group”), either to realise a work of art, or to conceive a technico-scientific project (such as Leonardo’s flying man). But Valéry very quickly became aware that with the mathematics he knew (a fairly elementary algebra and mechanics), he could not realise this ambitious programme. This is why he dreamed of an extension of mathematics, the “more subtle numbers” (abbreviated “N + S”), of which, apparently, the best definition is found in a notebook from 1929:
“— Under the secret name of nombres plus subtils, ‘(N + S)’, I meant this:
that there was a central mental attitude which took account of exchanges between the most different objects of knowledge — times, for example, and things that could be exchanged, being elements of a certain equivalence.
Which generalises ad infinitum (but qualitatively) equivalences like those of heat-work, mass-light, etc.
One thus arrives at conceiving a system of heterogeneities — and its equilibria — (Gibbs phases), but here one must conceive this system as maximum of heterogeneity — in its substitutions.
Whatever the nature of consciousness may be, it is substitution, and one can think that the elements which substitute themselves have at least between them, whatever they may be — a certain common property — a certain conformity or congruence with the unknown conditions of knowing — and therefore between themselves.
This relation has as particular cases what one calls space and time, relations, which both enclose heterogeneities — G₂, G₁ (and today G₄ᵢ).”
Translation by Thomas Murphy. From René Thom’s essay on Paul Valéry.
