The Elusive Proteus
Posted by David Corfield
Cassirer writes in Determinism and Indeterminism in Modern Physics:
At first glance the universality of the action principle seems by no means beyond question. This universality could be attained only at the cost of a circumstance which, from the purely physical point of view, led again and again to difficulties and doubts. For the more universally the principle was conceived, the more difficult it became to specify clearly its proper concrete content. It becomes finally a kind of Proteus, displaying a new aspect on each new level of scientific knowledge. If we ask what precisely that “something” might be to which the property of a maximum or a minimum is ascribed, we receive no definite and unambiguous answer. (p. 50)
Note the name ‘Proteus’, a Greek sea-god who
…can foretell the future, but, in a mytheme familiar from several cultures, will change his shape to avoid having to; he will answer only to someone who is capable of capturing him. From this feature of Proteus comes the adjective protean, with the general meaning of “versatile”, “mutable”, “capable of assuming many forms”: “Protean” has positive connotations of flexibility, versatility and adaptability.
Striking then how Saunders Mac Lane describes mathematics as ‘Protean’ in this paper.
The thesis of this paper is that mathematics is protean. This means that one and the same mathematical structure has many different empirical realizations. Thus, mathematics provides common overarching forms, each of which can and does serve to describe different aspects of the external world. (p. 3)
On the face of it, however, there’s a difference between Cassirer’s multiple-realisability and Mac Lane’s. The latter appears to be talking about a common structure reappearing with little disguise from one place to the next in the empirical world. But the examples Mac lane gives suggest he’s also thinking about the unpredictable unfolding of an idea within mathematics itself.
For the Galoisian idea he begins with Lagrange in 1770 finding resolvants for equations, and ends with ‘Various’ in 1990
One adjunction handles Galois and much more.
It is perhaps odd that he doesn’t include Poincaré and his deck transformations in this line of thought, although he does have Grothendieck in 1961 introducing Galois theory for covering spaces. And he might have said ‘covering spaces for number theory’.
So how’s that for a Protean principle in mathematics to rival the action principle in physics? Whatever that something is which has manifested itself most recently in -fibrations and fundamental groups in number theory, or can it be given a ‘definite and unambiguous form’?
Re: The Elusive Proteus
‘…the adjective protean, with the general meaning of “versatile”, “mutable”, “capable of assuming many forms”’
A complete aside but… Is there an etymological reason that protons are named as they are? The ancient alchemists considered Proteus as a symbol of First Matter. (There are references to that in Milton’s ‘Paradise Lost’.)