Rome by Different Roads

[In our basic essay on Consilience in Mathematics [which is under reconstruction, following a blogspot glitch; temporarily redirected to this post]  one topic was the way in which a given mathematical topic or object can be approached from quite different angles, or defined in superficially quite different ways, so that it is finally surprising that (after a lot of work and lucubration) you wind up at the same mathematical metropolis.   And just as with the phenomenon of parallax, or triangulation (which we learned in Boy Scouts), the result is a more multidimensional appreciation than we began with.]

We have discussed the notions of abstraction and of generalization as regards the overall ‘topology’ (or geography) of the  mathematical landscape.   Not the same as either of these are cases where two intuitively quite distinct notions turn out to imply each other -- to be provably logically equivalent.  As, the (growing) menagerie of propositions that are equivalent or weakly equivalent to the Axiom of Choice.

The closest notion to this that we have previously touched on is that of  “Rome by Different Roads”, which in the simplest case means merely that the same proposition, say of number theory, can be reached (proven) with distinct sets of tools, e.g. an heavy-machinery analytic proof versus an ‘unplugged’ proof (generally using quite different ideas) which ascetically restricts itself to using only elementary methods.   The proposition you arrive at remains unchanged, the variety of proofs  something of a curiosity.  (As, the many different ways of proving the Pythagorean Theorem, most elegantly involving no more than a single diagram -- a small square tiltedly inscribed within a larger.)   It’s like -- tiens!  this road leads to Rome as well.

Whereas in this deeper phenomenon, the different equivalences -- the different perspective views of the same central Thing -- actually wind up changing or rather sublating what one’s concept of Rome is.

It’s a bit like spooky action-at-a-distance in quantum physics, in which you can’t consider the particles to be quite independently existing, even though they are distinct.   There seems to be looming some supernatent notion, some eka-conception, of which each individual proposition from the pool of those interderivable, are like the sundry avatars of Vishnu, and not Vishnu himself.

The study of computability came to be known as recursion theory, because early formalizations by Gödel and Kleene relied on recursive definitions of functions.When these definitions were shown equivalent to Turing's formalization involving Turing machines, it became clear that a new concept – the computable function – had been discovered, and that this definition was robust enough to admit numerous independent characterizations.

-- Wiki, “Mathematical Logic”

Specifically, equivalent formulations of the intuitive notions of computability, independently arrived-at by different thinkers, include:  Church’s lambda-calculus; Kleene’s general recursive functions; Post’s automata; and Markov algorithms.

For the notion of “independent characterizations”, compare the Heisenberg formalism vs. the Schroedinger formalism  in quantum mechanics.   That one is a cause célèbre, and with few comparable situations in the history of physics;  whereas mathematics is chock-full of people discovering the same mountain via different slopes.

The Wikipedia article on “Representation theory” highlights “the diversity of approaches to representation theory.  The same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatoriecs  and topology.”

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For an application of this theme to the topic of uniform spaces,  consult this essay.