Never be misled by others. Whether you're facing inward
or facing outward, whatever you meet up with, just kill it!
If you meet a Buddha, kill the Buddha.
If you meet a patriarch, kill the patriarch.
If you meet an arhat, kill the arhat.
If you meet your parents, kill your parents.
If you meet your kinfolk, kill your kinfolk.
Then for the first time you will gain emancipation, will not be
entangled with things, will pass freely anywhere you wish to go. (Línjì Yìxuán)
Who am I?
Someone on internet knows me as Eotvos, others know me as tetrapharmakon. However my best avatar (grasping my whole intention and personality) is killing_buddha (as for my real name, it can be found in more bureaucratic pages).
I am an Italian (young, wannabe) mathematician who lived in Padua, Italy, until now, and then moved to Trieste beginning its first attempt to conquer the world. My scientific interests are pretty wide, even if I feel like the utmost ignorant on Earth. I spend much time reading about category theory, categorical algebra, differential geometry, algebraic geometry, homotopy theory, and geometry inspired by high energy physics. I've ever been fascinated by the epistemological problem of figuration of space: I still hope to find out a coherent geometrical model of Borges' Library of Babel.
I'm strenuously striving to get rid of any conceptual redundancy in (my) mathematical practice, pursuing the belief that "minimalism in assumptions goes together with maximalism in conceptual distinctions", and unravels the "real" behaviour of Mathematics under the probe of our mind. Ask yourself: how often, in the history of Science, formal analogies were covered by the "wrong" point of view? Here, "wrong" can be intended as the fragmented, edonistic tendency of a certain mathematical practice to concentrate collective efforts on solving a particular instance of a problem, instead of building a theory eroding your question through the millennia. I like to think I'm part of a cultural process whose end is neither near nor easy to reach, some kind of Human Genome Project applied to pure Mathematics; I also believe that this approach is the only possible way to reach a comprehensive, deep understanding of mathematical structures, in view of the superexponential grow-rate of their complexity and of the connectivity of the graph they form.
Why am I here?
At the beginning of my student life, I chose to study Geometry, and Algebra therefore, for a simple reason. When I was still a child, I dreamed to become an artist, a painter or a sculptor. Later I discovered Mathematics, and I found out the same feelings through the infinitely malleable and ideal shape of a manifold, or the polished and perfect matter which Riemann surfaces are made of. When I have a pencil or a chalk in my hand, when I write on a blank sheet or a blackboard, when I plot a graph or I draw a commutative diagram, then I feel the same artistic sensations.
With the passing of time, Mathematics revealed me the deep and majestic identity between the shape of an object (its geometrical nature, its physical and plastic properties) and its gist (its purest essence, its algebraic and axiomatic construction).
Following Klein's point of view, for example, I can perceive that what we call a geometry is nothing else than the result of applying the action of a suitable group over an appropriate set: on changing the shape of a space, we modify the relationships among the objects, not really caring about the objects themselves. Moving its early steps from a basic intuition (i.e., the identity principle), Mathematics elevates itself focusing on the relations among a wider spectrum of entities, whereas it is important the way in which the objects at hand are related to each other and not their particular nature.
What I do.So, pure mathematics is the art of recognizing patterns between objects, in such a way that two seemingly different entities (e.g. commutative rings and affine schemes, or pointed sets and sets with partial functions between them, or abelian simplicial groups and connective chain complexes of abelian groups) fall, in the end, in the same conceptual box (i.e. they are, in a suitable sense, different facets of the same "thing", whose real nature is hidden by the fog of our ignorance).
This point of view culminates, IMHO, in the mathematical theory of categories.
The leading idea wherewith Category Theory looks at Mathematics is subordinate the nature of the entities to the chance of linking them in a (sensible) web of mutual relations. Roughly speaking, one forgets about the "static" nature of objects, unraveling the "dynamical" ways it is modified by the action of external transformations (we call them morphisms). Starting from a very basic idea (the identity principle between entities in a "bunch"), we elevate to a more subtle one, the idea of relation between "things in the bunch". Mathematically speaking, we are witnessing a huge turnaround: the primeval ideas of set and element, introspective by their nature (a set is characterized by its elements), is no longer enough to completely describe those "systems" not fitting into this too introspective setting. Because of this we move studying "objects" in a leibnizian way, giving a number of external relations between indivisibles, in a family which gathers all the objects subject to a suitable intensional definition (in fact minding some foundational hardships like Russel's paradox).
Certainly this point of view may appear unnecessarily and unpleasantly smoky; the interested reader (if there is any) may profit of an analogy from linguistics.
The concrete side of a language is its semantics, the practical use of phonemes to recognize a precise object, astraying it from the context (the "universe"); synthax fulfils its abstract side, explaining relations between objects, minding their interdependence. What's worth in talking is not what words are, but how words relate each other.
As a matter of storiography, one could find in the seminal work of Eilenberg and Mac Lane General Theory of Natural Equivalences the birthday of the basic definition of category, functor, natural transformation et cetera. but (and it happens everytime one tries to date back an event) I likely believe that these ideas are much more ancient. One can in fact date back the origin of a "categorial" (=structuralist) point of view in Mathematics in Felix Klein's talk Vergleichende Betrachtungen über neuere geometrische Forschungen (Comparative observations on recent geometric research), with which (as I said before) we began to call a geometry the mere datum of a group action on a set of objects, whose nature is never investigated. Nothing matters but the orbit-spaces, and a "geometric property" in defined to be any property which is invariant under that action. Two objects are equal up to isomorphism if and only if the are linked by a suitable invertible transformation in the group which identifies our "geometry", or rather our structure (projective linear transformations, holomorphic maps, bijections, reflections, canonical transformations, monotone mappings, ...).
It's worth mentioning that even Poincaré said that "[...] les mathématiciens n'étudient pas des objets, mais des relations entre les objets; il leur est donc indifférent de remplacer ces objets par d'autres, pourvu que les relations ne changent pas." (La Science et l'hypothèse, 1902): so it's far from surprising that similar ideas had slowly but constantly permeated any other humanistic and scientific discipline, touching Sociology and Literature, with Propp's ''Morphology of the Folktale'', and even Philosophy, Linguistics and Psychology with the advent of the structuralist school.
The "minimalism" mentioned before together with the structuralist point of view offered by Category Theory, is simply the most natural way to implement this point of view in the practice of Mathematics: quoting an old post on MathOverflow: "If we didn't have category theory we would feel really stupid, constantly proving the same theorems about lots of 'different' objects."
(This is how I feel everyday when I'm doing Maths).
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