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rel: When We Cease To Understand the World by Labatut
NOTE
I have no practice here, nor reason to be reading this outside of curiousity. But I’ll try anyway.
Being Alone
“In those critical years I learned how to be alone. But even this formulation doesn’t really capture my meaning. I didn’t, in any literal sense, learn to be alone, for the simple reason that this knowledge had never been unlearned during my childhood. It is a basic capacity in all of us from the day of our birth. However these three years of work in isolation, when I was thrown onto my own resources, following guidelines which I myself had spontaneously invented, instilled in me a strong degree of confidence, unassuming yet enduring in my ability to do mathematics, which owes nothing to any consensus or to the fashions which pass as law..By this I mean to say: to reach out in my own way to the things I wished to learn, rather than relying on the notions of the consensus, overt or tacit, coming from a more or less extended clan of which I found myself a member. or which for any other reason laid claim to be taken as an authority. This silent consensus had informed me both at the lycee and at the university, that one shouldn’t bother worrying about what was really meant when using a term like” volume” which was “obviously self-evident”, “generally known,” ”in problematic” etc…it is in this gesture of ”going beyond to be in oneself rather than the pawn of a consensus, the refusal to stay within a rigid circle that others have drawn around one-it is in this solitary act that one finds true creativity. All others things follow as a matter of course.
Since then I’ve had the chance in the world of mathematics that bid me welcome, to meet quite a number of people, both among my “elders” and among young people in my general age group who were more brilliant, much more ‘gifted’ than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle–while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things I had to learn (so I was assured) things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates almost by sleight of hand, the most forbidding subjects.
In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still from the perspective or thirty or thirty five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve done all things, often beautiful things in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have to rediscover in themselves that capability which was their birthright, as it was mine: The capacity to be alone.”
On Grothendeick
Far removed from conventional mathematics, Grothendieck’s final studies were fixated on the problem of why evil exists in the world. His last recorded writing was a notebook logging the names of the deportees in his father’s convoy in August 1942. Matthieu believes his father’s plant distillations were linked with this attempt to explain the workings of evil: a form of alchemy through which he was attempting to isolate different species’ properties of resilience to adversity and aggression.
By summer 1989, the prophetic dreams had intensified into daily audiences, “absorbing almost all of my time and energy”, with an angel Grothendieck called either Flora or Lucifera, depending on whether she manifested as benevolent or tormenting.
Of his fully realised concepts, toposes were his furthest step in his quest to identify the deeper algebraic values at the heart of mathematical space, and in doing so generate a geometry without fixed points.
The magic of things
There seemed to be, within rhymes, a mystery which extended beyond words. I believed this until the day somebody told me that there was a simple “trick” to it; that rhyme was simply when one ends two consecutive spoken movements by the same syllable, so that, as if by magic, these phrases became verses.
In any case, when a problem “grabbed me”, I did not count the hours or days that I spent working on it, even if it meant losing track of everything else! (And such remains the case to this day… )
Omissions to questions he was asking himself:
As long as we were provided with recipes for all sorts of calculations, such as lengths, areas, volumes, through single, double, triple integrals (dimensions higher than 3 were carefully avoided… ), the problem of providing an intrinsic definition was omitted by both my professors and textbook authors.
The importance of being alone
Independently learning and converging:
I ended up learning, among many other things, that the work which I had been doing independently, and with the means at hand, was (essentially) what “everybody” knew as the “Lebesgue theory of measure and integration”. According to the two or three experts to whom I mentioned my work (or even showed a manuscript), I had just wasted my time redoing something “already known”. I actually do not recall being disappointed. At that moment, the idea of receiving “credit”, or even simply receiving approbation for the work that I was doing, must have still been foreign to my mind. Furthermore, my energy was completely taken by the process of familiarizing myself with an entirely different milieu and mostly learning what was considered in Paris to be the basic toolkit of the mathematician
It is in this act of “turning a blind eye”, of being oneself rather than the mere expression of the reigning consensus, of not to remain inscribed within the imperative circle to which they assign us - it is within this solitary act, above all else, that “creation” lies. Everything else comes after.
...technical terms. The latter will mostly go “above the heads”, not only of the “profane”, but also of the mathematical colleague who may not be completely “in” the field in question. One can of course feel free to skip the sections which seem too “involved”. Just as one can try to go through them, glimpsing as one goes, a shadow of the “mysterious beauty” (in the words of a non-mathematician friend of mine) of the universe of mathematical things, appearing as a multitude of “strange inaccessible Islands” in the vast moving waters of reflection
He belongs to a lineage of mathematicians whose spontaneous vocation and joy was to continuously construct new houses. In doing so, he cannot help but invent all of the tools required and furnishing. Yet once everything has been taken care of the builder rarely lingers on the premises, of which every stone and piece of wood carries a trace of the hand which shaped and placed it. The builder’s place lies not in the quietude of fully finished universes, however welcoming and harmonious they may be, whether they are a product of his own hands or those of his predecessors. His place is in the open air. He is friends with the wind, and does not fear solitude at work for weeks, years or, if need be, for an entire lifetime if no welcome succession presents itself. Just like everybody else, the builder only has two hands - but two hands which at each moment know what they need to do, which refuse neither the largest nor the most delicate tasks, and which never tire of comprehending, again and again, the multitude of things which become them.
Viewpoint and vision
I am led more towards the discovery of fertile viewpoints than towards the discovery of questions, notions, and statements, by my particular type of genius, which is constantly leading me to introduce, and more or less develop, entirely novel themes. It is this, I reckon, which is my most essential contribution to the mathematics of my time. In fact, these innumerable questions, notions, and statements which I just mentioned, only truly make sense for me once they are subjected to such a “viewpoint” - or more precisely they arise spontaneously from it; in the same way that a light (even a dim one) appearing in a pitch black night seems to invoke from the shadows contours which it suddenly reveals to us.
rel:Survey of Shadows
Sleepwalking:
The metaphor of the “sleepwalker” was inspired by the title of the wonderful book “the sleepwalkers” by Koestler (Calman L´evy), presenting an “Essay on the history of the conceptions of the universe”, starting from the origins of scientific thought, all the way to Newton. One of the facets of this history which struck Koestler, and which he highlights is the extent to which, often, the path from a given state of our understanding of the world, to some other state which (logically and with hindsight) seems very close, sometimes takes the most astounding detours, which appear to defy reason; and to think that yet, despite those thousand detours that could conceivably have lost them forever, and with the “certainty of sleepwalkers”, men who have gone on the quest for the “keys” of the Universe find, as if unintentionally and without even realizing it, other “keys” that they would never have thought of, and which nonetheless appear to be “the right ones”
Thought
I had thought about a similar thing Near Far Solution
Dream Fragments and digressions:
Moreover, the mode of mathematical expression which was professed and practiced by my elders, gave preeminence (to say the least) to the technical aspect of one’s work, and in no way encouraged the “digressions” that would have idled on the “motivations”; or even those which appeared to bring out of the mist some vision which perhaps was inspiring, but which, because it failed to be presented in the form of tangible constructions in wood, stone, or hard cement, was likened more to dream fragments, than to the craft of a conscientious and diligent artisan.
Form and structure
- number (arithmetic)
- size (metric)
- shape (geometric)
rel:Expression
The structure of a thing is not something which it is possible for us to “in- vent”. We can only patiently unravel it, humbly get to know it and “discover” it. If there is any ingenuity involved in this line of work, and if we sometimes take up the role of a blacksmith or that of a tireless builder, it is never to “model” or to “construct” “structures” - they didn’t have to wait for us to exist, and to be precisely what they are! Rather, it is to express, as faithfully as we can, those things which we are in the process of scanning and discovering, the structure that is reluctant to surrender and which we attempt to grasp, fumblingly, and through a perhaps fledgling language.
rel: The Zen of Seeing - Frederick Franck
What characterizes the value of the ingenuity and imagination of a researcher is the quality of his attention as he listens to the voice of things - for the things of the Universe never tire of talking about themselves and to reveal themselves to he who cares to listen. Thus, the most beautiful house, that in which the love of the builder is most evident, is not that which is larger or higher than the others. Rather, a house is beautiful if it faithfully reflects the structure and beauty hidden in things.
“Scheme” is a “magical fan” which links to one another several different branches, its “avatars” or “incarnations” in every possible characteristic.
Innocence: In our probing of things in the Universe (mathematical or otherwise), we dispose of a crucial rehabilitating power: innocence. By this I mean the original innocence which we have all received at birth and which rests within us, often the target of our scorn and of our deepest fears. It alone unites the humility and audacity which allow us to penetrate into the heart of things, while also allowing these things to penetrate into us and impregnate us with their meaning.
Topology
We have invented and created from the ground up all sorts of “meters” and “metersticks” that let us, against all odds, attached “measurements” of sorts (called “topological invariants”) to these tentacular “spaces” that previously seemed to escape all attempts to measure them - akin to an elusive mist.
What really matters in a topological space isn’t its “points” or its subsets of points (and the proximity relations between them, etc…); rather, it is the sheaves on that space that matter, and the category which they form.
Mutations of the notion of space - or breath and faith
Introduces the notion of topos, which is a metamorphosis of the notion of space, and with it the promise of renewal of topology and geometry. There are transpositions to the traditional spaces of yesteryear.
The heart within the heart
It is the theme of topoi, not that of schemes, which constitutes this “bed”, or “deep river”, in which geometry and algebra, topology and arithmetic, mathematical logic and category theory, the world of the continuous and that of “discontinuous” or “discrete” structures come to be wed
Schemes are the heart of the novel geometry, topoi is its dwelling.
“Yoga of motives”
As a mathematician, he sees two forces at play. The builder and the pioneer/explorer.
He continues to develop these traits to subtypes, like the builder-child.
The reflection thus led to the realization that this haughty “builder”, or (more modestly) the child-who-plays-at-building-houses, was but once of the two faces of the famous child-at-play. There is also the child-who-likes-exploring- things, who enjoys rummaging and burrowing away in the sand or the nameless muddy waters, in the most improbable and bizarre places… Probably in an attempt to put up a front (even if only for myself…), I started introducing him under the flamboyant title of “pioneer”, followed that of “explorer”, a more down to earth title that nonetheless remained charged with prestige. It begged the question of which title between “builder” and “pioneer-explorer” sounded more manly, more appealing! Heads or tail?
The pioneer is also a girl (who was dressing up as a boy) - a sister of the ponds, the rain, the drizzle and the night, silent and almost invisible as result of her tendency to fade into the shadows.
strange rel: Alterity Stranger
To become acceptable in earnest, the name would need to feature the other as well. Yet, strangely, “the other” doesn’t really have a name. The only one that somewhat does it is “explorer”.
A Letter
Thought
Directly addressing the reader here. This feels science fictional.
The text which I am hereby sending you, of which a limited number of copies were typed and printed by my university, is neither an off-print, nor a preprint. Its title, R´ecoltes et Semailles, makes this much clear. I am sending it to you the way I would send a long letter - including the personal dimension, indeed. If I have decided to send it to you, rather than await for you to learn about it some day (if your curiosity leads you to it) in the form of some publicly available volume in a library (if there even exists an editor crazy enough to engage in such an adventure…), it is because I am addressing this letter to you more than to others. I have thought of you more than once in the course of writing it - I must say that I have now been writing this letter for more than a year, and devoting all my energy to the task. It is a gift I am making you, and I took great care in the process to give out what I had best to offer (at any given moment). I do not know whether or not you will welcome this gift - until your response (or absence thereof) brings me the answer.
read it not only with your eyes and head but also with your heart, you will surely recognize yourself even in places where you are not explicitly named.
Gangrene - or the spirit of our times