Il Dio dei matematici

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Questa intervista al grandissimo Enrico Bombieri e’ molto profonda. A un certo punto B. dice:

“Il Dio che viene dal pensiero di Gauss, così come il riferimento ‘il cielo stellato sopra di me’ di Kant, che pur non essendo un riferimento a Dio rappresenta un pensiero di umiltà, presi da soli e non in un contesto più grande, ci danno solo un Dio astratto. Il problema dell’origine dell’universo, che chiaramente è di natura dinamica e non statica, appare in ogni cultura fin dalle origini dell’umanità. Il Big Bang dell’astrofisica moderna non solo ci fa pensare alla creazione biblica, ci dice anche che il tempo è stato creato insieme all’universo, un concetto che risale alla metafisica di sant’Agostino. La matematica è essenziale per dare consistenza a tutto questo, ma da sola non basta per dire che questa visione dell’origine dell’universo stellato di Kant sia esatta al 100 per cento. Il Dio dell’amore non c’è”

E ancora:

“Pascal e De Giorgi avevano compreso che Dio non è solo un Dio platonico, astratto, geometrico, aritmetico, o semplicemente creatore di un universo lasciato a se stesso. Essi avevano la visione di un Dio che è più difficile da comprendere, un Dio che è fatto non solo di potenza ma anche di amore infinito. Solamente così diventa possibile, con umiltà, accettare il concetto cristiano della Redenzione.”

Il problema e’ questo: non e’ per niente chiaro in che senso si dovrebbe essere portati a trascendere l’orizzonte del dio geometrico in una ottimistica visione dell’universo, un dio che non sia cioe’ puro caso, natura matrigna.

Nella lezione  sul concetto Spinoziano di Dio (sotto), dice Carlo Sini:

“sostanza, sive natura sive deus, non e’ una cosa, non si puo’ farne scienza.
Si puo’ fare scienza degli oggetti particolari – modi. Il mondo non e’ una cosa, e’ una superstizione, un modo di dire. Il mondo – la sostanza, non e’ una cosa. E le cose non sono sostanza, sono transiti, modi della sostanza.”

E ancora:

La conoscenza di secondo genere – la scienza matematica – ci porta a conoscere – calcolare- l’estenzione e il movimento. La filosofia, i.e. la conoscenza di terzo genere, constinua a vivere perche’ e’ un cammino di liberazione, un esercizio di liberazione.
[…]
La conoscenza di terzo genere, comprensione della infinita libera necessita’ del tutto- Dio. Questo trascende completemente il carattere socratico cristiano (platonico-cristiano) del pensiero occidentale.

Le parole di Bombieri sono platonico-cristiane. Il concetto di amore e’ fuori dell’orizzonte spinoziano. La fede e’ tutta qui. La fede – come dice Sini- e’ tutt’uno col carattere contingente dell’universo. La necessita’ libera del sive natura e’ al di fuori.

Una possibile risoluzione del paradosso consiste forse nell’interpretare il concetto di Rivelazione come una grande metafora dell’apprendimento in un universo caotico, dove il segnale e’ sempre sporco, “rumoroso”. La sofferenza che uno ha da portare e’ dovuta al processo di apprendimento, continuo, indefesso.

Male come incompletezza, e non male radicale, sarebbe allora implicazione di bonta’ nel senso di Bombieri. Negativo come elemento casuale, “vetro sporco”, mai male manicheo. Forse e’ questo il senso di (Kierkegaard, “Vangelo delle Sofferenze”, I):

“certamente il pellegrino non si riconosce dal bastone che porta in mano […]; mentre il chiamarsi credente indica evidentemente che si e’ in viaggio, poiche’ la fede significa propriamente che cio’ che io cerco non e’ qui, proprio per questo io lo credo. Fede significa precisamente l’inquietudine profonda, forte, beata che trascina il credente; poiche’ il credente non puo’ mettersi a sedere tranquillo, come ci si siede con un bastone in mano. Il credente cammina in avanti”.

Rimane in ogni caso il dubbio del perche’ infinita potenza e bonta’ (sic) abbiano prodotto un messaggio cosi’ ‘sporco’, cosi’ difficile da decodificare, cosi’ ‘noisy’.

Vedi anche qui.

Deflation of human agency

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One of the premier Art Deco buildings in London, the famed Senate House (above) keeps resonating in my mind every time I wander in it. Especially overnight.
Vastly different in concept but equally capable of impressing with its geometry, the Brutalist masterpiece by Ernő Goldfinger (Trellick Tower: below) gives a sort of shock on its first appearance.

Trellick Tower was an ideal-typical realization of Le Corbusier’s “Radiant City” concept but in “A Clockwork Orange” Kubrick placed Alex and his comrades in one of Britain’s most notorious environments, Tavy Bridge area of Thamesmead South: in the Brutalist architecture Kubrick sensed the scary vibrations coming from Skinner’s “Beyond Freedom and Dignity”. Who was right?

Norbert Wiener said that the fundamental assumption of 1930 dictatorships rested on a conceptual fallacy, i.e. the misrepresentation of the human nervous system:

[the] aspiration of the fascists for a human state based on the model of the ant results from a profound misapprehension both of the nature of the ant and of the nature of man. […]. The human individual, capable of vast learning and study, which may occupy almost half of his life, is physically equipped, as the ant is not, for this capacity. […] Those who would organize us according to permanent individual functions and permanent human restrictions condemn the human race to move at much less than half speed.
( The Human Use of Human Beings, pg.51-2)

Both the modernist utopia of the Swiss architect and the dystopia hinted at by Wiener rested on the same assumption: life is path-dependent, human being is intrinsically plastic, and can be cast in various shapes. In  “Variation of Animals and Plants” (1871),

Darwin had already found that the brains of hares and rabbits that grew up confined in boring hutches were 15 to 30 percent smaller than those of their wild counterparts. Conversely,  when animals are placed in an “enriched environment,” a large enclosure full of objects that are renewed each day and in which they can play with one another, their brains grow and develop more synapses. Children who are seriously neglected during their early development also have smaller brains […]; their intelligence and linguistic and fine motor control are permanently impaired, and they are impulsive and hyperactive.  (D. Swaab, “We are our brains”, 2014)

The question at stake in the dilemma above is really the interplay of freedom and necessity. This is the message that resonates from the Senate House and the Trellick Tower. We cannot escape this message:  we can only hope to be able to stand their presence, to endure their shadow.

The organization of society implied by those massive structures is a reminder that man is a social animal, thriving in the social organization of labor. At the same time their very conceptual and physical presence deflates human agency to the level of the ant: which corresponds in material terms to the epistemological pessimism of Dostoevsky’s “Great Inquisitor”, when he assumes that the elite has to control the common man as he is not able to discern good from evil for himself. Christ gave men liberty out of a misplaced thrust, the Inquisitor’s stance reads.

Truth is that today we cannot do without the towers and what they represent. We have to inhabit these imposing concrete islands. We have to pass through them, to rise to their level if we want to face our time. But we have to accomplish that without losing the touch with our uniqueness of human beings. We do not want to relinquish Enlightenment’s epistemological optimism, lest we end in Great Inquisitor’s trap.

There is no room to regress to previous modes of social interaction. We have to struggle for “the human use of human beings” while inhabiting the many towers of Modern times.

“It is not important what we cover, but what you discover” (Victor Weisskopf)

Otium, in ancient Roman time, was the time productively spent pursuing the agenda of your personal growth – be it the cultivation of your garden or the virtuous studium of arts & science. This concept was clearly coming from the greek “paideia”: in XIX century’s Germany it will be given a new incarnation in Wilhelm Von Humboldt’s foundation of the modern University.

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It is easily forgotten that, when one attends a research seminar, one is simply following on the footsteps of that venerable tradition.

A good metric for the rationality of a social system could in principle be based on what is the ratio of aggregate time spent by the economic agents in jobs perceived as conducive to personal growth as opposed to jobs perceived as mere toil.

The (social) target function to minimize would then involve not the time spent on the job market, but the toil itself, something Keynes was hinting at in his famous piece on the “Economic Possibilities for our Grandchildren” (1930).
Clearly a necessary condition for that is again and again that technology reaches a stage where all the toil is left to automata. The Human Use of Human Beings is just that.

“To enquire and to create”

There used to be an old concept in classical philosophy, the concept of the full and meaningful life, creative and inquisitive. German classical philosophy recovered that from the ancient Greek civilization, in particular from Ⅴ century Athens. Unfortunately, that was a hugely unequal society, where a minority of optimates could freely dispose of most of the other’s time and lives. The modern world put an end to this non-sense, with obvious repercussions (see here). Marx’s “Entfremdung” concept grew out of his acquaintance with Hegel’s reflections on classical Greece, and the perceived cultural inferiority of the modern world against the old polis. Can we recover the lost integrity of classical society in the industrial world which is obviously based on the division of labor? This is Marx starting point. Similar reflections were voiced earlier by Wilhelm von Humboldt

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one of the founders of modern higher education, which he based on a concept of humanity closely related to the ancient civilization paradigm (Paideia). The birth of Germany’s Bildung is here. The following excerpt from “Limits of State Action” (1792) is very explicit.

“To inquire and to create, these are the centers around which all human pursuits more or less directly revolve.” “But all moral cultures spring solely and immediately from the inner life of the soul and can never be produced by external and artificial contrivances. The cultivation of the understanding, as of any of man’s other faculties, is generally achieved by his own activity, his own ingenuity, or his own methods of using the discoveries of others.[…] Man never regards what he possesses as so much his own, as what he does, and the laborer who tends the garden is perhaps in a truer sense its owner than the listless voluptuary who enjoys its fruits. And since truly human action is that which flows from inner impulse, it seems as if all peasants and craftsmen might be elevated into artists, that is men who love their labor for its own sake, improve it by their own plastic genius and invented skill, and thereby cultivate their intellect, ennoble their character and exult and refine their pleasures, and so humanity would be ennobled by the very things which now, though beautiful in themselves, so often tend to be degraded. Freedom is undoubtedly the indispensable condition without which even the pursuits most congenial to individual human nature can never succeed in producing such salutary influences. Whatever does not spring from a man’s free choice, or is only the result of instruction and guidance, does not enter into his very being but remains alien to his true nature. He does not perform it with truly human energies, but merely with mechanical exactness. And if a man acts in a mechanical way, reacting to external demands or instruction, rather than in ways determined by his own interests and energies and power, we may admire what he does, but we despise what he is.” (W. von Humboldt, “Limits of State Action”)

Enter Chomsky now. His wavelength is the same.
In “Government in the Future”, Noam Chomsky outlines the core ideas of a plan which, in his own words, puts the burden of proof on the shoulders of authority.

The problem that Noam does not appear to discuss in his long talk is (of course) the Division of Labor and the specialization that it implies. It is not clear to me whether a technologically advanced society can do without.

Wiener on learning & Gödel

This is a marvellous paper by former colleague of Norbert Wiener, N. Levinson.
The bibliography of his papers is here.

“It is no coincidence that my first childish essay into philosophy,written when I was in high school and not yet eleven years old, was called The theory of ignorance. Even at that time I was struck with
the impossibility of originating a perfectly tight theory with the aid of so loose a mechanism as the human mind. And when I studied with Bertrand Russell, I could not bring myself to believe in the existence of a closed set of postulates for all logic, leaving no room for any arbitrariness in the system defined by them. Here, without the justification of their superb technique, I foresaw something of the critique of Russell which was later to be carried out by Gödel and his followers, who have given real grounds for the denial of the existence of any single closed logic following in a closed and rigid way from a body of stated rules.
“To me, logic and learning and all mental activity have always been incomprehensible as a complete and closed picture and have been understandable only as a process by which man puts himself en rapport with his environment. It is the battle for learning which is significant, and not the victory. Every victory that is absolute is followed at once by the Twilight of the gods, in which the very concept of victory is dissolved in the moment of its attainment.
“We are swimming upstream against a great torrent of disorganization, which tends to reduce everything to the heat-death of equilibrium and sameness described in the second law of thermodynamics.
What Maxwell, Boltzmann, and Gibbs meant by this heat-death in physics has a counterpart in the ethics of Kierkegaard, who pointed out that we live in a chaotic moral universe. In this, our main obligation
is to establish arbitrary enclaves of order and system. These enclaves will not remain there indefinitely by any momentum of their own after we have once established them. Like the Red Queen, we cannot stay where we are without running as fast as we can.
“We are not fighting for a definitive victory in the indefinite future.
It is the greatest possible victory to be, to continue to be, and to have been. No defeat can deprive us of the success of having existed for some moment of time in a universe that seems indifferent to us.
“This is no defeatism, it is rather a sense of tragedy in a world in which necessity is represented by an inevitable disappearance of differentiation. The declaration of our own nature and the attempt to
build up an enclave of organization in the face of nature’s overwhelming tendency to disorder is an insolence against the gods and the iron necessity that they impose. Here lies tragedy, but here lies glory too. These were the ideas I wished to synthesize in my book on cybernetics»
[“I am a mathematician”, pp. 323-325].

Engineering initial conditions (some words of thanks)

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The human civilization stumbled first upon metric geometry (the Greeks), then it invented projective geometry (Italian Renaissance) and finally conquered topology (the XIX century), as the study of continuous transformations. See the master himself, here.
The human being, in his/her development from infancy to adulthood, discovers first topology, stumbles upon projective geometry and finally conquers the metric concepts. Ontogenesis does not match phylogenesis. This question was studied first I believe by Jean Piaget: a debate on that (and other points) was hold in France in 1975 between the Cartesian Universal Grammar camp (Chomsky) and the Darwinian evolutionary epistemology camp (Piaget), see here for an index of that series of talks.

In order to develop spacial intuition in children, the MIT mathematician Seymour Papert came along.
Working under the influence of Piaget, he invented Logo (here a modern implementation client side).
Ideas in one of his books (Mindstorms Children, Computers, And Powerful Ideas) are now embedded in very interesting projects like Arduino or Raspberry Pi and more generally in all the talks about programming being part of children early education.

And now for my own history, similar to a karstic river. I had a very gifted teacher in my primary school, the best one I have ever had, who gave me incredibly advanced pointers. I was introduced to Tangram and Logo at a very early age, but somehow the lesson went lost in the grammar school I later attended, amidst Sophokles and Kant, Tacitus and Augustinus. Sort of Dead Poets Society experience.
But the spark he fired could not be forgotten, and I later recovered my passion in the form of programming and open source software, Backus-Naur grammars and kernel recompiling. Building a career as a numerical analyst and finally reaching for “PI in the Sky” (i.e. math), the true calling.

Years later, I stumbled again upon his name. He was working with Eduknoppix in the Italian school system and he had published a book on science (what else?). Then came to my mind that it would have been only a matter of time. He had engineered the initial conditions for the dynamical systems of our minds such that in time we would have converged. When I landed on Debian Linux by chance in one of the most fruitful encounters of my life that spark started firing again.

Why János Bolyai and Nikolai Ivanovich Lobachevsky landed on non-euklidean geometry at the same time (around 1830 that was), why Norbert Wiener and Andrei Kolmogorov landed on the same stochastic filtering equations at the same time (1941 it was) and so much more examples from science?
Is somehow the more stringent set of boundary conditions imposed on the human mind in the pursuit of mathematical knowledge enough to guarantee convergence if same initial conditions are given?

Thank you, Antonino

“What is the logical form of this?” on Sen, Gramsci, Sraffa and Wittgenstein

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In a perceptive book, Ray Kurzweil, chief engineer at Google, and responsible for the Hidden Markov Model that stays behind much of today speech and text recognition software (like Siri), restates  the position of the Logical Neopositivism as the following:
Amongst all possible worlds (Leibniz anyone?), in a multiverse where most realizations have faded into oblivion because they did not (randomly) stumbled upon information structures that could be algorithmically compressed, our world is such (from an anthropic cosmological principle) because it possess a recursive structure, hence information in it is susceptible of being encoded. For reason that are probably beyond investigation (recall Kant’s Ideen der Vernunft in the Dialectic of Reason, i.e. “God, the Soul, the World”: here of course, World is the idea of Reason).

To summarize:
1) Our world is such because it has a compressibility structure (based on recursion);
2) Our neocortex has a pattern structure based on recursion and
3) Our language has a recursive structure based on recursion (see this article by Chomsky).
(This is a brilliant way to restate, in three sentences, Wittgenstein.)

Then Antonio Gramsci came along, and as the Nobel prize Amartya Sen shows in this article, the ideas of the Italian marxist fed into the speculations of the great austrian philosopher. Nature versus Nurture, except there is not Nature per se, but the neocortex structure is the outcome of social processes as well as a (darwinian) adaptation to the world as it is.

It may be interesting to investigate the structure of power deep inside a concrete language.
The Grammars vs feedback control ideas in Artificial Intelligence is a restatement of the same dicotomy:
the Tractatus Logico-Philosophicus as a Cartesian meditation very close home to Noam Chomsky,
whilst the (Gramsci-inspired) Second Wittgenstein as a feedback control type of investigation.

The causality vs correlation debate really got started there. See also Gramsci Linguistic Turn

As a final aside, a set of contributions by Italian scholar LoPiparo (see this) have identified glottology (i.e. the original field of study of Gramsci) as the source of the powerful idea of hegemony.

So: hidden within language there is power.The discovery of the true nature of power (i.e. (cultural) hegemony) comes from the study of glottology. The circle closes.