a context of Gromov's program

Perhaps as gift for curious websurfers, Micha Gromov, geometer (more) at the IHES, posted shortly before last christmas parts of a fascinating essay discussing "real" and "artificial" intelligence, pattern formation and detection, learning, language, of course mathematics and lot's of other interesting topics on his website. Theme of his essay is a sketch of a kind of program for exploring a possible general background structure behind the invention of and dealing with "ideas" and concepts. As the idea of a special cognitive background structure as root of mathematical thinking can be traced back at least to the ancient Platonists, Gromov's essay motivates to take a small tour through it's history for looking if Gromov's thoughts fit to that. Many thanks to Matilde Marcolli and Yuri I. Manin, who kindly allow to include pieces of their art for illustrating some issues, of course this does not imply any sharing of the thoughts expressed here.

The mindset called "Platonism" received more attention recently with Yuri I. Manin's interview in the AMS Notices and his remarks in a following book review on the history of mathematics. Among mathematicians, "Platonism" is about meaningfull mathematical concepts and structures as being "preexistent" and "found", in contrast of being "made" or "invented" - hidden structures as being "revealed" in the course of research instead of being "constructed". Yuri Manin expresses the resulting mental image of mathematics with a beautifull picture:

"... a great castle and you gradually start seeing it's contours through the deep mist, and begin to investigate something. How you formulate what it is you've seen depends on your type of thinking and the scale of what you have seen. And so you begin to blow away the mists, to find appropriate telescopes, seek analogies with edifices that have been discovered before, create a language for the things you see so vaguely ..."

That may sound rather detached from immediate "reality" (whatever that is) and about something like a dodecahedral cloud floating in the air. However, the idea of a screwdriver illustrates the meaning of a "Platonic idea" probably much better. Even a superficial browsing of contemporary mathematics shows that "platonic" programs and concepts are among the leading forces for actual research. E.g. there was hardly ever a more "platonic" insight than Grothendieck's quest for the "mysterious functor" relating different ways to do geometry (="cohomology theories") in number theory, which became one of the most usefull tools in the applied business of cryptography. Many more examples could be given, like the quest of the "field with one element" in the context of treating geometry and number theory in a unified way, or that for the correct definition of "higher categories" as expressing what "spaces" really are. Ca. 20% of Manin/Pachishkin's Encyclopedia volume on number theory present "Analogies and Visions", Kato's beautifull survey on his groundbreaking work is framed in poetical expressions of a platonic mentality: “Mysterious properties of zeta values seem to tell us (in a not so loud voice) that our universe has the same properties: The universe is not explained just by real numbers. It has p-adic properties … We o u r s e l v e s may have the same properties.”

Despite being named after Plato, the "platonic mindset" goes back to far older ancient greek philosophers and there had never been a commonly accepted set of welldefined doctrines for it. Plato maps a road towards it in his dialogues, but carefully makes clear that his real teaching was an esoteric one and never explicitely written down. Perhaps he thought that to be unnecessary because it pervaded all ancient greek philosophy before him and consited more of a mentality instead of specific beliefs. Mentalities can be learned by practice, not by just reading dialogues. And this was his reason for advertizing the yoga of mathematics. Plato distinguished between three levels of thinking: sensation, reason and higher intuition. These levels of thinking correspond different layers of external reality and of internal psychology. The special feature of "ideas" as the reality counterparts of higher intuition was their identity with higher intuition's counterparts in psychology: "ideas" as common foundation of both external reality and internal psychology. They provided the connection between individual thinking and reality which made knowledge possible and consequently, the best knowledge is that which connects most directly with them. For the same reason, "ideas" were considered "more real" than everything else - in contrast to the current view of "ideas" as reduced versions of reality, they were perceived as enrichments. An analogy: Classical greek sculptures were about visualizing ‘abstract’ concepts; 19th century scholars concluded that this forces them to have been colourfree because ‘abstract concept’=’reduced’, ancient greeks made them probably for the same reason very colourfull because for them ‘abstract concept’=’enriched’. This way, "Platonism" solved the epistemological problem and avoided ontological troubles. But it run into other troubles: if higher intuition can't be reduced to rational thinking characterized by verbal reasoning, intuition becomes increasingly incommunicable the more interesting it becomes. Similar, if true knowledge only comes from turning towards the common basis of outside and internal reality, but language is based on distinguishing these and everything else, language itself becomes unfit. Worse - even having a localized, personal ego is incompatible with knowledge. Actually, some antique philosophers teached that knowledge is impossible, if possible then unexpressable, if expressable then incomprehensible. Others turned to expressing in paradoxa, irony and myths, what apparently became a trademark for deep thinking like some millenia later twisted language for german thinkers. An other resulting invariant of greek philosophy is it's insistance that real learning is accompanied by a separation of the personal ego from the egofree part of the mind which only connects to "ideas". Walter Benjamin describes that in his "Ursprung des deutschen Trauerspiels" with "Die Wahrheit ist der Tod der Intention" ("Truth is the death of intention"). He specifies the special relation of language to truth with: "Das ihr gemäße Verhalten ist nicht ein Meinen im Erkennen, sondern ein in sie Eingehen und Verschwinden" ("The language's appropriate behavior is not building opinions, but an immersion into truth and a vanishing in it"). The loss of this connection and the subsequent arbitraryness of normal language in naming and distinguishing things was for ancient philosophers like Parmenides the basic error of the human mind whose stabilization and ramifications created the world of discourse, doubt and pseudoknowledge we live in now. Acc. to Benjamin this original mistake shows up in the faint air of melancholy pervading the images of nature since then. It is the poet's and the philosopher's task to restore the lost connection for brief moments ...

... or to create:
"to name something was to give birth to a new entity .... Humans could exercise Free Will and put in perspective mathematics and philosophy", describe Graham and Kantor in their study on mathematics and religiousity in set theory research in Russia a century ago. Pavel Florensky, 'the Russian da Vinci', argued: "it is the word alone that makes the cognitive process possible, that makes objective what was still subjective, sums up our inward longing for reality and places before us the cognitive urge as a goal and value."

Yuri Manin wrote a very beautifull poem presenting this role of "arbitrary naming" in creating frames of meanings for further creative work. It has been stimulated by real ships he observed and produces instantly a strong image and the complexity of the relation naming/knowing comes without effort precisely when the barges' poetic associations set in. There is probably nothing more arbitrary than the names of the ships - but, well ... just read and notice what happens:

Barges on the Rhine


... I’ve read half–through the catalogue of ships

Ossip Mandelshtam

ANGELUS DEI, with coal humps on his back,
strides along as a forlorn camel that lost sight of his needle’s eye.

IMMACULATA shyly cuddles to the left bank
trying to cover her face with a veil of feeble smoke.

God cannot be comprehended, but can be named.
The Name of God is God himself,
so the Name Worshippers believed. The venerable Illarion,Pavel Florensky.
And infinity cannot be comprehended, but can be named(Georg Cantor’s Alephs).
Who can comprehend barges? Who names them?
He - or She - is beyond the comprehension of mortals as well ...

CURA DEI’s Diesel engine snorts scornfully at INNUENDO.

who fancies himself a kamikaze, the divine wind,
that will smite the dishonorable ones,
heaving in lacerated flames
Hiroshima, Bikini atoll, the World Trade Center, Königswinter.

TOLERANTIE only faintly pants.

DISSIDENTIA ran aground last night,
her clawed belly rumbled on wet stones,
a hundred meters from my balcony.
She will sail again – but whither do we sail, pray, tell me, ZEMBLA!

PANTA RHEI ... everything flows ... πάντα ῥεῖ ...

Translated from the Russian original by Yuri Manin

Angelus Dei (lat.): God's Angel
Immaculata (lat.): the Immaculate, Vrigin Mary
Cura Dei (lat.): God's care
Innuendo (lat.): insulting allusion
Sayonara (jap.): Fare Well!
Mon Désir (fr.): my desire
Banzai (jap.): battle call of japanese warriers
Kamikaze (jap.): "divine wind", military suicide aviators in World War II
Tolerantie (dutch): the tolerance
Dissidentie (lat.): disaccord
Zembla: Imaginary country in Vladimir Nabokov’s novel “Pale Fire”.

A friend just called attention to Eco's recent book "The vertigo of lists", dealing with the many roles that lists of names played in literature and art, from the catalog of ships in Homer to the list of Don Giovanni, touching on a lot of different aspects of abstract or disembodied "naming" too.

A surprising reference to platonic truth is in Werner Herzog's cineastic aesthetics, as the try to catch with the artist's tools the "extatic truth" behind the superficial "truth of the accountants". The method he teaches in his film school is "not for the faint hearted... For those who have a fire burning within." Herzog suggests: "Follow your vision. Form secretive Cells everywhere. At the same time, be not afraid of solitude." His "Bells from the deep" about russian mysticism gives a nice idea, incl. the unintended satire unavoidably attached to such docu.

At least since the mid 19th century, scholars traced these themes back to an even older tradition of shamanism. E.g. Nietzsche's Friend Rohde’s great work “Psyche. Seelencult und Unsterblichkeitsglaube der Griechen” and the small books on the greeks and Nietzsche by G. Colli. Dodds “The Greeks and the Irrational” contains a very interesting chapter on that. Before philosophers, "higher insights" showed in temple slaves, whose blood attracted spirits who then could be eavesdropped by the priests, and the priests then interpreted what they heard with “logos”. In philosophers, both roles became united in one person who performed extatic metempsychoses followed by rational exegesis. The books of Parmenides and Empedocles still start with tales of voyages of their souls. Here is a link conc. the reconstruction of Empedocles’ ideas acc. to Kranz (after he continued the Diels, Kranz edition of Presocratic fragments). Like the extases caused by visiting spirits in shamans, the early philosopher's job was not a voluntarily chosen one, but a vocation. As the separation of the personal ego from the egofree part of the mind connected with "platonism", this needs not to be an enjoyable experience. Yuri Manin tells here his fight with his first math book.

The very forcefull triptych (1 ,2 ,3 ) by Matilde Marcolli, parts of which are reproduced here, exposes this as agonizing condition.You see the painfull complexity there from belonging to more that one world and not to reduce to one's projections on both. The style of renaissance, when science emerged, with moments of surrealism is no coincidence, as is the theme "liberty".

Among the psychological attitudes related to shamanism are "staying vigil", "attentive listening" to the unpredictable whispers of the spirits, keeping the voice of the ego low, "looking for hints". Thomas McEvilley wrote a fascinating book (online scan, youtube interview, thanks to J.H.) about possible connections with ancient indian philosophy, it’s summaries of single theories are very excellent for themself: Sociologically, shamanism was rooted in tribal culture and small communities. Shamanism were one-man operations, restricted to a small clientele. Shamans were "fermions", independent and isolated from each other. When with emerging urban structures in Mesopotamia, temple bureaucracies and career priests in university-style hierarchical organisations (e.g. the Marduk temple employed ca. 7,000 people) stabilized the state, shamanistic practices became tamed to a homeopathic dose of “intrinsic motivation/curiosity in the things for themself”, predictable and finally substituted by an entirely different mindset and communication style. Imagine how career-priests probably felt towards those wandering orphics, fakirs, sorcerers etc.? Bureaucrats are "bosons". Of course, the shaman's relation to language was complex too and the possibility that instead of expressing truth, they misuse their skills to deceive was theme since ages. Yuri Manin analyses here the "trickster"-theme in ancient myths and the possible anthropological background of shaman/tricksters. The platonic concept of knowledge as becoming that what is to be known by an act of immersion and identification comes from shamanistic practices. Shamanistic traditions continue to be a vital part of everyday culture in East Asia, e.g. Siberia, Japan and Korea. In Europe, beyond antiquity and perhaps transmitted through neoplatonic communities in south france, this stayed alife among medieval scholars (beautifully told in M. Yourcenar's "L'Œuvre au noir") and turned into the selfconcept of early scientists. Giordano Bruno used in his essay (La Bibioteca Ideale di G. Bruno) on "heroic frenzies" the myth of Actaeon and Artemis as metapher for the philosopher's hunt for truth: During a hunt, Actaeon meets Artemis, the godess of hunting, and is transformed by her into the stag he runs after with his dogs, who then dismember him. Bruno's essay is highly recommended as selfdescription of the mentality of the early science community in the most crucial piece of it's formation.

L. Graham and J.-M. Kantor describe in "Naming Infinity. A true history of religious mysticism and Mathematical creativity" a very interesting re-emergence of similar traits in the Russian school of set theory in the first third of the XXth century. (google scan, an article by Moore, a review, Bourbaki's history book) From Freeman Dyson's review: "... the puzzling cultural dynamics that converted religious mysticism into mathematical insight. The authors particularly probe the surprising way that a religious heresy (Name Worshipping) emboldened the Russian mathematicians who finally surmounted the theoretical difficulties that had overwhelmed earlier pioneers in set theory." Here are links on Illarion the monk and Pavel Florensky.

In antiquity, the "higher intuition" sketched above was taken for granted and philosophy was largely about it's practical training and stabilization. Modern discussions of "Platonism" focus instead on the ontological status of abstract concepts and their relation with linguistics. Gromov's takes up again the antique route by starting from the existence of an anthropological invariant producing those "higher intuitions", provides plausible reasons aside intuitive evidence, but leaves ontological questions aside. Like the antique platonists. After Gromov singles out the specific mentality, he suggests the existence of a formal backgroundstructure behind it, gives that a name and sketches a program for dealing with it. This is the "modern part" of his program.