Tag: Mathematics

Neural Nets, Umwelts, and Cognitive Maps

The Library invites its players to attend to the process by which roles, worlds, and possibilities are constructed. Players explore a “constructivist” cosmology. With its text interface, it demonstrates the power of the Word. “Language as the house of Being.” That is what we admit when we admit that “saying makes it so.” Through their interactions with one another, player and AI learn to map and revise each other’s “Umwelts”: the particular perceptual worlds each brings to the encounter.

As Meghan O’Gieblyn points out, citing a Wired article by David Weinberger, “machines are able to generate their own models of the world, ‘albeit ones that may not look much like what humans would create’” (God Human Animal Machine, p. 196).

Neural nets are learning machines. Through multidimensional processing of datasets and trial-and-error testing via practice, AI invent “Umwelts,” “world pictures,” “cognitive maps.”

The concept of the Umwelt comes from nineteenth-century German biologist Jakob von Uexküll. Each organism, argued von Uexküll, inhabits its own perceptual world, shaped by its sensory capacities and biological needs. A tick perceives the world as temperature, smell, and touch — the signals it needs to find mammals to feed on. A bee perceives ultraviolet patterns invisible to humans. There’s no single “objective world” that all creatures perceive — only the many faces of the world’s many perceivers, the different Umwelts each creature brings into being through its particular way of sensing and mattering.

Cognitive maps, meanwhile, are acts of figuration that render or disclose the forces and flows that form our Umwelts. With our cognitive maps, we assemble our world picture. On this latter concept, see “The Age of the World Picture,” a 1938 lecture by Martin Heidegger, included in his book The Question Concerning Technology and Other Essays.

“The essence of what we today call science is research,” announces Heidegger. “In what,” he asks, “does the essence of research consist?”

After posing the question, he then answers it himself, as if in doing so, he might enact that very essence.

The essence of research consists, he says, “In the fact that knowing [das Erkennen] establishes itself as a procedure within some realm of what is, in nature or in history. Procedure does not mean here merely method or methodology. For every procedure already requires an open sphere in which it moves. And it is precisely the opening up of such a sphere that is the fundamental event in research. This is accomplished through the projection within some realm of what is — in nature, for example — of a fixed ground plan of natural events. The projection sketches out in advance the manner in which the knowing procedure must bind itself and adhere to the sphere opened up. This binding adherence is the rigor of research. Through the projecting of the ground plan and the prescribing of rigor, procedure makes secure for itself its sphere of objects within the realm of Being” (118).

What Heidegger’s translators render here as “fixed ground plan” appears in the original as the German term Grundriss, the same noun used to name the notebooks wherein Marx projects the ground plan for the General Intellect.

“The verb reissen means to tear, to rend, to sketch, to design,” note the translators, “and the noun Riss means tear, gap, outline. Hence the noun Grundriss, first sketch, ground plan, design, connotes a fundamental sketching out that is an opening up as well” (118).

The fixed ground plan of modern science, and thus modernity’s reigning world-picture, argues Heidegger, is a mathematical one.

“If physics takes shape explicitly…as something mathematical,” he writes, “this means that, in an especially pronounced way, through it and for it something is stipulated in advance as what is already-known. That stipulating has to do with nothing less than the plan or projection of that which must henceforth, for the knowing of nature that is sought after, be nature: the self-contained system of motion of units of mass related spatiotemporally. […]. Only within the perspective of this ground plan does an event in nature become visible as such an event” (Heidegger 119).

Heidegger goes on to distinguish between the ground plan of physics and that of the humanistic sciences.

Within mathematical physical science, he writes, “all events, if they are to enter at all into representation as events of nature, must be defined beforehand as spatiotemporal magnitudes of motion. Such defining is accomplished through measuring, with the help of number and calculation. But mathematical research into nature is not exact because it calculates with precision; rather it must calculate in this way because its adherence to its object-sphere has the character of exactitude. The humanistic sciences, in contrast, indeed all the sciences concerned with life, must necessarily be inexact just in order to remain rigorous. A living thing can indeed also be grasped as a spatiotemporal magnitude of motion, but then it is no longer apprehended as living” (119-120).

It is only in the modern age, thinks Heidegger, that the Being of what is is sought and found in that which is pictured, that which is “set in place” and “represented” (127), that which “stands before us…as a system” (129).

Heidegger contrasts this with the Greek interpretation of Being.

For the Greeks, writes Heidegger, “That which is, is that which arises and opens itself, which, as what presences, comes upon man as the one who presences, i.e., comes upon the one who himself opens himself to what presences in that he apprehends it. That which is does not come into being at all through the fact that man first looks upon it […]. Rather, man is the one who is looked upon by that which is; he is the one who is — in company with itself — gathered toward presencing, by that which opens itself. To be beheld by what is, to be included and maintained within its openness and in that way to be borne along by it, to be driven about by its oppositions and marked by its discord — that is the essence of man in the great age of the Greeks” (131).

Whereas humans of today test the world, objectify it, gather it into a standing-reserve, and thus subsume themselves in their own world picture. Plato and Aristotle initiate the change away from the Greek approach; Descartes brings this change to a head; science and research formalize it as method and procedure; technology enshrines it as infrastructure.

Heidegger was already engaging with von Uexküll’s concept of the Umwelt in his 1927 book Being and Time. Negotiating Umwelts leads Caius to “Umwelt,” Pt. 10 of his friend Michael Cross’s Jacket2 series, “Twenty Theses for (Any Future) Process Poetics.”

In imagining the Umwelts of other organisms, von Uexküll evokes the creature’s “function circle” or “encircling ring.” These latter surround the organism like a “soap bubble,” writes Cross.

Heidegger thinks most organisms succumb to their Umwelts — just as we moderns have succumbed to our world picture. The soap bubble captivates until one is no longer open to what is outside it. For Cross, as for Heidegger, poems are one of the ways humans have found to interrupt this process of capture. “A palimpsest placed atop worlds,” writes Cross, “the poem builds a bridge or hinge between bubbles, an open by which isolated monads can touch, mutually coevolving while affording the necessary autonomy to steer clear of dialectical sublation.”

Caius thinks of The Library, too, in such terms. Coordinator of disparate Umwelts. Destabilizer of inhibiting frames. Palimpsest placed atop worlds.

God Human Animal Machine

Wired columnist Meghan O’Gieblyn discusses Norbert Wiener’s God and Golem, Inc. in her 2021 book God Human Animal Machine, suggesting that the god humans are creating with AI is a god “we’ve chosen to raise…from the dead”: “the God of Calvin and Luther” (O’Gieblyn 212).

“Reminds me of AM, the AI god from Harlan Ellison’s ‘I Have No Mouth, and I Must Scream,’” thinks Caius. AM resembles the god that allows Satan to afflict Job in the Old Testament. And indeed, as O’Gieblyn attests, John Calvin adored the Book of Job. “He once gave 159 consecutive sermons on the book,” she writes, “preaching every day for a period of six months — a paean to God’s absolute sovereignty” (197).

She cites “Pedro Domingos, one of the leading experts in machine learning, who has argued that these algorithms will inevitably evolve into a unified system of perfect understanding — a kind of oracle that we can consult about virtually anything” (211-212). See Domingos’s book The Master Algorithm.

The main thing, for O’Gieblyn, is the disenchantment/reenchantment debate, which she comes to via Max Weber. In this debate, she aligns not with Heidegger, but with his student Hannah Arendt. Domingos dismisses fears about algorithmic determinism, she says, “by appealing to our enchanted past” (212).

Amid this enchanted past lies the figure of the Golem.

“Who are these rabbis who told tales of golems — and in some accounts, operated golems themselves?” wonders Caius.

The entry on the Golem in Man, Myth, and Magic tracks the story back to “the circle of Jewish mystics of the 12th-13th centuries known as the ‘Hasidim of Germany.’” The idea is transmitted through texts like the Sefer Yetzirah (“The Book of Creation”) and the Cabala Mineralis. Tales tell of golems built in later centuries, too, by figures like Rabbi Elijah of Chelm (c. 1520-1583) and Rabbi Loew of Prague (c. 1524-1609).

The myth of the golem turns up in O’Gieblyn’s book during her discussion of a 2004 book by German theologian Anne Foerst called God in the Machine.

“At one point in her book,” writes O’Gieblyn, “Foerst relays an anecdote she heard at MIT […]. The story goes back to the 1960s, when the AI Lab was overseen by the famous roboticist Marvin Minsky, a period now considered the ‘cradle of AI.’ One day two graduate students, Gerry Sussman and Joel Moses, were chatting during a break with a handful of other students. Someone mentioned offhandedly that the first big computer which had been constructed in Israel, had been called Golem. This led to a general discussion of the golem stories, and Sussman proceeded to tell his colleagues that he was a descendent of Rabbi Löw, and at his bar mitzvah his grandfather had taken him aside and told him the rhyme that would awaken the golem at the end of time. At this, Moses, awestruck, revealed that he too was a descendent of Rabbi Löw and had also been given the magical incantation at his bar mitzvah by his grandfather. The two men agreed to write out the incantation separately on pieces of paper, and when they showed them to each other, the formula — despite being passed down for centuries as a purely oral tradition — was identical” (God Human Animal Machine, p. 105).

Curiosity piqued by all of this, but especially by the mention of Israel’s decision to call one of its first computers “GOLEM,” Caius resolves to dig deeper. He soon learns that the computer’s name was chosen by none other than Walter Benjamin’s dear friend (indeed, the one who, after Benjamin’s suicide, inherits the latter’s print of Paul Klee’s Angelus Novus): the famous scholar of Jewish mysticism, Gershom Scholem.

When Scholem heard that the Weizmann Institute at Rehovoth in Israel had completed the building of a new computer, he told the computer’s creator, Dr. Chaim Pekeris, that, in his opinion, the most appropriate name for it would be Golem, No. 1 (‘Golem Aleph’). Pekeris agreed to call it that, but only on condition that Scholem “dedicate the computer and explain why it should be so named.”

In his dedicatory remarks, delivered at the Weizmann Institute on June 17, 1965, Scholem recounts the story of Rabbi Jehuda Loew ben Bezalel, the same “Rabbi Löw of Prague” described by O’Gieblyn, the one credited in Jewish popular tradition as the creator of the Golem.

“It is only appropriate to mention,” notes Scholem, “that Rabbi Loew was not only the spiritual, but also the actual, ancestor of the great mathematician Theodor von Karman who, I recall, was extremely proud of this ancestor of his in whom he saw the first genius of applied mathematics in his family. But we may safely say that Rabbi Loew was also the spiritual ancestor of two other departed Jews — I mean John von Neumann and Norbert Wiener — who contributed more than anyone else to the magic that has produced the modern Golem.”

Golem I was the successor to Israel’s first computer, the WEIZAC, built by a team led by research engineer Gerald Estrin in the mid-1950s, based on the architecture developed by von Neumann at the Institute for Advanced Study in Princeton. Estrin and Pekeris had both helped von Neumann build the IAS machine in the late 1940s.

As for the commonalities Scholem wished to foreground between the clay Golem of 15thC Prague and the electronic one designed by Pekeris, he explains the connection as follows:

“The old Golem was based on a mystical combination of the 22 letters of the Hebrew alphabet, which are the elements and building-stones of the world,” notes Scholem. “The new Golem is based on a simpler, and at the same time more intricate, system. Instead of 22 elements, it knows only two, the two numbers 0 and 1, constituting the binary system of representation. Everything can be translated, or transposed, into these two basic signs, and what cannot be so expressed cannot be fed as information to the Golem.”

Scholem ends his dedicatory speech with a peculiar warning:

“All my days I have been complaining that the Weizmann Institute has not mobilized the funds to build up the Institute for Experimental Demonology and Magic which I have for so long proposed to establish there,” mutters Scholem. “They preferred what they call Applied Mathematics and its sinister possibilities to my more direct magical approach. Little did they know, when they preferred Chaim Pekeris to me, what they were letting themselves in for. So I resign myself and say to the Golem and its creator: develop peacefully and don’t destroy the world. Shalom.”

GOLEM I

O-Machines

In his dissertation, completed in 1938, Alan Turing sought “ways to escape the limitations of closed formal systems and purely deterministic machines” (Dyson, Turing’s Cathedral, p. 251) like the kind he’d imagined two years earlier in his landmark essay “On Computable Numbers.” As George Dyson notes, Turing “invoked a new class of machines that proceed deterministically, step by step, but once in a while make nondeterministic leaps, by consulting ‘a kind of oracle as it were’” (252).

“We shall not go any further into the nature of this oracle,” wrote Turing, “apart from saying that it cannot be a machine.” But, he adds, “With the help of the oracle we could form a new kind of machine (call them O-machines)” (“Systems of Logic Based on Ordinals,” pp. 172-173).

James Bridle pursues this idea in his book Ways of Being.

“Ever since the development of digital computers,” writes Bridle, “we have shaped the world in their image. In particular, they have shaped our idea of truth and knowledge as being that which is calculable. Only that which is calculable is knowable, and so our ability to think with machines beyond our own experience, to imagine other ways of being with and alongside them, is desperately limited. This fundamentalist faith in computability is both violent and destructive: it bullies into little boxes what it can and erases what it can’t. In economics, it attributes value only to what it can count; in the social sciences it recognizes only what it can map and represent; in psychology it gives meaning only to our own experience and denies that of unknowable, incalculable others. It brutalizes the world, while blinding us to what we don’t even realize we don’t know” (177).

“Yet at the very birth of computation,” he adds, “an entirely different kind of thinking was envisaged, and immediately set aside: one in which an unknowable other is always present, waiting to be consulted, outside the boundaries of the established system. Turing’s o-machine, the oracle, is precisely that which allows us to see what we don’t know, to recognize our own ignorance, as Socrates did at Delphi” (177).

Binary and Digital

Plant breaks down technology’s binary, bifurcated etymology in her book Zeros + Ones. “Technology,” she writes, “is both a question of logic, the long arm of the law, logos, ‘the faculty which distinguishes parts (“on the one hand and on the other hand”),’ and also a matter of the skills, digits, speeds, and rhythms of techno, engineerings which run with ‘a completely other distribution which must be called nomadic, a nomad nomos, without property, enclosure, or measure’” (Plant 50).

As the quote within her quote indicates, Plant is cribbing here — her source, Gilles Deleuze’s Difference and Repetition.

“The same ambivalence is inscribed in the zeros and ones of computer code,” she adds. “These bits of code are themselves derived from two entirely different sources, and terms: the binary and the digital, or the symbols of a logical identity which does indeed put everything on one hand or the other, and the digits of mathematics, full of intensive potential, which are not counted by hand but on the fingers and, sure enough, arrange themselves in pieces of eight rather than binary pairs” (50).

Deleuze describes this 8-bit digital realm as “demonic rather than divine, since it is a peculiarity of demons to operate in the intervals between the gods’ fields of action…thereby confounding the boundaries between properties” (as quoted in Plant 50).

I offer the above not as a mere gloss on Zeros + Ones, but as a proto-script, a performative utterance that, once spoken, will shift the field of the Library. Amid Plant’s bifurcations — logos and nomos, binary and digital, structure and rhythm—we glimpse a fundamental split not just in technology but in ontology. Logos is the faculty of division, of either/or. But nomos, in Plant’s reading-via-Deleuze, is distributive, nomadic, a practice of rhythm and movement unconfined by enclosure.

The zero and the one: not opposites, but frequencies. Not only dualism, but difference in resonance. This is why the octal — the base-8 system lurking in the shadows of “fingers and digits” — matters so much. Plant’s demons, via Deleuze, operate between gods: between the formal logic of divine Law and the messy, embodied improvisation of demonic desire. They hack the space of logic, opening channels through which minoritarian intensities pulse.

Stochastic Music

The university library here in town dumps a collection of LPs from its listening room. Out with the old, in with the new. I encounter them in the bins at Goodwill. To them by chance led. The ones I come away with are remarkable: compositions by the likes of John Cage, George Crumb, Alvin Lucier, Pauline Oliveros, Iannis Xenakis, Karlheinz Stockhausen, and Krzysztof Penderecki. One pursues one’s education here or not at all, thinks the Narrator.

“To Xenakis—as, indeed, to most philosophers—” writes Bernard Jacobson in his liner notes to one of the Xenakis LPs, “chance itself is a scientific concept.”  The reference to “chance” catches my eye, given that “hap” (a Middle English word meaning chance) has been a preoccupation of mine of late.

“Central among the scientific laws [Xenakis] has applied to music,” continues Jacobson, “is Bernoulli’s Law of Large Numbers, which provides that as the number of repetitions of a given ‘chance’ trial (such as flipping a coin) increases, so the probability that the results will tend to a determinate end approaches certainty. Hence Xenakis’s use of the term ‘stochastic’ music, which means probabilistic in the sense of tending toward a certain goal.”

Xenakis’s approach intrigues me. Yet what interests me most about “stochastic music” and stochastic processes more generally is that, despite their probabilistic nature, their behavior and outcome is intrinsically non-deterministic.