Related reading: "Freud and the Future," by Thomas Mann (1936) —
"… these representations are winged with the strongest
and most sweeping powers of suggestion. But not only does
the dream psychology which Schopenhauer calls to his aid
bear an explicitly psychoanalytic character, even to the
presence of the sexual argument and paradigm …."
The "Change Arises" phrase in the previous post suggests a sort of
architectural plan and elevation for the structure of the I Ching —
A search for background on the previous post's Eliot Weinberger yields,
from Berlin . . .
"In 2000 he was the first US writer to be honoured with
the Order of the Aztec Eagle from the Mexican government."
— "2000 zeichnete ihn die mexikanische Regierung als ersten
US-amerikanischen Autor mit dem Order of the Aztec Eagle aus."
The Aztec Eagle with a serpent in its beak, landing on a prickly pear,
is pictured on the flag of Mexico.
See also Weinberger's work at Prickly Paradigm Press.
Related material: Other Log24 posts tagged Prickly.
Illustration, from a search in this journal for “Symplectic” —
.
Some background: Rift-design in this journal and …
"Plato thought nature but a spume that plays
Upon a ghostly paradigm of things"
— W. B. Yeats, "Among School Children"
This post’s title is from the tags of the previous post —
The title’s “shift” is in the combined concepts of …
Space and Number
From Finite Jest (May 27, 2012):
The books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.
For some details of the shift, see a Log24 search for Boole vs. Galois.
From a post found in that search —
“Benedict Cumberbatch Says
a Journey From Fact to Faith
Is at the Heart of Doctor Strange“
— io9 , July 29, 2016
” ‘This man comes from a binary universe
where it’s all about logic,’ the actor told us
at San Diego Comic-Con . . . .
‘And there’s a lot of humor in the collision
between Easter [ sic ] mysticism and
Western scientific, sort of logical binary.’ “
[Typo now corrected, except in a comment.]
Illustrations from a post of Feb. 17, 2011:
Plato’s paradigm in the Meno —
Changed paradigm in the diamond theorem (2×2 case) —
* For related remarks, see posts of May 26-28, 2012.
This post was suggested by Paradigms Lost
(a post cited here a year ago today),
by David Weinberger's recent essay "Shift Happens,"
and by today's opening of "The Raven."
David Weinberger in The Chronicle of Higher Education , April 22—
"… Kuhn was trying to understand how Aristotle could be such a brilliant natural scientist except when it came to understanding motion. Aristotle's idea that stones fall and fire rises because they're trying to get to their natural places seems like a simpleton's animism.
Then it became clear to Kuhn all at once. Ever since Newton, we in the West have thought movement changes an object's position in neutral space but does not change the object itself. For Aristotle, a change in position was a change in a quality of the object, and qualitative change tended toward an asymmetric actualization of potential: an acorn becomes an oak, but an oak never becomes an acorn. Motion likewise expressed a tendency for things to actualize their essence by moving to their proper place. With that, 'another initially strange part of Aristotelian doctrine begins to fall into place,' Kuhn wrote in The Road Since Structure ."
Dr. John Raven (of Raven's Progressive Matrices)—
"… these tools cannot be immediately applied within our current workplaces, educational systems, and public management systems because the operation of these systems is determined, not by personal developmental or societal needs, but by a range of latent, rarely discussed, and hard to influence sociological forces.
But this is not a cry of despair: It points to another topic which has been widely neglected by psychologists: It tells us that human behaviour is not mainly determined by internal properties— such as talents, attitudes, and values— but by external social forces. Such a transformation in psychological thinking and theorising is as great as the transformation Newton introduced into physics by noting that the movement of inanimate objects is not determined by internal, 'animistic,' properties of the objects but by invisible external forces which act upon them— invisible forces that can nevertheless be mapped, measured, and harnessed to do useful work for humankind.
So this brings us to our fourth conceptualisation and measurement topic: How are these social forces to be conceptualised, mapped, measured, and harnessed in a manner analogous to the way in which Newton made it possible to harness the destructive forces of the wind and the waves to enable sailing boats to get to their destinations?"
Before Newton, boats never arrived?
Continued from March 10, 2011 — A post that says
"If Galois geometry is thought of as a paradigm shift
from Euclidean geometry, both… the Kuhn cover
and the nine-point affine plane may be viewed…
as illustrating the shift."
Yesterday's posts The Fano Entity and Theology for Antichristmas,
together with this morning's New York Times obituaries (below)—
—suggest a Sunday School review from last year's
Devil's Night (October 30-31, 2010)—
|
Sunday, October 31, 2010 ART WARS – m759 @ 2:00 AM … There is a Cave – Paradise Lost , by John Milton
|
See also Ash Wednesday Surprise and Geometry for Jews.
(Continued from February 19)
The cover of the April 1, 1970 second edition of The Structure of Scientific Revolutions , by Thomas S. Kuhn—
This journal on January 19, 2011—
If Galois geometry is thought of as a paradigm shift from Euclidean geometry,
both images above— the Kuhn cover and the nine-point affine plane—
may be viewed, taken together, as illustrating the shift. The nine subcubes
of the Euclidean 3x3x3 cube on the Kuhn cover do not form an affine plane
in the coordinate system of the Galois cube in the second image, but they
at least suggest such a plane. Similarly, transformations of a
non-mathematical object, the 1974 Rubik cube, are not Galois transformations,
but they at least suggest such transformations.
See also today's online Harvard Crimson illustration of problems of translation—
not unrelated to the problems of commensurability discussed by Kuhn.
Harvard Science Review (Winter 1997) on Thomas Kuhn's
The Structure of Scientific Revolutions —
"…his language often portrays paradigms as cults
and the battle between paradigms as quasi-religious wars."
Related material: This journal's "Paradigms" on February 17th
and the following notes—
"These passages suggest that the Form is a character or set of characters
common to a number of things, i.e. the feature in reality which corresponds
to a general word. But Plato also uses language which suggests not only
that the forms exist separately (χωριστά ) from all the particulars, but also
that each form is a peculiarly accurate or good particular of its own kind,
i.e. the standard particular of the kind in question or the model (παράδειγμα )
[i.e. paradigm ] to which other particulars approximate….
… Both in the Republic and in the Sophist there is a strong suggestion
that correct thinking is following out the connexions between Forms.
The model is mathematical thinking, e.g. the proof given in the Meno
that the square on the diagonal is double the original square in area."
— William and Martha Kneale, The Development of Logic,
Oxford University Press paperback, 1985
Plato's paradigm in the Meno —
Changed paradigm in the diamond theorem (2×2 case) —
Aspects of the paradigm change* —
Monochrome figures to
colored figures
Areas to
transformations
Continuous transformations to
non-continuous transformations
Euclidean geometry to
finite geometry
Euclidean quantities to
finite fields
Some pedagogues may find handling all of these
conceptual changes simultaneously somewhat difficult.
* "Paradigm shift " is a phrase that, as John Baez has rightly pointed out,
should be used with caution. The related phrase here was suggested by Plato's
term παράδειγμα above, along with the commentators' specific reference to
the Meno figure that serves as a model. (For "model" in a different sense,
see Burkard Polster.) But note that Baez's own beloved category theory
has been called a paradigm shift.
Update from 10 minutes later:
Summary of the 1966 Landin paper in a Google AI Overview —
|
In his 1966 paper "The Next 700 Programming Languages," Peter Landin explored the potential for a large and diverse family of programming languages, arguing for a principled approach to language design focusing on well-defined frameworks and a "well-mapped space" of possible languages. He introduced ISWIM (If You See What I Mean), an abstract language that served as a foundational concept for functional programming. [1, 2]
Here's a more detailed explanation:
In essence, Landin's "Next 700 Programming Languages" paper was a seminal work that envisioned a future where programming languages would be designed more systematically and in a more principled manner, paving the way for the development of various functional and dataflow programming paradigms. [1, 2, 6]
AI responses may include mistakes.
|
Two links from the homepage of the Crary of the cmu.edu link above —
A Man For All Seasons
The Gods of the Copybook Headings .
Other material related to the name Crary —
A more specific account of
Outside/Inside box-thinking . . .
|
A connection discovered on April 1, 2013 — (Click to enlarge the image below.)
|
A screenshot today of a May 31 NY Times review of a book on hacking —
|
You, Xi-lin; Zhang, Peter. "Interality in Heidegger."
The term "interology" is meant as an interventional alternative to traditional Western ontology. The idea is to help shift people's attention and preoccupation from subjects, objects, and entities to the interzones, intervals, voids, constitutive grounds, relational fields, interpellative assemblages, rhizomes, and nothingness that lie between, outside, or beyond the so-called subjects, objects, and entities; from being to nothing, interbeing, and becoming; from self-identicalness to relationality, chance encounters, and new possibilities of life; from "to be" to "and … and … and …" (to borrow Deleuze's language); from the actual to the virtual; and so on. As such, the term wills nothing short of a paradigm shift. Unlike other "logoi," which have their "objects of study," interology studies interality, which is a non-object, a no-thing that in-forms and constitutes the objects and things studied by other logoi. |
Some remarks from this journal on April 1, 2015 —
Manifest O
|
| 83-06-21 | An invariance of symmetry The diamond theorem on a 4x4x4 cube, and a sketch of the proof. |
| 83-10-01 | Portrait of O A table of the octahedral group O using the 24 patterns from the 2×2 case of the diamond theorem. |
| 83-10-16 | Study of O A different way of looking at the octahedral group, using cubes that illustrate the 2x2x2 case of the diamond theorem. |
| 84-09-15 | Diamonds and whirls Block designs of a different sort — graphic figures on cubes. See also the University of Exeter page on the octahedral group O. |
The above site, finitegeometry.org/sc, illustrates how the symmetry
of various visual patterns is explained by what Zhang calls "interality."
The New York Times today has an obituary for
Kevin Lippert, the founder and publisher of
Princeton Architectural Press, who reportedly
died at 63 on March 29, 2022.
“'There was a space between the academic,
theory-heavy M.I.T. Press and the coffeetableism
of Rizzoli,' Mr. Lamster wrote, adding that
Princeton Architectural Press would fill the gap
with 'the voice of the young practitioner.'
Mr. Lippert championed emerging architects.
He published Steven Holl’s seminal architectural
manifesto, 'Anchoring,' in 1989, and wrote the
introduction to the book of the same name.
Mr. Holl, in a tribute to Mr. Lippert on his website,
called him 'a committed intellectual and impresario
for the culture of architecture.'”
— Katharine Q. Seelye, April 17, 2022, 2:21 p.m. ET
From the cited tribute to Lippert on Holl's website —
"An excerpt from his publisher’s foreword to Anchoring :
In its iconic simplicity, his work seems to be about
the language of architecture, not in the allusive sense
used by postmodernists nor in the paradigmatic sense
used by so-called 'deconstructivists' but at the level of
essences of tropes and morphs… He is the only
American architect of his generation to be directly
influenced by the main lines in modern philosophy and
music, that is to say, by the line leading from Husserl
through to Heidegger and by separate achievements
of Bartok and Schonberg ."
Actually, although the above "iconic simplicity" passage,
up to the ellipsis after "morphs," is from the foreword
by Lippert, the references that follow the ellipsis — to
Husserl, Heidegger, Bartok, and Schonberg — are not
from Lippert's foreword, but from the introduction by
one Kenneth Frampton —
From Google Books:
Bibliographic data —
Another architectural memorial, from the reported date of Lippert's death —
Ereignis in the Stanford Encyclopedia of Philosophy —
|
Further aspects of the essential unfolding of Being are revealed by what is perhaps the key move in the Contributions—a rethinking of Being in terms of the notion of Ereignis, a term translated variously as ‘event’ (most closely reflecting its ordinary German usage), ‘appropriation’, ‘appropriating event’, ‘event of appropriation’ or ‘enowning’. (For an analysis which tracks Heidegger's use of the term Ereignis at various stages of his thought, see Vallega-Neu 2010). The history of Being is now conceived as a series of appropriating events in which the different dimensions of human sense-making—the religious, political, philosophical (and so on) dimensions that define the culturally conditioned epochs of human history—are transformed. Each such transformation is a revolution in human patterns of intelligibility, so what is appropriated in the event is Dasein and thus the human capacity for taking-as (see e.g., Contributions 271: 343). Once appropriated in this way, Dasein operates according to a specific set of established sense-making practices and structures. In a Kuhnian register, one might think of this as the normal sense-making that follows a paradigm-shift. — Michael Wheeler, 2011 |
See as well "reordering" in Sunday evening's post Tetrads for McLuhan
and in a Log24 search for Reordering + Steiner.
For Oslo artist Josefine Lyche, excerpts
from a Google image search today —
Material related to Lyche's experience as an adolescent with a ZX Spectrum computer —
Click "Hello World" for a larger image.
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From "The Phenomenology of Mathematical Beauty," The Lightbulb Mistake . . . . Despite the fact that most proofs are long, and despite our need for extensive background, we think back to instances of appreciating mathematical beauty as if they had been perceived in a moment of bliss, in a sudden flash like a lightbulb suddenly being lit. The effort put into understanding the proof, the background material, the difficulties encountered in unraveling an intricate sequence of inferences fade and magically disappear the moment we become aware of the beauty of a theorem. The painful process of learning fades from memory, and only the flash of insight remains. We would like mathematical beauty to consist of this flash; mathematical beauty should be appreciated with the instantaneousness of a lightbulb being lit. However, it would be an error to pretend that the appreciation of mathematical beauty is what we vaingloriously feel it should be, namely, an instantaneous flash. Yet this very denial of the truth occurs much too frequently. The lightbulb mistake is often taken as a paradigm in teaching mathematics. Forgetful of our learning pains, we demand that our students display a flash of understanding with every argument we present. Worse yet, we mislead our students by trying to convince them that such flashes of understanding are the core of mathematical appreciation. Attempts have been made to string together beautiful mathematical results and to present them in books bearing such attractive titles as The One Hundred Most Beautiful Theorems of Mathematics . Such anthologies are seldom found on a mathematician’s bookshelf. The beauty of a theorem is best observed when the theorem is presented as the crown jewel within the context of a theory. But when mathematical theorems from disparate areas are strung together and presented as “pearls,” they are likely to be appreciated only by those who are already familiar with them. The Concept of Mathematical Beauty The lightbulb mistake is our clue to understanding the hidden sense of mathematical beauty. The stark contrast between the effort required for the appreciation of mathematical beauty and the imaginary view mathematicians cherish of a flashlike perception of beauty is the Leitfaden that leads us to discover what mathematical beauty is. Mathematicians are concerned with the truth. In mathematics, however, there is an ambiguity in the use of the word “truth.” This ambiguity can be observed whenever mathematicians claim that beauty is the raison d’être of mathematics, or that mathematical beauty is what gives mathematics a unique standing among the sciences. These claims are as old as mathematics and lead us to suspect that mathematical truth and mathematical beauty may be related. Mathematical beauty and mathematical truth share one important property. Neither of them admits degrees. Mathematicians are annoyed by the graded truth they observe in other sciences. Mathematicians ask “What is this good for?” when they are puzzled by some mathematical assertion, not because they are unable to follow the proof or the applications. Quite the contrary. Mathematicians have been able to verify its truth in the logical sense of the term, but something is still missing. The mathematician who is baffled and asks “What is this good for?” is missing the sense of the statement that has been verified to be true. Verification alone does not give us a clue as to the role of a statement within the theory; it does not explain the relevance of the statement. In short, the logical truth of a statement does not enlighten us as to the sense of the statement. Enlightenment , not truth, is what the mathematician seeks when asking, “What is this good for?” Enlightenment is a feature of mathematics about which very little has been written. The property of being enlightening is objectively attributed to certain mathematical statements and denied to others. Whether a mathematical statement is enlightening or not may be the subject of discussion among mathematicians. Every teacher of mathematics knows that students will not learn by merely grasping the formal truth of a statement. Students must be given some enlightenment as to the sense of the statement or they will quit. Enlightenment is a quality of mathematical statements that one sometimes gets and sometimes misses, like truth. A mathematical theorem may be enlightening or not, just as it may be true or false. If the statements of mathematics were formally true but in no way enlightening, mathematics would be a curious game played by weird people. Enlightenment is what keeps the mathematical enterprise alive and what gives mathematics a high standing among scientific disciplines. Mathematics seldom explicitly acknowledges the phenomenon of enlightenment for at least two reasons. First, unlike truth, enlightenment is not easily formalized. Second, enlightenment admits degrees: some statements are more enlightening than others. Mathematicians dislike concepts admitting degrees and will go to any length to deny the logical role of any such concept. Mathematical beauty is the expression mathematicians have invented in order to admit obliquely the phenomenon of enlightenment while avoiding acknowledgment of the fuzziness of this phenomenon. They say that a theorem is beautiful when they mean to say that the theorem is enlightening. We acknowledge a theorem’s beauty when we see how the theorem “fits” in its place, how it sheds light around itself, like Lichtung — a clearing in the woods. We say that a proof is beautiful when it gives away the secret of the theorem, when it leads us to perceive the inevitability of the statement being proved. The term “mathematical beauty,” together with the lightbulb mistake, is a trick mathematicians have devised to avoid facing up to the messy phenomenon of enlightenment. The comfortable one-shot idea of mathematical beauty saves us from having to deal with a concept that comes in degrees. Talk of mathematical beauty is a cop-out to avoid confronting enlightenment, a cop-out intended to keep our description of mathematics as close as possible to the description of a mechanism. This cop-out is one step in a cherished activity of mathematicians, that of building a perfect world immune to the messiness of the ordinary world, a world where what we think should be true turns out to be true, a world that is free from the disappointments, ambiguities, and failures of that other world in which we live. |
How many mathematicians does it take to screw in a lightbulb?
* A footnote in memory of a dancer who reportedly died
yesterday, August 29 — See posts tagged Paradigm Shift.
"Birthday, death-day — what day is not both?" — John Updike
Ellmann on Joyce and 'the perception of coincidence' —
"Samuel Beckett has remarked that to Joyce reality was a paradigm,
an illustration of a possibly unstatable rule. Yet perhaps the rule
can be surmised. It is not a perception of order or of love; more humble
than either of these, it is the perception of coincidence. According to
this rule, reality, no matter how much we try to manipulate it, can only
assume certain forms; the roulette wheel brings up the same numbers
again and again; everyone and everything shift about in continual
movement, yet movement limited in its possibilities."
— Richard Ellmann, James Joyce , rev. ed.. Oxford, 1982, p. 551
"But perhaps the desire for story
is what gets us into trouble to begin with."
— Sarah Marshall on June 5, 2018
"Beckett wrote that Joyce believed fervently in
the significance of chance events and of
random connections. ‘To Joyce reality was a paradigm,
an illustration of a possibly unstateable rule…
According to this rule, reality, no matter how much
we try to manipulate it, can only shift about
in continual movement, yet movement
limited in its possibilities…’ giving rise to
‘the notion of the world where unexpected simultaneities
are the rule.’ In other words, a coincidence … is actually
just part of a continually moving pattern, like a kaleidoscope.
Or Joyce likes to put it, a ‘collideorscape’."
— Gabrielle Carey, "Breaking Up with James Joyce,"
Sydney Review of Books , 15 June 2018
Carey's carelessness with quotations suggests a look at another
author's quoting of Ellmann on Joyce —
From the online New York Times this morning —
"Origin is Mr. Brown’s eighth novel. It finds his familiar protagonist,
the brilliant Harvard professor of symbology and religious iconography
Robert Langdon, embroiled once more in an intellectually challenging,
life-threatening adventure involving murderous zealots, shadowy fringe
organizations, paradigm-shifting secrets with implications for the future
of humanity, symbols within puzzles and puzzles within symbols and
a female companion who is super-smart and super-hot.
As do all of Mr. Brown’s works, the new novel does not shy away from
the big questions, but rather rushes headlong into them."
— Profile of Dan Brown by Sarah Lyall
See also yesterday's Log24 post on the Feast of St. Michael and All Angels.
Or: Two Rivets Short of a Paradigm
Detail from an author photo:
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From rivet-rivet.net:
The philosopher Graham Harman is invested in re-thinking the autonomy of objects and is part of a movement called Object-Oriented-Philosophy (OOP). Harman wants to question the authority of the human being at the center of philosophy to allow the insertion of the inanimate into the equation. With the aim of proposing a philosophy of objects themselves, Harman puts the philosophies of Bruno Latour and Martin Heidegger in dialogue. Along these lines, Harman proposes an unconventional reading of the tool-being analysis made by Heidegger. For Harman, the term tool does not refer only to human-invented tools such as hammers or screwdrivers, but to any kind of being or thing such as a stone, dog or even a human. Further, he uses the terms objects, beings, tools and things, interchangeably, placing all on the same ontological footing. In short, there is no “outside world.” Harman distinguishes two characteristics of the tool-being: invisibility and totality. Invisibility means that an object is not simply used but is: “[an object] form(s) a cosmic infrastructure of artificial and natural and perhaps supernatural forces, power by which our last action is besieged.” For instance, nails, wooden boards and plumbing tubes do their work to keep a house “running” silently (invisibly) without being viewed or noticed. Totality means that objects do not operate alone but always in relation to other objects–the smallest nail can, for example, not be disconnected from wooden boards, the plumbing tubes or from the cement. Depending on the point of view of each entity (nail, tube, etc.) a different reality will emerge within the house. For Harman, “to refer to an object as a tool-being is not to say that it is brutally exploited as a means to an end, but only that it is torn apart by the universal duel between the silent execution of an object’s reality and the glistening aura of its tangible surface.” — From "The Action of Things," an M.A. thesis at the Center for Curatorial Studies, Bard College, by Manuela Moscoso, May 2011, edited by Sarah Demeuse |
From Wikipedia, a programming paradigm:
See also posts tagged Turing's Cathedral, and Alley Oop (Feb. 11, 2003).
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