Friday, March 10, 2023
Hofstadter on Geometry
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Monday, June 26, 2017
Four Dots
Analogies — “A : B :: C : D” may be read “A is to B as C is to D.”
Gian-Carlo Rota on Heidegger…
“… The universal as is given various names in Heidegger’s writings….
The discovery of the universal as is Heidegger’s contribution to philosophy….
The universal ‘as‘ is the surgence of sense in Man, the shepherd of Being.
The disclosure of the primordial as is the end of a search that began with Plato….
This search comes to its conclusion with Heidegger.”
— “Three Senses of ‘A is B’ in Heideggger,” Ch. 17 in Indiscrete Thoughts
See also Four Dots in this journal.
Some context: McLuhan + Analogy.
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Thursday, March 2, 2017
Review
From a 2002 note, "The Shining of May 29" —
Related material: The remarks in this journal on April 1, 2013.
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Friday, November 27, 2015
Einstein and Geometry
(A Prequel to Dirac and Geometry)
"So Einstein went back to the blackboard.
And on Nov. 25, 1915, he set down
the equation that rules the universe.
As compact and mysterious as a Viking rune,
it describes space-time as a kind of sagging mattress…."
— Dennis Overbye in The New York Times online,
November 24, 2015
Some pure mathematics I prefer to the sagging Viking mattress —
Readings closely related to the above passage —
Thomas Hawkins, "From General Relativity to Group Representations:
the Background to Weyl's Papers of 1925-26," in Matériaux pour
l'histoire des mathématiques au XXe siècle: Actes du colloque
à la mémoire de Jean Dieudonné, Nice, 1996 (Soc. Math.
de France, Paris, 1998), pp. 69-100.
The 19th-century algebraic theory of invariants is discussed
as what Weitzenböck called a guide "through the thicket
of formulas of general relativity."
Wallace Givens, "Tensor Coordinates of Linear Spaces," in
Annals of Mathematics Second Series, Vol. 38, No. 2, April 1937,
pp. 355-385.
Tensors (also used by Einstein in 1915) are related to
the theory of line complexes in three-dimensional
projective space and to the matrices used by Dirac
in his 1928 work on quantum mechanics.
For those who prefer metaphors to mathematics —
Rota fails to cite the source of his metaphor.
|
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Wednesday, December 17, 2014
For Rilke’s Panther
The title refers to yesterday evening's remarks titled
"Free the Philosophical Beast" in The Stone , a NY Times weblog.
The January 2015 issue of the Notices of the American Mathematical Society
has an article by Michael J. Barany. From November 2012 remarks
by Barany :
"A highlight of the workshop was Cathryn Carson’s interpretation
of the transcendental phenomenology and historicism of Husserl,
Heidegger, Cassirer, and a few others, launched from a moving
reflection on the experience of reading Kuhn."
See Carson's paper "Science as Instrumental Reason: Heidegger, Habermas,
Heisenberg," Continental Philosophy Review (2010) 42: 483–509.
Related material: Monday's Log24 posts Rota on Husserl and Annals of Perception.
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Tuesday, December 9, 2014
Plan 9…
Or: Bullshit for Brit … continues.
From the new film "I Origins," starring Brit Marling —
Plan 9:
The protagonist of "I Origins" is led to the above billboard
by apparently chance encounters with 11 's — such as the
1111 on the following page —
Update of Dec. 10, 2014: The "bullshit" in the subtitle above refers
to the remarks of Joan Stambaugh, not those of Nicholas of Cusa.
The passage from Nicholas was added because it indicates a more
reliable source than Stambaugh, because it is relevant to lines
about the metaphorical significance of light in "I Origins," and
because it contains the number 1111.
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Tuesday, October 7, 2014
As Is
"That simple operator, 'as,' turns out to carry within its philosophical grammar
a remarkable complex field* of operations…."
— Charles Altieri, Painterly Abstraction in Modernist American Poetry,
Cambridge University Press, 1989, page 343
See also Rota on Heidegger (What "As" Is, July 6, 2010), and Lead Belly
on the Rock Island Line — "You got to ride it like you find it."
* Update of Oct. 10, 2014: See also "Complex + Grid" in this journal.
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Thursday, February 2, 2012
Die Nichtung
"It seems that Hilbert had no taste for philosophers….
Hans Rademacher told this reviewer that, after Heidegger
once lectured in Göttingen, Hilbert gleefully repeated
to everyone the phrase "Das Nichts nichtet die Nichtung ."
— Gian-Carlo Rota, Discrete Thoughts , 2nd ed., p. 233
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Tuesday, July 6, 2010
What “As” Is
or: Combinatorics (Rota) as Philosophy (Heidegger) as Geometry (Me)
“Dasein’s full existential structure is constituted by
the ‘as-structure’ or ‘well-joined structure’ of the rift-design*…”
— Gary Williams, post of January 22, 2010
Background—
Gian-Carlo Rota on Heidegger…
“… The universal as is given various names in Heidegger’s writings….
The discovery of the universal as is Heidegger’s contribution to philosophy….
The universal ‘as‘ is the surgence of sense in Man, the shepherd of Being.
The disclosure of the primordial as is the end of a search that began with Plato….
This search comes to its conclusion with Heidegger.”
— “Three Senses of ‘A is B’ in Heideggger,” Ch. 17 in Indiscrete Thoughts
… and projective points as separating rifts—
* rift-design— Definition by Deborah Levitt—
“Rift. The stroke or rending by which a world worlds, opening both the ‘old’ world and the self-concealing earth to the possibility of a new world. As well as being this stroke, the rift is the site— the furrow or crack— created by the stroke. As the ‘rift design‘ it is the particular characteristics or traits of this furrow.”
— “Heidegger and the Theater of Truth,” in Tympanum: A Journal of Comparative Literary Studies, Vol. 1, 1998
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Monday, June 7, 2010
Inspirational Combinatorics
According to the Mathematical Association of America this morning, one purpose of the upcoming June/July issue of the Notices of the American Mathematical Society is
"…to stress the inspirational role of combinatorics…."
Here is another contribution along those lines—
Eidetic Variation
from page 244 of
From Combinatorics to Philosophy: The Legacy of G.-C. Rota,
hardcover, published by Springer on August 4, 2009
(Edited by Ernesto Damiani, Ottavio D'Antona, Vincenzo Marra, and Fabrizio Palombi)
"Rota's Philosophical Insights," by Massimo Mugnai—
"… In other words, 'objectivism' is the attitude [that tries] to render a particular aspect absolute and dominant over the others; it is a kind of narrow-mindedness attempting to reduce to only one the multiple layers which constitute what we call 'reality.' According to Rota, this narrow-mindedness limits in an essential way even of [sic ] the most basic facts of our cognitive activity, as, for example, the understanding of a simple declarative sentence: 'So objectivism is the error we [make when we] persist in believing that we can understand what a declarative sentence means without a possible thematization of this declarative sentence in one of [an] endless variety of possible contexts' (Rota, 1991*, p. 155). Rota here implicitly refers to what, amongst phenomenologists is known as eidetic variation, i.e. the change of perspective, imposed by experience or performed voluntarily, from which to look at things, facts or sentences of the world. A typical example, proposed by Heidegger, in Sein und Zeit (1927) and repeated many times by Rota, is that of the hammer."
* Rota, G.-C. (1991), The End of Objectivity: The Legacy of Phenomenology. Lectures at MIT, Cambridge, MA, MIT Mathematics Department
The example of the hammer appears also on yesterday's online New York Times front page—
Related material:
From The Blackwell Dictionary of Western Philosophy—
Eidetic variation — an alternative expression for eidetic reduction
Husserl's term for an intuitive act toward an essence or universal, in contrast to an empirical intuition or perception. He also called this act an essential intuition, eidetic intuition, or eidetic variation. In Greek, eideo means “to see” and what is seen is an eidos (Platonic Form), that is, the common characteristic of a number of entities or regularities in experience. For Plato, eidos means what is seen by the eye of the soul and is identical with essence. Husserl also called this act “ideation,” for ideo is synonymous with eideo and also means “to see” in Greek. Correspondingly, idea is identical to eidos.
An example of eidos— Plato's diamond (from the Meno )—
For examples of variation of this eidos, see the diamond theorem.
See also Blockheads (8/22/08).
Related poetic remarks— The Trials of Device.
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Wednesday, May 19, 2010
Mathematics and Gestalt
"We acknowledge a theorem's beauty
when we see how the theorem 'fits'
in its place, how it sheds light around itself,
like a Lichtung, a clearing in the woods."
— Gian-Carlo Rota, Indiscrete Thoughts
Here Rota is referring to a concept of Heidegger.
Some context—
"Gestalt Gestell Geviert: The Way of the Lighting,"
by David Michael Levin in The Philosopher's Gaze
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Saturday, April 19, 2008
Saturday April 19, 2008
A Midrash for Benedict
On April 16, the Pope’s birthday, the evening lottery number in Pennsylvania was 441. The Log24 entries of April 17 and April 18 supplied commentaries based on 441’s incarnation as a page number in an edition of Heidegger’s writings. Here is a related commentary on a different incarnation of 441. (For a context that includes both today’s commentary and those of April 17 and 18, see Gian-Carlo Rota– a Heidegger scholar as well as a mathematician– on mathematical Lichtung.)
From R. D. Carmichael, Introduction to the Theory of Groups of Finite Order (Boston, Ginn and Co., 1937)– an exercise from the final page, 441, of the final chapter, “Tactical Configurations”–
“23. Let G be a multiply transitive group of degree n whose degree of transitivity is k; and let G have the property that a set S of m elements exists in G such that when k of the elements S are changed by a permutation of G into k of these elements, then all these m elements are permuted among themselves; moreover, let G have the property P, namely, that the identity is the only element in G which leaves fixed the n – m elements not in S. Then show that G permutes the m elements S into
n(n -1) … (n – k + 1)
____________________
____________________
m(m – 1) … (m – k + 1)
sets of m elements each, these sets forming a configuration having the property that any (whatever) set of k elements appears in one and just one of these sets of m elements each. Discuss necessary conditions on m, n, k in order that the foregoing conditions may be realized. Exhibit groups illustrating the theorem.”
This exercise concerns an important mathematical structure said to have been discovered independently by the American Carmichael and by the German Ernst Witt.
For some perhaps more comprehensible material from the preceding page in Carmichael– 440– see Diamond Theory in 1937.
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