Log24

Sunday, August 1, 2021

Freudenthal vs. Weyl

Filed under: General — Tags: — m759 @ 2:10 am

Hans Freudenthal in 1962 on the axiomatic approach to geometry
of Fano and Hilbert —

"The bond with reality is cut."

Some philosophical background —

For Weyl's "few isolated relational concepts," see (for instance)
Projective Geometries over Finite Fields , by
J. W. P. Hirschfeld (first published by Oxford University Press in 1979).

Weyl in 1932 —

Mathematics is the science of the infinite , its goal the symbolic comprehension of the infinite with human, that is finite, means. It is the great achievement of the Greeks to have made the contrast between the finite and the infinite fruitful for the cognition of reality. The intuitive feeling for, the quiet unquestioning acceptance of the infinite, is peculiar to the Orient; but it remains merely an abstract consciousness, which is indifferent to the concrete manifold of reality and leaves it unformed, unpenetrated. Coming from the Orient, the religious intuition of the infinite, the apeiron , takes hold of the Greek soul in the Dionysiac-Orphic epoch which precedes the Persian wars. Also in this respect the Persian wars mark the separation of the Occident from the Orient. This tension between the finite and the infinite and its conciliation now become the driving motive of Greek investigation; but every synthesis, when it has hardly been accomplished, causes the old contrast to break through anew and in a deepened sense. In this way it determines the history of theoretical cognition to our day. 

— "The Open World: Three Lectures on the Metaphysical Implications of Science," 1932

Tuesday, October 24, 2023

A Bond with Reality:  The Geometry of Cuts

Filed under: General — Tags: , , , — m759 @ 12:12 pm


Illustrations of object and gestures
from finitegeometry.org/sc/ —

Object

Gestures

An earlier presentation of the above
seven partitions of the eightfold cube:

Seven partitions of the 2x2x2 cube in a book from 1906

Related mathematics:

The use  of binary coordinate systems
as a conceptual tool

Natural physical  transformations of square or cubical arrays
of actual physical cubes (i.e., building blocks) correspond to
natural algebraic  transformations of vector spaces over GF(2).
This was apparently not previously known.

See "The Thing and I."

and . . .

Galois.space .

 

Related entertainment:

Or Matt Helm by way of a Jedi cube.

Friday, February 4, 2022

The Guralnik Cube

Filed under: General — Tags: — m759 @ 2:04 pm

New York Review of Books , Dec. 16, 2021 issue —
Lorrie Moore on the documentary series "Couples Therapy" —

"Few of the people sitting on the couch avoid the cliché of
one person (a man) playing fruitlessly with a plastic puzzle
while the other speaks tearfully and avails herself of a
Kleenex box. In season 1, there is literally a Rubik’s cube,
and no one ever solves it, an unfortunate but apt metaphor.
During one session, when the cube has been placed out of reach,
one of the husbands gets up to look for it, finding it on a shelf." 

See also . . .

"The bond with reality is cut." — Hans Freudenthal 

Monday, April 12, 2021

Models: A Return to Utrecht

Filed under: General — Tags: — m759 @ 3:41 pm

References to a 1960 conference paper by Freudenthal in this journal
suggest another paper from the same conference …

See as well other posts now tagged . . .

The Utrecht Models.

For my own work on models, see
Finite Geometry of the Square and Cube.

Saturday, October 17, 2020

Modernist Cuts

Filed under: General — Tags: , , — m759 @ 10:05 pm

"The bond with reality is cut."

— Hans Freudenthal, 1962

Indeed it is.

Related screenshot of a book review
from the November AMS Notices

Thursday, August 20, 2020

“One More Reality Show”

Filed under: General — Tags: — m759 @ 9:27 am

“The bond with reality is cut.”

— Hans Freudenthal, 1962

Indeed it is.

Sunday, September 22, 2019

Colorful Tale

Filed under: General — Tags: , , — m759 @ 7:59 pm

“Perhaps the philosophically most relevant feature of modern science
is the emergence of abstract symbolic structures as the hard core
of objectivity behind— as Eddington puts it— the colorful tale of
the subjective storyteller mind.”

— Hermann Weyl, Philosophy of  Mathematics and
    Natural Science 
, Princeton, 1949, p. 237

"The bond with reality is cut."

— Hans Freudenthal, 1962

Indeed it is.

From page 180, Logicomix — It was a dark and stormy night

http://www.log24.com/log/pix11/110420-DarkAndStormy-Logicomix.jpg

Monday, May 20, 2019

The Bond with Reality

Filed under: General — Tags: , , , , — m759 @ 10:00 pm


"The bond with reality is cut."

— Hans Freudenthal, 1962

Indeed it is.

Sunday, May 19, 2019

The Building Blocks of Geometry

From "On the life and scientific work of Gino Fano
by Alberto Collino, Alberto Conte, and Alessandro Verra,
ICCM Notices , July 2014, Vol. 2 No. 1, pp. 43-57 —

" Indeed, about the Italian debate on foundations of Geometry, it is not rare to read comments in the same spirit of the following one, due to Jeremy Gray13. He is essentially reporting Hans Freudenthal’s point of view:

' When the distinguished mathematician and historian of mathematics Hans Freudenthal analysed Hilbert’s  Grundlagen he argued that the link between reality and geometry appears to be severed for the first time in Hilbert’s work. However, he discovered that Hilbert had been preceded by the Italian mathematician Gino Fano in 1892. . . .' "

13 J. Gray, "The Foundations of Projective Geometry in Italy," Chapter 24 (pp. 269–279) in his book Worlds Out of Nothing , Springer (2010).


Restoring the severed link —

Structure of the eightfold cube

See also Espacement  and The Thing and I.
 

Related material —

 
 

Tuesday, April 2, 2013

Hermite

Filed under: General,Geometry — Tags: , — m759 @ 7:14 pm

A sequel to the quotation here March 8 (Pinter Play)
of Joan Aiken's novel The Shadow Guests

Supposing that one's shadow guests are
Rosenhain and Göpel (see March 18)

Hans Freudenthal at Encyclopedia.com on Charles Hermite:

"In 1855 Hermite took advantage of Göpel’s and Rosenhain’s work
when he created his transformation theory (see below)."

"One of his invariant theory subjects was the fifth-degree equation,
to which he later applied elliptic functions.

Armed with the theory of invariants, Hermite returned to
Abelian functions. Meanwhile, the badly needed theta functions
of two arguments
had been found, and Hermite could apply what
he had learned about quadratic forms to understanding the
transformation of the system of the four periods. Later, Hermite’s
1855 results became basic for the transformation theory of Abelian
functions as well as for Camille Jordan’s theory of 'Abelian' groups.
They also led to Herrnite’s own theory of the fifth-degree equation
and of the modular equations of elliptic functions. It was Hermite’s
merit to use ω rather than Jacobi’s q = eπω as an argument and to
prepare the present form of the theory of modular functions.
He again dealt with the number theory applications of his theory,
particularly with class number relations or quadratic forms.
His solution of the fifth-degree equation by elliptic functions
(analogous to that of third-degree equations by trigonometric functions)
was the basic problem of this period."

See also Hermite in The Catholic Encyclopedia.

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