Atdtda22: [43.8i] Crisis in mathematics, 632 #1

[Mark Kohut]

And Math is Awful in this section.....dungeon of coal-black steps, a residue of a nameless mineral from which all light had been removed---Cf. "History is the story
of our relation to light? "[inexact].....someone will send the corrected quote.....

Math, the foundation of the "practical" sciences......


----- Original Message ----
From: Paul Nightingale <isread at btinternet.com>
To: pynchon-l at waste.org
Sent: Saturday, November 24, 2007 10:45:44 AM
Subject: Atdtda22: [43.8i] Crisis in mathematics, 632 #1

[632.14-16] ... the little known but rewarding Museum der Monstrositaten, a
sort of nocturnal equivalent of Professor Klein's huge collection of
mathematical models ...

Well, if you like hagiography:

The spirit of geometry from at least 1872 to 1922 cannot be better or more
briefly described than in a famous sentence of Felix Klein. All the
astounding inventiveness and infinite variety of geometry during that
amazingly prolific half century is seen as one orderly, simple whole from
the commanding summit which Klein recognized as the proper point of view to
sweep in the whole of the past of geometry and to foresee much of its
future. Here is the famous sentence: 

"Given a manifold and a group of transformations of the same, to develop the
theory of invariants relating to that group."

From: ET Bell, The Queen of the Sciences, Williams & Wilkins, 1931, 69. 


Cf. earlier references to Felix Klein, eg Kit struggling to keep up with
Umeki when they have taken possession of the mysterious Q-weapon, on 565:

Trying to remember what he could of Felix Klein's magisterial Vorlesungen
uber das Ikosaeder, which had been required reading at Gottingen, but not
having much luck.

That "which had been required reading" perhaps suggests it no longer is; or
perhaps Kit is thinking back to his preparations for study there.

And then, on 593:

Gottlob ... had come to Gottingen from Berlin to study with Felix Klein, on
the strength of Klein's magisterial Mathematical Theory of the Top (1897)
...

An interesting repetition of "magisterial".


And so to:

[632.22-23] ... the current "Crisis" in European mathematics ...

Recall that until the later part of the nineteenth century, all of
continuous mathematics was ultimately based on geometrical concepts, because
this type of mathematics was founded on real numbers, which at that time
were conceived of as length of line segments existing in physical, Euclidean
space. This allowed mathematics to be identified with reality and its
methods justified metaphysically. However, the introduction of non-Euclidean
geometries into mathematics in the early part of the nineteenth century
greatly weakened this conceptual foundation for continuous mathematics,
because it allowed the Euclidean nature of physical space to be seriously
questioned. Prior to the introduction of non-Euclidean geometries, it was
generally believed that Euclidean geometry was not only a valid description
of physical space, but it was the only "thinkable" one, that is, it was
unthinkable that space was not Euclidean.

[...]

Uncertainties about the Euclidean nature of physical space produced
difficulties for the foundations of mathematics, because most of mathematics
outside of arithmetic was founded on Euclidean concepts. The
"arithmetization of analysis," which was completed in the late part of the
nineteenth century, resolved such difficulties by founding mathematics on
arithmetic--a subject that had unquestioned mathematical significance and no
philosophical doubts about its reality. After this "arithmetization" was
achieved, the traditional roles between analysis and geometry became
inverted: analysis now provided the basis for models of axiomatic geometry
both Euclidean and non-Euclidean. Such a change, however, produced serious
meaningfulness problems for geometry: In the geometries based on
analysis--henceforth called "analytic geometries"--it was difficult to
distinguish geometric concepts from non-geometric ones. Before, when
geometry was based on physical space, metaphysical principles about physical
reality could be invoked to define "geometrical." But in analytical geometry
no analogous program could be carried out, because arithmetic and its
metaphysics provided no insight into the geometrical nature of things. Many
new geometries were discovered, and it became very clear that since there
are so many different and varied types of geometries and only one physical
space, only a very few geometries--perhaps not even including Euclidean
geometry--could be naturally and directly interpreted in physical space. The
upshot of all of this was that if the concept of "geometrical" were to be
developed for the wide class of known geometries, then it had to be based on
principles other than metaphysical ones about physical or mathematical
reality. The mathematician Felix Klein (1849-1925) conceived of a program to
accomplish exactly this. 

In a famous address given at Erlangen in 1872, Klein identified geometries
with groups of transformations and the concept "geometrical" with invariance
under transformational groups. However, a satisfactory justification for
this position, in my opinion, was not given either in the Erlangen Address
or in Klein's subsequent publications on the subject. In fact, there is
noticeable lack of any serious effort by Klein and his followers to justify
it philosophically. Nevertheless the "Erlanger Program"-as it was
subsequently known-had a powerful and positive impact on the field of
geometry, and today is seen as one of the major advances in mathematics.
This is primarily because the identification of geometries with
transformation groups proved to be an enormously fruitful idea. Among other
things, it gave ideas about the possible range of geometries and about how
these were related to one another, and it provided techniques that
transformed subtle geometric questions into straightforward, easy-to-solve
group-theoretical ones. It also provided very interesting insights into
meaningfulness issues. 

From: Louis Narens, Theories of Meaningfulness, Lawrence Erlbaum, 2002,
23-24.


      ____________________________________________________________________________________
Be a better pen pal. 
Text or chat with friends inside Yahoo! Mail. See how.  http://overview.mail.yahoo.com/
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://waste.org/pipermail/pynchon-l/attachments/20071124/0b923d16/attachment-0001.html>