At present this web site is merely the beginnings of a small
library of papers and e-books
by the author (Michael Hugh Knowles).
mhk(at)mhknowles(dot)net
Special Notice:
On
March 10, 2012, MHK gave a presentation
(1079-03-89) titled
“Does
the Banach-Tarski Paradox have an Evil Twin?!” (this
links to unofficial copy of the official abstract)
(slideshows,
the actual paper with slightly easier to read abstract)
at the 2012 Southeastern Sectional Meeting (#1079) of the American
mathematical Society, at the University of South Florida in Tampa, FL, USA.
in the Session for Contributed Papers, I, Room 354,
Cooper Hall, USF.
See
http://www.ams.org/meetings/sectional/2188_progfull.html
________________________________________________________________
Request to teachers/others who make any use of the papers found here e.g. in
classes: could you please let me know what use they have been to you.
Please e-mail me at
mhk(at)mhknowles(dot)net
Thank you!
Table of Contents
e-papers:
Proof of
a New, Simple but Fundamental Bijection Theorem in Set Theory
Revised December 6, 2011
Opening
a
Community Deconstruction of Set Theory
ProgXML
‒
Programming-XML
Cosmology “Un-Survey” Of “Un-Discoveries”
e-books:
Newton’s
Great... Oversight
e-Papers
Proof of
a New, Simple but Fundamental Bijection Theorem in Set Theory
(71KB;
3 pages;
Revised, December 06, 2011)
In my LinkedIn Profile Contact
section, I refer to the following theorem I discovered (back in
the middle 1990s) in set theory:
THEOREM: Given a bijection
B(SP,SI)
from a pre-image set
SP onto
an image set
SI, where
SP
and
SI
have at least one element
EC
in common, then using only
simple bijectivity preserving operations one can construct a new bijection
BB
from
SP-{EC}
(the pre-image set
SP
with the common element
EC
removed) onto
SI-{EC}
(the image set
SI
with the common element
EC
removed), i.e. the
bijection
BB(SP-{EC},SI-{EC}).
You can find its proof (very simple, less than a full page) starting on the
first page of the paper with link to pdf file, above.
Summary commentary follows. If you have reasonable popular math skills, none of it will be
any big challenge. The
serious challenge comes when pondering the theorem’s theoretical consequences for set
theory, and for the 2/3 to 3/4 of modern math that depends on set theory and the
concept of “Dedekind-infinite”
for their theoretical foundations.
Perhaps Gauss, Poincaré, Kronecker, Brouwer and many other mathematicians were
right to condemn set theory as ultimately unsound.
MATHEMATICS TEACHERS: please feel free to use this paper in your classes.
Opening
a
Community Deconstruction of Set Theory
(406KB; 19 pages;
August 21, 2011)
In my LinkedIn Profile Contact
section, I refer to starting a LinkedIn Group that would initiate and carry out
a community “deconstruction” of set theory. “Deconstruction” (see
http://en.wikipedia.org/wiki/Deconstruction) has become a popular term, not as well
defined perhaps as it once was, for carefully analyzing something (in certain ways),
usually something in the “soft sciences” or in philosophy. Here it is intended
to be a
“forensic” attempt to re-raise the question of the theoretical soundness of the
set theory of Georg Cantor, best known through the axiomatization of it by
Zermelo and Fraenkel (and Skolem, who is usually forgotten in this context),
acronymed as ZF, or ZFC if the Axiom of Choice is included. I have started this
“deconstruction” by “forensically” examining the concept of
“Dedekind-infinite”, the concept that first formally summarized the paradoxes of
infinity so as to make them a part of mathematics, in particular the theory of
infinite sets founded (mostly) by Georg Cantor. The presentation (the pdf file
linked to above) is easy going,
so any pop mathematician should be able to handle it.
MATHEMATICS and PHILOSOPHY TEACHERS: please feel free to use this paper in your classes.
ProgXML
‒
Programming-XML
(308KB; ~41 small pages; June 21, 2011)
A
Concept Paper / “Light
Green Paper” proposing that standard High Level Languages be replaced
entirely (more
or less) by one or (probably) more “ProgXML”
type systems, using a general and naturally extensible programming variant of
XML to hold and manipulate the semantics of computer programs. This would allow much greater facility in presentation, entering and editing
of program semantics and pragmatics, and in linking among disparate systems, than our standard HLL approaches
currently
allow.
NB:
ProgXML is a completely different concept from what have become the usual
“XML-based programming languages”, which are more-or-less standard High Level
Languages, except that XML is used to implement them. They have only their
single, standard HLL mode for program entry-editing-presentation/viewing.
COMPUTER SCIENCE TEACHERS: please feel free to use this
Concept Paper / “Light
Green Paper” in your classes.
Cosmology “Un-Survey” Of “Un-Discoveries”
(355KB;
~46 small pages; October 15, 2012)
This is relatively serious attempt to bring to daylight many obvious but
overlooked ideas in cosmology, such as the theoretical necessity of multi-dimensional time in general relativity,
as well as in reality. Einstein said something
like “Imagination…is more important than
knowledge. Knowledge is limited. Imagination encircles the world.”
This paper will help you stretch your cosmological imaginations.
SCIENCE TEACHERS: please feel free to use this paper in your classes.
e-Books
e-book: Newton’s
Great... Oversight
(3.5MB;
152 pages;
November 7, 2011)
This is oriented toward gifted children (of any age). Newton overlooked that his
theory of gravity predicts that lighter and heavier bodies will fall at
different rates, with one fascinating exception! The trail leads us to...
Lagrange’s Trojan Points and Trojan Asteroids
with their Tadpole and Horseshoe Orbits
SCIENCE TEACHERS: please feel free to use this
e-book in your classes.