Axiom of
Choice
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HISTORIA MATEMATICA Tuesday, April 18 2000 Volume 02 : Number 021
~~~~~~~~~~~~~~~~~~~~~~~~~~ TABLE of SUBJECTS ~~~~~~~~~~~~~~~~~~~~~~~~~~
[HM] Early Set Theory Texts
Re: [HM] Kurt Grelling
Re: [HM] Kurt Grelling
Re: [HM] Kurt Grelling
[HM] Neue Fries'sche Schule
Re: [HM] Kurt Grelling
Re: [HM] Early Set Theory Texts
Re: [HM] Neue Fries'sche Schule
[HM] Midwest Conference-Call for Papers
Please see the end of this digest.
----------------------------------------------------------------------
Date: Mon, 17 Apr 2000 08:37:14 -0400
From: "J.M. Plotkin"
Subject: [HM] Early Set Theory Texts
In the first set theory book sweepstakes Gerhard Hessenberg's (very
interesting) 1906 book, Grundbegriffe der Mengenlehre, should not be
forgotten. The "Vorwort" ends with a special thanks to Zermelo for
his "Korrekturbogen" and for the permission to make use of as yet
unpublished results.
Jacob Plotkin
J.M. Plotkin
Department of Mathematics
Michigan State University
East Lansing, MI 48824-1027
Ph: (517) 353-8484
Fax: (517) 432-1562
email: plotkin@math.msu.edu
------------------------------
Date: Mon, 17 Apr 2000 15:52:47 +0100
From: "Dr. Volker Peckhaus"
Subject: Re: [HM] Kurt Grelling
Kurt Grelling wrote only one textbook in set theory, his 1924
"Mengenlehre" which was translated into Spanish (Mexico) in 1943
(I have never seen a copy of this translation). His dissertation on
"Axiome der Arithmetik" of 1910 is on set theory but definitely not a
textbook. Considerations on set theory can be found in his joint
paper with Nelson "Bemerkungen zu den Paradoxieen von Russell
und Burali-Forti" of 1908, where Zermelo attempts to avoid the
paradoxes were discussed.
Grelling was member of the Neue Fries'sche Schule, and he
definitely knew Hessenbergs "Grundbegriffe der Mengenlehre"
which was published in the Abhandlungen der Fries'schen Schule in
1906. This is a textbook, at least Hessenberg considered it as such.
Maybe Tarski and Grelling spoke about Hessenberg's book.
A (almost) complete online bibliography of Grelling's works can be
found on my Grelling webside:
http://www.phil.uni-erlangen.de/~p1phil/personen/peckhaus/grelling/id3.htm
Volker Peckhaus
*******************************************************************
Prof. Dr. Volker Peckhaus
Interdisziplinaeres Institut fuer
Wissenschaftstheorie und Wissenschaftsgeschichte
Universitaet Erlangen Nuernberg
Bismarckstr. 1
D-91054 Erlangen
E-mail: vrpeckha@phil.uni-erlangen.de
URL:
http://www.phil.uni-erlangen.de/~p1phil/personen/peckhaus/home.html
*******************************************************************
------------------------------
Date: Mon, 17 Apr 2000 12:23:36 +0200 (IST)
From: Avinoam Mann
Subject: Re: [HM] Kurt Grelling
What is "Grelling's paradox"?
Avinoam Mann
------------------------------
Date: Mon, 17 Apr 2000 16:56:29 +0200 (MEST)
From: Walter Felscher
Subject: Re: [HM] Kurt Grelling
On Mon, 17 Apr 2000, Avinoam Mann wrote:
>
> What is "Grelling's paradox"?
An adjective is called heteronomical if it does NOT have the property it
expresses. So 'heteronomical' is an adjective. Is it heteronomical ?
W.F
------------------------------
Date: Mon, 17 Apr 2000 10:49:06 -0400
From: zucker@mail.cas.McMaster.CA (Jeffery Zucker)
Subject: [HM] Neue Fries'sche Schule
Volker Peckhaus wrote:
> Grelling was member of the Neue Fries'sche Schule, and he
> definitely knew Hessenbergs "Grundbegriffe der Mengenlehre"
> which was published in the Abhandlungen der Fries'schen Schule in
> 1906. This is a textbook, at least Hessenberg considered it as such.
> Maybe Tarski and Grelling spoke about Hessenberg's book.
Could Dr Peckhaus (or anyone else) please say something about the
Neue Fries'sche Schule? I must confess, I know nothing about it.
Thank you
Jeff Zucker
------------------------------
Date: Mon, 17 Apr 2000 13:01:43 -0300
From: Julio Gonzalez Cabillon
Subject: Re: [HM] Kurt Grelling
Re: What is "Grelling's paradox"?
Dear Avinoam,
Just for the sake of my laziness may I copy a passage from one
of my textbooks (1991) for my students:
6. Paradoja de Grelling.
Esta paradoja inventada por Kurt Grelling en 1908 puede formularse
muy simplemente como sigue:
Consideremos todos los adjetivos del idioma espan~ol. En algunos
casos, el significado de estos adjetivos se aplica a los adjetivos
mismos; en la mayoria de los casos, en cambio, esto no ocurre. Asi,
por ejemplo, el adjetivo "breve" es breve, el adjetivo "polisilabico"
es polisilabico, el adjetivo "adjetivo" es adjetivo; en tanto que
"azul", "sublime", "paleontologico" no se autodescriben.
Clasifiquemos, pues, todos los adjetivos de nuestro idioma en dos
grupos: el de los que se describen a si mismos, que llamaremos
_autologicos_, y el de los que no se autodescriben, que denominaremos
_heterologicos_.
(i) A que grupo pertenece el adjetivo "heterologico"?
(ii) Explica por que el adjetivo "heterologico" es heterologico si
y solo si no lo es.
Sugerencia: De acuerdo con nuestra clasificacion de los adjetivos,
"heterologico" debe ser autologico o bien heterologico, pero no
ambas cosas a la vez.
Si consideramos que dicho adjetivo es heterologico, entonces el
enunciado
"heterologico" es heterologico
afirma que el adjetivo en cuestion se describe a si mismo, y en
consecuencia, debe ser autologico.
Por otra parte, si consideramos que el adjetivo "heterologico" es
autologico, entonces el enunciado
"heterologico" es autologico
afirma que el adjetivo en cuestion no se autodescribe, y por lo
tanto, debe ser heterologico.
Observacion: Los adjetivos "autologico" y "heterologico" no figuran
en el Diccionario de la Real Academia Espan~ola, pero, obviamente,
este hecho no modifica la paradoja en absoluto.
Best regards, Julio GC
------------------------------
Date: Mon, 17 Apr 2000 18:34:11 +0200 (MEST)
From: Walter Felscher
Subject: Re: [HM] Early Set Theory Texts
I concur with Mr. Plotkin that Hessenberg's Grundbegriffe der Mengenlehre
is a quite impressive book; it appeared both as such and as an article in
Nelson's Abhandlungen der Fries'schen Schule, N.S. 1 (1906), 4.Heft,
487-706 .
There are, partly earlier, book-like articles by Adolf Schoenflies which
report on the then state of Cantorian set theory:
Die Entwicklung der Lehre von den Punktmannigfaltigkeiten.
Jahresber.D.M.V. 8 (1900) , 2tes Heft , 1-250
Die Entwicklung der Lehre von den Punktmannigfaltigkeiten. Zweiter Teil.
Jahresber.D.M.V. Erg"anzungsband II . Leipzig 1908
Entwickelung der Mengenlehre und ihrer Anwendungen. Erste Haelfte.
Leipzig 1913
Returning to Hessenberg, his paper
Kettentheorie und Wohlordnung. Crelle 135 (1909) 81-133
can hardly be underestimated in its importance. Not that it was understood
by his contemporaries. But Hessenberg, analyzing Zermelos second proof of
the well ordering theorem, studied the general ways to construct well
ordered subsets of ordered sets - with the one restriction that order
always was inclusion and ordered sets were subfamilies of power sets. In
the course of this, Hessenberg stated and proved the fixpoint theorem
which thirty years later was rediscovered - for ordered sets now - by
Nicolas Bourbaki. The amazing thing is that Hessenberg's proof is
precisely the same as that given by Bourbaki ! (only that at one small
point a simpler argument can be used due to the circumstance that
Hessenberg's order is inclusion). For details, I refer to my article in
Archiv d.Math. 13 (1962) 160-165 and to my book Naive Mengen und Abstrakte
Zahlen from 1979 , p.200 ff .
W.F.
------------------------------
Date: Mon, 17 Apr 2000 15:40:14 -0300
From: Julio Gonzalez Cabillon
Subject: Re: [HM] Neue Fries'sche Schule
> Could ... anyone ... please say something about the
> Neue Fries'sche Schule? I must confess, I know nothing
> about it.
Jeff, for a quick snapshot take a look at
http://www.friesian.com/school.htm
Regards, Julio GC
------------------------------
Date: Mon, 17 Apr 2000 15:36:53 -0400
From: "Daniel J. Curtin"
Subject: [HM] Midwest Conference-Call for Papers
Dear List members,
I hope this is an appropriate use of the list.
Cheers
Dan Curtin
Call for Papers
Midwest History of Mathematics Conference
October 13-14, 2000
Kim Plofker, of Brown University, an expert in the mathematics and
astronomy of ancient and medieval India will give the keynote
address. The conference will be hosted by Northern Kentucky
University at a site convenient to the Greater Cincinnati Airport.
Twenty-minute contributed talks on any topic in the History of
Mathematics are being solicited, with the possibility that the
resulting papers will be published in a conference proceedings volume.
Titles and abstracts, along with the speaker's name, address,
affiliation, and Email address should be submitted for consideration
by July 14, 2000 to:
Daniel J. Curtin
Department of Mathematics and Computer Science
Northern Kentucky University
Highland Heights, KY 41099
curtin@nku.edu
------------------------------
End of HISTORIA MATEMATICA V2 #21
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HISTORIA MATEMATICA Wednesday, April 19 2000 Volume 02 : Number 022
~~~~~~~~~~~~~~~~~~~~~~~~~~ TABLE of SUBJECTS ~~~~~~~~~~~~~~~~~~~~~~~~~~
Re: [HM] Kurt Grelling
Re: [HM] Neue Fries'sche Schule
Re: [HM] Kurt Grelling
Re: [HM] Kurt Grelling
[HM] Mathematics in Latvia Through the Centuries
Re: [HM] historiography of mathematics
Re: [HM] Early Set Theory Texts
Please see the end of this digest.
----------------------------------------------------------------------
Date: Tue, 18 Apr 2000 11:47:41 +0200 (IST)
From: Avinoam Mann
Subject: Re: [HM] Kurt Grelling
Thanks, Julio and Walter, for explaining Grelling's paradox, and also
giving me incidentally a lesson in Spanish. I was not aware that this
paradox was attributed to Grelling. When I took a course in set theory
from A.A.Fraenkel, in my freshman year, he mentioned this paradox as one
equivalent to Russel's, if we identify an adjective with the set of
objects described by that adjective. Fraenkel used "predicate", not
"adjective", and accordingly "predicable" and "impredicable".
Avinoam Mann
------------------------------
Date: Tue, 18 Apr 2000 10:46:53 +0100
From: "Dr. Volker Peckhaus"
Subject: Re: [HM] Neue Fries'sche Schule
> Could Dr Peckhaus (or anyone else) please say something about the
> Neue Fries'sche Schule? I must confess, I know nothing about it.
When Leonard Nelson (1882-1927) started to study philosophy in Go"ttingen
in winter 1903 he immediately founded the Neue Fries'sche Schule with
the founding members Carl Brinkmann, Heinrich Goesch and Alexander Ru"stow.
He took up the older project of the Fries'sche Schule (1847-1849) with
Ernst Friedrich Apelt, Matthias Jakob Schleiden, Oscar Xavier Schlo"milch
and Oscar Schmidt in Jena. They tried to continue the philosophical
program of their teacher Jakob Friedrich Fries (1773-1843), i.e. adopt
Kant's critical philosophy to new developments in science.
Nelson joint this program very closely. In 1904, still as a student, he
founded the Abhandlungen der Fries'schen Schule. His co-editors were
Karl Kaiser and Gerhard Hessenberg (1874-1925). A successor of the
Abhandlungen is the journal Ratio. In 1913 Nelson founded the Jakob
Friedrich Fries-Gesellschaft, in 1918 the Internationaler Jugend-Bund
(IJB), 1925 the Internationaler Sozialistischer Kampfkreis (ISK). After
the war, students of Nelson became influential in the left wing of the
German Social Demokratic Party (SPD). Leonard Nelson was always the
driving force. Among the scholars standing close to Nelson were Kurt
Grelling, Otto Meyerhof, Paul Bernays, Richard Courant (for a while).
Karl Popper, Max Black and Paul Lorenzen were influenced.
Nelson aimed at a critical mathematics, i.e. a transcendental foundation
of Hilbert-style axiomatics. He therefore enjoyed Hilbert's support
against the philosophy professors in Go"ttingen.
The story is told in detail in my book
Hilbertprogramm und Kritische Philosophie, Vandenhoeck & Ruprecht:
Go"ttingen 1990.
A concise description can be found in my paper
"Fries in 'Hilbert's Go"ttingen': Die Neue Fries'sche Schule," in
Wolfgang Hogrebe/Kay Hermann (eds.) Jakob Friedrich Fries. Philosoph,
Naturwissenschaftler und Mathematiker, Peter Lang: Frankfurt a.M.
1999, 353-368.
********************************************************************
Prof. Dr. Volker Peckhaus
Interdisziplinaeres Institut fuer
Wissenschaftstheorie und Wissenschaftsgeschichte
Universitaet Erlangen Nuernberg
Bismarckstr. 1
D-91054 Erlangen
E-mail: vrpeckha@phil.uni-erlangen.de
URL:
http://www.phil.uni-erlangen.de/~p1phil/personen/peckhaus/home.html
********************************************************************
------------------------------
Date: Tue, 18 Apr 2000 18:46:09 +0200 (MEST)
From: Walter Felscher
Subject: Re: [HM] Kurt Grelling
Looking for an answer to Mr. Dawson's question from yesterday, it soon
becomes clear that it is not among mathematicians but rather among
positivist philosophers where some information about Kurt Grelling can
be found.
"The internet encyclopedia of Philosophy" in its article 'Logical
Positivism', www.utm.edu/research/ieo/l/logpos.htm, contains the
following short biography:
Kurt Grelling (1886 - ?). Logician and philosopher. He was a victim
of Nazist persecution and it is supposed that he died with his wife in
Auschwitz concentration camp during 1942, although it has been also
reported that Grelling was killed in 1941 at the border between France
and Spain while he was trying to escape in Spain. Hempel remembers
that Oppenheim made every effort to allow Grelling to immigrate in USA
but -- according to Hempel -- immigration officials were perplexed
by an alleged Grelling's propensity towards Communism; so there was
a delay that was fatal to Grelling, who was captured in France and
killed in a Polish concentration camp. The episode is reported in
Hempel, 'Autobiografia intellettuale' in Oltre il positivismo logico,
Armando : Rome, 1988 (this essay is the text of an interview Hempel
gave to Richard Noland in 1982, published for the first time in
Italian translation in 1988). Grelling was a teacher in secondary
school and was interested in logical problems. A semantic paradox is
named after him, the Grelling's paradox, formulated in 1908 by Grelling
and Leonard Nelson. There are some words which have the property they
express; for example 'short' is short. Those words are called
autological. The other words are called heterological; for example,
'long' is an heterological word -- it is not long. Now the question
is whether 'heterological' is heterological. If yes, then 'heterological'
is by definition an autological word and thus it is not heterological.
If no, then 'heterological' has the property it designate and therefore
it is heterological. Thus, 'heterological' is heterological if and only
if it is not heterological.
Grelling collaborated with Go"del and in 1936 he published an article
in which he defended Go"del's theorem of incompleteness against an
erroneous interpretation, according to which Go"del's theorem is indeed
a paradox like Russell's paradox ('Gibt es eine Go"delsche Antinomie?'
in Theoria, 3, 1936). Grelling was also interested in the analysis
of scientific explanation and in Gestalt approach.
Another informative piece of work is Dr.Peckhaus' complete bibliography
whose online address he supplied us with yesterday (www.phil.uni-erlangen.de/
~p1phil/personen/peckhaus/home.html) . From that it becomes clear
that during the years 1911-1922 Grelling published exclusively
journalistic articles in publications connected with the German Social
Democratic Party, from 1924 onwards his publications were exclusively
in the field of positivist philosophy. At the same website, Dr.Peckhaus'
article "Kantianer oder Neukantianer" contains some information about
Jakob Friedrich Fries and how his name gave rise to that of the
"Fries'sche Schule" of Grelling's coauthor Leonard Nelson.
W.F.
------------------------------
Date: Tue, 18 Apr 2000 21:44:46 +0300 (EET DST)
From: xpolakis@otenet.gr (Antreas P. Hatzipolakis)
Subject: Re: [HM] Kurt Grelling
Walter Felscher wrote (in part):
> "The internet encyclopedia of Philosophy" in its article 'Logical
> Positivism', www.utm.edu/research/ieo/l/logpos.htm, contains the
> following short biography:
The URL of Mauro Murzi's essay : _Logical Positivism_. The Internet
Encyclopedia of Philosophy, is:
http://www.utm.edu/research/iep/l/logpos.htm
More on-line documents related to Grelling can be found by a search via the
NOESIS Philosophy Search Engine:
http://noesis.evansville.edu/bin/index.cgi
> Kurt Grelling (1886 - ?). Logician and philosopher. He was a victim
> of Nazist persecution and it is supposed that he died with his wife in
> Auschwitz concentration camp during 1942, although it has been also
> reported that Grelling was killed in 1941 at the border between France
> and Spain while he was trying to escape in Spain.
An interesting historical article on mathematicians in Hitler Germany
was published in _Topology Atlas_
Here is a quote:
The last item is the List of German-Speaking Mathematicians Murdered or
Driven into Suicide by the Nazis. There are 14 of them: Ludwig Berwald,
Otto Blumenthal, Ludwig Eckart (suicide), Paul Epstein (suicide), Walter
Froehlich, Kurt Grelling, Gerhard Haenzel (suicide), Fritz Hartogs (suicide),
Felix Hausdorff (suicide), Margarete Kahn, Nelli Neumann (first wife of
Courant), Georg Pick, Robert Remak, Alfred Tauber. Nobody, no organization
was prepared and capable enough to save them from this terminal blow.
Terror and Exile and a Letter About it by Michael Golomb
http://at.yorku.ca/t/o/p/c/71.htm
Antreas
------------------------------
Date: Tue, 18 Apr 2000 17:26:33 -0300
From: Julio Gonzalez Cabillon
Subject: [HM] Mathematics in Latvia Through the Centuries
Dear List,
I thought it might be worth taking a look at the web page:
http://www.math.cornell.edu/~dtaimina/mathinlv.html
in which Ingrida Henina & listmember Daina Taimina have uploaded a
*draft* of their very informative paper on the history of mathematics
in Latvia.
Cheers,
Julio GC
------------------------------
Date: Tue, 18 Apr 2000 17:56:20 -0300
From: Julio Gonzalez Cabillon
Subject: Re: [HM] historiography of mathematics
At 08:52 a.m. 08/04/00 -0700, David Stump typed:
|
| It seems to me that the following discussion raises a general
| and very important question, namely how do we as historians
| determine the "major trends" of mathematics.
| Do we look at research, at teaching in graduate schools,
| undergraduate schools, mathematical associations, journals,
| all of the above? What is mathematics, in the phrase 'history
| of mathematics'?
|
Dear List,
Am I right to believe that you have nothing to say on this
important question? ... Or is it just a trivial question? ...
Cheers, Julio GC
------------------------------
Date: Wed, 19 Apr 2000 00:00:41 EDT
From: Bill Everdell
Subject: Re: [HM] Early Set Theory Texts
Thanks to Walter Felscher for reminding us of Schoenflies and his
"Die Entwicklung der Lehre von den Punktmannigfaltigkeiten I" in
Jahresbericht der DMV 8 (1900), 2tes Heft, 1-250. It looked to me
like an extremely important text for historians even before I found
Felscher's own remarkable account of the early history of set
theory in _Naive Mengen und Abstrakte Zahlen_ (1979) (and Herbert
Mehrtens's _Moderne Sprache Mathematik_); but now, with the book
long since returned to the library, I am wondering how I came to
record Schoenflies's first name as Arthur instead of Adolf. Are
they the same? And is the Schoenflies who wrote this >>erste
grosse Bericht u"ber die Mengenlehre<< (Mehrtens) the same as the
Schoenflies in whose contributions to set theory (and to topology)
L.E.J. Brouwer found "mistakes," (as Jeremy Gray reported in
Historia-Matematica on 20 December 1999)?
Bill Everdell, Brooklyn
------------------------------
End of HISTORIA MATEMATICA V2 #22
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HISTORIA MATEMATICA Saturday, April 22 2000 Volume 02 : Number 025
~~~~~~~~~~~~~~~~~~~~~~~~~~ TABLE of SUBJECTS ~~~~~~~~~~~~~~~~~~~~~~~~~~
Re: [HM] Truth tables
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Date: Sat, 22 Apr 2000 00:21:07 EDT
From: Bill Everdell
Subject: Re: [HM] Truth tables
John Conway wrote:
<>
I have not yet read McColl's series of papers on logic, but they do indeed
date from before 1900. My reference:
Hugh McColl, "The Calculus of Equivalent Statements I" in Proceedings of the
London Mathematical Society 9(1877) 9-20.
____ "The Calculus of Equivalent Statements II & III" in Proceedings of the
LMS 9(1878) & 10(1878)
____ "The Calculus of Equivalent Statements IV" in Proceedings of the LMS
(1880)
Does John Conway know if Mc(Mac)Coll's "bucket" notation comes from here?
Charles Sanders Peirce referenced McColl in 1880 in one of his own key papers
on logic and notation that followed on his "Logic of Relatives" (1870) and
included a definition of finite set:
Peirce, "On the Algebra of Logic: A Contribution to the Philosophy of
Notation" in American Journal of Mathematics 3(1880): 15-57 (Peirce,
Writings, Chronological edition, 4:163-209).
Peirce's notation (e.g. p-