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From: David Bernier on 29 Jan 2010 05:12
Chip Eastham wrote:
> On Jan 28, 11:44 am, "Dave L. Renfro" <renfr...(a)cmich.edu> wrote:
>> Gottfried Helms wrote:
>>> It is a very unusual word (to say the least). I suspect,
>>> he constructed it from some property, which is obscure
>>> to me so far. Do you have an online reference (for instance
>>> digitized in some math-journal-archive) or at least
>>> an abstract?
>> Below are some on-line references. I also have a math
>> question further down below for those more interested
>> in math than translation issues.
>>
>> According to a google word search, "Weitenbehaftungen"
>> appears in the following 1902 book of his 67 times:
>>
>> "Die grunds�tze und das wesen des unendlichen in der
>> mathematik und philosophie", 1902http://books.google.com/books?id=lBcQAAAAYAAJ
>>
>> See also:
>>
>> pp. 583-584 of "Kant-Studien", Volume 10, 1905http://books.google.com/books?id=PNhDAAAAIAAJ&pg=PA583
>>
>> p. 614 of "The journal of philosophy, psychology and scientific
>> methods", Volume 2, 1905http://books.google.com/books?id=p8MGXJGHrAgC&pg=PA614
>>
>> pp. 65-74 [= 181-190 of google file] of "Lehrproben und Lehrg�nge
>> f�r die Praxis der Schulen", Volume 22, 1906http://books.google.com/books?id=pGIVAAAAIAAJ&pg=PA181
>>
>> For more, on the very small chance anyone wants more,
>> this search brings up many others:
>>
>> http://books.google.com/books?q=Kurt+Geissler+Weitenbehaftungen&as_brr=1
>>
>> I'm also interested in a rough idea of what kind of
>> mathematical idea/analysis he's carrying out on the
>> last page of his paper
>>
>> "Die Asymptote und die Weitenbehaftungen", Zeitschrift f�r
>> Mathematischen und Naturwissenschaftlichen Unterricht 34
>> (1903), 313-324.http://books.google.com/books?id=L7IWAQAAIAAJ&pg=PA324
>>
>> Dave L. Renfro
>
> After a bit of poking around in the book Die grundsatze
> und das wesen des unendlichen in der mathematik und
> philosophie (roughly, The foundations and essence of
> infinity in mathematics and philosphy), I suspect that
> the "prefix" Weiten (width) is being used in a sense
> of space. For instance, there's a dialog starting
> on p. 335 of the book called "The concept of Behaftung
> as an interim solution to the problem", cast as a
> dialog "with a young man", in which appears the phrase
> "zwei r�umliche Behaftungen" (p. 340). The adjective
> r�umliche means spatial (or space-like). I also saw
> the word Weitenbehaftung paired with area or territory
> in some places in the book.
>
> The following section of the book (after dialog with a
> young man) is called Do the Behaftungen lead back to
> the old difficulties? (p. 344). So I submit that the
> crux of the mystery is what Geissler meant by Behaftung.
>
> In addition to stickiness/adhesion, haftung can mean
> liability or legal responsibility. The fact that Geissler
> does so much "work" trying to elucidate the concept is
> something of a clue as to an esoteric meaning.

I retrieved the attached *.doc file that Dave Renfro left at
the Math Forum here:
< http://mathforum.org/kb/message.jspa?messageID=6960240&tstart=0 >

In the reply where he begins with:
"I'm attaching to this post (made at Math Forum) what
hopefully will be a digital file [...]"

Geissler speaks at length about how the tangents to the part of
the hyperbola y^2 - x^2 = 1 (say) in the 1st Quadrant
"approach" the asymptote y - x = 0 as x, y -> +oo
in that
(a) the distance from (x, sqrt(x^2 + 1)) to the asymptote y-x = 0
tends to 0 as x -> +oo ,
(b) the angle theta(x) between the tangent to the hyperbola at
(x, sqrt(x^2 + 1)) and the asymptote y-x = 0 has as
property: lim_{x -> +oo} theta(x) = 0.


From looking at:
< http://de.wikipedia.org/wiki/Grenzwert_(Funktion) >
and using Babelfish,

it seems "Grenzwert" is limit-value or limit, and Grenz means
border, "frontiere" in French, close in meaning to "boundary".

_Grenz_ appears very often in Geissler's writing in the
*.doc file.

If we go outside the accepted treatment of limits in real analysis, one
could ask:
"Is it true that as as x -> +oo in the above
[following-up on (a) and (b)], the limit of the tangent to
the hyperbola at (x, sqrt(x^2 + 1)) *is* the asymptote,
i.e. y-x = 0 ?"

I'm not familiar with a contemporary (elementary) notion of
limit of parameterized lines. You mention the German
"haftung" as sometimes meaning stickiness/adhesion.

Topologically in R^2, I think it's right that
h = { (x, y) s.t. y^2 - x^2 = 1} is closed,
ell = { (x, y) s.t. y-x = 0} is closed,

h /\ ell = {0}, and that h and ell have disjoint boundaries.


If we think of a similar set-up in the projective real plane,
adding a point at infinity, actually using the Riemann sphere
and the stereographic projection, then it seems clear that
adding the North pole to both figures makes sense projectively
and then the projection of the hyperbola and of the asymptotes
meet at the point at infinity, which is the North pole.

This suggests "stickiness" between h and ell, which is
false in the Euclidean plane, but in some sense true
if we think of the projective plane and adding a
point at infinity to h and ell.

With respect to (Spatial) and Behaftungen, it sure seems
that, as you say, the crux of the mystery is what Geissler
meant by Behaftung, and that it could be esoteric ...

Regards,

David Bernier
From: Rob Johnson on 29 Jan 2010 10:36
In article <9950f785-1a3c-4548-8cc6-f6e716ea65d5(a)a5g2000yqi.googlegroups.com>,
"Dave L. Renfro" <renfr1dl(a)cmich.edu> wrote:
>I am seeking an English word or phrase equivalent for
>the German word "Weitenbehaftungen" that was used by
>the German mathematician/philosopher
>
>Friedrich Jacob [Jakob] Kurt Geissler [Gei=DFler] (1859-1941)
>
>I am not particularly interested in his theories on infinity
>and infinitesimals (which were widely criticized around 1903
>through the 1910s on several grounds, one of which was his lack
>of knowledge of then modern mathematical developments that
>related to his work), but rather I simply want to come up with
>a reasonable English version for the title of one of his papers:
>
>Die Asymptote und die Weitenbehaftungen

I don't know if this helps, but in

<http://mind.oxfordjournals.org/cgi/reprint/XVI/63/468-a.pdf>

I found this quote:

>Kurt Geissler. 'Das Willensproblem : Historische
>Ubersicht u. Darstellung durch Weitenbehaftungen.' [Twenty-two pages
>of historical survey. Fourteen pages devoted to show the importance
>for this problem of certain concepts of the higher mathematics. Concludes
>that the human will may be really free through qualities accruing to it
>from its connexion with a higher order (Weitenbehaftungen).]

Rob Johnson <rob(a)trash.whim.org>
take out the trash before replying
to view any ASCII art, display article in a monospaced font
From: Ask me about System Design on 29 Jan 2010 17:43
On Jan 28, 6:53 am, "Dave L. Renfro" <renfr...(a)cmich.edu> wrote:
> I am seeking an English word or phrase equivalent for
> the German word "Weitenbehaftungen" that was used by
> the German mathematician/philosopher
>
> Friedrich Jacob [Jakob] Kurt Geissler [Geißler] (1859-1941)
>
> I am not particularly interested in his theories on infinity
> and infinitesimals (which were widely criticized around 1903
> through the 1910s on several grounds, one of which was his lack
> of knowledge of then modern mathematical developments that
> related to his work), but rather I simply want to come up with
> a reasonable English version for the title of one of his papers:
>
> Die Asymptote und die Weitenbehaftungen
>
> Thanks,
>
> Dave L. Renfro

In contrast to the informed, researched answers posted
elsewhere in this thread, I give an intuitionistic
spark. The notion of equivalence class struck me while
reading the thread, so there might be some notion of
identification happening from some infinite process,
this identification being subtle and hard to obtain.

(I use intuitionistic in manner possibly similar to
that in which Geissler uses Behaftung; as a suggestive
but made-up term; I do not intend intuitionistic to be
interpreted mathematically, however.)

Gerhard "Ask Me About System Design" Paseman, 2010.01.28
From: Chip Eastham on 29 Jan 2010 19:21
On Jan 29, 5:12 am, David Bernier <david...(a)videotron.ca> wrote:
> Chip Eastham wrote:
> > On Jan 28, 11:44 am, "Dave L. Renfro" <renfr...(a)cmich.edu> wrote:
> >> Gottfried Helms wrote:
> >>> It is a very unusual word (to say the least). I suspect,
> >>> he constructed it from some property, which is obscure
> >>> to me so far. Do you have an online reference (for instance
> >>> digitized in some math-journal-archive) or at least
> >>> an abstract?
> >> Below are some on-line references. I also have a math
> >> question further down below for those more interested
> >> in math than translation issues.
>
> >> According to a google word search, "Weitenbehaftungen"
> >> appears in the following 1902 book of his 67 times:
>
> >> "Die grundsätze und das wesen des unendlichen in der
> >> mathematik und philosophie", 1902http://books.google.com/books?id=lBcQAAAAYAAJ
>
> >> See also:
>
> >> pp. 583-584 of "Kant-Studien", Volume 10, 1905http://books.google.com/books?id=PNhDAAAAIAAJ&pg=PA583
>
> >> p. 614 of "The journal of philosophy, psychology and scientific
> >> methods", Volume 2, 1905http://books.google.com/books?id=p8MGXJGHrAgC&pg=PA614
>
> >> pp. 65-74 [= 181-190 of google file] of "Lehrproben und Lehrgänge
> >> für die Praxis der Schulen", Volume 22, 1906http://books.google.com/books?id=pGIVAAAAIAAJ&pg=PA181
>
> >> For more, on the very small chance anyone wants more,
> >> this search brings up many others:
>
> >>http://books.google.com/books?q=Kurt+Geissler+Weitenbehaftungen&as_brr=1
>
> >> I'm also interested in a rough idea of what kind of
> >> mathematical idea/analysis he's carrying out on the
> >> last page of his paper
>
> >> "Die Asymptote und die Weitenbehaftungen", Zeitschrift für
> >> Mathematischen und Naturwissenschaftlichen Unterricht 34
> >> (1903), 313-324.http://books.google.com/books?id=L7IWAQAAIAAJ&pg=PA324
>
> >> Dave L. Renfro
>
> > After a bit of poking around in the book Die grundsatze
> > und das wesen des unendlichen in der mathematik und
> > philosophie (roughly, The foundations and essence of
> > infinity in mathematics and philosphy), I suspect that
> > the "prefix" Weiten (width) is being used in a sense
> > of space.  For instance, there's a dialog starting
> > on p. 335 of the book called "The concept of Behaftung
> > as an interim solution to the problem", cast as a
> > dialog "with a young man", in which appears the phrase
> > "zwei räumliche Behaftungen" (p. 340).  The adjective
> > räumliche means spatial (or space-like).  I also saw
> > the word Weitenbehaftung paired with area or territory
> > in some places in the book.
>
> > The following section of the book (after dialog with a
> > young man) is called Do the Behaftungen lead back to
> > the old difficulties? (p. 344).  So I submit that the
> > crux of the mystery is what Geissler meant by Behaftung.
>
> > In addition to stickiness/adhesion, haftung can mean
> > liability or legal responsibility.  The fact that Geissler
> > does so much "work" trying to elucidate the concept is
> > something of a clue as to an esoteric meaning.
>
> I retrieved the attached *.doc file that Dave Renfro left at
> the Math Forum here:
> <http://mathforum.org/kb/message.jspa?messageID=6960240&tstart=0>
>
> In the reply where he begins with:
> "I'm attaching to this post (made at Math Forum) what
>   hopefully will be a digital file [...]"
>
> Geissler speaks at length about how the tangents to the part of
> the hyperbola  y^2 - x^2 = 1 (say)  in the 1st Quadrant
> "approach" the asymptote  y - x = 0  as x, y -> +oo
> in that
> (a) the distance from (x, sqrt(x^2 + 1)) to the asymptote  y-x = 0
>       tends to 0 as x -> +oo ,
> (b) the angle theta(x) between the tangent to the hyperbola at
>      (x, sqrt(x^2 + 1)) and the asymptote y-x = 0  has as
>      property:  lim_{x -> +oo} theta(x) = 0.
>
>  From looking at:
> <http://de.wikipedia.org/wiki/Grenzwert_(Funktion) >
> and using Babelfish,
>
> it seems "Grenzwert" is limit-value or limit, and Grenz means
> border, "frontiere"  in French, close in meaning to "boundary".
>
> _Grenz_ appears very often in Geissler's writing in the
> *.doc file.
>
> If we go outside the accepted treatment of limits in real analysis, one
> could ask:
> "Is it true that as as x -> +oo in the above
> [following-up on (a) and (b)], the limit of the tangent to
> the hyperbola at  (x, sqrt(x^2 + 1)) *is* the asymptote,
> i.e. y-x = 0 ?"
>
> I'm not familiar with a contemporary (elementary) notion of
> limit of parameterized lines. You mention the German
> "haftung" as sometimes meaning stickiness/adhesion.
>
> Topologically in R^2, I think it's right that
> h = { (x, y) s.t. y^2 - x^2 = 1} is closed,
> ell = { (x, y) s.t. y-x = 0} is closed,
>
> h /\ ell = {0}, and that h and ell have disjoint boundaries.
>
> If we think of a similar set-up in the projective real  plane,
> adding a point at infinity, actually using the Riemann sphere
> and the stereographic projection, then it seems clear that
> adding the North pole to both figures makes sense projectively
> and then the projection of the hyperbola and  of the asymptotes
> meet at the point at infinity, which is the North pole.
>
> This suggests "stickiness" between h and ell, which is
> false in the Euclidean plane, but in some sense true
> if we think of the projective plane and adding a
> point at infinity to h and ell.
>
> With respect to (Spatial) and Behaftungen, it sure seems
> that, as you say, the crux of the mystery is what Geissler
> meant by Behaftung, and that it could be esoteric ...
>
> Regards,
>
> David Bernier

Behaftung does seem in the majority of passages
where it occurs to connect with some limiting
process, as in the "dialog with a young man"
where the opening seems to concern a secant
line approaching a tangent line, forming some
triangles whose "Behaftungen" are related to
finite and infinitesimally small angles.

Much earlier in the book (maybe p. 50 or so)
there is a passage which seems to "justify"
the "reinvention" of a concept (Weitenbehaftung)
for his philosophical purpose at hand. So I
assume the use of the word was expected to
pose something of a mystery even to his
contemporaneous readers.

My initial suspicion in looking through the
passages (of "Die grundsätze und das wesen des
unendlichen in der mathematik und philosophie")
was that Behaftung would turn out to mean a
kind of "intrinsic" attribute. My German
reading comprehension is like looking through
a keyhole at the sweep of Geissler's thought,
but the relatively concise statements about
Behaftung which I could follow suggest it
has more to do with limits, perhaps "clustering"
or accumulation points as we say in topology.

regards, chip
From: Chip Eastham on 29 Jan 2010 19:57
On Jan 28, 9:53 am, "Dave L. Renfro" <renfr...(a)cmich.edu> wrote:
> I am seeking an English word or phrase equivalent for
> the German word "Weitenbehaftungen" that was used by
> the German mathematician/philosopher
>
> Friedrich Jacob [Jakob] Kurt Geissler [Geißler] (1859-1941)
>
> I am not particularly interested in his theories on infinity
> and infinitesimals (which were widely criticized around 1903
> through the 1910s on several grounds, one of which was his lack
> of knowledge of then modern mathematical developments that
> related to his work), but rather I simply want to come up with
> a reasonable English version for the title of one of his papers:
>
> Die Asymptote und die Weitenbehaftungen
>
> Thanks,
>
> Dave L. Renfro

Best guess (see discussion elsewhere on thread):

Asymptotes and Spatial Clusterings

[Obscure, yes, but serviceable?]

regards, chip
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