A few weeks ago I exchanged some emails with Stanislaw Domoradzki, and
especially with Walter (Felscher), and I thought it would be fine to air
this correspondence on the [HM] list for the benefit of all interested
readers.
Cheers, Julio
----------------------------------------------------------------------
Date: Sat, 02 Oct 1999 19:53:30 -0200
To: Walter Felscher <walter.felscher@uni-tuebingen.de>
From: Julio Gonzalez Cabillon <jgc@adinet.com.uy>
Subject: Re: on Hilbert's papers
Dear Walter,
Many thanks for your most informative email. When I submitted my original
post to [HM] I suspected that there was much more than the eye can meet
- and for that reason I wrote: "I append below some __quick__ comments"...
I believe it might be worth sending your reply to the forum, appended with
your original private query (which would provide the right context),
Mertens' reference and, perhaps, Hilbert's review. What do you think?
By the way, if I am remembering things correctly, Mertens wrote other
articles on *this* subject. I should look that up.
Un saludo cordial,
Julio
...
----------------------------------------------------------------------
Date: Mon, 25 Oct 1999 17:24:08 +0200 (MEST)
From: Walter Felscher <walter.felscher@uni-tuebingen.de>
To: Julio Gonzalez Cabillon <jgc@adinet.com.uy>
Subject: Re: on Hilbert's papers
Dear Julio,
thank you for your letter from October 2nd. Reading it only now, after
the return from my travels, I quite agree to your proposition
> I believe it might be worth sending your reply to the forum,
> appended with your original private query (which would provide
> the right context), Mertens' reference and, perhaps, Hilbert's
> review. What do you think?
- provided you suppose that now, after three weeks, the readers still
recall that topic of discussion. If not, we should preserve our
communications until it comes up again.
...
With best wishes
your
Walter
----------------------------------------------------------------------
Since I (like Walter) am not quite sure whether most listmembers still
recall the thread under discussion I include my shooting posting right
below:
Date: Thu, 23 Sep 1999 04:48:37 -0300
To: xpolakis@otenet.gr (Antreas P. Hatzipolakis)
From: Julio Gonzalez Cabillon <jgc@adinet.com.uy>
Subject: Re: [HM] Mertens Theorem
Cc: historia-matematica@chasque.apc.org
Re: Does anyone know who Mertens was?
Estimado Antreas:
I append below some __quick__ comments. I hope others will join in
providing thoughtful remarks on Mertens. With a few notable exceptions,
many subscribers of [HM] seem to have forgotten, alas, what interesting
and informative and helpful an international 700-forum might/should be.
Of course, I warmly agree with a distinguished listmember when she remarks
[private mail]: "I would prefer if people would confine themselves to
things about which they have thought deeply."
_WE_ are the forum, dear colleagues! And yes, I know, we ALL are _VERY_
busy. None the less ... I urge some __SCHOLARS__ to take a more active
attitude towards [HM] - i.e. I would be very pleased if they submit just
one posting (say) every six months ;-)
Mertens is the well-known number theorist Franciszek Mertens who was born
on March 20, 1840, in Schroda, Posen (a former Prussian province, now
Sroda, in Poland), and died in Vienna, on March 5, 1927.
As we all know, Gauss died in 1855, and that very year Dirichlet, who had
been teaching at the University of Berlin since 1828, went to Go"ttingen
to succeed Gauss. Therefore, Dirichlet's chair became vacant in 1855,
and Kummer came to Berlin to be appointed professor, a position he held
until he retired in 1883. As we also know, Kronecker was a student of
Kummer, and it seems apparent that it was due to him that Kronecker became
interested in number theory.
In his youth, Mertens moved to Berlin where he became a student at Berlin
University, and where he studied under Kronecker and Kummer.
I find somehow unfair to state, as I've read, that "Mertens was a number
theorist who is best remembered for his elementary proof of the Dirichlet
theorem which appears in most modern textbooks". As a matter of fact,
Franz Mertens published a long string of important papers on many topics.
For instance, this is the very Mertens who, in 1886, re-proved Gordan's
theorem for binary systems by an inductive method.
According to Morris Kline [MT from A to MT, p.929], Mertens
"assumed the theorem to be true for any given set of binary forms and
then proved it must still be true when the degree of one of the forms
is increased by one. He did not exhibit explicitly the finite set of
independent invariants and covariants but he proved that it existed.
The simplest case, a linear form, was the starting point of the
induction and such a form has only powers of itself as covariants.
"Hilbert, after writing a doctoral thesis in 1885 on invariants, in
1888 also re-proved Gordan's theorem that any given system of binary
forms has a finite complete system of invariants and covariants. His
proof was a modification of Mertens's. Both proofs were far simpler
than Gordan's. But Hilbert's proof also did not present a process for
finding the complete system.
...
"Hilbert's existence proof was so much simpler than Gordan's laborious
calculation of a basis that Gordan could not help exclaiming, 'This is
not mathematics; it is theology'. However, he reconsidered the matter
and said later, 'I have convinced myself that theology also has its
advantages'. In fact he himself simplified Hilbert's existence proof."
[1]
As an anecdote, I recall that on October 21, 1881, Heine died, and a
replacement was needed to fill the chair at Halle. Cantor thought of
three mathematicians for the new available position. Dedekind was first
in Cantor's list, followed by Heinrich Weber, and Franz Mertens. But, as
it seems, in 1882 Cantor got surprised when Dedekind declined the offer,
and the shock was even worse when Weber, and also Mertens declined.
Franz Mertens first worked in Cracow, and then moved to Austria.
Incidentally, Ernst Fischer and Schro"dinger, for instance, were
students of Mertens at the University of Vienna.
[1] Nachrichten Ko"nig. Gesellschaft der Wissenschaften zu Go"ttingen,
1899, pp 240-242.
[2] Dick, Auguste:
"Franz Mertens: 1840-1927: eine biographische Studie", mit einer Einl.
von Edmund Hlawka, Graz: Berichte der mathematisch-statistischen Sektion
im Forschungszentrum Nr. 151, 38 pages, 1981.
[3] te Riele, Herman J.J.:
"Some Historical and Other Notes about the Mertens Conjecture and its
Recent Disproof, _Nieuw Archief voor Wiskunde_ Vierde Serie, vol 3,
no 2, pp 237-243, 1985.
This article answers some questions about the disproof of the Mertens
conjecture by A.M. Odlyzko and H.J.J. te Riele [cf. "Disproof of the
Mertens conjecture", _J. Reine Angew. Math._ vol 357, pp 138-160,
1985]. This conjecture (1897) attracted lots of interest in its almost
100 years of existence because its truth would imply the truth of the
Riemann hypothesis. The disproof relies on extensive computations with
the zeros of the zeta-function, and does not provide an explicit
counterexample. In his paper [2], the roles of Stieltjes and Mertens
are sketched in their historical perspective.
[4] Domoradzki, Stanislaw:
"Franciszek Mertens (1840-1927)" (in Polish. English summary),
_Opuscula Mathematica_ vol 13, pp 109-115, 1993.
Springtime has arrived, here!
Julio Gonzalez Cabillon
With best regards, Julio
----------------------------------------------------------------------
Date: Thu, 07 Oct 1999 13:14:25 +0200
From: Stanislaw Domoradzki <domoradz@atena.univ.rzeszow.pl>
To: Julio Gonzalez Cabillon <jgc@adinet.com.uy>
Subject: Re: Franciszek Mertens (1840-1927)
Dear Professor Julio Gonzalez Cabillon,
Thank you very much for your letter.
I would like to inform you that:
in the period when the Jagiellonian University was officially German
(under rule of Austrian authorities) Mertens was registered with the
name Franz (which corresponds to the Polish name Franciszek). He signed
Polish documents without a name: just "Dr Mertens". Also I would like
to draw your attention that he signed his Polish publications (he wrote
about 20) with the name Franciszek, even in the period when he worked
in Graz and Vienna. Thus in Poland, Mertens remained with the name
Franciszek (he worked 20 years, he merited a lot for Polish mathematics
and the Jagiellonian University) in the rest of the world he was know
(in particular in Austria) with the name Franz.
I have not seen his birth certificate. I guess that he was registered
there as Franz, because Sroda where he was born belonged to Germany at
that time.
It would be very important if you could state that, in Poland, he is
known as Franciszek.
Sincerely yours
Stanislaw Domoradzki
----------------------------------------------------------------------
Date: Sun, 26 Sep 1999 15:44:57 +0200 (MEST)
From: Walter Felscher <walter.felscher@uni-tuebingen.de>
To: Julio Gonzalez Cabillon <jgc@adinet.com.uy>
Subject: Re: [HM] Mertens Theorem
Dear Julio,
can you give me the location at which Mertens published his proof
of which you write
> For instance, this is the very Mertens who, in 1886, re-proved
> Gordan's theorem for binary systems by an inductive method.
>
> Hilbert, after writing a doctoral thesis in 1885 on invariants,
> in 1888 also re-proved Gordan's theorem that any given system
> of binary forms has a finite complete system of invariants and
> covariants. His proof was a modification of Mertens's.
I want to look at Mertens' proof since, as I remarked on FOM [*],
Hilbert's proof appeared to me as the first one to use conceptual
methods which we now call set-theoretical.
Best wishes
your
W.F.
[*] For handy reference I append the FOM message at the end of this
"fat" posting, JGC.
----------------------------------------------------------------------
Date: Sun, 26 Sep 1999 14:12:36 -0200
To: Walter Felscher <walter.felscher@uni-tuebingen.de>
From: Julio Gonzalez Cabillon <jgc@adinet.com.uy>
Subject: Re: [HM] Mertens Theorem
Dear Walter,
The precise reference is:
Mertens, F.:
"Beweis, dass alle Invarianten und Covarianten eines Systems bina"rer
Formen ganze Functionen einer endlichen Anzahl von Gebilden dieser Art
sind" (1886), Journal de Crelle, vol 100 (1887), pp. 223-230.
Kind regards,
Julio
----------------------------------------------------------------------
Date: Sun, 26 Sep 1999 14:51:17 -0200
To: Walter Felscher <walter.felscher@uni-tuebingen.de>
From: Julio Gonzalez Cabillon <jgc@adinet.com.uy>
Subject: Hilbert's review
Dear Walter,
If you have the time and the energy it would be nice if you report
back what transpires after reading Mertens' paper. It is a long
time since I put my hands on the original sources.
For your information I append Hilbert's own review (1886) of Mertens
paper "Beweis, dass alle Invarianten und ..." for _Jahrbuch u"ber
die Fortschritte der Mathematik_:
Giebt es fu"r ein System von bina"ren Grundformen f, f', f", ... eine
endliche Anzahl von Invarianten und Covarianten, durch welche sich
alle anderen Invarianten und Covarianten in gazer und rationaler
Weise ausdru"cken lassen, so gilt, wie man ohne Schwierigkeit
erkennt, das Gleiche auch fu"r dasjenige System von Grundformen,
welches entsteht, wenn man den Formen f, f', f", ... noch eine
Linearform p hinzufu"gt. Es wird nun durch Lo"sung einer diophantischen
Gleichung bewiesen, dass es stets moeglich ist, aus den Invarianten
und Covarianten des Formensystems f, f', f", ... in ganzer und
rationaler Weise ein endliches System von solchen besonderen
Invarianten und Covarianten zusammenzusetzen, welche in den
Coefficienten der Grundform f und in denjenigen der Linearform
p von gleichem Grade sind und welche u"berdies insofern ein in sich
abgeschlossenes System bilden, als durch dieselben eine jede
Invariante oder Covariante von gleicher Eigenschaft ganz und rational
dargestellt werden kann. Setzt man schliesslich fu"r die Form f von
der n^ten Ordnung das Product qr...s ihrer n Linearfactoren ein
und vertauscht die Linearform p successive mit q, r, ... s, so
la"sst sich durch geeignete Combination der entstehenden Bildungen
fu"r das Grundformensystem f', f", ... , p, q, r, ..., s ein
vollsta"ndiger Bestand von solchen besonderen Invarianten und
Covarianten ableiten, welche symmetrisch sind in Bezug auf die
Coefficienten von p, q, r, ..., s durch die Coefficienten des
Productes g=pqr...s zu ersetzen; es ergiebt sich dann ein
vollsta"ndiger Bestand von Invarianten und Covarianten fu"r das
Formensystem g, f', f", ..., d. i. fu"r ein Formensystem, welches
entsteht, wenn man in dem urspru"nglichen Formensystem an Stelle
der Grundform f von der n^ten Ordnung eine Form g von der
(n+1)^ten Ordnung einsetzt. Indem man nun von einem nur aus
Linearformen bestehenden Grundfomensysteme ausgeht, erkennt man in
Obigem einen allgemeinen und von den Hu"lfsmitteln der
Symbolik unabha"ngigen Beweis fu"r den Gordan'schen Fundamentalsatz.
Un fuerte abrazo, Julio GC
----------------------------------------------------------------------
Date: Thu, 30 Sep 1999 17:31:59 +0200 (MEST)
From: Walter Felscher <walter.felscher@uni-tuebingen.de>
To: Julio Gonzalez Cabillon <jgc@adinet.com.uy>
Subject: on Hilbert's papers
Dear Julio,
I went to the mathematical library, and here is what I found.
There are two relevant articles by Hilbert, both reprinted in volume 1
of his collected papers:
"Uber die Endlichkeit des Invariantensystems f"ur bin"are Grundformen
Math.Ann. 33 (1889) 223-226 .
"Uber die Theorie der algebraischen Formen.
Math.Ann. 36 (1890) 473-534 .
1.
The content of the first article is a third proof of Gordan's
theorem
for any system M of binary forms [homogenous polynomials in n=2
variables], there is a finite set of invariants of M such that
every other such invariant is a linear combination of them with
integers as coefficients.
The first proof was Gordan's in his Vorlesungen "Uber
Invariantentheorie, vol.2 , 1885 , the second had been Mertens'
in Crelle 100 (1887) 223-230. In the second sentence of his article,
Hilbert acknowledges his dependence on these predecessors: "In what
follows, another proof for this fundamental theorem will be
given, which shows close analogies to Gordan's original method,
while on the other hand the line of thought proceeds parallel to
the proof of Mertens." As a further remark: I do not agree with
Kline's statement which you quote
But Hilbert's proof also did not present a process for finding
the complete system.
Hilbert explicitly describes the finite basis by the formulas
numbered (6) in his article, and Merten's proof as well contains
a recursive construction of the basis.
2.
Hilbert's second article contains, in the first of its four parts,
what is usually called 'Hilbert's finite basis theorem', HFBT, not
as one of its numbered theorems but, as the italicized statement
Aus einem jeden beliebig gegebenen Formensystem [in n
variables; to be called M below] l"asst sich stets eine
endliche Zahl von Formen derart ausw"ahlen, dass jede
andere Form des Systems durch lineare Kombination jener
ausgew"ahlten Formen erhalten werden kann.
Here the coefficients of the linear combination are forms in the
same variables, but not necessarily belonging to the given
system. For later use, I shall call the set of those selected
forms an F-basis of M .
Hilbert's proof proceeds in two steps. First he shows as Theorem I
Given an infinite sequence F_1, F_2,... of forms in n variables,
then there is a number m such that every form in this sequence
can be written as a linear combination of the first m forms in
this sequence with suitable other forms in n variables as
coefficients.
Its proof is by induction on n where the cases n=1 and n=2 are
shown explicitly, and the reduction of the case n to the case n-1
is carried out by an explicit calculation, This first step is
perfectly constructive.
The second step is highly inconstructive. The system M does not
need to be given in form a sequence, but is assumed be defined by a
decidable law. Assume HFBT to be wrong for M . Then
choose F_1 in M not identical zero,
choose F_2 in M not representable as a product A_1 F_1 for some
form A_1 in n variables,
choose F_3 in M not representable as A_1 F_1 + A_2 F_2 for
forms A_1 , A_2 in n variables,
and so on.
Then the sequence F_1, F_2, ... is infinite and such that none of
its members is a linear combination of the previous ones. This
contradicts Theorem 1.
This concludes Hilberts proof of HFBT ; it uses what in set
theory is called now the Axiom of Dependent Choices. It is this
proof, and its use for the following HFBIT, which provoked
Gordan's dictum about theology. (I should mention that later
proofs of HFBT avoid both the indirect argument and the use of
dependent choices.)
In the fourth part of his article, Hilbert proves as Theorem V
the finite basis theorem for invariants, HFBITn , stated by him as:
given a first system (System von Grundformen) H of forms
in n variables, together with a system of linear
transformations of the sets of variables occurring in
them, then there is finite number of integral and rational
invariants of M such that every other invariant is an integer
linear combination of them.
The proof begins with an application of HFBT to the system M of
invariants of H . This produces an F-basis of M such that every
member of M becomes a linear combination of elements of the
F-basis with suitable forms as coefficients. It then remains to
be shown that every such coefficient can be replaced by an
appropriate invariant without changing the value of the linear
combination; The case n=2 is handled with help of a theorem from
Hilbert's doctoral dissertation [also in his "Uber eine
Darstellungsweise der invarianten Gebilde im bin"aren Formengebiete,
Math.Ann.30 (1887) 15-29]; the case n=3 is treated with help of an
argument said to be essentially due to Gordan (Vorlesungen "uber
Invariantentheorie, vol.2 (1885) para 9), Clebsch and Mertens
("Uber invariante Gebilde tern"arer Formen. Sitzungsber.Akad.Wiss.
Wien, Math.-phys.Kl. 95 ]. It then is shown that the treatment
given for the case n=3 can be carried over to the general case.
3.
You continued your quotations from Kline by
Hilbert's existence proof was so much simpler than Gordan's
laborious calculation of a basis that Gordan could not help
exclaiming ...
This is quite true, but it should be kept in mind that your first
quotation from Kline referred to (Hilbert's proof of) Gordan's
HFBIT2 , while here Kline speaks of the set-theoretical proof of HFBT.
In so far, the comparition made by Kline here can easily mislead:
Gordan proved HFBIT2 and could not dream of attacking HFBITn . That
requiered HFBT (though not necessarily with an inconstructive proof
such as Hilbert's).
Returning now to Mertens, we can say that he was a predecessor for
Hilbert's proof of HFBIT2 in 1887 . But he was in no way a predecessor
for Hilbert's HFBT and its 'theological' reasoning.
Best regards
your
Walter
----------------------------------------------------------------------
To: Martin Davis <martin@eipye.com>
Subject: FOM: mathematics versus theology
From: Walter Felscher <walter.felscher@uni-tuebingen.de>
Date: Wed, 4 Aug 1999 22:23:17 +0200 (MEST)
cc: fom@math.psu.edu
Mr. Davis, in his "Re: FOM: history, funding etc." from August
3rd, quoted from a forthcoming book of his and mentioned the
German mathematician "Paul Gordon" which spelling he consistently
repeated nine times. But the man's name was
Paul [Albert] Gordan - with "a" instead of "o" ;
he was born in Breslau on April 27, 1837, and died in Erlangen on
December 21, 1912 . He never had completed the Gymnasium, yet
having attended a lecture of Kummer's in 1855/56 he chose to
study mathematics, and later held a professorship at Erlangen
from 1875 to 1910 . Usually an obvious misspelling is unimportant,
but recent experiences make me uncertain how obvious to American
readers such misspelling actually is.
Mr. Davis is quite right to observe that Gordan is well remembered
for his dictum "This is not mathematics, this is theology"
referring to Hilbert's first proof of his finite basis theorem
(Math. Ann. 36 (1890) 473 ff ). I do not have a source for this
dictum; it does not seem to appear in Gordan's writings, and so
it has the status of hearsay. It seems hopeless to ask which
additional explanations Gordan may have given in connection with
this dictum.
But it is not illegitimate to ask what Gordan may have MEANT with
it. Particularly since here we are concerned with foundations,
Mr. Davis' correct, but somewhat roundabout phrase about the
"power of abstract thought" used in Hilbert's proof, leaves open
the question in which form this power was actually employed.
Simply saying that Hilbert's proof was just sooo different from
what one was accustomed to, will not seem explain Gordan's
contrasting of mathematics versus theology.
The key, it appears, is Hilbert's description of his proof on p.
478 of the quoted article, viz.
Um ein solches Formensystem festzulegen, denke man sich ein
Gesetz gegeben, verm"oge dessen ausnahmslos f"ur eine jede
beliebig angenommene Form entschieden werden kann, ob sie dem
System zugeh"oren soll oder nicht. Wir nehmen nun an, es sei
nicht m"oglich, aus dem gegebenen Formensystem eine endliche
Zahl von Formen derart auszuw"ahlen, dass jede andere Form des
Systems durch lineare Combination jener ausgew"ahlten Formen
erhalten werden kann. Dann w"ahlen wir nach Willk"ur aus dem
System eine nicht identisch verschwindende Form aus und
bezeichnen dieselbe mit F1 ; ... Entsprechend sei F4 eine Form
des Systems, welche sich nicht in die Gestalt A1F1 + A2F2 +
A3F3 bringen l"asst. ...
So Hilbert starts from a system of forms, a set in our words,
described by some property which either holds for a form or does
not. Next he assumes that it is not the case that there is a finite
basis. Then he takes an F1 and continues, choosing (in view of
his assumption) F2, F3, ... such that Fn+1 is not in the span of
the previous Fi ...
So there is a set of forms, and if A(B,f) abbreviates that a set
B is a finite basis with f in its span, then from the assumption
that it is not the case
there is B : for every f : A(B,f)
Hilbert concludes
for every B : there is f : not A(B,f)
and then applies this successively to the B's he constructs one
after the other. Thus Hilbert forms the negation of a quantifier
ranging over sets.
Clearly, the habit to speak about sets originated with Dedekind
(rather than with Cantor), yet it is one thing to form infinite
sets of numbers and to give names to them, but quite another one
to consider the (2nd order) set of all sets B's AND to perform
quantifier-negating arguments on it which necessarily assumes it
to be a "completed" infinite totality.
Here, then, seems to be the place where for Gordan mathematics
touched theology.
Of course, there are also are infinitely many dependent choices
of the forms f occurring in Hilbert's argument. But if we recall
how often uses of AC at that time went unnoticed [e.g. when
'constructing' a convergent sequence, in a point set A, for an
epsilon-delta-adherence point of A ], then we may safely assume
that it was not the sequence of infinitely many dependent choices
that caused the concerns about an employment of 'theological'
methods: it was the use of completed 2nd order totalities,
accessible to the pure thought's acting when negating quantifiers,
I may add that Hilbert during the years to follow published at
least two more proofs of his finite basis theorem, and that even
Gordan produced a further proof of his own - and none of these
later proofs used the negation of quantifiers in the provocative
form as it had appeared in Hilbert's first proof from 1890 .
While Dedekind formed ideals and cuts as sets, and probably would
not have hesitated to speak in passing about the system of all
ideals in some ring, or of all cuts on the rationals, I am not
aware that anyone before Hilbert used arguments forming negations
of quantifiers which range over sets.
W.F.