FOM: mathematics versus theology

Walter Felscher walter.felscher at
Wed Aug 4 16:23:17 EDT 1999

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.


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