About existence-as-consistency

Marcel Ertel marcelertel at gmail.com
Fri Jul 2 12:09:35 EDT 2021

 Dear Giovanni,

I think this quote from Cantor's 1883 *Grundlagen* *einer allgemeinen
Mannigfaltigkeitslehre* *(Foundations of a general theory of manifolds*),
paragr. 8, may be relevant:

Mathematics is in its development entirely free and is only bound in the
> self-evident respect that its concepts must both be consistent with each
> other and also stand in exact relationships, established by definition, to
> those concepts which have previously been introduced and are already at
> hand and established. In particular, in the introduction of new numbers it
> is only obligated to give definitions of them which will bestow such
> determinacy and, in certain circumstances, such a relationship to the older
> numbers that they can in any given instance be precisely distinguished. As
> soon as a number satisfies all these conditions it can and must be regarded
> in mathematics as existent and real. (Translation by W. W. Tait)

German original (in: G. Cantor, *Gesammelte Abhandlungen*, edited by E.
Zermelo, Springer, 1932, p. 182):

Die Mathematik ist in ihrer Entwickelung völlig frei und nur an die
> selbstredende Rücksicht gebunden, daß ihre Begriffe sowohl in sich
> widerspruchslos sind, als auch in festen durch Definitionen geordneten
> Beziehungen zu den vorher gebildeten, bereits vorhandenen und bewährten
> Begriffen stehen. Im besondern ist sie bei der Einführung neuer Zahlen nur
> verpflichtet, Definitionen von ihnen zu geben, durch welche ihnen eine
> solche Bestimmtheit und unter Umständen eine solche Beziehung zu den
> älteren Zahlen verliehen wird, daß sie sich in gegebenen Fällen unter
> einander bestimmt unterscheiden lassen. Sobald eine Zahl allen diesen
> Bedingungen genügt, kann und muß sie als existent und real in der
> Mathematik betrachtet werden.

In fact, the first sentence might be more accurately translated as follows:

... only bound in the self-evident respect that its concepts must both
be *consistent
> within themselves *and also stand ...

Note that in the same paragraph, Cantor distinguishes two types of
existence: the "immanent" and the "transient" reality of concepts.
According to Tait's interpretation (with which I tend to agree), the former
corresponds to purely mathematical existence, the latter to a concept being
instantiated in physical or psychological reality. The latter is explicitly
said *not *to be necessary for mathematical existence, although Cantor had
faith that the two *de facto *always occur together (for what seem to be
mystic religious motives, the "Unity of the All", *ibid.*). See Tait's
article "Cantor's *Grundlagen *and the Paradoxes of Set Theory" (reprinted
in *The Provenance of Pure Reason*), where he also discusses a disagreement
with Michael Hallett on this matter.

Best wishes
Marcel Ertel
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