About existence-as-consistency

[Mikhail Katz]

Dear Giovanni,

It seems to me the title of your post could have been 'Consistency as
existence', which is the direction of greater novelty.  This is of
course usually attributed to David Hilbert (who was motivated in part
by a battle against Brouwer over the meaning of mathematical
"existence" and was therefore interested in softening up/lowering
expectations with regard to the latter), but in fact rather detailed
investigations in this direction are already found in Leibniz.

Leibniz was confronted with the uncomfortable situation with regard to
his new mathematics which involved entities beyond the traditional
material found in Euclid and Greek geometry: negative numbers, surds,
imaginary roots, infinitesimals, some of which seemed not only
impossible but quite heretical to some of his contemporaries.

Some of them still seem heretical today though with regard to a
different faith; more on this below.

Leibniz's strategy was to soften up the concept of 'impossibility' by
introducing a clever distinction between 'absolute impossibility' and
'accidental impossibility'.  Leibniz defines 'absolute impossibility'
as one involving a contradiction, but 'accidental impossibility'
involves merely a feature of the particular world we are familiar
with.  This ties up with the Leibnizian doctrine of 'possible worlds'
but comparing this to modern concepts of 'distinct models' would be
far-fetched.  At any rate the idea of a mathematical object being
possible without having a counterpart in the physical world ('natura
rerum') is definitely there in Leibniz.

How successful Leibniz was in convincing his contemporaries can be
judged from George Berkeley's reaction a few decades later :-) In
fact, only about 130 years ago, George Cantor was acting like a raving
lunatic with regard to (more precisely, against) infinitesimals way
before he was committed to a physical asylum.

Cheers, Misha Katz

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On Tue, Jun 29, 2021 at 1:50 AM <sambin at math.unipd.it> wrote:

> Dear Fomers,
>
>
>
> I am deeply interested in the historical origin and explanation of the
> principle by which consistency of an axiomatic theory T (typically ZFC) is
> sufficient to  justify it and derive that what it speaks about exists (in
> the case of ZFC, sets satisfying the properties described by its axioms). I
> call this principle: existence-as-consistency, shortly EaC.
>
> I am thinking for instance of the appearance of EaC in Hilbert's program
> in the 1920s. I suspect that the standard model theoretic explanation of
> EaC (by which T is consistent iff it has a model) came later.
>
>
>
> A related question is: is there a way to avoid assuming EaC while keeping
> classical logic (and hence validity of LEM)?
>
>
>
> I thank in advance for any information and comments.
>
> Giovanni Sambin
>
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