About existence-as-consistency

Colin McLarty colin.mclarty at case.edu
Fri Jul 2 09:54:54 EDT 2021

The earliest place I know of Poincare insisting that, in mathematics,
existence just means consistency is his 1893 essay 'Le continu
mathematique', *Revue de Metaphysique et de Morale *I, 26-34.  He follows
Tannery in using Dedekind cuts to define the continuum.  He stresses this
"true mathematical continuum" is different from the physicist's intuitive
continuum, and from the metaphysical continuum, and this is the correct one
for mathematics.  He notes it requires intercalating infinitely many
new points among the rational numbers and says:

no one will doubt the possibility of the operation, unless from forgetting
> that possible, in the language of geometers, simply means free from
> contradiction' (1893, page 27, also on page 44 of the 1921 collection of
> translated essays,  *The foundations of science*).


On Thu, Jul 1, 2021 at 11:34 PM Curtis Franks <Curtis.D.Franks.7 at nd.edu>

> I do not have references handy, but Poincaré's writings (e.g., the essays
> selected in Bill Ewald's sourcebook) contain many instances of the
> existence-as-consistency idea. I do not recall seeing the idea in
> Poincaré's writings before 1900, so it might not predate Hilbert's own
> expressions. But it is interesting that Poincaré was some sort of
> constructivist, dissassociting the idea somewhat from classical logic.
> Curtis Franks
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