# FOM: 'constructivism' as 'minimalistic platonism'

Stephen G Simpson simpson at math.psu.edu
Thu Jun 1 22:58:16 EDT 2000

Schuster Thu, 1 Jun 2000 18:55:11 +0200 (MET DST):

> The reason why to choose intuitionistic logic is then the insight
> due to Brouwer that tertium non datur can hardly be applied in
> infinite contexts: who is legitimated to decide whether there is
> any odd perfect number?

Let me play devil's advocate for a moment.

>From the realist point of view, there is nothing wrong with tertium
non datur.  If P states something unambiguous about something real,
then necessarily P is either so or not so, i.e., we can confidently
assert ``P or not P'', even if we don't know which of the two is the
case.  And this reasoning would seem to apply across the board, even
if P involves an (actually or potentially) infinite sequence of
natural numbers, provided the number sequence is real, as your
``minimalist Platonism'' assumes.

However, perhaps you want to read ``P or not P'' differently,
interpreting ``P'' as ``we already know that P is so'', and ``not P''
as ``we already know that P is not so''.  With that reading, we cannot
assert ``P or not P'' until we have decided P, i.e., until we have
proved P or proved not P.

Is this why you reject tertium non datur?  You seem to confirm this
interpretation by saying

> the choice of the logic is directed by the assumption that there is
> no a priori knowledge of such matters.

Thus classical logic seems to be the logic of what really exists,
while intuitionistic logic seems to be the logic of our knowledge as
it develops over time.  Is this correct?

If this is correct, then why can't classical and intuitionistic logic
peacefully coexist, in the same system?  All we need is a modal
operator distinguishing ``P'' from ``we know P''.  Right?  Or perhaps
a modal/temporal operator, ``we know P at time t''.

However, a conflict arises if you deny that ``P'' has meaning apart
from ``we know P''.  In other words, a conflict arises if you insist
on subjectivism, i.e., if you deny the objective principle, that
reality exists independently of our knowledge of it.

Where do you stand on this?

-- Steve