FOM: 'constructivism' as 'minimalistic platonism'

[Peter Schuster]

There's only little to add to Allen Hazen's comments 
[Hazen Fri, 02 Jun 2000 19:52:02 +0800]
on Steve Simpson's advocatus diaboli reply
[Simpson Thu, 1 Jun 2000 22:58:16 -0400 (EDT)]
to my posting on `constructivism as minimalistic platonism'
[Schuster Thu, 1 Jun 2000 18:55:11 +0200 (MET DST)]. 

Let me nevertheless try to answer some of Steve Simpson's questions. 

>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.

Realism and tertium non datur are indeed one body forever [Dummett]. 
By the way, the common use of `either ... or ...' does not seem to 
obeye any restriction like `even if we do not know which of the two 
is the case'; this appears to obtain also for classical mathematics. 

However, any realism powerful enough to `apply across the board' from 
the finite to the infinite has to be a full-blown one, which not only 
assumes all integers to exist but also that `we must know, we will 
know' [Hilbert] everything about them. The `minimalistic platonism' 
I have in mind shall, of course, only be based on the former 
assumption, namely, that the number sequence (plus induction) is 
something like `God-given' [Kronecker], period. The case is just as 
for predicative mathematics: even if some set is fine, already its 
power set has to be handled with care. In other words, even supposing 
the existence of the number sequence as a `completed infinity' 
(whatever this might be) does not necessarily involve that all 
subsets/properties of it are detachable/decidable. 

>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.

As soon as one takes serious epistemic and/or temporal matters, one 
indeed risks that tertium non datur goes out of the window, 
an observation which could be made also by most realists 
(except the omniscient ones). The question, however, is 
whether mathematicians are willing to let their logic depend on such 
issues; my impression is that most of them are not. By the way, I would 
like to know whether there are attempts to relate intuitionistic logic 
with temporal logic, too, just as with epistemic logic. 

>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?

Classical logic ought rather to be viewed as the logic of what really 
exists AND will entirely be known, sometimes. Because nobody can know 
what some yet unknown knowledge will be like and whether it will 
eventually be known (unless it IS already known), one must invoke 
some strong omniscience principle to rescue classical logic, since 
otherwise it would be restricted to the knowledge of the day. 

>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''.

Don't they already peacefully coexist, intuitionistic logic being a 
subsystem (sic!) of classical logic?  Such modal/temporal operators 
might therefore be cut off by Ockham's razor, notwithstanding that 
they might be of much theoretical interest.  

>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.

Which conflict? Who denies that `there is some odd perfect number' 
has a perfect meaning apart from being inserted in sentences like 
`we now know that ...'? Isn't the case just as for axiom schemata 
with unbounded variables? We know what `number', `odd', and `perfect' 
means, and even those of us who usually indentify `there is some' 
with `there cannot be no one' would presumably not be content with 
any proof of this statement that did not exhibit a witness, i.e., 
a concrete odd perfect number. 



Name: Peter M. Schuster
Position: Wissenschaftlicher Assistent
Instituition: University of Munich, Mathematics Department
Research interest: constructive mathematics
http://www.mathematik.uni-muenchen.de/~pschust