[FOM] Logically determined

[Irving]

Paul Budnik wrote:

> 2+2=4 is true in all universes with cardinality of at least 4 assuming
> one is using those symbols to refer to standard integers. ...
>

> The consistency of a formal system is logically determined  by its
> axioms. The axioms are either consistent or inconsistent. G?del proved
> that one may not be able to decide this question from those axioms. Thus
> the intuitive idea of logically determined must mean something more than
> deducible from axioms. G?del proved the incompleteness of logic as well
> as mathematics.
>
>

Prof. Budnik goes on to state:

> I suggest that logically determined statements are those that make an
> objective statement about a recursively enumerable sequence of events.
> The catch is defining which statements (and thus which relationships
> between events) are objective. Since logic is provably incomplete this
> cannot be precisely defined. It is a philosophical idea not a
> mathematical assertion. It is a suggested refinement to the notion of a
> Platonic ideal reality. It is based on contemporary mathematics and
> computer science and on the old idea that infinity refers to an
> unlimited and thus unrealizable potential.

I think that, from the standpoint of philosophy, it is needful not only 
to distinguish logically determined statements from philosophical 
claims, but to also distinguish metaphysically or epistemologically 
determined claims. Recall Bertrand Russell's query to Leon Henkin 
whether Goedel's incompleteness theorems permitted 2 + 2 = 4.001 in 
"school-boy" arithmetic. As I have sated here before, I do not believe, 
as a I told Henkin when he asked me, that Russell was merely joking 
when he asked Henkin this question. (I can provide the bibliographic 
data from Henkin, Russell, Goedel, and Dawson regarding these and 
related material, for those who may like them and cannot find them in 
the FOM archives from my previous posts.) But, from the philosophical 
perspective, to assert that some logical relations are objective, we 
might want to avoid the psychologism of the early Husserl and the 
phenomenologism of the later Husserl, if we are to avoid allowing 
ourselves Russell's 2 + 2 = 4.001 as an objective mathematical truth 
just because we can think, say, or write it.



Irving H. Anellis
Visiting Research Associate
Peirce Edition, Institute for American Thought
902 W. New York St.
Indiana University-Purdue University at Indianapolis
Indianapolis, IN 46202-5159
USA
URL: http://www.irvinganellis.info