FOM: Realism and formalism
V.Sazonov at doc.mmu.ac.uk
Mon Jun 12 15:21:27 EDT 2000
Fred Richman wrote:
> "V. Sazonov" wrote (in response to Fred Richman):
> >> If I believe in the objective existence of the natural number series,
> >> am I committed to the proposition "either there exists an odd perfect
> >> number, or all perfect numbers are even"?
> > And what does it mean at all "to believe in the
> > objective existence of the natural number series"
> > if it is not just a play with words?
> If there is no defensible realist position, then, of course, it is
> pointless for me to wonder whether the law of excluded middle follows
> from it. I had thought that there was a defensible realist position,
> and that believing in the objective existence of the natural number
> series was more or less the minimal part of it.
Of course my question was not related
with the law of excluded middle. I just doubt that *even* this
"minimal part" makes a sense. First, what does it mean to believe
in the objective existence of "the" natural number series? I do not
understand this "the". Thus, natural numbers defined by Primitive
Recursive Arithmetic (we have at least some, may be vague
intuition on them) constitute presumably a *shorter* series of
numbers than those defined by Peano Arithmetic. The latter is
closed under Ackerman function, while the former is probably not.
Is there a *longest* series consisting of *all* natural numbers?
What does it mean *all*? However who *believes* seems even does
not admit any questions concerning his beliefs.
Just believe and no questions!
On the other hand, in other sciences we can have beliefs of quite
different character, e.g. in Newton laws. Here we always admit
that they could in principle be violated by some experiments
(e.g. for very high speed motions). Also we not only can believe
in Newton laws, but can explain *what do these lows mean* and
*what do this belief mean*. Thus, my second question even on the
"minimal part" of so called "realistic position on mathematics"
is what does it mean "to belive"?
> > And also, why such unrealistic way of thinking
> > to believe in the *objective* existence of evidently
> > *unrealistic* things is called "realism"?
> I can sympathize with this attitude. Bishop called such a way of
> thinking "idealism". Mycielski has said that idealism in mathematics
> is the belief that mathematical objects are real, and realism in
> mathematics is the belief that mathematical objects are ideal. But I
> didn't think that the terms were being used in that way in the
> discussions here.
Yes, I know. But the problem is not only in terminology. From a
***consistent*** realist (unlike idealist) position concerning
material world (I would call this *scientific realism*; is this
term reasonable? ) it seems impossible to *believe* that
mathematical objects are real. I think that the opinions of
Bishop and Mycielski are also based on such consideration. They
just give *proper and more general* philosophical characteristics
of corresponding views on mathematics. It is not just renaming
or play with words.
Moreover, as I understand, scientific realist hardly would *ground*
a philosophy of mathematics on some beliefs. Such kind of beliefs
seems to be something outside of science. Scientific realist should
find out, or discover (not just call) what is real in mathematics
in the proper philosophical/scientific sense of this word. As
mathematical objects are numbers, etc., there is a strong temptation
to call *them* real (what else?), but of course not for scientific
realist who strongly distinguishes objects of the real world from
ideal objects (even illusions) of our imagination. However, it is
also clear that there are also quite different objects in
mathematics - *formalisms*, axioms, rules and rigorous proofs
which are sufficiently real, even implementable by computers
(the analogue of computer software). Formal rules and axioms are
(syntactical) objects of a different kind, playing in mathematics
different role than, say, numbers. Only in *meta*mathematical
considerations they become analogous to numbers (via G"odel numbering
or the like).
But this should (I think, inevitably) lead us to formalist
philosophy of mathematics. Thus, formalism may be considered
as scientific realist view on mathematics which probably needs
in further development besides simple declaration.
(i) realism in mathematics understood in some unfortunate traditional
sense of word and most often used here in FOM seems to be inconsistent
with scientific realism (concerning our real world of physical objects)
(ii) scientific realism in mathematics reduces to formalist philosophy
(to what else?).
Is anything incorrect here? And what is so bad in formalist view on
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