Re: Platonism

From: J.E. (troubled6man_at_yahoo.com)
Date: 11/29/04 

Date: 29 Nov 2004 14:39:25 -0800

tchow@lsa.umich.edu wrote in message news:<41aa2a59$0$566$b45e6eb0@senator-bedfellow.mit.edu>...
> In article <39d6e584.0411280712.56f2b972@posting.google.com>,
> J.E. <troubled6man@yahoo.com> wrote:
> >Most mathematicians will retreat to
> >ZFC if you press them either (1) hard for the basis for their claims
> >or (2) really hard about the meaning of their claims. Just because in
> >informal practise you consciously ignore the alleged basis for your
> >beliefs doesn't mean they are not the basis.
>
> You've changed your tune here. What you said before was:
>
> >Most mathematicians when they say "exist" have an axiomatic system
> >(like ZFC) in mind, and mean that they are considering a formla with
> >the symbol E flipped backwords in it.
>
> Actually having ZFC in mind when you say "exist" is very different
> from pulling out ZFC as a magic amulet when cornered into defending
> your beliefs.

Apples and oranges. If the only time that you discuss "what you
really meant" is when cornered, then what you say then about your past
intent is the only information available to outsiders about your past
intent. We have introsepction about ourselves, but if we discuss the
intentions of others, we have to go by the best available evidence.

> It's probably fair to say that a lot of mathematicians use ZFC as a
> magic incantation without having much more than a foggy idea of what
> they are saying. That's far from saying that they are really formalists.

It means that if they only time that they give consideration to the
meaning of their words is in terms of ZFC, that that is their intended
meaning of their words. And if all you have is formalism, then you
seem like a formalist. A formalist doesn't have anything else other
than formalism.
 
> >I doubt it, seriously. I agree it depends on how it crumbles, but
> >your claim that it would always survive is simply baseless.
>
> Not at all. ZFC isn't the only foundation around. Work in the field known
> as "reverse mathematics" has demonstrated that much weaker systems than ZFC
> suffice for much of mathematics. If ZFC is found to be inconsistent, most
> likely the inconsistency won't percolate down to the weaker systems. Some
> mathematicians might get worried and confused for a while when they read the
> news about ZFC's inconsistency, but after the logicians reassure them that
> they can just use a different magic spell, they will go back to doing
> business as usual. This sort of thing has already been witnessed
> historically; various "crises in foundations" haven't caused mathematics
> as a whole to do more than hiccup.

That's an interesting viewpoint, full on ontological content. If
natural numbers are thought of as finite ordinals, then the modern
theorems about natural numbers are different than old platonic
theorems because they are about sets, specifically ordinals. And the
newer theorems about natural numbers are about sets. If you replaced
sets with nets for lack of a better word, then you could again define
natural numbers to be particular nets, but then all the theorems would
again be about NEW objects, so the THEOREMS would be different. The
abreviated an informal english (german, french, russian, whatever)
proofs might be the same for these new nets, but the theorem itself
would be about a different thing. To say that things would "be the
same" is to elevate the informal proofs about the subjects of the
proofs as "the real things" to mathematicians. Which is an
interesting view, and I can see it's appeal. It does seem strange to
put a higher value on the informal than the formal.

And regarding history I think we do disagree, I think many sloppy
things were going on preceeding the foundational crises and fixing
them was more than a hiccup, because among other things it was an
important acomplishment. It might be more accurate to say that the
sloppy things were fixed more by the different proof methods and
definitions, such as delta epsilon arguements. And so one way to
judge changes in foundations could be usefulnesss for changing how
mathematicians practise what they do. New abilities to describe,
define and prove is more important than avoiding contradictions per
se. Is this a fair reading of your statement?

Relevant Pages

  • Re: Platonism
    ... Retreating to ZFC is unlikely. ... My assessment of the beliefs of mathematicians comes from conversations ... reason to be suspicious of formalism. ... and not a problem with the mathematicians concept of reals or of ...
    (comp.theory)
  • Re: Platonism
    ... Retreating to ZFC is unlikely. ... My assessment of the beliefs of mathematicians comes from conversations ... reason to be suspicious of formalism. ... and not a problem with the mathematicians concept of reals or of ...
    (sci.math)
  • Re: How do We Know that ZF is the Axiomatization that Proves everything Provable?
    ... For example, the axioms of PA prove facts about arithmetic, ... the same theorems. ... level formalism includes that and other sets ... So lets stick to FOL and ZFC, eh voila there are *already* some gaps. ...
    (sci.logic)
  • Re: Platonism
    ... > from pulling out ZFC as a magic amulet when cornered into defending ... And if all you have is formalism, ... > mathematicians might get worried and confused for a while when they read the ... theorems because they are about sets, ...
    (sci.math)
  • Re: Platonism
    ... >>believe it might be called formalism, and it is fairly common IMO. ... > copy of ZFC in a book somewhere on their shelves. ... Most mathematicians will retreat to ... ZFC if you press them either hard for the basis for their claims ...
    (sci.math)
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