Re: existencism

On 26 Sep, 04:21, Michael Siemon <mlsie...@xxxxxxxxx> wrote:
In article <Bgfvm.3$bP...@xxxxxxxxxxxx>,
 Garamond Lethe <cartographi...@xxxxxxxxxxxxxx> wrote:

Hmmm...

...

But plenitudinous platonism of the kind Balaguer promotes gets pretty
close, and with Ed Zalta you have one straddling both fields. see in
particular Balaguer 1998 "Platonism and Anti-Platonism in Mathematics,"

Oh, cool, the library has a copy.  I had been rooting around in the
philosophy of mathematics section a few weeks ago but I didn't come up
with anything that looked this interesting.  Many thanks.

where he does indeed postulate a multiplicity of mathematical universes,
each corresponding to a consistent mathematical theory.

This strikes me as extremely odd. A (localized, as I assume must be the
case here) "consistent mathematical theory" is just math-as-usual. If
there is something in the "consistent theory" which is _not_ consistent
with _other_ math, I would be hard-pressed to understand what is going
on. Yes, one gets non-standard analysis, or (at the other extreme) some
forms of constructivist theorizing which (one way or the other) don't
immediately emerge from "standard" [e.g. "Bourbaki" :-)] presentations
of the encyclopedic body of math -- but good, consistent math is good
math. Period. So I just don't understand the putative issue.

you criticism is very similar to the one Donald Martin made in
“Multiple universes of sets and indeterminate truth values”, Topoi, 20
(1): 5–16.2001 - Essentially, all these different mathematical
theories can be shown to "collapse"in one mega theory.

And yes, there are big problems with this view an it is not one I
would use for mathematics myself - though or our topic, i was always
tempted to use it in my religion. gives me even more deities to
believe in ;o)

Balaguer's position is philosophically interesting because as far as I
can see, he takes the exact opposite of the intuitions that normally
leads mathematicians to platonism, and build a very weak form of
platonism on its back. John W. might like it - it gets very close to
flatus vocis, or purely semantic, platonism..

One rationale you gave yourself: Goedel's results also show that our
axiomatic set theories cannot rule out non-standard set theoretical
universes. But do these non standard universes exist? Platonists would
intuitively say no; In fact, it is the whole point of being a
platonist to be able to say that say NBG or ZF only describe a part of
the set theoretical reality, a shortcoming of our theories/language.
"Non-standard universes" are a consequence of this shortcoming, they
are a pathology. But why is the standard universe the right one? Isn't
this just "set theoretical racism"? ;o) In his view, the standard
universe and the non standard universes all exist equally and on a
par.

Related to this, and more prominently used, is this argument: Start
with ZF. This gives you certain formal objects. We know Choice is
independent from ZF, so add this to ZF. We now have a universe of
sets in which the hypothesis is true and another in which it is false.
Now add even more and stronger reflection schemes. You get larger and
largr cardiinals. Again, they exist or do not exist depending of the
axiomatic system chosen. But If you have say a subtle Cardinal in one
universe and not in the other, _all_ formal objects in the two
universes will have different properties (those expressed as a
relation to this Cardinal) Leibniz identity, and hey presto you have a
multitude of universes, one for each axiomatic system.

One alleged consequence of this approach is that consistency is the
only criteria you need to decide if a mathematical object exists, you
do not need any appeal to semantic intuitions that platonists tend to
use to justify the inclusion or exclusion of axioms additional to ZF

.

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