Re: Jech's Set Theory

aatu.koskensilta@xxxxxxxxx wrote:
Yes, that is the appeal of the kind of formalism you seem to find
congenial. It's neat and tidy, and sidesteps all the messy conceptual
issues.

Indeed, the purpose is to sidestep (better to say, postpone)
philosophical problems so that I can get on with the business of
learning some mathematics, as the mathematics (during this phase of my
studies) is of more interest to me than is philosophy of mathematics.
And I wonder whether realism might not sidestep but rather reduces
explanation too drastically (I mentioned in another post why I don't
find realism to be explanatorily satisfying). I am sympathetic to
realism as justified by indispensibility and naturalism arguments. But
I incorporate that, for my own thinking, as working AS IF the objects
exist platonistically without committing to saying that they DO exist
platonistically. And I find at least the general idea of Hilbert's
contentual/ideal distinction (to the admittedly limited extent I
understand all the technicalities) to fit with the idea of acting "as
if". Yes, I am rather cagy about all this, since, as I mentioned, my
agenda is basically to postpone arriving at philosophical positions
while I learn more of the mathematics itself. And, again, this is just
for my own purpose as I do not necessarily espouse that other people
adopt the pieces of formalism, structuralism, fictionalism,
if-then-ism, and naturalism I have cobbled together to serve
PROVISIONALLY for myself as a view that is reasonable, to myself, for
the purpose of making sense, for myself, of mathematics.

I don't know that I have the kind of direct mathematical perception that Godel
mentioned.

Very few people do. I certainly don't.

But I don't see how one can commit to realism (except by way of such
arguments as indispensibility and naturalism) without that perception.
That is, unless one accepts the word of other people that they do have
such perception.

My endless bickering about these
issues is rather prompted by the observation - forcefully made by
Kreisel on several occasions - that all philosophical explanations put
forth to this day of what mathematical assertions amount to, how they
are justified, and so forth, are much more dubious than the perfectly
clear mathematical assertions themselves.

That's an interesting tack. But what priority does that give to a
realist philosophical explanation as opposed to any other philosophical
explanation? Also, I wonder if it's not question begging. Why must one
suppose that assertions about the mathematical infinite are any more
clear than certain philosophical principles (except, tellingly, the
mathematical assertions are formalized).

I prefer to take mathematics
at face value even when trying to understand it in some philosophical
sense, rather than somehow trying to pretend I don't know what
perfectly ordinary mathematical assertions mean, or not to accept this
or that piece of perfectly ordinary mathematics.

Right, but how can I say whether an assertion about something infinite
is true or not? For the empirical and for the abstract yet finite, I
can say what I mean by an assertion of truth. But this is not so easy
for an assertion such as the generalized continuum hypothesis. I accept
ordinary mathematics about the infinite for a number of reasons. But I
don't find that doing so conflicts with the kind of provisional
philosophical views I've mentioned, as meanwhile I just can't say with
a straight face, "The generalized continuum hypothesis. Yeah, it's
either platonistically true or platonistically false and I hope some
day to directly perceive some platonistic truth that entails whether
the generalized continuum hypothesis is true or false."

To lend more authrity
to my bald claims, and to show off with my learning, let me quote Otto
Neurath:

We are like sailors who on the open sea must reconstruct their ship
but are never able to start afresh
from the bottom.

I like that. Doesn't Quine say something similar even more generally
about scientific knowledge? (I think it also relates to the idea of
coherence that I mention latter in this post.)

So, at least for now, I demur from such strong mathematical realism as
Godel's view of direct mental perception, and instead I am, at least
for now, content to say that when I make a mathematical assertion, I'm
merely reporting upon theoremhood.

That's your prerogative - I was there once, too. Getting out of there
is a bit of a baffler, and involves asking onself certain questions,
and taking them seriously. As an appetizer, you might reflect on the
following attempt of mine to take seriously the idea that set
theoretical assertions are to be interpreted as assertions about this
or that being provable (in, say, ZFC)
(<cFXIf.672$Yg6.112@xxxxxxxxxxxxxxxxxxxxxxx>):

http://groups.google.com/group/sci.logic/msg/ed7be27b4456c35a?hl=fi&;

Thanks for that link. I'll read it this weekend.

The fundamental problem with the sort of formalism you seem to espouse

Please be careful. Don't forget that I've been clear that I am not an
advocate of any particular philosophy or even of my own philosophical
hodgepodge, so I do not espouse on these matters, but rather I'm only
reporting the way I've provisionally made sense of these matters for
myself.

is that people are not (usually) consistent in it, and tend to take,
perhaps without realizing it, such assertions as "it is provable in ZFC
that P" or "ZFC is consistent" at face value, and not as assertions
about this or that being provable in some theory.

I very much appreciate that point. Personally, I wish not to be
inconsistent in that regard. But I recognize that eventually the
regress (escalation, I call it) of formalisms has to stop so that we
then have to take certain informal principles as governing. And I am
content, at least for now, to take these principles as the principles
of ZFC but at an informal level. So the realist might retort, "But how
can you take them as governing if they're not true; and truth now is
platonistic since you've admitted that you've stopped the escalations
to formal model theoretic truth." My answer is that I take the
principles as governing in a sense of being reasonable, and as
importantly, as "reverberating" in COHERENCE (related to, but not
confined to consistency) with mathematics and the notion of a set, and
better than, or at least more convientlty (for me personally), any
other principles, and especially as the only principle regarding
infinity that I must adopt is that there is a successor-closed set,
which is eminently reasonable and, again, coherent for me even though I
cannot even say what I would MEAN by asserting that it is
platonistically true.

In fact, it seems
most people interpret at least all arithmetical assertions in the
obvious way, and not as being about provability of anything in
anything.

That's fine, but it is not entailed that I ought to view this as do
most people. I suspect that most mathematicians aren't even concerned
with such questions. I would think it is just as glib to think that
common, everyday, philosophically naive mathematical notions of
meaning trump other views.

Again, as a good antidote to 'glib formalism', I recomment
Torkel's thesis _Provability and Truth_.

I've worked on understanding that paper. I'll return to it when I'm
better educated. Oh, how I wish Franzen were still with us. How lucky
we were that through posting we could talk with him whenever we wanted
to.

Of course, we can again
paraphrase Wittgenstein and note that perhaps only those who have
thought these thoughts already are ready to appreciate them. Further,
we can also note that as to the practice of mathematics these issues
are obviously utterly irrelevant; set theory done by a formalist is
indistinguishable from set theory done by a rabid Platonist as
mathematics.

Yes, that indistinguishability is exactly what I mentioned when I
responded to your post a few months ago in which you asked how one can
be said to be "doing" set theory while thinking of it as nothing more
than syntactical proving. (And, please keep in mind that,, anyway, I
don't at all view the study of set theory as NOTHING more than
syntactical proving.)

At a certain point, after one has put the formalisms in place, one can say,
"Okay, from now on, when I say something in English (or whatever natural
language or even semi-formal language), it is too render my formulas, whereas
previously I used formulas to render my ideas couched in English."

Why? What is gained by saying anything like that?

It tends to keep things more exact for me, and it reduces my
commitments in discussions. I am stipulating that what I mean when I
say certain things is to report on the formulations and derivations
rather than having what I say be regarded as assertions about
platonistic objects and what is or is not true about them.

MoeBlee

.

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