Re: Jech's Set Theory

MoeBlee wrote:
Yes, that seems like mathematical platonism/realism to me. So my
question would be how for a NON-realist, mathematical sentences are
plain true or plain false (irrespective of proof or even of model
theoretic test) but also dependent on mind to conceive the abstract
states of affairs at issue.

Before answering your longer reply to my latest post, let me briefly
(ha-ha!) address the issue of Platonism. Earlier I said I find
Platonism either an empty metaphor or simply false, depending on how it
is conceived. In order to explain what I mean by this it is absolutely
necessary that I describe my personal take on what philosophy of
mathematics is about, and how we should go about waxing philosophical
about mathematics. First of all, we have the actual mathematical
practice of thinking about mathematics, proving stuff, devising new
concepts, finding connections betweem diverse areas of mathematics,
organizing some body of results into a coherent exposition, writing
papers, trying to get a daft student to understand some concept or mode
of argument, trying to wrap our heads around complex proofs, and so on
and so on. As I see it, the purpose of philosophy of mathematics is to
understand this practice, and in particular explicate and articulate
what sort of implicit assumptions underlie its various aspects. This,
of course, is not the only possibility, and classically it has been
customary to think that philosophy of mathmatics should justify this
practice in some sense, and even, if no such justification is to be
found, lead to a revision of the practice. Now, I believe this to be an
error, for the simple reason that an analysis of mathematical practice
is bound to be more general and abstract - and hence necessarily more
doubtful - than concrete bits of that practice itself. Hence we can't
hope to provide a justification by philosophical means, and even less
can we hope to define mathematics by means of such a general analysis.

Obviously we must also admit that there is interplay between the
practice of mathematics and philosophical ideas. In proof theory, in
set theory, and historically, in analysis, say, philosophical arguments
have played a role in deciding what sorts of arguments and principles
are mathematically acceptable. However, this is an exceptional
situation, and in ordinary mathematical practice philosophical ideas
can safely be left for the Sunday tea.

Given this conception of the nature of philosphy of mathematics, let's
consider the question of realism and Platonism. As I said elsewhere,
virtually everyone is a mathematical realist in the minimal sense that
they do in fact, at least when they're not (having a try at)
philosophizing, say stuff like "there exists a prime greater than two",
"reals form a complete ordered field" and so forth, which clearly imply
the existence of mathematical objects, since prims and reals are
mathematical objects. Concerning naturals at least most are also
realists in the sense that they treat (most) questions about naturals
as determined by mathematical fact, and not as something we can decide
by stipulation. For example, most people don't feel free to just
stipulate that Goldbach's conjecture is true, or false, according to
personal whimsy. Some statements, of course, can be decided by
stipulation alone. The empty set exists just because we say it does,
and there is apparently nothing more to its existence than that; or, at
least, it is very difficult to imagine any other considerations or
arguments for either the existence or non-existence of the empty set -
of course, we do find some people arguing for example that the empty
set can't exist given the way we talk of collections in natural
language, but such arguments, while however revealing about natural
language, are not relevant to the mathematical understanding of sets.

It appears that Platonism, as often understood, wants to say something
additional about the existence of mathematical objects, and about the
objectivity of mathematical facts about this objects. It is difficult
to pinpoint exactly what this additional metaphysical reality and
objectivity amounts to, exactly, but as a metaphorical expression of
the natural tendency to treat facts about naturals as independent of
the whimsy of fantasies of any mathematician, there is nothing wrong in
it. But such a metaphor has no explanatory value, and adds nothing to
our understanding of the mathematical practice not already covered by
the observations alluded to in the previous paragraph. Another
alternative is to take Platonism literally, as a claim about the
metaphysical make-up of the world (in the wide sense of the word).
According to this view there is, in addition to the physical realm -
and perhaps the realm of subjective thoughts and cultural artefacts -
an independent, but real, realm inhabited by mathematical objects. Some
extreme Platonists go even further and claim there is some
pseudo-causal link between these objects and the human mind, so that
through a third eye of mathematical intuition we have something like a
direct access to the mathematical domain. If we disregard this
extremity, and it is easy to do so as there appears to be no evidence
for such a third eye, the problem remains that it is difficult to see
what the point of postulating this realm is. If it is indeed entirely
independent of us, what's there to rule out the possibility that the
cumulative hierarchy stops at V_omega+374 and after that we have just a
haphazard collection of sets that would belong to V_omega+375, or that
all the naturals up to 42387423 exist but, accidentally, 42387424 is
missing. In the literature we find a cornucopia of such problems. As
always with such philosophical conundrums, philosophers have produced
an endless stream of answers, counter-examples and counter-arguments,
counter-counter-answers and so forth, but in the end we are left with a
convoluted abstract model that has very little apparent connection to
the actual practice of mathematics, a fine intellectual exercise, to be
sure.

Now, of course, it is indeed a very interesting question whether or not
the objectivity of mathematical facts - at least about naturals or
finite structures - is just a brute fact about how we conceive
mathematical objects when we do mathematics, or whether there is some
interesting explanation for this attitude. However, this question is
not answered in any interesting way by postulating some metaphysical
entities and abstract relations between them.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

.

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