FOM: realism and bivalence
Stephen Ferguson
srf1 at st-andrews.ac.uk
Sun Feb 1 10:23:43 EST 1998
On Fri, 30 Jan 1998, Neil Tennant wrote:
> John Steel answers Harvey Friedman as follows:
>
> >Realism asserts that there are sets, and hence ... "there are sets" is true.
> > ... Whether this is Godelian naive realism I don't know.
>
> >I would subscribe to "either there is a measurable cardinal or there is
> >no measurable cardinal".
>
> Comment 1. One needs to distinguish ontological realism from semantic realism.
> The first of Steel's claims just quoted expresses a commitment only to
> ontological realism. Semantic realism would go further. According to the
> semantic realist, every sentence is either true or false of "the world",
> i.e. (in this case) the intended model of set theory, independently of our
> means of coming to know what its truth-value is. Saying that there are sets
> is not yet enough to get you determinate truth-values for all statements
> about sets.
...
>
> Philosophically, we have:
>
> ontological realism < semantic realism < naive realism
N.T.'s distinction between ontological realism and semantic realism is
helpful; I'd like to say a little about what I think showing each of these
to be would consist in, and try to make it clear what effect I think fom
can have on these sorts of discussions in phom. I'd then like to say
something about what I think some of the components are, in the other
direction, i.e. the effect of phom on fom.
1 ontological realism: sometimes called Realism in Ontology
(eg Shapiro _Philosophy of mathematics, Structure and Ontology_1997)
Often this is taken to arise largely from a face value construal of
mathematical language (Resnik Nous, 1981) and the truth of mathematical
statements.
But, one could put forward this much and still not be an ontological
realist: this is common in say, the philosophy of science (I'm thinking of
Arthur Fine's NOA) or more generally in the philosophy of language (e.g.
Horwich's deflationism)
Crispin Wright has argued that, in effect, realism in ontology is achieved
by beefing up this *minimal* notion, by what he calls realism relevant
properties. This is an entirely general approach to realism, but is quite
applicable to phom.
These two properties are what he calls cognitive command, and width of
cosmological role. I'm sure some of you are familiar with them, but I'll
make some quick comments anyway (See Wright Truth and Objectivity 1992)
cognitive command holds of a discourse if it is a priori that disputes
over the truth value of a statement of the discourse must be due to
vagueness, ambiguity, differences in personal evidential threshholds, or
cognitive shortcoming.
So for example, humor fails to have cognitive command, as you and I may
have different senses of humor, and we neither think that this is due to
jokes being vague, or to one of us making a mistake. It looks like
mathematics is quite a good candidate for cognitive command, or at least,
large areas of it might be.
Any discourse with cognitive command will be more objective than one
without it; of course, this does not tell you *why* the discourse is
objective, it merely gives you a unified approach to showing that a
discourse is objective.
If a discourse has width of cosmological role, it must be even more
objective, even more robust: this is a measure of the range of uses (e.g.
explanations) that appeal to the subject matter of a discourse can be put
to, other than simply being believed.
Anyone can beleive in the subject matter of a discourse - what would make
it objective would be that this subject matter could be appealed to ina
number of other ways, e.g. in explanations. So we might think that for
example, quantum mechanics had width of cosmological role, as it is
appealed to to explain the formation of complex ions, such as copper
sulphate (the 3d-3s level electron splitting explains why copper sulphate
turns blue when you put it into water)
In the mathematical case, this might involve working out what it is for an
application of mathematics to count as exlanatory.
So if some area of mathematics has both the minimal levels of assertoric
content (meaning) and these realism relevant properties ( cognitive
command and width of cosmological role ) then ontological realism will be
appropriate.
2 Semantic realism is the combination of ontological realism with
realism in truth value.
So what is realism in truth value?
Just what N.T. explained it was in his original post - this commitment to
Bivalence which is so prevalent, for example, in Dummett's interpretation
of intuitionism. BUT, suppose that in a dicourse, the following constraint
on truth holds quite generally:
If P is true, then there is evidence for it.
i.e. if P is true, P is knowable.
Call this the Epistemic Constraint. (EC)
Now, EC will imply Bivalence fails, or at least, is highly implausible.
But, this need not cause the downfall of certain types of moderate
realism.
Call semantic realism the strong position, which takes ontological realism
and adds this notion of realism in truth value, i.e. Bivalence holds. This
is the sort of view the historical Frege held, and is sometimes also
called platonism (with a little p)
But what happens if we take ontological realism, and add EC. Then there
are two options, along the lines (some say) of Locke's distinction between
primary (shape) and secondary (colour) concepts.
Are statements of mathematics true because provable
OR
Statements of mathematics are provable because they are true.
i.e. is it a case of detecting (tracking) the truth, or is it something to
do with projection (conventions?)
I take it that the views of Frege (semantic realism) Hilbert (optimism -
there is no ignorance: detection) and the later Wittgenstein/ Lakatos/
(Hersh?) (projection/ conventions) might fall into these three categories.
Go/"del's stronger epistemological Platonism (with a capital P) or naive
relaism as N.T. puts it, is so much stronger than even semantic realism,
that it always strkes me as too strong to be plausible.
2 What can fom tell us about mathematics, and can it help provide
evidence that mathematics fits into any ofthese philosophical categories?
On one level, the sort of fom which deals in conceptual priority etc.
looks like it is ina good position to argue for cognitive command. Maybe
a better argument might come from trying to generalise Lakatos' stufff on
heuristic falsifiers (in the Tymoczko volume New Directions in mathematics
recently referred to ). If mathematics were like humor, we would not take
counter-examples so seriously, so from the fact that we can basea
methodology on the way we treat counter-examples, mathematics (or some
portions of it) will have cognitive command.
Maybe some of the discussions as to what mathematics is really for, might
help explain the difference between application and mathematical
explanation. Maybe explanation is as Detlefsen attributes to Hilbert, why
ontological commitments need only be to 'real' mathematics, not ideal
mathematics, as it is real mathematics that explains the use of ideal
mathematics. (Detlefsen 1986 Hilbert's Program)
Maybe some of the discussions over the independence of axioms, CH etc
clear of the distinction between whether semantic realism holds for set
theory, or whether EC plus optimism is what we work with.
3 I hope some folk are still with me - this is all very much more
programatic than it is fleshed out: (you might guess these are some of the
ideas I kick about in more detail in my dissertation)
If the influence of fom on phom is to be provide clear insight into what
it would be for one of these distinctions, rather than another, to be the
most appropriate for mathematics (or maybe to show that some areas of
mathematics, e.g. number theory and set theory, are *more* objective than
other areas, e.g. satisfy full semantic realism, while other bits only are
real enough to support optimistic realism, say), what can be said about
the relationship in the other direction?
This is where my vision fails me - for I see very few connections between
phom and fom. One connection I doo see clearly, is the distinction, which
Shapiro makes explicit, between valuing the study of foundations, and the
philosophical doctrine of foundationalism.
Another connection I see, is where fom can provide technical / formal
precise statements of properties which are relevant to phom: maybe this is
just a different shade on what I wrote above.
Finally, I see some connection to do with new avenues of interest -for
example, recent philosophical work on Frege seems to have led to work on
Hume's Principle, and there to work in general on abstraction principles.
I realise that this posting is more philosophical than foundational, but I
hope that I ahve managed to offer something of a plausible refinement to
N.T. distinction between ontological and semantic realism, and also some
way that could lead to clean lines of discussion of issues such as
objectivity.
(e.g. would someone such as Hersh accept cognitive command and width of
cosmological role for mathematics, but argue that the cultural components
lead to a failure for semantic realism?)
Yours
Stephen
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