[FOM] FTGI1:CPC & 'Ordinary Language'

[Sean C Stidd]

I think that the account that Wittgenstein provided of the propositional
calculus in the Tractatus is still of some interest. It runs more or less
like this: given that you have a pair of propositions, either of which
can be true or false, there are sixteen different ways in which the
truth-value of the combined proposition can depend on the truth-values of
its constituents. The five classical connectives are five of these;
others include such curiosities as Tautology and Contradiction operators
which produce a true or false proposition irrespective of the
truth-values of its constituents. This makes it appear as though there is
nothing more in the propositional connectives 'really' than what was
already there in the possibility of a proposition's being true or false:
this is one reason Wittgenstein says that 'the logical constants do not
represent'. Logic for the early W. then 'reduces' to the combinatorics of
these bivalent propositions.

Of course, everything that Wittgenstein says in the Tractatus about
quantification, class theory, etc. is useless for the philosophy of
modern mathematics, because, as Russell already pointed out in his
introduction, "No logic can be considered adequate until it has been
shown to be capable of dealing with transfinite numbers," and
Wittgenstein does not do this; furthermore, pace Russell, there are
principled reasons why it could not be done in the system of the
Tractatus. Nominalists have been trying to recover from the apparent
commitments to abstract objects in set theory for over a century now, and
despite a long history of failure they still haven't given up: this is
one central issue in the philosophy of mathematics today. But I think
it's useful to see Wittgenstein as offering a 'nominalist' interpretation
of the propositional calculus (one which entails no abstract commitments
beyond those already present in the commitment to propositions, if any
(W. wouldn't have thought there were any - trying to show why is part of
what his theory of the proposition is doing)), and one which may well be
successful: the problem is that he tries to do modality and
quantification in the same way, and given what we know now it seems
highly doubtful that they can be.

It could be replied on LW's behalf that he might not have thought
advanced mathematics did mean anything, that he was trying to provide a
logic which handled ordinary factual assertions. To decide whether the
Tractatus succeeds in this we'd have to know to what degree ordinary
factual assertions are already connected up with various conceptions of
the infinite. This is the issue raised by the indispensability arguments,
another central concern of contemporary philosophy of mathematics, at
least if we allow 'ordinary factual assertions' to include physics.
(Which we'd better, because ordinary language physics is at least in part
demonstrably false. W certainly would have in the T.)

Since I brought up LW anyway I'll make a few comments on this
interminable debate over ordinary concepts and mathematics as well.
Though there are a few interesting and wonderful exceptions where
scientific and mathematical research produce explanations that really can
be made clear to people with little or no training, most advanced work is
not like this. One has to study some advanced science or mathematics even
to understand most of what is being said, and only on the basis of such
an understanding are you in a position to try to see how it fits together
with concepts from other spheres of life, if indeed it does at all.

I can see no principled reason to object to this: the same thing is true
of any disciplined activity. Even in the arts the sorts of things artists
say to other artists about what they are doing are different from the
sorts of things they say to a general public about what they are doing.
More tendentiously, the demand that advanced practitioners 'make sense'
in some everyday sort of way is rather like going to a foreign country
and complaining that its natives don't speak to you in English.

That said, there are some cases where it becomes extremely important to
try to work towards this kind of translation. Scientists sometimes make
claims about political matters or about general phiosophical issues (e.g.
determinism and free will) on the basis of their scientific beliefs. In
these cases there is a great need for an understanding of the scientific
considerations that motivate these claims so that we can ascertain
whether they really provide anything like a good argument or not.
Unfortunately, this is getting harder and harder to do well as research
becomes more and more advanced and specialized in all fields.

Also, there is an ordinary/theoretical distinction which can be drawn in
FOM and is of some interest to the community. This is the difference
between ordinary mathematics, which includes advanced research in
algebra, number theory, geometry, analysis, topology, etc., and
foundational studies, including logic, set theory, model theory, category
theory, etc. The mathematician committed to ordinary mathematics, say the
working number theorist, may well be skeptical about the importance of
foundational studies: "why do I need a 'foundation' for what I'm talking
about when I'm engaged in a discipline that has clearer norms of rational
conduct, a clearer conception of its objects, and better agreement on
results than 99.999% of everything any human being has ever said"? And
then, if we don't think that the ordinary understanding of the working
mathematician is sufficient for some reason, we will try to convince him
or her that this isn't quite enough, either by reminding him of certain
basic philosophical and logical points about reasoning, the nature of
justification, etc., or by trying to show that our foundational calculus
can do things for him that he can't do himself within his own way of
working, and in addition provides a powerful unification of his
discipline with other parts of mathematics. There are arguments on both
sides here, but it seems to me that insofar as there is a serious
ordinary vs. theoretical debate in FOM it is this one, the understanding
of the ordinary mathematician of their subject(s) as against our
mathematical, logical, and philosophical reconstructions of it. 

This is not to say that observations concerning concepts more central to
our everyday lives are never relevant to the philosophy of mathematics -
sometimes they are of great use in illuminating some complex form of
reasoning by analogy, e.g.. But to use them in a serious scientific, and
especially in a critical way, one must also understand what the
scientists are trying to do on their end: especially why they think they
understand something that there are some prima facie reasons from our
ordinary conceptual life for thinking that they can't understand.  You
can't do conceptual therapy if you don't understand what concepts are in
play.

Regards,

Sean C. Stidd
Lecturer in Philosophy
Wayne State University