FOM: Comments on a paper by Hartry Field
Torkel Franzen
torkel at sm.luth.se
Wed Feb 25 09:24:35 EST 1998
WHAT THIS POSTING IS ABOUT
Hartry Field has written a paper entitled "Which Undecidable
Mathematical Sentences Have Determinate Truth Values?" which he has
commendably made available on the web, thus increasing its readership
by at least one. The title of the paper has a fomish ring to it, and
on reading it, I find it relevant to matters that have been discussed
on the list. Hence the comments below.
The paper can be found at
http://www.nyu.edu/gsas/dept/philo/faculty/field/papers/truthvalues.html
There are two main claims argued for in the paper:
(i) "there is a natural account of how our practices might give
determinate content to undecidable but f-decidable sentences that
does not extend to the typical undecidable sentences of set
theory"
(ii) "consideration of Godel's theorem gives no decisive reason for
thinking that some undecidable sentences have determinate truth
value".
Here an "undecidable mathematical sentence" is explained as "a
mathematical sentence such that neither it nor its negation follows
in first order logic from our fullest mathematical theory", and
an f-decidable sentence is one that is "a semantic consequence (in
this sense) of our fullest mathematical theory".
(i) and (ii) naturally prompt many questions about "our fullest
mathematical theory", "semantic consequence", "determinate", and other
concepts. To avoid getting bogged down in such issues, I will state some
objections to (i) and comments on (ii) in fairly general terms.
(1) COSMOLOGICAL ASSUMPTIONS AND THE QUANTIFIER "THERE ARE FINITELY MANY".
It's a common view, discussed earlier on the list, that the
continuum problem - and in particular the question of the truth or
falsity of the continuum hypothesis - doesn't necessarily have any
answer in the (mathematical) nature of things, because of a lack of
determinacy in the concept "subset of the natural numbers". Because of
this indeterminate character of the notion of (arbitrary) subset of
the natural numbers, statements about the totality of such subsets
need not have any determinate truth value.
This view is often coupled with a different view of the totality of
natural numbers. Unlike the totality of subsets of the natural
numbers, this is fully determinate, and therefore also statements
quantifying over the natural numbers - arithmetical statements - have
a determinate truth value.
In Field's paper, this common distinction between set theory and
arithmetic is accepted as a starting point. (He doesn't, and I won't,
consider the intuitionist point of view that the natural numbers
forming a fully determinate totality doesn't in itself entail that
statements about the natural numbers have a determinate truth value.)
Field proposes a justification of the distinction.
The basic argument is that if there is a specifiable omega-sequence
in (physical) nature, then the determinacy of the mathematical concept
of finiteness is secured through the (assumed) determinacy of the
physical concepts used in a characterization of finiteness. He gives
an example: If we assume
(A) there are arbitrarily long finite sequences, but no infinite
sequences, of events with a first and a last member (in time)
and a separation in time between any two of its members of at
least one second
then it follows that the meaning of "there are finitely many x such
that A(x)" is determinate, through the equivalence
(B) there are finitely many x such that A(x) iff there is an injective
function from the set of x such that A(x) to some set of events
which has an earliest and a latest member, and is such that any two
of its members occur at least one second apart.
Now, in considering the merits of this sort of thing, one needs some
idea of in what sense the quantifier "there exist finitely many" might
be held to be "indeterminate". Field mentions vagueness as one type of
indeterminacy, using "bald" as an example, but as has been commented
on earlier on the list, vagueness in this sense is hardly at issue in
connection with standard mathematical concepts. It's not as though we
had some numbers 0,1,2,.. which are clearly finite numbers, and a
number omega which is clearly not, and a range of uncertainty in
between, where we can't say definitely whether a proposed item is a
finite number or not. Such vagueness does arise if we consider
"feasible numbers", but notions of feasibility are explicitly excluded
from classical arithmetic.
People who suggest that there is an indeterminacy in "finitely many"
most often, in my experience, base this view on the existence of nonstandard
models of first order theories of arithmetic. They suggest that we
can't say or do anything to single out a standard model of arithmetic,
or equivalently, a standard meaning of the quantifier "finitely many",
and that the very meaning of "finitely many" is therefore indeterminate.
(This is the kind of argument that Field mentions in his paper in
connection with Putnam's "Models and Reality".)
Without at all considering whether such a view is justified, the question
then is whether Field's argument should convince somebody who holds this
view that "finitely many" is in fact determinate if (A) is true.
Field says that the argument from (A) to "'finitely many' is
determinate" need not itself presuppose that "finitely many" or (A)
has a definite meaning, since it is quite possible to carry out
conclusive arguments using concepts that are not determinate. While
this is true enough as a general observation (and well illustrated by one
of Kreisel's old favorites, the argument by which validity in an
informal and indefinite sense is proved to coincide with validity in
the set-theoretical sense), it's not clear that it applies in the
present case. The idea that (A) implies - via (B) - that
"finitely many" is determinate rests on the assumption that "our
physical vocabulary is quite determinate". We may accept that our
physical vocabulary is quite determinate, so that no misgivings
analogous to those in the mathematical case arise concerning the
meaning of "event" or "time" or "precedes in time" or "one second
apart". But this doesn't extend to the use of the indeterminate
"finite sequence", whether we are talking about sequences
of events or of numbers. So why should (B) lead us to the conclusion
that "finitely many" is determinate?
Field's attempt to substantiate this conclusion from (B) is by way
of models. Let the theory S include (impure) set theory and (A). (B)
is provable in S. Then, assuming (A), for any model of S in which
"event" etc are given their standard and determinate interpretations,
if "there are finitely many x such that A(x)" is true in the model,
then A(x) is in fact true in that model only for finitely many x. (Thus
number-theoretic sentences have the same truth value in every standard
model of S, which yields the "undecidable sentences with determinate
truth values" of the title.)
There is nothing wrong with the form of this argument, but it
contains nothing to convince somebody who regards "there are finitely
many" as indeterminate that the cosmological assumption (A) implies
that "there are finitely many" is determinate. Events are indeed (we
may assume) quite determinately referred to in our speech, but this
doesn't make statements such as "there are finitely many events such
that..." any more determinate than "there are finitely many natural
numbers such that...". The whole implication "if (A) then 'finitely
many' means 'finitely many' in every model of S" doesn't tell us
anything about what we ourselves mean or could mean by "finitely
many", however determinate the realm of events. It only tells us that
on any reasonable understanding of the indeterminate "finitely many",
it will be true that if (A) holds on that understanding, then
"'finitely many' means the same in every model of S" will hold on that
same understanding.
In short, my conclusion regarding (i) is that Field hasn't actually
exhibited any "practices" that "give determinate content" to
"finitely many" on the assumption that (A) holds. (This doesn't mean
that I take the view that there is anything indeterminate about
"finitely many". On the contrary, I would argue that "finitely many"
isn't indeterminate at all.)
(2) GODEL'S THEOREM AND "OUR FULLEST MATHEMATICAL THEORY"
Here I'll be brief. It's a natural reaction when somebody talks about "our
fullest mathematical theory" to say that Godel's theorem shows that
there isn't any such, since any theory which we recognize as correct
can be extended to a stronger theory which we still recognize as
correct. In response, Field again invokes vagueness:
This objection seems to me misguided: the most one can get
from Godel's theorem is that 'our fullest mathematical
theory' is vague, and that for each consistent and
recursively enumerable theory M that is a pretty good
candidate for its denotation, M U {G(M)} (or a theory that
includes that) is also not bad as a candidate for
its denotation. Compare 'bald': under the usual crude
idealization that baldness depends only on the number of
hairs on the head, we know that if {x|x has less than n hairs}
is a pretty good candidate for the extension of 'bald', then
{x|x has less than n+1 hairs} isn't a bad candidate either.
This is somewhat startling. After all, it wouldn't be very convincing
to say that the proof that there is no largest natural number strictly
speaking only shows that for any pretty good candidate for "largest
natural number" N, the number N+1 will also be a pretty good candidate.
To understand why Field invokes vagueness, we need to look at his
explanation of "our fullest mathematical theory". He says that by
"our fullest mathematical theory" is meant "something like 'the set
consisting of all our explicit mathematical beliefs, plus perhaps
those mathematical sentences we could easily be brought to
explicitly believe, plus perhaps their logical consequences'".
This description is indeed a bit vague, in view of the "easily be
brought to believe", but it's also unclear, since people disagree on
how "belief" enters into the use of e.g. the axioms of ZFC in
mathematics, and since "brought to explicitly believe" is open to
various interpretations. It emerges later that Field also presupposes
that to the extent that we speak of the sentences of "our fullest
mathematical theory" as true, axioms for this truth predicate will
also be part of "our fullest mathematical theory". Since he doesn't
actually suggest any candidate for "our fullest mathematical theory",
I'm not prepared to dispute his assertion that Godel's theorem does
not yield any true arithmetical statement not provable in "our fullest
mathematical theory". Rather, it's unclear to me that it makes any
sense to speak of a Godel sentence for "our fullest mathematical
theory".
Thus, what the second part of the paper demonstrates, to my mind, is
that the notion of "our fullest mathematical theory" and of
decidability in such a theory are such that the claim (ii) is not very
controversial. It's simply not clear that anybody would want to claim
that "consideration of Godel's theorem give a decisive reason for
thinking that some undecidable sentences have determinate truth value"
in Field's sense. However, this dubious character of the notion of
"our fullest mathematical theory" need not play any significant role in
our evaluation of (i).
---
Torkel Franzen, Computer Science, Lulea technical university
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