Re: Name the thesis: "Formal sentences capture informal ones"

From: Stephen Harris (cyberguard1048-usenet_at_yahoo.com)
Date: 02/03/05 

Date: Thu, 03 Feb 2005 17:25:50 GMT

"Jamie Andrews; real address @ bottom of message" <me@privacy.net> wrote in
message news:36a8vlF4v8123U1@individual.net...
>>> In comp.theory tchow@lsa.umich.edu wrote:
>>>> (*) Formal sentences (in PA or ZFC for example) adequately express
>>>> their informal counterparts.
>>>> Any candidates for a catchy name for (*)?
>
>> "Jamie Andrews; real address @ bottom of message" <me@privacy.net> wrote
>> in
>> message news:367nq8F4qcoivU1@individual.net...
>>> Is this not the central tenet of "logicism"?
>
> In comp.theory Stephen Harris <cyberguard1048-usenet@yahoo.com> wrote:
>> Logicism is the theory that mathematics is an extension of
>> logic and therefore all mathematics is reducible to logic.
>> Kurt Gödel's incompleteness theorem ultimately undermined
>> the purpose of the project. The attempted resurrection of
>> this theory is styled neo-logicism.
>
> Please cite Wikipedia when you copy from it.
>
> I think the Wikipedia entry might not be entirely accurate.

Point well taken. Some sources more likely to be accurate:

http://www.helsinki.fi/collegium/eng/Raatikainen/godelfinal.pdf

"There has been some dispute on the issue as to whether Godel's
theorems conclusively refute logicism, that is, the claim that
mathematics can be reduced to logic, as endorsed, for instance,
by Frege and Russell. Obviously this issue dependsheavily on how
one understands the essence of logicism. Clearly Gödel's theorems
show that all arithmetical truths are not reducible to the
standard first-order logic, or indeed, to any recursively
axiomatizable system. On the other hand, one may restrict the
logicist thesis to some class of mathematical truths (such as
known truths, or humanly knowable ones), and/or extend the
scope of logic. There is, though, the threat that the issue
becomes trivial or wholly verbal. ...

Sternfeld (1976) and Rodríguez-Consuegra (1993), on the other
hand, argue that it is possible to defend logicism even after
Godel's theorems. Sternfeld and Rodríguez-Consuegra appeal to
the fact that Godel's theorems do not provide an absolutely
undecidable statement, but only a relative one. This is
certainly true. Yet this defense apparently collapses logicism
into the view that every mathematical truth is derivable in
some formal system. This, however, makes the thesis completely
trivial. Furthermore, would this not imply that not only
mathematics but also all empirical facts are "logically true"?

Geoffrey Hellman (Hellman 1981, see also Reinhard 1985) has
analyzed the bearing of Gödel's theorems on logicism in more
detail. Hellman focuses only on the thesis that knowable
mathematical truth can be identified with derivability in
some formal system. Logicism so understood cannot be directly
refuted by Gödel's first theorem.

Hellman subsequently gives a considerably more complicated
argument which leans on Gödel's second theorem, and breaks
the argument down into two cases. First, he concludes that
no finitely axiomatizable logicist system exists. Second, he
considers non-finitely axiomatizable systems, and here the
claim is weaker: such logicist systems may exist, but Godel's
second theorem prohibits our being able to know of any
particular system that it is one of them.5 Hellman's argument
has the advantage of not depending on any particular
restrictive way of drawing the controversial line between
logic and non-logic."
----------------------------------------------------------

http://math.stanford.edu/~feferman/papers/logiclogicism.pdf

(page 12) "One should be aware that the notion of
absoluteness is itself relative, and is sensitive
to a background set theory, hence again to the
question of what entities exist. For examples of
absolute operations which are patently set-theoretical
yet come out as logical on the Tarski-Sher thesis,
see the just mentioned reference. ...

It is undeniable that the relation of identity has a
"universal", accepted and stable logic (at least in the
presence of totally defined predicates and functions,
as is usual in the PC with =), and that argues for giving
it a distinguished role in logic even if it should not
turn out to be logical on its own under some cross-domain
invariance criterion, such as under homomorphisms.
Of course, even if a form of the latter is accepted as a
criterion for logicality, one is still free to consider
the operations which are defined from I by those provided
in Theorem 6. That of course buys one the quantifiers E_K
for ? finite, but not those for ? infinite, whose loss is
discussed separately, next." (pages 21 & 22)

It seems to me clear that the cardinality quantifiers E_?
for ? uncountable belong to mathematics (specifically, set
theory) and not to logic; they are all excluded by the
homomorphism invariance condition, along with the E_? for
? countable. As just remarked, the finite ones are recovered
once one includes the identity I. The quantifier"there exist
infinitely many", for ? = Aleph_0 is a borderline case to
which intuition and experience do not provide a clearcut
answer as to its status. It can, however, be assimilated to
logical notions under the homomorphism invariance criterion
simply by restricting one's consideration to those operations
which are invariant over infinite domains M_0, without
thereby including the E_? for ? uncountable. The
"completeness" argument for logicality (suggested by Quine
in the case of =) here gives quite anomalous results, since
one has a complete logic for E_? for the case that ? = Aleph_1
by the work of Keisler (1970) while, as is well known, there
is no such logic for the case that ? = Aleph_0.

I also agree with Quine (1986, pp. 64 ff) that second-order
and higher order quantification go beyond the bounds of logic.
He takes these (famously) to be "set theory in sheep's
clothing", and it is certainly true that the understood
meaning of such quantifiers depends on what sets exist, or
alternatively - if such quantifiers are regarded as binding
predicate variables- of what predicates exist.

6 Tarski and Boolos on logicism.

In his "What are logical notions?" lecture that was the
starting point for this paper, Tarski concluded with a
discussion of its relevance to the logicist program, as
follows:

"The question is often asked whether mathematics is a part
of logic. Here we are interested in only one aspect of this
problem, whether mathematical notions are logical notions,
and not, for example, in whether mathematical truths are
logical truths, which is outside our domain of discussion.
(Tarski 1986, p. 151)"

SF: His answer is, curiously: "As you wish"! The argument
is that since "the whole of mathematics can be constructed
within set theory, or the theory of classes", and since
"all usual set-theoretical notions" can be defined in terms
of the relation of membership, the determination comes down
to whether membership is a logical notion. *11 But -Tarski
goes on- two methods have been provided for the foundations
of set theory following the discovery of paradoxes in that
subject, namely the theory of types as exemplified in
Principia Mathematica (which he takes implicitly in unramified
form), and axiomatic set theory as formulated by Zermelo,
et al. If one follows the method of the theory of types then
membership is a part of logic, since it is invariant under
the extension to higher types of any permutation of the domain
of individuals. On the other hand, if axiomatic set theory is
followed, there is "only one universe of discourse and the
membership relation between its individuals is an undefined
relation, a primitive notion." On that account, membership is
not a logical notion, since as Tarski had shown earlier, there
are only four permutation-invariant relations between
individuals, the universal relation, the empty relation, the
identity relation and its complement.

{*11. It is also curious that Tarski ignores the fact due to
his fundamental result on the nondefinability of truth-in-L
within a language L, that the mathematical notion of truth
of sentences of the language of set theory cannot be defined
within set theory (and similarly for type theory).}

Tarski winds up these considerations as follows:

"This conclusion ["As you wish!"] is interesting, it seems
to me, because the two possible answers correspond to two
different types of mind. Am onistic conception of logic,
set theory, and mathematics, where the whole of mathematics
would be a part of logic, appeals, I think, to a fundamental
tendency of modern philosophers. Mathematicians, on the other
hand, would be disappointed to hear that mathematics, which
they consider the highest discipline in the world, is a part
of something so trivial as logic; and they therefore prefer
a development of set theory in which set-theoretical notions
are not logical notions. The suggestion which I have made
does not, by itself, imply any answer to the question of
whether mathematical notions are logical."(Tarski 1986,p.153)

SF: Though Tarski's consideration only of the question
"whether mathematical notions are logical notions" and not
of "whether mathematical truths are logical truths" appears
at first sight to be a reasonable one, it is not clear that
the two can be separated so neatly. For, any argument one
way or the other about the first question must necessarily
invoke assumptions about various properties of the notions
involved, and those lead one into the second question.

By contrast, Boolos shows that FA, which does not use
extensions, is consistent. Then, following the lead of
Wright (1983), he shows that "(o)nce Hume's principle
is proved, Frege makes no further use of extensions."
(Boolos 1998, p. 191). In his discussion of the
significance of this work, Boolos comes to the following
provocative conclusions (op. cit., p. 200): "(1) Numbers
is no logical truth; and therefore (2) Frege did not
demonstrate the truth of logicism in the Foundations of
Arithmetic. (3) Logic is synthetic if mathematics is,
because (4) there are many interesting, logically true
conditionals with antecedent Numbers whose mathematical
content is not appreciably less than that of their
consequents." And he adds to these: "(5) Since we have
no understanding of the role of logic or mathematics in
cognition, the failure of logicism is at present quite
without significance for our understanding of mentality."

In view of my working identification of logic with the
first-order predicate calculus PC, I am in agreement with
(1) and (2). I am more or less in disagreement with (3),
though I don't have strong feelings about what being
synthetic amounts to. I don't see (4) since all results
of mathematics can be represented as logical consequences
of mathematical hypotheses. As to (5), I agree with the
conclusion, but not the premise; it seems to me that we
do have some understanding of the role of logic, and to
some extent of mathematics, in cognition, though we
surely have much farther to go in both respects. To
reiterate my introductory remarks, I think that the
theoretical study of what a logical operation is, and
hence of what the scope of logic is, must be connected
with the more empirical study of the role of logic in
the exercise of human rationality. I am optimistic that
a better understanding of either will inform the other."

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