[FOM] Gödel's theorems and abstract notions

[Olivier Souan]

Having an interest in mathematical platonism, I am currently studying
Gödel's own
interpretation of his theorems (especially the second incompleteness
theorem), w.r.t. the foundation of
arithmetics.

§1. The disjunctive conclusion
------------------------------

Its most striking expression can be found
in the Gibbs Lecture (1951), where, after noticing the lack of formalization
of
most philosophical concepts, he says :

"The following disjunctive conclusion is inevitable : either mathematics is
incompletable in this sense, that its
evident axioms can never be comprised in a finite rule, that is to say, the
human mind (even within the realm of pure mathematics)
infinitely surpasses the powers of any finite machine, or else there exist
absolutely unsolvable diophantine problems of the type
specified (were the case that both terms of the disjunction are true is not
excluded, so that there are, strictly speaking, three
alternative.)" (Gibbs Lecture, CW III, p.310)

Taking this interpretation as a given, I still have some trouble figuring
out its precise philosophical and technical consequences, all the more as
Gödel is sometimes quite reluctant to admit the existence of "absolutely
unsolvable diophantine problems".

But let's focus instead on the first disjunct only (antimechanism). It reads
:

(G1a) "the evident axioms of mathematics can never be comprised in a finite
rule"

(G1b) "that is to say, the human mind (even within the realm of pure
mathematics) infinitely surpasses the powers of any finite machine."

The transition from (G1a) to (G1b), i.e. from the Foundations of Mathematics
properly speaking to the philosophy of mind, is not immediate. It is made
"by combining the proof of this result with Turing's theory of computing
machines", as Gödel writes in a draft letter to Times (1963) (Communicated
by Mark van Atten)

However (G1a) also deals with the epistemology of mathematics, since Gödel
speaks of  e v i d e n t  axioms. That means that mathematical intuition
("evidence"), which deals with the abstract notions contained in the axioms
of mathematics (cf "What's Cantor's
continuum problem?"), cannot be formalized the way Hilbert wants to.

Thus, according to Gödel, mathematical intuition include a non mechanical
and non computational element. It consists in   a b s t r a c t   n o t i o
n s, which are irreducible to Hilbert's concrete and finitist formalization.
What are those elements?


§2. Abstract notions and mathematical axioms
----------------------------------------------

Gödel explicitely links those notions with the axioms of mathematics and the
consistency of arithmetics (and beyond). He even gives a list of those
notions :

"in order to prove the consistency of classical number theory (and 'a
fortiori' of all stronger systems) certain 'abstract' concepts (and the
directly evident axioms referring to them) must be used, where 'abstract'
means concepts which do not refer to sense objects" ; note 27 : "examples of
such abstract concepts are, for example, 'set', 'function of integers',
'demonstrable' (the latter in the non-formalistic sense of 'knowable to be
true'), 'derivable', etc., or finally 'there is', referring to all  p o s s
i b l e  combinations of symbols."(Gibbs Lecture, 1951, CW III, p.318)

I guess that his 'antimechanism' rests on the mental existence of such
abstract notions, which are irreducible to concrete hilbertian-style
elements. But
a new problem arises : are those notions finitary or not?

On the one hand Gödel writes that his results were established using a
"highly
transfinite notion of mathematical truth" (To Wang, dec 7, 1967). He further
asserts that "in mathematical reasoning the non-computational (i.e.
intuitive element consists in intuitions of higher and higher
infinities)"(To Tillich, june 1963 - thanks to Mark van Atten for having
pointed this one to me).

On the other hand, he did promote the use of *abstract concepts*, and,
possibly, of *finitary abstract concepts.*  The Dialectica paper, version of
1972, note b, reads :  "there is nothing in the term 'finitary' which would
suggest a
restriction to concrete knowledge. Only Hilbert's special interpretation of
it makes this restriction." Indeed, Gödel has tried to find non-infinitary,
non-concrete,
abstract, finitary concepts beyond PRA which provides an extension of
finitism so as to give the consistency of arithmetics.

It is then not so easy to conclude that the mind
uses infinitary concepts or schemes of reasoning, insofar as Gödel focused
its efforts on proving the indispensibility of *abstract* concepts, rather
than proving
directly the indispensibility of *infinitary* ones.


§3. Abstract notions and the consistency of arithmetics.
-------------------------------------------------------

As it is well known, Gödel tried to solve the consistency of arithmetics by
using the framework of an "extended
finitism" using 'abstract notions.'

Here is my question : are those 'abstract notions' infinitary, at least in
the way they are used in the foundations of arithmetics? The answer seems to
be obviously 'yes', but the demonstration is still pending.

In fact, Gödel, at last in the 30s, considers those 'abstract notions' to be
constructivist. In the 1933 Cambridge Lectures (1933o in the CW), he
contemplates the
existence of intermediary layers between strict hilbertian finitism and
full-blown platonism.

In his Zilsel lectures (1938?), Godël hints to three different ways of
extending finitism :

a- Transfinite induction up to epsilon_0 (Gentzen)
b- The Modal-Logical interpretation
c- The Functional interpretation (Dialectica)

Historically, those three solutions take root in the constructivist
viewpoint. Initially, Gentzen's proof had a strong constructivist flavour,
and the functional interpretation presupposes intuitionism. This is not so
surprising, as hilbertian finitism and intuitionistic constructivism were
regarded in the early 30s, as the two only possible frameworks for the
foundation of mathematics.

Let's have a quick look at those three ways.

§3a Epsilon_0
----------------

As for (a), the status of  epsilon_0 (and of transfinite induction in
general) is not clear. Is it really infinitary or not? Here are some lines
of thought. Your comments are very welcome.

- Richard Zack (The Practice of Finitism, 2002), examining Ackermann 1924,
shows that transfinite induction was
already in use in the hilbertian school. But as far as I can see, Ackerman
only uses omega ^omega^omega, and not at all inaccessible ordinals, the
first one of them being
epsilon_0. Thus the use of transfinite induction by the Hilbertian school is
not sufficient to prove the 'finiteness' of epsilon_0.

- In the Zilsel Lectures, Gödel affirms that within finitary
number theory (PRA) some ordinals of the second number class (up to
omega^omega) can be defined and used, as well as proofs relying on them. But
epsilon_0 is out of reach of PRA.

- The ordinal numbers can be understood as canonical representatives for
well orderings (up to isomorphism). They are only notational, and not really
infinitary.

- On the contrary, Kreisel ("Ordinal logics and the
characterization of informal notions of proof" (1958)), clearly considers
epsilon zero as the first 'really' infinitary ordinal, and puts it
beyond the reach of the idealization of concrete processes.

So what are we to think of epsilon_zero and Gentzen's proof? Do we remain
in finitism (although a non hilbertian version) or not? Is there some
literature on that question?

§3b. The 'modal logical' solution
--------------------------------

 The
Modal Logical route
offers a consistency proof for PA relatively to HA (Heyting Arithmetics).
But what is the proof
theoretical ordinal of HA? Anyway, as for the modal logical route, Gödel
complains that it requires the
infinitary notion of "totality of all possible proofs". ("The present
situation in the foundation of mathematics"(1933(o) Cambridge Lecture))

§3c. The functional interpretation.
----------------------------------

Gödel also introduces the abstract notion of computable fonctional of finite
type to give an interpretation of HA. This interpretation is more precise
(definite) than the intuitionistic notion of proof, and claims to be
finitary. But Gödel freely uses the a universal quantification on all finite
types, as well as impredicativity. It then seems that his functional
solution uses
infinitary concepts.

At the end of his lecture at Zilsel, Gödel adds that "the epistemological
importance, as a better foundation, is very diminished by the fact that
those different systems are not contained in the finitary theory of
numbers"(ie, PRA). But 'finitary', here, means 'finitary in Hilbert's
sense', not infinitary per se, and leaves open the possibility of an
abstract
extension of finitism, in a constructivist (and perhaps intensional) sense.

Anyway, it seems to me that those pretensions to finitism are somewhat
unnatural, and cover some hidden ontological commitment to infinitary
notions. If abstract concepts can be finite, infinitary abstract
notions are required for the foundation of arithmetics. I would like to have
more arguments and more conceptual refinement to prove that position. What
do you think?


--------------------------------------------
Olivier Souan
Writing a PhD on "Mathematical Platonism"
Université Paris IV Sorbonne
Paris, France
---------------------------------------------