Russell vs Hilbert

[Richard Kimberly Heck]

On 12/15/20 4:47 PM, Joe Shipman wrote:
> One of my Twitter friends claims that the view that all mathematical
> propositions are decidable from a few axioms was held by Russell prior
> to Hilbert, and that Russell was claiming, in his books, not only to
> have reduced all mathematical reasoning to a few logical principles,
> but also to have claimed that those principles were sufficient to
> settle all questions.
>
> This is based on his reading of the following 1903 passage:
>
> "THE present work has two main objects. One of these, the proof that
> all pure mathematics deals exclusively with concepts definable in
> terms of a very small number of fundamental logical concepts, and that
> all its propositions are deducible from a very small number of
> fundamental logical principles" 
>
> In my opinion, by “propositions”, Russell meant “theorems”, not “true
> statements”, and therefore one may not jump to the conclusion that
> Russell failed to achieve his object and did not prove what he claimed
> to be trying to prove.
>
> Can any Russell experts shed light on this question?

I'm not an expert on Russell specifically, but I do know something about
this period, and there's a distinction I'd like to emphasize.

But first: Russell lived for almost 40 years after Gödel's proof of the
incompleteness theorem. I'd be stunned if he never had something to say
about it---and hopefully actual Russell experts will chime in on that.
I'm hopeful, too, that, if he did have something to say, it was more
sensible than what Wittgenstein had to say (though various apologists,
no less than Hilary Putnam, have tried to make some sense of what he did
say).

So, first, Russell's usage of 'proposition' is notoriously ambiguous.
Second, what Russell thought in 1903 (indeed, in some month in 1903) is
not necessarily indicative of what he thought at any other time. His
views changed rapidly then, and for a long time after. Third, I strongly
suspect myself that you are right, Joe, that what Russell meant was that
all /known/ theorems of mathematics could be derived from the axioms of
PM. That is what he actually sets out to show. Fourth, certainly by
/Principia/ (a decade later), Russell was aware that some /proofs/ of
accepted results depended upon what he called the 'multiplicative axiom'
(the axiom of choice), and he surely would have understood that this
dependence might be essential. So, fifth, it's hard for me to see how
Russell could not have understood, at least as an abstract possibility,
that there might be established theorems of mathematics that could not
be proven given the specific axioms he had identified. That is what we
might call 'boring incompleteness'. The obvious response would be to add
some new axioms, though it might take real work to find out which axioms
it would make the most sense to add. (Frege considers such a
possibility, explicitly, in "Mr Peano's Conceptual Notation and My Own",
which was one of several off-prints that he sent to Russell in response
to Russell's first letter to him.)

What Russell surely did not anticipate was /essential/ incompleteness.
Even Gödel does not properly formulate that result in his 1931 paper, as
he acknowledges in a famous footnote: Until Turing's work, we lack a
good account of what a 'formal' theory is and so a good account of
essential incompleteness.

That's the crucial distinction. And I assume it must also apply to
Hilbert. Surely he would have understood, at least as an abstract
possibility, that the specific axioms he'd identified might be
insufficient to decide all arithmetical questions. Sp just add some new
ones! But what he may not have fully appreciated was the possibility
that this insufficiency might be irresolvable. It takes some thinking to
understand how that can be so.

Riki

-- 
----------------------------
Richard Kimberly (Riki) Heck
Professor of Philosophy
Brown University

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