Russell vs Hilbert

[Ignacio Añón]

I'm no Russel expert, merely an admirer.

In his autobiography, Russell mentions how disapointed he was, when meeting
Godel, Einstein, and Pauli, in finding them so biased towards german
metaphysics. He evidently felt that for him logic, was more of an
axiomatic, rigourous, technical thing, than it was for Godel, whom he calls
"an unadulterated platonist..."

Godel, who had studied Russell's technical work, in his typical
obsessive-maniacal fashion, after reading this coment by russell, said,
interestingly enough, in an unsent letter, that he was no more of an
"unadulterated platonist" than Russell was in the 1920's...

In the early days, Russell was evidently closer to meaning "theorems", than
"true statements", though these terms were not so well discriminated then,
it seems...

It was only with Rosser and Church, for instance, that primitive recursive
statements, were distinguished from reucrsive ones.

The relevant aspect to bear mind, when coming to understand the natural
growth of the axiomatic aproach, is that Godel came to prominence, as one
of the few world wide experts, who had digested, at the same time, both the
principia, and hilbert's work as presented in the grundzuge der
theoretische logik(in godel's original incompleteness paper, hilbert is one
of the few references used, if I recall correctly...)

Whether Godel destroyed any ilussion Russel and Hilbert supossedly had,
about finding a simplistic decision procedure, based on a set of axioms for
all math, is one of those questions, that smells like a waste of time, and
seems to spring from the desire to project a false, simplistic drama, into
the ambiguos, and complex, historical growth of modern logic...

I therefore agree with you, that Russell was not refuted by Godels results,
as you say:

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?

— JS



El mar., 15 de diciembre de 2020 21:56, Joe Shipman <joeshipman at aol.com>
escribió:
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?

— JS


El mar., 15 de diciembre de 2020 21:56, Joe Shipman <joeshipman at aol.com>
escribió:

> 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?
>
> — JS
>
>
>
> Sent from my iPhone
>
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