Russell vs Hilbert

[Timothy Y. Chow]

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?

The quote comes from Russell's book, "The Principles of Mathematics."  In 
the same book, Russell says that pure mathematics is the class of all 
propositions of the form "p implies q."  This view, that the entire 
content of mathematics consists of assertions of implications, is usually 
called logicism.  The *conceptual* reduction of mathematics to logic is 
what Russell is concerned with in the book, not the claim that all 
mathematical truths can be *algorithmically* generated from a few axioms 
via some sequent calculus.  I think that it's anachronistic to read the 
phrase "its propositions are deducible from" in algorithmic terms; I would 
instead parse it as something like "its propositions can be reduced to" or 
"its propositions are analyzable in terms of".

But you can judge for yourself by reading the preface and the first 
chapter of Russell's book.

https://people.umass.edu/klement/pom/pom.pdf

Tim