What Russell meant by "traditional pure mathematics"

Heck, Richard Kimberly richard_heck at brown.edu
Wed Sep 7 01:50:17 EDT 2022

What if what Russell meant by "the theory of natural numbers" was *higher
order* arithmetic? I.e., more or less the theory Goedel calls P? Russell
was not particularly concerned with first order anything. That's a later
development. And it more or less fits with how his reduction of analysis
actually goes. You get the reals a type up from the naturals, which are at
type 2.

That's related to the 'real mathematical result' in PM that Hardy mentions:
2^2^k is Dedekind infinite if k is infinite.


On Wed, Sep 7, 2022, 1:39 AM Timothy Y. Chow <tchow at math.princeton.edu>

> There is currently a historical discussion happening on MathOverflow, and
> a side issue that has come up is what Russell meant by "traditional pure
> mathematics" in his "Introduction to Mathematical Philosophy":
> https://mathoverflow.net/a/429823/
> My impression is that during that era, people tended to think of
> mathematics as consisting of three "layers": arithmetic, analysis, and
> infinite set theory.  For example, after Gentzen proved the consistency of
> arithmetic, one of his next goals was to prove the consistency of
> analysis.  So when I see Russell talking about reducing all "traditional
> pure mathematics to the theory of the natural numbers" and then about
> Peano providing axioms adequate for the natural numbers, I interpret him
> as equating "traditional pure mathematics" with "arithmetic."  In
> particular, I don't interpret Russell as saying that Peano's axioms
> sufficed to derive everything known at the time about real and complex
> analysis.
> But am I misreading Russell?  Did he think that real and complex analysis
> had been reduced to Peano's axioms?  I feel pretty sure that he didn't
> think that all of infinite set theory had been reduced to Peano's axioms,
> even though nowadays we would say that "pure mathematics" includes
> infinite set theory.
> Come to think of it, was there a sharp notion of what "analysis" meant
> back then?  When Gentzen was trying to prove the consistency of analysis,
> what precisely did he have in mind by "analysis"?
> Tim
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