What Russell meant by "traditional pure mathematics"

Timothy Y. Chow tchow at math.princeton.edu
Tue Sep 6 11:37:43 EDT 2022

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":


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 

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"?


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