FOM: Tragesser on fom and sheep

J P Mayberry J.P.Mayberry at
Sat Feb 21 13:21:28 EST 1998

	It seems to me that Robert Tragesser, in his challenge to Steve 
Simpson, Harvey Friedman, et al., has firmly grasped the wrong end of 
the stick. He writes

	"The point is that most mathematics operates within
	conceptual frames that are phenomenologically quite
	distinct from set-constructs"

	Now I'm not entirely sure what he means by 
"phenomenologically", but there is clearly a sense in which what he 
says is true. Each branch of mathematics has its own ethos: its own 
collection of central concepts and techniques of argument. But that is 
quite beside the point here. For in every case those concepts are 
defined, ultimately, in terms of the concepts of set theory, and those 
techniques of argument are justified, ultimately, in terms of the basic 
principles of set theory. Set theory is the *foundation* on which those 
branches of mathematics rest, not a "probe" that we use to investigate 
otherwise autonomous theories.
	Maybe Tragesser had something like real analysis in mind when 
he spoke about mathematics that "operates within a conceptual frame 
that is phenomenologically quite distinct from set-constructs". Isn't 
real analysis an autonomous theory? Wasn't it already a going concern 
long before set theorists appeared on the scene? But we all know *that* 
story. The "arithmeization" of analysis - the purging of its 
foundations of any *logical* dependence on old fashioned intuitive 
(anschaulich) geometry - was a project that the greatest mathematicians 
of the last century took part in.
	How do you cut the Calculus free of any logical dependence on 
old fashioned geometry. It isn't easy. It required great effort by 
mathematicians of the stature of Weierstrass, Dedekind, Cantor, 
Russell, Zermelo, . . . to accomplish this. And when they had finished, 
Lo and Behold, most of the results of their predecessors found a 
place, in new guises, in the radically transformed theory.
	Nowadays when we teach begining analysis to our students our 
principal task is to get them to think analytically, that is, to get 
them to see how to convert naive arguments in anschaulich geometry into 
rigorous arguments based on real algebra and set theory. And, of 
course, the algebra of real numbers must itself be developed using the 
axiomatic method which, in turn rests on set theory. So set theory lies 
at the base of the whole theory of real analysis.
	To be sure, when professional mathematicians do real analysis 
they *think* in naive geometry (with a mixture of real algebra and set 
theory thrown in). This is "phenomenologically" distinct from a purely 
structuralist ("set-theoretical") view of the subject. No doubt it is 
also a necessary psychological and heuristic aid to thinking about 
difficult matters. But the logical basis for their proofs must be 
purely analytical (based on real algebra and set theory), not 
geometrical in the old fashioned sense of anschaulich geometry. 

J P Mayberry
J.P.Mayberry at

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