FOM: Shipman on introducing concepts

[John Mayberry]

	Joe Shipman wrote in his posting of 30 April 99 "It should be 
possible in a foundational work like 'On Numbers and Games' simply to 
INTRODUCE a concept like ordered pair with the appropriate notation and 
properties, without finding an equivalent in the language with only the 
epsilon relational symbol."
 	It seems to me, however, that it is precisely in foundations 
that this sort of "introduction" of concepts is completely 
illegitimate. Why should mathematicians be granted license to 
"introduce" objects and concepts that no one would dream of granting to 
theologians, say?
 	In his subsequent reply to Martin Davis, Shipman appeals to the 
distinction between *meta*-mathematics, which deals with foundational 
matters, and *ordinary* mathematics where it is ok just to "introduce" 
primitive notions. But surely the foundations of mathematics are a part 
of mathematics ("ordinary" is redundant here), in fact, the part of 
mathematics which explains, among other things, how the axiomatic 
method works, and therefore how it is that geometers can "introduce" 
points at infinity, algebraists can extend their fields by 
"introducing" roots of polynomial equations, and Conway can "introduce" 
his pairing relation. This means that in foundations we cannot just 
carry on using familiar axiomatic techniques as if they required 
neither analysis nor justification. For they do require analysis and 
justification, and therefore we cannot use the conventional axiomatic 
method to *establish* the foundations of mathematics, although, of 
course, we can use it to *study* those foundations mathematically, once 
they have been established (eg, we can formalise set theory with a view 
to discovering limitations on what we can prove).
	 It is precisely when we are dealing with *foundations* that we 
come up against basic concepts which cannot themselves be defined, but 
upon which all mathematical definitions ultimately rest, and basic 
principles (axioms, in the true and original sense of "axiom") which 
cannot themselves be proved, but to which all mathematcal proofs 
ultimately appeal. Because as mathematicians we deal with rigorous 
proof and rigouous definition, it is not open to us lay down our basic 
concepts as "primitive notions" without worrying what they mean, or our 
basic principles as "formal axioms" without worrying about whether they 
are true. If our basic concepts are not meaningful, then none of our 
concepts are meaningful, and if our basic principles are not true then 
none of our theorems are true either.
 	

John Mayberry
School of Mathematics
University of Bristol

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John Mayberry
J.P.Mayberry at bristol.ac.uk
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