The proposition that "mathematical structure is given by data and conditions" is so broad as to be vacuous from a foundational standpoint. While there may be disagreement between mathematical camps over whether algebraic frameworks rest on logical or vice versa, common to both is the idea that one starts with a (non-logical) language and equips it with a theory. I don't see how "data and conditions" can be interpreted as giving undue weight to either the equational or first-order starting points. The sentiment could just as validly preface a graduate course on first order model theory. Vaughan Pratt On 5/25/2012 3:09 PM, Ellis D. Cooper wrote:
I was permitted to audit a graduate course on category theory guided by Sammy at Columbia University in the early 1960s. I recall his insistence that mathematical structure is given by data and conditions. Is that idea implicit or explicit in Bourbaki? Has that idea been superceded? How does it relate to the development of algebraic theories as understood by Lawvere, Linton, Barr-Wells, the Elephant, and so on?
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