Dear Professors: Street, Rosebrugh, Lemay, Taylor et al., Thank you very much for positng my working-question (Lemay :) I'll write to you again after thinking through the relations between mathematical methods, models, theories, and examples, especially from your perspective (as it appears from your response, Lemay ;) I'll also write again after carefully studying Professor Street's presentation, which is about (the elemental?) natural transformations (as in: natural transformation is required to define functor which, in turn, is required to define category). For now, in the spirit of full disclosure, natural transformation, in the sense of structure-respecting maps, appear to account for the effectiveness of mathematics in natural sciences, along the following lines (open to their fate ;) 1. We are given 'change', which we objectify (e.g., physical constrasts (particulars) are sensed by featherless biped brains ;) objects are perceived; geometric objectification of objects as structures is made possible thanks to our minds (mental concepts i.e., properties along with their mutual determinations). 2. Given that a concept (abstract general) that is invariant across a given category of experiences (planned perceptions) is given in the given (change), surely, the given makes it possible to objecfity (the invariant of a category of the given changes). Isn't it yet another reason to reorient science/mathematics towards "the given" and away from its (pathalogical ;) fixation on) "exits" (see Rosebrugh & Lawvere, Sets for Mathematics, p. 240)? I look forward to your corrections (unvarinshed ;) Happy Weekend :) Thanking you, Yours truly, posina P.S. Professor Street, I recently started working my way, inspired by Professor by F. William Lawvere's Perugia Notes (https://conceptualmathematics.substack.com/p/perugia-notes-prof-f-w-lawvere, pp. 101-116), through the relation between Cayley (that you alluded to) and Yoneda (barely a baby-step: https://conceptualmathematics.substack.com/p/monoid ;) On Sun, Oct 29, 2023 at 12:01 PM Ross Street wrote: > > ================================================ > "Yoneda showed that maps in any category can be > represented as natural transformations" (Lawvere & Schanuel, > Conceptual Mathematics, p. 378). Isn't this reason enough to think of > category theory as the theory of naturality? > ================================================ > > That would be like saying group theory is the theory of permutations > (because of the Cayley theorem). > > Perhaps my little colloquium talk entitled > > ``The natural transformation in mathematics'' > > at > > http://science.mq.edu.au/~street/MathCollMar2017_h.pdf > > would be of some interest in this connexion. I am sure lots of us have > given similar talks. The goal of the paper considered the first in category > theory was to define natural transformation. That required functor, and > that required category. > > Ross You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files | Leave group | Learn more about Microsoft 365 Groups