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 <ross.street@mq.edu.au> 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
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