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charset="utf-8" Content-Transfer-Encoding: quoted-printable 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-lawver= e, 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=E2=80=AFPM Ross Street wrote: > > =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D > "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? > =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D > > 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 catego= ry > 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 mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | = Learn more about Microsoft 365 Groups --MCBoundary=_12310292112300491 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=UTF-8 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.substa= ck.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.substa= ck.com/p/monoid ;)

On Sun, Oct 29, 2023 at 12:01=E2=80=AFPM Ross Street <ross.street@mq.edu= .au> wrote:
>
> =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
> "Yoneda showed that maps in any category can be
> represented as natural transformations" (Lawvere & Schanuel,<= br> > Conceptual Mathematics, p. 378). Isn't this reason enough to think of<= br> > category theory as the theory of naturality?
> =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
>
> 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 cat= egory
> theory was to define natural transformation. That required functor, an= d
> that required category.
>
> Ross
 
 
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