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Comment on On elements in category theory by Conceptual Mathematics
https://matteocapucci.wordpress....Professor F. William Lawvere discusses (on p. 12, following the above quote) how to bust the myth based on nailed-down descriptions of special objects 1, 2, and 3, which serve as domains for the arrows that are the objects, maps, and commutativities; the corresponding structure made of these component objects along with the structural maps between them is displayed in Fig. 8; https://zenodo.org/records/7164047). We also need another component object 4 (associativity), in addition to the above 1, 2, and 3 (https://github.com/mattearnshaw/lawvere/blob/master/pdfs/1966-category-of-categories-as-a-foundation-for-mathematics.pdf, p.8).
24.1.2024 05:11Comment on On elements in category theory by Conceptual Mathematics
https://matteocapucci.wordpress....Comment on I’ve been posting on LocalCharts by Conceptual Mathematics
https://matteocapucci.wordpress....My hearty congratulations Sir! I'm very happy to see you focus on reflective subcategories; one of these days I'll write to you requesting your help in constructing a category of Reflecting (with structure-semantics adjoints as objects) to complement the more familiar categories of Being (unity / reflexive graphs) and of Becoming (change / dynamical systems). In the absence of a/the category of Reflecting, the so-called cognitive science will continue to be a failed enterprise (https://drive.google.com/file/d/1hqPzhQhUWaIm5X_L1mvzVVLegeR7s3g4/view?usp=sharing).
24.1.2024 03:33Comment on I’ve been posting on LocalCharts by Conceptual Mathematics
https://matteocapucci.wordpress....Comment on No, the Yoneda lemma doesn’t solve the problem of qualia. by Keith E Peterson
https://matteocapucci.wordpress....In reply to <a href="https://matteocapucci.wordpress.com/2023/07/15/no-the-yoneda-lemma-doesnt-solve-the-problem-of-qualia/#comment-152">mattecapu</a>.
My point is that you need to have all mutations of altered color-perception AND instinctual responses that appropriately respond to this change of color-perception, which is extremely unlikely. Having an inverted spectrum of perception, but only instinctual responses to non-inverted perception wont activate the response in that individual, putting them under negative evolutionary pressure. Likewise, having a non-inverted spectrum of perception, but only instinctual responses to inverted perception wont activate the response in that individual either, also putting them under negative evolutionary pressure. Thus, these instinctual responses to color-perception are enough to fix an (admittedly probabilistic) equalizer from a space of instinctual responses and color-perception, with only probabilistic isomorphic arrows between spaces instinctual responses and probabilistic isos between spaces of color perception. Unless you believe biological variation behaves like magic, this does fix or 'overdetermine' an (admittedly probabilistic) equality given by the induced equalizers. And equalizers are indeed able to be described via Yoneda. So your assertion that Yoneda lemma can only discern at the level of isos for this particular category is ill-posed, at least probabilistically.
Yes, you could claim that an individual could have all the mutations so that color-perception and all corresponding color-dependent instinctual responses that are iso to those in another person are present, but I'll counter that the series of mutations required to make such a square to commute is so improbable, it is not really worthy of consideration. Its like arguing any key can in theory open any locked door, because in theory keys and door-locks can be switched, even though the probability is exceedingly low.
Interestingly, this even gives an experiment test for the possibility of altered color-perception between individuals: search for individuals whose color-dependent normal or even supernormal instinctual responses mismatches that of their wider group, preferably clones, controlling for color-blindness. Search for altered genes or gene expression in those individuals genomes and see if when those genes are put in other individuals, the mismatch persists. If so, then there is grounds to think color-perception is truly mutable. So now it's an empirical question, not a mere philosophical one.
23.8.2023 21:23Comment on No, the Yoneda lemma doesn’t solve the problem of qualia. by Keith E Peterson
https://matteocapucci.wordpress....Comment on No, the Yoneda lemma doesn’t solve the problem of qualia. by mattecapu
https://matteocapucci.wordpress....In reply to <a href="https://matteocapucci.wordpress.com/2023/07/15/no-the-yoneda-lemma-doesnt-solve-the-problem-of-qualia/#comment-151">Keith E Peterson</a>.
You confuse subjective experience with objective experience. Replacing qualia with 'isomorphic' (whatever this means) ones, by definition, doesn't change their functional effects. Hence there wouldn't be any evolutionary advantage difference.
Also, as I argue in the post 'relations [...] overdetermine color perception' is not what Yoneda lemma says.
23.8.2023 08:20Comment on No, the Yoneda lemma doesn’t solve the problem of qualia. by mattecapu
https://matteocapucci.wordpress....Comment on No, the Yoneda lemma doesn’t solve the problem of qualia. by Keith E Peterson
https://matteocapucci.wordpress....*Knock knock*
“Who is there?”
“It’s evolutionary biology barging in to point out that if color perception is altered to some isomorphic remapping of colors, then evolved behaviours involving color would put those with altered perceptiond at an evolutionary disadvantage, i.e. seeing red as green would cause them to increase their likelihood of dying from handling poisonous creatures, plants, or fungi, due to a diminished avoidance response.”
In other words, the relations between the colors of things in nature, psychological response crafted by the past survival of humans and their ancestors, and the space of color perception, overdetermine color perception enough to falsify the ‘inverted color problem’. So yes, the Yoneda lemma does solve the issue.
23.8.2023 01:55Comment on No, the Yoneda lemma doesn’t solve the problem of qualia. by Keith E Peterson
https://matteocapucci.wordpress....Comment on Mathematicians don’t care about foundations by Madeleine Birchfield
https://matteocapucci.wordpress....In reply to <a href="https://matteocapucci.wordpress.com/2022/12/21/mathematicians-dont-care-about-foundations/#comment-145">Madeleine Birchfield</a>.
"One doesn’t need the additional hardship of telling classical mathematicians that they have to give up on LEM and AC, that in the foundations the intermediate value theorem cannot be proven, the fundamental theorem of algebra cannot be proven, that all real numbers have an infinite decimal expansion cannot be proven, and that all vector spaces are free cannot be proven, all facts that a classical mathematician would take to simply be true."
On second thought, while constructive mathematicians cannot prove that these statements are true in constructive mathematicians, they still can prove that "LEM implies the intermediate value theorem", "LEM implies the fundamental theorem of algebra", et cetera, in the sense that one could construct a function with domain a type representing LEM for a universe and codomain a type representing the IVT or the FTA, et cetera.
9.8.2023 14:36Comment on Mathematicians don’t care about foundations by Madeleine Birchfield
https://matteocapucci.wordpress....Comment on Mathematicians don’t care about foundations by Madeleine Birchfield
https://matteocapucci.wordpress....“Many people seem to believe mathematicians work in non-constructive, non-structural, battered foundations because they love their Platonic realm and have a kink for AC and LEM.”
The notions of non-constructive and non-structural are rather independent of each other. Most mathematicians will likely continue to work with AC and LEM even if they switch to a structural set theory like ETCS or a dependent type theory like the one in Lean. On the other hand, constructivists are and have in the past been very comfortable using non-structural set theories like IZF or CZF.
“In fact, I’m sure if at the start of an undergrad mathematical curriculum we provided students with a good ‘naive type theory’, mathematicians would just grow to use it. They’d still won’t care, but they’d happily credit Martin-Löf for giving legitimacy to their mathematics instead of Cantor, Zermelo and Fraenkel.”
Personally, I’ve come to the conclusion that if naive dependent type theory is to succeed at replacing naive set theory as the default syntax that mathematicians use, dependent type theorists would have to present their theories with universes satisfying LEM and (the set-theoretic) AC, so that the subject could be made more accessible to classical mathematicians. Compared to the general population of mathematicians, constructive mathematicians are a small minority, and predicative constructive mathematics even smaller of a minority; the population of classical mathematicians dwarf them both.
In practice, the dominant approaches to dependent type theory, MLTT and cubical type theory, are primarily constructive and predicative in nature, which means that not only is there a material-structural gap in understanding, there is also a gap on what statements are provable in the foundations. It is hard enough to get a mathematician who is used to membership being a proposition which could be negated and sets being elements of other sets to adjust to the notion that sets and elements are completely different things and membership is a typing judgment with no truth value. One doesn’t need the additional hardship of telling classical mathematicians that they have to give up on LEM and AC, that in the foundations the intermediate value theorem cannot be proven, the fundamental theorem of algebra cannot be proven, that all real numbers have an infinite decimal expansion cannot be proven, and that all vector spaces are free cannot be proven, all facts that a classical mathematician would take to simply be true.
8.8.2023 04:36Comment on Mathematicians don’t care about foundations by Madeleine Birchfield
https://matteocapucci.wordpress....