Jon, hi and thank you for this. I guess two inadequate replies from me, one to you and one to Glen, because I don’t think I have the content that would be needed for any substantive contribution.
I have seen these maps from particular computational systems to categories, and I work with somebody who is enthusiastic for that project and tries to do it for various stochastics. I can kind of, as an outsider, imagine what that would look like, and given elementary examples, I can follow them in a babyish way, once step at a time. But having spent no meaningful time and effort with Category Theory, I am not really able to “think” about anything in it, so I am kind of far behind and passive toward anyone trying to do such actual work. But at a common-language impressionistic way, the description you give below seems right to package things I said (in “imperative” rather than “functional” language, to say it metaphorically). The implication from symmetries in the laws of nature, and the transition from states of matter that have the same symmetries, and those that don’t, to new phenomena such as the emergence of a new class of sound waves, is called Goldstone’s Theorem, if that saves you time in looking at the literal algebraic construction being used. On magnetic fields’s “doing no work”; I don’t know the history of Griffith’s characterization, so it doesn’t sound weird to me. I also don’t see how it would require a move past the classical electrodynamic description to a quantum field theoretic one. It seems that, with all the standard limitations of classical theories, Maxwellian classical electromagnetism and General Relativity work together consistently and quite well as classical theories of these fields in spacetime. It is no problem that magnetic fields do no work, since the thing we associate with a force is always perpendicular to the direction of some displacement associated with them. But since, upon changing a reference frame (a “boost” within the group of Lorentz transformations), linear combinations of electric and magnetic fields change labelings, we have no trouble accounting for work where it is done. I remember reading Edward Purcell’s undergraduate book on Electricity and Magnetism, which I still regard as one of the best-written textbooks for its level that I have seen in any subject, and thinking suddenly that all was perfectly sensible and trackable. The other texts had never done that for me. Jackson’s graduate text, which gets quickly into the hazing ritual of graduate E&M, didn’t make any such effort at exposition. The issue of formal systems, and their role in bringing into being habits of thought that, later, we wrongly suppose to be “natural attitudes”, remains a subject of high interest to me. Though like many such subjects, the amount of serious work I put into them is usually negligible. Oh well. Eric > On Aug 18, 2024, at 2:29, Jon Zingale <jonzing...@gmail.com> wrote: > > Eric, > > Apologies right off, the following analysis is for myself and probably should > be kept to myself. On the one hand, I am struggling to learn a particular > formalism. On the other hand, I am struggling to get better at understanding > what it is that people I admire appear to be doing. The formalism I am > continuing to explore is the adjunction (Con ⊣ Lang) relating formal type > theories to their categories of models[⊣]. > > You begin your exposition with the embodied description of a paddlist > building up a set of experiences and (via signal-boosting?) arriving at a > reasonably stable system of relations (restoring forces and spatial > relations, say). That is, you extract the internal logic of some phenomenon > from a principle *model* to a *theory* of types, relations and deductive > rules. The derived type theory comes equipped with group theoretic relations > capable of distinguishing what we define as a purely formal SOLID type from a > purely formal LIQUID type. > > Isolation of a formal theory provides leverage reflected in the category of > models: > 1. The theory provides a means for producing a generic model, free of surplus > meaning and yet preserving desired logical consequences such as P and S waves > when reinterpreted in the principle model. > > 2. One can study the *shape* of the interpretations of the theory via > morphisms from the generic model into the principle model. > > 3. As a corollary, group theoretic deductions of the theory are consistently > embodied in the model, correctly assigning properties like solid and liquid > to the intended scoped-patterns. As I understand you, "framed within a very > partial description of nature." > > You then proceed to perform certain calculations within the context of the > theory, creating types A and B (presented as propositions in the logic of the > theory) and then pointing to the sorts of deductive exercises one might hope > to perform with these types. Relative to the principle model, so long as our > interpretations are valid we don't particularly care whether we are speaking > of "a hockey puck, or an Evangelical Who Knows the Glory of God, or a > heathen, or a psychologist". What matters from this perspective are the > theory-preserving morphisms from our generic and arguably privileged POV > model to our principle model. > > I feel like I am still not grokking what I can from your discussion of > causality, so I will leave that (very interesting) exposition aside for now. > One thing that strikes me as being meaningful is something about the nature > of the generic model. For instance, there is Griffith's *famous grey box* in > his text on *classical* electrodynamics wherein he states that *magnetic > forces do no work*. What sometimes frustrates students of this theory[B] is > that models constructed from classical electrodynamics give rise to insanely > complex epicycle-like thought experiments involving electric forces actually > *doing the work*. This isn't a criticism, of course, because what seems to > satisfy physics students is then to enrich the classical theory to a quantum > theory. > > All said of course, if one were to argue that you did none of the formal > things I describe above, I fully accept that too. > > Jon > > [⊣] https://ncatlab.org/nlab/show/syntactic+category > <https://linkprotect.cudasvc.com/url?a=https%3a%2f%2fncatlab.org%2fnlab%2fshow%2fsyntactic%2bcategory&c=E,1,FN1R_gyHRJGrAYUkdtwEaJUqFC_50md5dCwm8tzFYNhtQ0dSuVVa107Yka30h-GFL2hmxdpfNfitGCGmX5MNYeadql4aRaZ5jT8SS17k9iYkj9M,&typo=1> > [B] https://www.youtube.com/watch?v=fHG7qVNvR7w > -. --- - / ...- .- .-.. .. -.. / -- --- .-. ... . / -.-. --- -.. . > FRIAM Applied Complexity Group listserv > Fridays 9a-12p Friday St. Johns Cafe / Thursdays 9a-12p Zoom > https://linkprotect.cudasvc.com/url?a=https%3a%2f%2fbit.ly%2fvirtualfriam&c=E,1,C5roF09L53AFQ5OOOPMsvwTPkcm_xTSdFH7yXK3fLfz2qc82JiUlsGjO-s9wmqVHvcvixLdbxG3D_8a53Prh0TBXt417k1ZamJp16F5In9YuHBeSrik,&typo=1 > to (un)subscribe > https://linkprotect.cudasvc.com/url?a=http%3a%2f%2fredfish.com%2fmailman%2flistinfo%2ffriam_redfish.com&c=E,1,Ts6-iwwZ-VD6imxDRmBfyGM60WT8zsYNMv6saXCY7dD7PabwBweusCZvekrTeUB8vaMbQ_BR8grAJlxtyVs4uWVSb5etHDpst1e706D1&typo=1 > FRIAM-COMIC > https://linkprotect.cudasvc.com/url?a=http%3a%2f%2ffriam-comic.blogspot.com%2f&c=E,1,akpwfoBQPn_a6DIpgl9mTpvkUIG6m2Bfy4rbRRjyCRMbrr3kyrBfJ-Hg552Bg3uju7pn2DnZpeCsX0192bhLA6VLOjy8S4eeNVLF8iuO&typo=1 > archives: 5/2017 thru present > https://linkprotect.cudasvc.com/url?a=https%3a%2f%2fredfish.com%2fpipermail%2ffriam_redfish.com%2f&c=E,1,ppX7rBFr_48RSftniIrE-NYapWgI9Lr4OVJsbPGb0nSvcEQL4BuTRzddYlChmrN7rvksFMIwEIa22c_u2p_KcLHL191RTZ4s8KbsRVGfNrqG&typo=1 > 1/2003 thru 6/2021 http://friam.383.s1.nabble.com/
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