Since I'm likely to read more from this dude, and have already posted a bit on him elsewhere, I figured I'd give him his own thread instead of cluttering up the others. I'll post below what has already been posted elsewhere in the forum. I also want to point out inthesaltmine's blog posts on Zalamea: Peirce's Playground and Sheaf Theory Part 1.

The forum excerpts:

This post referenced Zalamea’s mathematical sheaf logic. From this link:

"The point of this seminar is not only to acquaint us with the vibrant landscape of contemporary mathematics – and the field of sheaf logic and category theory, in particular – but to show us how this landscape’s powerful new concepts can be deployed in the fields of philosophy and cultural production. Its aim is nothing less than to ignite a new way of thinking about universality and synthesis in the absence of any absolute foundation or stable, pre-given totality – a problem that mathematics has spent the better part of the last fifty years thinking its way through, and which it has traversed by means remarkable series of conceptual inventions – a problem which has also animated philosophical modernity and its contemporary horizon.

"Our guide in this endeavour will be Fernando ZALAMEA, a Columbian mathematician, philosopher and novelist whose work seeks to explore the life of contemporary mathematics while redeploying its concepts and forces beyond their native domain. In an incessant, pendular motion, he weaves the warp of post-Grothendieckian mathematics through a heterogeneous weft of materials drawn from architecture and fiction, sculpture and myth, poetry and music.

"Just as analytic philosophy emerged from the shockwaves of the explosion of classical logic and set theory onto the scene in the early 20th century, the conceptual force of mathematics after Grothendieck holds the potential to spawn a new, 'synthetic' vision of mathematically-conditioned philosophy for the present age, one which Zalamea foreshadows under the rubrics of transitory ontology, epistemological sheaves, and universal pragmaticism."

It turns out Zalamea is a big fan of Peirce. Here's one of his articles on Peirce. The first thing of interest is that in exploring Peirce's Logic Notebook Zalamea appreciates that it is a chronological diary of the development of Peirce's thought. I.e., it shows the process of how Peirce came to his conclusions, the brainstorming, the multifarious and scattered ideas that only later became refined into his formulations. I've oft said that is IPS, how the threads and dialogs explore various topics, and how we then come to more solid positions. It's a media in res, continually.

Which is another of Zalamea's appreciations in the above. How Peirce mixes and matches various fields and paradigms, finding those interstices of intersection between domains and categories. And which Zalamea himself uses in his own interlacing Vennish mixes of sheath logic and category theory.

Also see Zalamea's essay on transmodernism here. From the Intro:

"Transmodernity' –both diachronic and methodological- hopes to reintegrate many awkward postmodern differentials, to balance some supposed breaks with more in-depth sutures, to counter relativism with a topological logic where some 'universal relatives' provide invariants beyond the flux of transformations [....] [with a] merging [of] reason and sensibility which must explore the borders (TRANS) of thought [....] to reinterpret universals as partial invariants of a logic of change, where the borders of reason and sensibility appear as objects of reason in their own right. The important crisis revealed by Postmodernism (impossibility of unique perspectives, impossibility of cutting out antinomies, impossibility of stable hierarchies, etc.) can nevertheless be well understood using a continuous geometrical logic of reason and sensibility, open both to changes and invariances. This short article is intended as a programmatic one, pointing out the possible relevance that some non-standard pragmatic thinking (Peirce’s 'pragmaticism', Latin America’s 'razonabilidad') may have for our Transmodern epoch."

This is interesting, from pp. 3-4 of the last citation:

"Broadening these precepts to the general context of semiotics, for knowing a given arbitrary sign (the context of the actual) we must run through the multiple contexts of interpretation that can interpret the sign (the context of the possible), and within each context, we must study the practical (imperative) consequents associated with each of those interpretations (the context of the necessary). In this process the relations between the possible contexts (situated in a global space) and the relations between the fragments of necessary contrastation (placed in a local space) take a fundamental relevance; this underscores the conceptual importance of the logic of relations, which was systematized by Peirce himself. Thus the pragmaticist maxim shows that knowledge, seen as a logico-semiotic process, is preeminently contextual (as opposed to absolute), relational (as opposed to substantial), modal (as opposed to determinate), and synthetic (as opposed to analytic). The maxim filters the world through three complex webs that enable us to differentiate the one in the many, and, inversely, to integrate the many in the one: the modal web already mentioned, a representational web and a relational web.

"One of the virtues of Peircean pragmati(ci)sm, and, in particular of the fully modalized pragmaticist maxim, consists, however, in making possible it to reintegrate anew the multiple in the one, thanks to the third-relational-web. Indeed, after decomposing a sign into subfragments within the several possible contexts of interpretation, the correlations between the fragments give rise to new forms of knowledge, which were hidden in the first perception of the sign. The pragmatic dimension stresses the connection of some possible correlations, discovering analogies and transferences between structural strata that were not discovered until the process of differentiation had been performed. Thus, although the maxim detects the fundamental importance of local interpretations, it also encourages the reconstruction of the global approaches by way of adequate gluing of the local. The pragmaticist maxim should accordingly be seen as a kind of abstract differential and integral calculus, which can be applied to the general theory of representations, i.e. to logic and semiotics as understood, in a more generic way, by Peirce."

And this is nice, resonant with my boundary reveries here and following:

"Peirce’s modal, multipolar and topological system investigates then the study of transferences of information around regions and borders on such a continuum. The TRANS motto is a crucial one for Peirce. His many classifications of the sciences show how one can 'tincture'the regions of knowledge using his cenopythagorical categories (1-3), and Peirce’s most creative ideas [...] lie precisely on the
borders of regions where information is being transferred" (4).

I'm also reminded of our prior discussion of Peirce in this post and following. Above that I was discussing Marks-Tarlow on boundaries, and below relating it to my later concept of rhetaphor.

On p. 15 I again brought up Zalamea. Let's now take a peek at his book "Peirce's Continuum." A few excerpts:

"It should be obvious that a given model alone (actual, determinate) cannot, in principle, capture all the richness of a general concept (possible, indeterminate). [...] The existence of multiple ways of representing and modeling should avoid any identification of a mathematical concept with a mathematical object. [...]  It turns out that continuity is a protean concept, which [...]  can be modeled in several diverse ways. [...] The continuum (general)  can only be approached by its different signs (particular models) in representational contexts" (4 - 5).

More from Peirce's Continuum, linked above. In section 1.3 he talks of the general very much like Bryant's or DeLanda's withdrawn or virtual. E.g., "whatever is free of particularizing attachments, determinative, existential or actual. The general is what can live in the realm of possibilia" (10). He sees the continuum as in the general category which he relates to Peirce's thirdness. It is woven with secondness (determinacy and actuality) and firstness (indetermination and chance). However, apparently unlike Bryant's withdrawn and more like DeLanda's virtual, the general continuum is "homogenized and regularized, overcoming and melting together all individual distinctions" (10).

He quotes Peirce on 12, seeming to further support the above, where the continuum is supermultitudinous and as such "individuals are no longer distinct from one another. [...] They have no existence [...] except in their relations to one another. They are no subjects, but phrases expressive of the properties of the continuum."

Near the end of chapter one he's talking about different logics. The law of the excluded middle does not apply to a general logic of possibilia, whereas it does to the particular logic of the actual. However there is an intermediate kind he calls neighborhood logic which seems closer to the general kind, in that it has more to do with those boundaries where something is and is not of a particular kind. And it is here that multitudinous points in possibilia are described as "infinitesimal monads" (24). I'm not sure if this is something like attractors that exist in the virtual; or actual, individual suobjects of the Bryant kind. And/or both, in that any given substantive suobject is a mix of virtual and actual, yet an autonomous individual nonetheless. But only in this intermediate "neighborhood?"

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Before we get too deep in intelligentsia, and it will get deep, I'll kick off the comments with a lighter touch. In this song Lady Gaga calls out Fernando's name.

More from PC:

"Veronese's continuum -- intuitive, prelogical, pretopological -- starts from a non-set theoretic notion of emptiness, a weaving and amalgamating synthetic notion which can be viewed as a smooth fluid, both finite and unlimited, in which parts melt naturally with the whole" (31). 

On 32 he discusses Brouer's continuum, where human minds can both mark the continuum and observe it, sounding a lot like Bryant's use of Spencer-Brown. Brouer calls this 'two-oneness' (32) (akin to our notions of Buddhist emptiness as 'not one, not two'), and how the general possibilia of continuum moves into the actual particular (33). This leads to a theory of 'boundaries unfolding' (34).

Starting on 36 he talks of Vopenka's alternative set theory, which "breaks the usual set theoretic equivalence between intensionality and extensionality." I'm again reminded of Bryant's notions of the difference between the two, where the virtual withdrawn of a suobject's endo-structure is intensional, and the actual exo-relations in its parts are extensional. Zalamea also notes that "very little of the indefinite and infinite range of intensional possibilia can be effectively actualized" (36). For Peirce, "the realm of possibilia and the intensionality of real potentials reign over the actual extensionality of existence" (37).

Note: All of which are discussed from other vantages in this thread (shameless plug).

On 38 of PC he's talking about how the continuum, itself a "vessel and bridge between the general and the particular," can be understood with the mathematical theory of categories which contain "real universals." While he here starts to get technical with the math jargon, what is not explored is the basic categorical image schema, which in this post notes that some of the most complex mathematical structures are re-imagined based on the most early stages of human development in the vaguely differentiated "primordial continuum." Image schema, as we know, are also a "vessel and bridge between the general and the particular," and real categorical human universals.

In the linked reference in the last post the most complex mathematical structures discussed were "continuous topological transformations." I'm not sure if that's the same kind of math Zalamea is discussing in the referenced section, but he does use a lot of topological terms.

In this piece Zalamea talks of an extended reason, one that includes and integrates imagery (eidos) as well as language (logos).

"One of the greatest strengths of images, and, consequently, of an expanded reason [...] consists in the peculiar capacity of the eidolon to capture simultaneously [...] an interior, an exterior and a border." (Can you find a more basic image schema?)

It is the border between reason and imagination where he finds "some of the best creative manifestations." He also finds a relation of this to Plato's "middle way" between the sensible and the intelligible. (Remember my gal Khora?) Therein we can unite with the "continuous universal [...] beyond the superficial cognitive levels of the mind." (Remember differance as hyperobject?) He gives examples of how this is so in art, but I've yet to see him go below this into image schema, which are exactly the imaginative rationality he discusses. Recall this post:

"Recall this from L&J (Metaphors We Live By) on imagination and reason:

'What we are offering in the experientialist account of understanding and truth is an alternative which denies that subjectivity and objectivity are our only choices. We reject the objectivist view that there is absolute and unconditional truth without adopting the subjectivist alternative of truth as obtainable only through the imagination, unconstrained by external circumstances. The reason we have focused so much on metaphor is that it unites reason and imagination. Reason, at the very least, involves categorization, entail-ment, and inference. Imagination, in one of its many aspects, involves seeing one kind of thing in terms of another kind of thing—what we have called metaphorical thought. Metaphor is thus imaginative rationality. Since the categories of our everyday thought are largely metaphorical and our everyday reasoning involves metaphorical entailments and inferences, ordinary rationality is therefore imaginative by its very nature. Given our understanding of poetic metaphor in terms of metaphorical entailments and inferences, we can see that the products of the poetic imagination are, for the same reason, partially rational in nature' (138-9)."

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