Pierce’s Playground: Zalamea on Sheaf Theory and Dialogue

Razonabilidad leading the way

I am reading through Fernando Zalamea’s Synthetic Philosophy of Contemporary Mathematics for the first time and have come across a more clear understanding of the idea of “sheaf theory”. My original, underlying understanding of sheaves has not changed, but the manner in which I will articulate my thoughts has assuredly been cleaned up a bit.

Preliminary edit: I’ve also gone through, after writing this post, essays that were sent to me by Trent Knebel which gave rise to three powerful synchronicities. I’ve added the relevant excerpts, both despite and because of their sudden disruption of my train of thought.

In particular, Zalamea characterizes this superposed idea from category theory as a “methodological sheaf” (see lecture here), or a “sheaf of filters for the decantation of reality” which resonates exceptionally well with my own posts. Moreover, he organizes his work in a pragmatic, otherwise mathematical style as opposed to this Novalis-intoxicated prose-poetical one, which may amount to a greater transmission of information.

 EDIT1: First synchronicity - Here’s a gem from Zalamea on Novalis (article here):

In Novalis’ Allgemeines, the stage is carefully set to an investigation of fluxions of consciousness, both in their differential and integral trends, with all sorts of remarks elucidating the tensions between (“modern”) relative fabrics with invariants and (“postmodern”) residues and singularities. Novalis, as many other Romantic geniuses, was indeed an early explorer of the TRANS phenomena: all his work, both philosophical and poetical, focuses on motion and studies knowledge as transformation.

In any case, I want to interlace (or “weave”) some of his work with mine in order to strike the balance between styles for the reader. I am the first to admit my  style is notoriously “wild” by comparison (with “wilderness theology” possibly reflecting the sense of this “hazy zone”, or “protoplasmic medium”, or “magma of creation”) provided especially that I taught myself this methodology by continuing to listen to my personal unconscious while stumbling around in the darkness along the way. Having only recently finished reading his book, however, I was pleased to find that my thoughts could be articulated much more clearly than I could have ever done at the time.

Zalamea writes, on page 122-123, the following (links added):

In this way, a quadruple synthetic strategy takes shape in category theory.

First of all, internally, in each concrete category, we seek to characterize certain special constructions in terms of their environmental properties in the given class (see here, “Fragmentation”). Then, externally, in the general field of abstract categories, we seek out certain universal constructions that can account for the characterizations obtained in the concrete categories (see here, “Fragmented Body”. In the third stage, in a remarkable weaving between concrete and abstract categories, we go on to define adequate functors of differentiation and reintegration (see here, “Continuous Integration”. Finally, the same functors become the object of investigation from a synthetic point of view, and their osmoses and obstructions (= “natural transformations”) are studied systematically (see here, “Sheaf Theory”).

[…] Indeed, if the philosophy of mathematics could make use of the synthetic lessons on differentiation and reintegration codified in both the Peircean pragmaticist maxim and in the functorial processes of category theory, many of the fundamental problems in philosophy of mathematics might acquire new glints and twists that, we believe, could enrich philosophical dialogue (see here, “Radical Paralysis”).

I have embedded my relevant posts as “ornaments” in his quote in order to demonstrate the development of my learned strategy of approach in relation to his otherwise succinct summary.

EDIT2: Second synchronicity - Zalamea on the subject of “paralysis” and our “Vision beyond vision”, with continued reference to Pavel Florensky (see here):

The comfortable separation of knowledges- a rigid mathematics, eternal, rational, deductive; a poetry, plastic, dynamic, emotional, inventive–has produced an unnecessary and frustrating paralysis. If we have to return to understanding by “seeing with the mind and with the heart” we already have many examples that transgress borders and reintegrate with extreme exactitude and sensibility, rationality and imagination, control and inventiveness.

Furthermore, I have classified (see here) each of these steps on “my journey” as an ascent up the Tree of Life or descent into the Inferno of Dante. In particular, Zalamea’s general approach from #1-3 constitutes for me the experiential axis from Beauty to Anger, which is appropriate given a generalized categorization of his obviously immense passion for the discipline of mathematics (“Beauty”) and his response to mainstream mathematics (“Anger”). Furthermore, the spill-over into sheaf theory (#4) marks the beginnings of “Strength” (such as his own exposition) and “Heresy” (again, also such as his own exposition). This all seems appropriate, as he is led to the possibility of “glints and twists” which suggest the illusion of the dead-end (hence, paralysis) known as “cutting-edge” thought, but thereupon opens the essential “dialogical” component for future development and progress.

EDIT 3: Third synchronicity – Zalamea in a brief discussion on Dante (ibid.), and possibly a seg-way into questions of non-violence as such as related to New Life.

“Violence against yourselves” (title of an illustration by Rauschenberg) has sectioned the unitary wealth of human creativity. Nevertheless, from real experiences of suffering in the limits, from systematic works open to the study of borders, from true monuments of transit in the form and in the underlying movement of the world, as we have been exploring in this essay, an extended reason can achieve “new life”. The tenor of New Life in Dante –the first full conjunction of critical reason and of imaginary construction in occidental literature, with the union of a compilation of poems a commentary in prose on the poetry, this by the same author and in the same work can perhaps then look to a new revival of reason.

Yet, the book unfortunately ends here, with the humble gesture towards a potential dialogue. Though it begins with a survey of past dialogues, it does not end by having this dialogue itself. And appropriately so, as a dialogue is difficult to capture in book-form and I cannot bring myself to expect more from an already explosive performance on his part. I say unfortunately not out of disgust at any short-comings, but because he has left me wanting more. As such, I have nothing but hope for Zalamea’s personal development in the future, and the deepest appreciation for his work.

I am left asking, perhaps prematurely “Where, if anywhere, could he go from here?” just as Novalis might ask “Where are we really going? Always home.” If my “invisible adventure” is a testimony to or otherwise indicates anything significant all, then I may have some ideas with which I could begin to engage him in this dialogue. Therefore, if I could interview any living person at this stage, it would likely be Fernando Zalamea. I would likely begin by asking him to answer the question “Where or what do you understand by home?” through the sheaf of filters he has presented so elegantly.

I have said it twice already that if our theory is one of sheaves or of superposition, then our “generic practice” is thus one of being after-the-harvest as gleaners or participating observers (in the QM sense). While we may of course wait for the harvest to pass, for the next great mathematician to leave her mark so we may pick up a few new fragments to creatively synthesize with our “methodological sheaf”, this generic practice is not satisfying in and of itself for the unique self. It may minimally sustain life as does water, but it does not fill it with meaning as does having salt on the table. I must engage Zalamea therefore not as a mathematician, but as an ordinary individual living presumably in the city of Bogota, Colombia – which is to say from a non-mathematical perspective – if we are to make any progress in conversation.

Essentially, I must place “his work” at a distance in order to understand the man himself. I must play the game as though I do not know the rules (and generally speaking I do not actually know the rules of advanced mathematics), and ask him repeatedly the frustrating  two-word question “Why bother?” until I receive an adequate answer, possibly accepting no single answer as “sufficient”. Why bother with the domain of “mathematics” in the first place? Moreover, if we do choose to go beyond (mas alla) “elementary mathematics” to advanced mathematics”, why bother with this contemporary and synthetic approach? This is the misreading of wisdom (see here).

The latter question he may perhaps answer more or less “sufficiently” in the otherwise vague form of “enriching, new glints and twists”, but does he have anything to say in the former case? I want to know, specifically, the role mathematics plays in his very form of life, how it relates to the enhancement of his life, the continuance of his life, the lives of those around him, and so forth. The focus, of course, being on Life as such. I suspect initially that he is more right than he knows by unleashing or otherwise enleashing the prefix “-en” from its dormancy. This is the enfolding of wisdom (see here).

If I am to take Zalamea’s “methodological sheaf” as an object of investigation in its own right, we get closer to understanding what is meant by “Construct awareness” (see here).

On this subject he writes, page 121:

The Peircean web of webs, in effect, opens onto the modal realms in their entirety, and systematically attends to contrasting given facts (within the phenomenological world) and necessary behaviors (within well-defined contextual systems), so as to then reintegrate them in an extended spectrum of possible signs. 

This is an important question because this sheaf theory may itself become weaponized, or our thought may otherwise crystallize around it in ways which are destructive.

Ideally, we are to rise out of this paralysis, concluding the segment on Hersey moving into a space of Mercy and engaging questions of Violence and Trauma as such. Ideally, we are sent out into the mysteries of “Nature” into the worlds of “Sentient intelligence” (see here) for further experimentation in the “Web of webs”.  Yet, we cannot play here if our strength is  too great and may harm others unknowingly, in a kind of “You must be this tall to ride” kind of way. A certain construct awareness is thus that which ensures that we can proceed without violence and harm. This is the wisdom of security (see here).

We are given with this awareness to ask questions about the use of our new “method” of sheaf theory; the way in which we “make use” of it matters. It is roughly here where the term “mental gymnastics” (see here, “Non-Acrobatics”) becomes applicable, as though playing on the web of a jungle jim at “Pierce’s Playground”, not at any point beforehand. We associate this at once with Understanding and Fraud, for obvious reasons. I think, without knowing the man personally, that Zalamea is fundamentally honest, open, and hospitable. That is, he has it seems a “high degree of trustworthiness” (see his article on Pierce and Kant, here). As such, he should be led next to consider the life of the man behind the application of the sheaf. A discussion on the “phenomenological world” and “necessary behaviors” in relation to our vision of the Body is of course forthcoming in this dialogue.

The manner in which each of the four steps play out is dependent upon the agent on stage (e.g. “gluing correctly” or not), is dependent in this case upon Zalamea-the-man in his flesh. Accordingly, I must seek out an understanding of that Spirit which animates him before proceeding.

For Zalamea, I wonder what is the unseen knowledge (gnosis) he has come across in the fleeting experience of life? What is the point, what are the tacit values, lurking behind all of these beautifully-crafted pages? What voice of things does he hear in the silent-noise? There stands the man, but what is his message for humanity in the last instance? How, for example, would he like to be remembered in history? As simply a “great mathematician” or “something else” entirely (“something very distinct”)? What impact does he seek to have on the world? Which fragments are we to pull from his work, and in what ways? These are all decidedly non-mathematical questions.

It bears some light on this question that Alexander Grothendieck is an avowed pacifist, who insists upon “listening to the voice of things” just as I have done in the central chamber of the salt mine (see here, “Altar-ations”) — or perhaps in this, the “wild heart of mathematics”.  It seems there is a certain intensity which spans between his work and his form of life. This HEBEL character of Life in general flows through the story of Grothendieck (see .pdf here), leading him to active non-violence. For Zalamea, what life-affirming “thing” hides in the “wild heart of mathematics”? Why bother, Dr. Zalamea? What does a careful synthesis of these fragments reveal to you as a human? After all, he writes “in honor of the human spirit” — what precisely does this mean to him?

The simplicities of counting and complexities of advanced mathematics which lie behind the thinking through and enacting of non-violence become more easily taken to heart the more one points to dialogue. I have no hesitation in saying that Fernando Zalamea’s book  Synthetic Philosophy of Contemporary Mathematics is the most important one today in the field of philosophy of mathematics — not so much for its uniquely Goethean discoveries and sheaf theory — but for its explicit call to dialogue and exploration of the wild heart of life whose name is Satyagraha (see here) of which the world of mathematics is but a especially-significant microcosm.

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3 thoughts on “Pierce’s Playground: Zalamea on Sheaf Theory and Dialogue”

  1. Great post! It somehow helps me shape my own reading of Zalamea. About Novalis: Zalamea edited a volume with papers by several people from Bogotá, called Rondas en Sais (alluding to the disciples of Sais). If you haven’t seen it, it might be interesting for you. The essays all bear on Mathematics and Contemporary Culture – there are writings about Florensky (F. Vargas), about Galoisian Philosophy of Mathematics (A. Cruz), Alexander Grothendieck (G. Restrepo), Shelah and Zilber (A. Villaveces), literary narrative and mathematics (J. Moreno), Cinema and Math (A. Martín), Mathematics and Contemporary Art (F. Zalamea). There is also a little anthology of contemporary mathematicians, writing about their doing math (Grothendieck, Shelah, Connes, Gromov, Zilber), fragments on literary invention and mathematics (by Musil, Broch, Borges, Novalis, etc.). The book is in Spanish – I wrote here (http://atsmi.wordpress.com/2013/03/24/rondas-en-sais/) a longer description of the book, in Spanish. I also quoted this post of yours in a small post about dialogue with Zalamea (http://atsmi.wordpress.com/2013/03/24/rondas-en-sais/). Best regards, Andrés Villaveces

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