Mathematics is Metaphysics
World philosophy has been stuck for decades in a scientific vision that cannot explain reality any more. Thus to get beyond Kant and reawaken thought and culture.
Fernando Zalamea
I propose here a pamphlet in favor of a synthetic thought at the beginning of the 21c — “a pamphlet [opuscolo] with an aggressive character” according to a celebrated dictionary — implying a combat with light weapons but for assault; the thought implicates every form of intersection between sciences, arts, philosophy, essay, critique, and struggles systematically against stagnant and tired compartments; the synthesis counterposes analyses, according to well defined polarities: compositions and decompositions, relations versus elements, exterior versus interior, impurity versus sterilization, (mathematical) category versus set theory; the 21c invites us to a reflection on our contemporary intellectual spirit [spettro]. In the 20c, there was such a strong influence (in some moments brilliant, in others deplorable) of the analytic linguistic turn that it is now time to formulate a counter-proposal for a new non-dogmatic opening of thought.
Giancarlo Rota underlined in many polemical articles the existence of a “perilous influence of mathematics on philosophy.” The expression refers to the influence of a restricted mathematical logic (classical first order logic) and a fundamentalist perspective (Cantorian set theory) that, taken too seriously by philosophers, uncritically gave space to a certain “hard” analytic philosophy, guided by logical and linguistic considerations with which — so it was thought — one could eliminate metaphysical digression [divagazione] or phenomenological or aesthetic imprecision. All in all, that expression was in its time ironic and paradoxical because the real mathematics (of figures such as Galois, Riemann, Hilbert or Grothendieck, from the middle of the 19c to the end of the 20c) developed without curing itself at all of the subtle logical and linguistic considerations of analytic philosophy. In reality, there was indeed through a profound ignorance of the “real mathematics” of the beginning of the 20c (algebraic and analytic number theory, abstract algebra, topology, complex variables, functional analysis, etc.) and through an (un)conscious construction of myth that it was possible to attain the project of analytic philosophy, which was only apparently rooted in mathematical studies, whereas the real situation was completely opposite.
The multiple references to a supposed “Fregean revolution” in the foundations of mathematics at the beginning of the 20c showed a good example of this mythology which obstinately endured. You ask yourselves, of any logic active in the beginning of the 20c in what consisted this supposed “Fregean revolution” you discover that treats a simply inexistent event, a myth created by “standard” philosophies and histories of logic. In reality, it is well known that logic includes three fundamental branches (model theory, recursion, sets) of which the first, model theory, generated the major logical advances in the last decades (Shelah, Zilber, Hrushovskii), which had very clear ancestors (Peirce, Löwenheim, Tarski) and where Frege glittered by his absence. Recursion theory began with Hilbert, Skolem, Gödel, and set theory with Cantor and Zermelo and, although in these two lines one could insert in part also Frege, certainly he could not compare (be regarded) as “precursor” much less as “revolutionary.” Thus, in the development of mathematical logic, the special position of Frege constituted only a tenacious myth. On the other hand, and more broadly, if one thinks of the development of real mathematics, the figure of Frege is entire out of the running. In addition there is [Altra cosa è registrare] Frege’s central and essential impetus for Russell and for the development of analytic philosophy. But this confirms only that “analytic philosophy” and “real mathematics” have always followed divergent paths.
Therefore we must explode the prime grand myth of analytic philosophy based on mathematics. The “dangerous influence of mathematics on philosophy,” in Rota’s phrase [calembour], must be sharpened as the “dangerous influence of mathematical logic and Cantorian theory of sets on analytic philosophy.” From the moment that mathematics is infinitely vaster than the pair classical logic + Cantorian sets, arises a new influence of real mathematics on philosophical thought. Our concept in this article is summarized by suggesting that this influence is blossoming but, seeing as real mathematics is practically unknown, it has not arrived at the point of being “dangerous.” It would be good if after some decades, a critic at the level of Rota could observe the “dangerous influence of topology, of complex variables, of logic of sheaves and category theory on philosophical thought.” By then, however, the term “philosophy” will be discovered a new continent, a sort of “synthetic philosophy” and the spirit of mathematical, philosophical, and critical methodology (that) can be put in motion for opening new fields of thought.
Many great masters at the end of the 19c and the beginning of the 20c already profoundly explored the studies of the limits of contradiction (Peirce, Florenskij), and adopted a “geological critique” of art and culture (Warburg Benjamin, Auerbach). In a sense of opening and of opposed similar [contrapposizione simile], it can be said that the Italian school was particularly attentive to the alternative, sia a partire starting from Peirce (with figures on world rank like Rosella Fabbrichesi and Giovanni Maddalena) sia a partire from Florenskij and Warburg. It is true to say that in mathematics as in arts, the “two principal modes with which humanity thinks,” as Francastel said, the fundamental source of invention comes from a staircase of contradiction, obstruction, blind spots, a list that has taken place outside of analytic sterilization. One of the prime objectives of which one must call “synthetic philosophy” consists of taking on board this network of shadows and boundaries forgotten by the “normal” currents of analytic philosophy.
The most important of the “shadows” in the history of philosophy, excluded with dubious pride from analytic philosophy, is doubtless Metaphysics. This “other to Physics,” which is to say also the “other to language,” becomes for the fastidious analytic philosophers [analysts] tantamount to the contradiction “other to logic.” However one recalls that the greatest advances in science and the arts, and therefore the most creative moments in history of humanity are found outside language and outside logic. For philosophy to leave aside the conceptual study of a Riemann, a Mahler, a Monet, is an academically accepted barbarism, as if [analytic] philosophy wants to incestuously limit its own work to a primary, secondary, tertiary… n-ary discussion of self-contained philosophical systems.
Creativity — which is oscillation, rupture, disparate prospectives — can be comprehended only starting from a synthetic conglomeration of convolution which explains the sparse attention from analytic philosophy toward the creative potential of the human. Mathematical creativity, like artistic creativity, is none other than incessant convolution. The invention of Galois groups revolves around the obstructions obtained from the local analysis of [integral] equations; Galois urged literally a “metaphysics” of equations. Riemann surfaces revolve around the problem of the multivalence of certain complex functions and permitted structurally including the Multiple into the One. The “rising tide” of Grothendieck covers an analytically incomprehensible object mediating a category of cycles [circoli] that permit understanding it (the object) synthetically with respect to its ambient. In many cases, only a synthetic vision, thanks to sophisticated network of cycles, permits an advance in mathematics.
The situation, therefore, obviously swings back and forth between the analytic and synthetic prospectives. The work of future generations of philosophers is immense. First, one needs to get rid of [sbarazzarsi] analytic obfuscation and expose the other side of the balance; at times this other side was denoted “Continental Philosophy,” but it’s worthwhile now to re-order the panorama in a more precise mode (systemic and systematic), introducing the terminology Analytic Philosophy / Synthetic Philosophy. Second, one needs to construct the body of synthetic doctrine in a very ample way, putting in play great architects (Peirce, late Whitehead, Cassirer, Merleau-Ponty, Deleuze, etc.) and great critics (Warburg, Florenskij, Benjamin, Batjin, Blumenberg, etc.) who have addressed reasoning as territory of frontiers and passages [transiti]. Third, one needs to re-construct culture as infinite fragmentation of an alternative tertiary, a network of residues and traversals [traversi], like places in incessant exchange (Serres). In a word, one must rewrite, 200 years later, the Quaderno generale of Novalis, grand precursor of which is TRANS, thanks to the infinite variety of passages which appeared in the 19c and 20c.
The object of Synthetic Philosophy (as Continental Philosophy understood it at least in part) must confront many fragments of knowledge that Analytic Philosophy considered as intractable: contradictions, blind points, vague borders, obscure foundations of truth, imprecise shadows where creativity explodes, aesethetic potentiality, etc. Metaphysics, far from dead, has never been so alive, thanks to the horror generated by those who would kill it. The great profundities of Greek philosophy, the intractable dimensions of a Lullo or a Leibniz, resurrected untamed. In the star of Grothendieck contemporary mathematics discovers the multitude of archetypes unthinkable a few decades ago (classifying topoi, motives, Zilber or Gromov groups, Simpson’s inverse mathemtatics, etc.). In the star of Weinberg, contemporary cosmology is capable of discovering the structural archetypes of the origin of the universe. In the star of Petitot contemporary neurogeometry discovers the neuronal archetypes that could permit the naturalization of phenomenology. In the star of Kiefer, contemporary art encounters the archetypes of destruction and beauty according to the zigzagging cycles of civilization. Everything tends to show that form (the beginning of the 20c), structure (the middle of the 20c) and process (the beginning of the 21c) are much more important than the linguistic and logical turns proposed as the only reason by the analyticists.
As Benjamin said in “Passages” of Paris, we must reawaken.
NOTES
His work:
Fernando Zalamea’s most important work is Synthetic Philosophy of Contemporary Mathematics, Urbanomic London 2012, in which the lovers of mathematics and philosophy can find a profound reconstruction of 20c mathematics. The italics in the article are the author’s.
Zalamea, a “global mind” who redescribes philosophy with mathematics
Giovanni Maddalena
The encounter with Fernando Zalamea is always an event: mathematician with extremely vast knowledge, philosophy, art critic, writer, essayist, a man of culture in all fields, citizen of Rome, Barcelona, Paris, Boston, and above all, of his beloved Bogotà, where he teaches Mathematics at the National University, Zalamea is true universal thinker. A recently announced book of Gianluigi Ricuperati, indicated via a complex algorithm the 100 global minds, the most influential in the world. (“100 Global Minds,” Roads Publishing. Above all, Zalamea was more than any [colpisce piu ancora] the master. When he affably encounters students and young researchers, Zalamea listens, directs, urges always through pertinent notes that valorize that which they propose. Zalamea inserts each’s studies in a painting that is so vast and with a perspective so ample that each can find his own work in philosophy different from that which it / he would propose to start from “real” mathematics. One can glimpse a slice of this immense cultural panorama in the small essay, a miniature polemical pamphlet, that Zalamea has written exclusively for Il Foglio.
The idea is simple. World philosophy is certified [attestato] and blocked in an analytic that rests its foundations on mathematical understandings tied to programs already superceded wtih Gödel (1932) and crystallized in a scientistic and naturalistic version that has little to do with [che poco ha a vedere] the development of mathematics and of science of the last 80 years. Also, in his present writing, Zalamea contests the importance of some strongholds of this project.
The critique of analytic philosophy has often been carried by the traditional continental philosophy (hermeneutics, phenomenolgy, etc.). Critiques, above all, always seem naive and external, conducted by thoughts that appear to escape a strict logical confrontation. The idea of Zalamea, instead, is to criticize analytic starting from logic, and re-establish continental philosophy on the basis of the experience of real mathematics, pardoxically closer to the study of phenomenology, semiotics, aesthetics and hermeneutics than that of analytic development. In this sense, it is no longer geography (continentale) that counts but the type of reasoning (synthetic) that for now is entirely unexplored, except for the works of a few great precursors like Peirce and Florenskij.
With his proposal, Zalamea contests in fact the distinction between the sciences of nature and the sciences pf the spirit, and more profoundly, impregnated with Kantian knowledge, re-inserts metaphyics in the rational compass. I also think that the synthetic perspective will entirely depart from the Kantian schema, as I have tried to do with the theory of gesture, which is also the fruit of continuous dialogue with Fernando Zalamea. I hope that this article helps put into discussion a cultural framework heretofore dated and impoverished.
[ Translated by Sha Xin Wei ]