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Peirce's cenopythagorean categories, Merleau-Ponty's chiasmatic entrelacs and Grothendieck's Résumé
Progress in Biophysics and Molecular Biology xxx (2015) 1e5
Contents lists available at ScienceDirect
Progress in Biophysics and Molecular Biology journal homepage: www.elsevier.com/locate/pbiomolbio
Peirce's cenopythagorean categories, Merleau-Ponty's chiasmatic entrelacs and Grothendieck's R esum e Fernando Zalamea Universidad Nacional de Colombia, Colombia
a r t i c l e i n f o
a b s t r a c t
Article history: Available online xxx
We present Peirce's cenopythagorean categories and Merleau-Ponty's entrelacs and chiasma, as universal phenomenological tools, particularly useful for a better understanding of dynamic, non-classical, nonseparated contemporary mathematics. As a case study, we revisit Grothendieck's R esum e, and we explore its extremely rich mathematical, semiotical and phenomenological entanglements. © 2015 Published by Elsevier Ltd.
Keywords: Functional analysis Banach spaces Category theory Mathematics Semiotics Philosophy Phenomenology
The R esum e de la th eorie m etrique des produits tensoriels topologiques (Grothendieck, 1956) can be now assessed as one of the greatest mathematical articles of the XXth century. A gigantic contribution, little understood beyond specialists, it must nevertheless reach broader audiences. Written in 1953, while Grothendieck was in his first postdoc year in Sao Paulo, received in June 1954 and published two years later in the Boletim of the Mathematical Society of Sao Paulo, it lay in obscurity, before it came to be recognized as anticipating by thirty years the study of the fine structure of Banach spaces (Diestel et al., 2008). Grothendieck's inventiveness in the R esum e goes well beyond what is usually analyzed in texts dealing with mathematical creativity, and can profit from different perspectives, both semiotical and phenomenological, which reveal better the mathematical and methodological richness of Grothendieck's achievement. On one side, Peirce's cenopythagorean categories (Peirce, 1886) blend freshness (caeno) with mathematics (Pythagorean), and its dense multivel superpositions and reflections can be used as a wonderful tool to unravel Grothendieck's multivel creativity. On another side, Merleau-Ponty's entrelacs and chiasme (Merleau-Ponty, 1964a, 1964b) emphasize the continuity of experience, along a dynamic web of projections and translations which help to explain Grothendieck's category-theoretic smoothness paths. In this article we will address the question of understanding Grothendieck's R esum e through phenomenological perspectives, an E-mail address: [email protected]. URL: http://www.docentes.unal.edu.co/fzalameat/
exercise which enters into the more general realm of approaching mathematics through phenomenology (Rosen 2006). In particular, the connection between Grothendieck, Peirce and Merleau-Ponty is here observed for the first time. This yields a natural understanding of deep mathematical strategies (projectivity and injectivity in Hilbert and Banach spaces, unified topological-algebraic-functional analysis thought, emergence of applied category-theoretic methods), in which smoothness and multiplication (keys for Grothendieck) can be better revealed thanks to Peirce's thirdness and Merleau-Ponty's flesh and entrelacs. In the first section we briefly present Peirce's (phenomenological) categories, in the second section we deal with Merleau-Ponty's entrelacs and chiasme, and in the third section we apply both perspectives in order to provide an analysis of Grothendieck's R esum e and of the implicit emergence of mathematical categories in his work. 1. Peirce's three cenopythagorean categories Phaneroscopy, or the study of the phaneron, that is the complete collective spectrum present to the mind, includes the doctrine of Peirce's cenopythagorean categories, which observe the universal modes (or “tints”) occurring in phenomena. Peirce's three categories are vague, general and indeterminate, and can be found simultaneously in every phenomenon. They are intricated in several levels, but can be prescised (distinguished, separated, detached) following a recursive layer of interpretations, in progressively more and more determined contexts. A dialectics between the One and the Many, the universal and the particular, the general
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Please cite this article in press as: Zalamea, F., Peirce's cenopythagorean categories, Merleau-Ponty's chiasmatic entrelacs and Grothendieck's Resume, Progress in Biophysics and Molecular Biology (2015), http://dx.doi.org/10.1016/j.pbiomolbio.2015.07.011
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and the concrete is multilayered along a dense variety of theoretical and experimental fibers. Peirce's Firstness detects the immediate, the spontaneous, whatever is independent of any conception or reference to something else. Secondness is the category of facts, mutual oppositions, existence, actuality, material fight, action and reaction in a given world (two uses of the term “category” should not be confused in this paper: “category” alone will refer to a philosophical category, following Aristotle; “category-theoretic”, “category theory”, or “mathematical category” will refer to its technical mathematical sense). Thirdness proposes a mediation beyond clashes, a third place where the “one” and the “other” enter a dialogue, the category of sense, representation, synthesis. As (Peirce, 1886, p. 300) reckons, By the Third, I understand the medium which has its being or peculiarity in connecting the more absolute first and second. The end is second, the means third. A fork in the road is third, it supposes three ways. ( … ) The first and second are hard, absolute, and discrete, like yes and no; the perfect third is plastic, relative, and continuous. Every process, and whatever is continuous, involves thirdness. ( … ) Action is second, but conduct third. Law as an active force is second, but order and legislation third. Sympathy, flesh and blood, that by which I feel my neighbor's feelings, contains thirdness. Every kind of sign, representative, or deputy, everything which for any purpose stands instead of something else, whatever is helpful, or mediates between a man and his wish, is a Third. Peirce's vague categories can be “tinctured” with keywords as follows: (1) Firstness: immediacy, first impression, freshness, sensation, unary predicate, monad, chance, possibility. (2) Secondness: actionereaction, effect, resistance, alterity, binary relation, dyad, fact, actuality. (3) Thirdness: mediation, order, law, continuity, knowledge, ternary relation, triad, generality, necessity. The three peircean categories interweave recursively and produce a nested hierarchy of interpretative modulations. A series of modes and tones enter the analysis, and, as we shall see, when applied to Grothendieck's musical ear the series helps to explain the profound complexity of the R esum e. The interest of Peirce's method lies in the permanent iterative possibility of his categorical analysis (sequences of the form n.m.p.q … with n, m, p, q ranging through 1, 2, 3 e see examples below in the R esum e). The iteration allows, in each new contextual level (p, q, …), further and further refinements of previous distinctions obtained in prior levels (m, n, …). Dynamic knowledge yields progressive precision through progressive prescision. Intelligence grows with the definition of more and more contexts of interpretation, and the association of finer and finer cenopythagorean tinctures inside each context. The conceptual and practical back-and-forth between diverse layers is governed by the pragmatic(ist) maxim, which intertwines naturally with Peirce's categories. The maxim asserts that we can only attain knowledge after conceiving a wide range of representation possibilities for signs (firstness), after perusing activeereactive contrasts between sub-determinations of those signs (secondness), and after weaving recursive information between the observed semiotic processes (thirdness). The maxim acts as a sheaf with a double support function for the categories: a contrasting function (secondness) to obtain local distinctive hierarchies, and a mediating function (thirdness) to globally unify the different perspectives. In fact, a very broad, conceptual differential and integral calculus seems in act, to be a universal lattice of forms of
reintegrating the Many into the One. It is a strategy which also recalls the basic goals of mathematical category theory, where the apparently different description of objects in diverse concrete mathematical categories is reintegrated through their universal behavior in abstract mathematical categories. 2. Merleau-Ponty's chiasmatic entrelacs When faced with contemporary mathematics we cannot escape a certain transitory ontology that, at first, terminologically speaking, seems self-contradictory. Nevertheless, though the Greek ontotet^ es sends us, through Latin translations, to a supposedly atemporal “entity” or an “essence” that ontology would study, there is no reason (besides tradition) to believe that those entities or essences should be absolute and not asymptotic, governed by partial gluings in a correlative evolution between the world and knowledge. Bimodality, in the sense of Petitot, that is, dynamic movement both in physical and morphological-structural space, is related to such a state of things, where in fact, “things” are to be replaced by “processes” (functors, natural transformations and adjunctions in a precise category-theoretic setting). Both prefixes (trans-, bi-) provide a suitable ground to understand the wanderings of contemporary mathematics (circumscribing modern mathematics to the period 1830e1950, from Galois to Grothendieck, and contemporary mathematics to the period 1950-today). The philosophical basis of such a dynamic ontology can be found in Merleau-Ponty's theory of shifting, both in the general realm of knowledge and in the particular realm of visuality, and in Badiou's specific transitory ontology for mathematics (Badiou, 1998) (going back to Novalis provides other forgotten foundations e see (Margantin, 1998), (Kassenbrock, 2009)). For Merleau-Ponty, the “height of reason” consists in feeling the shifting of the soil (MerleauPonty, 1964a, 92), in detecting the movement of our beliefs and supposed claims of knowing: “each creation changes, alters, clarifies, deepens, confirms, exalts, recreates or creates by anticipation all the others” (Merleau-Ponty, 1964a, 92). A complex and mobile tissue of creation surges into view, full of “detours, transgressions, slow encroachments and sudden drives”, and in the contradictory coats of sediment emerges the force of creation entire. In Eye and Mind, Merleau-Ponty describes the body operative in the domains of knowing as a “sheaf of functions interlacing vision and movement” (Merleau-Ponty, 1964a, 16). That sheaf serves as an inter la Serres) between the real and the imaginary, between change (a discovery and invention, and allows us to capture the continuous transformation of an image into its obverse, through the various visions of interpreters. Two of the major theses of (Merleau-Ponty, 1964a) combine the necessity of both thinking the recto/verso dialectic and thinking in a continuous fashion: A. What is proper to the visible is to possess a fold of invisibility, in the strict sense. B. To unfold the world without separating thought is, precisely, modern ontology. There is thus a compelling need to understand the obverse (Zalamea, 2013), to study knowledge without artificial divisions (yielding some of the main mathematical achievements of Grothendieck's second period, 1958e1970), and to explore the corresponding entanglement between the positive and the negative (with deep mathematical examples such as RiemanneRoch's harmony between holomorphic and meromorphic dimensions related to genera, or Peirce's harmony between recto and verso of the assertion page related to modal calculi e we will also see some wonderful harmonies between opposites in Grothendieck's
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R esum e). In The Visible and the Invisible, Merleau-Ponty codifies this urge through a double metaphor: entrelacs and chiasme (MerleauPonty, 1964b, 170e201). Interlacing (entrelacs) and chiasma (chiasme, partial crossing of the optic nerves in the brain) serve to codify the “world without separating thought”, the continuum everywhere present: “the concomitant fabric of interior and exterior horizons”, “the knot in the weft of the simultaneous and the successive”, “the double crossing between visible and tangible”, “the ramifications of body and world” through “many leaves and faces” (insinuating Riemann surfaces), an “interlacing” which “renounces to think through planes and perspectives”, etc. (MerleauPonty, 1964b, 171, 172, 175, 177, 180). On the other hand, a characteristic perspective of Merleau-Ponty's final writings is the emphasis on flesh as a natural frontier between Mind and Nature. Concrete embodiment (along visibility) enters a dialogue with abstract conceptuality (along invisibility), and the chiasmatic entrelacs acquires a fine material quality which fosters the multiplicity and connectivity of knowledge. Merleau-Ponty's chiasmatic entrelacs recalls Peirce's thirdness, and both may be understood as conscious strategies to open the imagination to non-classical logics (where the law of excluded middle fails and properties and their opposites can hold at the same time), non-separated geometries (where spaces of sheaves are not Hausdorff, well beyond the sheaf of holomorphic functions closer to our “usual” geometries), non-isolated objects (where structures are not determined by their points, as in non-extensional Grothendieck topos). Peirce obtained a glimpse of such a huge panorama with his triadic logic and the existential graphs (Zalamea, zanne and Proust some 2012). Merleau-Ponty discovered in Ce profound artistic anticipations (Merleau-Ponty, 1964a, 1964b). Grothendieck refounded algebraic geometry on the idea of nonseparation (preceded by many Ehresmann ideas, in particular (Ehresmann, 1952)), through outstanding discoveries (archetypes, structures) and inventions (types, languages): schemes, topos, motives. Generic points in algebraic varieties and the Heyting algebra structure of subobject classifiers in an arbitrary elementary topos (a la Lawvere, intended as mathematical categories of variable sets) show the natural emergence of non-separation and non-classical logics in the heart of mathematical thought. Even if they are little known for the moment, these contributions have changed our classical conception of number (discrete) and magnitude (continuous), of arithmetic and geometry, eliminating their artificial separation, and making them merge as forms of a unified, higher-level mathematical category-theoretic view. We will now see how Grothendieck's R esum e ea rare gem with a strong influence on eight Fields medalistse encompasses in a nutshell this non-separated trend of thought. 3. Grothendieck's R esum e revisited Grothendieck's Ph. D. Thesis (defended 28 February 1953) invented nuclear spaces as locally convex topological vector spaces (lctvs) with a harmonic balance describable globally: (1) a lctvs space E is nuclear iff the two natural topologies on E5F (from projective and injective seminorms) coincide for every lctvs F; or, equivalently, locally: (2) a lctvs space is nuclear iff the family of seminorms which define the topology possesses a telescopic Fredholm property (Grothendieck, 1955). The classical examples of nuclear spaces are spaces of holomorphic functions; smoothness is the key functional-analysis idea sought to be axiomatized by Grothendieck. When Grothendieck writes his R esum e, in Sao Paulo, only a semester after his Ph. D., some of the general ideas present in the Thesis explode, yielding an understanding of the structural properties of classes (not yet called mathematical “categories”) of Hilbert and Banach spaces, from the triple perspective of tensor
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norm inequalities, analytical structures and algebraic factorizations (Grothendieck, 1956). The three main structures-m ere of Bourbaki enter simultaneously into the picture eOrder, Topology, Algebrae both to solve outstanding open problems (BanacheMazur) and to open deep, unsuspected new roads (Grothendieck's inequality, see below). Since (Grothendieck, 1956) is very little known and even more scarcely read, we present in Fig. 1 a sketch of its contents (pages refer to the transcription of the R esum e available through www. ime.usp.br; Figs. 2e4 are from the original article). Grothendieck sets himself to the task of exploring all “reasonable” norms (for which the canonical bilinear forms are of norm less than 1) on the tensor product of two Banach spaces (Grothendieck, 1956, 8). The strategy is beautiful: (i) detect canonical comparison norms, (ii) make transformations of those norms, (iii) cover the complete panorama with structural cleavages. The three stages can be explained through many levels of under la Bourbaki, we have (i) order, (ii) topology, (iii) standing: (A) a algebra; (B) semiotically, we have (i) semantics, (ii) syntactics, (iii) pragmatics (the Peircean order of the categories is here reversed: secondness, firstness, thirdness); (C) phenomenologically, following Merleau-Ponty, we have (i) continuity, (ii) separation, (iii) entrelacs; (D) musically, we have (i) theme, (ii) variations, (iii) fugue. In particular, in the first stage (i), we are able to observe one of Grothendieck's major obsessions at work. Since the Ph. D. Thesis, and all the way through his proof of generalized RiemanneRoch (Borel and Serre, 1958), Grothendieck was able to see projectivity and injectivity everywhere: projective and injective products, in the Thesis; projective and injective norms, in the R esum e; existence of enough injectives as the key construction in mathematical abelian categories with a generator, in order to reconstruct cohomology ^hoku (Grothendieck, 1957); injecfrom derived functors, in the To tion and projection subcases of proof, in the handling of extended RiemanneRoch. Projectivity and injectivity are in fact specializations (¼ spatialisations) of a wider look, which will become central in Grothendieck's second period (1958e1970), and which will search for profound archetypes of mathematical knowledge. Multiplicity, diversity, types are then to be understood as projected from the archetypes, or, dually, injected into them. In Merleau-Ponty's terms, flesh (types) echoes the invisible (archetypes). The strategy works wonders in the case of norms on tensor products of Banach spaces. The greatest and smallest reasonable norms turn out to be precisely the projective and injective norms (see Fig. 2), and we have a semantical order from where mathematics can proceed, a continuum theme to be developed through all sort of topological and algebraic variations. Grothendieck's imagination unleashes with the formation on new tensor norms (Fig. 3). For any 5-norm a, that is, an application which gives, for each pair of normed spaces, a reasonable bounded norm on their tensor product (Grothendieck, 1956, 8e9), Grothendieck produces its optimal canonical variations:/a, a\, \a, a/, the first two being the greatest left and right injectives below a, the second two being the smallest left and right projectives above a. As Grothendieck explains, the syntactic modifier (/, \) (“trait rateur”) codifies iconically (in Peirce's sense) the very semanalte tical content of the sign: bottom inclination (object embedded) referring to injectivity, upward inclination (object prolonged) referring to projectivity. The concreteness of icons can also be seen as an embodiment of Merleau-Ponty's flesh: the specific form of the signs refer to invisible (algebraic, topological) properties that only a deep category-theoretic analysis will reveal. Further (Fig. 3, bottom), a calculus of transpositions, duality and “inclinations” yields a sort of musical harmonics of 5-norms. The variations, or alterations, can be applied to the main norms ∨ (injective), ∧ (projective), and a wonderful series of twelve
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sume . Fig. 1. Sketch of contents of the Re
Fig. 2. Semantic level: Grothendieck's projectivity and injectivity (Grothendieck, 1956, 27).
(“Banach”) 5-norms appears syntactically: ∨, ∧, \∨, ∨/,/∧, ∧\, \∨/,/∧\,/\∨, ∨/\, \/∧, ∧\/. Adding two other norms coming from Hilbert space ideas (Grothendieck, 1956, 40e41), produces a list of fourteen natural 5-norms. Then the “miracle” explodes: (1) the list covers (up to equivalence) all possible natural norms, (2) the norms are ordered in a symmetric lattice, (3) each entry in the lattice is completely characterized by exact algebraic factorization properties of the norm between types of classical spaces (L, integral bounded; C, continuous bounded on compacts; H, Hilbert bounded) (see Fig. 4). We are thus in presence of a true pragmatic achievement (again in Peirce's sense): the iconic signs enter into the realm of relational logic, and their meaning coincides with their action along the entire analytic, algebraic and semiotic web. Merleau-Ponty's entrelacs is here attained in a wonderful musical harmonics, a blend of (1) syntactic freshness and economy, (2) semantic complexity, and (3) pragmatic architectonics. Merleau-Ponty's chiasma, a metaphor coming from the crossing of optics nerves in the brain, can be seen in turn ein the R esum ee as the deep crossing of Bourbaki's nerves (structures-m ere: Order, Topology, Algebra) in Grothendieck's brain. The chiasmatic entrelacs possesses several entanglement levels, if we follow Peirce's cenopythagorean categories: (2) initial ∨, ∧ norms; (2.1) diagonal alterations (icons are forms of firstness); (2.2) transposition and duality (indexes are forms of secondness); (2.3) lattice of intermediate norms (symbols are forms of thirdness); (2.3.1) algebraic
Fig. 3. Syntactic level: Grothendieck's notations (Grothendieck, 1956, 28).
characterizations through factorizations (finiteness is a form of firstness); (2.3.3) topological characterizations (continuity is a form of thirdness), etc. On another hand, the dual role of Merleau-Ponty's flesh, understood as a bridge between Mind and Nature, corresponds, in Grothendieck's category-theoretic thinking, to the
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outstanding mathematical tradition converges in Grothendieck, to further diverge in new, unexpected constructions. In his R esum e, Grothendieck was able to encompass, in a unified vision, functional analysis, topology and algebra, to think in terms of general classes of norms, to foresee the structure of classes (¼ mathematical categories) of Banach and Hilbert spaces, and, further, to discover a mysterious constant which seemed to codify (¼ project) the complete panorama of tensor products in Banach spaces. A rare gem, the R esum e can be seen as the germ of Grothendieck's incredible imagination in the period 1955e1970, after his Kansas “turn to geometry” (Grothendieck, 1986, 01/26-27). Acknowledgments The author thanks the extremely useful comments of Steven Rosen who helped to strenghten the phenomenological/mathematical connection between Merleau-Ponty and Grothendieck, two other anonymous referees who helped to correct obscurities both of form and content, and two other final referees who added references and precised some of the mathematics involved. References
Fig. 4. Pragmatic level: Grothendieck's fourteen natural norms (Grothendieck, 1956, 37).
natural bridge between Abstract and Concrete mathematical categories, which gives to the theory all its universal force. Grothendiecks's inequality (Grothendieck, 1956, 59e60) asserts that the projective norm ∧ is bounded by the Hilbert norm modulo a universal constant. One can then obtain a “best” constant, which depends on the base field (complex or real, different in each case), and which has not yet been completely determined. In a profound stratum, beyond the lattice of norms, lies thus a new archetype (Grothendieck's constant), which governs in an unsuspected way, many applications in widely different fields: fine local structure of Banach spaces, C*-algebras, non-commutative geometry, quantum mechanics, graph theory, computer science, etc. e see (Pisier, 2012). The dialectics archetype/types, One/Many, universal/concrete, propels Grothendieck's main achievements, nurturing a long French and German tradition in modern mathematics (Galois, , Cantor, Hilbert), for which new foundations and Riemann, Poincare new problems will emerge. It is interesting to observe that such an
d'ontologie transitoire. Seuil, Paris. Badiou, A., 1998. Court traite ore me de Riemann-Roch (d'apre s des re sultats Borel, A., Serre, J.P., 1958. Le the dits de A. Grothendieck). Bull. Soc. Math. Fr. 86, 97e136. ine Diestel, J., Fourie, J.H., Swart, J., 2008. The Metric Theory of Tensor Products. Grosume Revisited. American Mathematical Society, Providence. thendieck's Re Ehresmann, C., 1952. Structures locales (Conference, Rome 1952). In: tes et commente es, Ehresmann, A.C. (Ed.), C. Ehresmann, Oeuvres comple Amiens, Evrard, 1984, vol. 1, pp. 411e420. aires. Grothendieck, A., 1955. Produits tensoriels topologiques et espaces nucle American Mathematical Society, Memoirs 16, Providence. sume de la the orie me trique des produits tensoriels Grothendieck, A., 1956. Re topologiques. Bol. Soc. Mat. Sao Paulo 8, 1e79. bre homologique. To ^hoku Math. J. Grothendieck, A., 1957. Sur quelques points d'alge 9, 119e221. Grothendieck, A., 1986. R ecoltes et Semailles, manuscript, p. 1252. Kassenbrock, B.W., 2009. Novalis and the Two Cultures: the Chiasmic Discourse of Mathematics, Philosophy and Poetics. Ph. D. Thesis. New York University. me mine ralogique et cosmologique chez Novalis, ou les Margantin, L., 1998. Syste plis de la terre. L'Harmattan, Paris. Merleau-Ponty, M., 1964a. L'Oeil et l'Esprit. Gallimard, Paris (reed. Folio, 2004). Merleau-Ponty, M., 1964b. Le visible et l'invisible. Gallimard, Paris (reed. Folio, 2004). Peirce, C.S., 1886. One, two, three: an evolutionist speculation. In: C.S. Peirce, Writings, A Chronological Edition, vol. 5. Indiana University Press, Bloomington, pp. 300e301. Pisier, G., 2012. Grothendieck's theorem, past and present. Bull. Amer. Math. Soc. 49, 237e323. Rosen, S., 2006. Topologies of the Flesh. Ohio University Press, Athens. Zalamea, F., 2012. Peirce's Logic of Continuity. Docent Press, Boston. n. Las fuentes contradictorias de la Zalamea, F., 2013. Antinomias de la creacio n en Vale ry. Fondo de Cultura Econo mica, Warburg, Florenski. Santiago invencio de Chile.
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Contents lists available at ScienceDirect
Progress in Biophysics and Molecular Biology journal homepage: www.elsevier.com/locate/pbiomolbio
Peirce's cenopythagorean categories, Merleau-Ponty's chiasmatic entrelacs and Grothendieck's R esum e Fernando Zalamea Universidad Nacional de Colombia, Colombia
a r t i c l e i n f o
a b s t r a c t
Article history: Available online xxx
We present Peirce's cenopythagorean categories and Merleau-Ponty's entrelacs and chiasma, as universal phenomenological tools, particularly useful for a better understanding of dynamic, non-classical, nonseparated contemporary mathematics. As a case study, we revisit Grothendieck's R esum e, and we explore its extremely rich mathematical, semiotical and phenomenological entanglements. © 2015 Published by Elsevier Ltd.
Keywords: Functional analysis Banach spaces Category theory Mathematics Semiotics Philosophy Phenomenology
The R esum e de la th eorie m etrique des produits tensoriels topologiques (Grothendieck, 1956) can be now assessed as one of the greatest mathematical articles of the XXth century. A gigantic contribution, little understood beyond specialists, it must nevertheless reach broader audiences. Written in 1953, while Grothendieck was in his first postdoc year in Sao Paulo, received in June 1954 and published two years later in the Boletim of the Mathematical Society of Sao Paulo, it lay in obscurity, before it came to be recognized as anticipating by thirty years the study of the fine structure of Banach spaces (Diestel et al., 2008). Grothendieck's inventiveness in the R esum e goes well beyond what is usually analyzed in texts dealing with mathematical creativity, and can profit from different perspectives, both semiotical and phenomenological, which reveal better the mathematical and methodological richness of Grothendieck's achievement. On one side, Peirce's cenopythagorean categories (Peirce, 1886) blend freshness (caeno) with mathematics (Pythagorean), and its dense multivel superpositions and reflections can be used as a wonderful tool to unravel Grothendieck's multivel creativity. On another side, Merleau-Ponty's entrelacs and chiasme (Merleau-Ponty, 1964a, 1964b) emphasize the continuity of experience, along a dynamic web of projections and translations which help to explain Grothendieck's category-theoretic smoothness paths. In this article we will address the question of understanding Grothendieck's R esum e through phenomenological perspectives, an E-mail address: [email protected]. URL: http://www.docentes.unal.edu.co/fzalameat/
exercise which enters into the more general realm of approaching mathematics through phenomenology (Rosen 2006). In particular, the connection between Grothendieck, Peirce and Merleau-Ponty is here observed for the first time. This yields a natural understanding of deep mathematical strategies (projectivity and injectivity in Hilbert and Banach spaces, unified topological-algebraic-functional analysis thought, emergence of applied category-theoretic methods), in which smoothness and multiplication (keys for Grothendieck) can be better revealed thanks to Peirce's thirdness and Merleau-Ponty's flesh and entrelacs. In the first section we briefly present Peirce's (phenomenological) categories, in the second section we deal with Merleau-Ponty's entrelacs and chiasme, and in the third section we apply both perspectives in order to provide an analysis of Grothendieck's R esum e and of the implicit emergence of mathematical categories in his work. 1. Peirce's three cenopythagorean categories Phaneroscopy, or the study of the phaneron, that is the complete collective spectrum present to the mind, includes the doctrine of Peirce's cenopythagorean categories, which observe the universal modes (or “tints”) occurring in phenomena. Peirce's three categories are vague, general and indeterminate, and can be found simultaneously in every phenomenon. They are intricated in several levels, but can be prescised (distinguished, separated, detached) following a recursive layer of interpretations, in progressively more and more determined contexts. A dialectics between the One and the Many, the universal and the particular, the general
http://dx.doi.org/10.1016/j.pbiomolbio.2015.07.011 0079-6107/© 2015 Published by Elsevier Ltd.
Please cite this article in press as: Zalamea, F., Peirce's cenopythagorean categories, Merleau-Ponty's chiasmatic entrelacs and Grothendieck's Resume, Progress in Biophysics and Molecular Biology (2015), http://dx.doi.org/10.1016/j.pbiomolbio.2015.07.011
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and the concrete is multilayered along a dense variety of theoretical and experimental fibers. Peirce's Firstness detects the immediate, the spontaneous, whatever is independent of any conception or reference to something else. Secondness is the category of facts, mutual oppositions, existence, actuality, material fight, action and reaction in a given world (two uses of the term “category” should not be confused in this paper: “category” alone will refer to a philosophical category, following Aristotle; “category-theoretic”, “category theory”, or “mathematical category” will refer to its technical mathematical sense). Thirdness proposes a mediation beyond clashes, a third place where the “one” and the “other” enter a dialogue, the category of sense, representation, synthesis. As (Peirce, 1886, p. 300) reckons, By the Third, I understand the medium which has its being or peculiarity in connecting the more absolute first and second. The end is second, the means third. A fork in the road is third, it supposes three ways. ( … ) The first and second are hard, absolute, and discrete, like yes and no; the perfect third is plastic, relative, and continuous. Every process, and whatever is continuous, involves thirdness. ( … ) Action is second, but conduct third. Law as an active force is second, but order and legislation third. Sympathy, flesh and blood, that by which I feel my neighbor's feelings, contains thirdness. Every kind of sign, representative, or deputy, everything which for any purpose stands instead of something else, whatever is helpful, or mediates between a man and his wish, is a Third. Peirce's vague categories can be “tinctured” with keywords as follows: (1) Firstness: immediacy, first impression, freshness, sensation, unary predicate, monad, chance, possibility. (2) Secondness: actionereaction, effect, resistance, alterity, binary relation, dyad, fact, actuality. (3) Thirdness: mediation, order, law, continuity, knowledge, ternary relation, triad, generality, necessity. The three peircean categories interweave recursively and produce a nested hierarchy of interpretative modulations. A series of modes and tones enter the analysis, and, as we shall see, when applied to Grothendieck's musical ear the series helps to explain the profound complexity of the R esum e. The interest of Peirce's method lies in the permanent iterative possibility of his categorical analysis (sequences of the form n.m.p.q … with n, m, p, q ranging through 1, 2, 3 e see examples below in the R esum e). The iteration allows, in each new contextual level (p, q, …), further and further refinements of previous distinctions obtained in prior levels (m, n, …). Dynamic knowledge yields progressive precision through progressive prescision. Intelligence grows with the definition of more and more contexts of interpretation, and the association of finer and finer cenopythagorean tinctures inside each context. The conceptual and practical back-and-forth between diverse layers is governed by the pragmatic(ist) maxim, which intertwines naturally with Peirce's categories. The maxim asserts that we can only attain knowledge after conceiving a wide range of representation possibilities for signs (firstness), after perusing activeereactive contrasts between sub-determinations of those signs (secondness), and after weaving recursive information between the observed semiotic processes (thirdness). The maxim acts as a sheaf with a double support function for the categories: a contrasting function (secondness) to obtain local distinctive hierarchies, and a mediating function (thirdness) to globally unify the different perspectives. In fact, a very broad, conceptual differential and integral calculus seems in act, to be a universal lattice of forms of
reintegrating the Many into the One. It is a strategy which also recalls the basic goals of mathematical category theory, where the apparently different description of objects in diverse concrete mathematical categories is reintegrated through their universal behavior in abstract mathematical categories. 2. Merleau-Ponty's chiasmatic entrelacs When faced with contemporary mathematics we cannot escape a certain transitory ontology that, at first, terminologically speaking, seems self-contradictory. Nevertheless, though the Greek ontotet^ es sends us, through Latin translations, to a supposedly atemporal “entity” or an “essence” that ontology would study, there is no reason (besides tradition) to believe that those entities or essences should be absolute and not asymptotic, governed by partial gluings in a correlative evolution between the world and knowledge. Bimodality, in the sense of Petitot, that is, dynamic movement both in physical and morphological-structural space, is related to such a state of things, where in fact, “things” are to be replaced by “processes” (functors, natural transformations and adjunctions in a precise category-theoretic setting). Both prefixes (trans-, bi-) provide a suitable ground to understand the wanderings of contemporary mathematics (circumscribing modern mathematics to the period 1830e1950, from Galois to Grothendieck, and contemporary mathematics to the period 1950-today). The philosophical basis of such a dynamic ontology can be found in Merleau-Ponty's theory of shifting, both in the general realm of knowledge and in the particular realm of visuality, and in Badiou's specific transitory ontology for mathematics (Badiou, 1998) (going back to Novalis provides other forgotten foundations e see (Margantin, 1998), (Kassenbrock, 2009)). For Merleau-Ponty, the “height of reason” consists in feeling the shifting of the soil (MerleauPonty, 1964a, 92), in detecting the movement of our beliefs and supposed claims of knowing: “each creation changes, alters, clarifies, deepens, confirms, exalts, recreates or creates by anticipation all the others” (Merleau-Ponty, 1964a, 92). A complex and mobile tissue of creation surges into view, full of “detours, transgressions, slow encroachments and sudden drives”, and in the contradictory coats of sediment emerges the force of creation entire. In Eye and Mind, Merleau-Ponty describes the body operative in the domains of knowing as a “sheaf of functions interlacing vision and movement” (Merleau-Ponty, 1964a, 16). That sheaf serves as an inter la Serres) between the real and the imaginary, between change (a discovery and invention, and allows us to capture the continuous transformation of an image into its obverse, through the various visions of interpreters. Two of the major theses of (Merleau-Ponty, 1964a) combine the necessity of both thinking the recto/verso dialectic and thinking in a continuous fashion: A. What is proper to the visible is to possess a fold of invisibility, in the strict sense. B. To unfold the world without separating thought is, precisely, modern ontology. There is thus a compelling need to understand the obverse (Zalamea, 2013), to study knowledge without artificial divisions (yielding some of the main mathematical achievements of Grothendieck's second period, 1958e1970), and to explore the corresponding entanglement between the positive and the negative (with deep mathematical examples such as RiemanneRoch's harmony between holomorphic and meromorphic dimensions related to genera, or Peirce's harmony between recto and verso of the assertion page related to modal calculi e we will also see some wonderful harmonies between opposites in Grothendieck's
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R esum e). In The Visible and the Invisible, Merleau-Ponty codifies this urge through a double metaphor: entrelacs and chiasme (MerleauPonty, 1964b, 170e201). Interlacing (entrelacs) and chiasma (chiasme, partial crossing of the optic nerves in the brain) serve to codify the “world without separating thought”, the continuum everywhere present: “the concomitant fabric of interior and exterior horizons”, “the knot in the weft of the simultaneous and the successive”, “the double crossing between visible and tangible”, “the ramifications of body and world” through “many leaves and faces” (insinuating Riemann surfaces), an “interlacing” which “renounces to think through planes and perspectives”, etc. (MerleauPonty, 1964b, 171, 172, 175, 177, 180). On the other hand, a characteristic perspective of Merleau-Ponty's final writings is the emphasis on flesh as a natural frontier between Mind and Nature. Concrete embodiment (along visibility) enters a dialogue with abstract conceptuality (along invisibility), and the chiasmatic entrelacs acquires a fine material quality which fosters the multiplicity and connectivity of knowledge. Merleau-Ponty's chiasmatic entrelacs recalls Peirce's thirdness, and both may be understood as conscious strategies to open the imagination to non-classical logics (where the law of excluded middle fails and properties and their opposites can hold at the same time), non-separated geometries (where spaces of sheaves are not Hausdorff, well beyond the sheaf of holomorphic functions closer to our “usual” geometries), non-isolated objects (where structures are not determined by their points, as in non-extensional Grothendieck topos). Peirce obtained a glimpse of such a huge panorama with his triadic logic and the existential graphs (Zalamea, zanne and Proust some 2012). Merleau-Ponty discovered in Ce profound artistic anticipations (Merleau-Ponty, 1964a, 1964b). Grothendieck refounded algebraic geometry on the idea of nonseparation (preceded by many Ehresmann ideas, in particular (Ehresmann, 1952)), through outstanding discoveries (archetypes, structures) and inventions (types, languages): schemes, topos, motives. Generic points in algebraic varieties and the Heyting algebra structure of subobject classifiers in an arbitrary elementary topos (a la Lawvere, intended as mathematical categories of variable sets) show the natural emergence of non-separation and non-classical logics in the heart of mathematical thought. Even if they are little known for the moment, these contributions have changed our classical conception of number (discrete) and magnitude (continuous), of arithmetic and geometry, eliminating their artificial separation, and making them merge as forms of a unified, higher-level mathematical category-theoretic view. We will now see how Grothendieck's R esum e ea rare gem with a strong influence on eight Fields medalistse encompasses in a nutshell this non-separated trend of thought. 3. Grothendieck's R esum e revisited Grothendieck's Ph. D. Thesis (defended 28 February 1953) invented nuclear spaces as locally convex topological vector spaces (lctvs) with a harmonic balance describable globally: (1) a lctvs space E is nuclear iff the two natural topologies on E5F (from projective and injective seminorms) coincide for every lctvs F; or, equivalently, locally: (2) a lctvs space is nuclear iff the family of seminorms which define the topology possesses a telescopic Fredholm property (Grothendieck, 1955). The classical examples of nuclear spaces are spaces of holomorphic functions; smoothness is the key functional-analysis idea sought to be axiomatized by Grothendieck. When Grothendieck writes his R esum e, in Sao Paulo, only a semester after his Ph. D., some of the general ideas present in the Thesis explode, yielding an understanding of the structural properties of classes (not yet called mathematical “categories”) of Hilbert and Banach spaces, from the triple perspective of tensor
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norm inequalities, analytical structures and algebraic factorizations (Grothendieck, 1956). The three main structures-m ere of Bourbaki enter simultaneously into the picture eOrder, Topology, Algebrae both to solve outstanding open problems (BanacheMazur) and to open deep, unsuspected new roads (Grothendieck's inequality, see below). Since (Grothendieck, 1956) is very little known and even more scarcely read, we present in Fig. 1 a sketch of its contents (pages refer to the transcription of the R esum e available through www. ime.usp.br; Figs. 2e4 are from the original article). Grothendieck sets himself to the task of exploring all “reasonable” norms (for which the canonical bilinear forms are of norm less than 1) on the tensor product of two Banach spaces (Grothendieck, 1956, 8). The strategy is beautiful: (i) detect canonical comparison norms, (ii) make transformations of those norms, (iii) cover the complete panorama with structural cleavages. The three stages can be explained through many levels of under la Bourbaki, we have (i) order, (ii) topology, (iii) standing: (A) a algebra; (B) semiotically, we have (i) semantics, (ii) syntactics, (iii) pragmatics (the Peircean order of the categories is here reversed: secondness, firstness, thirdness); (C) phenomenologically, following Merleau-Ponty, we have (i) continuity, (ii) separation, (iii) entrelacs; (D) musically, we have (i) theme, (ii) variations, (iii) fugue. In particular, in the first stage (i), we are able to observe one of Grothendieck's major obsessions at work. Since the Ph. D. Thesis, and all the way through his proof of generalized RiemanneRoch (Borel and Serre, 1958), Grothendieck was able to see projectivity and injectivity everywhere: projective and injective products, in the Thesis; projective and injective norms, in the R esum e; existence of enough injectives as the key construction in mathematical abelian categories with a generator, in order to reconstruct cohomology ^hoku (Grothendieck, 1957); injecfrom derived functors, in the To tion and projection subcases of proof, in the handling of extended RiemanneRoch. Projectivity and injectivity are in fact specializations (¼ spatialisations) of a wider look, which will become central in Grothendieck's second period (1958e1970), and which will search for profound archetypes of mathematical knowledge. Multiplicity, diversity, types are then to be understood as projected from the archetypes, or, dually, injected into them. In Merleau-Ponty's terms, flesh (types) echoes the invisible (archetypes). The strategy works wonders in the case of norms on tensor products of Banach spaces. The greatest and smallest reasonable norms turn out to be precisely the projective and injective norms (see Fig. 2), and we have a semantical order from where mathematics can proceed, a continuum theme to be developed through all sort of topological and algebraic variations. Grothendieck's imagination unleashes with the formation on new tensor norms (Fig. 3). For any 5-norm a, that is, an application which gives, for each pair of normed spaces, a reasonable bounded norm on their tensor product (Grothendieck, 1956, 8e9), Grothendieck produces its optimal canonical variations:/a, a\, \a, a/, the first two being the greatest left and right injectives below a, the second two being the smallest left and right projectives above a. As Grothendieck explains, the syntactic modifier (/, \) (“trait rateur”) codifies iconically (in Peirce's sense) the very semanalte tical content of the sign: bottom inclination (object embedded) referring to injectivity, upward inclination (object prolonged) referring to projectivity. The concreteness of icons can also be seen as an embodiment of Merleau-Ponty's flesh: the specific form of the signs refer to invisible (algebraic, topological) properties that only a deep category-theoretic analysis will reveal. Further (Fig. 3, bottom), a calculus of transpositions, duality and “inclinations” yields a sort of musical harmonics of 5-norms. The variations, or alterations, can be applied to the main norms ∨ (injective), ∧ (projective), and a wonderful series of twelve
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sume . Fig. 1. Sketch of contents of the Re
Fig. 2. Semantic level: Grothendieck's projectivity and injectivity (Grothendieck, 1956, 27).
(“Banach”) 5-norms appears syntactically: ∨, ∧, \∨, ∨/,/∧, ∧\, \∨/,/∧\,/\∨, ∨/\, \/∧, ∧\/. Adding two other norms coming from Hilbert space ideas (Grothendieck, 1956, 40e41), produces a list of fourteen natural 5-norms. Then the “miracle” explodes: (1) the list covers (up to equivalence) all possible natural norms, (2) the norms are ordered in a symmetric lattice, (3) each entry in the lattice is completely characterized by exact algebraic factorization properties of the norm between types of classical spaces (L, integral bounded; C, continuous bounded on compacts; H, Hilbert bounded) (see Fig. 4). We are thus in presence of a true pragmatic achievement (again in Peirce's sense): the iconic signs enter into the realm of relational logic, and their meaning coincides with their action along the entire analytic, algebraic and semiotic web. Merleau-Ponty's entrelacs is here attained in a wonderful musical harmonics, a blend of (1) syntactic freshness and economy, (2) semantic complexity, and (3) pragmatic architectonics. Merleau-Ponty's chiasma, a metaphor coming from the crossing of optics nerves in the brain, can be seen in turn ein the R esum ee as the deep crossing of Bourbaki's nerves (structures-m ere: Order, Topology, Algebra) in Grothendieck's brain. The chiasmatic entrelacs possesses several entanglement levels, if we follow Peirce's cenopythagorean categories: (2) initial ∨, ∧ norms; (2.1) diagonal alterations (icons are forms of firstness); (2.2) transposition and duality (indexes are forms of secondness); (2.3) lattice of intermediate norms (symbols are forms of thirdness); (2.3.1) algebraic
Fig. 3. Syntactic level: Grothendieck's notations (Grothendieck, 1956, 28).
characterizations through factorizations (finiteness is a form of firstness); (2.3.3) topological characterizations (continuity is a form of thirdness), etc. On another hand, the dual role of Merleau-Ponty's flesh, understood as a bridge between Mind and Nature, corresponds, in Grothendieck's category-theoretic thinking, to the
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outstanding mathematical tradition converges in Grothendieck, to further diverge in new, unexpected constructions. In his R esum e, Grothendieck was able to encompass, in a unified vision, functional analysis, topology and algebra, to think in terms of general classes of norms, to foresee the structure of classes (¼ mathematical categories) of Banach and Hilbert spaces, and, further, to discover a mysterious constant which seemed to codify (¼ project) the complete panorama of tensor products in Banach spaces. A rare gem, the R esum e can be seen as the germ of Grothendieck's incredible imagination in the period 1955e1970, after his Kansas “turn to geometry” (Grothendieck, 1986, 01/26-27). Acknowledgments The author thanks the extremely useful comments of Steven Rosen who helped to strenghten the phenomenological/mathematical connection between Merleau-Ponty and Grothendieck, two other anonymous referees who helped to correct obscurities both of form and content, and two other final referees who added references and precised some of the mathematics involved. References
Fig. 4. Pragmatic level: Grothendieck's fourteen natural norms (Grothendieck, 1956, 37).
natural bridge between Abstract and Concrete mathematical categories, which gives to the theory all its universal force. Grothendiecks's inequality (Grothendieck, 1956, 59e60) asserts that the projective norm ∧ is bounded by the Hilbert norm modulo a universal constant. One can then obtain a “best” constant, which depends on the base field (complex or real, different in each case), and which has not yet been completely determined. In a profound stratum, beyond the lattice of norms, lies thus a new archetype (Grothendieck's constant), which governs in an unsuspected way, many applications in widely different fields: fine local structure of Banach spaces, C*-algebras, non-commutative geometry, quantum mechanics, graph theory, computer science, etc. e see (Pisier, 2012). The dialectics archetype/types, One/Many, universal/concrete, propels Grothendieck's main achievements, nurturing a long French and German tradition in modern mathematics (Galois, , Cantor, Hilbert), for which new foundations and Riemann, Poincare new problems will emerge. It is interesting to observe that such an
d'ontologie transitoire. Seuil, Paris. Badiou, A., 1998. Court traite ore me de Riemann-Roch (d'apre s des re sultats Borel, A., Serre, J.P., 1958. Le the dits de A. Grothendieck). Bull. Soc. Math. Fr. 86, 97e136. ine Diestel, J., Fourie, J.H., Swart, J., 2008. The Metric Theory of Tensor Products. Grosume Revisited. American Mathematical Society, Providence. thendieck's Re Ehresmann, C., 1952. Structures locales (Conference, Rome 1952). In: tes et commente es, Ehresmann, A.C. (Ed.), C. Ehresmann, Oeuvres comple Amiens, Evrard, 1984, vol. 1, pp. 411e420. aires. Grothendieck, A., 1955. Produits tensoriels topologiques et espaces nucle American Mathematical Society, Memoirs 16, Providence. sume de la the orie me trique des produits tensoriels Grothendieck, A., 1956. Re topologiques. Bol. Soc. Mat. Sao Paulo 8, 1e79. bre homologique. To ^hoku Math. J. Grothendieck, A., 1957. Sur quelques points d'alge 9, 119e221. Grothendieck, A., 1986. R ecoltes et Semailles, manuscript, p. 1252. Kassenbrock, B.W., 2009. Novalis and the Two Cultures: the Chiasmic Discourse of Mathematics, Philosophy and Poetics. Ph. D. Thesis. New York University. me mine ralogique et cosmologique chez Novalis, ou les Margantin, L., 1998. Syste plis de la terre. L'Harmattan, Paris. Merleau-Ponty, M., 1964a. L'Oeil et l'Esprit. Gallimard, Paris (reed. Folio, 2004). Merleau-Ponty, M., 1964b. Le visible et l'invisible. Gallimard, Paris (reed. Folio, 2004). Peirce, C.S., 1886. One, two, three: an evolutionist speculation. In: C.S. Peirce, Writings, A Chronological Edition, vol. 5. Indiana University Press, Bloomington, pp. 300e301. Pisier, G., 2012. Grothendieck's theorem, past and present. Bull. Amer. Math. Soc. 49, 237e323. Rosen, S., 2006. Topologies of the Flesh. Ohio University Press, Athens. Zalamea, F., 2012. Peirce's Logic of Continuity. Docent Press, Boston. n. Las fuentes contradictorias de la Zalamea, F., 2013. Antinomias de la creacio n en Vale ry. Fondo de Cultura Econo mica, Warburg, Florenski. Santiago invencio de Chile.
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