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Fernando Zalamea

Fernando Zalamea Traba is a mathematician at Universidad Nacional de Colombia at Bogotá with main interests in categorical logic, non-classical logic, history and philosophy of mathematics. He is also writing essays in cultural studies.

On the difference between modern and contemporary mathematics

Just as Albert Lautman identified 5 features characteristic of advanced (post mid-19th century) mathematics (1-5 in this list), Zalamea identifies an additional set of 5 characteristics (6-10 in this list) which emerge only in the mid 20th century and (in addition to 1-5, which are conserved) define contemporary mathematics:

  1. the complex hierarchisation of various theories, irreducible to systems of intermediate deduction;

  2. the richness of the models, irreducible to linguistic manipulation;

  3. the unity of structural methods and of conceptual polarites, beyond the effective multiplicity of models;

  4. the dynamics of the creative activity, in a permanent back-and-forth between freedom and saturation, open to the Platonic division and the Platonic dialectic;

  5. the mathematically demonstrable relation between what is multiple on a given level and what is singular on another, through a sophisticated lattice of mixed ascents and descents.

  6. the structural impurity of arithmetic (Weil’s conjectures, Langlands program, the theorems of Deligne, Faltings and Wiles etc.)

  7. the systematic geometrization of all environments of mathematics (sheaves, homologies, cobordisms, geometrical logic etc.)

  8. the schematization and the liberation from set theoretical, algebraic, and topological restrictions (groupoids, categories, schemas, topoi, motifs etc..)

  9. the fluxion and deformation of the usual boundaries of mathematical structures (nonlinearity, noncommutativity, nonelemantarity, quantization etc)

  10. the reflexivity of theories and models onto themselves (classification theory, fixed-point theorems, monstrous models, elementary/nonelementary classes etc..)

On reintegrating our dispersed culture

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A programmatic construction of a sheaf of partial cultural gluings could then be articulated around three main pieces of information: (I) Methodological forces: topological and transformational thinking, universal relatives, logic of sheaves, residuation theory, pendulum weaving, etc. (II) Cultural realizations: critical theory, metaphoric sedimentation, contaminating strata, mediating hierarchies, etc. (III) Projective goals: description of reflective residues (gluing local and global), dense fabrics (joining multipolar threads), plastic generic forms (allowing continuity and dislocating dualisms), etc. It is our contention that (i) Peirce’s system and many mathematical tools, both modern (Riemann, Galois) and contemporary (Grothendieck), provide all the necessary theoretical background to support (I); (ii) Germany’s critical dialectical tradition and Latin America’s TRANS essayists give good examples of how to deal with (II); and (iii) the very “end” of Postmodernism as such, with its reformulations within Modernism and Transmodernism, show the imperative of the integrated relativity, plasticity and contamination sought in (III). Precise labours on these problems will take years, but, with many non-standard tools at hand, pragmaticism and “razonabilidad” can lead the way.

References

Last revised on December 10, 2014 at 05:58:32. See the history of this page for a list of all contributions to it.