Hilbert's Program Then and Now

Hilbert's program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to "dispose of the foundational questions in mathematics once and for all, "Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, "finitary" means, one should give proofs of the consistency of these axiomatic systems. Although Godel's incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial successes, and generated important advances in logical theory and meta-theory, both at the time and since. The article discusses the historical background and development of Hilbert's program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s.Comment: 43 page

`Third' Quantization of Vacuum Einstein Gravity and Free Yang-Mills Theories

Based on the algebraico-categorical (:sheaf-theoretic and sheaf cohomological) conceptual and technical machinery of Abstract Differential Geometry, a new, genuinely background spacetime manifold independent, field quantization scenario for vacuum Einstein gravity and free Yang-Mills theories is introduced. The scheme is coined `third quantization' and, although it formally appears to follow a canonical route, it is fully covariant, because it is an expressly functorial `procedure'. Various current and future Quantum Gravity research issues are discussed under the light of 3rd-quantization. A postscript gives a brief account of this author's personal encounters with Rafael Sorkin and his work.Comment: 43 pages; latest version contributed to a fest-volume celebrating Rafael Sorkin's 60th birthday (Erratum: in earlier versions I had wrongly written that the Editor for this volume is Daniele Oriti, with CUP as publisher. I apologize for the mistake.

Finitary Topos for Locally Finite, Causal and Quantal Vacuum Einstein Gravity

Previous work on applications of Abstract Differential Geometry (ADG) to discrete Lorentzian quantum gravity is brought to its categorical climax by organizing the curved finitary spacetime sheaves of quantum causal sets involved therein, on which a finitary (:locally finite), singularity-free, background manifold independent and geometrically prequantized version of the gravitational vacuum Einstein field equations were seen to hold, into a topos structure. This topos is seen to be a finitary instance of both an elementary and a Grothendieck topos, generalizing in a differential geometric setting, as befits ADG, Sorkin's finitary substitutes of continuous spacetime topologies. The paper closes with a thorough discussion of four future routes we could take in order to further develop our topos-theoretic perspective on ADG-gravity along certain categorical trends in current quantum gravity research.Comment: 49 pages, latest updated version (errata corrected, references polished) Submitted to the International Journal of Theoretical Physic

`Iconoclastic', Categorical Quantum Gravity

This is a two-part, `2-in-1' paper. In Part I, the introductory talk at `Glafka--2004: Iconoclastic Approaches to Quantum Gravity' international theoretical physics conference is presented in paper form (without references). In Part II, the more technical talk, originally titled ``Abstract Differential Geometric Excursion to Classical and Quantum Gravity'', is presented in paper form (with citations). The two parts are closely entwined, as Part I makes general motivating remarks for Part II.Comment: 34 pages, in paper form 2 talks given at ``Glafka--2004: Iconoclastic Approaches to Quantum Gravity'' international theoretical physics conference, Athens, Greece (summer 2004