On Hilbert's Sixth Problem

This book explores the premise that a physical theory is an interpretation of the analyticocanonical formalism. Throughout the text, the investigation stresses that classical mechanics in its Lagrangian formulation is the formal backbone of theoretical physics. The authors start from a presentation of the analyticocanonical formalism for classical mechanics, and its applications in electromagnetism, Schrödinger's quantum mechanics, and field theories such as general relativity and gauge field theories, up to the Higgs mechanism. The analysis uses the main criterion used by physicists for a theory: to formulate a physical theory we write down a Lagrangian for it. A physical theory is a particular instance of the Lagrangian functional. So, there is already an unified physical theory. One only has to specify the corresponding Lagrangian (or Lagrangian density); the dynamical equations are the associated EulerLagrange equations. The theory of Suppes predicates as the main tool inthe axiomatization and examples from the usual theories in physics. For applications, a whole plethora of results from logic that lead to interesting, and sometimes unexpected, consequences. This volume looks at where our physics happen and which mathematical universe we require for the description of our concrete physical events. It also explores if we use the constructive universe or if we need settheoretically generic spacetimes.

New work by two of the most renowned philosophers from Brazil Explores which mathematical universe is required for the description of concrete physical events Stresses that classical mechanics in its Lagrangian formulation is the formal backbone of theoretical physics

Auteur
Francisco Antonio Doria is Professor Emeritus, UFRJ. PhD (mathematical physics, advisor Leopoldo Nachbin, 1977, CBPF, Rio Brazil). With Newton da Costa settled the 1976 Arnol'd Hilbert Symposium Problems; proved that chaos in dynamical systems theory is undecidable. Newton C. A. da Costa is a well-known expert in the field of nonclassical logics, developed the theory of paraconsistent logics.

Résumé
"This book is a compilation, 'an essay', of the bulk of their work from 1990 to the present. This 191 page essay includes some historical background and lots of snippets and parts of da Costa and Doria's work on the meta-mathematics of mathematical physics. It starts with a primer on graduate-level basic physics ... ending with a consideration of hypercomputation." (Deborah Konkowski, zbMATH 1494.00005, 2022)

Contenu
Foreword.- 1. Preliminary.- Part I. Physics: A Primer. 2. Classical mechanics.- 3. Variational calculus.- 4. Lagrangian formulation.- 5. Hamilton's equations.- 6. HamiltonJacobi theory.- 7. Where the action is.- 8. From classical to quantum.- 9. Field theory.- 10. Electromagnetism.- 11. Special relativity.- 12. General relativity.- 13. Gauge field theories.- Part II. Axiomatics. 14. Axiomatizations in ZFC.- Part III. Technicalities. 15. Hierarchies.- Part IV. More applications. 16. Arnol'd's 1974 problems.- 17. Forcing and gravitation.- 18. Economics and ecology.- Part V. Computer science. 19. Fastgrowing functions.- Part VI. Hypercomputation. 20. Hypercomputation.- References.