Abstract
This paper is divided into two parts. The first proposes a philosophical frame and it “uses” for this a recent book on a phenomenological approach to the foundations of mathematics. Gödel’s 1931 theorem and his subsequent philosophical reflections have a major role in discussing this perspective and we will develop our views along the lines of the book (and further on). The first part will also hint to the connections with some results in Mathematical physics, in particular with Poincaré’s unpredictability (three-body) theorem, as an opening towards the rest of the paper. As a matter of fact, the second part deals with the “incompleteness” phenomenon in Quantum physics, a wording due to Einstein in a famous joint paper of 1935, still now an issue under discussion for many. Similarities and differences w.r. to the logical notion of incompleteness will be highlighted. A constructivist approach to knowledge, both in mathematics and in physics, underlies our attempted “unified” understanding of these apparently unrelated theoretical issues.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
A. Aspect, P. Grangier and G. Roger: Experimental Realization of the Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell’s Inequalities. Phys. Rev. Let. 9 (1982) p 91
F. Bailly and G. Longo: Mathématiques et sciences de la nature. La singularité physique du vivant (Hermann, Paris 2006). (English introduction, downloadable)
F. Bailly and G. Longo: Randomness and Determination in the interplay between the Continuum and the Discrete. Mathematical Structures in Computer Science 17, 2 (2007)
H. Barreau and J. Harthong (eds): La mathmatique non standard (Ed. CNRS 1989)
J. Barrow-Green: Poincaré and the three body Problem. History of Mathematics XI (AMS 1997)
J. L. Bell: A Primer of Infinitesimal Analysis (Cambridge Univ. Press, Cambridge 1998)
J. S. Bell: On the Einstein-Podolsky-Rosen Paradox. Physics 1 (1964) p 195
J. Binney, N. J. Dowrick, A. J. Fisher and M. E. J. Newman: The Theory of Critical Phenomena: An Introduction to the Renormalization Group (Oxford Univ. Press, Oxford 1992)
M. Bitbol: L’aveuglante proximité du réel (Flammarion, Paris 2000)
D. Bohm: The Paradox of Einstein, Rosen and Podolsky. Quantum Th. (1951) p 611
V. Brattka and G. Presser: Computability on subsets of metric spaces. Theoretical Computer Science, 305 (2003)
V. Brattka and K. Weihrauch: Computability on subsets of Euclidean space I: Closed and compact subsets. Theoretical Computer Science 219 (1999)
C. Calude: Information and Randomness: An Algorithmic Perspective (Springer, Berlin 2002)
C. Calude and M. Stay: From Heisenberg to Gödel via Chaitin. International Journal of Theoretical Physics 44,7 (2005) pp 1053–1065
P. Collins: Continuity and computability of reachable sets. Theoretical Computer Science 341 (2005)
A. Connes: A. Non-commutative Geometry (Academic Press, New York 1994)
S. Dehaene: The Number Sens (Oxford Univ. Press, Oxford 1997)
R. L. Devaney: An introduction to Chaotic Dynamical Systems (Addison-Wesley, Reading 1989)
F. Diacu: Singularities of the N-Body Problem (Publications CRM, Montreal 1992)
A. Einstein, B. Podolsky and N. Rosen: Can Quantum-Mechanical Description of Physical Reality be Considered complete? Phys. Rev. 41 (1935) p 777
B. van Fraassen: Lois et symetries (Vrin, Paris 1994)
G. Frege: The Foundations of Arithmetic (1884) (Evanston 1980)
K. Gödel: Remark About the Relationship between Relativity Theory and Idealistic Philosophy. In: Albert Einstein: Philosopher-Scientist, ed by P. A. Schilpp (Open Court, LaSalle Ill.) pp 557–562. Reprinted with corrections and additions in Collected Works, ed by S. Feferman et. al., Vol. 2 (Oxford University Press, Oxford 1990)
L. Harrington et al. (eds): H. Friedman’s Research on the Foundations of Mathematics (North-Holland, Amsterdam 1985)
D. Hilbert: Les fondements de la géométrie (1899) (Dunod, 1971)
M. Hoyrup: Dynamical Systems: Stability and Simulability. Mathematical Structures in Computer Science 17,2 (2007)
M. Hoyrup, A. Kolcak and G. Longo: Computability and the Morphological Complexity of some dynamics on Continuous Domains. Invited survey TCS (2008) To appear
T. Jech: Set Theory (Springer, Berlin 1997)
H. J. Jensen: Self-Organized Criticality, Emergent Complex Behavior in Physical and Biological Systems (Cambridge Lectures in Physics 1998)
J. Laskar: Large scale chaos in the Solar System. Astron. Astrophysics 287, L9–L12, (1994)
G. Longo: Laplace (A note on incompleteness). An item of J-Y. Girard’s Logic Dictionnary, at the end of “Locus Solum”, MSCS, vol. 11, n.3, (Cambridge Univ. Press, Cambridge 2001)
G. Longo: Reflections on Incompleteness or On the proofs of some formally unprovable propositions and Prototype Proofs in Type Theory (invited Lecture: Types for Proofs and Programs Durham UK, Dec. 2000). In: Lecture Notes in Computer Science 2277 ed by Callaghan et al. (Springer, Berlin 2002) pp 160–180
G. Longo: The Cognitive Foundations of Mathematics: human gestures in proofs and mathematical incompleteness of formalisms. In: Images and Reasoning, ed by M. Okada et al. (Keio University Press, Tokio 2005)
G. Longo: Laplace, Turing and the “imitation game” impossible geometry: randomness, determinism and programs in Turing’s test. In: The Turing Test Sourcebook, ed by R. Epstein, G. Roberts and G. Beber (Kluwer, Dordrecht 2007)
G. Longo and P. E. Tendero: The causal incompleteness of Programming Theory in Molecular biology, Foundations of Science (to apprear). French version in: Evolution des concepts fondateurs de la biologie du XXIe siècle, ed by Miquel (DeBoeck, Paris 2007)
G. Longo and T. Paul: The Mathematics of Computing between Logic and Physics. In: Computability in Context: Computation and Logic in the Real World, ed by Cooper and Sorbi (Imperial College Press, World Scientific 2008) To appear
P. Martin-Löf: The definition of random sequences. Information and Control 9 (1966) pp. 602–619
L. Nottale: La relativité dans tous ses états (Hachette, Paris 1999)
J. Paris and L. Harrington: A mathematical incompleteness in Peano Arithmetic. In: Handbook of Mathematical Logic, ed by J. Barwise (North-Holland, Amsterdam 1978)
F. Patras: La penseé mathématique contemporaine (PUF 2001)
K. Petersen: Ergodic Theory (Cambridge Univ. Press, Cambridge 1990)
J-L. Petit: Solipsisme et Intersubjectivité (Cerf 1996)
S. Y. Pilyugin: Shadowing in dynamical systems (Springer, Berlin 1999)
M. B. Pour-El and J. I. Richards: Computability in analysis and physics. Perspectives in mathematical logic (Springer, Berlin 1989)
J-M. Salanskis: L’hermneutique formelle (Ed. CNRS 1991)
J. Tappenden: Geometry and generality in Frege’s philosophy of Arithmetic. Synthese 102,3 (1995)
R. Tieszen: Phenomenology, Logic, and the Philosophy of Mathematics (Cambridge Univ. Press, Cambridge 2005)
H. Weyl: Philosophy of Mathematics and of Natural Sciences (Princeton University Press, Princeton 1949)
K. Weihrauch: Computable Analysis. In: Texts in Theoretical Computer Science (Springer, Berlin 2000)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Italia
About this chapter
Cite this chapter
Bailly, F., Longo, G. (2008). Phenomenology of Incompleteness: From Formal Deductions to Mathematics and Physics. In: Lupacchini, R., Corsi, G. (eds) Deduction, Computation, Experiment. Springer, Milano. https://doi.org/10.1007/978-88-470-0784-0_13
Download citation
DOI: https://doi.org/10.1007/978-88-470-0784-0_13
Publisher Name: Springer, Milano
Print ISBN: 978-88-470-0783-3
Online ISBN: 978-88-470-0784-0
eBook Packages: Humanities, Social Sciences and LawPhilosophy and Religion (R0)