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Incomputability in Physics and Biology


Computability has its origins in Logic within the framework formed along the original path laid down by the founding fathers of the modern foundational analysis for Mathematics (Frege and Hilbert). This theoretical itinerary, which was largely focused on Logic and Arithmetic, departed in principle from the renewed relations between Geometry and Physics occurring at the time. In particular, the key issue of physical measurement, as our only access to ‘reality’, played no part in its theoretical framework. This is in stark contrast to the position in Physics, where the role of measurement has been a core theoretical and epistemological issue since Poincaré, Planck and Einstein. Furthermore, measurement is intimately related to unpredictability, (in-)determinism and the relationship with physical space–time. Computability, despite having exact access to its own discrete data type, provides a unique tool for the investigation of ‘unpredictability’ in both Physics and Biology through its fine-grained analysis of undecidability – note that unpredictability coincides with physical randomness in both classical and quantum frames. Moreover, it now turns out that an understanding of randomness in Physics and Biology is a key component of the intelligibility of Nature. In this paper, we will discuss a few results following along this line of thought.

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F. Bailly and G. Longo (2011) Mathematics and Natural Sciences. The physical singularity of Life, Imperial College Press. (The original book was in French: Hermann, Paris, 2006; the Introduction and Longo's papers are available from

F. Bailly and G. Longo (2008) Extended Critical Situation. Journal of Biological Systems 16 (2) 309336.

F. Bailly and G. Longo (2009) Biological Organization and Anti-Entropy. Journal of Biological Systems 17 (1) 6396.

M. Braverman and M. Yampolski (2006) Non-computable Julia sets. Journal of the American Mathematical Society 19 551578.

C. Calude (2002) Information and randomness, Springer-Verlag.

B. S. Cooper and P. Odifreddi (2003) Incomputability in Nature. In: B. S. Cooper and S. S. Goncharov (eds.) Computability and Models – Perspectives East and West, Kluwer Academic/Plenum137160.

J. Costa , B. Lo and J. Mycka (2009) A foundation for real recursive function theory. Annals of Pure and Applied Logic 160 (3) 255288.

N. da Costa and F. Doria (1991) Undecidability and incompleteness in classical mechanics. International Journal of Theoretical Physics 30 10411073.

P. Gacs , M. Hoyrup and C. Rojas (2011) Randomness on Computable Probability Spaces – A Dynamical Point of View. In: S. Albers and J.-Y. Marion (eds.) Preface: Special Issue on Theoretical Aspects of Computer Science (STACS). Theory of Computing Systems 48 (3) 465485.

M. Kaern , T. C. Elston , W. J. Blake and J. J. Collins (2005) Stocasticity in gene expression: from theories to phenotypes. Nature Review Genetics 6 451464.

J. Krivine , R. Milner and A. Troina (2008) Stochastic Bigraphs. Proceedings of MFPS XXIV: Mathematical Fondations of Programming Semantics. Electronic Notes in Theoretical Computer Science 218 7396.

T. S. Kuhn (1961) The Function of Measurement in Modern Physical Science. Isis 52 161193.

J.-J. Kupiec (2009) On the lack of specificity of proteins and its consequences for a theory of biological organzation. Progress in Biophysics and Molecular Biology 102 (1) 4552.

G. Longo (2002) Reflections on Concrete Incompleteness: Invited Lecture. In: P. Callaghan , Z. Luo , J. McKinna and R. Pollack (eds.) Types for Proofs and Programs. Springer-Verlag Lecture Notes in Computer Science 2277 160180. (Revised version in Philosophia Mathematica (2011) 19 (3) 255–280.)

G. Longo (2009) From exact sciences to life phenomena: following Schroedinger and Turing on Programs, Life and Causality. Information and Computation 207 (5) 543670.

G. Longo (2010a) Incomputability in Physics. In: F. Ferreira , B. Löwe , E. Mayordomo and L. Mendes Gomes (eds.) Programs, Proofs, Processes: Proceedings 6th Conference on Computability in Europe, CiE 2010. Springer-Verlag Lecture Notes in Computer Science 6158 276285.

G. Longo and M. Montévil (2011) From Physics to Biology by Extending Criticality and Symmetry Breakings. Progress in Biophysics and Molecular Biology 106 (2) 340347.

G Longo and P.-E. Tendero (2007) The differential method and the causal incompleteness of Programming Theory in Molecular Biology. Foundations of Science 12 337366.

C. Moore (1990) Unpredictability and undecidability in dynamical systems. Physical Review Letters 64 (20) 23542357.

M. Mossio , G. Longo and J. Stewart (2009) Computability of closure to efficient causation. Journal of Theoretical Biology 257 (3) 489498.

H. Poincaré (1903) Review of Hilbert's Foundations of geometry. Bulletin of the American Mathematical Society 10 (1) 123.

M. B. Pour-El and J. I. Richards (1989) Computability in analysis and physics. Perspectives in Mathematical Logic 1, Springer-Verlag.

A. Raj and A. van Oudenaarden (2008) Nature, nurture, or chance: stochastic gene expression and its consequences. Cell 135 216226.

K. Svozil (1993) Randomness and undecidability in Physics, World Scientific.

A. M. Turing (1950) Computing Machines and Intelligence. Mind LIX (236) 433460.

L. J. Young , R. Nilsen , K. G. Waymire , G. R. MacGregor and T. R. Insel (1999) Increased affiliative response to vasopressin in mice expressing the V1a receptor from a monogamous vole Nature 400 766768.

H. Weyl (1985) Axiomatic Versus Constructive Procedures in Mathematics (edited by T. Tonietti ). The Mathematical Intelligencer 7 (4).

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Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
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