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Formal Universes

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Logic, Rewriting, and Concurrency

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9200))

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Abstract

This essay addresses the concerns of the foundations of mathematics of the early 20th century which led to the creation of formally axiomatized universes. These are confronted with contemporary developments, particularly in computational logic and neuroscience. Our approach uses computational models of mental experiments with the infinite in set-theory and symbol-manipulation systems, in particular models of combinatory logic.

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Notes

  1. 1.

    In memory and appreciation of a quarter century of discussion on the foundation and philosophy of mathematics and computer science with José Meseguer.

  2. 2.

    With respect to the “naive” set theory in which its structure is discussed.

  3. 3.

    Cf. German “rechnen” = to compute.

  4. 4.

    An ongoing project accessible online at HomotopyTypeTheory.org.

  5. 5.

    For a critique cf. [11].

  6. 6.

    Among others by my late colleague, the quantum chemist H. Primas, in [1].

  7. 7.

    The Plotkin-Scott-Engeler model.

  8. 8.

    Understanding the model may be helped by considering sets of expressions \( \alpha \longrightarrow x\) as partial and many-valued function from G(A) to G(A), namely as sets of pairs of arguments and values in G(A). (If we so wish, we may also see these expressions as lists with head x and tail \(\alpha \)).

  9. 9.

    Reviewed in a recent survey [15].

  10. 10.

    Worked out in a 1971/78 paper by the author, reprinted in his collection Algorithmic Properties of Structures, World Publ.Co., 1990, pp. 87–95.

  11. 11.

    Cf. the review by an originator of the idea of a number sense [7].

  12. 12.

    To interpret expression in X as cascades, transform subexpressions such as \(\{a \} \longrightarrow (\{b, c\} \longrightarrow d)\) successively by absorption into \(\{a,b,c\} \longrightarrow d\).

  13. 13.

    Diverse papers available online from the author’s website, cf. “Neural Algebra”.

  14. 14.

    Proof of theorem 66; dismissed by the critical comment of Emmy Noether, one of the editors.

  15. 15.

    J.W.v.Goethe, Faust, Der Tragödie erster Teil, Nacht.

  16. 16.

    Free translation by this author. Original: “Du gleichst dem Geist den Du begreifst, nicht mir”.

  17. 17.

    In a letter to A. Robinson, [10].

References

  1. Agazzi, E. (ed.): The Problem of Reductionism in Science. Kluwer Academic Publishers, Dordrecht (1991)

    Google Scholar 

  2. Crick, F.: The Astonishing Hypothesis. Simon and Schuster Ltd., London (1994)

    Google Scholar 

  3. Curry, H.B.: Grundlagen der kombinatorischen Logik, Am. J. Math. 51, 509–536, 789–834 (1930)

    Google Scholar 

  4. van Dantzig, D.: Is \(10^{10^{10}}\) a finite number? Dialectica 9, 273–277 (1955)

    Article  MathSciNet  Google Scholar 

  5. Davis, M.: Pragmatic Platonism. In: Friedman, M., Tennant, N. (eds.) Foundational Adventures Essays in Honor of Harvey. College Publications, London (2014)

    Google Scholar 

  6. Dedekind, R.: Was sind und was sollen die Zahlen? reprinted in Dedekind’s collected works, Braunschweig (1932)

    Google Scholar 

  7. Dehaene, S., et al.: Arithmetic and the brain. Curr. Opin. Neurobiol. 14, 218–224 (2004)

    Article  Google Scholar 

  8. Engeler, E., et al.: The Combinatory Programme. Birkhäuser Basel, Boston (1995)

    Book  MATH  Google Scholar 

  9. Feferman, S., Gödel, K. (eds.): Collected Works, vol. III, p. 50. Oxford University Press, Oxford (2003). comments by S.F. p. 39

    Google Scholar 

  10. Feferman, S., Gödel, K. (eds.): Collected Works, vol. V, p. 204. Oxford University Press, Oxford (2003)

    Google Scholar 

  11. Gray, L.: A mathematician looks at Wolfram’s new kind of science. Notices AMS 50, 200–211 (2003)

    MATH  Google Scholar 

  12. Hilbert, D.: Ueber das unendliche. Math. Ann. 95, 161–190 (1926)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jäger, G., et al.: Universes in explicit mathematics. Ann. Pure Appl. Logic 109, 141–162 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Meseguer, J.: Twenty years of rewriting logic. J. Logic Algebraic Program. 81, 721–778 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Pelayo, A., Warren, M.A.: Homotopy type theory and Voevodsky’s univalent foundations. Bulletin A.M.S. 51, 597–648 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Welti, E.: Die Philosophie des Strikten Finitismus. Peter Lang Verlag, Bern (1981)

    MATH  Google Scholar 

  17. Wolfram, S.: A New Kind of Science. Wolfram Media, Champaign (2002)

    MATH  Google Scholar 

  18. Zermelo, E.: Ueber Grenzzahlen und Mengenbereiche. Fund. Math. 16, 29–47 (1930)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. ETH Zurich, Zürich, Switzerland

    Erwin Engeler

Authors
  1. Erwin Engeler
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Correspondence to Erwin Engeler .

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Editors and Affiliations

  1. Universidad Complutense de Madrid, Madrid, Spain

    Narciso Martí-Oliet

  2. University of Oslo, Oslo, Norway

    Peter Csaba Ölveczky

  3. SRI International, Menlo Park, California, USA

    Carolyn Talcott

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Engeler, E. (2015). Formal Universes. In: Martí-Oliet, N., Ölveczky, P., Talcott, C. (eds) Logic, Rewriting, and Concurrency. Lecture Notes in Computer Science(), vol 9200. Springer, Cham. https://doi.org/10.1007/978-3-319-23165-5_12

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  • DOI: https://doi.org/10.1007/978-3-319-23165-5_12

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