Abstract
Mario Bunge forcefully argues for Dual Axiomatics, i.e., an axiomatic method applied to natural sciences which explicitly takes into account semantic aspects of the concepts involved in an axiomatization. In this paper we will discuss how dual axiomatics is equally important in mathematics; both historically in Hilbert and Bernays’s conception as well as today in a set-theoretical environment.
This work is partially supported by the Portuguese Science Foundation, FCT, through the projects UID/MAT/00297/2019 (Centro de Matemática e Aplicações) and PTDC/MHC-FIL/2583/2014 (Hilbert’s 24th Problem), and by the Udo Keller Foundation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The labels are adapted for later references. The fifth (“it exhibits the legitimate components of the theory as well as their deductive organization, and with it the logical status of each constituent”) and seventh (“it eases the understanding and memorization of theories”) have to taken for granted in mathematics; the sixth one (“it facilitates the empirical test of theories”) is, however, specific for empirical theories only.
- 2.
The distinction of characterizing and generalizing axiomatics is not new; one may find it in variations and under different names at several place. We like to mention Salt (1971) who uses, cum grano salis, interpreted axiomatic theory for our characterizing axiomatics and abstract theory for generalizing axiomatics; he also refers to Bernays’s respective distinction of material or pertinent axiomatics and descriptive axiomatics (Bernays 1967). It is not by accident that Salt can use this distinction to defend Bunge against Freudenthal (1970) who apparently misread Bunge (1967b) when he took interpreted axiomatic theories as abstract ones.
- 3.
See, for instance, Corry (2002).
- 4.
- 5.
This is very nicely illustrated by the following recollection of Saunders MacLane:
In Hilbert’s original work on integral equations, a point in a Hilbert space was an infinite sequence of complex numbers yn with the sum of squares convergent. But von Neumann’s lecture began with: ‘Take a Hilbert space—an infinite-dimensional vector space over the complex numbers complete in a positive definite norm.’ At the end of the lecture Hilbert asked Professor von Neumann, ‘I would like to know, just what a Hilbert Space is.’ Hilbert thought of his spaces concretely, not axiomatically. (MacLane 2005, p. 49f)
- 6.
For instance, a lecture of Hilbert of 1913 (Hilbert 1913) contains a chapter entitled “axioms of algebra”; but it does not axiomatizes Algebra in the modern sense, but just the real numbers.
- 7.
See Hilbert and Bernays (1934, pp.1 and 2f.); the corresponding sentences read in German:
Eine Verschärfung, welche der axiomatische Standpunkt in HILBERTs „Grundlagen der Geometrie“ erhalten hat, besteht darin, daß man von dem sachlichen Vorstellungsmaterial, aus dem die Grundbegriffe einer Theorie gebildet sind, in dem axiomatischen Aufbau der Theorie nur dasjenige beibehält, was als Extrakt in den Axiomen formuliert ist, von allem sonstigen Inhalt aber abstrahiert.
and
Andererseits können wir bei der inhaltlichen Axiomatik deshalb nicht stehenbleiben, weil wir es in der Wissenschaft, wenn nicht durchweg, so doch vorwiegend mit solchen Theorien zu tun haben, die gar nicht vollkommen den wirklichen Sachverhalt wiedergeben, sondern eine vereinfachende Idealisierung des Sachverhaltes darstellen und darin ihre Bedeutung haben.
- 8.
Formal axiomatics requires contentual axiomatics as a necessary supplement. It is only the latter that provides us with some guidance for the choosing the right formalism, and with some instructions on how to apply a given formal theory to a domain of actuality. (Hilbert and Bernays 2011, p. 2)
- 9.
Weyl (1944, p. 640f.) comments on this:
But how to make sure that the “game of deduction” never leads to a contradiction? Shall we prove this by the same mathematical method the validity of which stands in question, namely by deduction from axioms? This would clearly involve a regress ad infinitum. It must have been hard on Hilbert, the axiomatist, to acknowledge that the insight of consistency is rather to be attained by intuitive reasoning which is based on evidence and not on axioms.
And further on:
Incidentally, in describing the indispensable intuitive basis for his Beweistheorie Hilbert shows himself an accomplished master of that, alas, so ambiguous medium of communication, language. With regard to what he accepts as evident in this “metamathematical” reasoning, Hilbert is more papal than the pope, more exacting than either Kronecker or Brouwer. […] Elementary arithmetics can be founded on such intuitive reasoning as Hilbert himself describes, but we need the formal apparatus of variables and “quantifiers” to invest the infinite with the all important part that it plays in higher mathematics. Hence Hilbert prefers to make a clear cut: he becomes strict formalist in mathematics, strict intuitionist in metamathematics.
- 10.
Recall Hilbert’s “Am Anfang ist das Zeichen.” (In the beginning is the sign.) and his discussion around this citation (Hilbert 1922, p. 163).
- 11.
See, for instance, Kahle (2015).
- 12.
Even less, he could be considered as a naive formalist, as already Bernays (1975, p. 2) stressed: “Yet it was certainly not Hilbert’s intention that mathematics should consist only of proof theory (though some of his statements describing his attitude to metamathematics might suggest this view).” See also the discussion in Kahle (2019).
- 13.
As semantics should provide meaning, this is in accordance with Georg Kreisel, who proposed to translate “inhaltlich” by “meaningful” (in a personal communication to Jan von Plato).
- 14.
Thus, it would be ahistorical to read Hilbert’s “inhaltlich” just as semantic in terms of set-theoretical structures—still, such a reading will probably match to a large extent with Hilbert’s understanding of the term.
- 15.
- 16.
By Kikuchi and Kurahashi (2016) they are called, quite correctly, insane models of PA.
- 17.
Georg Kreisel liked to point to this formula as a false one, revealing in this way a quite strong semantic bias; see, for instance, Kreisel (1986, p. 143).
- 18.
- 19.
In all these newer results a certain imperfection of our contemporary mathematics is expressed; and these results cause at least that, next to the formal systems of the classical theories, the contentual, not formalized set theory, as it is applied in semantics, upholds its importance.
References
Bernays, P. (1967). Scope and limits of axiomatics. In M. Bunge (Ed.), Delaware seminar in the foundations of physics (pp. 188–191). Berlin/New York: Springer.
Bernays, P. (1975). Mathematics as a domain of theoretical science and of mental experience. In H. Rose & J. Shepherdson (Eds.), Logic colloquium ’73 (Studies in logic and the foundations of mathematics, Vol. 80, pp. 1–4). Elsevier.
Bernays, P. (1978). Nachwort. Jahresbericht der Deutschen Mathematiker-Vereinigung, 81(1), 22–24.
Blumenthal, O. (1935). Lebensgeschichte. In: David Hilbert: Gesammelte Abhandlungen (Vol. III, pp. 388–429). Berlin: Springer.
Bourbaki, N. (1950). The architecture of mathematics. The American Mathematical Monthly, 57(4), 221–232.
Bunge, M. (1967a). Analogy in quantum mechanics: From insight to nonsense. British Journal for the Philosophy of Science, 18, 265–286.
Bunge, M. (1967b). Foundations of physics (Springer tracts in natural philosophy, Vol. 10). Berlin: Springer.
Bunge, M. (1967c). The structure and content of a physical theory. In M. Bunge (Ed.), Delaware seminar in the foundations of physics (pp. 15–27). Berlin/New York: Springer.
Bunge, M. (1976). Review of Wolfgang Stegmüller’s the structure and dynamics of theories. Mathematical Reviews, 55, 33.
Bunge, M. (1980). The mind–body problem. Oxford: Pergamon.
Bunge, M. (2009). The failed theory behind the 2008 crisis. In M. Cherkaoui and P. Hamilton (Eds.), Raymond Boudon: A life in sociology (Vol. I, pp. 127–142). Oxford: Bardwell.
Bunge, M. (2017a). Doing science. New Jersey: World Scientific.
Bunge, M. (2017b). Why axiomatize? Foundations of Science, 22(4), 695–707.
Clavius, C. (1574). Euclidis Elementorum Libri XV. Romae.
Corry, L. (2002). Modern algebra and the rise of mathematical structures (2nd revised ed.). Basel: Birkhäuser.
Freudenthal, H. (1970). What about foundations of physics. Synthese, 21(1), 93–106.
Grattan-Guinness, I. (2000). The search for mathematical roots, 1870–1940. Logic, set theories and the foundations of mathematics from Cantor through Russell to Gödel. Princeton: Princeton University Press.
Hilbert, D. (1899). Grundlagen der Geometrie. In Festschrift zur Feier der Enthüllung des Gauss-Weber-Denkmals in Göttingen, herausgegeben vom Fest-Comitee (pp. 1–92). Teubner.
Hilbert, D. (1900). Über den Zahlbegriff. Jahresbericht der Deutschen Mathematiker-Vereinigung, 8, 180–184.
Hilbert, D. (1913). Elemente und Prinzipien der Mathematik. Vorlesung Sommersemester 1913, Universität Göttingen, available at the Max Planck Institute for the History of Science, Berlin.
Hilbert, D. (1918). Axiomatisches Denken. Mathematische Annalen, 78(3/4), 405–415. Lecture delivered on 11 September 1917 at the Swiss Mathematical Society in Zurich.
Hilbert, D. (1922). Neubegründung der Mathematik. Abhandlungen aus dem Mathematischen Seminar der Hamburgischen Universität, 1, 157–177.
Hilbert, D. (1928). Probleme der Grundlegung der Mathematik. In Atti del Congresso Internazionale dei Matematici, Bologna, settembre 3–19, 1928, Nicola Zanichelli, offprint.
Hilbert, D., & Bernays, P. (1934). Grundlagen der Mathematik I, (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Vol. 40). Springer; 2nd edition 1968.
Hilbert, D., & Bernays, P. (1939). Grundlagen der Mathematik II (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Vol. 50). Springer; 2nd edition 1970.
Hilbert, D., & Bernays, P. (2011). Grundlagen der Mathematik I/Foundations of Mathematics I. College Publications, bilingual edition of Prefaces and §§1–2 of Hilbert and Bernays (1934).
Hilbert, D., & Cohn-Vossen, S. (1932). Anschauliche Geometrie (Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. XXXVII). Springer; English translation: Hilbert and Cohn-Vossen (1952).
Hilbert, D., & Cohn-Vossen, S. (1952). Geometry and the imagination. AMS Chelsea Publishing; English translation of Hilbert and Cohn-Vossen (1932).
Kahle, R. (2015). Gentzen’s theorem in context. In R. Kahle & M. Rathjen (Eds.), Gentzen’s centenary: The quest for consistency (pp. 3–24). Heidelberg: Springer.
Kahle, R. (2018). Structure and structures. In M. Piazza & G. Pulcini (Eds.), Truth, existence and explanation (Boston studies in the philosophy and history of science, Vol. 334, pp. 109–220). Springer.
Kahle, R. (2019). Is there a “Hilbert Thesis”? Studia Logica, 107(1), 145–165.
Kikuchi, M., & Kurahashi, T. (2016). Illusory models of Peano Arithmetic. The Journal of Symbolic Logic, 81(3), 1163–1175.
Kreisel, G. (1986). Proof theory and synthesis of programs: Potentials and limitations. In Eurocal ’85 (Lecture notes in computer science, Vol. 203, pp. 136–150). Springer.
MacLane, S. (2005). A mathematical autobiography. Wellesley: AK Peters.
Moschovakis, Y. (2006). Notes on set theory (Undergraduate texts in mathematics, 2nd ed.). New York: Springer.
Popper, K. R. (1945). The open society and its enimies (The spell of Plato, Vol. I). London: Routledge.
Salt, D. (1971). Physical axiomatics: Freudenthal vs. Bunge. Foundations of Physics, 1(4), 307–313.
Scriba, C. J., & Schreiber, P. (2010). 5000 Jahre Geometrie (Vom Zählstein zum Computer, 3rd ed.). Berlin/Heidelberg: Springer.
Spinoza, B. (1677). Ethica, ordine geometrico demonstrata.
Weyl, H. (1944). David Hilbert and his mathematical work. Bulletin of the American Mathematical Society, 50(9), 612–654.
Yasugi, M., & Passell, N. (Eds.). (2003). Memoirs of a proof theorist. World Scientific; English translation of a collection of essays written by Gaisi Takeuti.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Kahle, R. (2019). Dual Axiomatics. In: Matthews, M.R. (eds) Mario Bunge: A Centenary Festschrift. Springer, Cham. https://doi.org/10.1007/978-3-030-16673-1_35
Download citation
DOI: https://doi.org/10.1007/978-3-030-16673-1_35
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-16672-4
Online ISBN: 978-3-030-16673-1
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)