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Searches for the origins of the epistemological concept of model in mathematics

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Abstract

When did the concept of model begin to be used in mathematics? This question appears at first somewhat surprising since “model” is such a standard term now in the discourse on mathematics and “modelling” such a standard activity that it seems to be well established since long. The paper shows that the term— in the intended epistemological meaning—emerged rather recently and tries to reveal in which mathematical contexts it became established. The paper discusses various layers of argumentations and reflections in order to unravel and reach the pertinent kernel of the issue. The specific points of this paper are the difference in the epistemological concept to the usually discussed notions of model and the difference between conceptions implied in mathematical practices and their becoming conscious in proper reflections of mathematicians.

Zusammenfassung

Wann begann der Begriff des Modells in der Mathematik benutzt zu werden? Diese Frage mag auf den ersten Blick Erstaunen auslösen, weil ,Modell’ heute im Diskurs über die Mathematik einen solch selbstverständlichen Ausdruck und ,Modellieren’ eine solche Standard-Aktivität bildet, dass man ihn als Begriff für schon lange etabliert hält. Der Artikel zeigt, dass dessen Ursprünge—in der hier intendierten epistemologischen Bedeutung—dagegen relativ jung sind und versucht, die mathematischen Kontexte aufzudecken, in denen der Begriff etabliert wurde. Der Artikel führt durch mehrere Schichten von Argumentationen und Reflexionen, um den tatsächlichen Kern der Fragestellung herauszuschälen und ihn so zu erreichen. Der spezifische Ansatz des Artikels betrifft den Unterschied zwischen der epistemologischen Bedeutung von Modell zum üblichen Verständnis von Modell sowie den Unterschied zwischen in mathematischen Praktiken implizierten Konzeptionen und dem Bewusstwerden in eigenen Reflexionen von Mathematikern.

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Notes

  1. Besides research on recursion theory and on graph theory, a specialty of Carstens’s research had always been model theory.

  2. It would be rewarding to study the history of this historiography: since when did histories of mathematics attribute the use of ‘model’ to one or more mathematicians?

  3. Throughout the preparations, Freudenthal had used the term “notion”. Only for the Proceedings, he changed to “concept”,

  4. In his letters, Tarski consistently spoke of “notion of model”.

  5. This paragraph is based on the part of the Freudenthal Nachlass regarding the preparation of the 1960 conference. The folders of this part are not numerated and their sheets not paginated. The final financial account was of 28 March 1960.

  6. sous quelque aspect qu’on le prenne, un modèle fait toujours fonction de médiateur entre un champ théorique dont il est une interprétation et un champ empirique dont il est une formalisation.

  7. Bachelard does not study the model notion with the logicists, as maintain Armatte and Dalmedico (2004, p. 249, note 9), but on the contrary excludes it from her focus (Bachelard 1979, p.9).

  8. Actually, this entry had been my first finding making me aware of the different role of models for theoretical physics and for mathematics.

  9. In fact, there has been an extended discussion in theoretical physics since about the last third of the nineteenth century, which is interpreted by historians of physics as the “precursor” of the model notion—namely under the term of ,analogy’, as shown by Don Howard in his presentation “Maxwell, Boltzmann, and Hertz on Models and Analogies in Physics“ at the Seven Pines Symposium, Stillwater, Minnesota, 2012; retrieved from http://pitp.physics.ubc.ca/confs/7pines2012/talks/HOWARD_on%20Models%20and%20Analogies%20SPXVI.pdf on May 19th, 2014. Particularly noteworthy were the debates at the 1895 Naturforscherversammlung in Lübeck on energetics and Boltzmann’s contributions there.

  10. An excellent recent study on mathematical models in the meaning of material objects is the paper by Rowe (2013).

  11. di preparare gli elementi geometrici d’una costruzione materiale, possibilmente facile ed esatta, della superficie stessa.

  12. Boi, too, consistently speaks of ,,notion de modèle“.

  13. A certain specification will be made below regarding the 1928 revision.

  14. Es gibt also eine Anzahl von Maßbestimmungen, die sich logisch gleichberechtigt neben die euklidische Geometrie stellen.

  15. Klein had introduced his notion of “Maßbestimmung” to characterise the various geometries in his seminal paper of 1871. There he had introduced it by referring to Arthur Cayley’s sixth memoir upon quantics (1859) where Cayley introduced the notion of metric to study projective space (Klein 1871, p. 573).

  16. Klein had dealt with Riemann’s approaches and contributions extensively, both in the 1893 lectures and in the 1928 revision. Helmholtz, however, was discussed in 1893 in a proper section—already in a reserved manner, presenting him as “not a professional mathematician”—whereas he is mentioned only briefly in 1928, and in a rather negative manner. Schoenflies, too, presented Riemann at length, contrary to Helmholtz. But he remarked on both: “the mathematical treatment was yet quite strongly amalgamated with speculative reasoning” (Schoenflies 1919, p. 289).

  17. On pourrait dire que la vérité de la géométrie d’Euclide n’est pas incompatible avec celle de la géométrie de Lovatchewski, puisque l’existence d’un groupe n’est pas incompatible avec celle d’un autre groupe.

  18. Ainsi, un modèle est une représentation concrète dans une théorie familière d’énoncés et de relations, qui sont d’abord perçus comme purement formels. [...] C’est l’attribution d’un sens déterminé à des entités a priori sans signification tel que les énoncés formels soient vérifiés. Si l’on rappelle qu’«attribuer un sens» c’est «interpréter», on voit qu’un modèle est une interprétation de ces énoncés dans laquelle ceux-ci sont vrais.

  19. See the remarks in the introduction about Grattan-Guinness’ seminal paper (Grattan-Guinness 2004). In historiography, there is an extensive methodological discussion about it as Whig history, see Fried (2001).

  20. See the section on theoretical physics.

  21. The original shows the misprint ,,if“.

  22. What Kant really knew were the attempts of Johann Schultz, professor at the same university and Kant’s counsellor in mathematical questions, to prove the Parallel Postulate—in publications from 1780 on (Schubring 1982, Stark 1993, p. 42).

  23. Tinne Hoff Kjeldsen (Copenhagen) drew my attention to a use of the model notion in a publication on theoretical biology in 1934: in a symposium on quantitative biology, Nicolas Rashevsky had presented the paper: physico-mathematical aspects of cellular multiplication and development. The ensuing discussion, also documented in the published Proceedings, was lively and extensive. A contribution there was by “Dr. Cole” who used the model notion in a manner as if it was in that context quite common: “No mention has been made of any electrical effects in the cell model, but I would be interested in knowing how large the electrical energy of some living cell might be in comparison with the factors which have been considered” (Rashevsky 1934, p. 197).

  24. Vierteljahresschrift der naturforschenden Gesellschaft Zürich, 86 (1941), pp. XIX–XX. (Is mathematics still the model of all theoretical sciences?).

  25. Mathematische Annalen, 118 (1941), 251-262. (Model of a differential calculus with actually infinitesimal quantities of first order).

  26. Die Philosophie hat diese letzte Frage noch nicht in objektiv zwingendem Sinne beantwortet.

  27. The term “inhaltlich” is much liked in German, but there is no good equivalent translation. The main meaning is expressed by: with regard to contents. The translation of this opus magnum of Hilbert and Bernays begun recently and having so far achieved a first part of volume I uses “contentual” (Hilbert and Bernays 2011, p. 3 and passim).

  28. The translation 2011 uses the term “some domain of reality” (p. 4).

  29. “... theories that get their significance from a simplifying idealisation of an actual state of affairs rather than from a complete reproduction of it. A theory of this kind cannot get a foundation through a reference to either the evident truth of its axioms or to experience; rather such a foundation can only be given when the idealisation performed, i.e. the extrapolation through which the concept formations and the principles of the theory come to overstep the reach either of intuitive evidence or of the data of experience, is understood to be consistent” (Hilbert and Bernays 2011, p.4).

  30. I learned from Sinaceur’s paper on Tarski that Kurt Gödel (1906–1978) used the term “model” in 1929 (Sinaceur 2001, p. 51), in his paper Über die Vollständigkeit des Logikkalküls. In fact, he says there regarding a method for achieving proofs to be free of contradiction: “sie würde ja eine Garantie dafür bieten, daß diese Methode in jedem Fall zum Ziele führt, d.h. daß entweder ein Widerspruch sich herstellen oder die Widerspruchslosigkeit durch ein Modell sich beweisen lassen muß” (Gödel 1929, p. 60). As Sinaceur remarks, Gödel does not define there what he means by “model”. But she does not mention that he refers to Brouwer. He continues there namely: “Daß man aus der Widerspruchslosigkeit eines Axiomensystems nicht ohneweiters auf die Konstruierbarkeit eines Modells schließen kann, darauf hat besonders L. E. Brouwer mit Nachdruck hingewiesen”. Actually, I did not succeed in finding such affirmations by Brouwer in volume I of his Collected Works (Brouwer 1975); it might also have been a personal communication. Anyhow, the Gödel quote clearly refers to Hilbert’s meta-mathematical program.

  31. The strict proof that non-Euclidean geometry is absolutely consistent in itself had yet to follow. This resulted almost of itself in the further development of non-Euclidean geometry. As often happens, the simplest way of proving this was not discovered at once. It was discovered by Klein as late as 1870 and depends on the construction of a Euclidean model for non-Euclidean geometry (Weyl 1922, pp. 117–118).

  32. The English translation is slightly different: “Klein himself interpreted his construction in this sense, namely as endowment of projective space with a Lobatschewskyan metric, not as construction of a model by means of metric Euclidean space” (Weyl 1949, p. 68).

  33. The terms used in the English translation are not quite exact: “Leerform” should be ,empty mold’ and “inhaltlich” is given as “concrete”, while it should be “contentual” (see above).

  34. The translation of that statement rather obscures: when “designata have been exhibited for the names of the basic concepts, on the basis of which the axioms become true propositions.”

  35. [\(\ldots \)] la technique de construction des modèles pour prouver la compatibilité ou indépendance mutuelle de certains axiomes. Ce faisant, il opère un décrochement important dans la compréhension de la notion de modèle. En effet, les modèles qu’il fabrique sont non pas des visualisations dans l’espace euclidien de théories plus abstraites ou moins familières mais des modèles formels, construits à partir de résultats algébriques ou arithmétiques assez abstaits.

  36. Wie man erkennt, gibt es unendlichviele Geometrien, die den Axiomen I-IV, \(\hbox {V}_{1}\) genügen.

  37. This is also Weyl’s meaning of 1944 in his Hilbert obituary; he might have read Coxeter’s book before.

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  1. IDM, Fakultät für Mathematik, Universität Bielefeld, Universitätsstrasse 25, 33615, Bielefeld, Federal Republic of Germany

    Gert Schubring

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Communicated by: Menso Folkerts.

To the memory of Hans-Georg Carstens (1945–2012).

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Schubring, G. Searches for the origins of the epistemological concept of model in mathematics. Arch. Hist. Exact Sci. 71, 245–278 (2017). https://doi.org/10.1007/s00407-017-0188-5

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