Dogmas and the changing images of foundations

Philosophia Scientiæ CS 5 (2005) Fonder autrement les mathématiques ................................................................................................................................................................................................................................................................................................ José Ferreirós Dogmas and the Changing Images of Foundations ................................................................................................................................................................................................................................................................................................ Avertissement Le contenu de ce site relève de la législation française sur la propriété intellectuelle et est la propriété exclusive de l'éditeur. 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Référence électronique José Ferreirós, «Dogmas and the Changing Images of Foundations», Philosophia Scientiæ [En ligne], CS 5|2005, mis en ligne le 01 août 2008, consulté le 13 octobre 2012. URL: http://philosophiascientiae.revues.org/91; DOI: 10.4000/philosophiascientiae.91 Éditeur : Université Nancy 2 http://philosophiascientiae.revues.org http://www.revues.org Document accessible en ligne sur : http://philosophiascientiae.revues.org/91 Ce document est le fac-similé de l'édition papier. Tous droits réservés Dogmas and the Changing Images of Foundations José Ferreirós Universidad de Sevilla Abstract: I offer a critical review of several different conceptions of the activity of foundational research, from the time of Gauss to the present. These are (1) the traditional image, guiding Gauss, Dedekind, Frege and others, that sees in the search for more adequate basic systems a logical excavation of apriori structures, (2) the program to find sound formal systems for so-called classical mathematics that can be proved consistent, usually associated with the name of Hilbert, and (3) the historicist alternative, guiding Riemann, Poincaré, Weyl and others, that seeks to perfect available conceptual systems with the aim to avoid conceptual limitations and expand the range of theoretical options. I shall contend that, at times, assumptions about the foundational enterprise emerge from certain dogmas that are frequently inherited from previous, outdated images. To round the discussion, I mention some traits of an alternative program that investigates the epistemology of mathematical knowledge. Resumen: Ofrecemos una revisión crítica de varias concepciones de la investigación sobre los fundamentos de la matemática, desde los tiempos de Gauss hasta el presente. Se trata de (1) la imagen tradicional, que guió a Gauss, Dedekind, Frege y otros, y que ve en la búsqueda de sistemas básicos más adecuados una excavación lógica de estructuras apriori, (2) el programa de encontrar sistemas formales correctos para la llamada matemática clásica que puedan demostrarse consistentes, habitualmente asociado al nombre de Hilbert, y (3) la alternativa historicista, que guió a Riemann, Poincaré, Weyl y otros, la cual busca perfeccionar los sistemas conceptuales disponibles a fin de evitar limitaciones conceptuales y ampliar el abanico de opciones teóricas. Philosophia Scientiæ, cahier spécial 5, 2005, 27–42. 28 José Ferreirós Defenderé que, en ocasiones, se encuentran supuestos acerca del trabajo sobre fundamentos que emergen de ciertos dogmas, frecuentemente heredados de imágenes previas ya superadas. Para completar la discusión, menciono algunos rasgos de un programa alternativo, que investiga la epistemología del conocimiento matemático. Confronted with my title, the reader may have thought that it is an unhappy idea to put together the notion of foundations and the word “dogma.” After all, foundational research consists of logical and/or mathematical results formulated and proved in the most rigorous possible way. Thus we are talking of a domain of objective results, unaffected by beliefs or vogues. To put it simply: we are talking about logic, not about any aspect of culture that may be affected by dogmas or by historical shifts. Only the fact that we have been living in this intellectual atmosphere of postmodernism — you may have reflected — can explain why such titles are taken even a bit seriously. What can I reply? The difficult thing for me would be to convince you that those points include some subtleties that are far from being “evident,” and, at the same time, that granting this does not throw us in the arms of postmodern thinking. That, as I say, would be a difficult argument, and I will not try to make it here. So let me begin in a different way. Let me underscore that the word “images” is in my title for some reason. My aim is to reflect on the activity of foundations research. Once we conceive of it as an activity, a practice, it is automatic that foundations research does not come down to a bunch of theorems — not even a multilayered set of problems, methods and theories. Following Leo Corry [1997],1let us call those theories, methods and results the body of foundational knowledge. As all practitioners know, the practice of foundations research is also guided by certain images of this enterprise, images which may vary from researcher to researcher, and certainly from time to time. These images tell us about the goals one pursues when doing foundational research, about important and irrelevant problems, acceptable and unacceptable, promising and unpromising ways of approaching these problems, and so on. What my title suggests is not that the body of foundational results is 1Corry talks about the modern structural approach in the images of mathematics and in the body of mathematics (e.g., with Bourbaki, or with category theory). Dogmas and the Changing Images of Foundations 29 affected by dogmas, just that some images of foundations are. Put this way, I am sure more than one reader will be relieved. In fact, examples of dogmatic attitudes abound. A noteworthy one can be found in Quine’s writings on set theory. I do not mean his distrust of the Zermelo–Fraenkel system for axiomatic set theory (ZFC) and his desire to consider alternative systems. What seems dogmatic to me is his philosophical evaluation of the paradoxes, the so-called “bankruptcy theory.” Here, Quine was fully in agreement with Russell: the paradoxes had shown commonsense logic to be contradictory. This view is put forward not only in his Mathematical Logic [Quine 1940, § 29], but also in the much later Set Theory and its Logic [Quine 1963, Introduction]. Here he writes that the “only natural attitude” towards the notion of class, which is so fundamental to thought, is the Comprehension Principle: that every open sentence in one variable determines a class. The effect of the paradoxes was to discredit this natural attitude, to show that “commonsense is bankrupt,” and “intuition” is not to be trusted. Quine was dismayed to find that most logicians were “retraining their intuition” by immersing themselves in the system of ZFC set theory. In his view, one ought to consider the whole variety of possible systems (type theory, ZFC, von Neumann–Bernays–Gödel set theory, Quine’s own systems known as NF and ML,2and so on), treating none of them as standard. For it would be imprudent to consider one single system as the standard, natural one [Quine 1963, pp. viii, 1, 5]. In my opinion, Quine’s views on what he calls “commonsense” and “bankruptcy” are worthless as a philosophy. There is no rationale for thinking that the Comprehension Principle is more “natural” as a hypothesis than any other, say Zermelo’s Axiom of Separation. As a piece of history, however, Quine’s views are very interesting and revealing — they display before us the picture of a generation of logicians whose “intuitions,” to use his word, had been trained in the logic of classes based on Comprehension.3One is tempted to apply here Kuhn’s terminology, speaking of a paradigm shift (from the Comprehension logic of classes to axiomatic systems of set theory), of Quine as a member of the old generation, unable to absorb the impact of the revolution, and so on. But this is not our topic today. 2The abbreviations come from the titles of the works in which he proposed those systems: ‘New Foundations’ (1937) and Mathematical Logic [Quine 1940]. 3The notion of intuition that emerges here is very far from the Kantian one, but perhaps close to the teachings of Felix Klein. 30 José Ferreirós 1. Generalities about dogma. In what follows I intend to discuss several different images of foundations, and a few dogmas that are in fact interrelated. Let me open my mind and mention at least one of these right away. We all have some image of what the relations are between foundations, logic, and human reason (a noteworthy triangle). This frequently belongs to our cherised intimate convictions, and it may well happen that we do not want to discuss them with strangers. I beg your pardon, but as a philosopher it is my business to thematize such intimate convictions and reflect about them, eventually trying to see if they depend too much on traditional beliefs. Just in case somebody doubts that the connections between logic, reason and foundations can really be found in noteworthy contributions to this field, let me offer an example. Recall the well-known Hilbert program. It is interesting to reflect on some of the more philosophical opinions that Hilbert expressed (and actually he was generous in offering such opinions). For instance, in his address ‘Axiomatisches Denken’ of 1918 we can read: [. . . ] the most important mathematical thinkers [. . . ] have always cultivated the relations to the neighboring sciences, especially the great empires of physics and epistemology, above all for the benefit of mathematics itself. The essence of these relations and the reasons for their fruitfulness will be most clear [... by] describing [...] the axiomatic method. [Hilbert 1935, 146] In recent years, historians such as Volker Peckhaus [1991] and Leo Corry [1997] have made clear the extent to which this was not propaganda, but the expression of a serious concern of Hilbert’s with physics (especially its axiomatization) and with epistemology. In the celebrated 1900 lecture on ‘Mathematische Probleme’, Hilbert already expresses his concern with deductive finitism, and says as follows: This requirement of logical deduction by means of a finite number of processes is simply the requirement of rigour in reasoning. Indeed the requirement of rigour, which has become proverbial in mathematics, corresponds to a universal philosophical necessity of our understanding (Verstand). [Ewald 1996, vol. 2, 1099] As Michael Hallett [1994] has made clear in a paper devoted to connections between Hilbert’s axiomatic method and the “laws of thought,” his finitism was thus meant to express a key (though weak) principle about the workings of the mind. In 1928, Hilbert made the point by saying nothing less than the following: Dogmas and the Changing Images of Foundations 31 The fundamental idea of my proof theory is none other than to describe the activity of our understanding, to make a protocol of the rules according to which our thinking actually proceeds. [van Heijenoort 1967, 475] In these sentences, the word Verstand seems to have been chosen to comply with Kant’s terminology,4but, being a bit less literal, we could perfectly well translate it by “reason.” In sum, according to Hilbert there is a very close link between foundational research and the study of thinking and reason, i.e., epistemology. To be sure, Hilbert did not pretend that his Beweisstheorie would exhaust that study and supplant epistemology. Many important epistemological matters would remain open even after the satisfactory completion of his program (see Peckhaus [1991]). But the foundational results of Beweisstheorie certainly had, in his view, quite direct epistemological implications. Is this a conception that one must share? Or, does it depend on certain assumptions that are open to questioning? A short general reflection may suggest the latter. In both scientific and mathematical research, ten years are a very long time, but, as you know, in the realm of human beliefs and intellectual traditions a century may not be a very long time. It is only in the last century and a half that we have started to think about human beings and minds in terms of evolution.5We belong to a culture that has almost always regarded men and women as having an eternal essence, and most of us have been raised in this belief. Different names for that essence, obviously with different connotations, have been “the soul” and “reason.” Kant’s philosophical ideas, to be sure, were formulated long before any need could be felt to even faint that such a belief should be questioned. So one might well employ the following rule of thumb: whenever you see a foundational researcher that is deeply influenced by Kant, beware and consider the possibility that he may be equating epistemology with foundations in a way that is unwarranted, i.e., that is based on questionable assumptions. Intimately entangled with the idea of a special link between logical foundations and human reason, one can find other noteworthy beliefs. When we do research on the foundations of mathematics, we look for systematic foundations. Of course it is very healthy and interesting to 4His distinction of the more dry faculty of the Verstand (understanding) from the idealizing faculty of the Vernunft (reason). 5I mean here the fact of biological evolution on earth, and leave open the thorny question of the right theory of evolution. 32 José Ferreirós search for unification and system, and I could have nothing against it. But, when coupled with the above-mentioned belief in a transcendent essence, one can easily be led into thinking that the systematic foundations one obtains tell us something about human reason. The very idea of systematicity may thus, coupled with other beliefs, give rise to a second dogma. We might go on trying to locate and reconsider beliefs that could be dogmatic. Some come to my mind rather quickly. The idea that mathematics and logic are marked out by enjoying absolute certainty, by consisting of results built on the most solid rock (not “on sand,” as Weyl once wrote). Or perhaps our habit of taking logic to be a(even the) kernel of human knowledge, and our associated tendency to assume that any act of thinking ought to correspond to some kind of logic (say, e.g., inductive logic). Also the high expectations we place on the capacity of formal systems to capture all kinds of subtle concepts and relations. But enough. At this point we shall stop talking generalities and begin to examine some particular, historically given images of foundations. This will allow me to review some key stages in the evolution of foundational ideas, and also to analyze the emergence and persistence of some dogmatic beliefs. We shall concentrate on the first two mentioned above: the idea of a special link between foundations and epistemology, and the epistemological interpretation of systematicity. 2. Two traditional images: Gauss to Hilbert. Let me begin with Gauss, just 200 years ago. As a young man, Gauss was interested in philosophy and read Kant and other philosophers with some care. We are fortunate to have a few statements of his, expressing thoughts about the philosophical foundations of mathematics — what he called, using now obsolete terminology, the “metaphysics of mathematics.”6Occasionally, Gauss felt the need to enter into a discussion of such philosophical topics in order to clear the way to novel mathematical developments. Thus he ended up touching on the “metaphysics” of number, of magnitudes, and of space. Particularly important were his parallel reflections on the epistemological foundations of geometry and of arithmetic. These reflections were rich in mathematical content, since they related to Gauss’s evolving views on non-Euclidean geometry and differential geometry, on number theory and function theory. Their final 6Fragments from 1800, 1816–17, 1825, 1831. See [Ferreirós, forthcoming]. Dogmas and the Changing Images of Foundations 33 outcome was synthesized by the learned man in the form of a Greek motto:  “God does arithmetic,” his thoughts consist in numbers and numberrelations, even when we cannot follow them. The significance of this statement can only be understood if we take into account that Plato is reported to have said, “God does geometry.” The Gaussian motto documents the end of the millenary domination of geometry in Western images of mathematical knowledge, and starts a historical period in which pure mathematics would be “under the sign of number” [Hilbert 1897]. The idea was spelled out in letters of 1817 and 1830; I quote the 1830 letter to Bessel: According to my most intimate conviction, the theory of space has a completely different position with regards to our knowledge apriori, than the pure theory of magnitudes. Our knowledge of the former lacks completely that absolute conviction of its necessity (and therefore of its absolute truth) which is characteristic of the latter. We must humbly acknowledge that, if number is just a product of our minds, space also has a reality outside our minds, and that we cannot prescribe its laws apriori. [Gauss 1900, 201] The language is strikingly Kantian, certainly more so than the language we find in Poincaré’s allusions to Kantian epistemology. Gauss was not doing “metaphysics” in our current sense of the word. What he did was to search for new systematic accounts of several mathematical topics, to elaborate new mathematical theories, hoping to find thereby the philosophical and epistemological basis of mathematical knowledge. But his results and theories did not possess philosophical meaning in and off themselves. They acquired epistemological significance because Gauss viewed them against the background of Kantian (and Leibnizian) epistemology. He was guided by the old image of human knowledge as a combination of elements derived from two sources: Reason and the senses, the rational and the empirical. In Gauss’s view, agreeing with Leibniz and Kant, mathematical knowledge has a strictly rational core, which is an a priori product of pure Reason. Geometry did not belong in that core, which he now identified with “arithmetic” in a broad sense — the theory of the complex number system in all its aspects. Though some of these reflections became available to the public as of 1831, for the most part they remained in the hands of a few friends until 34 José Ferreirós they finally were published in the 1860s. From this time on, it became customary among relevant German authors to conceive of “arithmetic” as another name for pure mathematics, and to exclude geometry from this domain.7To give just an example, when Dedekind gave expression to the view that pure mathematics is logic, 4 years after Frege but independently of him, he wrote: “arithmetic (algebra, analysis) is just a part of logic.” Interestingly, Dedekind chose to synthesize his new view with a motto that prolonged the Plato–Gauss tradition:     i.e., “man ever arithmetizes.” With this new move, the pure theory of numbers and their relations ceases to be a godly matter, to become human, very human. In the preface to his epoch-making Was sind und was sollen die Zahlen?, Dedekind made clear his opinion that arithmetic and pure mathematics are a matter of logic, an immediate product of the reine Denkgesetze, the pure laws of thought. They rest solely on the notions of set and mapping, and therefore on primitive abilities of the mind without which no thinking at all is possible. Thus they are available to anyone in possession of a gesunder Menschenverstand, a sane common understanding. Dedekind went as far as to propose the view that the chain of proofs that he presented in detail takes place actually in the mind of his readers as soon as they employ numbers; only it takes place unconsciously, and so our consciousness extends merely to some byproducts, complex arithmetical truths that we usually mistake for simple, intuitive evidences. To me, the situation with Dedekind is reminiscent of what we have seen apropos of Quine. I admire very much Dedekind’s work, but his belief that the notions of set and mapping are primitive ones, that they are engraved in our minds from the time of conception — this I can only regard as an unwarranted belief, and in all likelihood a false one. It is surprising to find that so many authors, including good philosophers of mathematics in recent years, still consider the notion of set as an intuitive one, as epistemologically primitive.8Of course this point is easier for us to grasp with hindsight, thanks to our knowledge of issues like the 7There is an early exception, Martin Ohm, who identified Zahlenlehre with pure mathematics from as early as 1819. He is certainly relevant because his textbooks and his views enjoyed wide diffusion in the Gymnasien. 8Choosing only among first-rate authors, an example can be found in Maddy [1992]. Dogmas and the Changing Images of Foundations 41 Peano arithmetic involves quantification over infinite domains, and thus what Hilbert and his followers called a “transfinite axiom” [Hilbert 1925, 382]. What a genetic epistemology (in the above sense) should do, is to analyze the epistemological roots of these different practices and theories, and to understand the links between them (be they cognitive, historical, social, or what not). It should also emphasize the shifts and displacements which distance them, and search for the factors that help explain those shifts. In doing so, one must carefully avoid the temptation of being over-systematic. So let me take exception to Quine’s dogma, and invite you again, in the spirit of Riemann, to reconsider received ideas on all of these issues. References Benoit,P.,Chemla,K.,Ritter,J.eds. 1992 Histoire de fractions, fractions d’histoire, Basel: Birkhäuser, 1992. Bernays, Paul 1976 Abhandlungen zur Philosophie der Mathematik,Darmstadt: Wissenschaftliche Buchgesellschaft, 1976. Corry, Leo 1997 Modern algebra and the rise of mathematical structures,Basel/ Boston: Birkhäuser, 1997. Dedekind, Richard 1888 Was sind und was sollen die Zahlen?, Braunschweig: Vieweg, 1888; reprinted in Gesammelte mathematische Werke,NewYork: Chelsea, 1969; Eng. trans. in [Ewald 1996, vol. 2]. Ewald, William, ed. 1996 From Kant to Hilbert, Oxford: Oxford University Press, 2 vols. Ferreirós, José forthcoming    : The rise of pure mathematics as arithmetic with Gauss, in C. Goldstein, N. Schappacher, J. Schwermer, eds., The Shaping of Arithmetic: Number theory after Carl Friedrich Gauss’s Disquisitiones Arithmeticae, Berlin: Springer. Frege, Gottlob 1893 Grundgesetze der Arithmetik, vol. 1, Jena: Pohle, 1893; reprint: Hildesheim: Olms, 1969. 42 Gauss, Carl F. 1900 Werke, vol. 8, Göttingen: Dieterich, 1900; reprint Hildesheim: Olms, 1973. Hallett, Michael 1994 Hilbert’s Axiomatic Method and the Laws of Thought, in A. George, Mathematics and Mind, Oxford: Oxford University Press, 1994, 158–200. Van Heijenoort, Jean 1967 From Frege to Gödel, Harvard Univ. Press, 1967; reprinted in 2002. Hilbert, David 1897 Bericht über die Theorie der algebraischen Zahlen,Jahresbericht der DMV, 4, 1897; reprint in Gesammelte Abhandlungen, Berlin: Springer, vol. 1, 1932. 1935 Gesammelte Abhandlungen, vol. 3, Berlin: Springer, 1935; reprinted in 1970. Maddy, Penelope 1992 Realism in mathematics, Oxford: Clarendon Press, 1992. Peckhaus, Volker 1991 Hilbertprogramm und kritische Philosophie, Göttingen: Vandenhoeck & Ruprecht, 1991. Quine, Willard Van O. 1940 Mathematical Logic, New York: Norton, 1940. 1963 Set theory and its logic, Harvard Univ. Press, 1963. 1969 Epistemology naturalized, in Ontological Relativity and other essays, Columbia Univ. Press, 1969. Riemann, Bernhard 1854 Über die Hypothesen, welche der Geometrie zu Grunde liegen, in Gesammelte Werke, Berlin: Springer, 1991; Eng. trans. in Ewald, op. cit., vol. 2. (See also Riemann’s Fragmente philosophischen Inhalts, in Gesammelte Werke.) Sieg, Wilfried 1999 Hilbert’s programs: 1917–1922, The Bulletin of Symbolic Logic, 5, 1–44.