Philosophies of Intuitionism: Why We Need Them

teorema Vol. XXVI/1, 2007, pp. 73-82

Miriam Franchella

RESUMEN Desde su primera aparición en el mundo de la matemática, el intuicionismo ha planteado el problema de si resulta apropiado presentarse en tanto que fundamentado en una Weltanschauung (como de hecho lo fue, al menos en la mente de su padre fundador) o de si habría sido mejor explicarlo como una práctica matemática autosuficiente. En es- te sentido, traigo a colación las posiciones de Brouwer, Griss y Heyting y defiendo que el hecho de que Brouwer hubiera basado el intuicionismo en su Weltanschauung es un buen argumento para responder al desafío neo-naturalista de Maddy. Tal respuesta po- dría solucionar también dos problemas que habían quedado abiertos: la tarea de la filoso- fía y la diferencia de estatuto epistémico entre la astrología y la matemática.

ABSTRACT From its first appearance in the world of mathematics, intuitionism has raised the issue of whether it is appropriate to present intuitionism as grounded in a Weltan- schauung (as indeed it was, at least in the mind of its originator) or whether it would have been better to explain intuitionism as a self-standing mathematical practice. At this remove, I recall the positions of Brouwer, Griss, and Heyting and then argue that the very fact that Brouwer grounded intuitionism in his Weltanschauung is a good argument to respond to Maddy’s neonaturalist challenge. Such a response might also solve two problems left open: the task of philosophy and the difference in epistemic status between astrology and mathematics.

I. TWO WELTANSCHAUUNGEN TOWARDS INTUITIONISM.

In 1907 intuitionism was introduced into the philosophy of mathematics by L.E.J. Brouwer in his Ph.D. thesis On the Foundations of Mathematics. Brouwer was able to include only a few philosophical excerpts of his planned work because his thesis advisor, D.K. Korteweg, let on that such ideas would not have been well received by the mathematicians to which they were ad- dressed. Only traces of Brouwer’s philosophical thought are to be found: his belief that natural numbers originate from two-ity and some references to Kant and Russell. But there is no trace of the mysticism that underpins his acceptance of evidence as a truth criterion of mathematics. Only later did these frag- ments appear, during some lectures, to be rediscovered and published well after Brouwer’s death by W.P. van Stigt, who showed the world how Brouwer’s

73 74 Miriam Franchella mathematical thought was grounded in his mystical thought. Specifically, according to Brouwer’s mysticism [see van Stigt (1990) and van Atten (2003)], all attempts to reach something outside the self are incomplete and, therefore, frustrating. On this basis, mysticism asserts that the possibility for man to be happy lies in his inner self. So long as man is imprisoned in his body, he con- tinues to perceive and, as a result, cannot completely and definitively reach his inner self. He must, however, seek it out as much as possible by trying to abstain from all kinds of domination. Moreover, mathematics, because it is science and language, is clearly an example of moving towards the exterior and away from the self. On this basis, Brouwer aimed to reconstruct a mathematics that could be evaluated as morally positive. There were two ba- sic conditions: mathematics has to be developed 1) in the intellect (and not «on paper» for which he blamed the formalists), and 2) without any intention of applying it. First, Brouwer needed to search for an inner something from which mathematics might spring. Secondly, he had to elaborate a theory both of mathematical existence and of truth that was consistent with this position. Temporal intuition could plausibly be accepted as its starting point: the ca- pacity (intrinsic to man) for isolating an instant, conserving it in memory, isolating another instant from the former, and so on, made it possible to produce natural numbers. Although time was the first step outwards, the free unfolding of intuition was morally safe because it was «not bound to the exterior world, and thereby to finiteness and responsibility» [Brouwer (1948), p. 1239]. In developing mathematics internally, Brouwer could preserve neither a theory of mathematical entities as independent of man, nor a classical theory of truth as referring to something outside of man. Brouwer maintained that it was more appropriate both to affirm that mathematical entities exist only if they can be constructed mentally and to define truth as the inner experience of mental evidence. We now know that there was another, alternative, strong philosophical approach that led to mathematical intuitionism: that of G.F.C. Griss. Griss’ adherence to intuitionism depends on the Weltanschauung he explains in his 1946 book Idealistische Filosofie. It is based on the original content that consciousness grasps when it reaches its own fullness: the subject distinguishes himself from the object, but one has no meaning without the other. This datum is the condition of all experience, and can be considered from three standpoints: remarking the entities produced by their splitting and the relations among them, remarking the unity between subject and object, or considering both these aspects as being linked. In the first case we are practicing mathematics, in the second case, mysticism, and, in the third case, philosophy. Let us begin with philosophy: because it grasps the original datum, it is also able to show us the aim of human life. Hence, such philosophy can establish ethics. Our purpose consists of developing the product of a full awareness of the subject-object link: it yields to the abolition of every form of Philosophies of Intuitionism: Why We Need Them 75 dualism and to the harmony with the all. This implies three aspects: 1) the indi- vidual grows only through contacts with his fellows; 2) each man is responsible for the others; and 3) spiritual growth does not occur without material growth. Mysticism, Griss affirms, lives out what philosophy both realizes and lives. It feels the union between man and the totality of the world, between man and God. Mathematics is a way to analyze the original datum. This has two significant consequences. First, the subject-object distinction it starts from can be considered the schema for producing new entities: therefore it can be stated that the foundation of natural numbers lies precisely in the original datum. Second, mathematical constructions cannot be conceived independently by the mathematician, because in general the product cannot be conceived without its producer. Therefore, something can be said to exist as a mathematical entity only if it is built mentally and can be considered true only if it is evident to the intellect. This was Griss’ path towards intuitionist mathematics.

II. WITHOUT A PHILOSOPHY?

Heyting, one of Brouwer’s most brilliant pupils, was the first intuitionist to keep the philosophical grounding of their way of doing mathematics hidden. This approach was dictated both by Heyting’s aims and by the relations among foundational schools of the period. Heyting began his activity in the late 1920s at a time when logicism was in decline, there was scarce adherence to intuitionism, and the schism between Brouwer and Hilbert had to be reckoned with. Brouwer’s expanding influence had exasperated Hilbert [see van Dalen (1990)] and Brouwer had already managed to bring Hilbert’s pupil Weyl over to this way of thinking. As for what Heyting himself explicitly affirmed [Heyting (1978), p. 15], his aim was to make intuitionism known so that his colleagues would take an interest in it and develop some parts of mathematics using intuitionist methods. The split between Brouwer and Hilbert had caused the relationship between intuitionists and formalists to turn cold, thus preventing theoretical debate, let alone effective collaboration, ab initio. Given this situation, Heyting’s plan to draw mathematicians’ attention to intuitionism was subordinated to settling the conflictual relationships between intuitionists and formalists. Heyting therefore devised his strategy of popularizing intuitionism. As a firm basis on which to find agreement with formalists, he chose the fact that both foundational schools put themselves forward as antimetaphysical. By this he meant that: 1) they both rejected the classical (Platonist) belief in the transcendent existence of entities, that is, the belief that they exist independently of men; and 2) in order to do mathematics according to their respective methods, there was no need for preliminary acceptance of a particular philosophical standpoint. 76 Miriam Franchella

From 1931, Heyting separated what he later called Brouwer’s «theory of knowledge» from his «theory of values» [Heyting (1968), p. 312]. It was therefore clear to Heyting that in Brouwer there is a theory of value. Yet, because mathematics according to Brouwer consists of developing human mental possibilities, Heyting could take up this result and stress its antimetaphysical nature by separating it from the theory of value that had given rise to Brouwer’s theory. Heyting (1956) [p. 2] claimed he had no objection to a mathematician privately admitting any metaphysical theory, but Brouwer’s program entailed studying mathematics as something simpler and more immediate than metaphysics, i.e. mental constructions. He was conscious that «in order to construct mathematical theories no philosophical preliminaries are needed, but the value we attribute to this activity will depend upon our philosophical ideas» [Heyting (1956), p. 9] and he actually had his own theory of value, which he revealed in some unpublished papers, kept in his archive in Haarlem. A recollection of his philosophical thought can be found in Franchella’s (1995) «Like a bee on a window-pane» and only a few salient points are taken up again here. Heyting started with the content of our consciousness, our first experience, which is total indefiniteness: not even the self is distinguished from the rest of existence. This content is so rich and intense that man could not survive if man experienced it for a long time. As a result, some form of protection is needed: individualization takes place, i.e. sensations are separated from each other and are distinguished from the self. Once objects have been identified, they need to be linked. This is where spatialization and temporalization come in. If we verify the content of our consciousness, we realize at once that we perceive that other persons exist, through an exchange of thoughts that enrich ourselves. The other certainty that we have is that the outer world exists. This certainty is characterized by the fact that one does not expect sudden changes in reality, but only gradual variation: this is a general idea, a model of the world that we have. Our belief in the outer world is direct, while our concept of an outer world is an abstraction. And abstractions can be placed on a scale of evidence topped by «God, real numbers, and large cardinal numbers». It is clear that Heyting’s theory of value considers evidence fundamental, for it starts with the evidence of the content of our consciousness.

III. ABOUT NATURALISM

Why consider these viewpoints again, after so many years? I believe that they afford an opportunity to answer the epistemological challenge of Maddian neonaturalism. Let us (very briefly) recall that Penelope Maddy began her academic output as a set-theoretical realist, believing in the possibility of ac- Philosophies of Intuitionism: Why We Need Them 77 cepting the existence of (autonomous) sets in a way comparable to how we accept the existence of physical objects (see for instance her 1990 Realism in Mathematics). She shared Quine’s naturalism, i.e. «The recognition that it is within science itself, and not in some prior philosophy, that reality is to be identified and described» [Quine (1981), p. 21]. And again: «the abandon- ment of the goal of a first philosophy. It sees natural science as an inquiry into reality, fallible and corrigible, but not answerable to any supra-scientific tribunal and not in need of any justification beyond observation and the hypo- thetic-deductive method» [Quine (1975), p. 72]. She also shared his indispensability argument for the existence of mathematical objects (which was consistent with naturalism), though she deemed further specification neces- sary to avoid subordinating mathematics to physics. Later, she stressed [Maddy (1992)] that even in physics it is not enough for an entity to be part of a good theory in order to be accepted: by way of example she recalled that atoms were accepted by the community of physicists only when they attained some sort of «visibility». She then quoted [Maddy (1992)] case studies in which set theorists decline to wait for physical results that might void their questions going on, rather, with their research (for instance, only space-time continuity in physics gives meaning to the quest for new axioms for the 1 Lebesgue measurability of sets whose complexity is ). Later, she ¦2 stressed that physicists do not consider mathematical hypotheses on a par with physical hypotheses [Maddy (1995)] and that they require visibility in order for their entities to be accepted whereas mathematicians require only consistency and fruitfulness [Maddy (1997 and 1998)]. Furthermore, she pointed out [Maddy (2001)] that mathematical concepts (like that of group) did not become established and commonly used within the mathematical community until they had shown themselves to be useful tools for mathematics. Along the way, she became convinced that «naturalism», extended to mathematics, and so called neonaturalism should mean considering mathematics independently of physics, taking only its practice into account while leaving mathematics to establish its ends and its methods on its own. Specifi- cally, in the case of set theory, this means that criteria for defining axioms should be based on the aims of set theory itself. From this viewpoint, then, she investigated whether ontological questions played a role in the develop- ment of mathematics. She concluded [Maddy (1997), p. 191] that

the historical record gives a fairly clear indication of what did finally resolve the methodological debates: impredicative definitions are needed for a classical theory of real numbers (among other things); the Axiom of Choice is so fruitful in many branches of mathematics that mathematicians refused to give it up. In other words, these debates were decided on straightforwardly mathematical grounds. 78 Miriam Franchella

About the possible role of ontology in set-theory, she declared [Maddy (1997), p. 192]:

For now, what we have learned is that mathematical practice itself gives us little ontological guidance. The methods of natural science […] also tell us that ordi- nary physical objects and many unobservable exist, that they do so objectively and spatio-temporally; but the methods of mathematics —the methods used to select its axiom systems, structure its proofs, determine its research directions— tell us no more than that certain mathematical objects exist. They tell us nothing about the nature of that existence —is it objective? Is it spatio-temporal?— indeed, nothing seems to preclude even Fictionalist or Formalist interpretations.

Hence, all discussion about mathematical ontology can only disturb mathematical practice: it is the Wittgensteinian «philosophical fog». Nor is the question of «truth» relevant to the practice of mathematics: therefore, Maddy calls her latest position «arealism» [Maddy (2005b), p. 364]. She also responded to two possible criticisms: the uncriticizability of sciences that naturalism in general brings with it and the possibility of considering astrology on a par with mathematics from a naturalistic perspective. To the first criticism she replied: «But we’ve already seen that this is not true. Natural science itself is a self-critical enterprise that develops and debates its own methodological norms» [Maddy (1997), p. 181]. And to the second criticism she replied: «Should we then move to astrological naturalism? Again the answer is no. The reason is simple: mathematics is staggeringly useful, seemingly indispensable, to the practice of natural science while astrology is not» [Maddy (1997), pp. 204-205]. Finally, she concluded on philosophical tasks [Maddy (1996), p. 503]:

This is not an anti-philosophical view: nothing is commended to the flames. […] This is not to say that philosophy itself is useless. […] Philosophical views can be extremely helpful, even essential, in a heuristic sense, and, in fact, though I have no example at hand to cite, I suppose an erstwhile philosophical view might migrate and eventually become so thoroughly entangled with actual scientific or mathematical practice as to become a legitimate methodological maxim. From his point of view, philosophy should be encouraged, much in the way pure mathematics should be encouraged even by those whose only interest is in ultimate applications: there’s no predicting what will turn out to be useful one day. (Emphasis mine).

In general, about Maddy:

1) in order to destroy a milestone of the indispensability argument, the acceptability of physical entities on the basis that they are part of a «good theory», she alleges the example of atoms, which were Philosophies of Intuitionism: Why We Need Them 79

accepted by the community of physicists only after atoms somehow became visible. Still, we must note that the first scientists to propose atoms (who were not believed by the community), the real innovators of physics, did not see atoms but introduced them for theoretical purposes: Ostwald wrote a warning textbook, but Dalton had introduced the atomic hypothesis!

2) the simple fact that some mathematicians accept a mathematical notion not for its applicability in physics but only for its usefulness in mathematics cannot be considered something tried and fixed that might give rise to a general criterion for acceptability. It may be a sign of an acrimonious division between mathematicians and physicists; it may represent a lost opportunity for challenges and suggestions among different fields of science; it may be transitory and subject to change at the hands of a government’s grant policy (for example, if grants are given only for applied mathematics). I do not intend to affirm that mathematics must be studied for its applicability. I only want to stress that, within Maddy’s argument, the historical fact that some mathematicians are unconcerned with mathematics’ applicability to physics cannot be deemed a reason for them to behave so always.

In particular, on Maddy and intuitionism:

1) Maddy does not very often consider intuitionism during her scientific production. She discarded it in 1989 because she considered set theory more interesting (and perhaps essential in founding the rest of the issue), after having distinguished between the philosophies of constructivists and constructivism as a branch of mathematics. She expresses her appreciation for the latter —quoting van Dalen and Troelstra as authors— without acknowledging the fact that, at least in the case of intuitionism, that kind of constructivism was generated by Weltanschauungen. Namely, she wrote [Maddy (1989), p. 1122]:

None of this applies of course to constructivism as a branch of mathematics, now a growth industry on the boundary of logic and computer science.

Where «none of this» referred to the just stated opinion that

philosophies of mathematics bear the same relation to mathematics as philosophy of science does to physical science: that is, its goal is to describe and account for mathematics as it is practiced. […] This is not to say that philosophy 80 Miriam Franchella

never performs a critical function, but it’s hard to imagine a philosophical theory being so well-supported that the most rational resolution of a conflict between it and a significant piece of science would be to chuck the science. Philosophical constructivist reject the same approach to mathematics, insist- ing that mathematics must go when it conflicts with the philosophy, but I confess that I don’t see their justification. [Maddy (1989), p. 1122]

In her 1990 book, she qualified intuitionism a form of verifica- tionism (hence, clearing referring to Dummett’s approach to intuitionism). In her 2002 paper «A Naturalist Look at Logic» Maddy mentioned intuitionist logic as a form of rejection of the law of bivalence, which is, according to her Kantian (updated through Frege) account of our reasoning machinery, something not rooted in our understanding of the world.

Nearly everyone admits that bivalence does sometimes fail —for instance, in the case of vague predicates, or in talk of fictional entities [Maddy (2002), p. 80],

so there is no problem with such refutation:

Intuitionists understand mathematical entities as mental constructions […] and conclude that the idealization of bivalence is inappropriate in this case, as well. We may disagree on the advisability of this approach to mathematics, but logically speaking, it is simply a case of avoiding full classical logic in a context where one of its idealizations is purportedly not appropriate! [Maddy (2002), p. 80]

In the footnote to this paragraph, she specified: «I’m speaking here of the early intuitionism of Brouwer and Heyting, which focused on the nature of mathematical entities». Here, she deems Heyting to share Brouwer’s ontological view (mathematical entities are mental constructions), while she does not notice that Heyting shared with her –– albeit on different grounds–– a kind of «commitment phobia».

1) By separating constructivism from philosophies of constructivism that generated it (as we saw above), she missed the chance to witness the historical evidence of a positive influence philosophy has had upon mathematics. That is why I believe it is now important to recall the philosophical basis of intuitionism. Maddy always shows herself as very attentive and open not only to the actual practice of the sciences, but also their past practice. So, a presentation of intuitionism à la Brouwer (or à la Griss) could show her how in the past some philosophies produced a form of mathematics that is still alive and practiced. While she can see Philosophies of Intuitionism: Why We Need Them 81

actual philosophies as forms of intrusions into mathematics (for instance, Dummett’s reasons for intuitionism stemming from philosophy of language) in the name of her neonaturalism, she cannot say anything about the past. She can only, for the sake of consistency, take note of the fact that some «philosophies» existed that caused great changes inside mathematics and that these later flourished. This would also solve both the problem of philosophy’s task that she cannot explain (as we saw in the above quotation) and the problem of re-using the applied nature of mathematics to differentiate mathematics’ epistemic status from that of astrology. A discipline’s epistemic status cannot depend on something outside the discipline itself.

There can be a new form of philosophical modesty. Considering naturalism a possibility, alongside opposing forms of «first philosophy», some people may be more inclined to accept naturalism and others more inclined to assign philosophy a stronger role.

Dipartimento di Filosofia Universitá degli Studi di Milano Via Festa del Perdono 7, 20122 Milano, Italia E-mail: [email protected]

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