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Most physics theories are deterministic, with the notable exception of quantum mechanics which, however, comes plagued by the so-called measurement problem. This state of affairs might well be due to the inability of standard mathematics to "speak" of indeterminism, its inability to present us a worldview in which new information is created as time passes. In such a case, scientific determinism would only be an illusion due to the timeless mathematical language scientists use. To investigate this possibility it is necessary to develop an alternative mathematical language that is both powerful enough to allow scientists to compute predictions and compatible with indeterminism and the passage of time. We argue that intuitionistic mathematics provides such a language and we illustrate it in simple terms., First presented at the workshop "Experiencing Reality Directly" (Jerusalem, May 22-24, 2019)

Husserl’s lifelong interest in Kant eventually becomes a preoccupation in his later years when he finds his phenomenology in competition with Neokantianism for the title of transcendental philosophy. Some issues that Husserl is concerned with in Kant are bound up with the works of Lambert. Kant believed that the role played by principles of sensibility in metaphysics should be determined by a “general phenomenology” on which Lambert had written. Kant initially believed that man is capable only of symbolic cognition, not intellectual intuition. Lambert saw an increasing need in mathematics for symbolic cognition as exemplified in his proofs of the irrationality of π and e. Kant takes from Leibniz and Lambert an unrestricted notion of construction, allowing him to view mathematics as constructing its concepts in intuition, while Lambert’s proofs convince him that all mathematical problems are eventually solvable. Husserl criticizes Kant’s intuitionism for its inadequate accommodation of meaning to intuition, which he redresses with his theory of categorial intuition. This may improve on Kant’s intuition of the axiom of parallels but not so clearly on his spatial intuition. Husserl opposes Kant’s view of space as the form of outer intuition with his own view of it as the form of things. Husserl’s exploration of the geometry of visual space, which involves his earliest uses of epoche and reduction, converges however with Hilbert’s logical analysis of Kantian spatial intuition in leading to Euclidean spatial judgments. Hilbert’s analysis leads him to affirm the solvability of all well posed mathematical problems, a thesis complicated by the outbreak of logical paradoxes. Untroubled by such paradoxes, Husserl develops a supramathematics of all possible deductive systems whose completeness implicitly would also provide solutions of all such problems. Husserl’s full transcendental turn coincides with his realization that in effecting his Copernican turn, Kant was really the first to detect the “secret longing” of modern philosophy for a phenomenological clarification of being. Husserl now finds that Kant’s transcendental deductions presuppose a pure ego not adequately analyzed by Kant that survives the phenomenological reduction. Husserl’s idealism leads him to develop his own intuitionism, which adds to Kant’s, a “method of clarification” of mathematical concepts intended to clarify difficult impossibility proofs, but neither Husserl nor Kant base arithmetic on time. Husserl’s critique of the room left in Kant’s idealism, for things in themselves, leads to his own monadological solution of the problem of intersubjectivity. Husserl’s final judgment on Kant is that his form of transcendental idealism did not enable him to achieve absolute subjectivity through a genuine transcendental reduction.

Husserl’s first work formulated what proved to be an algorithmically complete arithmetic, lending mathematical clarity to Kronecker’s reduction of analysis to finite calculations with integers. Husserl’s critique of his nominalism led him to seek a philosophical justification of successful applications of symbolic arithmetic to nature, providing insight into the “wonderful affinity” between our mathematical thoughts and things without invoking a pre-established harmony. For this, Husserl develops a purely descriptive phenomenology for which he found inspiration in Mach’s proposal of a “universal physical phenomenology.” To account for applications to any domain, Husserl envisages a theory of all possible deductive systems, which he develops extensively in his Gottingen lectures wherein he engages with Hilbert’s work on deductive systems for geometry, real arithmetic, and physics. This leads Husserl to formulate claims of decidability and proofs of completeness for various arithmetics that result from his analysis of Kronecker’s general arithmetic. Careful attention to these proofs seem to show that Husserl was not oblivious to problems that underlie our incompleteness theorems, namely that of showing that some inversions of his algorithmic arithmetic are undefined. His growing preoccupation with the issue of a pre-established harmony between mathematical thought and reality motivate his pursuit of a “supramathematics” of all possible complete theory forms to demystify such harmony, by having such a form on hand for describing any empirical domain. He soon decides that a transcendental idealism of nature will reveal the wonderful affinity of thoughts and things comprising such harmony, to be a wonderful “parallelism of objective unities and constituted manifolds of consciousness.” But the paradoxes of logic and set theory cloud the clarity of mathematics, which Weyl would restore with Brouwer’s intuitionism and Hilbert with his metamathematics. Husserl informed Weyl that his student Becker had formulated a phenomenological foundation not only for Weyl’s generalization of relativity theory but also for the Brouwer-Weyl continuum. But Weyl eventually rejected much of Becker’s work, especially when it became clear that his phenomenological intuitionism could not account for the success of Hilbert’s transfinite mathematics in quantum physics. Becker responded to this “crisis of phenomenological method” with his mantic phenomenology celebrating the magic of mathematical mysticism, which Husserl finally rejects in favor of a pluralistic phenomenology of mathematics and nature.

The ongoing discussion regarding the propriety of cognitive enhancement is, at its core, biopolitical. In an attempt to understand the underlying issues, it is instructive to view the subject through the perspectives of those at the extremes – the Transhumanists who enthusiastically support the development of cognitive enhancement and the Bioconservatives who view the enterprise with disapprobation. I argue that many of the positions undertaken by proponents of either view are influenced to a considerable degree by a variety of pre-existing political positions as much as, if not more so than, by deeper philosophical reflection, a phenomenon very much in keeping with Jonathan Haidt’s social intuitionist model of moral reasoning (Haidt, Psychol Rev 108:814–834, 2001). Despite the vocal nature of the extremes, there also exists a broad middle ground which I term The view from reasonableness which is much less of a polemic and more of a conversation. Given that the ultimate outcome rests more on how the populace responds than the views of experts, I suggest that supplementing normative claims driven by biopolitical intuitions with empirical data on public attitudes towards cognitive enhancement might allow us to move beyond polemics to deepen our appreciation of how to move forward.

The spring of 1980 I spent as visiting lecturer at Utrecht. The volume of Heyting’s Collected Papers had not yet been put together, and his philosophical papers could not be found at Oxford. Accordingly, I availed myself of the opportunities offered by Dutch libraries and read the relevant papers. A couple of years earlier, I had learned about Constructive Type Theory from Per Martin-Lof, and Michael Beeson, who had just written a paper on a theory of constructions, was an eager sparring partner in almost daily discussions at Utrecht. The outcome of these ponderings was this chapter on which you are reading now as an afterword. It was ready toward the end of the summer 1981, and my Oxford Professor Dana Scott suggested to me that I should submit it to Richmond Thomason, the editor of the Journal of Philosophical Logic, at a meeting of authors for the Handbook of Philosophical Logic at Bad Homburg. I did so and the paper was readily accepted; however, a special issue on intuitionism was being prepared, and Thomason suggested that I might want to wait in order to have it appear in that issue. Thus, the paper appeared only in 1983 but had circulated rather widely in the intervening time.

Godel’s incompleteness theorems had a dramatic impact on Hilbert’s foundational program. That is common lore. For some, e.g. von Neumann and Herbrand, they undermined the finitist consistency program; for others, e.g. Godel and Bernays, they left room for a fruitful development of proof theory. This paper aims for a nuanced and deepened understanding of how Godel’s results effected a transformation of proof theory between 1930 and 1934. The starting-point of this transformative period is Godel’s announcement of a restricted unprovability result in September of 1930; its end-point is the completion of Gentzen’s first consistency proof for elementary number theory in late 1934. Hilbert, surprisingly, is the initial link between starting-point and end-point. He addressed Godelian issues in two strikingly different papers, (1931a) and (1931b), without mentioning Godel. In (1931a) he takes on the challenge of the restricted unprovability result; in (1931b) he responds to the second incompleteness theorem concerning the unprovability of consistency for a system S in S. He does so by bringing in semantic considerations and by pursuing novel, but also highly problematic directions: he argues for the contentual correctness of a constructive (for him, finitist) theory that includes intuitionist number theory. Gentzen followed the new directions in late 1931, addressed methodological issues and metamathematical problems in ingenious ways, while building on ideas and techniques that had been introduced in proof theory. Most distinctive is Gentzen’s struggle with contentual correctness and its relation to consistency. That is reflected in a sequence of notes Gentzen wrote between late-1931 and late-1934, but it is also central in his classical paper (1936). The immediate lessons seem to be: (i) there is real continuity between Hilbert’s proof theory and Gentzen’s work, and (ii) there is deepened concern for interpreting intuitionist arithmetic (and thus understanding classical arithmetic) from a more strictly constructive perspective. Ironically, the latter concern deeply influenced Godel’s functional interpretation of intuitionist arithmetic.

Is logic in its essence epistemological or ontological? This question was the starting point of Per Martin-Lof’s lecture at the conference at which the contributions to this volume were presented. In this essay I shall limit myself to the more specific question whether the concepts of truth and proof are epistemological or ontological. That proof is an epistemic concept is of course normally not in doubt, whereas opinions differ concerning truth.

Book review on intuitionism

A critical examination of notions associated with mathematical intuitionism, with attention to mathematical practice, internal vs. external negations, constructive meaning, weak counterexamples, and theories of constructions. The article includes a theory of constructions based on nonstandard realizability.

Intuitionistic logic encompasses the principles of reasoning which were used by L. Brouwer in developing his intuitionistic mathematics, beginning in [Bro07]. Intuitionistic logic can be succinctly described as classical logic without the law ϕ ∨ ¬ϕ of excluded middle. Brouwer observed that this law was abstracted from finite situations and its application to statements about infinite collections is not justified. One of the consequences of the rejection of the law of excluded middle is that every intuitionistic proof of an existential sentence can be effectively transformed into an intuitionistic proof of an instance of that sentence. More precisely, if a formula of the form ∃xϕ(x) without free variables is provable in the intuitionistic predicate logic, then there is a term t without free variables such that ϕ(t) is provable. In particular, if ϕ ∨ ψ is provable, then either ϕ or ψ is provable. In that sense intuitionistic logic may provide a logical basis for constructive reasoning. A formal system of intuitionistic logic was proposed in [Hey30]. A relationship between the classical propositional logic PC and intuitionistic propositional logic INT was proved by Glivenko in [Gli29], namely a PC-formula ϕ is PC-provable if and only if ¬ ¬ϕ is INT-provable. Kripke semantics for intuitionistic logic was developed in [Kri65].