Intuitionism and Constructivism

This paper scrutinizes the debate over logical pluralism. I hope to make this debate more tractable by addressing the question of motivating data: what would count as strong evidence in favor of logical pluralism? Any research program should be able to answer this question, but when faced with this task, many logical pluralists fall back on brute intuitions. This sets logical pluralism on a weak foundation and makes it seem as if nothing pressing is at stake in the debate. The (...) present paper aims to improve this situation by looking at a promising case study and drawing general lessons about the kind of evidence that would support logical pluralism. I argue that the best motivation for logical pluralism will ultimately be rooted in certain kinds of performative data. (shrink)

In this paper I argue that constructivism in mathematics faces a dilemma. In particular, I maintain that constructivism is unable to explain (i) the application of mathematics to nature and (ii) the intersubjectivity of mathematics unless (iii) it is conjoined with two theses that reduce it to a form of mathematical Platonism. The paper is divided into five sections. In the first section of the paper, I explain the difference between mathematical constructivism and mathematical Platonism and I outline my argument. (...) In the second, I argue that the best explanation of how mathematics applies to nature for a constructivist is a thesis I call Copernicanism. In the third, I argue that the best explanation of how mathematics can be intersubjective for a constructivist is a thesis I call Ideality. In the fourth, I argue that once constructivism is conjoined with these two theses, it collapses into a form of mathematical Platonism. In the fifth, I confront some objections. (shrink)

It is usual to identify initial conditions of classical dynamical systems with mathematical real numbers. However, almost all real numbers contain an infinite amount of information. I argue that a finite volume of space can’t contain more than a finite amount of information, hence that the mathematical real numbers are not physically relevant. Moreover, a better terminology for the so-called real numbers is “random numbers”, as their series of bits are truly random. I propose an alternative classical mechanics, which is (...) empirically equivalent to classical mechanics, but uses only finite-information numbers. This alternative classical mechanics is non-deterministic, despite the use of deterministic equations, in a way similar to quantum theory. Interestingly, both alternative classical mechanics and quantum theories can be supplemented by additional variables in such a way that the supplemented theory is deterministic. Most physicists straightforwardly supplement classical theory with real numbers to which they attribute physical existence, while most physicists reject Bohmian mechanics as supplemented quantum theory, arguing that Bohmian positions have no physical reality. (shrink)

Resumo Neste artigo, será discutida a noção de “infinitude cardinal” – a qual seria predicada de um “conjunto” – e a noção de “infinitude ordinal” – a qual seria predicada de um “processo”. A partir dessa distinção conceitual, será abordado o principal problema desse artigo, i.e., o problema da possibilidade teórica de uma infinitude de estrelas tratado por Dummett em sua obra Elements of Intuitionism. O filósofo inglês sugere que, mesmo diante dessa possibilidade teórica, deveria ser possível predicar apenas infinitude (...) ordinal. A questão principal surge do fato de que parece ser problemático predicar ordinalmente infinitude de “estrelas”. Mesmo diante dessa possibilidade, Dummett sugere que o intuicionista poderia apenas reinterpretar infinitude cardinal como sendo infinitude ordinal. Ora, iremos mostrar que, se Dummett não fornece razões extras que sustentem essa posição, então será difícil interpretar um caso empírico infinitário como sendo também um caso ordinal ou potencial de infinitude. Para resolver esse problema de Dummett, em Brouwer se encontram alguns pressupostos idealistas necessários para argumentar em favor da ideia de que, mesmo em um contexto empírico, como o de uma infinitude de estrelas, poderíamos predicar infinitude ordinal. Então, depois de discutir as duas noções de “infinitude” e apresentar o problema de Dummett, será apresentada a abordagem idealista de Brouwer – a qual pelo menos explicaria de modo mais plausível as razões que poderiam motivar um intuicionista a predicar infinitude ordinal até mesmo de um caso empírico e espacial. (shrink)

The aim of the present thesis is twofold. First we propose a constructive solution to Frege's puzzle using an approach based on homotopy type theory, a newly proposed foundation of mathematics that possesses a higher-dimensional treatment of equality. We claim that, from the viewpoint of constructivism, Frege's solution is unable to explain the so-called ‘cognitive significance' of equality statements, since, as we shall argue, not only statements of the form 'a = b', but also 'a = a' may contribute to (...) an extension of knowledge. Second, we study this higher-dimensional account of equality from a constructive (computational) standpoint and, based on these considerations, we offer a new perspective to the project proposed by Peter Aczel of developing conventions and notations for an informal style of doing mathematics in type theory. To that end, we adopt the cubical type theory of Angiuli, Favonia and Harper, a framework that can be seen as constructive refinement of the ideas of homotopy type theory. (shrink)

ABSTRACT Theories of sets such as Zermelo Fraenkel set theory are usually presented as the combination of two distinct kinds of principles: logical and set-theoretic principles. The set-theoretic principles are imposed ‘on top’ of first-order logic. This is in agreement with a traditional view of logic as universally applicable and topic neutral. Such a view of logic has been rejected by the intuitionists, on the ground that quantification over infinite domains requires the use of intuitionistic rather than classical logic. In (...) the following, I consider constructive set theories, which use intuitionistic rather than classical logic, and argue that they manifest a distinctive interdependence or an entanglement between sets and logic. In fact, Martin-Löf type theory identifies fundamental logical and set-theoretic notions. Remarkably, one of the motivations for this identification is the thought that classical quantification over infinite domains is problematic, while intuitionistic quantification is not. The approach to quantification adopted in Martin-Löf’s type theory is subtly interconnected with its predicativity. I conclude by recalling key aspects of an approach to predicativity inspired by Poincaré, which focuses on the issue of correct quantification over infinite domains and relate it back to Martin-Löf type theory. (shrink)

Constructive set theory a' la Myhill-Aczel has been extended in (Cantini and Crosilla 2008, Cantini and Crosilla 2010) to incorporate a notion of (partial, non--extensional) operation. Constructive operational set theory is a constructive and predicative analogue of Beeson's Inuitionistic set theory with rules and of Feferman's Operational set theory (Beeson 1988, Feferman 2006, Jaeger 2007, Jaeger 2009, Jaeger 1009b). This paper is concerned with an extension of constructive operational set theory (Cantini and Crosilla 2010) by a uniform operation of Transitive (...) Closure, \tau. Given a set a, \tau produces its transitive closure \tau a. We show that the theory ESTE of (Cantini and Crosilla 2010) augmented by \tau is still conservative over Peano Arithmetic. (shrink)

The eta rule for a set A says that an arbitrary element of A is judgementally identical to an element of constructor form. Eta rules are not part of what may be called canonical Martin-Löf type theory. They are, however, justified by the meaning explanations, and a higher-order eta rule is part of that type theory. The main aim of this paper is to clarify this somewhat puzzling situation. It will be argued that lower-order eta rules do not, whereas the (...) higher-order eta rule does, accord with the understanding of judgemental identity as definitional identity. A subsidiary aim is to clarify precisely what an eta rule is. This will involve showing how such rules relate to various other notions of type theory, proof theory, and category theory. (shrink)

Cantor’s proof that the reals are uncountable forms a central pillar in the edifices of higher order recursion theory and set theory. It also has important applications in model theory, and in the foundations of topology and analysis. Due partly to these factors, and to the simplicity and elegance of the proof, it has come to be accepted as part of the ABC’s of mathematics. But even if as an Archimedean point it supports tomes of mathematical theory, there is a (...) question that demands clarification: What, exactly, does Cantor’s proof show? One of few places where this question is addressed is Appendix II of Wittgenstein’s Remarks on the Foundations of Mathematics. This essay is devoted to clarifying Wittgenstein’s remarks in that section. (shrink)

Straightforwardly and strictly intuitionistic inferences show that the Brouwer– Heyting–Kolmogorov interpretation, in the presence of a formulation of the recognition principle, entails the validity of the Law of Testability: that the form ¬ f V ¬¬ f is valid. Therefore, the BHK and recognition, as described here, are inconsistent with the axioms both of intuitionistic mathematics and of Markovian constructivism. This finding also implies that, if the BHK and recognition are suitably formulated, then Brouwer’s original weak counterexample reasoning was fallacious. (...) The results of the present article extend and refine those of McCarty, C.. Antirealism and Constructivism: Brouwer’s Weak Counterexamples. The Review of Symbolic Logic. First View. Cambridge University Press. (shrink)

Strictly intuitionistic inferences are employed to demonstrate that three conditions—the existence of Brouwerian weak counterexamples to _Test_, the recognition condition, and the _BHK_ interpretation of the logical signs—are together inconsistent. Therefore, if the logical signs in mathematical statements governed by the recognition condition are constructive in that they satisfy the clauses of the _BHK_, then every relevant instance of the classical principle _Test_ is true intuitionistically, and the antirealistic critique of conventional logic, once thought to yield such weak counterexamples, is (...) seen, in this instance, to fail. (shrink)

This is a survey of Gödel's perennial preoccupations with the limits of finitism, its relations to constructivity, and the significance of his incompleteness theorems for Hilbert's program, using his published and unpublished articles and lectures as well as the correspondence between Bernays and Gödel on these matters. There is also an important subtext, namely the shadow of Hilbert that loomed over Gödel from the beginning to the end.

We approach the philosophy of mathematics via a discussion of the differences between classical mathematics and constructive mathematics, arguing that each is a valid activity within its own context.

We prove the Banach-Steinhaus theorem for distributions on the space [MATHEMATICAL SCRIPT CAPITAL D] within Bishop's constructive mathematics. To this end, we investigate the constructive sequential completion equation image of [MATHEMATICAL SCRIPT CAPITAL D].

In his paper "Finitism", W.W. Tait maintains that the chief difficulty for everyone who wishes to understand Hilbert's conception of finitist mathematics is this: to specify the sense of the provability of general statements about the natural numbers without presupposing infinite totalities. Tait further argues that all finitist reasoning is essentially primitive recursive. In this paper, we attempt to show that his thesis "The finitist functions are precisely the primitive recursive functions" is disputable and that another, likewise defended by him, (...) is untenable. The second thesis is that the finitist theorems are precisely the universal closures of the equations that can be proved in PRA. /// En su articulo "Finitism", W.W. Tait sostiene que la dificultad principal para quien quiere comprender la concepción hilbertiana de la matemática finitista es ésta: especificar el sentido de la demostrabilidad de enunciados generales sobre los números naturales sin presuponer totalidades infinitas. Además, Tait argumenta que todo razonamiento finitista es esencialmente primitivo recursivo. En este artículo tratamos de mostrar que su tesis "Las funciones finitistas son precisamente las funciones primitivas recursivas" es discutible y que otra, también defendida por él, resulta insostenible. La segunda tesis es que los teoremas finitistas son precisamente las clausuras universales de las ecuaciones que pueden demostrarse en PRA. (shrink)

In this paper it is argued, firstly, that Kuhnian revolutions in mathematics are logically possible, in the sense of not being inconsistent with the nature of mathematics; and, secondly, that Kuhnian revolutions are actually possible, in the sense that a Kuhnian paradigm for mathematics can be exhibited which would, if accepted by the mathematical community, produce a full Kuhnian revolution. These two arguments depend on first proving that a shift from a classical conception of mathematics to an intuitionist conception would (...) be incommensurable, that is, that some classical statements, possessing meanings which cannot be preserved in the intuitionist language, would become unintelligible. The vague but intriguing thesis is then tentatively advanced that Kuhnian revolutions are even historically possible, in the sense that only what we might call ‘accidental’ historical factors may have prevented mathematics from undergoing just such a Kuhnian revolution in the early years of the twentieth century. (shrink)

This is a long-awaited new edition of one of the best known Oxford Logic Guides. The book gives an introduction to intuitionistic mathematics, leading the reader gently through the fundamental mathematical and philosophical concepts. The treatment of various topics, for example Brouwer's proof of the Bar Theorem, valuation systems, and the completeness of intuitionistic first-order logic, have been completely revised.

ABSTRACTAccording to a famous argument by Dummett, the concept of set is indefinitely extensible, and the logic appropriate for reasoning about the instances of any such concept is intuitionistic, not classical. But Dummett's argument is widely regarded as obscure. This note explains how the final chapter of Rumfitt's important new book advances our understanding of Dummett's argument, but it also points out some problems and unanswered questions. Finally, Rumfitt's reconstruction of Dummett's argument is contrasted with my own preferred alternative.

The dissertation provides an analysis and elaboration of Michael Dummett's proof-theoretic notions of validity. Dummett's notions of validity are contrasted with standard proof-theoretic notions and formally evaluated with respect to their adequacy to propositional intuitionistic logic.

It is often said that a correct logical system should have no counterexample to its logical rules and the system must be revised if its rules have a counterexample. If a logical system (or theory) has a counterexample to its logical rules, do we have to revise the system? In this paper, focussing on the role of counterexamples to logical rules, we deal with the question. -/- We investigate two mutually exclusive theories of arithmetic - intuitionistic and paraconsistent theories. The (...) paraconsistent theory provides a (strong) counterexample to Ex Contradiction Quodlibet (ECQ). On the other hand, the intuitionistic theory gives a (weak) counterexample to the Double Negation Elimination (DNE) of the paraconsistent theory. If any counterexample undermines the legitimate use of logical rules, both theories must be revised. -/- After we investigate a paraconsistent counterexample to ECQ and the intuitionist’s answer against it, we arrive at the unwelcome conclusion that ECQ has both a justification and a counterexample. Moreover, we argue that if a logical rule were abolished whenever it has a counterexample, a promising conclusion would be logical nihilism which is the view that there is no valid logical inference, and so a correct logical system does not exist. Provided that the logical revisionist is not a logical nihilist, we claim that not every counterexample is the ground for logical revision. While logical rules of a given system have a justification, the existence of a counterexample loses its role for logical revision unless the rules and the counterexample share the same structure. (shrink)

Two mereological theories are presented based on a primitive apartness relation along with binary relations of mereological excess and weak excess, respectively. It is shown that both theories are acceptable from the standpoint of constructive reasoning while remaining faithful to the spirit of classical mereology. The two theories are then compared and assessed with regard to their extensional import.

Gentzen's three consistency proofs for elementary number theory have a common aim that originates from Hilbert's Program, namely, the aim to justify the application of classical reasoning to quantified propositions in elementary number theory. In addition to this common aim, Gentzen gave a “finitist” interpretation to every number-theoretic proposition with his 1935 and 1936 consistency proofs. In the present paper, we investigate the relationship of this interpretation with intuitionism in terms of the debate between the Hilbert School and the Brouwer (...) School over the significance of consistency proofs. First, we argue that the interpretation had the role of responding to a Brouwer-style objection against the significance of consistency proofs. Second, we propose a way of understanding Gentzen's response to this objection from an intuitionist perspective. (shrink)

: Bereft of the illusion of an epistemic vantage point external to science, what should be our commitment towards the categories, concepts and terms of that very science? Should we, despaired of the possibility to found these concepts on rock bottom, adopt empiricist skepticism? Or perhaps the inexistence of external foundations implies, rather, immunity for scientific ontology from epistemological criticism? Philosophy's "realism debate" died out without providing a satisfactory answer to the dilemma, which was taken over by the neighboring disciplines. (...) The "symmetry principle" of the "StrongProgramme" for the sociology of science-the requirement that truth and error receive the same kind of causal explanations-offered one bold metaphysical answer, under the guise of a methodological decree. Recently, however, it has been argued that this solution is not bold enough, that the social constructivists replaced the naïve presumption of an independent nature which adjudicates our beliefs with a mirror-image presumption of a sui generis society which furnishes these beliefs autonomously. The proper metaphysics for a foundationless epistemology, argues Bruno Latour, is one which grants nature and society, object and subject, equal roles in the success and failure of science and technology; one in which history of society merges with a history of things-in-themselves. The paper analyzes the philosophical and methodological motivations and ramifications of this extraordinary suggestion. (shrink)

The notion of potential infinity dominated in mathematical thinking about infinity from Aristotle until Cantor. The coherence and philosophical importance of the notion are defended. Particular attention is paid to the question of whether potential infinity is compatible with classical logic or requires a weaker logic, perhaps intuitionistic.

This paper is about Poincaré’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincaré, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincaré’s core commitment in the philosophy of geometry: the view that geometry is the study (...) of groups of operations. In place of the established view I offer a revised view, according to which Poincaré held that axioms in geometry are in fact assertions about invariants of groups. Groups, as forms of the understanding, are prior in conception to the objects of geometry and afford the proper definition of those objects, according to Poincaré. Poincaré’s view therefore contrasts sharply with Kant’s foundation of geometry in a unique form of sensibility. According to my interpretation, axioms are not definitions in disguise because they themselves implicitly define their terms, but rather because they disguise the definitions which imply them. (shrink)

The recent discovery of an interpretation of constructive type theory into abstract homotopy theory suggests a new approach to the foundations of mathematics with intrinsic geometric content and a computational implementation. Voevodsky has proposed such a program, including a new axiom with both geometric and logical significance: the Univalence Axiom. It captures the familiar aspect of informal mathematical practice according to which one can identify isomorphic objects. While it is incompatible with conventional foundations, it is a powerful addition to homotopy (...) type theory. It also gives the new system of foundations a distinctly structural character. (shrink)

I introduce a formalization of probability which takes the concept of 'evidence' as primitive. In parallel to the intuitionistic conception of truth, in which 'proof' is primitive and an assertion A is judged to be true just in case there is a proof witnessing it, here 'evidence' is primitive and A is judged to be probable just in case there is evidence supporting it. I formalize this outlook by representing propositions as types in Martin-Lof type theory (MLTT) and defining a (...) 'probability type' on top of the existing machinery of MLTT, whose inhabitants represent pieces of evidence in favor of a proposition. One upshot of this approach is the potential for a mathematical formalism which treats 'conjectures' as mathematical objects in their own right. Other intuitive properties of evidence occur as theorems in this formalism. (shrink)

On the basis of Martin-Löf’s meaning explanations for his type theory a detailed justification is offered of the rule of identity elimination. Brief discussions are thereafter offered of how the univalence axiom fares with respect to these meaning explanations and of some recent work on identity in type theory by Ladyman and Presnell.

In the early 1920s, the German mathematician David Hilbert (1862–1943) put forward a new proposal for the foundation of classical mathematics which has come to be known as Hilbert's Program. It calls for a formalization of all of mathematics in axiomatic form, together with a proof that this axiomatization of mathematics is consistent. The consistency proof itself was to be carried out using only what Hilbert called “finitary” methods. The special epistemological character of finitary reasoning then yields the required justification (...) of classical mathematics. Although Hilbert proposed his program in this form only in 1921, various facets of it are rooted in foundational work of his going back until around 1900, when he first pointed out the necessity of giving a direct consistency proof of analysis. Work on the program progressed significantly in the 1920s with contributions from logicians such as Paul Bernays, Wilhelm Ackermann, John von Neumann, and Jacques Herbrand. It was also a great influence on Kurt Gödel, whose work on the incompleteness theorems were motivated by Hilbert's Program. Gödel's work is generally taken to show that Hilbert's Program cannot be carried out. It has nevertheless continued to be an influential position in the philosophy of mathematics, and, starting with the work of Gerhard Gentzen in the 1930s, work on so-called Relativized Hilbert Programs have been central to the development of proof theory. (shrink)

An important open problem in Reverse Mathematics is the reduction of the first-order strength of the base theory from IΣ1IΣ1 to IΔ0+expIΔ0+exp. The system ERNA, a version of Nonstandard Analysis based on the system IΔ0+expIΔ0+exp, provides a partial solution to this problem. Indeed, weak Königʼs lemma and many of its equivalent formulations from Reverse Mathematics can be ‘pushed down’ into ERNA, while preserving the equivalences, but at the price of replacing equality with ‘≈’, i.e. infinitesimal proximity . The logical principle (...) corresponding to weak Königʼs lemma is the universal transfer principle from Nonstandard Analysis. Here, we consider the intermediate and mean value theorem and their uniform generalizations. We show that ERNAʼs Reverse Mathematics mirrors the situation in classical Reverse Mathematics. This further supports our claim from Sanders [19] that the Reverse Mathematics of ERNA plus universal transfer is a copy up to infinitesimals of that of WKL0. We discuss some of the philosophical implications of our results. (shrink)

The main result of this paper is a weak constructive version of the uniform continuity theorem for pointwise continuous, real-valued functions on a convex subset of a normed linear space. Recursive examples are given to show that the hypotheses of this theorem are necessary. The remainder of the paper discusses conditions which ensure that a sequentially continuous function is continuous. MSC: 03F60, 26E40, 46S30.