Intuitionism and Constructivism

I discuss problems with Martin-Löf's distinction between analytic and synthetic judgments in constructive type theory and propose a revision of his views. I maintain that a judgment is analytic when its correctness follows exclusively from the evaluation of the expressions occurring in it. I argue that Martin-Löf's claim that all judgments of the forms a : A and a = b : A are analytic is unfounded. As I shall show, when A evaluates to a dependent function type (x : (...) B) → C, all judgments of these forms fail to be analytic and therefore end up as synthetic. Going beyond the scope of Martin-Löf's original distinction, I also argue that all hypothetical judgments are synthetic and show how the analytic-synthetic distinction reworked here is capable of accommodating judgments of the forms A type and A = B type as well. Finally, I consider and reject an alternative account of analyticity as decidability and assess Martin-Löf's position on the analytic grounding of synthetic judgments. (shrink)

Modal logic has been used to analyze potential infinity and potentialism more generally. However, the standard analysis breaks down in cases of divergent possibilities, where there are two or more possibilities that can be individually realized but which are jointly incompatible. This paper has three aims. First, using the intuitionistic theory of choice sequences, we motivate the need for a modal analysis of divergent potentialism and explain the challenges this involves. Then, using Beth–Kripke semantics for intuitionistic logic, we overcome those (...) challenges. Finally, we apply our modal analysis of divergent potentialism to make choice sequences comprehensible in classical terms. (shrink)

We provide an application of a sequent calculus framework to the formalization of definite descriptions. It is a continuation of research undertaken in [20, 22]. In the present paper a so-called free description theory is examined in the context of different kinds of free logic, including systems applied in computer science and constructive mathematics for dealing with partial functions. It is shown that the same theory in different logics may be formalised by means of different rules and gives results of (...) varying strength. For all presented calculi a constructive cut elimination is provided. (shrink)

Brouwer’s intuitionism was a far-reaching attempt to reform the foundations of mathematics. While the mathematical community was reluctant to accept Brouwer’s work, its response to later-developed brands of intuitionism, such as those presented by Hermann Weyl and Arend Heyting, was different. The paper accounts for this difference by analyzing the intuitionistic versions of Brouwer, Weyl, and Heyting in light of a two-tiered model of the body and image of mathematical knowledge. Such a perspective provides a richer account of each story (...) and points to a possible connection between the community’s reaction and the changes each mathematician had proposed. (shrink)

I argue against the interpretation of propositions as intentions and proof-objects as fulfillments proposed by Heyting and defended by Tieszen and van Atten. The idea is already a frequent target of criticisms regarding the incompatibility of Brouwer’s and Husserl’s positions, mainly by Rosado Haddock and Hill. I raise a stronger objection in this paper. My claim is that even if we grant that the incompatibility can be properly dealt with, as van Atten believes it can, two fundamental issues indicate that (...) the interpretation is unsustainable regardless: (1) it is hard to determine, without appealing to propositional intentions on pain of circularity, what intention a proof-object should be understood as a fulfillment of; (2) due to a difficult fulfillment dilemma, it is unclear, at best, what the object of an intention corresponding to a proposition is. (shrink)

The mathematician G.F.C. Griss is known for his program of negationless intuitionistic mathematics. Although Griss’s rejection of negation is regarded as characteristic of his philosophy, this is a consequence of an executability requirement that mental constructions presuppose agents’ executing corresponding mental activity. Restoring Griss’s executability requirement to a central role permits a more subtle characterization of the rejection of negation, according to which D. Nelson’s strong constructible negation is compatible with Griss’s principles. This exposes a ‘holographic’ theory of negation in (...) negationless mathematics, in which a full theory of negation is ‘flattened’ in a putatively negationless setting. (shrink)

This book publishes for the first time notes from two notebooks of Gödel which formed the basis of a course on intuitionism Gödel delivered at Princeton in the.

Brouwer’s intuitionistic program was an intriguing attempt to reform the foundations of mathematics that eventually did not prevail. The current paper offers a new perspective on the scientific community’s lack of reception to Brouwer’s intuitionism by considering it in light of Michael Friedman’s model of parallel transitions in philosophy and science, specifically focusing on Friedman’s story of Einstein’s theory of relativity. Such a juxtaposition raises onto the surface the differences between Brouwer’s and Einstein’s stories and suggests that contrary to Einstein’s (...) story, the philosophical roots of Brouwer’s intuitionism cannot be traced to any previously established philosophical traditions. The paper concludes by showing how the intuitionistic inclinations of Hermann Weyl and Abraham Fraenkel serve as telling cases of how individuals are involved in setting in motion, adopting, and resisting framework transitions during periods of disagreement within a discipline. (shrink)

In this paper we analyse in the framework of constructive mathematics (BISH) the validity of Farkas' lemma and related propositions, namely the Fredholm alternative for solvability of systems of linear equations, optimality criteria in linear programming, Stiemke's lemma and the Superhedging Duality from mathematical finance, and von Neumann's minimax theorem with application to constructive game theory.

In this article I present a disagreement between classical and constructive approaches to predicativity regarding the predicative status of so-called generalised inductive definitions. I begin by offering some motivation for an enquiry in the predicative foundations of constructive mathematics, by looking at contemporary work at the intersection between mathematics and computer science. I then review the background notions and spell out the above-mentioned disagreement between classical and constructive approaches to predicativity. Finally, I look at possible ways of defending the constructive (...) predicativity of inductive definitions. (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)

We provide a framework for understanding agnosticism. The framework accounts for the varieties of agnosticism while vindicating the unity of the phenomenon. This combination of unity and plurality is achieved by taking the varieties of agnosticism to be represented by several agnostic stances, all of which share a common core provided by what we call the minimal agnostic attitude. We illustrate the fruitfulness of the framework by showing how it can be applied to several philosophical debates. In particular, several philosophical (...) positions can be aptly conceived of as instances of agnosticism whilst retaining their differences and distinguishing features. (shrink)

This book is dedicated to V.A. Yankov’s seminal contributions to the theory of propositional logics. His papers, published in the 1960s, are highly cited even today. The Yankov characteristic formulas have become a very useful tool in propositional, modal and algebraic logic. The papers contributed to this book provide the new results on different generalizations and applications of characteristic formulas in propositional, modal and algebraic logics. In particular, an exposition of Yankov’s results and their applications in algebraic logic, the theory (...) of admissible rules and refutation systems is included in the book. In addition, the reader can find the studies on splitting and join-splitting in intermediate propositional logics that are based on Yankov-type formulas which are closely related to canonical formulas, and the study of properties of predicate extensions of non-classical propositional logics. The book also contains an exposition of Yankov’s revolutionary approach to constructive proof theory. The editors also include Yankov’s contributions to history and philosophy of mathematics and foundations of mathematics, as well as an examination of his original interpretation of history of Greek philosophy and mathematics. (shrink)

We investigate which part of Brouwer’s Intuitionistic Mathematics is finitistically justifiable or guaranteed in Hilbert’s Finitism, in the same way as similar investigations on Classical Mathematics already done quite extensively in proof theory and reverse mathematics. While we already knew a contrast from the classical situation concerning the continuity principle, more contrasts turn out: we show that several principles are finitistically justifiable or guaranteed which are classically not. Among them are: fan theorem for decidable fans but arbitrary bars; continuity principle (...) and the axiom of choice both for arbitrary formulae; and $\Sigma _2$ induction and dependent choice. We also show that Markov’s principle MP does not change this situation; that neither does lesser limited principle of omniscience LLPO ; but that limited principle of omniscience LPO makes the situation completely classical. (shrink)

The paper examines Yankov’s contribution to the history of mathematical logic and the foundations of mathematics. It concerns the public communication of Markov’s critical attitude towards Brouwer’s intuitionistic mathematics from the point of view of his constructive mathematics and the commentary on A.S. Esenin-Vol’pin program of ultra-intuitionistic foundations of mathematics.

The paper explores Hermann Weyl’s turn to intuitionism through a philosophical prism of normative framework transitions. It focuses on three central themes that occupied Weyl’s thought: the notion of the continuum, logical existence, and the necessity of intuitionism, constructivism, and formalism to adequately address the foundational crisis of mathematics. The analysis of these themes reveals Weyl’s continuous endeavor to deal with such fundamental problems and suggests a view that provides a different perspective concerning Weyl’s wavering foundational positions. Building on a (...) philosophical model of scientific framework transitions and the special role that normative indecision or ambivalence plays in the process, the paper examines Weyl’s motives for considering such a radical shift in the first place. It concludes by showing that Weyl’s shifting stances should be regarded as symptoms of a deep, convoluted intrapersonal process of self-deliberation induced by exposure to external criticism. (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)

What is the relationship between the world and logic, between intuition and language, between objects and their quantitative determinations? Rationalists, on the one hand, hold that the world is structured in a rational way. Representationalists, on the other hand, assume that language, logic, and mathematics are only the means to order and describe the intuitively given world. In World and Logic, Jens Lemanski takes up three surprising arguments from Arthur Schopenhauer’s hitherto undiscovered Berlin Lectures, which concern the philosophy of language, (...) logic, and mathematics. Based on these arguments, Lemanski develops a new position entitled ‘rational representationalism’: the world is always structured by human beings according to linguistic, logical, and mathematical principles, but the basic vocabulary of these structural descriptions already contains metaphors taken from the world around us. (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.

An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details of proofs worked out and examples and exercises to aid the reader's understanding. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The first half covers topics in structural proof theory, including the Gödel-Gentzen translation of classical into intuitionistic logic, natural deduction and the normalization theorems, the sequent calculus, including cut-elimination and mid-sequent theorems, and various applications of these (...) results. The second half examines ordinal proof theory, specifically Gentzen's consistency proof for first-order Peano Arithmetic. The theory of ordinal notations and other elements of ordinal theory are developed from scratch, and no knowledge of set theory is presumed. The proof methods needed to establish proof-theoretic results, especially proof by induction, are introduced in stages throughout the text. Mancosu, Galvan, and Zach's introduction will provide a solid foundation for those looking to understand this central area of mathematical logic and the philosophy of mathematics. (shrink)

In this paper, we give an introduction to intuitionistic logic and a defense of it from certain formal logical critiques. Intuitionism is the thesis that mathematical objects are mental constructions produced by the faculty of a priori intuition of time. The truth of a mathematical proposition, then, consists in our knowing how to construct in intuition a corresponding state of affairs. This understanding of mathematical truth leads to a rejection of the principle, valid in classical logic, that a proposition is (...) either true or false (put symbolically, a ∨ ~a). The rejection of this principle leads to a different system of formal logic. This logic has been critiqued as being three-valued in such a way that it is self-contradictory. That this is a misunderstanding of intuitionistic logic can be proven formally on the basis of Heyting's axioms and rules of inference for intuitionistic logic. A proposition that is neither true nor false does not, on an intuitionist view, have some third truth value, but lacks any truth value whatsoever. In the process of proving that this is the case we will also prove several other theorems which will give us some insight into the formal similarities and differences between intuitionistic and classical mathematics, specifically with regard to the validity of different proof techniques. (shrink)

In this paper, I revisit Frege's theory of sense and reference in the constructive setting of the meaning explanations of type theory, extending and sharpening a program–value analysis of sense and reference proposed by Martin-Löf building on previous work of Dummett. I propose a computational identity criterion for senses and argue that it validates what I see as the most plausible interpretation of Frege's equipollence principle for both sentences and singular terms. Before doing so, I examine Frege's implementation of his (...) theory of sense and reference in the logical framework of Grundgesetze, his doctrine of truth values, and views on sameness of sense as equipollence of assertions. (shrink)

The aim of this paper is to establish a phenomenological mathematical intuitionism that is based on fundamental phenomenological-epistemological principles. According to this intuitionism, mathematical intuitions are sui generis mental states, namely experiences that exhibit a distinctive phenomenal character. The focus is on two questions: what does it mean to undergo a mathematical intuition and what role do mathematical intuitions play in mathematical reasoning? While I crucially draw on Husserlian principles and adopt ideas we find in phenomenologically minded mathematicians such as (...) Hermann Weyl and Kurt Gödel, the overall objective is systematic in nature: to offer a plausible approach towards mathematics. (shrink)

This book publishes the previously unpublished Wittgenstein-Skinner Archive held in Trinity College Cambridge Wren Library. The principal Editor is Arthur Gibson, joined by the Editor Niamh O'Mahony in the editing project. The manuscripts were transcribed by Arthur Gibson, checked and edited by Niamh O'Mahony and Arthur Gibson, with additional assistance from Kelsey Gibson. The Chapters that reproduce the Archive, including the Preface, and Part I (chapters 1 and 2) are authored by Arthur Gibson. Arthur Gibson and Niamh O'Mahony jointly edited (...) the Appendices and variously authored parts of the Appendices' introductions and notes. The Foreword was contributed by Professor Brian McGuinness. (shrink)

According to L.E.J. Brouwer, there is room for non-definable real numbers within the intuitionistic ontology of mental constructions. That room is allegedly provided by freely proceeding choice sequences, i.e., sequences created by repeated free choices of elements by a creating subject in a potentially infinite process. Through an analysis of the constitution of choice sequences, this paper argues against Brouwer’s claim.

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)

This edited work presents contemporary mathematical practice in the foundational mathematical theories, in particular set theory and the univalent foundations. It shares the work of significant scholars across the disciplines of mathematics, philosophy and computer science. Readers will discover systematic thought on criteria for a suitable foundation in mathematics and philosophical reflections around the mathematical perspectives. The first two sections focus on the two most prominent candidate theories for a foundation of mathematics. Readers may trace current research in set theory, (...) which has widely been assumed to serve as a framework for foundational issues, as well as new material elaborating on the univalent foundations, considering an approach based on homotopy type theory (HoTT). The further sections then build on this and are centred on philosophical questions connected to the foundations of mathematics. Here, the authors contribute to discussions on foundational criteria with more general thoughts on the foundations of mathematics which are not connected to particular theories. This book shares the work of some of the most important scholars in the fields of set theory (S. Friedman), non-classical logic (G. Priest) and the philosophy of mathematics (P. Maddy). The reader will become aware of the advantages of each theory and objections to it as a foundation, following the latest and best work across the disciplines and it is therefore a valuable read for anyone working on the foundations of mathematics or in the philosophy of mathematics. (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)

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.

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)

This paper sketches a way of supplementing classical mathematics with a motivation for a Brouwerian theory of free choice sequences. The idea is that time is unending, i.e. that one can never come to an end of it, but also indeterminate, so that in a branching time model only one branch represents the ‘actual’ one. The branching can be random or subject to various restrictions imposed by the creating subject. The fact that the underlying mathematics is classical makes such perhaps (...) delicate issues as the fan theorem no longer problematic. On this model, only intuitionistic logic applies to the Brouwerian free choice sequences, and there it applies not because of any skepticism about classical mathematics, but because there is no ‘end of time’ from the standpoint of which everything about the sequences can be decided. (shrink)

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.

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)

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)

In this article we examine the natural interpretation of a ramified type hierarchy into Martin-Löf type theory with an infinite sequence of universes. It is shown that under this predicative interpretation some useful special cases of Russell’s reducibility axiom are valid, namely functional reducibility. This is sufficient to make the type hierarchy usable for development of constructive mathematical analysis in the style of Bishop. We present a ramified type theory suitable for this purpose. One may regard the results of this (...) article as an alternative solution to the problem of the proliferation of levels of real numbers in Russell’s theory, which avoids impredicativity, but instead imposes constructive logic. The intuitionistic ramified type theory introduced here also suggests that there is a natural associated notion of predicative elementary topos. (shrink)

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)

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.

In this paper it is argued that the understanding of Brouwer as replacing truth conditions with assertability or proof conditions, in particular as codified in the so-called Brouwer-Heyting-Kolmogorov Interpretation, is misleading and conflates a weak and a strong notion of truth that have to be kept apart to understand Brouwer properly: truth-as-anticipation and truth- in-content. These notions are explained, exegetical documentation provided, and semi-formal recursive definitions are given.

The derivative is a basic concept of differential calculus. However, if we calculate the derivative as change in distance over change in time, the result at any instant is 0/0, which seems meaningless. Hence, Newton and Leibniz used the limit to determine the derivative. Their method is valid in practice, but it is not easy to intuitively accept. Thus, this article describes the novel method of differential calculus based on the double contradiction, which is easier to accept intuitively. Next, the (...) geometrical meaning of the double contradiction is considered as follows. A tangent at a point on a convex curve is iterated. Then, the slope of the tangent at the point is sandwiched by two kinds of lines. The first kind of line crosses the curve at the original point and a point to the right of it. The second kind of line crosses the curve at the original point and a point to the left of it. Then, the double contradiction can be applied, and the slope of the tangent is determined as a single value. Finally, the meaning of this method for the foundation of mathematics is considered. We reflect on Dehaene’s notion that the foundation of mathematics is based on the intuitions, which evolve independently. Hence, there may be gaps between intuitions. In fact, the Ancient Greeks identified inconsistency between arithmetic and geometry. However, Eudoxus developed the theory of proportion, which is equivalent to the Dedekind Cut. This allows the iteration of an irrational number by rational numbers as precisely as desired. Simultaneously, we can define the irrational number by the double contradiction, although its existence is not guaranteed. Further, an area of a curved figure is iterated and defined by rectilinear figures using the double contradiction. (shrink)

Revision sequences are a kind of transfinite sequences which were introduced by Herzberger and Gupta in 1982 as the main mathematical tool for developing their respective revision theories of truth. We generalise revision sequences to the notion of cofinally invariant sequences, showing that several known facts about Herzberger’s and Gupta’s theories also hold for this more abstract kind of sequences and providing new and more informative proofs of the old results.

We show that, by choosing definitions carefully, a version of Frege's theorem can be proved in intuitionistic logic.