Church's thesis

There seems to be a view that intuitionists not only take the Axiom of Choice (AC) to be true, but also believe it a consequence of their fundamental posits. Widespread or not, this view is largely mistaken. This article offers a brief, yet comprehensive, overview of the status of AC in various intuitionistic and constructivist systems. The survey makes it clear that the Axiom of Choice fails to be a theorem in most contexts and is even outright false in some important contexts. Of the systems surveyed, only intensional type theory renders AC a theorem, but the extent of AC in that theory does not include, for instance, real analysis. Only a small amount of extensionality is required in order for the obvious proof an intuitionist might offer for AC to break down.

Brouwer's papers after 1945 are characterized by a technique, known as the method of the creating subject. It has been supposed that the method was radically new in his work, since Brouwer seems to introduce an idealized mathematician into his mathematical practice. A newly opened source, the unpublished text of a lecture of Brouwer from 1934, fully supports the conclusions of our analysis that:-There is no idealized mathematician involved in the method.-The method was not new at all.-Brouwer uses an incomplete sequence, also known as choice sequence, in this method, which is special. The method does not take its place in the standard works on choice sequences.

Gödel regarded the Dialectica interpretation as giving constructive content to intuitionism, which otherwise failed to meet reasonable conditions of constructivity. He founded his theory of primitive recursive functions, in which the interpretation is given, on the concept of computable function of finite type. I will 1) criticize this foundation, 2) propose a quite different one, and 3) note that essentially the latter foundation also underlies the Curry-Howard type theory and hence Heyting's intuitionistic conception of logic. Thus the Dialectica interpretation (in so far as its aim was to give constructive content to intuitionism) is superfluous.

Par des suites de choix, nous comprenons des suites qui ne sont pas déterminées complètement par une loi arithmétique. Elles sont des objets caractéristiques de l'intuitionnisme de Brouwer. Nous prétendons qu'à partir de 1927, l'utilisation par Brouwer de suites de choix particulières n'est pas reconnu comme tel. Nous prétendons que l'utilisation de ces suites dans la méthode du sujet créatif, après la seconde guerre mondiale, n'a pas à être mis en relation avec l'utilisation de celles-ci dans les années vingt et qu'elles sont mal interprétées. Nous montrons où se trouvent ces suites de choix dans l'oeuvre de Brouwer et comment elles doivent être traitées.

Subject of this paper is the method of the creative subject, as used by Brouwer after 1945. We shall interpret it in such a way that, in our opinion, it arises naturally from intuitionistically accepted concepts. The reconstruction will be carried out in a Kripke model. 1NTRODUCTlON In a number of papers published after 1945 Brouwer introduced a new device. In constructing counterexamples for classical theorems he seemed to refer explicitly to the activity of an idealized mathematician. This so-called method of the creative subject is controversial, also within intuitionism. The first attempt to formalize the new method was done by Kreisel ([Kr 671); it was further elaborated by Myhill ([My 681) and Troelstra ([Tr 691). The resulting theory of the creative subject, TCS for short, is strong enough to obtain Brouwer's result. But the interpretation of its terms is somewhat vague, it does not fully explain Brouwer's argument and there is a paradox, discovered by Troelstra, which has not been resolved satisfactorily. These questions play a minor role in the work of two more recent authors on this subject, Martin0 ([Ma 821, [Ma 851) and Posy ([PO 761, [PO 771). Their main interest is the application of the TCS in other parts of Brouwer's work. We want to show that Brouwer's method can be interpreted in such a way that it arises naturally from intuitionistically accepted concepts. This reconstruction suggests an interpretation of the basic term I-, of the TCS, which

Brouwer's papers after 1945 are characterized by a technique, known as the method of the creating subject. It has been supposed that the method was radically new in his work, since Brouwer seems to introduce an idealized mathematician into his mathematical practice. A newly opened source, the unpublished text of a lecture of Brouwer from 1934, fully supports the conclusions of our analysis that:-There is no idealized mathematician involved in the method.-The method was not new at all.-Brouwer uses an incomplete sequence, also known as choice sequence, in this method, which is special. The method does not take its place in the standard works on choice sequences.

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The series Boston Studies in the Philosophy and History of Science was conceived in the broadest framework of interdisciplinary and international concerns. Natural scientists, mathematicians, social scientists and philosophers have contributed to the series, as have historians and sociologists of science, linguists, psychologists, physicians, and literary critics. The series has been able to include works by authors from many other countries around the world. The editors believe that the history and philosophy of science should itself be scientific, self-consciously critical, humane as well as rational, sceptical and undogmatic while also receptive to discussion of first principles. One of the aims of Boston Studies, therefore, is to develop collaboration among scientists, historians and philosophers. Boston Studies in the Philosophy and History of Science looks into and reflects on interactions between epistemological and historical dimensions in an effort to understand the scientific enterprise from every viewpoint.

Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of which number is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of notation. The purpose of this article is to explore the relationship between the notations acceptable for computation, the usual idealizations involved in theories of computability, flowing from Alan Turing’s monumental work, and de re propositional attitudes toward numbers and other mathematical objects.

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.

In the philosophy of mathematics one speaks about Formalism, Logicism, Platonism and Intuitionism. Actually one should add also Calculism. These foundational views can be given a clear technological meaning in the context of Computer Mathematics, that has as aim to represent and manipulate arbitrary mathematical notions on a computer. We argue that most philosophical views overemphasize a particular aspect of the mathematical endeavor.

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.

The series Boston Studies in the Philosophy and History of Science was conceived in the broadest framework of interdisciplinary and international concerns. Natural scientists, mathematicians, social scientists and philosophers have contributed to the series, as have historians and sociologists of science, linguists, psychologists, physicians, and literary critics. The series has been able to include works by authors from many other countries around the world. The editors believe that the history and philosophy of science should itself be scientific, self-consciously critical, humane as well as rational, sceptical and undogmatic while also receptive to discussion of first principles. One of the aims of Boston Studies, therefore, is to develop collaboration among scientists, historians and philosophers. Boston Studies in the Philosophy and History of Science looks into and reflects on interactions between epistemological and historical dimensions in an effort to understand the scientific enterprise from every viewpoint.

Define a system to be a collection of objects with certain relations. A basketball defence is a system of people under various positioning and defence-role relations; an extended family is a system of people under blood and marital relationships; and a chess configuration is a system of pieces under certain spatial and ‘possible move’ relations. A structure is the abstract form of a system, which ignores or abstracts away from any features of the objects that do not bear on the relations. The slogan of structuralism is that mathematics is the science of structure. The idea is that the subject matter of a given branch of mathematics is a structure, or a kind of structure. Typically, the structures studied in mathematics are free-standing, in the sense that the various positions or places in the structure are characterized only with respect to each other, and anything at all can play the various roles, and stand in the various relations. Define a ‘natural number system’ to be a counta...

3. As Kleene showed (Kleene and Vesley 1965, pp. 87-88), a condition is that the bar is decidable. Brouwer does not make this condition explicit, but in his proofs of 1924 (Brouwer 1924D2) and 1927 (Brouwer 1927B), it is satisfied, because there the bar is defined by an application of the continuity principle for choice sequences. The proof from 1954 (Brouwer 1954A), however, in which the bar is neither defined so as to be decidable, nor explicitly required to be, is incorrect. This seems to have been an oversight on Brouwer's part, rather than an overgeneralization, as the method he gives to construct the well-ordering depends on the decidability of the bar. See on this point also Veldman 2006. 4. Here we note and accept an ambiguity. In the primary sense, a demonstration is the act by and in which knowledge is acquired; this corresponds to Brouwer's 'Beweis'. But we also use 'demonstration' to translate Brouwer's 'Beweisführung', which is not an act but an object, namely the act objectified.

The first edition of this book appeared as the second volume in the series, Oxford Logic Guides, in 1977 and became the standard introduction to the intuitionistic philosophy of mathematics very soon. The present edition satisfies the highest standards of publishing quality, compared to the rather poor features of the former printing. The second edition preserves the original structure of the content adopted by the author for the first edition. Chapter 1 outlines preliminaries concerning constructive proofs, the meaning of logical constants, a sample of logical laws typical for intuitionism, and functional completeness. Elementary intuitionistic mathematics (arithmetic, real numbers, order, the axiom of choice) is discussed in Chapter 2. More advanced topics (infinity, the fan theorem, bar induction, the continuity principle, the Bar Theorem and its proof by Brouwer, continuous functionals and their representation, the uniform continuity theorem) are presented in Chapter 3. The next chapter is devoted to logical matters (natural deduction, the sequent calculus, cut-elimination, decidability of intuitionistic sentential logic, normalization). Chapter 5 contains an exposition of semantics for intuitionistic logic (valuation systems, lattices and finite model property, topological spaces, Beth trees, the semantics for intuitionistic predicate logic, the completeness of intuitionistic predicate logic, generalized Beth tree, compactness). Intuitionistic formal systems, realizability, and the Creative Subject (an idealized mathematician used in thought-experiments for motivating axioms and basic principles of mathematical reasoning) are discussed in Chapter 6 (under the title "Some further topics" ). Concluding philosophical remarks (philosophical foundations of constructive mathematics, the notion of proof, partial functions, logical constants as represented on Beth trees,

Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of which number is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of notation. The purpose of this article is to explore the relationship between the notations acceptable for computation, the usual idealizations involved in theories of computability, flowing from Alan Turing’s monumental work, and de re propositional attitudes toward numbers and other mathematical objects.

We answer the question of computational reasons for epistemic hardness of certain class of philosophically interesting mathematical concepts. We justify the statement that mathematical knowability may be identified with algorithmic learnability. We present framework of experimental logics equivalent to the notion of learnability. Then we prove the main result. By adjoining the minimal possible set of undecidable sentences to recursive axiomatization of arithmetics and closing it under logical consequence, we obtain a non-learnable theory. This gives an explanation to the fact that undecidable arithmetical sentences are cognitively difficult. We conclude that cognitively accessible mathematical concepts are exactly within the scope of learnablity. 1 Knowability as algorithmic learnability Emergence and development of recursion theory and computer science enable us to rigorously address the question of characterising the class of mathematical concepts that are cognitively accessible t...

Abstract. Brouwer’s papers after 1945 are characterized by a technique, known as the method of the creating subject. It has been supposed that the method was radically new in his work, since Brouwer seems to introduce an idealized mathematician into his mathematical practice. A newly opened source, the unpublished text of a lecture of Brouwer from 1934, fully supports the conclusions of our analysis that:-There is no idealized mathematician involved in the method.-The method was not new at all.-Brouwer uses an incomplete sequence, also known as choice sequence, in this method, which is special. The method does not take its place in the standard works on choice sequences. 1

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Tradition is classical. Surely, nothing could be more pleonastic than that? The logical tradition, certainly, was squarely classical from Bolzano to Carnap, with, say, Frege, Moore, Russell and the Wittgenstein of the Tractatus as intermediaries. Propositions are construed as being in themselves true-or-false. Indeed, in this tradition, a declarative sentence S expresses a proposition (or is a proposition, depending on what version of the theory that is adopted) by being true-or-false. So the meaningfulness of a sentence consists in its being true-or-false. But S is true-orfalse, or so they say, only when S is true, or when S is false. On the classical account the presumption of bivalence is built into the very notion of meaningfulness: there is no difference between asserting that Λ is a proposition and asserting that A is true-or-false. The matter came to the fore in the foundations of set theory. In his first attempt at giving an application criterion for sets Cantor noted: Eine Mannigfaltigkeit (ein Inbegriff, eine Menge) von Elementen, die irgendwelcher Begriffssphäre angehören, nenne ich wohldefiniert, wenn auf Grund ihrer Definition und infolge des logischen Prinzips vom ausgeschlossenen Dritten es als intern bestimmt angesehen werden muss, sowohl ob irgendein derselben Begriffssphäre angehöriges Objekt zu der gedachten Mannigfaltigkeit gehört oder nicht, wie auch, ob zwei zur Mannigfaltigkeit gehörige Objekte trotz formaler Unterschiede in der Art des Gegebenseins einander gleich sind oder nicht.' Here Cantor's reference to the Law of the Excluded Third is, in my opinion, of the above kind, where meaningfulness, rather than logicality, is at issue: when α is a set, and a an object from the right 'concept sphere', α e α has to be a well-put statement, that is, it must be determinately true-or-false. 2 The passage continues: Im allgemeinen werden die betreffenden Entscheidungen nicht mit den zu Gebote stehenden Methoden oder Fähigkeiten in Wirklichkeit sicher und genau ausführbar sein; daraufkommt es aber hierdurch nicht an, sondern allein auf die interne Determination, welche in konkreten Fällen, wo die Zwecke fordern, durch Vervollkommnung der Hilfsmittel zu einer aktuellen (externen) Determination auszubilden ist. The issue concerning the actual execution of methods of evaluation and decision was a bone of contention between Cantor and Kronecker. The Cantorian point of view was followed by Zermelo in his axiomatization, where definite Eigenschaften were used in the formulation of the Axiom der Aussonderung? Similarly, Frege imposed a very strict sharpness condition of complete determination on his propositional functions ("concepts"). Also he allowed for the possibility that human agents might not be able to execute the required decision. 4 Wittgenstein stressed similar points hi the Tractatus. A sentence must have trueor-false bipolarity in order to be able to say anything. The possibility of human 135

In 1907 L.E.J.Brouwer, then an unknown Dutch mathematician, stepped into the running debate on the origin and certainty of mathematics with his thesis On the foundations of mathematics (Brouwer 1907). In his view mathematics consists of mental constructions only; there are no mathematical truths outside the human mind. The constructions are built up from the natural numbers N and an intuitively given continuum. With N Brouwer constructed the integers Z and the rational numbers Q, but he had no satisfactory way to introduce the real numbers. It would take more than ten years before Brouwer would present a solution of this problem. In 1918 Brouwer, now a famous topologist, presented a complete new reconstruction of mathematics along lines set out in his thesis (Brouwer 1918). He called his reconstruction intuitionism, the existing mathematical practice classical. His foundational problem, reaching the power of the continuum from the discrete, he solved by the introduction of sequences...

The introduction of computable (alternately, recursive) function theory by Post, Church, Kleene, Godel, Turing, Malcev made it possible to analyse the computability of mathematical notions and constructions within the context of classical mathematics. Quite separately, the constructiveness of algebra was a principal concern of Kronecker in the late nineteenth century, and the constructiveness of analysis was a principle concern of Brouwer in the early twentieth century. Brouwer's work motivated the denition of rst order intuitionistic logic as introduced by his disciple Heyting. Kroneckerian eld theory was reworked as computable eld theory by F r/"oelich and Shepherdson in the 1950's, [4]. Systematic study of recursive algebra and recursive

A bounded monotone sequence of reals without a limit is called a Specker sequence. In Russian constructive analysis, Church's Thesis permits the existence of a Specker sequence. In intuitionistic mathematics, Brouwer's Continuity Principle implies it is false that every bounded monotone sequence of real numbers has a limit. We claim that the existence of Specker sequences crucially depends on the properties of intuitionistic decidable sets. We propose a schema (which we call ED) about intuitionistic decidability that asserts "there exists an intuitionistic enumerable set that is not intuitionistic decidable" and show that the existence of a Specker sequence is equivalent to ED. We show that ED is consistent with some certain well known axioms of intuitionistic analysis as Weak Continuity Principle, bar induction, and Kripke Schema. Thus, the assumption of the existence of a Specker sequence is conceivable in intuitionistic analysis. We will also introduce the notion of double Specker sequence and study the existence of them.

Subject of this paper is the method of the creative subject, as used by Brouwer after 1945. We shall interpret it in such a way that, in our opinion, it arises naturally from intuitionistically accepted concepts. The reconstruction will be carried out in a Kripke model. 1NTRODUCTlON In a number of papers published after 1945 Brouwer introduced a new device. In constructing counterexamples for classical theorems he seemed to refer explicitly to the activity of an idealized mathematician. This so-called method of the creative subject is controversial, also within intuitionism. The first attempt to formalize the new method was done by Kreisel ([Kr 671); it was further elaborated by Myhill ([My 681) and Troelstra ([Tr 691). The resulting theory of the creative subject, TCS for short, is strong enough to obtain Brouwer's result. But the interpretation of its terms is somewhat vague, it does not fully explain Brouwer's argument and there is a paradox, discovered by Troelstra, which has not been resolved satisfactorily. These questions play a minor role in the work of two more recent authors on this subject, Martin0 ([Ma 821, [Ma 851) and Posy ([PO 761, [PO 771). Their main interest is the application of the TCS in other parts of Brouwer's work. We want to show that Brouwer's method can be interpreted in such a way that it arises naturally from intuitionistically accepted concepts. This reconstruction suggests an interpretation of the basic term I-, of the TCS, which

Par des suites de choix, nous comprenons des suites qui ne sont pas déterminées complètement par une loi arithmétique. Elles sont des objets caractéristiques de l'intuitionnisme de Brouwer. Nous prétendons qu'à partir de 1927, l'utilisation par Brouwer de suites de choix particulières n'est pas reconnu comme tel. Nous prétendons que l'utilisation de ces suites dans la méthode du sujet créatif, après la seconde guerre mondiale, n'a pas à être mis en relation avec l'utilisation de celles-ci dans les années vingt et qu'elles sont mal interprétées. Nous montrons où se trouvent ces suites de choix dans l'oeuvre de Brouwer et comment elles doivent être traitées.

Par des suites de choix, nous comprenons des suites qui ne sont pas déterminées complètement par une loi arithmétique. Elles sont des objets caractéristiques de l'intuitionnisme de Brouwer. Nous prétendons qu'à partir de 1927, l'utilisation par Brouwer de suites de choix particulières n'est pas reconnu comme tel. Nous prétendons que l'utilisation de ces suites dans la méthode du sujet créatif, après la seconde guerre mondiale, n'a pas à être mis en relation avec l'utilisation de celles-ci dans les années vingt et qu'elles sont mal interprétées. Nous montrons où se trouvent ces suites de choix dans l'oeuvre de Brouwer et comment elles doivent être traitées.

Brouwer's papers after 1945 are characterized by a technique known as the method of the creating subject. It has been supposed that the method was radically new in his work, since Brouwer seems to introduce an idealized mathematician into his mathematical practice. A newly opened source, the unpublished text of a lecture of Brouwer from 1934, fully supports the conclusions of our analysis that:– There is no idealized mathematician involved in the method;– The method was not new at all;– Brouwer uses an incomplete sequence, also known as choice sequence, in this method, which is special. The method does not take its place in the standard works on choice sequences.

This paper presents a defense of Epistemic Arithmetic as used for a formalization of intuitionistic arithmetic and of certain informal mathematical principles. First, objections by Allen Hazen and Craig Smorynski against Epistemic Arithmetic are discussed and found wanting. Second, positive support is given for the research program by showing that Epistemic Arithmetic can give interesting formulations of Church's Thesis.

Turing and Church formulated two different formal accounts of computability that turned out to be extensionally equivalent. Since the accounts refer to different properties they cannot both be adequate conceptual analyses of the concept of computability. This insight has led to a discussion concerning which account is adequate. Some authors have suggested that this philosophical debate-which shows few signs of converging on one view-can be circumvented by regarding Church's and Turing's theses as explications. This move opens up the possibility that both accounts could be adequate, albeit in their own different ways. In this paper, I focus on the question of whether Church's thesis can be seen as an explication in the precise Carnapian sense. Most importantly, I address an additional constraint that Carnap puts on the explicative power of axiomatic systems-an axiomatisation explicates when it is clear which mathematical entities form the theory's intended model-and that implicitly applies to axiomatisations of recursion theory used in Church's account of computability. To overcome this difficulty, I propose two possible clarifications of the pre-systematic concept of "computability" that can both be captured in recursion theory, and I show how both clarifications avoid an objection arising from Carnap's constraint.

A ideia de objectos matemáticos que estão em permanente desenvolvimento no tempo foi pela primeira vez avançada por L.E.J. Brouwer. Na matemática intuicionista estes objectos são concebidos como sequência infinitas de números naturais que em qualquer estágio do seu crescimento têm apenas um número finito de valores, além disso, tais valores podem ser livremente escolhidos, no sentido em que a sua produção não necessita de ser determinada por nenhuma regra matemática definida. Tais objectos são denominados de sequências de escolha. O presente trabalho tem como objectivo fornecer uma resposta à sequinte questão: são as sequências de escolha legítimos objectos matemáticos? A resposta que iremos propor e à qual iremos argumentar favoravelmente é a seguinte: tais objectos não podem ser considerados objectos matemáticos legítimos. Com esta tese em vista, iremos discutir as propriedades intrínsecas das sequências de escolha relativamente à maneira como são incorporadas no contexto matemático e as suas implicações. Seguindo esta metodologia pretendemos atingir um cabal entendimento filosófico das consequências em que incorremos ao aceitarmos sequências de escolha como objectos da ontologia matemática e das razões que temos para não as aceitarmos como tal.

Errett Bishop's work in constructive mathematics is overwhelmingly regarded as a turning point
for
mathematics based on intuitionistic logic.
It brought new life to this form of mathematics and prompted
the development of new areas of research which
witness  today's depth and breadth of constructive mathematics.
Surprisingly, notwithstanding the extensive mathematical progress  since the publication of (Bishop 1967),
there has been no corresponding advances in the philosophy
of constructive mathematics Bishop style.

The aim of this paper is to foster the philosophical debate about this form of mathematics. I begin by considering key elements of philosophical remarks by Bishop, especially focusing on Bishop's assessment of Brouwer.
I then compare these remarks with ``traditional'' philosophical arguments for intuitionistic logic and argue that the latter
are in tension with Bishop's views. ``Traditional'' arguments for intuitionistic logic turn out to be also in conflict with significant recent developments in constructive mathematics.
This rises pressing questions for the philosopher of mathematics, especially with regard to the possibility of offering alternative philosophical arguments for constructive mathematics. I conclude with the suggestion
to look anew at Bishop's  own remarks for inspiration.

The Turing machine is one of the simple abstract computational devices that can be used to investigate the limits of computability. In this paper, they are considered from several points of view that emphasize the importance and the relativity of mathematical languages used to describe the Turing machines. A deep investigation is performed on the interrelations between mechanical computations and their mathematical descriptions emerging when a human (the researcher) starts to describe a Turing machine (the object of the study) by different mathematical languages (the instruments of investigation). Together with traditional mathematical languages using such concepts as ‘enumerable sets’ and ‘continuum’ a new computational methodology allowing one to measure the number of elements of different infinite sets is used in this paper. It is shown how mathematical languages used to describe the machines limit our possibilities to observe them. In particular, notions of observable deterministic and non-deterministic Turing machines are introduced and conditions ensuring that the latter can be simulated by the former are established.

Our regimentation of Goodman and Myhill’s proof of Excluded Middle revealed among its premises a form of Choice and an instance of Separation. Here we revisit Zermelo’s requirement that the separating property be definite. The instance that Goodman and Myhill used is not constructively warranted. It is that principle, and not Choice alone, that precipitates Excluded Middle. Separation in various axiomatizations of constructive set theory is examined. We conclude that insufficient critical attention has been paid to how those forms of Separation fail, in light of the Goodman–Myhill result, to capture a genuinely constructive notion of set.

A bounded monotone sequence of reals without a limit is called a Specker sequence. In Russian constructive analysis, Church's Thesis permits the existence of a Specker sequence. In intuitionistic mathematics, Brouwer's Continuity Principle implies it is false that every bounded monotone sequence of real numbers has a limit. We claim that the existence of Specker sequences crucially depends on the properties of intuitionistic decidable sets. We propose a schema (which we call ED) about intuitionistic decidability that asserts "there exists an intuitionistic enumerable set that is not intuitionistic decidable" and show that the existence of a Specker sequence is equivalent to ED. We show that ED is consistent with some certain well known axioms of intuitionistic analysis as Weak Continuity Principle, bar induction, and Kripke Schema. Thus, the assumption of the existence of a Specker sequence is conceivable in intuitionistic analysis. We will also introduce the notion of double Specker sequence and study the existence of them.

The aim of this paper is to take a look at Péter's talk Rekursivität und Konstruktivität delivered at the Constructivity in Mathematics Col-loquium in 1957, where she challenged Church's Thesis from a constructive point of view. The discussion of her argument and motivations is then connected to her earlier work on recursion theory as well as her later work on theoretical computer science.

The present work tries to answer a question concerning the validity of Charles Parsons’ proposal as a possible solution to Benacerraf’s dilemmas. Through the analysis of the author’s main works, of his advocates’ defense and his detractors’ attacks, it tries to shed some light on the pros and cons of an ontological-metaphysical and epistemological inquiry on the foundations of mathematics based on the combination of mathematical intuition and noneliminative structuralism.
Chapter 1 is a brief exam, through Parsons’ interpretation, of the three main figures on which he relies to support his view: Kant’s analysis of the intuitive faculty; Hilbert’s finitism; Gödel’s conception of mathematical intuition. The aim of this first section is to highlight the elements, coming from the three philosophers, that set the base for the formulation of Parsons’ two main ideas that can stand as a solution to Benacerraf’s work: mathematical intuition and noneliminative structuralism.
The latter idea is the core of Chapter 2, which faces the ontological problem: the argument of What Numbers Could Not Be. The chapter begins with a brief summary both of the dilemma and Benacerraf’s solution, that is, an eliminative structuralism. After that, the focus moves on Parsons’ position, which both criticize the eliminative perspective and presents as an alternative a noneliminative structuralism. Among other things, a peculiar emphasis is put on the analysis of the so-called ‘quasi-concrete objects’, the specific objects of intuition and one of the main elements of Parsons’ proposal. In the end, it comes the exam of the difficulties, and the eventual solutions, that a position like that of Parsons is forced to face.
A similar structure is adopted in Chapter 3, which faces the epistemological problem: the argument of Mathematical Truth. After a brief summary of the dilemma, the section focuses on a detailed exam of mathematical intuition as Parsons conceives it. Particular emphasis is put on the scientific rigor of Parsons’ analysis, at the core of which one can find ideas such as the splitting of the intuitive faculty into intuition of and intuition that and the analogy between intuition and sense perception. Moreover, crucial to the dilemma, the interaction between intuition and quasi-concrete objects is the other main issue, since the latter can be seen as a bridge between the pure abstractness of numbers and the concrete. The end of the chapter discusses the main objection, along with possibile answers or adjustments, moved against the attempt of justifying Dedekind-Peano axioms and PRA through mathematical intuition.
Chapter 4 contains some final remarks on this survey over the possibility to interpret Parsons’ position as an answer to Benacerraf’s dilemmas with some suggestions in order to improve the proposal from a Kantian perspective. The point that the work tries to bring out is that, although, to the present day, Parsons’ ideas do not, in fact, provide a complete solution to the problems that Benacerraf’s articles brought up, nevertheless, an intuition-based noneliminative structuralism can provide interesting clues for further studies on this matter. Besides, such a suggestion can be a good starting point for a line of thought that aims to unravel one of the main issues in the field of philosophy of mathematics by using the potential of a faculty that has been ignored and mistreated for a long time.

This work (hereafter OCP) is comparable in many ways to the Cambridge Dictionary of Philosophy (CDP). Both are modeled on the dictionary format; both are multi-authored; both are very popular; both are in second edition. For many purposes, OCP and CDP are interchangeable. However, Cambridge charges about half of the price Oxford wants. I have spent many happy hours with both. I own both editions of each. Each has its excellent and useful entries, and each has its mediocre or useless entries. In OCP, ignoratio elenchi exemplifies the former; axiom the latter.
The axiom entry is one of the shortest in the book, 70 words; the entry Montague, Richard is twice as long. Some of the entries are more than ten times as long. The axiom entry gives a dogmatic version of the evasive conventionalist view with no mention of epistemic issues such as certainty or self-evidence, nothing on various axiom/postulate distinctions, and nothing on the views of Aristotle, Frege, or Hilbert. Its sole reference is the 1974 edition of Quine’s Methods of Logic, hardly a locus classicus on the topic.
After comparing many of the logic entries, I would conclude that those in the CDP are usually better. The two entries on Church's thesis are written by equally qualified experts. Both imply correctly that Church's thesis is not a proposition admitting of mathematical proof or disproof in the usual sense: it is a proposal to "identify" the pre-theoretic intuitive concept of "effectively calculable function" with the mathematically precise number-theoretic property "recursiveness". The CDP entry is several times as long as the OCP and it is much more informative concerning the historical and philosophical importance of Church's thesis. A somewhat different comparison applies to the entries titled Church, Alonzo. Again, the CDP entry is much longer and much more informative than the OCP. Both entries are flawed.

Mathematical structuralism is a theory in the philosophy of mathematics which argues that mathematical objects are defined solely by their place in mathematical structures.

Mathematical structuralism adopts a position that's common to most other philosophical structuralisms: it denies that there are any “intrinsic properties” of objects. It even denies the very existence of objects apart from structures.

This means that we don't have objects (or things): we only have structures and relations. In terms of mathematical structuralism only: we don't have numbers until we also have structures and relations. In other words, numbers are born of their structures and relations.

In terms of Paul Benacerraf's initial reasons for formulating mathematical structuralism.

Benacerraf firstly noted that algebraic theorists had no position on the ontology of mathematical objects. Such theorists were only concerned with their “structure”. Thus Benacerraf asked himself whether or not what is true of algebraic theories is also true of other mathematical theories.

Two obvious points need to be stated here:

i) Is it objects which have these relations to one another?
ii) Is it objects which are part of a structure?

It's worth noting that “objective truth” (or at least truth) isn't rejected or denied by mathematical structuralists: it's just the account of how that truth comes about which is different to other accounts. Put simply, mathematical objects don't bring about objective truth: abstract structures do. Another way of putting this is to say that nothing is said about any mathematical object other than its place in a structure. Thus it seems to follow that there is no ontology of mathematical objects offered by mathematical structuralists.

In his famous paper, An Unsolvable Problem of Elementary Number Theory , Alonzo Church (1936) identified the intuitive notion of effective calculability with the mathematically precise notion of recursiveness. This proposal, known as Church's Thesis, has been widely accepted. Only a few papers have been written against it. One of these is László Kalmár's An Argument Against the Plausibility of Church's Thesis from 1959. The aim of this paper is to present Kalmár's argument and to fill in missing details based on his general philosophical thoughts on mathematics.

This is a set of slides originally given as a short talk at Keele University in 2014.

The talk outlines some of the existing ideas about choice sequences in 2014 and is aimed at a non-specialist audience.  I feel the title itself proves to be a sufficient abstract as to what the talk addresses.