# Mark van AttenFrench National Centre for Scientific Research | CNRS

Mark van Atten

PhD (Utrecht, 1999) and Habilitation à diriger des recherches (Paris, 2010)

## About

139

Publications

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576

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Citations since 2016

Introduction

Additional affiliations

October 2011 - present

December 2003 - September 2011

## Publications

Publications (139)

English translation of M. van Atten, “Les multiplicités définies de Husserl et les Théorèmes d’incomplétude de Gödel”, in : J. Farges et D. Pradelle (éds), Phénoménologie et fondements des sciences, Paris, Hermann, 2019, pp. 87–104.

Leibniz described imaginary roots, negatives, and infinitesimals as useful fictions. But did he view such 'impossible' numbers as mathematical entities? Alice and Bob take on the labyrinth of the current Leibniz scholarship.

By way of commenting on the prior literature, it is argued that both the first and the second of Gödel's Incompleteness Theorems have a bearing on the view on mathematics that Husserl presents in Formale und transzendentale Logik, and that this bearing is not small.

Guided by a passage in Kreisel, this is a discussion of the relations between the phenomena in the title, with special attention to the method of analysis and synthesis in Greek geometry, fixed point theorems, and Kreisel's contact with Gödel.

As Weyl was interested in infinitesimal analysis and for some years embraced Brouwer’s intuitionism, which he continued to see as an ideal even after he had convinced himself that it is a practical necessity for science to go beyond intuitionistic mathematics, this note presents some remarks on infinitesimals from a Brouwerian perspective. After an...

In ‘The philosophical basis of intuitionistic logic’, Michael Dummett discusses two routes towards accepting intuitionistic rather than classical logic in number theory, one meaning-theoretical (his own) and the other ontological (Brouwer and Heyting's). He concludes that the former route is open, but the latter is closed. I reconstruct Dummett's a...

In 'The philosophical basis of intuitionistic logic', Michael Dummett discusses two routes towards accepting intuitionistic rather than classical logic in number theory, one meaning-theoretical (his own) and the other ontological (Brouwer and Heyting's). He concludes that the former route is open, but the latter is closed. I argue that Dummett's ob...

Kripke's Schema (better the Brouwer-Kripke Schema) and the Kreisel-Troelstra Theory of the Creating Subject were introduced around the same time for the same purpose, that of analysing Brouwer's 'Creating Subject arguments'; other applications have been found since. I first look in detail at a representative choice of Brouwer's arguments. Then I di...

Kripke's Schema (better the Brouwer-Kripke Schema) and the Kreisel-Troelstra Theory of the Creating Subject were introduced around the same time for the same purpose, that of analysing Brouwer's 'Creating Subject arguments'; other applications have been found since. I first look in detail at a representative choice of Brouwer's arguments. Then I di...

As Weyl was interested in infinitesimal analysis and for some years embraced Brouwer's intuitionism, which he continued to see as an ideal even after he had convinced himself that it is a practical necessity for science to go beyond intuitionistic mathematics, this note presents some remarks on infinitesimals from a Brouwerian perspective. After an...

As Weyl was interested in infinitesimal analysis and for some years embraced Brouwer's intuitionism, which he continued to see as an ideal even after he had convinced himself that it is a practical necessity for science to go beyond intuitionistic mathematics, this note presents some remarks on infinitesimals from a Brouwerian perspective. After an...

A common objection to the definition of intuitionistic implication in the Proof Interpretation is that it is impredicative. I discuss the history of that objection, argue that in Brouwer's writings predicativity of implication is ensured through parametric polymorphism of functions on species, and compare this construal with the alternative approac...

I argue that Brouwer’s notion of the construction of purely mathematical objects and Husserl’s notion of their constitution by the transcendental subject coincide. Various objections to Brouwer’s intuitionism that have been raised in recent phenomenological literature (by Hill, Rosado Haddock, and Tieszen) are addressed. Then I present objections t...

Kant held that under the concept of √2 falls a geometrical magnitude, but not a number. In particular, he explicitly distinguished this root from potentially infinite converging sequences of rationals. Like Kant, Brouwer based his foundations of mathematics on the a priori intuition of time, but unlike Kant, Brouwer did identify this root with a po...

A list of corrections to the book Essays on Gödel's Reception of Leibniz, Husserl, and Brouwer (Springer, 2015).

A prominent problem for the Theory of the Creating Subject is Troelstra's Paradox. As is well known, the construction of that paradox depends on the acceptability of a certain impredicativity, of a kind that some intuitionists accept and others do not. After a presentation of the Theory of the Creating Subject and the paradox, I argue that the para...

In Mathematical Thought and Its Objects, Charles Parsons argues that our knowledge of the iterability of functions on the natural numbers and of the validity of complete induction is not intuitive knowledge ; Brouwer disagrees on both counts. I will compare Parsons' argument with Brouwer's and defend the latter. I will not argue that Parsons is wro...

Réédition, traduite en allemand et avec des notes supplémentaires par W. Howard, de “Mysticism and mathematics : Brouwer, Gödel, and the Common Core Thesis”, in : W. Deppert et M. Rahnfeld, Klarheit in Religionsdingen, Leipzig, Leipziger Universitätsverlag, 2003, pp. 145-160.

We present a new English translation of L.E.J. Brouwer's paper ‘De onbetrouwbaarheid der logische principes’ (The unreliability of the logical principles) of 1908, together with a philosophical and historical introduction. In this paper Brouwer for the first time objected to the idea that the Principle of the Excluded Middle is valid. We discuss th...

We present a new English translation of L.E.J. Brouwer's paper `De
onbetrouwbaarheid der logische principes' (The unreliability of the
logical principles) of 1908, together with a philosophical and
historical introduction. In this paper Brouwer for the first time
objected to the idea that the Principle of the Excluded Middle is
valid. We discuss th...

In Mathematical Thought and Its Objects, Charles Parsons argues that our knowledge of the iterability of functions on the natural numbers and of the validity of complete induction is not intuitive knowledge; Brouwer disagrees on both counts. I will compare Parsons' argument with Brouwer's and defend the latter. I will not argue that Parsons is wron...

In Mathematical Thought and Its Objects, Charles Parsons argues that our knowledge of the iterability of functions on the natural numbers and of the validity of complete induction is not intuitive knowledge; Brouwer disagrees on both counts. I will compare Parsons' argument with Brouwer's and defend the latter. I will not argue that Parsons is wron...

I look at Gödel’s relation to Brouwer and show that, besides deep disagreements, there are also deep agreements between their philosophical ideas. This text was originally written in French and published in a special issue on logic of Pour la Science, the French edition of Scientific American. This accounts for its introductory character and the ab...

In an envelope of material relating to his work on the translation and revision of the Dialectica paper in 1968, Gödel kept a note that is in shorthand but in which one immediately notices the longhand name ‘Leibniz’. When transcribed and put into context, the note allows one to show that Leibniz was a source of inspiration for Gödel’s revision of...

Chapter 1. Introduction.- Part I Godel and Leibniz.- Chapter 2 A note on Leibniz's argument against infinite wholes.- Chapter 3. Monads and sets: on Godel, Leibniz, and the Reflection Principle.- Chapter 4. Godel's Dialectica Interpretation and Leibniz.- Part II Godel and Husserl.- Chapter 5. Phenomenology of mathematics.- Chapter 6. On the philoso...

This is an introduction to the phenomenology of mathematics, written for phenomenologists.

We compare Gödel’s and Brouwer’s explorations of mysticism and its relation to mathematics.

These are the preface and introduction to my book Essays on Gödel's Reception of Leibniz, Husserl, and Brouwer (Dordrecht: Springer, 2015).

« On the unreliability of the logical principles » : A new French translation of L. E. J. Brouwer’s 1908 article with annotations and commentary
In his seminal paper « On the unreliability of the logical principles » of 1908, L. E. J. Brouwer draws for the first time the revisionistic consequences of the general view on logic that he had presented...

This article shows that an apparent puzzle in finance and accounting is resolved by changing from classical to constructive (more specifically, intuitionistic) mathematics. Our position is that it is unproblematic if real-world actors behave inconsistently with nonconstructive mathematical results. Thus the solution to our puzzle lies in a deeper c...

Traduction française commentées des cahiers de K.Gödel, PUP.

Kant held that under the concept of √2 falls a geometrical magnitude, but not a number. In particular, he explicitly distinguished this root from potentially infinite converging sequences of rationals. Like Kant, Brouwer based his foundations of mathematics on the a priori intuition of time, but unlike Kant, Brouwer did identify this root with a po...

Leibniz had a well-known argument against the existence of infinite wholes that is based on the part-whole axiom: the whole is greater than the part. The refutation of this argument by Russell and others is equally well known. In this note, I argue (against positions recently defended by Arthur, Breger, and Brown) for the following three claims: (1...

Kurt Gödel (1906–1978) did groundbreaking work that transformed logic and other important aspects of our understanding of mathematics, especially his proof of the incompleteness of formalized arithmetic. This book on different aspects of his work and on subjects in which his ideas have contemporary resonance includes papers from a May 2006 symposiu...

I argue that Brouwer’s notion of the construction of purely mathematical objects and Husserl’s notion of their constitution by the transcendental subject coincide. Various objections to Brouwer’s intuitionism that have been raised in recent phenomenological literature (by Hill, Rosado Haddock, and Tieszen) are addressed. Then I present objections t...

Die Mathematik und das synthetische Apriori. Erkenntnistheoretische Untersuchungen über den Geltungsstatus mathematischer Axiome - Wille Matthias . Die Mathematik und das synthetische Apriori. Erkenntnistheoretische Untersuchungen über den Geltungsstatus mathematischer Axiome. Mentis, Paderborn, 2007, 234 pp. - Volume 15 Issue 1 - Mark van Atten

Gödel once offered an argument for the general reflection principle in set theory that took the form of an analogy with Leibniz’ monadology. I discuss the mathematical and philosophical background to Gödel’s argument, reconstruct the proposed analogy in detail, and argue that it has no justificatory force. The paper also provides further support fo...

Issu de deux conférences tenues à "Uppsala university", Suède, en août 2004

The Dutch mathematician and philosopher L. E. J. Brouwer (1881--1966)developed a foundation for mathematics called `intuitionism'. Intuitionism considers mathematics to consist in acts of mental construction based on internal time awareness. According to Brouwer, that awareness provides the fundamental structure to all exact thinking. In this note,...

Brouwer’s demonstration of his Bar Theorem gives rise to provocative questions regarding the proper explanation of the logical
connectives within intuitionistic and constructivist frameworks, respectively, and, more generally, regarding the role of
logic within intuitionism. It is the purpose of the present note to discuss a number of these issues,...

The article comments on the foregoing article by Christophe Sigwart on concepts of number and compares Sigwart's text to the "intuitionistic" mathematical work developed by Dutch logician and philosopher L. E. J. Brouwer. The two men's positions are compared on such concepts as the intersubjective validity of mathematics, inner time consciousness,...

With logicism and formalism, intuitionism is one of the main foundations for mathematics proposed in the twentieth century; and since the seventies, notably its views on logic have become important also outside foundational studies, with the development of theoretical computer science. The aim of the book is threefold: to review and complete the hi...

With logicism and formalism, intuitionism is one of the main foundations for mathematics proposed in the twentieth century; and since the seventies, notably its views on logic have become important also outside foundational studies, with the development of theoretical computer science.
The aim of the book is threefold: to review and complete the hi...

Connecting Phenomenology and MathematicsTranscendental Phenomenology as a Foundation of MathematicsExamples

On the intended interpretation of intuitionistic logic, Heyting's Proof Interpretation, a proof of a proposition of the form p -> q consists in a construction method that transforms any possible proof of p into a proof of q. This involves the notion of the totality of all proofs in an essential way, and this interpretation has therefore been object...

The purpose of this section is threefold. First, to provide a standard to evaluate Husserl’s and Brouwer’s original positions by. Secondly, to let this standard be the methodological clue to the reconstruction of choice sequences in chapter 6. Thirdly, to justify the revisionism implied in that reconstruction; which is all the more urgent since, as...

The aim is to use phenomenology to justify Brouwer’s choice sequences as mathematical objects

The purpose of this chapter is to bring out a conflict between Brouwer’s and Husserl’s philosophies of mathematics. I will begin by locating a particular point of disagreement, and pin down what it consists in. Disagreement by itself does not warrant speaking of a conflict. A further condition should be fulfilled, namely, the presence of mutual pre...

In chapter 5, I explained Husserl’s principle that transcendental phenomenology provides the ontology for the a priori sciences. What the basic objects, and the axioms governing them, in an a priori science are, is to be disclosed in phenomenological analysis. Now I want to show this idea in action: the constitution analysis of choice sequences, un...

In section 5.1, it was argued that with regard to pure mathematics, phenomenology is capable of ontological judgements. Generally, complete justification for asserting the existence of a supposed object consists in giving a strict constitution analysis; what is specific to the case of pure mathematics is that the laws governing strict constitution...

Can the straight line be analysed mathematically such that it does not fall apart into a set of discrete points, as is usually done but through which its fundamental continuity is lost? And are there objects of pure mathematics that can change through time?
The mathematician and philosopher L.E.J. Brouwer argued that the two questions are closely...

Accounting theory treats the residual income and discounted dividends models as equivalent,yet empirical researchers obtain different results under the two approaches. This paper offers a theoretical explanation for this apparent anomaly, by allowing the expectations of remote future earnings to be non-unique. In this setting, both models yield ran...

After a brief survey of Gödel’s personal contacts with Brouwer and Heyting, examples are discussed where intuitionistic ideas had a direct influence on Gödel’s technical work. Then it is argued that the closest rapprochement of Gödel to intuitionism is seen in the development of the Dialectica Interpretation, during which he came to accept the noti...

## Projects

Project (1)

To explore the current viability of pre-1930 style "atavistic" use of contentual formal languages, especially regarding the notions of inference and axioms, within the Philosophy of Logic and the Philosophy of Mathematics.