Mark van Atten | Centre National de la Recherche Scientifique / French National Centre for Scientific Research (original) (raw)
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The Mathematical Intelligencer
HAL (Le Centre pour la Communication Scientifique Directe), 2022
By way of commenting on the prior literature, it is argued that both the first and the second of ... more 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.
HAL (Le Centre pour la Communication Scientifique Directe), Mar 7, 2022
Guided by a passage in Kreisel, this is a discussion of the relations between the phenomena in th... more 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.
Inquiry, 2019
In 'The philosophical basis of intuitionistic logic', Michael Dummett discusses two routes toward... more 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 argument against the ontological route and argue that it fails. Call a procedure 'investigative' if that in virtue of which a true proposition stating its outcome is true exists prior to the execution of that procedure; and 'generative' if the existence of that in virtue of which a true proposition stating its outcome is true is brought about by the execution of that procedure. The problem with Dummett's argument then is that a particular step in it, while correct for investigative procedures, is not correct for generative ones. But it is the latter that the ontological route is concerned with.
Algemeen Nederlands Tijdschrift Voor Wijsbegeerte, 2003
Bernays has suggested that the different positions in the foundations of mathematics are not so m... more Bernays has suggested that the different positions in the foundations of mathematics are not so much contradicting each other as acknowledging different types and degrees of evidence. Although this move indeed renders the different positions compatible, there may still be principled ways to privilege one position over another. As an example, an argument is advanced why Godel's argument that the classical powerset of N lends itself to an (idealized) intuitive overview, though perhaps not contradictory, in an important sense fails.
Logic, Epistemology, and the Unity of Science, 2014
Logic, Epistemology, and the Unity of Science, 2015
ABSTRACT - Draws extensively on Gödel's archive - Analyses historical and systematical as... more ABSTRACT - Draws extensively on Gödel's archive - Analyses historical and systematical aspects of Gödel's philosophical project - Evaluates Gödel's use of phenomenology and challenges recent interpretations This volume tackles Gödel's two-stage project of first using Husserl's transcendental phenomenology to reconstruct and develop Leibniz' monadology, and then founding classical mathematics on the metaphysics thus obtained. The author analyses the historical and systematic aspects of that project, and then evaluates it, with an emphasis on the second stage. The book is organised around Gödel's use of Leibniz, Husserl and Brouwer. Far from considering past philosophers irrelevant to actual systematic concerns, Gödel embraced the use of historical authors to frame his own philosophical perspective. The philosophies of Leibniz and Husserl define his project, while Brouwer's intuitionism is its principal foil: the close affinities between phenomenology and intuitionism set the bar for Gödel's attempt to go far beyond intuitionism. The four central essays are `Monads and sets', `On the philosophical development of Kurt Gödel', `Gödel and intuitionism', and `Construction and constitution in mathematics'. The first analyses and criticises Gödel's attempt to justify, by an argument from analogy with the monadology, the reflection principle in set theory. It also provides further support for Gödel's idea that the monadology needs to be reconstructed phenomenologically, by showing that the unsupplemented monadology is not able to found mathematics directly. The second studies Gödel's reading of Husserl, its relation to Leibniz' monadology, and its influence on his published writings. The third discusses how on various occasions Brouwer's intuitionism actually inspired Gödel's work, in particular the Dialectica Interpretation. The fourth addresses the question whether classical mathematics admits of the phenomenological foundation that Gödel envisaged, and concludes that it does not. The remaining essays provide further context. The essays collected here were written and published over the last decade. Notes have been added to record further thoughts, changes of mind, connections between the essays, and updates of references. Content Level » Research Keywords » Edmund Husserl - Gottfried Wilhelm von Leibniz - Kurt Gödel - L.E.J. Brouwer - Platonism - intuitionism - monadology - phenomenology of mathematics - philosophy of mathematics Related subjects » Epistemology & Philosophy of Science - Mathematics - Philosophical Traditions
History of Philosophy of Science, 2002
The Dutch mathematician and philosopher L.E.J. Brouwer (1881–1966) is well known for his ground-b... more The Dutch mathematician and philosopher L.E.J. Brouwer (1881–1966) is well known for his ground-breaking work in topology and his iconoclastic philosophy of mathematics, intuitionism. What is far less well known is that Brouwer mused on the philosophy of the natural sciences as well. Later in life he also taught courses in physics at the University of Amsterdam.
Brouwer Meets Husserl, 2007
Brouwer Meets Husserl, 2007
Logic, Epistemology, and the Unity of Science, 2014
I look at Godel’s relation to Brouwer and show that, besides deep disagreements, there are also d... more I look at Godel’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 absence of references and footnotes. The translation and slight revision are my own.
The Mathematical Intelligencer
HAL (Le Centre pour la Communication Scientifique Directe), 2022
By way of commenting on the prior literature, it is argued that both the first and the second of ... more 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.
HAL (Le Centre pour la Communication Scientifique Directe), Mar 7, 2022
Guided by a passage in Kreisel, this is a discussion of the relations between the phenomena in th... more 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.
Inquiry, 2019
In 'The philosophical basis of intuitionistic logic', Michael Dummett discusses two routes toward... more 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 argument against the ontological route and argue that it fails. Call a procedure 'investigative' if that in virtue of which a true proposition stating its outcome is true exists prior to the execution of that procedure; and 'generative' if the existence of that in virtue of which a true proposition stating its outcome is true is brought about by the execution of that procedure. The problem with Dummett's argument then is that a particular step in it, while correct for investigative procedures, is not correct for generative ones. But it is the latter that the ontological route is concerned with.
Algemeen Nederlands Tijdschrift Voor Wijsbegeerte, 2003
Bernays has suggested that the different positions in the foundations of mathematics are not so m... more Bernays has suggested that the different positions in the foundations of mathematics are not so much contradicting each other as acknowledging different types and degrees of evidence. Although this move indeed renders the different positions compatible, there may still be principled ways to privilege one position over another. As an example, an argument is advanced why Godel's argument that the classical powerset of N lends itself to an (idealized) intuitive overview, though perhaps not contradictory, in an important sense fails.
Logic, Epistemology, and the Unity of Science, 2014
Logic, Epistemology, and the Unity of Science, 2015
ABSTRACT - Draws extensively on Gödel's archive - Analyses historical and systematical as... more ABSTRACT - Draws extensively on Gödel's archive - Analyses historical and systematical aspects of Gödel's philosophical project - Evaluates Gödel's use of phenomenology and challenges recent interpretations This volume tackles Gödel's two-stage project of first using Husserl's transcendental phenomenology to reconstruct and develop Leibniz' monadology, and then founding classical mathematics on the metaphysics thus obtained. The author analyses the historical and systematic aspects of that project, and then evaluates it, with an emphasis on the second stage. The book is organised around Gödel's use of Leibniz, Husserl and Brouwer. Far from considering past philosophers irrelevant to actual systematic concerns, Gödel embraced the use of historical authors to frame his own philosophical perspective. The philosophies of Leibniz and Husserl define his project, while Brouwer's intuitionism is its principal foil: the close affinities between phenomenology and intuitionism set the bar for Gödel's attempt to go far beyond intuitionism. The four central essays are `Monads and sets', `On the philosophical development of Kurt Gödel', `Gödel and intuitionism', and `Construction and constitution in mathematics'. The first analyses and criticises Gödel's attempt to justify, by an argument from analogy with the monadology, the reflection principle in set theory. It also provides further support for Gödel's idea that the monadology needs to be reconstructed phenomenologically, by showing that the unsupplemented monadology is not able to found mathematics directly. The second studies Gödel's reading of Husserl, its relation to Leibniz' monadology, and its influence on his published writings. The third discusses how on various occasions Brouwer's intuitionism actually inspired Gödel's work, in particular the Dialectica Interpretation. The fourth addresses the question whether classical mathematics admits of the phenomenological foundation that Gödel envisaged, and concludes that it does not. The remaining essays provide further context. The essays collected here were written and published over the last decade. Notes have been added to record further thoughts, changes of mind, connections between the essays, and updates of references. Content Level » Research Keywords » Edmund Husserl - Gottfried Wilhelm von Leibniz - Kurt Gödel - L.E.J. Brouwer - Platonism - intuitionism - monadology - phenomenology of mathematics - philosophy of mathematics Related subjects » Epistemology & Philosophy of Science - Mathematics - Philosophical Traditions
History of Philosophy of Science, 2002
The Dutch mathematician and philosopher L.E.J. Brouwer (1881–1966) is well known for his ground-b... more The Dutch mathematician and philosopher L.E.J. Brouwer (1881–1966) is well known for his ground-breaking work in topology and his iconoclastic philosophy of mathematics, intuitionism. What is far less well known is that Brouwer mused on the philosophy of the natural sciences as well. Later in life he also taught courses in physics at the University of Amsterdam.
Brouwer Meets Husserl, 2007
Brouwer Meets Husserl, 2007
Logic, Epistemology, and the Unity of Science, 2014
I look at Godel’s relation to Brouwer and show that, besides deep disagreements, there are also d... more I look at Godel’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 absence of references and footnotes. The translation and slight revision are my own.
- Draws extensively on Gödel's archive - Analyses historical and systematical aspects of Gödel's ... more - Draws extensively on Gödel's archive
- Analyses historical and systematical aspects of Gödel's philosophical project
- Evaluates Gödel's use of phenomenology and challenges recent interpretations
This volume tackles Gödel's two-stage project of first using Husserl's transcendental phenomenology to reconstruct and develop Leibniz' monadology, and then founding classical mathematics on the metaphysics thus obtained. The author analyses the historical and systematic aspects of that project, and then evaluates it, with an emphasis on the second stage.
The book is organised around Gödel's use of Leibniz, Husserl and Brouwer. Far from considering past philosophers irrelevant to actual systematic concerns, Gödel embraced the use of historical authors to frame his own philosophical perspective. The philosophies of Leibniz and Husserl define his project, while Brouwer's intuitionism is its principal foil: the close affinities between phenomenology and intuitionism set the bar for Gödel's attempt to go far beyond intuitionism.
The four central essays are `Monads and sets', `On the philosophical development of Kurt Gödel', `Gödel and intuitionism', and `Construction and constitution in mathematics'. The first analyses and criticises Gödel's attempt to justify, by an argument from analogy with the monadology, the reflection principle in set theory. It also provides further support for Gödel's idea that the monadology needs to be reconstructed phenomenologically, by showing that the unsupplemented monadology is not able to found mathematics directly. The second studies Gödel's reading of Husserl, its relation to Leibniz' monadology, and its influence on his published writings. The third discusses how on various occasions Brouwer's intuitionism actually inspired Gödel's work, in particular the Dialectica Interpretation. The fourth addresses the question whether classical mathematics admits of the phenomenological foundation that Gödel envisaged, and concludes that it does not.
The remaining essays provide further context. The essays collected here were written and published over the last decade. Notes have been added to record further thoughts, changes of mind, connections between the essays, and updates of references.
Content Level » Research
Keywords » Edmund Husserl - Gottfried Wilhelm von Leibniz - Kurt Gödel - L.E.J. Brouwer - Platonism - intuitionism - monadology - phenomenology of mathematics - philosophy of mathematics
Related subjects » Epistemology & Philosophy of Science - Mathematics - Philosophical Traditions
Guided by a passage in Kreisel, this is a discussion of the relations between the phenomena in th... more 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, Lawvere's Fixed Point theorem, and Kreisel's contact with Gödel.
By way of commenting on the prior literature, it is argued that both the first and the second of ... more 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 th... more 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.
Guided by a passage in Kreisel, this is a discussion of the relations between the phenomena in th... more 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.
In 'The philosophical basis of intuitionistic logic', Michael Dummett discusses two routes toward... more 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 objection to the ontological route fails.
Kripke's Schema (better the Brouwer-Kripke Schema) and the Kreisel-Troelstra Theory of the Creati... more 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 discuss the original use of the Schema and the Theory, their justification from a Brouwerian perspective, and instances of the Schema that can in fact be found in Brouwer's own writings. Finally, I defend the Schema and the Theory against a number of objections that have been made. Brouwer's views may be wrong or crazy (e.g. self-contradictory), but one will never find out without looking at their more dubious aspects.
As Weyl was interested in infinitesimal analysis and for some years embraced Brouwer's intuitioni... more 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 introduction and a look at Robinson's and Nelson's approaches to classical nonstandard analysis, three desiderata for an intuitionistic construction of infinitesimals are extracted from Brouwer's writings. These cannot be met, but in explicitly Brouwerian settings what might in different ways be called approximations to infinitesimals have been developed by early Brouwer, Vesley, and Reeb. I conclude that perhaps Reeb's approach, with its Brouwerian motivation for accepting Nelson's classical formalism, would have suited Weyl best.
A common objection to the definition of intuitionistic implication in the Proof Interpretation is... more 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 predica-tivity of implication is ensured through parametric polymorphism of functions on species, and compare this construal with the alternative approaches to predicative implication of Goodman, Dummett, Prawitz, and Martin-Löf.
Abstract : This work presents the first complete German transcription of the Max Phil X notebook ... more Abstract : This work presents the first complete German transcription of the Max Phil X notebook of Kurt Gödel, offered in an open access version to the scholars of the Gödelian community. The original notebook was written in Gabelsberger from March 1943 to January 1944 and it belongs to a series of fifteen notebooks. Some excerpts of these notebooks were transcribed and published in the third volume of Kurt's Gödel Collected Works. The Gabelsberger text of the Max-Phil notebooks is now available on the web site of the IAS (https://library.ias.edu/godelpapers). Concerning the content of Max-Phil X, it shows an overall unity that can understood only considering the lapse of time in which it was written: there are three facts that occupies Gödel’s mind at this time. His preparation of the Russell paper, his conversations with Einstein, which were current from 1942, and his intensive study on Leibniz began in 1943. The remarks of Max-Phil X can mostly related to these three events, with some exceptions. The notebook offers therefore a set of philosophical remarks from logic to cosmology, pursued in an intimate dialog, with Gödel's own published work, with authors from the past and contemporaries.
A prominent problem for the Theory of the Creating Subject is Troelstra's Paradox. As is well kno... more 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 paradox moreover depends on Markov's Principle, in a
form that no intuitionist should accept. A postscript discusses a new
version of the paradox that Troelstra has proposed in reaction to my
argument.
slides for Intuitionistic inductive definitions, Nantes 2022, 2022
Slides for a talk on the grammar and pragmatics of intuitionistic inductive definitions in Nantes... more Slides for a talk on the grammar and pragmatics of intuitionistic inductive definitions in Nantes, October 2022.