1. Curriculum Vitae
  2. Experience and Skills
  3. Selected Academic Publications
  4. Selected Other Publications (in Dutch)
  5. Structures for Everyone
  6. Additional Links

1. Curriculum Vitae

Life. I was born in Amsterdam, the Netherlands (Europe), in January 1962. The head-master of my primary school perceived talent for music in my person when he heard me sing. History has proved him wrong. I wanted to become a zoologist anyway. Then I discovered there are no wild animals in Holland. The boring biology-lessons in secondary school, with their emphasis on seeds, plants and the Holy Environment, annihilated my ambition to become a zoologist.
In my final examination of secondary school I ended as the top of my class (May 1978). From September 1978 to August 1983, I followed a vocational course, obtaining qualifications that allow me to teach physics and mathematics at intermediate level (secondary school). When I was about 18 years old, I sensed that my destiny was to become a philosopher-scientist and acted accordingly. Occasionally I also hear the voice of a literary vocation calling, which manages to perturb my worldline at times --- see for this Het Bloedbad and De Muur.

Undergraduate Studies. From September 1983 to August 1989, I followed a six year programme at the Free University of Amsterdam, comprising: theoretical physics (main subject; passed examinations with distinction --- cum laude); elements of mathematics; philosophy of science; history of the natural sciences; history of philosophy; basic Latin. Some of my undergraduate work on theoretical nuclear physics has been published. I partly funded myself with money earned by teaching private pupils, by teaching in a private school and by organising gambling games.

More Life. The first half of 1990 I spent in Africa, mostly in Tunisia, where among other things I earned some money by performing simple manual labour, such as slaughtering chickens. I did not return to Holland empty-handed, for in August 1990 I married a Tunisian woman (in Amsterdam) I had met in Tunis; our offspring consists of two sons, Souleyman (1993-2010), who was killed in a traffic accident at the age of 17 years and 1 week, Faysal (1994), and one daughter, Olfa (1998) --- no pictures of me and them on this webpage, as if they were trophees of some kind, or as suspicious visual testimonies of Look At Me Being A Cute Loving Father.

Advanced Studies. Since October 1991 I have been active as a graduate student in the foundations and philosophy of science at Utrecht University, busily engaged in fulfilling my destiny. I followed courses in logic, philosophy of quantum theory and philosophy of space and time (mostly relativity theory). In November 1998 I publicly defended my omnious and multidisciplinary Thesis Structures for Everyone. Contemplations and proofs in the foundations and philosophy of physics and mathematics . Part of my Thesis is published in two parts in Studies in the History and Philosophy of Modern Science B and my Thesis is published in its entirety as a 600 page book.

In the beginning of 1999 I gave a private course `Philosophy and Quantum Mechanics' in Amsterdam (co-organised with the theologian-philosopher-thief J.A.G.M. Rutten); it was attended by fifteen persons. A short version I have given at the International School of Philosophy (see below).

From March 2000 until June 2004 I was post-doctoral researcher, funded by the Dutch Science Organisation (NWO), for a period of 3 years and 9 months (4 days a week). My working home base therefore remains Utrecht University. Since July 2005 I work officially at the Philosophy Faculty of Erasmus University Rotterdam; in Utrecht I am a `guest-researcher'.

Areas of Specialisation, Competence and Interest.

Beliefs. I believe that life is absurd, horrible and good for a few laughs and more tears. I believe that when we die, our world comes to an end --- the most disturbing prospect we all have to come to terms with in one way or another. I believe that reason and imagination are the quintessential attributes of our species, which somehow emerge from those staggering constellations of quarks & leptons that we are, and which has evolved, slowly and haphazardly, from green slime that once drifted in the primal soup on planet Earth.

Back to Contents.

2. Experience and Skills

  1. Taught at a private school in the years 1985-1988 and prepared pupils (aged 12-18) for their secondary school examinations in physics, mathematics and chemistry.

  2. Led a number of reading groups with undergraduate students in the philosophy of science in the academic years 19921994 at Utrecht University and supervised the writing of short papers.

  3. Assisted in the Lecture courses on the foundations of quantum mechanics and on the foundations of relativity theory in the academic years 1992-1994 at Utrecht University.

  4. Organised and led, in 1993 at Utrecht University, a small reading group of graduate students, researchers and professors, who were studying J. von Neumann's Mathematische Grundlagen der Quantenmechanik (1932).

  5. Taught a series of lectures on mathematical Measure Theory to a small group of undergraduate and graduate students, researchers and professors, in 1994 at Utrecht University.

  6. Described my work over the past few years to international conference audiences in Cambridge, Oxford, Amsterdam, Florence, Castiglioncello (Italy) and Cologne. I am generally considered to be a lively and engaging speaker.

  7. Written quite a number of book reviews (in Dutch), for the official periodical of the Dutch Physical Society and for the Dutch quality paper NRC Handelsblad, and a number of essays in prose on physics, philosophy and literature, for the Dutch literary magazine Hollands Maandblad.

  8. Languages: Dutch (native language), English (fluent), French and German (semi-fluent), Latin (basic), and Arabic (some oral ability).

  9. Assisted in the translation of a few books from English into Dutch, including the lectures on The nature of space and time given by S. Hawking and R. Penrose at the Newton Institute in Cambridge (Prometeus, 1996).

  10. From November 1991 until 1995 at Utrecht University, I initiated and organised, together with a few undergraduate students, the Foundations of Physics Student Colloquium.

  11. Initiated and established contact between the Utrecht group and the Cambridge group of M.L.G. Redhead in 1993 --- now dissolved and re-assembled partly in Oxford.

  12. Taught the course `Foundations and Philosophy of Quantum Mechanics' for 3rd/4th year students of Utrecht University (September-December 2000).

  13. Taught a few times (2000, 2001) a course in mathematics at the Faculty of Mathematics of the University of Amsterdam in preparation of the Colloquium Doctum Examination, which is an examination that future-students need to pass who do not have the required secondary school diploma (VWO Wiskunde B) but want to study mathematics, physics or computer science.

  14. Taught a course at the ISW (International School for Philosophy) in Leusden, Holland (2001), an intensive course (1.5 day) called `Philosophy and Quantum Mechanics' (in Dutch). This is for people interested in philosophy who have no knowledge of quantum mechanics but have secondary-school acquaintance with mathematics (complex numbers, functions, vectors, probability).

  15. Organised (2000-now) national reading group in the philosophy of science and of physics.

  16. Assisted H. Philipse in organising a national Epistemology Seminar (2002-2004), every two weaks held in Utrecht.

  17. Taught a Bachelors courses on Philosophy of Matter and Philosophy of Science at Erasmus University Rotterdam (Fall 2004, Fall 2005)

  18. Taugth a Bachelors course on Philosophy of Mind at Erasmus University Rotterdam (Fall 2006, Fall 2007)

  19. Taught a Masters course `Unity and Disunity of Science', together with dr. Caterina Marchionni at Erasmus University Rotterdam (Spring 2007, Fall 2007)

  20. Taught a Masters course `Truth' at Erasmus University Rotterdam (Spring 2007), with Simon Blackburn from Cambridge University visiting.

  21. Participated in the Bachelors course Film & Philosopy at Erasmus University Rotterdam (Fall 2005, Fall 2006, Fall 2007)

  22. Received VIDI-funding of the National Foundation for Scientific Research (Dutch: NWO) for `A world of stuctures'. (Budget: 600.000 Euro, 2006--2010).

  23. Since January 2010 president of the Dutch Society for the Philosophy of Science (DSPS), in Dutch: Nederlandse Vereniging voor WetenschapsFilosofie (NVWF).

  24. Organised with Tim de Mey a masters course Analytic Metaphysics at Erasmus University Rotterdam.

  25. Orgainised a new masters course Philosophy of Matter at Erasmus University Rotterdam.

Back to Contents.

3. Selected Academic Publications

  1. `The Dyson Equation. An application of quantum field-theoretic techniques to the many-body problem in nuclear physics', undergraduate thesis (partly published in the next item), July 1988.

  2. `Fragmentation of Single-Particle Strength and the Validity of the Shell Model', Nuclear Physics A531 (1991) 253--284 (co-author).

  3. `On the Principle of Relativity', Foundations of Physics Letters 5 (1992) 591-596.

  4. `Worldlines are Growing! On Ontological Fatalism, Temporal Becoming and the Special Theory of Relativity', May 1992, unpublished; discussed by R. Clifton and M. Hogarth in Synthese 103 (1995) 355-387.

  5. `Philosophy of Physics for Pedestrians', Studies in the History and Philosophy of Modern Science 25 (1994) 505-509.

  6. `On Stochastic Einstein Locality in Algebraic Quantum Field Theory', International Journal of Theoretical Physics 33 (1994) 91-102.

  7. `Is Lorentz-covariant Quantum Field Theory Stochastic Einstein Local?' `Is Lorentz-covariant Quantum Field Theory Stochastic Einstein Local?' Philosophy of Science 61 (1994) 457-474.

  8. `Fixing A Hole', Foundations of Physics Letters 8 (1995) 549-562.
    [Count the number of allusions to songs of The Beatles in this publication.]
    Concerns The Hole Argument.

  9. `The Equivalence Myth of Quantum Mechanics', published in two parts in Studies in the History and Philosophy of Modern Physics 28 (1997) 35-61, 219-247, and an Addendum in 30 (1999) 543-545.
    Part 1
    Part 2

  10. `The Locality Scandal of Quantum Mechanics', invited contribution to proceedings of International Conference on Logic, Methodology, and Philosophy of Science, Florence, 1995, in Language, Quantum, Music , M. Dalla Chiara et al. (eds.), Dordrecht: Kluwer, 1999, 241-248.

  11. Structures for Everyone. Contemplations and proofs in the foundations and philosophy of physics and mathematics, (PhD-Thesis published as book, November 1998).

  12. `Sets, Classes and Categories', British Journal of the Philosophy of Science 52 (2001) 539-573.

  13. `Disunity in Unity', Erkenntnis 55 (2001) 132-143.
    [Review essay of Margaret Morrison's Unifying Scientific Theories. Physical Concepts and Mathematical Structures (2000)]

  14. `Wetenschapsfilosofische vooruitgang' Algemeen Nederlands Tijdschrift voor Wijsbegeerte 2002.
    [Review essay of Th.A.F. Kuipers' From Instrumentalism to Constructive Realism. On Some Relations between Confirmation, Empirical Progress and Truth Approximation (2000)]

  15. `Refutability Revamped: How Quantum Mechanics Saves the Phenomena', Erkenntnis 56 (2003) 189-211.

  16. Review of Patrick Suppes' Representation and Invariance in Scientific Structures, Studies in the History and Philosophy of Modern Physics 35 (2004) 713-720

  17. `The Implicit Definition of the Set-Concept' , Synthese 138 (2004) 417-451.

  18. `Maxwell's Lonely War', Studies in the History and Philosophy of Modern Physics 35 (2004) 109-119.

  19. `Deflating Skolem', Synthese 138 (2005) 223-253.

  20. `Can Constructive Empiricism Adopt the Concept of Observability?', Philosophy of Science 71 (2004) 637-654.

  21. `The Deep Black Sea: Observability and Modality Afloat', British Journal for the Philosophy of Science 56 (2005) 61-99.

  22. `In Defence of Constructive Empiricism: Metaphysics versus Science', General Journal for the Philosophy of Science 39 (2008) 131-156.
    Includes a critical analysis of A.N. Maxwell's argument that science presupposes metaphysics, as expounded in his The Comprehensibility of the Universe (Oxford: Clarendon Press, 1998) and in numerous other papers of his.

  23. `De Waarneembare Wereld', Algemeen Nederlands Tijdschrift voor Wijsbegeerte 2005.

  24. `De Denkbewegingen van Harry Mulisch', Algemeen Nederlands Tijdschrift voor Wijsbegeerte 2006.

  25. `Is Quantum Mechanics Technologically Inadequate?', British Journal for the Philosophy of Science 58 (2007) 595-604.

  26. `Inconsistency in Classical Electrodynamics?', Philosophy of Science 74 (2007) 253-277.

  27. `Discerning Fermions' (co-authored with S.W. Saunders), British Journal Philosophy of Science 59 (2008) 499-548.

  28. `How to talk about unobservables' (co-authored with B.C. van Fraassen), Analysis 68.3 (2008) 197-205.

  29. `Discerning Elementary Particles' (co-authored with M.P. Seevinck), Philosophy of Science 76 (2009) 179-200.

  30. `The Insidiously Enchanted Forrest' (Review Essay of B.C. van Fraassen's Scientific Representation, OUP, 2008),
    Studies in the History and Philosophy of Modern Physics 40 (2009) 268-272.

  31. `Whithering Away, Weakly',
    Synthese 180 (2011) 223-233.

  32. `Reflections on a Revolution at Stanford',
    to appear in: Synthese 18? (2011).

  33. `The Characterisation of Structure: Definition versus Axiomatisation', in: The Present Situation in the Philosophy of Science, F. Stadler et al. (eds.), Dordrecht: Springer Verlag, 2010.

  34. `Kant en Keus. Een Ontogenese van de Paradox van Banach & Tarski', Algemeen Nederlands Tijdschrift voor Wijsbegeerte (2010), Nr. 2.

  35. `Cantor-Von Neumann Set-Theory', Logique et Analyse 213 (2011) 31-48.

  36. `How to Defeat Wuthrich's Abysmal Embarrassment Argument against Space-Time Structuralism', to appear in: Philosophy of Science (2011), PSA 2010 Proceedings.

Submissions or work in progress that will be submissions:

  1. `A Logical Approach to Physical Systems'
  2. `Cantor-Von Neumann set-theory'
  3. `A Decent Description of Aspect's Experiment'
  4. `Intentionality and Constructive Empiricism' [with F. Buekens], re-re-submitted to Erkenntnis, 2011.
  5. `Understanding with and without Explanation' [with A. Nounou], submitted to Synthese, 2011.
  6. `Space-Time Structuralism', in preparation.
  7. `The Rise of Relationals', re-submitted to Mind, 2011.
  8. `Circular Discernment in Completely Extensive Structures and How to Avoid such Circles Generally', submitted to Studia Logica, 2011.
  9. Cantor's Paradise and Von Neumann's Theory (book)
  10. Identity for Philosophers (book)


  1. Het gebruik van voorletters ten gunste van voornamen

Back to Contents.

4. Selected Other Publications (in Dutch)

  1. `De Tao van Capra',
    Hollands Maandblad
    7/8, 9 (1987);
    `De Tao van Frida'
    Hollands Maandblad 1 (1988)

  2. `De Literaire Oorlog. Over vals en echt in de polemiek'
    Hollands Maandblad, 4, 5/6 (1989).

  3. `Supersnaren' (met F.A. Bais), Natuur & Teckniek (1991)

  4. In Nederlands Tijdschrift voor Natuurkunde:
    `Can Schin's description of the EPR-paradox be considered complete?' (met H.W. de Regt) 58 (1992);
    `Krenten uit Princeton' 60 (1994);
    `Dick's Doolhof' 61 (1995);
    `De Quantisatie-Controverse' 67 (2001) 110-115;
    `Roeren in Rust' 67 (2001) 334-335;

  5. `Stephen Hawking, orakel tussen de wielen'
    Hollands Maandblad 1 (1996).

  6. `De Geniale Denker'
    Hollands Maandblad 8/9 (1996).
    [Bevat een definitie van `denkgenie']

  7. `Het Bloedbad' [kort verhaal], Hollands Maandblad 11/12 (1997).

  8. `Apoproegmena'
    [Zeer sterk verkorte versie van een reactie op Maarten 't Hart's `Over de risico's van de filosofie', door Maarten Franssen en mijzelf]
    Hollands Maandblad 5 (1998).

  9. In Amsterdamse Boekengids :
    `The Force of Symmetry', 10 (Juni 1997);
    [Vervriendelijkte versie van een aanvankelijk zeer kritische boekbespreking van Vincent Icke's The Force of Symmetry, dat overal gekraakt is behalve in het Nederlands Tijdschrift voor Natuurkunde; bevat begripsfouten en onwaarheden. Tijdschrift heet tegenwoordig de Academische Boekengids, zie 6. Additional Links.]
    `Quantum, Escher, Bub', 11 (September 1997)
    [Bespreking van Interpreting the Quantum World (1997) van Jeffrey Bub.]

  10. `Doctor Eenoog en de Taalheks. Over Hermans over Wittgenstein en versus Kazemier'
    Hollands Maandblad (Augustus 1999)
    [Dit is een bewerking van een gedeelte van een als boekje bedoeld typoscript, Het wezen van de onzin. Over Wittgenstein en Hermans ]

    `Licht en Donder. Over God en Allah in Nederland'
    Hollands Maandblad (April 2000)

    `Dagen met Jagdish Mehra'
    [Verslag van het bezoek dat de schrijver van het 2000 bladzijden tellende standaardwerk The Historical Development of Quantum Theory aan Utrecht bracht in 1991; naast de auteur maakt ook Gerard 't Hooft zijn opwachting.]
    Hollands Maandlbad (Augustus 2000)

    `Logika en Zonde. De Theologisering van Ludwig Wittgenstein'
    Hollands Maandblad (November 2000)

    `Een apofantische oerknal. Over de noodzaak van waarheid en betekenis'
    Hollands Maandblad (April 2001)

    `De grijnzende filosoof'
    Hollands Maandblad (Oktober 2001)

    `Het Verbod. Over tegenspraken, paraconsistente logica en een uitgekomen voorspelling van een filosoof'
    Hollands Maandblad (Januari 2002)

    `Harry Mulisch, 75. Over de ketelmuziek van een wereldcomponist'
    Hollands Maandblad (Oktober 2002)
    Hierin tracht ik aannemelijk te maken dat Mulisch niet kan redeneren en derhalve geen systematische wijsbegeerte bedrijft in zijn wijsgerige wonderwerk De compositie van de wereld (1980), in weerwil van zijn apodictische aankondiging in het Voorwoord.
    Deze publicatie bevat hinderlijke zetfouten: de apostrof bij 'patafysika is weg, namen van dagen en maanden zonder hoofdletter, `Aconsonant@' moet `consonant' zijn, etc.

    `Een geval van transcenditis. De ziektegeschiedenis van George Steiner'
    Hollands Maandblad (2003)

  11. `De Muur'
    [zeer kort Kafkaesk verhaal]
    Hollands Maandblad 44 (2003).

    `Filosofie op de voorpagina'
    Hollands Maandblad 45 (2004).

    Reactie op een artikel van A. Hoogland , niet verschenen in Hollands Maandblad

    Na talrijke wijzigingen stelselmatig geweigerd door redacteur Bastiaan Bommelje. Rudy Kousbroek onthulde in NRC Handelsblad dat A. Hoogland medefinancier is van Hollands Maandblad. In een ingezonden brief in NRC Handelsblad gaf Bommelje een partijtje waarheidsverdraaiing ten beste, dat hem een week later op een driedubbele afdroogpartij kwam te staan door A. Gerits, C. Andriesse en mijzelf. Zijn weerwoord werd niet meer geplaatst. In een Redactioneel van Hollands Maandblad is nooit een mea culpa van Bommelje verschenen --- wel rancuneuze en verongelijkte toespelingen. Reden voor mij om met hem te breken. Adieu Bas.

  12. In NRC Handelsblad:
    `Zcherven' (Februari 1991);
    `Spreken, Zien en Zwijgen' (18 April 1992);
    `Na de Oerknal' (13 Juni 1992);
    `Bestaan quarks werkelijk?' (20 April 1995);
    `Gewauwel, maar veelzeggend gewauwel' (2 November 1995);
    `Medicijn voor een krankzinnige theorie' (13 Juni 1996);
    `Vonken van de gelovige wetenschapper' (1 Augustus 1996);
    `Kat in de zak. De paradox van Schrödinger is nog altijd niet opgelost' (29 Maart 1997); en `Incoherentie en Intolerantie' (19 April 1997)
    [Wanhopige poging om Vincent Icke te redden van zichzelf]
    `Wijsbegeerte als dienstmaagd. Nederlandse filosofie vertoont trekken van een zwarte kousenkerk' (1 November 1997);
    `Dominee noch dienstmaagd' (1 Augustus 1998);
    `Muizen en Kikkers' (Mei 1999);
    `Onwaarneembare Elektronen' (19 Juni 1999);
    `Abstracte Nonsens. Categorieleer als nieuwe grondslagen van de wiskunde' (24 September 1999);
    `Tegen het gewauwel. Logisch Positivisme terug als Constructief Empirisme' (23 December 1999);
    `Kwantummechanica of Quantummechanika?' (13 Januari 2001);
    [Over een achterlijkheid van de jongste spellingsverandering]
    `Diep nadenken over waarheid' (7 April 2001);
    `Metafysika moet' (28 December 2002);
    `De voltooiing. Monumentale geschiedenis quantumtheorie niet zonder smetten' (2 November 2003);
    Reactie op een artikel van Willem Drees Jr Jr over de verhouding tussen wetenschap en geloof (Mei 2003)
    `Klokken en Kaarten' (niet verschenen);
    `Praten en Puberen te Cambridge', Bespreking van Klaas Landsman's Requiem voor Newton, verschenen bij uitgeverij Contact, 2005 (2 April 2005);
    `Waardeloze Wijsheid' door Menno Lievers, en een reactie: `Waardevolle Wijsheid' (Februari 2005)
    In debat met Professor J. de Mul ver Analytische versus Continentale Wijsgebeerte (December 2006-Januari 2007).

  13. In voorheen Tijdschrift voor de Geschiedenis der Geneeskunde, Natuurwetenschappen, Wiskunde en Techniek (Gewina) (sinds fusering met Belgische zustertijdschriften: Studium :
    Boekbespreking van Intellectueel Bedrog door Sokal & Bricmont; 25 (2002)
    Proefschriftbespreking van Einstein's Unification: General Relativity and the Quest for Mathematical Naturalness door J.A.E.F. van Dongen; 26 (2003)
    Laatstgenoemde werd geweigerd door Nederlands Tijdschrift voor Natuurkunde omdat er kritiek in voorkomt en er in de Redactie van dit blad kennelijk mensen zitten met lange tenen. In een telefoongesprek met een redactielid, om deze weigering toe te lichten, werd mij verzekerd dat dit redactielid wel voor plaatsing was. Jeroen van Dongen won met dit proefschrift overigens een prijs voor het beste fysisch historische proefschrift van dat jaar.

  14. In Akademische BoekenGids:
    `Een Zee van Tijd'
    Wat verscheen is niet deze versie maar een redationeel verminkte versie met bespottelijke toevoegingen en onbegrijpelijke weglatingen.

    In De Gids :
    `De onthoofding van Clio's stiefkind' (Maart 2005)

    `Gissen en Missen. Over de Psyche en de Persoonlijkheid van Karl Popper en de Geschiedenis van de XXste Eeuw' (November 2008)

Back to Contents.

5. Structures for Everyone

This is my thesis for doctorate published as a book by A. Gerits & Son (Amsterdam, 1998). For the small amount of 75 Dutch guilders (49 US $, 25 GB Pounds, 34 Euros) you can be the proud owner of a copy. Visit a bookshop that has an extensive collection of science and mathematics; or send an e-mail to the official publisher: (or visit his office: Prinsengracht 446, Amsterdam; usually open Monday-Friday, 9.00-17.00 hours).
Some information about the contents follows next: an overview of the Chapters and a Summary of the book. We mention that the book is (almost entirely) self-contained.
Several typo's and a few incorrect formulations so far have been discovered (contact the author via e-mail to receive a list).

Table of Contents

    1. Goals
    2. Philosophy of Science and Mathematics
    3. Prospectus and Contributions
    1. Prelude
    2. Standard Set Theory
    3. Set Structures
    4. The Structural View
    5. Set Models
    6. The Semantic View and the Translation View
    7. Appendix: More Set Structures
    1. Prospectus
    2. Pre-Mereological Investigations
    3. Mereological Investigations
    4. Meta-Mereological Investigations
    5. Conclusions
    6. Appendix: Proofs
    1. Prospectus
    2. Introductio Logico-Historicus
    3. The Practice of Physics
    4. Physical Theories
    5. The Sea of Stories
    6. Four Grand Physical Theories
    7. Structural Realism
    1. Introduction
    2. Transfinite Matrices and Complex Waves
    3. The Equivalence-Proof
    4. An Architecture of Quantum Mechanics
    5. Interpretations of Quantum Mechanics
    1. Plotting the Course and Reading the Chart
    2. Life in the Domain of Discourse
    3. Cantor's Paradise and Von Neumann's Constitution
    4. Category Theory
    5. Sets and Classes
    6. Structuralism
  7. Opera Consulta
  8. Summary (English) / Samenvatting (Dutch)
    Text of the English Summary follows below.


I Prelude

This slightly overcomprehensive, multidisciplinary Thesis moves on different fields of enquiry. The author's judgement of taste reads that it has become quite a castle: mathematical stone of a firm but not too abstract kind, ancient and modern, various physical towers, a number of rooms of philosophy of science and of history of science, logical-axiomatic foundations and a number of flags from the philosophy of language. It is a castle for the adventurous mind, for those who do not want to dig up continuously in the piece of earth inside the perimeter of their footsoles, but those who once in a while want to lift up their heads, dare to look around and take a stroll. Anyone who takes such an attitude seriously, like yer author, is destined to go beyond the official four-year term to write a Thesis. So it happened. Homo universalis is extinct, that is very old news, but will perhaps re-originate when we can implant CD-rommies. In the mean time homo plantaris is the measure of all things.

In this Thesis, the author has explained an answer to the question what kind of language suffices and which assumptions suffice to formulate and to shore, respectively, the whole of physical and mathematical theories --- in so far as his cerebral cortex overviews this. In brief, this Thesis is an old-fashioned quest for a founding language and a founding theory for physical and mathematical knowledge. Prior to the question whether we ought to do this lies the question whether we can do this, because if we answer the last-mentioned question negatively, then to address the afore-mentioned question is like clapping wings in a vacuum: one does not advance one scintilla. The author claims to show that it can be done. Whether something ought or ought not to be done, lies beyond the realm of theorem and proof. Parenthetically, the author thinks it ought to be done, because beauty is imperative. This is an aesthetic judgement.

The author also presents an analysis of: the founding concepts, ie class, set, structure, number, physical system and physical magnitude, of the view that guides the quest, ie the structural view, and of the philosophical thesis that motivates the quest and interprets the treasures to be discovered, ie structuralism.

II Mathematics & Philosophy

As a founding language for mathematics suffices, claims the author, a 1st-order subject-predicate language with class-variables and two logical constants: the familiar dyadic in-predicate and a monadic set-predicate (sets are a particular kind of classes). As a fouding theory, a set theory of Wilhelm Ackermann (from 1956) suffices, when extended with (a) an extension of Ackermann's comprehension schema for classes, which generates classes that contain sets only, to a comprehension schema analogous to Zermelo's axiom of separation (1908) but now for classes, and with (b) the following axioms for sets: John von Neumann's axiom of regularity and the Zermelo-Russell axiom of choice. This founding theory is baptised ARC. [The theory of Ackermann extended with Regularity en Choice.] The standard founding theory of mathematics, which is Zermelo's axiomatisation (1908) enriched with Regularity and the Von Neumann-Fraenkel-Skolem Axiom of Replacement (ZFC), has been able to carry the whole of mathematics. Until Category Theorie emerged. This branch of mathematics, created in the late 1940ies, was originally intended as a terminology to smoothen talk about algebraic-topological structures. But in the subsequent decades it became a mature mathematical theory to which F.W. Lawere (1968) ascribed foundational prospects. The methods of Category Theory, such as arguments by means of commutative diagrams, which somewhat resemble calculations by means of Feynman-diagrams in quantum field theory, have shown their face in many branches of mathematics. Today there is neither universal agreement about whether Category Theory has lived up to its founding ambitions nor about which axioms to adopt --- which is due to the absence of `natural' axioms (as in the case of Set Theory). Nevertheless most category-theoreticians are convinced that its founding capabilities are beyond dispute. The founding theory ARC is a synthesis of a logicist class theory and the Cantorian set theory, when re-interpreted as the author proposes. He has proved, on the basis of ZFC extended with an axiom asserting the existence of a non-denumerable strongly inaccessible cardinal number (i1), that ARC is consistent. (The proof of the consistency of ZFC in ARC is trivial but, conversely, the consistency of ARC is not provable in ZFC due to Gödel's Consistency Theorem: no consistent theory that encompasses arithmetic can prove its own consistency). With the aid of Albert Visser [Member of the Faculty of Philosophy of Utrecht University.] he has proved that ARC is a progressive extension of ZFC, which means that not all theorems from ARC about sets are provable in ZFC. In order to appreciate the equiconsistency-result, one needs to bear the following considerations in mind.

All set-theoretical foundations for Category Theory hitherto proposed, like the one of Saunders MacLane (one of its founding fathers) and the one of Solomon Feferman, haul i1 as a set in the domain of discourse of ZFC, as a result of which two objections rear their heads: (i) the erection of a unimaginably more encompassing cumulative hierarchy on top of the one of ZFC is comparable to killing an insect with a zillion Hydrogen bombs (we call to mind that a fly-swatter will do the job), and (ii) there is no reason why category-theoreticians ought to sing and dance in fear of the cardinality-guns of the set-theoreticians by being forced to grab in the cumulative hierarchy and observe i1. MacLane has recognised these objections --- and presumably held they are insuperable. ARC is the only theory that stands tall in the face of these objections. The pillar of this claim is the existence of so-called Levy-Vaught classes; they are obtained by arbitrary but finite iteration of the power-class operator on the class of all sets. The existence of these classes, in Ackermann's original axiomatisation (enriched with Regularity), has been proved already in 1961 by Levy and Vaught, more-or-less parenthetically as a corollary; it has been lying dormant since in a remote corner of Logic. The author kisses this corollary awake and claims that only now do we begin to understand, from a foundational perspective, its full significance. And this is how it came to be that ARC pridely parades with a face cleanly shaved by Ockam whilst the competition trips over its inaccessibly long beard.

The author pays much attention to the heuristic principles that led Georg Cantor in the XIXth century to set theory and his theory of the infinite, because the concept of a set still is the heart of the founding theory ARC. He argues that Von Neumann's axiomatisation (1922-1929, VNB) is the superior constitution of what Hilbert called `Cantor's Paradise', and not the standard axiomatisation (ZFC), as widely held in circles of mathematicians and beyond --- provided VNB is stripped of the logicist costume that Paul Bernays so ably designed for it. The author proves `half' of Von Neumann's Axiom of Limitation, interprets Von Neumann's ultimate sets als combinatorial inept, and criticises, on the basis of VNB, the interpretation of Cantor's absolute-infinite sets in the light of Principles of Reflection. He also provides a sharpened formulation of Ackermann's heuristic Limitation of Sharpness doctrine, and argues that this principle performs better with respect to Ackermann's set theory (and hence with respect to ARC), than Cantor's Limitation of Size doctrine with respect to ZFC. Finally he indicates that from Wittgenstein's perspective in the philosophy of language, the problem of trinitarian wholism, evoked by the concept of a set, dissolves. He also makes a modest attempt to calm down those who are in a state of panic after having received the message that the victorious ontological reduction of mathematical objects to sets is accompanied by heavy epistemological losses.

The author wishes to see these constributions to the foundations and philosophy of mathematics as a partial response and supplement to Micheal Hallett's epoch-making monograph Cantorian Set Theory and Limitation of Size (1984).

Furthermore, the author expounds the philsophical thesis of structuralism, which has its roots in Richard Dedekind's axiomatisation of the natural numbers in Was sind und was sollen die Zahlen? (1887), which was first clearly formulated by Bertrand Russell in The Principles of Mathematical Philosophy (1919) --- admittedly limited to ordering structures ---, and which was promulgated and practised by the magnificent Nicolas Bourbaki in his structural architecture of mathematics (from the 1930ies onwards). The extensional character of the language of mathematics in general and a fortiori of its founding theories (ZFC, VNB, ARC or some Category Theory), has as a consequence that it is, spoken from a Tarskian semantic perspective, impossible to distinguish from within a theory the isomorfic modells that make it true --- they demonstrably are semantically indistinguishable. The author proves that every set can act as referent in a model of (almost) any given theory. So there is a maximal indeterminacy of the referents of mathematical terms --- maximal ambiguity of reference, if you like. The sensational Lowenheim-Skolem Theorem (1922) teaches us that in addition the number of referents per model is maximally indeterminate (in a 1st-order language), unless finite. When we call to mind that the acceptance of a mathematical theory and of mathematical axioms never can be more than a free decision, then the young Russell's conclusion seems inescapable that mathematical knowledge is the kind of knowledge ``of which we neither know that it it is true nor what it is about''. Unless we embrace, with the mature Russell, the thesis of structuralism, according to which the subject-matter of mathematics is structure. The extremely general concept of structure of Bourbaki's accommodates a defence of this thesis as meaningful and plausible for the whole of mathematics. The awesome set-theoretical architecture of mathematics realised by Bourbaki over the decades is however transportable without much ado to a class-theoretical architecture of the whole of extant mathematical knowledge, on the basis of a class-theoretical extension of Bourbaki's set-theoretical concept of structure. This signals the move from VNB or ZFC to ARC.

The author demonstrates the existence of this architecture like Diogenes demonstrated the existence of locomotion: by moving about continuously. After a mental marathon he leads us by the hand through Bourbaki's general theory of structure. He also proves that `Cantor's paradise' exists as a class-theoretical structure among the other class-theoretical structures in his founding theory ARC; in this manner Cantor's Paradise is positioned in the new architecture of mathematics which encompasses Bourbaki's architecture.

III Physics & Philosophy

Physical theories are drenched in mathematics. The mathematical theories the physicist employs live however without exception in a first droplet of the cumulatieve hierarchy Psi[Om] of all sets in ZFC. But physical theories are not identical to mathematical theories. To express that physical theories are about physical reality, the author proposes a founding language that includes, besides sets and in, also physical system-variables and a dyadic subsystem-predicate. Many concepts from physics, such as a composite system, indistinguishable physical systems, a subsystem and physical magnitude are then rigourously definable. Physical systems one thus adds to the domain of discourse of ZFC as Zermelonian Urelementen, and they can, in combination with subsys, be axiomatised with so-called mereological axioms (we jointly denote them by M or Mp). One can postulate there are in the domain of discourse as many physical systems as there are sets -- which is a bit too much of the good stuff --, or that there is at least a set containing frakp physical systems, where frakp is an appropriately chosen cardinal number. The author proves that both ensuing deductive extensions of ZFC, denoted by ZFCM and ZFCMp, respectively, are equiconsistent to, and conservative over ZFC. These results demonstrate that taking physical systems seriously is logically impeccable.

The motivation for these mereological extensions of Set Theory simply is that physical systems, from kaons to the cosmos, from oxygen molecules to the the oceans of the world, from semi-conductors to binary stars, et cetera ad libitum, form the subject-matter of physics as we know it. Without physisical systems there is no physics. We remark that `chemical systems' and `biological systems' (organisms) also are physical systems. Whether for these monadic physical system-predicates one must add new axioms is an open question (presumably only for `biological' physical systems). There is no reaon why the foundation programme here promulgated would run afoul when we reach the frontier of physics.

Of course we can also add the mereological axioms to ARC. Then we obtain a founding language for mathematics and physics: a 1st-order predicate-logical language with class-variables, physical system-variables, the dyadic in-predicate (to the right of which we always find a class), the monadic set-predicate for class and the dyadic subsys-predicate between physical systems only. Like it is possible to characterise mathematical theories with a class-theoretical structure-predicate, this is also possible for scientific theories. In the 1950ies, Patrick Suppes was the first to see this and promulgated a research programme to axiomatise scientific theories with set-theoretical predicates. This method evades a formalisation of the theoretical language. This is the first characterisation of scientific theories alternative to the formal-linguistic characterisation of the Logical-Positivists (Rudolf Carnap cs). The addition of physical systems accommodates a rigourous definition of the concept of a physical structure and to characterise physical theories with structure-predicates. The author presents as examples: Newtonian particle mechanics, Newton's theory of universal gravitation, Einstein's special and general theory of relativity, algebraic relativistic quantumfield theory, and more extensively quantum mechanics. He calls attention to a few inconveniences (contradictions) in the impressive treatise The Logical Structure of Mathematical Physics (1971) about the structural view on physical theories of Suppes' pupil Joseph Sneed. The author also attempts to capture general notions of the philosophy of science rigourously in the founding language, such as: confirmation, refutation and `this experiment is relevant for that theory'.

The thesis of structuralism applied to physics reads that physics provides us with knowledge of the kind of physical structures that inhabit the world. This thesis originated in Russell's mind, caused by his discovery that we never can know more than structure. The mathematician Max Newman gave an example to illustrate that we cannot know the structure of unobservable physical systems on the basis of experimental data alone. The author proves the general theorem, connects this result with Putnam's `model-argument' (slightly different than William Demopoulos en Michael Friedman have done, in 1985), and proposes to dissolve the ensuing reference paradox linguistically by pointing out that the use of language is constitutive for the determination of the referents of referring terms.

A separate Chapter is devoted to the application of the structural view to quantum mechanics. By applying this view to matrix mechanics of Max Born, Pascual Jordan and Werner Heisenberg (1924-1925), to wave mechanics of Erwin Schrödinger (1926) and to quantum mechanics as axiomatised by Von Neumann (1932), the author reaches the conclusion that Schrödinger's (and Carl Eckart's) famous equivalenc-proof (1926) of matrix mechanics and wave mechanics is not foolproof, and that moreover, at the time, these two theories were not `equivalent' (isomorphic), in contradistinction to what is generally held among physicists and among historians of quantum mechanics. The analysis of the relevant `old' papers also brings into the lime-light that wave mechanics, in addition to Schrödinger's `interpretation' of the wave-function as a charge density, characterises physical systems in a significantly different manner in comparsion to matrix mechanics. This explains why Schrödinger employed a non-standard notion of `equivalence'. If we adopt Schrödinger's non-standard notion of `equivalence', the equivalence-proof remains invalid. The author argues that Schrödinger's proof breaks down when it hits the so-called moment problem, which involves the determination of a wave function by means of a denumerable sequence of definite Stieltjes-integrals. Only after Von Neumann entered the stage, everything turned out to be all right, because Von Neumann simply did not encounter this problem (due to the fact that relevant notion of `equivalence' here is the standard-notion of isomorphism). Von Neumann's celebrated Spectral Theorem (1929) plays the leading part; it makes a theory possible of unbounded Hilbert-space operators, which are indispensible for quantum mechanics --- like any mathematical physicists will have expected. Von Neumann actually proposed, like Dirac, a different, quantum-mechanical characterisation of physical systems, which is standaard ever since: the so-called state-observable characterisation.

So, under, around and above the equivalence-proof and Von Neumann's successful endeavour to make quantum mechanics mathematically respectable, subtle motions are performed in the field of physics and mathematics. The author claims to be the first to have historically charted these motions and their interaction. He also sketches, building on his internal-historical analysis, how to erect on Von Neumann's postulates for quantum mechanics, whenever desired with certain adjustments as proposed in recent decades, an architecture of quantum-mechanical structures, and how one should think of `an interpretation of quantum mechanics' from the perspective of the structural view.

Finally it is notewordy to remark the following corollary of the proved consistency of the the mereological extensions ZFCM and ZFCMp of ZFC relative to ZFC: the consistency of Newtonian particle mechanics, Newton's theory of universal gravitation, Einstein's special and general theory of relativity, algebraic relativistic quantumfield theory, and quantum mechanics has now been proved, because the set-extensions of the structure-predicates that characterise these physical theories in ZFCM or in ZFCMp, nota bene with fysical systems, are not identical to the empty set.

© F.A. Muller and A. Gerits & Son b.v.

Back to Contents.

6. Additional Links

Besides the links of the current master home page, here are a few other ones.

Back to Contents.