- Curriculum Vitae
- Experience and Skills
- Selected Academic Publications
- Selected Other Publications
- Structures for Everyone
- Additional Links
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
Het Bloedbad and
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
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
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
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
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.
AOS: foundations and philosophy
of physics, of mathematics and of logic,
philosophy of science.
AOC: general philosophy, history of philosophy, analytic
philosophy, physics, mathematics.
AOI: literature, music, art, aesthetics.
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.
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.
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.
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.
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).
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.
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.
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
Languages: Dutch (native language),
French and German (semi-fluent), Latin (basic),
and Arabic (some oral ability).
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).
From November 1991 until 1995 at Utrecht University, I
initiated and organised,
together with a few undergraduate students, the
Foundations of Physics Student Colloquium.
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.
Taught the course `Foundations and Philosophy of
Quantum Mechanics' for 3rd/4th year students of
Utrecht University (September-December 2000).
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.
- Taught a course at the
(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).
- Organised (2000-now) national reading group in
the philosophy of science and of physics.
- Assisted Prof.dr.mr. H. Philipse in organising
a national Epistemology Seminar (2002-2004), every
two weaks held in Utrecht.
Taught a Bachelors courses on Philosophy of Matter and
Philosophy of Science at
Erasmus University Rotterdam (Fall 2004, Fall 2005)
Taugth a Bachelors course on Philosophy of Mind at
Erasmus University Rotterdam (Fall 2006, Fall 2007)
Taught a Masters course `Unity and Disunity of Science', together
with dr. Caterina Marchionni at Erasmus University Rotterdam
(Spring 2007, Fall 2007)
Taught a Masters course `Truth' at Erasmus University Rotterdam
(Spring 2007), with Simon Blackburn from Cambridge University
Participated in the Bachelors course Film & Philosopy
at Erasmus University Rotterdam
(Fall 2005, Fall 2006, Fall 2007)
Received VIDI-funding of the National Foundation for
Scientific Research (Dutch: NWO) for `A world of stuctures'.
(Budget: 600.000 Euro, 2006--2010).
Since January 2010 president of the Dutch Society for the Philosophy of Science (DSPS), in
Dutch: Nederlandse Vereniging voor WetenschapsFilosofie (NVWF).
Organised with Tim de Mey a masters course Analytic Metaphysics at Erasmus University
- Orgainised a new masters course Philosophy of Matter at Erasmus University Rotterdam.
Back to Contents.
`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),
`Fragmentation of Single-Particle Strength and the Validity of
the Shell Model',
Nuclear Physics A531 (1991) 253--284 (co-author).
`On the Principle of Relativity',
Foundations of Physics Letters 5 (1992) 591-596.
`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.
`Philosophy of Physics for Pedestrians', Studies
in the History and Philosophy of Modern Science
25 (1994) 505-509.
Stochastic Einstein Locality in Algebraic Quantum Field
Theory', International Journal of Theoretical Physics
33 (1994) 91-102.
`Is Lorentz-covariant Quantum Field Theory Stochastic Einstein
`Is Lorentz-covariant Quantum Field Theory Stochastic Einstein
Local?' Philosophy of Science 61
`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.
`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.
`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.
Structures for Everyone. Contemplations
and proofs in the foundations and philosophy of
physics and mathematics,
(PhD-Thesis published as book, November 1998).
`Sets, Classes and Categories',
British Journal of the Philosophy of Science
52 (2001) 539-573.
`Disunity in Unity',
Erkenntnis 55 (2001) 132-143.
[Review essay of Margaret Morrison's
Unifying Scientific Theories. Physical Concepts and
Mathematical Structures (2000)]
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)]
`Refutability Revamped: How Quantum Mechanics Saves the Phenomena',
Review of Patrick Suppes' Representation and Invariance in
Scientific Structures, Studies in the History and Philosophy
of Modern Physics 35 (2004) 713-720
`The Implicit Definition of the Set-Concept' ,
Synthese 138 (2004) 417-451.
`Maxwell's Lonely War', Studies in the History and Philosophy of
Modern Physics 35 (2004) 109-119.
`Deflating Skolem', Synthese 138 (2005) 223-253.
`Can Constructive Empiricism Adopt the Concept of Observability?',
Philosophy of Science 71 (2004) 637-654.
`The Deep Black Sea: Observability and Modality Afloat',
British Journal for the Philosophy of Science 56 (2005) 61-99.
`In Defence of Constructive Empiricism: Metaphysics versus Science',
General Journal for the Philosophy of Science 39 (2008)
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.
`De Waarneembare Wereld',
Algemeen Nederlands Tijdschrift voor Wijsbegeerte 2005.
`De Denkbewegingen van Harry Mulisch',
Algemeen Nederlands Tijdschrift voor Wijsbegeerte 2006.
`Is Quantum Mechanics Technologically Inadequate?',
British Journal for the Philosophy of Science 58 (2007) 595-604.
`Inconsistency in Classical Electrodynamics?',
Philosophy of Science 74 (2007) 253-277.
`Discerning Fermions' (co-authored with S.W. Saunders),
British Journal Philosophy of Science 59 (2008) 499-548.
`How to talk about unobservables' (co-authored with B.C. van Fraassen),
Analysis 68.3 (2008) 197-205.
`Discerning Elementary Particles' (co-authored with M.P. Seevinck),
Philosophy of Science 76 (2009) 179-200.
`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.
`Whithering Away, Weakly',
Synthese 180 (2011) 223-233.
`Reflections on a Revolution at Stanford',
to appear in: Synthese 18? (2011).
`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.
`Kant en Keus. Een Ontogenese van de Paradox van Banach & Tarski',
Algemeen Nederlands Tijdschrift voor Wijsbegeerte (2010), Nr. 2.
`Cantor-Von Neumann Set-Theory', Logique et Analyse 213 (2011) 31-48.
`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:
`A Logical Approach to Physical Systems'
`Cantor-Von Neumann set-theory'
- `A Decent Description of Aspect's Experiment'
- `Intentionality and Constructive Empiricism' [with F. Buekens], re-re-submitted to Erkenntnis,
- `Understanding with and without Explanation' [with A. Nounou], submitted to Synthese, 2011.
- `Space-Time Structuralism', in preparation.
- `The Rise of Relationals', re-submitted to Mind, 2011.
- `Circular Discernment in Completely Extensive Structures and How to Avoid such Circles Generally',
submitted to Studia Logica, 2011.
- Cantor's Paradise and Von Neumann's Theory (book)
- Identity for Philosophers (book)
Het gebruik van voorletters ten gunste van voornamen
Back to Contents.
`De Tao van Capra',
Hollands Maandblad 7/8, 9 (1987);
`De Tao van Frida'
Hollands Maandblad 1 (1988)
`De Literaire Oorlog. Over vals en echt in de polemiek'
Hollands Maandblad, 4, 5/6 (1989).
`Supersnaren' (met F.A. Bais), Natuur & Teckniek
Nederlands Tijdschrift voor Natuurkunde:
`Can Schin's description of the EPR-paradox be
considered complete?' (met H.W. de Regt)
`Krenten uit Princeton' 60 (1994);
`Dick's Doolhof' 61 (1995);
`De Quantisatie-Controverse' 67 (2001) 110-115;
`Roeren in Rust' 67 (2001) 334-335;
`Stephen Hawking, orakel tussen de wielen'
Hollands Maandblad 1 (1996).
`De Geniale Denker'
Hollands Maandblad 8/9 (1996).
[Bevat een definitie van `denkgenie']
`Het Bloedbad' [kort verhaal],
Hollands Maandblad 11/12 (1997).
[Zeer sterk verkorte versie van een reactie
op Maarten 't Hart's `Over de risico's van de
Maarten Franssen en mijzelf]
Hollands Maandblad 5 (1998).
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.]
`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
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
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)
[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
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.
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
`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'
In debat met Professor J. de Mul ver Analytische versus Continentale Wijsgebeerte (December 2006-Januari 2007).
In voorheen Tijdschrift voor de Geschiedenis der Geneeskunde, Natuurwetenschappen,
Wiskunde en Techniek (Gewina) (sinds fusering met Belgische zustertijdschriften:
Boekbespreking van Intellectueel Bedrog door Sokal & Bricmont;
Proefschriftbespreking van Einstein's Unification: General Relativity and
the Quest for Mathematical Naturalness door J.A.E.F. van Dongen;
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.
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.
This is my thesis for doctorate published as a book
by A. Gerits & Son
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
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
- Philosophy of Science and Mathematics
- Prospectus and Contributions
- SET STUCTURES
- Standard Set Theory
- Set Structures
- The Structural View
- Set Models
- The Semantic View and the Translation View
- Appendix: More Set Structures
- MEREOLOGICAL STUCTURES
- Pre-Mereological Investigations
- Mereological Investigations
- Meta-Mereological Investigations
- Appendix: Proofs
- PHYSICAL STUCTURES
- Introductio Logico-Historicus
- The Practice of Physics
- Physical Theories
- The Sea of Stories
- Four Grand Physical Theories
- Structural Realism
- V. QUANTUM STUCTURES
- Transfinite Matrices and Complex Waves
- The Equivalence-Proof
- An Architecture of Quantum Mechanics
- Interpretations of Quantum Mechanics
- SETS, CLASSES AND CATEGORIES
- Plotting the Course and Reading the Chart
- Life in the Domain of Discourse
- Cantor's Paradise and Von Neumann's Constitution
- Category Theory
- Sets and Classes
- Opera Consulta
- Summary (English) / Samenvatting (Dutch)
Text of the English Summary follows below.
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,
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,
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
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
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
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
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
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
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
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
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
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
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.
links of the
current master home page, here are a few other ones.
- General Sites
- Some home pages of or dedicated to individuals
Polemic and prolific Popperian from Israel
Philosopher most often referred to.
If philosophy is a collection of footnotes to someone's work, this is it.
Staggering home page of a staggeringly productive
mathematical physicist, about all conceivable aspects of physics,
and about the `reverse Sokal-hoax'
The International Berkeley Society
Esse est percipi
Mainly philosopher of language
Referred to as `the Mersenne of philosophy of physics'
Criterion: a paper is a paper belonging to the philosophy of physics iff Jeremy
Butterfield is in the acknowledgements.
Luminary of the Vienna Circle
Philosopher of mind who has discovered a position that no one believes
The Noam Chomsky Archive
Also an engaging and self-sufficient critic of the policy
of every single government of the US.
Daniel Dennett and some
Philospher of mind who does not believe in consciousness but does
not admit it
The Einstein Archive and
Einstein on line
Physicist who became an icon of what he was: a genius.
For his two epoch-making contributions to physics he did not
receive the Nobel-prize.
The Everett FAQ
The Feynman Web Ring
B.C. van Fraassen
Creator of Constructive Empiricism, a view of science that drives
realists against the wall because they cannot confute it.
The Galileo Project
The scientist who separated science from religion to protect the last-mentioned
from the afore-mentioned. We hold him fully responsible.
The Kurt Gödel Society
Gerard 't Hooft
Inventor of Nobel-prize winning renormalisation procedure for Non-Abelian gauge theories
has a Sesame-Street-like homepage.
The Immanuel Kant site
The Kubrick Site
- Joseph K.'s
Kafka Page en Leni's
Franz Kafka Page.
For mathematical physics.
Dutch analytical philosopher; publishes mainly in the newspaper
Comment. This site includes papers on Lucas' Gödelian argument which purports
to show that human consciousness cannot operate like a computer, i.e. does not run
according to some algorithm. The physicist
Roger Penrose achieved world-wide popularity with this argument
in his The Emperor's New Mind (1989), although it was already discussed at
some length ten years earlier, by Douglas
Hofstaedter in his Gödel, Escher, Bach: an eternal golden braid
life and a
site which focuses on his contributions
to the computer.
Newton Resources and
Virtual Newton Museum
Friedrich Nietzsche Society and Helmut
Julius Robert Oppenheimer
The virtual Pauli exhibition
Charles S. Peirce
The Karl Popper Web.
believe that Popper was the greatest philosophers since
critical assessment of Popper.
Founding father of time logic
One of the best 20th-century philosophers of the USA;
changed his mind many times
Huge site, maintained by his son.
The Ayn Rand Institute
Henk W. de Regt
Dutch philosopher, lives on the border of
general philosophy of science and philosophy of physics
A provocative Dutch rationalist of sorts
The Bertrand Russell Society
Philosopher of mind who believes in consciousness and admits it
Philosopher of mathematics
Philosopher of many subjects and
confutator of claims of believers.
Full of responses to critics of his by now
Mathematical physicist. Advice on which topics not to study.
Simon Wolfe Saunders
Philosopher of physics
The Voltaire Society of America
structural realist from Cyprus, PhD under John Worrall (LSE),
roaming around on the European continent.
My impossible co-author.
Theoretical physicist who also writes about the history of physics
and issues of general interest (eg. the `Science Wars')
Process Philosophy of A.N. Whitehead.
Ludwig Wittgenstein, includes links to other
Wittgenstein-sites. Summaries of
books on Kripke's provoking analysis of Wittgenstein.
Austrian Wittgenstein Society.
Darwin Awards Archive
Contains stories of specimen of our species who sacrificed
themselves to enhance the quality of the human gene pool.
abe-books (global network of antiquarian book sellers)
are the places to look for books which are out
The American Philosophical Association
is the place to look for
philosophy jobs in the United States of America.
Atheism is not a system of beliefs, but the sensible rejection
of a wide variety of beliefs in the existence of Supernatural
Beings, Almighty Powers, Benign Creators and
similar figments of the imagination.
A wide variety of arguments in favour of atheism and
against religion are explained on
More like a system beliefs is
review site ,
another review sit
De Brakke Hond is a Dutch Literary Journal
which sometimes contains funny angry reviews of
De Academische Boekengids is supposed to be a Dutch
The New York Review of Books
Advertisement of Dutch cultural-literary periodical
Hollands Maandblad .
This is the only Dutch
cultural-literary journal which is not subsidised;
occasionally I publish
in it (see Selected Other Publications).
It contains (very) short stories, poems, drawings, polemics, essays
on a wide variety of topics and short notes, which usually
comment on current cultural-literary affairs. It was founded in 1959
by the late K.L. Poll, a literary critic and poet who also invented
the weekly cultural-literary supplement to
the newspaper where he worked (NRC Handelsblad, today
every respectable newspaper has such a supplement);
since 1994, Bastiaan Bommelje is editor in chief.
Carl Sagan's Baloney Detection Kit.
- Some useful sites
Back to Contents.