Preface to Festschrift Vintage Enthusiasms
In
2009 I was elected a Fellow of the Royal Society of Canada.
My
Erdös number is 3. According to the Mathematics Genealogy Project, I am
one of the 30000 or so mathematical descendants of Gauss and 740 of G. H. Hardy.
Translation of Grothendieck’s
The New Universal Church
My
research interests and published work have been wide-ranging, embracing
mathematical logic, philosophy and foundations of mathematics, classical set
theory, sets and classes as many, model theory, constructive mathematics,
Boolean algebras, lattice theory, category theory, topos
theory, restricted second-order and infinitary languages, large cardinals, the
axiom of choice, functional analysis, incompleteness, local set theory, type
theory, smooth infinitesimal analysis, the development of the continuum
concept, the logic of perception, quantum logic, foundations of quantum theory,
causal sets in spacetime theory,
the thought of Hermann Weyl, history and conceptual development of
mathematics, Frege’s theorem, type-reducing correspondences, oppositions
and paradoxes, mathematics and aesthetics, philosophy in literature, the nature
of consciousness.
1956-57 Drew College
Preparatory School, San Francisco, CA, USA
1957-58 Lick-Wilmerding
High School, San Francisco, CA, USA
1958-61 Millfield
School, Street, Somerset, UK
1962-65 Open Scholar, Exeter College,
Oxford. B.A., Mathematics, 1965
1965-68 Senior Scholar, Christ Church,
Oxford. D.Phil., Mathematics, 1969
12. Oppositions
and Paradoxes: Philosophical Perplexities in Science and Mathematics,
Broadview Press, 2016.
11. Intuitionistic Set
Theory. College Publications, 2014.
10. Perpetual Motion: The Making
of a Mathematical Logician. Llumina Press,
2010.
9. The Axiom of Choice. Front Matter. College
Publications, 2009.
8. The Continuous and the Infinitesimal in Mathematics and
Philosophy. Polimetrica, 2005.
7. (With D. DeVidi and G. Solomon†)
Logical
Options: An Introduction to Classical and Alternative Logics. Broadview
Press, 2001.
6. The Art of
the Intelligible: An Elementary Survey of Mathematics in its Conceptual
Development. Kluwer,
1999.
5. A
Primer of Infinitesimal Analysis. Cambridge University Press, 1998. Second
Edition, 2008. Review by W.A. J.
Luxemburg.
4. Toposes
& Local Set Theories: An Introduction. Clarendon Press, Oxford, 1988. Reprinted by Dover,
2008. Reviews
(?) thereof.
3. Boolean-Valued
Models and Independence Proofs in Set Theory. Clarendon Press, Oxford,
1977. 2nd edition, 1985. 3rd edition, 2005. Paperback
edition, 2011.
2. (With M. Machover). A
Course in Mathematical Logic. North-Holland, Amsterdam, 1977. 4th
printing, 2003.
1. (With A. B. Slomson). Models and
Ultraproducts: An Introduction. North-Holland, Amsterdam, 1969.
Reprinted by Dover, 2006.
Unpublished
books:
The Exploits of Joel Bennhall, Chrononaut and Scholar
Philosophy in Literature: A
Survey of the Novel of Ideas.
82. (With G. Badia) A Parametrized Axiomatization for a Large Number of Restricted Second-Order Languages. Journal of Logic and Computation, 5 August 2023. https://academic.oup.com/logcom/advance-article/doi/10.1093/logcom/exad050/7237691
80.
The
Continuum and the Evolution of the Concept of Real Number. Handbook
of the History and Philosophy of Mathematical Practice, pp. 1-91. Springer,
2021
79.
Reflections on
the Notion of “Structure” and Categories, in Structures Mères: Semantics, Mathematics and Cognitve Science, A. Peruzzi and S. Zipoli,
eds. Springer, 2020, pp. 1-17.
77. Categorical
Logic and Model Theory, in Categories for the Working
Philosopher, E. Landry, ed. Oxford University Press, 2017, pp. 113-136.
76.
Contribution to Philosophy of Logic: 5 Questions,
Adajian and Lupher, eds, Automatic Press, 2016.
75. Reflections on Mathematics and
Aesthetics. Aisthesis. Pratiche, linguaggi
e saperi dell’estetico,
[S.l.], v. 8, n. 1, p. 159-179, May 2015.
74. The Axiom of Choice in an Elementary Theory of
Operations and Sets, in Analysis and Interpretation in the
Exact Sciences: Essays in Honour of William Demopoulos. M. Frappier, D.
Brown and R. DiSalle, eds., Springer 2012
73. Types, Sets and Categories, in Sets
and Extensions in the 20th Century, A.Kanamori, D.
Gabbay and J. Woods, eds., Handbook of the History of Logic, Elsevier 2012.
72. (As Joel Bennhall) Inscrutable Harmonies: The Continuous and
the Discrete in the Playing of Jascha Heifetz and Glenn Gould, in Vintage Enthusiasms:
Essays in Honour of John L. Bell, D. Devidi, P.
Clark and M. Hallett, eds., Springer 2011.
71. The Axiom of Choice in the
Foundations of Mathematics, in Foundational Theories of Classical
and Constructive Mathematics, Giovanni Sommaruga, ed., Springer, 2011.
70. Cohesiveness, Intellectica, 41, 2009.
69. (With H. Korté) Hermann Weyl, Stanford Encyclopedia of
Philosophy, 2009
68.
The Axiom of Choice and the
Law of Excluded Middle in Weak Set Theories, Mathematical Logic
Quarterly, 54, no. 2, 2008.
67.
The Axiom of Choice,
Stanford Encyclopedia of Philosophy, 2008.
66.
Contribution to Philosophy of Mathematics: 5
Questions, Hendricks and Leitgeb, eds, Automatic Press, 2007
65.
Incompleteness in a
General Setting. Bulletin of Symbolic Logic 13, 2007.
64.
Cover Schemes, Frame-Valued Sets and Their Potential
Uses in Spacetime Physics, in Spacetime Physics Research Trends, Horizons in World Physics, Volume 248,
Nova Science Publishers, New York, 2007.
62.
Absolute and Variable Sets in Category Theory,
in What is Category Theory?
Polimetrica 2006
61.
Choice Principles in
Intuitionistic Set Theory , in A Logical Approach to Philosophy,
Essays in Honour of Graham Solomon, D. DeVidi
and T. Kenyon, eds., Springer, 2006.
59.
Divergent Concepts of the
Continuum in 19th and Early 20th Century Mathematics and
Philosophy, Axiomathes 15,
2005.
58.
The Development of Categorical Logic,
in Handbook of Philosophical Logic, Volume 12. Springer, 2005.
57. Continuity and Infinitesimals,
Stanford Encyclopedia of Philosophy, 2005.
56.
Oppositions and Paradoxes in
Mathematics and Philosophy, Axiomathes 15, 2005.
55.
Observations on Mathematics, in Mathematics as Story,
Proceedings of 2003 Fields Institute Conference, UWO, 2004.
53.
Russell’s Paradox and Diagonalization in a
Constructive Context, in 100 Years of Russell’s
Paradox, Munich 2001, Walter de Gruyter, 2004.
52.
Hermann Weyl's Later Philosophical Views: His
Divergence from Husserl, in Husserl and the Sciences, R.
Feist, ed. U. of Ottawa Press,
2003.
51.
Some New Intuitionistic
Equivalents of Zorn’s Lemma, Archive for Mathematical
Logic, 42, Number 8, 2003.
50.
Time and Causation in Gödel's Universe,
Transcendent Philosophy 3, 2002.
49. Observations on Category Theory, Axiomathes 12, 2001.
48.
The Continuum in Smooth Infinitesimal Analysis,
in Reuniting the Antipodes-Constructive and Nonstandard Views of the
Continuum. Symposion Proceedings, San Servolo/Venice, Italy, 1999. U. Berger, H. Osswald and
P. Schuster, eds. Kluwer, 2001.
47.
Continuity and the Logic of
Perception, Transcendent Philosophy 1, no. 2, 2000.
46.
Hermann Weyl on Intuition and
the Continuum, Philosophia Mathematica (3), 8, 2000.
45. Sets and Classes as
Many, Journal of Philosophical Logic, 29, 2000.
44.
Infinitary Logic, Stanford
Encylopedia of Philosophy, 2000
43.
Finite Sets and Frege Structures, Journal
of Symbolic Logic, 64, no. 4,1999.
42.
Frege's Theorem in a Constructive Setting,
Journal of Symbolic Logic,
64, no. 2, 1999.
41.
Boolean Algebras and Distributive Lattices Treated
Constructively, Math. Logic Quarterly 45, 1999.
40. Boolean
Algebras, Routledge Encyclopedia of Philosophy, 1998.
39.
Zorn’s Lemma and Complete Boolean Algebras in
Intuitionistic Type Theories, Journal of Symbolic Logic 62, no.
4, 1997.
38.
(With S. Gebellato) Precovers, Modalities, and Universal Closure Operators in
a Topos, Math. Logic Quarterly 42,
1996.
37.
Polymodal Lattices and Polymodal Logic, Math. Logic Quarterly 42,
1996.
36.
(With W. Demopoulos) Elementary Propositions and Independence, Notre
Dame J. of Formal Logic, 37, no. 1, 1996.
35.
Logical Reflections on the Kochen-Specker Theorem,
in Perspectives on Quantum Reality, R. Clifton, ed., Kluwer, 1996.
34.
(With R. Clifton†) QuasiBoolean
Algebras and Simultaneously Definite Properties in Quantum Mechanics. Int. J. of Throretical
Physics 34, 12, 1995.
33.
Infinitesimals and the Continuum, Mathematical
Intelligencer , 17, no. 2, 1995.
32.
Type-Reducing Correspondences and Well-Orderings:
Frege's and Zermelo's Constructions Re-examined,
Journal of Symbolic Logic, 60, no. 1, 1995.
31.
Frege's Theorem and the Zermelo-Bourbaki
Lemma. Appendix to Frege's Philosophy of Mathematics, W.
Demopoulos, ed. Harvard U.P., 1995
30. Fregean Extensions of First-Order Theories,
Math. Logic Quarterly, 40, 1994. (Also reprinted in W.
Demopoulos, ed. Frege's Philosophy of Mathematics, Harvard U.P. 1995)
29.
Hilbert's Epsilon Operator in Intuitionistic Type
Theories,
Math. Logic Quarterly, 39, 1993.
28.
(with W. Demopoulos) Frege's Theory of Concepts and
Objects and the Interpretation of Second-Order Logic, Philosophia
Mathematica, (3), 1, 1993.
27.
Hilbert's Epsilon-Operator and Classical Logic, Journal of Philosophical Logic,
22, 1993.
26. Some
Propositions Equivalent to the Sikorski Extension Theorem for Boolean Algebras,
Fundamenta Mathematicae 130, 1988.
25. Infinitesimals, Synthese,
75, 1988.
24.
Logic, the Paradoxes, and the Foundations of Mathematics, LSE
Quarterly Vol.I, No.3, 1987.
23. From Absolute to Local Mathematics, Synthese 69, 1986.
22. A New Approach to Quantum Logic, Brit.
J. Phil. Sc., 37, 1986.
21. Orthospaces and Quantum
Logic. Foundations of Physics 15, 1985.
20.
Orthologic, Forcing and the Manifestation
of Attributes, Proceedings of 1981 S.E. Asian Conference in Mathematical
Logic. North Holland, Amsterdam, 1983.
19.
The Strength of the Sikorski Extension Theorem for
Boolean Algebras, Journal of Symbolic
Logic 48, 1983.
18.
(With M.F. Hallett), Logic, Quantum Logic, and
Empiricism, Philosophy of Science 49, 1982.
17. Categories,
Toposes and Sets, Synthese,
51, No.3, 1982.
16.
Some Aspects of the Category of Subobjects
of Constant Objects in a Topos, Journal
of Pure and Applied Algebra 24, 1982.
15.
Category Theory and the Foundations of Mathematics,
Brit.J.Phil.Sci. 32, 1981.
14.
Isomorphism of Structures in S-Toposes, Journal of
Symbolic Logic, 46, 1981.
13.
The Infinite Past Regained: A Reply to Whitrow, Brit.J.Phil.Sci.
Sci, 1979
12.
Boolean Extensions as Toposes, Bull. de la
Soc. Francaise de Logique, Methodologie et Phil.des Sci.
6, 1979.
11.
Uncountable
Standard Models of ZFC + V = L, in Set Theory and Hierarchy Theory,
a Memorial Tribute to Andrzej Mostowski, Springer
Lecture Notes in Math. 537,1976.
10.
A Note on Generic Ultrafilters, Zeitschr. f. Math.Logik
und Grund.der Math. 22, 1976.
9.
Universal Complete Boolean Algebras and Cardinal
Collapsing, Zeitsch. f. Math.Logik und Grund. der Math. 22, 1976.
8.
A Characterization of Universal Complete Boolean
Algebras, J. London Math.Soc. (2),
12, 1975.
7.
On Compact Cardinals, Zeitschr.f.Math.Logik
und Grund.der Math. 20.1974.
6. (With D.H. Fremlin)
A Geometric Form of the Axiom of Choice, Fund.
Math. 77, 1972.
5.
(With D.H. Fremlin), The Maximal Ideal Theorem for Lattices of Sets,
Bull. London Math. Soc., 4, 1972.
4.
On the Relationship between Weak Compactness and
Restricted Second- Order Languages, Arch. Math. Logik 15,
1972.
3.
Some Remarks on Current Mathematical Practice, in Proceedings of the
Bertrand Russell Memorial Logic Conference, Denmark, 1971. Reprinted with commentary,
Philosophy of Mathematics Education Journal, No. 38, Dec. 2021
2.
(With F. Jellett). On the Relationship between the Boolean Prime Ideal
Theorem and Two Principles of Functional Analysis, Bull. de l'Acad. Pol. des Sci., XIX, No.3, 1971.
1.
Weak Compactness in Restricted Second-Order Languages,
Bull. de l'Acad. Pol. des Sci., No.3, 1970.
Gregory
H. Moore. Zermelo’s Axiom of Choice: Its Origins, Development,
and Influence. Mineola,
N.Y.: Dover Publications, 2013. ISBN 978-0-48648841-7 (pbk).
Pp. 448 Philosophia Mathematica (2014) 22 (1): 131-134 doi:10.1093/philmat/nkt038
Mark van Atten, Pascal Boldini, Michel
Bourdeau, and Gerhard Heinzmann, eds., One Hundred Years of Intuitionism
(1907–2007): The Cerisy Conference. Basel,
Boston, Berlin: Birkhäuser, 2008. ISBN
978-3-7643-8652-8. Pp. xiii + 422 Philosophia Mathematica (2013) 21
(3): 392-399 first published online March 19, 2013 doi:10.1093/philmat/nkt004
Jean-Pierre Marquis. From a Geometrical Point of View: A Study of the History and Philosophy
of Category Theory. By:
KAO, MOLLY; FILLION, NICOLAS; BELL, JOHN. Philosophia Mathematica,
Jun 2010, Vol. 18 Issue 2.
A. Kock, Synthetic Differential Geometry, 2nd
edition, Bulletin of Symbolic Logic 13, 2, 2007.
P. Rusnock, Bolzano’s Philosophy
and the Emergence of Modern Mathematics, Philosophia Mathematica 14,
3, 2006.
F. W. Lawvere and R. Rosebrugh, Sets for Mathematics, Featured Review, Mathematical Reviews,
2003.
B.
Jacob, Categorical Logic
and Type Theory, Studia Logica, 69, 2001.
F.
W. Lawvere and S. Schanuel, Conceptual
Mathematics: A First Introduction to Categories, Minds and Machines,
5, 1995.
B.
Rotman, Ad Infinitum: The Ghost in
Turing's Machine, Philosophia Mathematica, 3, 1995.
C.
McLarty, Elementary Categories, Elementary Toposes,
Journal of Symbolic Logic, 58, 1993.
J.
Chapman and F. Rowbottom, Relative
Category Theory and Geometric Morphisms, Bull. London Math. Soc.,
25, 1993.
J.
Lambek and P.J. Scott, Introduction to
Higher-Order Categorical Logic ,Journal of Symbolic Logic, 54, 1989.
Hao
Wang, Reflections on Kurt Gödel,
Philosophical Quarterly, 39, 1989.
P.
Gibbins, The Limits of Quantum Logic,
Philosophical Quarterly, 38, 1988.
Kurt
Gödel, Collected Works I, Philosophical
Quarterly, 37, 1987.
P.
T. Johnstone, Stone Spaces, Bull.
London Math. Soc., 19, 1987.
A.
P. Morse, A Theory of Sets, Bull.
London Math. Soc., 19, 1987.
E.
Bishop and D. Bridges, Constructive
Analysis, Bull. London Math. Soc., 18, 1986.
A.
G. Hamilton, Numbers, Sets, and Axioms,
Times Higher Ed. Supp.,13 May 1983.
G.
Moore, Zermelo's Axiom of Choice, Bull. London Math.
Soc. ,15, 1983.
R.
Goldblatt, Topoi: The Categorial Analysis
of Logic, Brit. J. Phil. Sci., 38, 1982.
J.
Barwise (ed.), Handbook of Mathematical Logic , Brit.
J. Phil. Sci., 30, 1979.
Gone to the Dogs, Times Lit. Supp.,
8 July 1977.
F.
Drake, Set Theory and T. Jech, The Axiom of Choice, Brit. J. Phil.
Sci., 27, 1975.
A.
Fraenkel, Y. Bar-Hillel and A. Levy, Foundations
of Set Theory, Brit. J. Phil. Sci., 26, 1975.
S.
W. P. Steen, Mathematical Logic, Brit.
J. Phil. Sci., 23, 1972.
J.
B. Rosser, Simplified Independence
Proofs: Boolean-Valued Models of Set Theory, Bull. London Math. Soc.,
3, 1971.
Commutative Rings as
Algebras of Intensive Quantities
What is the Source of the
Commutativity of the Basic Arithmetical Operations?
Quantum Incompatibility and
Noncommutativity
The Bibliography Paradox
Revisited
Googol Plus One: A Reminiscence
My Collaboration with Bill
Demopoulos
Recollections of Logicians,
Mathematicians, and Philosophers
Answers to a
Mathematical Questionnaire
The Consolations of Zorn’s
Lemma and the Escape into Maximality
Thoughts on the Incipient Loss
of Sight through Macular Hole
Fred Hoyle: Cosmologist Extraordinaire Under “Recollections” may be found
a reproduction of my handwritten notes on Hoyle’s cosmology lectures,
Cambridge 1962.
Nuclear Weapons and Climate
Change
"Edward
Hubert Linfoot" (Obituary) Bull. London Math. Soc., 16, 1984.
"Iterated
Boolean Extensions and the Consistency of Souslin's
Hypothesis, Lecture Notes No. 10, Dept. of Mathematics, National University
of Singapore, 1982.
English
translation of "Groupes Algebriques",
by M. Demazure and P. Gabriel as "Introduction
to Algebraic Geometry and Algebraic Groups", North-Holland,
Amsterdam, 1980.
Editor
(with J. Cole, G. Priest, and A. Slomson)
"Proceedings of the Bertrand Russell Memorial logic Conference",
Uldum, Denmark, 1971, published in Leeds, 1972.
English
translation of "La Geometrie dans le Monde
Sensible" in "Geometry and Induction", by J. Nicod, Routledge
and Kegan Paul, London, 1969.
"Model-Theoretic
Axiomatization Results for Certain Restricted Second-Order Languages",
Doctoral Dissertation, Oxford 1969.
"Infinitary
Languages", Diploma Dissertation, Oxford 1966.
(With
A. B. Slomson) "Introduction to Model
Theory", Mathematical Institute, Oxford, 1965
"A
Short Survey of Phrase-Structure Grammars", Elliott Computers Technical
Report 65/122, 1965.
https://www.flickr.com/photos/152639850@N05/
Newton and Einstein as Musicians
University of Colorado Lecture
Poster
Before emigrating to
Canada in 1989, I gave invited lectures at seminars and conferences in (select
listing only) Oxford, Cambridge, London, Bristol, Sussex, Leeds, Manchester,
Open University, York, St. Andrews, Glasgow, Paris, Florence, Warsaw,
Singapore, Montreal, Milano, and Chicago.
Since emigrating to Canada I have given the following invited lectures:
Mathematics
and Aesthetics. World
Logic Day, Jan i2, 2024
The Labyrinth of the Continuum and the Concept of the
Infinitely Small. Bard
College, April 2023.
Infinitesimals and the Labyrinth of the Continuum. 2018 William
Reinhardt Memorial Lecture, Philosophy Department, University of Colorado at
Boulder, Oct. 2018.
Reflections on the Axiomatic Approach
to Continuity. Conference on Axiomatic Thinking: One
Hundred Years since Hilbert’s Address in Zurich. University of Zurich, Oct. 2017.
Reflections on Bourbaki’s
Notion of Structure and Categories. Conference on Structures Méres in Semantics,
Mathematics and Cognition. University of Florence, May 2017.
Challenging the Presuppositions of Classical Set Theory. CUNY Graduate Center,
Oct. 2016.
Hermann Weyl and Constructivism, PhilMath Intersem
6, University of Paris 7-Diderot, June 2015.
Challenging the Logical Presuppositions of Classical Set
Theory. Mathematics
Department, University of Georgia, Feb. 2015.
Hermann
Weyl's Views
on the Foundations of Mathematics
and Physics as Informed by his Philosophical Outlook, International
Conference on Hermann Weyl and the New Physics, Université
Paris-Diderot, Dec. 2014.
Reflections on Mathematics and Aesthetics, International
Conference on Aesthetics in Mathematics, University of East Anglia, Dec. 2014.
Causal Sets and Frame-Valued Set Theory, Perimeter
Institute, May 2012.
On the Cohesiveness of the Continuum, Philosophy
Department, McMaster University, November 2011.
The Continuous and the Infinitesimal, Keynote Address,
Western Canadian Philosophical Association Conference, U. of Lethbridge,
October 2011.
ETHOS: An Elementary
Theory of Operations and Sets, Mathematics Department, University of Padova,
June 2011.
“The Axiom of Choice
in a Constructive Setting”, Philosophy
Department, McGill University, April 2010; Mathematics Department, University
of Cambridge, May 2010; Mathematics Department, University of Oxford, May 2010;
Philosophy Department, University of Florence, June 2010.
“Incompleteness
in a General Setting”, Mathematics Department,
University of Manchester, May 2010.
“On the
Cohesiveness of the Continuum”, Philosophy
Department, University of Bristol, May 2010.
“Continuity and
Infinitesimals”, Mathematics Department, London
School of Economics, May 2010.
“Russell’s Paradox and
Diagonalization in a Constructive Setting” Philmath
Intersem 2010, Foundations of Mathematics: What and Why?, University of Paris, June 2010.
“What is Categorical Logic?”,
International Conference on
Philosophy and Model Theory, University of Paris, June 2010.
“Intuitionistic
Logic, Frame-Valued Sets and Evolving Spacetime.” New Directions in Foundations of Physics Conference,
Washington D. C., May 2009.
“Cosmic Origins”. Philosophy
Department, University of Ohio, October 2008
“Cohesiveness”. Philosophy
Department, University of Bucharest, July 2008
“Infinitesimals
and the Continuum in the 17th Century”. Bucharest Colloquium in Early Modern Philosophy, July 2008.
“Infinitesimals
and the Continuum in Smooth Infinitesimal Analysis”. Congress for the Humanities and Social
Science, Vancouver, June 2008.
“Dissenting
Voices: Divergent Conceptions of the Continuum in 19th and Early 20th
Century Mathematics and Philosophy”,
Ramifications of Category Theory, International Conference, University of
Florence, November 2003.
“Synthetic
Differential Geometry as a Framework for Spacetime”,
Workshop on Sheaves and Topoi in Theoretical Physics, Imperial College, London,
July 2003.
“Russell’s
Paradox and Cantor’s Diagonalization in a Constructive Setting”, A Logical Approach to Philosophy, A Workshop in
Philosophical Logic in Memory of Graham Solomon, University of Waterloo, May
2003.
“Causal
Sets and Frame-Valued Set Theory”, Perimeter
Institute for Theoretical Physics, Waterloo, Ontario, March 2003.
“Oppositions
and Paradoxes in Mathematics”, Distinguished
Guest Lecture, Ontario Philosophical Society Meeting, University of Waterloo,
November 2002.
“Infinitesimals
and the Continuum”, Philosophy Department, University of Minnesota, October 2002.
8
lectures delivered at Mini-Workshop on Foundational Theories in Mathematics,
Mathematics Department, University of Trento, September 2002.
"Comparing
the Smooth and Dedekind Reals in Smooth Infinitesimal Analysis",
Conference on Nonstandard Methods and Applications in Mathematics, Pisa, June
2002
"Cosmological
Theories and the Question of the Existence of a Creator", Symposium on
Science, Religion, and Philosophy, University of Toronto, May 2002
"Infinitesimals
and the Continuum", Department of Philosophy, University of Lethbridge,
March 2002
"Sets
and Classes as Many", Departments of Philosophy, Mathematics and Computer
Science, University of Calgary, March 2002
"Infinitesimals
and the Continuum", Department of Philosophy, University of Alberta, March
2002
"Russell's
Paradox and Diagonalization in a Constructive Context", 100 Years of
Russell's Paradox, International Conference, Munich, June 2001.
"An
Invitation to Smooth Infinitesimal Analysis", Mathematics Department,
Instituto Superiore Tecnico,
Lisbon, May 2001.
"Boolean
Algebras and Distributive Lattices Treated Constructively", Logic Group,
Instituto Superiore Tecnico,
Lisbon, May 2001.
"Time
and Causation in Gödel's Universe", 2nd International Conference on
Mulla Sadra and Comparative Philosophy, School of Oriental and African Studies,
University of London, May 2001.
"The
Status of Some Principles and Theorems of Classical Mathematics in Constructive
Set Theory", Department of Philosophy, Indiana University, March 2001.
"Hermann
Weyl's Later Philosophical Views: His Divergence from Husserl", Conference
on Husserl and the Sciences, University of Ottawa, October 2000.
"The
Natural Numbers in Constructive Set Theories", Department of Philosophy,
University of Glasgow, May 2000.
"Smooth
Infinitesimal Analysis: An Introduction", Department of Mathematics,
University of Manchester, May 2000.
"Continuity
and the Logic of Perception", International Conference on Perception
according to Mulla Sadra and Western Schools of Philosophy, Birkbeck College,
University of London, May 2000.
"Boolean
Algebras and Distributive Lattices Treated Constructively", Department of
Mathematics, University of Munich, May 2000.
"The
Natural Numbers in Constructive Set Theories", 2nd PvView
Conference on Logic and Formal Topology, Department of Mathematics, University
of Padova, April 2000.
"Boolean
Algebras and Distributive Lattices Treated Constructively", Department of
Mathematics, University of Paris VII, April 2000.
"The
Incredible Shrinking Manifold: Spacetime from the Synthetic Point of
View", Conference on Spacetime, University of Western Ontario, April 2000.
"Smooth
Infinitesimal Analysis: An Introduction", Conference on Infinitesimals:
Concepts and Applications, University of Western Ontario, October 1999.
"Hermann
Weyl on Intuition and the Continuum", Conference on Intuition in
Mathematics and Physics, McGill University, September 1999.
"The
Continuum in Smooth Infinitesimal Analysis", Symposium on Constructive and
Nonstandard Views of the Continuum, Venice international University, May 1999
"Elementary
Toposes", four lectures, Mathematics Department,
UWO, November 1998.
"Boolean
Algebras and Distributive Lattices Treated Constructively", Mathematics
Department, University of Siena, June 1998.
"Boolean
Algebras and Distributive Lattices in a Constructive Setting", Mathematics
Department, University of Padova, June 1998.
"Whole
and Part in Mathematics", Bolzano Conference on Whole and Part, June 1998.
"Mathematics
and Physics in the Smooth World", UC Irvine Philosophy Dept. Conference on
Philosophy of Math. in Physics, March 1998.
"The
Rehabilitation of Infinitesimals", Philosophy Department, University of
St. Andrews, November 1997.
"The
Rehabilitation of Infinitesimals in Mathematics and Physics", Philosophy
Department, University of Pittsburgh, July 1997.
"Remarks
on Category Theory", CSHPM meeting on Ontological Issues in Category
Theory, Brock University, May 1996.
"The
Rehabilitation of Infinitesimals", Philosophy Department, University of
Toronto, December 1995.
"The
Rehabilitation of Infinitesimals", Philosophy Department, University of
Waterloo, November 1995.
"The
Philosophical Thought of Hermann Weyl", Canadian Philosophical Association
Annual Meeting, Calgary, July 1994.
"Type
Reducing Correspondences", Philosophy Department, Universityof
Padova, June 1994.
"How
to Make Intuitionistic Set Theory Classical", Math Department, University
of Padova, June 1994.
"Frege's
and Zermelo's Constructions Re-Examined",
Philosophy Department, University of Florence, June 1994.
"Category
Theory and Philosophy of Mathematics", Panel Discussion, Joint ASL/APS
Meeting, Kansas City, May 1994.
"Modalized
Distributive Lattices", Math. Department, McMaster University, March 1994.
"Fregean
Extensions of 1st Order Theories", One-day conference on Frege, St.
Andrews University, June 1993.
"Infinitesimals
and the Continuum", Philosophy Department, London School of Economics,
June 1993.
"Infinitesimals
and the Continuum", 19th Annual Meeting of the CSHPM, Carleton University,
Ottawa, May 1993.
"Hilbert's
epsilon-Calculus and Classical Logic", Philosophy
Department, University of Toronto, January 1993.
"How
to Make Intuitionistic Set Theory Classical". Mathematics Department,
University of Michigan, Ann Arbor, December 1992.
"Infinitesimals",
Physics Department, UWO, September 1992.
"How
to Make Intuitionistic Set Theory Classical", Mathematics Department,
McMaster University, February 1992.
"Variability
and Logic", Philosophy Department, University of Padova, June 1991.
"Local
Set Theories", Mathematics Department, University of Siena, May 1991.
"Hilbert's
epsilon-Calculus and Classical Logic", Philosophy
Dept., University of Florence, May 1991.
"Toposes and Local Set Theories” (a series of 15
lectures), Mathematics Department, University of Padova, May-June, 1991.
"Hilbert's
epsilon-Calculus and Classical Logic", Logic
Colloquium, SUNY at Buffalo, April 1991.
"Infinitesimals",
Mathematics Department, UWO, March 1991.
"The
Sikorski Extension Theorem for Boolean Algebras", Mathematics Department,
McMaster University, February 1991.
Teaching and Research Appointments
·
École
Polytechnique, Paris, 2007: Visiting Directeur de Recherche, CNRS.
At
the University of Western Ontario, 1989-2019
· Undergraduate
At
the London School of Economics 1968-89
Undergraduate
lecture courses on: algebra, multivariable calculus, analysis, mathematical
logic, set theory, history and philosophy of
mathematics.
Advanced
lecture courses on: set theory, model theory, Boolean algebras, functional
analysis, category theory.
At
Oxford, 1965-68
Tutorials in
mathematical logic, set theory, analysis, algebra, topology. Lecture courses in
model theory, Boolean algebras, set theory.
THESIS SUPERVISION:
At the University of Western Ontario
Emerson Doyle (Ph.D., Philosophy,
2013): The Methodological Roles of
Tolerance and Conventionalism in the Philosophy of Mathematics: Reconsidering Carnap’s Logic of Science.
Robert Moir[1] (Ph.D., Philosophy,
2013): Structures in Real Theory Application:
A Study in Feasible Epistemology.
Nicolas Fillion[2] (Ph.D., Philosophy, 2013): The Reasonable Effectiveness of Mathematics in the Natural Sciences.
Darren McDonald (Ph.D., Philosophy,
2012): Anti-Foundational Categorical
Structuralism.
Leo Jordao (Ph.D., Philosophy,
2010): Elements of a New Constructional
System.
Greg Andres (Ph,D,. Philosophy, 2007): The Metaphysical Basis of Logic
Eric Snyder (Ph.D., Philosophy,
2000): The Philosophy of Mathematics of
Wittgenstein’s Tractatus Logico-Philosophicus.
Richard Feist (Ph.D., Philosophy,
1999): The Mathematical Intuitionism of
Hermann Weyl
Elaine Landry (Ph.D., Philosophy,
1997): Category -Theoretic Realism: A
Linguistic Approach to the Philosophy of Mathematics
Gregory Hagen (Ph.D., Philosophy,
1996): Leibniz’s Puzzle and the
Smooth Continuum
David DeVidi (Ph.D., Philosophy, 1994): Term-Forming Operators in First-Order Logic
At the University of London
Samuel Fendrich (Ph.D., Philosophy,
1987): From Axiomatization to
Generalization of
Set Theory
Dennis Mentzeniotis[3] (Ph.D., Philosophy,
1986): Continuity and Infinitesimals
Enrique Hernandez[4] (Ph.D., Mathematics,
1984): Automorphisms of Models of Set
Theory
Michael Hallett (Ph.D., Philosophy,
1979): Cantorian Set Theory and
Limitation of Size.
Graham Priest (Ph.D., Mathematics,
1972): Type Theory in which Variables
Range over Predicates.
John Lake (Ph.D., Mathematics,
1972): Ackermann Set Theory.