Bill Lawvere

Biography in F. William Lawvere
The Unity of Mathematics
by Renato Betti pp223-230 of C. Bartocci et al. (eds.), Mathematical Lives,
DOI 10.1007/978-3-642-13606-1_33, Springer-Verlag Berlin Heidelberg 2011

title = {Mathematical Lives: Protagonists of the Twentieth Century From Hilbert to Wiles},
author = {Umberto Bottazzini (auth.), Claudio Bartocci, Renato Betti, Angelo Guerraggio, Roberto Lucchetti (eds.)},
publisher = {Springer-Verlag Berlin Heidelberg},
isbn = {3642136052,9783642136054,9783642136061},
year = {2011},
series = {},
edition = {1},
volume = {},
url = {}}


Lawvere’s conceptual guide is the rejection of the ideological perspective according to which theory is more fundamental than practice. Declaring his own debt to Hegel’s philosophy and Engel’s considerations, Lawvere calls this guide the logic of mathematics, distinct from and comprising mathematical logic: it is how our thinking develops, like the science of the shape of space and quantitative relations, as need arises. Mathematical work is thus linked to the investigation of the general laws of thinking, applied to the study of particular subjects.


It is my belief that in the next decade and in the next century the technical advances forged by category theorists will be of value to dialectical philosophy, lending precise form with disputable mathematical models to ancient philosophical distinctions such as general vs. particular, objective vs. subjective, being vs. becoming, space vs. quantity, equality vs. difference, quantitative vs. qualitative etc. In turn the explicit attention by mathematicians to such philosophical questions is necessary to achieve the goal of making mathematics (and hence other sciences) more widely learnable and useable.8

8 F. William Lawvere, “Categories of Space and Quantity”, pp. 14­30 in: Structures in Mathematical Theories: Reports of the San Sebastian International Symposium, September, 25­29, 1990, A. Dı´ez, J. Echeverrı´a, A. Ibarra (eds.), De Gruyter, 1992, p. 16.

Todo: Link above and all other footnotes. Add other “dialectical” quotes, references to Engels etc.

Link to Grassmann.

See nLab which has LOTS including more accessible links to papers.

Collected works at github.


Home Page

Papers Chronological

Papers by Subject Classification


9. Adjointness in Foundations, Dialectica 23 (1969), 281-296.

30. Some Thoughts on the Future of Category Theory, Category Theory, Proceedings Como 1990. Springer Lecture Notes in Mathematics 1488, Springer-Verlag (1991) 1-13.

31. Categories of Space and of Quantity, International Symposium on Structures in Mathematical Theories (1990), San Sebastian, Spain; The Space of Mathematics: Philosophical, Epistemological and Historical Explorations, DeGruyter, Berlin (1992), 14-30.

32. Tools for the Advancement of Objective Logic: Closed Categories and Toposes, J. Macnamara & G. E. Reyes (Eds). The Logical Foundations of Cognition, Oxford University Press (1994), 43-56.

33. Cohesive Toposes and Cantor’s “lauter Einsen”, Philosophia Mathematica, The Canadian Society for History and Philosophy of Mathematics, Series III, Vol. 2 (1994), 5-15.

36. Unity and Identity of Opposites in Calculus and Physics, Proceedings of ECCT 1994 Tours Conference, Applied Categorical Structures, 4: 167-174 Kluwer Academic Publishers, 1996.

38. Kinship and Mathematical Categories, Language, Logic, and Conceptual Representation (in memory of John Macnamara): 411-425, MIT Press 1999, Ed. P. Bloom, R. Jackendoff, and K. Wynn.

39. Categorie e Spazio: Un Profilo, Lettera matematica PRISTEM 31, Springer, Italy, (1999), 35-50. [Reprinted in La Mathematica a cura di Bartocci, Claudio, Giulio Einaudi editore (2010) vol. 4, 107-135.]

40. Comments on the Development of Topos Theory, Development of Mathematics 1950-2000, Ed. by J-P Pier, Birkhauser Verlag, Basel, 2000.

47. Foundations and Applications: Axiomatization and Education, The Bulletin of Symbolic Logic, vol. 9, No. 2, (2003), 213-224.

57. Long version with new commentary by the author and by Colin McLarty: An Elementary theory of the category of sets (1964) (long version) Reprints in Theory and Applications of Categories, online publication, No. 11, (2005), pp. 1-35 (link)

58. Reprint with new author commentary Diagonal Arguments and Cartesian Closed Categories (1969) Reprints in Theory and Applications of Categories, online publication, No. 15, (2006), pp. 1-13. (link)

59. Reprint with new author commentary Adjointness in Foundations (1969) Reprints in Theory and Applications of Categories, online publication, No. 16, (2006), 1-16. (link)

64. Euler’s Continuum Functorially Vindicated, Logic, Mathematics, Philosophy: Vintage Enthusiasms, Essays in Honour of John L. Bell, D. DeVidi et al. (Eds), Western Ontario Series in Philosophy of Science 75, (2011).

“Unity and identity of opposites in calculus and physics”, Lawvere, F.W. Appl Categor Struct (1996) 4: 167. (available Sci-Hub via DOI).

doi = {10.1007/bf00122250},
title = {Unity and identity of opposites in calculus and physics},
author = {F. William Lawvere},
publisher = {Springer},
journal = {Applied Categorical Structures},
issnp = {0927-2852},
issne = {1572-9095},
year = {1996},
volume = {4},
issue = {2-3},
page = {167–174},
url = {},

(Received: 5 December 1994; accepted: 6 May 1995)
Abstract. A significant fraction of dialectical philosophy can be modeled mathematically through the use of “cylinders” (diagrams of shape \Delta_1) in a category, wherein the two identical subobjects (united by the third map in the diagram) are “opposite”. In a bicategory, oppositeness can be very effectively characterized in terms of adjointness, but even in an ordinary category it may sometimes be given a useful definition. For example, an effective basis for teaching calculus is a ringed category satisfying the Hadamard-Marx property. The description in engineering mechanics of continuous bodies that can undergo cracking is clarified by an example involving lattices, raising a new questions about the foundations of topology.
Key words: cracking, dialectics, differentiation, Hadamard lemma, poll bodies, van der Waals fluids.


In early 1985, while I was studying the foundations of homotopy theory, it
occurred to me that the explicit use of a certain simple categorical structure
might serve as a link between mathematics and philosophy. The dialectical phi-
losophy, developed 150 years ago by Hegel, Schleiermacher, Grassmann, Marx,
and others, may provide significant insights to guide the learning and develop-
ment of mathematics, while categorical precision may dispel some of the mystery
in that philosophy. In any case, the structures described in the definition below
occur frequently enough in mathematics to suggest their systematic study…

Near the end of his life, Karl Marx wrote about the foundations Of differen-
tial calculus. The essence of his line of thought, later rigorously established by
Hadamard, yields an effective and simple basis for learning and developing the
subject if made explicit. The problem may be stated as follows: Presupposing
those laws of algebra which are equally valid for variable and constant quantities,
what is additionally required in order to determine the derivatives of genuinely
variable quantities and to establish the laws of the derivative? The answer is the
unity and identity of opposites permitting a single variable to be split into two
like variables and later collapsed again to one. How can we make this conclusion
into precise mathematics?

Walter Noll recently proposed a generalization of the usual concept of continuous
body, designed to describe motions during which the body can undergo cracking
and self-contact. The usual concept is that of a single topological space which at
each time is embedded in geometrical space; however, if during motion a body
undergoes cracking, neither the point set nor the open-set frame of the body
remains the same. Noll’s proposal is that all these various topological spaces
for various times nonetheless have something in common as a result of being
the “same” body, namely the Boolean algebra B of “parts” of the body…

1. Grassmann, H.: Ausdehnungslehre von 1844, Chelsea, 1969.
2. Grothendieck, A.: Pursuing Stacks, 400 p. Manuscript, 1983.
3. Kelly, G. M. and Lawvere E W.: On the complete lattice of essential localizations, Bull. Soc. Math. de Belgique (Serie A) 41 (1989), 289-319.
4. Lawvere, F. W.: Introduction to Categories in Continuum Physics, Springer LNM 1174, 1986, 1-16.
5. Lawvere, F. W.: Intrinsic co-heyting boundaries and the Leibniz rule in certain toposes, Category Theory Proceedings Como 1990, Springer LNM 1488, 1991, 279-281.
6. Lawvere, F. W.: Categories of space and of quantity, in Echeverria et al. (eds), The Space of Mathematics, de Gruyter, 1992, pp. 14–30.
7. Lawvere, F. W.: Cohesive toposes and Cantor’s “lauter Einsen”, Philosophia Mathematica (Series 111) 2 (1994), 5-15.
8. Marx, K.: The Mathematical Manuscripts, London New Park Publications Ltd., 1983.
9. Noll, Walter: The geometry of contact, separation, and reformation of continuous bodies, Arch. Rational Mech. Anal. 122 (1993), 197-212.
10. Quillen, D. G.: Homotopical Algebra, Springer LNM 43, 1967.

Toward the description in a smooth topos of the dynamically possible motions and deformations of a continuous bodyCahiers de topologie et géométrie différentielle catégoriques, tome 21, no 4 (1980), p. 377-392

It is an honor to participate in the commemoration of the work of Charles Ehresmann because he, like other great French geometers of our times, realized clearly that in order to make possible the learning, development, and use of concrete infinite-dimensional differential geometry, it is necessary to reconstruct it as a concept, and that this reconstruction is only possible on the basis of a sharp determination of the decisive abstract
general relation ( DAGR ) of the subject, and that in order to succeed in the latter determination, it is mandatory to develop the theory of categories.
Now that category theory has indeed been advanced to a very great extent, we can show our appreciation for what we have learned of it from these geometers and others, by taking up the physics of continuous bodies and fields,
which was after all the primary source of the geometry developed by their teachers such as E. Cartan. The recognition of that source, just as much as the internal axioms which we labor to perfect, is a DAGR of the subject ( see Karl Marx, Critique of Political Economy, Section 3 «on Method», for the explanation of the role of DAGR in the reconstruction of the concrete as a concept ).
According to Lenin, the scientific world-picture is a picture of matter-that-moves and matter-that-thinks », and moreover the special role of matter-that-thinks is to reflect the decisive relations in the world in order
to provide theory as a guide to action. This materialist world-picture is in opposition to the anti-scientific world-pictures of subjective idealism and objective idealism respectively. Subjective idealism was concocted by Plato, Berkeley, etc. in order to prepare the way for the acceptance of objective idealism, and this destructive and anti-scientific work was carried on more recently by Mach, Russell, Brouwer, Heisenberg, etc. Note that all these idealists made special distortions of the science of mathematics as one of the bases of their attempts to get the public to accept their philosophy that the world is a figment of imagination (whether ours or « god’s»).

(…17pp. Important diagrams also omitted…)


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