William Lawvere

Francis William Lawvere(/lɔːˈvɪər/;February 9, 1937 – January 23, 2023) was an American mathematician known for his work incategory theory,topos theoryand thephilosophy of mathematics.

William Lawvere
Lawvere inFlorence,2003
Born	Francis William Lawvere (1937-02-09)February 9, 1937 Muncie, Indiana,U.S.
Died	January 23, 2023(2023-01-23)(aged 85) Chapel Hill, North Carolina,U.S.
Alma mater	Columbia University
Known for	Contributions totopostheory andphilosophy of mathematics Lawvere theory Lawvere's fixed-point theorem
Awards	"Premio Giulio Preti", awarded by theRegional Council of Tuscany(2010)
Scientific career
Fields	Mathematics
Institutions	SUNY-Buffalo
Doctoral advisor	Samuel Eilenberg

Biography

Born inMuncie, Indiana,and raised on a farm outside Mathews, Lawvere received his undergraduate degree in mathematics fromIndiana University.

Lawvere studiedcontinuum mechanicsandkinetic theoryas an undergraduate withClifford Truesdell.He learned of category theory while teaching a course onfunctional analysisfor Truesdell, specifically from a problem inJohn L. Kelley's textbookGeneral Topology.

Lawvere found it a promising framework for simple rigorous axioms for the physical ideas of Truesdell andWalter Noll.Truesdell supported Lawvere's application to study further withSamuel Eilenberg,a founder of category theory, atColumbia Universityin 1960.

Before completing the Ph.D. Lawvere spent a year inBerkeleyas an informal student ofmodel theoryandset theory,following lectures byAlfred TarskiandDana Scott.In his first teaching position atReed Collegehe was instructed to devise courses incalculusandabstract algebrafrom a foundational perspective. He tried to use the then current axiomatic set theory but found it unworkable for undergraduates, so he instead developed the first axioms for the more relevant composition of mappings of sets. He later streamlined those axioms into theElementary Theory of the Category of Sets(1964), which became an ingredient (the constant case) of elementarytopos theory.

Lawvere died on January 23, 2023, in Chapel Hill, N.C., after a long illness at the age of 85.

Mathematical work

Lawvere completed hisPh.D.atColumbiain 1963 with Eilenberg. His dissertation introduced thecategory of categoriesas a framework for the semantics ofalgebraic theories.From 1964 to 1967 at the Forschungsinstitut für Mathematik at theETH in Zürichhe worked on the category of categories and was especially influenced byPierre Gabriel's seminars atOberwolfachonGrothendieck's foundation ofalgebraic geometry.He then taught at the University of Chicago, working withMac Lane,and at the City University of New York Graduate Center (CUNY), working withAlex Heller.Lawvere's Chicago lectures on categorical dynamics were a further step toward topos theory and his CUNY lectures on hyperdoctrines advancedcategorical logicespecially using his 1963 discovery that existential and universalquantifierscan be characterized as special cases ofadjoint functors.

Back in Zürich for 1968 and 1969 he proposed elementary (first-order)axiomsfor toposes generalizing the concept of the Grothendieck topos (seeHistory of topos theory) and worked with thealgebraic topologistMyles Tierneyto clarify and apply this theory. Tierney discovered major simplifications in the description ofGrothendieck "topologies".Anders Kocklater found further simplifications so that a topos can be described as a category withproductsandequalizersin which the notions of map space andsubobjectare representable. Lawvere had pointed out that a Grothendieck topology can be entirely described as anendomorphismof the subobject representor, and Tierney showed that the conditions it needs to satisfy are justidempotenceand the preservation of finite intersections. These "topologies" are important in bothalgebraic geometryandmodel theorybecause they determine the subtoposes as sheaf-categories.

Dalhousie Universityin 1969 set up a group of 15Killam-supported researchers with Lawvere at the head; but in 1971 it terminated the group. Lawvere was controversial for his political opinions, for example, his opposition to the 1970 use of theWar Measures Act,and for teaching the history of mathematics without permission.But in 1995 Dalhousie hosted the celebration of 50 years of category theory with Lawvere and Saunders Mac Lane present.

Lawvere ran a seminar in Perugia, Italy (1972–1974) and especially worked on various kinds ofenriched category.For example, ametric spacecan be regarded as an enriched category.From 1974 until his retirement in 2000 he was professor of mathematics atUniversity at Buffalo,often collaborating withStephen Schanuel.In 1977 he was elected to the Martin professorship in mathematics for five years, which made possible the meeting on "Categories in Continuum Physics" in 1982. Clifford Truesdell participated in that meeting, as did several other researchers in the rational foundations ofcontinuum physicsand in thesynthetic differential geometrythat had evolved from the spatial part of Lawvere's categorical dynamics program. Lawvere continued to work on his 50-year quest for a rigorous flexible base for physical ideas, free of unnecessary analytic complications. He wasprofessor emeritusof mathematics and adjunct professor emeritus of philosophy at Buffalo.

Mathematical work related to physics

A central motivation for Lawvere's work is the search for a good mathematical (rigorous) foundations ofphysics,specifically of (classical)continuum mechanics(or at least somekinematical aspects thereof,Lawvere does not seem to mentionHamiltonians,Lagrangiansor action functionals).

In an interview (page 8) he recalled:

I had been a student at Indiana University from 1955 to January 1960. I liked experimental physics but did not appreciate the imprecise reasoning in some theoretical courses. So I decided to study mathematics first. Truesdell was at the Mathematics Department but he had a great knowledge in Engineering Physics. He took charge of my education there.... in 1955 (and subsequently) had advised me on pursuing the study of continuum mechanics and kinetic theory. In Summer 1958, I studied Topological Dynamics with George Whaples, with the agenda of understanding as much as possible in categorical terms.... Categories would clearly be important for simplifying the foundations of continuum physics. I concluded that I would make category theory a central line of my study.

Then in the same interview (page 11) he said about the early 1960s:

I felt a strong need to learn more set theory and logic from experts in that field, still of course with the aim of clarifying the foundations of category theory and of physics.

The title of the early text "Toposes of laws of motion", which is often cited as the text introducingsynthetic differential geometry,clearly witnesses the origin and motivation of these ideas inclassical mechanics.

In an interview, William F. Lawvere reflects on his time as an assistant professor at theUniversity of Chicagoin 1967. He mentions that he and Mac Lane co-taught a course onmechanics,which led him to consider the justification of older intuitive methods ingeometry,eventually coining the term "synthetic differential geometry"This course was based on Mackey's bookMathematical Foundations ofQuantum Mechanics,indicating Mackey's influence oncategory theory.

Further in the interview, he discusses the origins of synthetic differential geometry, noting that the idea for the joint course on mechanics came from a suggestion by Chandra. This course was the first in a series, and Mac Lane later gave a talk on theHamilton-Jacobi equationat the Naval Academy in 1970, which was published inThe American Mathematical Monthly.He explains that he began applying Grothendieck topos theory, learned from Gabriel, to simplify the foundations ofcontinuum mechanics,inspired by Truesdell's teachings, Noll's axiomatizations, and his own efforts in 1958 to categorize topological dynamics.

A more detailed review of these ideas and their relation to physics can be found in the introduction to the book collectionCategories in Continuum Physics,which is the proceedings of a meeting organized by Lawvere in 1982.

In his 1997 talk "Toposes of Laws of Motion", Lawvere remarks on the longstanding program ofinfinitesimal calculus,continuum mechanics,anddifferential geometry,which aims to reconstruct the world from the infinitely small. He acknowledges the skepticism around this idea but emphasizes its fruitful outcomes over the past 300 years. He believes that recent developments have positioned mathematicians to make this program more explicit, focusing on how continuum physics can be mathematically constructed from "simple ingredients".

In the same talk, Lawvere mentions that the essential spaces required forfunctional analysisand physical field theories can be found in anytoposwith an appropriate object (T).

In his 2000 article "Comments on the Development of Topos Theory", Lawvere discusses his motivation for simplifying and generalizing Grothendieck's concept of topos. He explains that his interest stemmed from his earlier studies in physics, particularly the foundations ofcontinuum physicsas inspired by Truesdell, Noll, and others. He notes that while the mathematical apparatus used in this field is powerful, it often does not fit the phenomena well. Lawvere questions whether the problems and necessaryaxiomscould be stated more directly and clearly, potentially leading to a simpler yet rigorous account. These questions led him to apply the topos method in his 1967 Chicago lectures on categorical dynamics. He realized that further work on the notion oftoposwas necessary to achieve his goals. His time spent with Berkeley logicians in 1961-62, listening to experts on foundations, also influenced his approach.

Lawvere highlights that several books on simplifiedtopos theory,including the recent and accessible text by MacLane and Moerdijk, along with three excellent books onsynthetic differential geometry,provide a solid foundation for further work in functional analysis and the development of continuum physics.

Mathematical work related to philosophy

William Lawvere has also proposed formalizations incategory theory,categorical logicandtopos theoryof concepts which are motivated fromphilosophy,notably inGeorg Hegel'sScience of Logic(see there for more).

This includes for instance definitions of concepts found there such as objective and subjective logic, abstract general, concrete general, concrete particular,unity of opposites,Aufhebung,being, becoming, space and quantity, cohesion, intensive and extensive quantity... and so on.

In his work "Categories of Space and Quantity" fromThe Space of Mathematics(1992), William Lawvere expresses his belief that the technical advancements made by category theorists will significantly benefitdialectical philosophyin the coming decades and century. He argues that these advancements will provide precise mathematical models for age-old philosophical distinctions, such as general versus particular, objective versus subjective, and being versus becoming. He emphasizes that mathematicians need to engage with these philosophical questions to make mathematics and other sciences more accessible and useful. This, he notes, will require philosophers to learn mathematics and mathematicians to learn philosophy.

A precursor to this undertaking isHermann Grassmannwith hisAusdehnungslehre.

Political affiliations

The category theorist William Lawvere was a committedMarxist-Leninist;at one point he gave a talk called "Applying Marxism-Leninism-Mao Tse-Thung Thought to Mathematics & Science".

According to Anders Kock's obituary, in 1971:

the [Dalhousie] university administration refused to renew the contract with [Lawvere], due to his political activities in protesting against the Vietnam war and against the War Measures Act proclaimed by Trudeau, suspending civil liberties under the pretext of danger of terrorism.

As per the obituary on theCommunist Party of Canada (Marxist–Leninist)site:

More than 1,000 students rallied in the lobby of the Dal Student Union Building to oppose the arbitrary dismissal of Professor Lawvere.

He saw his political commitments as related to his mathematical work in sometimes surprising and unexpected ways: for instance, here's a passage fromQuantifiers and Sheaves(1970):

When the main contradictions of a thing have been found, the scientific procedure is to summarize them in slogans which one then constantly uses as an ideological weapon for the further development and transformation of the thing. Doing this for "set theory" requires taking account of the experience that the main pairs of opposing tendencies in mathematics take the form of adjoint functors, and frees us of the mathematically irrelevant traces (∈) left behind by the process of accumulating (∪) the power set (P) at each stage of a metaphysical "construction".

In the earlier sections of the paper, he discusses the "unity of opposites" between logic and geometry. He clarifies that his discussion of contradiction, ideology, and opposition is rooted in theMarxist tradition,referencing Mao's "On Contradiction"(1937) in the bibliography. Additionally, he connects various mathematical concepts toHegel's DialecticandLenin's theory of knowledgein other parts of his work.

Awards and honors

• In 2010 he received the "Premio Giulio Preti", awarded by theRegional Council of Tuscany
• In 2012 he became a fellow of theAmerican Mathematical Society.

Selected books

• 1986Categories in Continuum Physics(Buffalo, N.Y. 1982), edited by Lawvere andStephen H. Schanuel(with Introduction by Lawvere pp 1–16), Springer Lecture Notes in Mathematics 1174.ISBN3-540-16096-5;ebook
• 2003 (2002)Sets for Mathematics(withRobert Rosebrugh). Cambridge Uni. Press.ISBN0-521-01060-8
• 2009Conceptual Mathematics: A First Introduction to Categories(with Stephen H. Schanuel). Cambridge University Press, 2nd ed.ISBN978-0521719162;1997 pbk edition

See also

• Category theory
• Topos theory
• History of topos theory
• Synthetic differential geometry
• Philosophy of mathematics
• Dialectic
• Marxism–Leninism
• Lawvere–Tierney topology

External links

• A 2007 interview published on the Bulletin of the International Center for Mathematics of Coimbra, Portugal (Part I,Part II;both parts in one file)
• Reprints in Theory and Applications of Categories.Includes reprints of eight of Lawvere's fundamental articles, among them his dissertation and his first full treatment of the category of sets. Those two had circulated only as mimeographs.
• William F. Lawvere nLab entry
• Repository of William F. Lawvere's collected workshosted onGitHub.
• William Lawvereat theMathematics Genealogy Project
• Photograph