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Nicolaas Govert de Bruijn (1918–2012)Mathematician, computer scientist, logician

Fig. 1. Dick de Bruijn, 2003.

1. Introduction

On 17 February 2012, Nicolaas Govert (Dick) de Bruijn passed away in his home villageNuenen, the Netherlands, at the age of 93 years. He was a productive researcher with wide-ranging interests, and he was respected world-wide for his ground-breaking work in a variety ofareas. Our scientific community lost an eminent mathematician, computer scientist, and logician.

Dick de Bruijn was born on 9 July 1918 in the Hague, the Netherlands, into a family witheight children, Dick included. He attended high school (the Dutch HBS, Hogere Burger School)from 1930 to 1934, completing the five-year curriculum in only four years.

The societal climate was grim in these years of depression. There was a scarcity of jobs, andgrants for students did not then exist. Going to a university with a money loan and the ensuingprospect of future debts was hardly appealing. Consequently, the young de Bruijn was led toauto-didactical study and a general attitude of independence. A fortuitous circumstance was thatin the family home there were several books on mathematics floating around, belonging to hisbrother. Dick became fascinated by the contents of these books, and started to study for the so-called MO-diplomas K1 and K5, qualifications that were usually acquired by aspiring high schoolteachers (MO stands for Middelbaar Onderwijs, education at the intermediate level). Although

0019-3577/$ - see front matterhttp://dx.doi.org/10.1016/j.indag.2013.09.004

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these qualifications were not proper university degrees, this route via the MO-diplomas was notunusual, and was taken by several Dutch mathematicians who later rose to eminence, includingHendrik Kloosterman, David van Dantzig, and Bartel van der Waerden. One book that he foundparticularly inspiring was written by Fred Schuh, Professor at the Technische Hogeschool Delft,nowadays called Technical University Delft (TUD) [27]. De Bruijn related that he used to findthe various proofs first himself, to compare them subsequently with the ones in the book.

In his younger years, de Bruijn was of a rather shy nature, as he remembered in his lectureat the occasion of his ninetieth birthday (see Bibliography, symposium), looking back on hislife. And he added that in his own view this never really changed later on. Also, his memory, hethought, was moderate at best. As a compensation he developed the habit of associative thinking.This mental mechanism of associative thinking intrigued him all his life, and after his emeritatehe developed a cognitive theory of the brain, featuring a model for associative memory [9,17].

Fig. 2. Dick, 18 years old, studying for the K5 diploma.

2. De Bruijn’s career in a nutshell

At the age of eighteen, in 1936, in part by virtue of his MO-diplomas, Dick succeeded inacquiring a grant to study at a university, and he started his study of mathematics at LeidenUniversity in the Netherlands. Here, he was greatly inspired by the young professor (actually,lector in the Dutch academic function system of those years) Hendrik Kloosterman, who startedto build his courses on topology, group theory, and number theory from first principles, an ap-proach that was close to the heart of Dick de Bruijn. Kloosterman and de Bruijn shared in theirmathematical style a predilection for a very precise formulation and presentation, straight to thepoint and without circumlocutions.

In this period, de Bruijn was a student assistant at the Technische Hogeschool in Delft, fromSeptember 1939 to June 1944. His research was conducted under the supervision of Kloosterman.However, due to the restrictions induced by the war, it was not possible to obtain his doctor’sdegree (Ph.D.) in Leiden, and therefore the ceremonial defence of the dissertation was held atthe Vrije Universiteit in Amsterdam (nowadays called VU University Amsterdam), on 26 March1943, with Jurjen Koksma as promotor. The thesis was titled Over modulaire vormen van meer

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veranderlijken (On modular forms of several variables). The doctor’s title was granted with thejudicium cum laude.

With his doctor’s title acquired, Dick de Bruijn married Elizabeth (Bep) de Groot. Fourchildren were born from their marriage.

The young doctor started his research career at Philips Research Laboratory (NatuurkundigLaboratorium) in Waalre near Eindhoven, from 1944 to 1946, recruited thereto by ProfessorBalthasar van der Pol. De Bruijn flourished in this environment, and he cherished the contactswith colleagues from various disciplines, among them physicists and technicians, that he held inhigh esteem.

The next step in his career was his appointment to full professor in mathematics at the Tech-nische Hogeschool Delft [6], from October 1946 to September 1952, and subsequently at theUniversiteit van Amsterdam (UvA), from September 1952 to September 1960, where he foundEvert Willem Beth and Arend Heyting among his colleagues. In this period, de Bruijn was ap-pointed as a member of the KNAW, the Royal Netherlands Academy of Arts and Sciences, in theSection Mathematics.

One might have expected that the presence of leading logicians such as Beth and Heytingat the UvA would have drawn him towards research in the logical foundations of mathematics,but that turned out not to happen. Much of de Bruijn’s best mathematical work stems from thisperiod, including his important book from 1958, Asymptotic Methods in Analysis [7].

In spite of the eminent status of mathematics at the UvA, de Bruijn chose to continue hiscareer at the Technische Hogeschool in Eindhoven, being persuaded to make this choice by JaapSeidel (1919–2001), a friend and former fellow student from Leiden [18]. Seidel just had foundeda new Department of Mathematics in Eindhoven, and the challenge to cooperate in this freshendeavour appealed very much to de Bruijn, also because of the ensuing freedom in the selectionof subjects to study and teach. This new Department of Mathematics rose steadily to prominence,as witnessed by the fact that in 1972 four out of the ten members of the Section Mathematicsof the Royal Netherlands Academy of Arts and Sciences originated from the Department ofMathematics in Eindhoven. In addition to de Bruijn, C.J. (Chris) Bouwkamp, E.W. (Edsger)Dijkstra, and J.H. (Jack) H. van Lint were members.

De Bruijn was affiliated until his emeritate in 1984 with the Technische Hogeschool Eind-hoven, now Technical University Eindhoven (TU/e). From 1960 until his emeritate he was alsoa consultant for Philips Research Laboratory.

Later, de Bruijn recounted that he always was happy with his transition to Eindhoven. Oneof his reasons for the career switch was his expectation that mathematics was on the verge ofplaying an important role in industrial companies. In the 1970s, he was approached by theAmerican university Caltech, offering him a professorship in the United States, but de Bruijnopted definitively for Eindhoven.

After his emeritate, de Bruijn stayed active to a very advanced age, and in 2008 a symposiumwas organized at the TU/e, celebrating his contributions to mathematics and computer science[29].

3. De Bruijn’s research method

De Bruijn was a prolific researcher. His first paper was published in 1937. In his retrospectivetalk at his birthday symposium in 2008, De Bruijn recounted with some pride that even inhindsight this first paper was not so bad. By March 2004, his list of publications covered 196journal publications and approximately 120 technical (internal) reports. Furthermore, there were

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contributions to various books, and there is the aforementioned well-known book from 1958,Asymptotic Methods in Analysis, that has been reprinted many times, the last time in 1981 byDover.

De Bruijn stated that he never considered himself to be a ‘scholar’. Coming from a big familywith eight children, he learned from an early age onwards to act and think in an independentway. Much of his scientific education was auto-didactic. He did not read extensively in themathematical literature, also because of his memory capacity, which he described as modest.He had similar feelings of doubt as a young man about his social skills. But at the same time hewas endowed with a curious playful attitude, inclined to work out matters for himself from firstprinciples. His self-appraisal of initial shyness and social restrictions is in remarkable contrastwith the deep gratitude and affection that various prominent scientists all over the world felt forhim, described in their letters with congratulations to the nonagenarian in the ‘book of letters’offered in honour to him in 2008. He was an influential mentor in their careers and lives, theywitnessed. This book of letters is available at http://www.win.tue.nl/debruijn90/video/debruijn.html [29].

In his own perception, de Bruijn was not the architect of towering mathematical buildings,building on the work of others. He used to let himself be guided by his playfulness, by serendipity,and sometimes by chance.

Nor was de Bruijn the founding father of a substantial school. He supervised 11 promotions.Yet he maintained a world-wide network of numerous contacts, also resulting from several guestprofessorships. In the summer of 1959, he was Carl Friedrich Gauss professor in Gottingen, inthe summer of 1965 he was a guest professor at the University of Paris in Orsay, in the spring of1973 he was Sherman Fairchild Distinguished Scholar at the California Institute of Technologyin Pasadena, and in April 1984 he was guest professor at the University of Tel Aviv.

4. Areas

Dick de Bruijn left his mark in various areas in mathematics, logic, and computer science. Hecontributed principally in the following areas; the list is not exhaustive, but clearly demonstratesthe impressive width of the spectrum of his interests.

• Classical analysis: complex function theory, asymptotics, Fourier theory (De Bruijn–Newmanconstant), Wigner distributions.

• Functional analysis: generalized functions.• Number theory: R without Q, smooth numbers, analytical number theory, complementary sets

in abelian groups.• Discrete mathematics: de Bruijn sequences, partition problems, Polya theory, De Bruijn–

Erdos theorem, aperiodic plane tilings, quasicrystals.• Proof checking and mathematical languages: Automath, mathematical vernacular.• Logic: propositional logic, typed lambda calculus, de Bruijn indices [10], propositions-as-

types, term rewriting systems (weak diamond properties, decreasing diagrams).• Theory of the brain: a model of associative memory.

In this issue of this journal we have collected several papers pertaining to de Bruijn’s work inthe areas as listed.

5. Automath

In de Bruijn’s own opinion, he did not construct large mathematical buildings. Nowadaysone cannot agree with his opinion, however. He was the originator of a grand building where

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mathematics, logic, language, and computer science converge: the Automath project. Here, wewill restrict ourselves with a short description; for a thorough description we refer to the articleby Geuvers and Nederpelt in the present issue of this journal.

Fig. 3. Dick de Bruijn, around 1968.

In 1968, de Bruijn started the ground-breaking Automath project, initially designed for theverification of mathematical texts. A milestone was the proof of correctness of the book byEdmund Landau Grundlagen der Analysis; there was only one, innocent, error in the book [5,24].1

Much later, Automath turned out to have a far more encompassing scope of applications.With the Automath project, de Bruijn was at least a decade ahead of international developments.Initially, this resulted in a lack of appreciation, but later this was superseded by a world-widerecognition of his pioneering role. It was his deep insight to use lambda calculus with types as thefoundation for automated systems of verification. In computer science, de Bruijn can thereforebe considered as the founding father of applied type theory [2,4,22,23]. Automath also led to thepossibility to guarantee that both the hardware and the software respect their specification, whichis of crucial importance for the security of complex systems, ranging from mobile telephones tomedical instruments and traffic systems.

The Automath framework, consisting of a family of languages, was far ahead of its time, and itintroduced several concepts that were rediscovered later on, such as various typed lambda calculi,calculi for ‘explicit substitution’, ‘dependent types’ [3], and many more. Automath was the first

1 In 1930, Edmund Landau’s Grundlagen der Analysis was published by the Akademische Verlagsgesellschaft M.B.H.,Leipzig. In 1951, an English translation of Landau’s work was published by the Chelsea Publishing Company. The fulltitle was Foundations of Analysis: the Arithmetic of Whole, Rational, Irrational and Complex Number: A Supplement toText-Books on the Differential and Integral Calculus. (From http://www-history.mcs.st-and.ac.uk.)

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system that constituted an application of the famous Curry–Howard isomorphism, independentlyfound and formulated by de Bruijn, and nowadays also often called Curry–Howard–De Bruijnisomorphism (or correspondence) [21,28].

Dick de Bruijn had strong convictions about the formalization of mathematics versus anintuitive approach. The most eloquent formulation of his credo is due to de Bruijn himself2:

[. . . ] And apart from the computer’s qualities in precision and in speed, it has its influencein forcing us into an absolutely rigorous form of formalization. If we are unable to leavesomething to a computer, then it has not yet been sufficiently formalized. [. . . ] What one isforced to learn anyway is to draw a strict borderline between language and metalanguage.Mixing language and metalanguage is a well known source of errors and paradoxes. Thelanguage is the only thing the verification system checks, the metalanguage helps us tounderstand what we are doing. [. . . ] Many mathematicians dislike pushing formalizationto the extreme. The idea is that it kills intuitive thinking. I do not entirely agree. It maybe true that unnatural formalization replaces intuitive thinking by an entirely differentprocess of formula manipulation, but natural formalization supports intuition rather thandestroying it. Formalization and intuition should be each others best friends rather thanenemies. But part of what we call intuitive thinking is not of the kind that can be refined toproofs. That part cannot be formalized. Our brain processes are not based on logic or anyother foundation of mathematics, and nevertheless they produce wonderful things. But allmathematicians agree that the results of intuitive thinking have to be justified by rigorousreasoning, even though there may be different opinions about the level of formality.

In his retrospective talk at his birthday symposium in 2008, Dick de Bruijn related how, afterreaching the essential insights that are at the roots of Automath, he experienced an intense feelingof happiness with the realization of the purity of mathematics, independent of any a prioriphilosophical approach. The papers and technical reports that were produced in the Automathproject can be found in the Automath Archive [1]; see the short Bibliography for its location.

For a more extensive description of Automath, see [11,20] and the book [26]. Further papersin this issue in the direction created by the Automath project are the one by Pollack, Sakurai,Sato and Schwichtenberg, and the paper by Coquand and Danielsson.

6. Beyond work

The playful nature of Dick de Bruijn also manifested itself in his capacity as an amateurmagician, in particular concerning card games, where he combined playfulness with amathematical view. In the aforementioned collection of letters from 2008, Gerard Huet, affiliatedto the French Institut National de Recherche en Informatique et en Automatique (INRIA) andmember of the Academie des Sciences, talks about de Bruijn’s analysis of the riffle shuffletrick [13], based on a non-trivial property of sequences of zeros and ones. The formal proof inHuet’s proof assistant “Coq” [19] was a stiff challenge. The underlying combinatorial propertyturned out to be relevant for applications concerning quasicrystals, which were introduced byde Bruijn as a mathematical notion [12,14,15] building on the discovery by Roger Penrose(with subsequent additions by John Conway) of aperiodic tilings of the plane. Later, thesequasicrystals, with their five-fold symmetry, whose physical existence was for a long time

2 This quotation is contained in the note [16].

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deemed impossible, were encountered in nature, a discovery for which Daniel Shechtmanreceived the Nobel prize in 2011. In 2006, de Bruijn was still working on this theme, in ageneralization to what he called Gilbreath sequences. And Huet continued his letter, stating hisadmiration:

This interweaving between recreational activities, mathematical abstractions, andapplications to physics and other sciences is the mark of a great mind, curious aboutinterrelations between the real world and the world of mathematics.

De Bruijn also gave an analysis of various games such as Solitaire (see, e.g., [8]). He likedword-plays, bridging language and playfulness. Visiting a conference in the company of Dickde Bruijn was a pleasure. He always surprised and amused his accompanying colleagues withan incessant sequence of astute observations and questions about everyday phenomena, theconsecutive image inversions in an overhead projector, or the average number of equal symbols ina car number plate: everything was a subject of wonder and led him to questions that he engagedwith and solved.

Dick de Bruijn has left an indelible impression in everyone who lived and worked with him.In the past few years, even after he turned 90, Dick regularly attended the meetings of theSection Mathematics of the KNAW, the Royal Netherlands Academy of Arts and Sciences. JaapKorevaar, a member of the Section Mathematics and a friend of Dick de Bruijn, gives in his letterto Dick de Bruijn, included in the present issue of this journal, a detailed account of their jointinterests and experiences in the early years of their careers.

7. Honours and Awards

Dick de Bruijn was the recipient of several honours and awards. As mentioned, in 1957 hewas appointed member of the Royal Netherlands Academy of Arts and Sciences (KNAW), in theSection Mathematics. In 1970, he was one of the keynote lecturers at the International Congressof Mathematicians. In 1981, he received the Royal Decoration Knight of the Order of the Lionof the Netherlands (Ridder in de Orde van de Nederlandse Leeuw). In 1985, he received theSnellius medal, awarded every nine years. In 1988, he became honorary member of the RoyalNetherlands Mathematical Society (KWG). He received the AKZO prize in 1991, and a LifetimeAchievement Award in 2003 of the Netherlands Association for Theoretical Computer Science(NVTI). In the same year the satellite workshop of the international conference ICALP ’03,Mathematics, Logic and Computation was organised in his honour [25]. On his 90th birthday,in 2008, he was appointed honorary member of the Dutch VvL (Vereniging voor Logica, theNetherlands Association for Logic).

Acknowledgements

In this text, we have used sources composed or written by Jos Baeten, Francien Dechesne,Jan de Graaf, Rob Nederpelt, Henk Barendregt, Jaap Korevaar, Roel de Vrijer, and Frans deBruijn. Thanks to them all. A shorter version, in Dutch, has appeared in the collection ofLevensberichten 2012 of the KNAW, the Royal Netherlands Academy of Arts and Sciences, Am-sterdam University Press, November 2012. The present version appeared in Dutch, with minorchanges, in Nieuw Archief voor Wiskunde NAW 5/14 nr.1, pp. 18–21, March 2013. That issuealso contains contributions in Dutch by Guido Janssen, Pieter Moree, Nico Temme, Ton Kloksand Rob Tijdeman, Roel de Vrijer, and Henk Barendregt, on pp. 22–44.

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Bibliography

The following succinct bibliography gives some references to papers, books, or repositoriesmentioned in this article. A more complete bibliography is contained in this issue after this article.

References

[1] The Automath Archive. Hosted at Eindhoven University of Technology: http://www.win.tue.nl/automath/.[2] H.P. Barendregt, Lambda calculi with types, in: S. Abramsky, et al. (Eds.), Handbook of Logic in Computer Science,

Vol. 2, Oxford University Press, 1992, pp. 117–309. Chapter 2.[3] H.P. Barendregt, H. Geuvers, Proof-assistants using dependent type systems, in: J.A. Robinson, A. Voronkov (Eds.),

Handbook of Automated Reasoning, Vol. 2, Elsevier and The MIT Press, 2001, pp. 1149–1238. Chapter 18.[4] H.P Barendregt, W. Dekkers, R. Statman (Eds.), Lambda calculus with types, in: Perspectives in Logic,

ASL/Cambridge University Press, 2013.[5] L.S. van Benthem Jutting, Checking Landau’s “Grundlagen” in the Automath system, Ph.D. Thesis, Technische

Hogeschool Eindhoven, 1977.[6] N.G. de Bruijn, Eenige beschouwingen over de waarde der wiskunde (‘Some observations on the value of

mathematics’), Inaugural speech as professor of pure and applied mathematics and theoretical mechanics at DelftUniversity of Technology, 1946.

[7] N.G. de Bruijn, Asymptotic Methods in Analysis, North Holland Publishing Company and P. Noordhoff, DoverPublications, Inc., New York, 1958.

[8] N.G. de Bruijn, Programmeren van de Pentomino Puzzel (‘Programming the Pentomino Puzzle’), Euclides 47(1971) 90104. 1971/2.

[9] N.G. de Bruijn, Wiskundige modellen voor het levende brein (‘Mathematical models for the living brain’), Reportof the common meeting of the physics section (Afdeling Natuurkunde) of the KNAW, 83, No. 10, 1974.

[10] N.G. de Bruijn, Lambda calculus with namefree formulas involving symbols that represent reference transformingmappings, Indagationes Mathematicae 40 (1978) 348–356.

[11] N.G. de Bruijn, The mathematical language Automath, its usage, and some of its extensions, in: Symposiumon Automatic Demonstration (Versailles, December 1968), in: Lecture Notes in Mathematics, vol. 125, SpringerVerlag, 1970, pp. 29–61. Reprinted in [26], pp. 73–100. 1981.

[12] N.G. de Bruijn, Algebraic theory of Penrose’s non-periodic tilings of the plane, Kon. Nederl. Akad. Wetensch. Proc.Ser. A 84 (1981) pp. 38–52 and pp. 53–66 (= Indagationes Mathematicae 43).

[13] N.G. de Bruijn, A riffle shuffle card trick and its relation to quasicrystal theory (Dedicated to O. Bottema), 1985.http://alexandria.tue.nl/repository/freearticles/597580.pdf.

[14] N.G. de Bruijn, Quasicrystals and their fourier transform, in: Kon. Nederl. Akad. Wetensch. Proc. Ser. A, 89 (1986)123–152. (= Indagationes Mathematicae 48).

[15] N.G. de Bruijn, Algebraic theory of Penrose’s non-periodic tilings of the plane, Nederl. Akad. Wetensch. Proc. Ser.A 84 (1981) 38–66. (= Indagationes Math. 43). Reprinted in: P.J. Steinhardt and S. Ostlund (Eds.), The Physics ofQuasicrystals, World Scientific Publ. Comp., Singapore, pp. 673–700.

[16] N.G. de Bruijn, Notices of the American Mathematical Society 38 (1991) 8–15.[17] N.G. de Bruijn, Can people think? Journal of Consciousness Studies 3 (1996) 425–447.[18] N.G. de Bruijn, Jaap Seidel, a friend, Nieuw Archief voor Wiskunde 5/2 (3) (2001) 204–206.[19] The Coq Development Team, The Coq Proof Assistant, Reference Manual, Version 8.3. http://coq.inria.fr/refman/.[20] F. Dechesne, R.P. Nederpelt, N.G. de Bruijn (1918–2012) and his Road to Automath, the Earliest Proof Checker,

in: The Mathematical Intelligencer, 34 (4) (2012) 4–11.[21] W.A. Howard, The formulas-as-types notion of construction, in: J.P. Seldin, J.R. Hindley (Eds.), To H.B. Curry:

Essays on Combinatory Logic, Lambda calculus and Formalism, Academic Press, 1980, pp. 479–490.[22] F. Kamareddine, Thirty-five years of automating mathematics, in: Workshop Proceedings, Kluwer Academic

Publishers, Dordrecht, Boston, 2003.[23] F. Kamareddine, T. Laan, R. Nederpelt, A modern perspective on type theory: from its origins until today,

in: Applied Logic, vol. 29, Kluwer, 2004.[24] E. Landau, Grundlagen der Analysis, 1930, 1960. First ed.: Leipzig, 1930. Third ed.: Chelsea Publ. Comp. New

York, 1960.[25] Mathematics, logic and computation, A satellite workshop of ICALP03, in honour of N.G. de Bruijn’s 85th

anniversary, 4–5 July 2003. http://www.cedar-forest.org/forest/events/Bruijn03/.[26] R.P. Nederpelt, J.H. Geuvers, R.C. de Vrijer (Eds.), Selected papers on Automath, in: Studies in Logic and the

Foundations of Computer Science, vol. 133, Elsevier, 1994.

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[27] F. Schuh, Leerboek der Elementaire Theoretische Rekenkunde (‘Textbook on Elementary Theoretical Arithmetic’),Noordhoff. Part 1: De gehele getallen (‘The Integers’), 1919. Part 2: De meetbare Getallen (‘The MeasurableNumbers’), 1921.

[28] M.H. Sorensen, P. Urzyczyn, Lectures on the Curry–Howard isomorphism, in: Studies in Logic and the Foundationsof Mathematics, vol. 149, Elsevier Science, 2006.

[29] Symposium for de Bruijn’s 90th anniversary: http://www.win.tue.nl/debruijn90/video/debruijn.html.

Jan Willem Klop∗

Department of Computer Science,VU University Amsterdam,

NetherlandsE-mail address: j.w.klop@vu.nl.

∗ Correspondence to: Vrije Universiteit, Department of Theoretical Computer Science, DeBoelelaan 1081a, 1081 HV Amsterdam, Netherlands.