Per Martin-Löf
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Per Erik Rutger Martin-Löf (; ; born 8 May 1942) is a
Swedish Swedish or ' may refer to: Anything from or related to Sweden, a country in Northern Europe. Or, specifically: * Swedish language, a North Germanic language spoken primarily in Sweden and Finland ** Swedish alphabet, the official alphabet used by ...
logician Logic is the study of correct reasoning. It includes both Mathematical logic, formal and informal logic. Formal logic is the science of Validity (logic), deductively valid inferences or of logical truths. It is a formal science investigating h ...
, philosopher, and
mathematical statistician Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical an ...
. He is internationally renowned for his work on the foundations of
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
, statistics, mathematical logic, and
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...
. Since the late 1970s, Martin-Löf's publications have been mainly in
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premis ...
. In
philosophical logic Understood in a narrow sense, philosophical logic is the area of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophical ...
, Martin-Löf has wrestled with the philosophy of logical consequence and judgment, partly inspired by the work of Brentano, Frege, and Husserl. In
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
, Martin-Löf has been active in developing intuitionistic type theory as a constructive foundation of mathematics; Martin-Löf's work on type theory has influenced
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...
. Until his retirement in 2009, Per Martin-Löf held a joint chair for Mathematics and Philosophy at
Stockholm University Stockholm University ( sv, Stockholms universitet) is a public research university in Stockholm, Sweden, founded as a college in 1878, with university status since 1960. With over 33,000 students at four different faculties: law, humanities, s ...
.Member profile
Academia Europaea The Academia Europaea is a pan-European Academy of Humanities, Letters, Law, and Sciences. The Academia was founded in 1988 as a functioning Europe-wide Academy that encompasses all fields of scholarly inquiry. It acts as co-ordinator of Europea ...
, retrieved 2014-01-26.
His brother
Anders Martin-Löf Anders Martin-Löf (born 16 March 1940) is a Swedish physicist and mathematician. He has been a professor at the Department of Mathematics of Stockholm University. Martin-Löf did his undergraduate studies at the KTH Royal Institute of Technolo ...
is now emeritus professor of mathematical statistics at Stockholm University; the two brothers have collaborated in research in probability and statistics. The research of Anders and Per Martin-Löf has influenced statistical theory, especially concerning exponential families, the expectation-maximization method for missing data, and model selection. Per Martin-Löf received his PhD in 1970 from
Stockholm University Stockholm University ( sv, Stockholms universitet) is a public research university in Stockholm, Sweden, founded as a college in 1878, with university status since 1960. With over 33,000 students at four different faculties: law, humanities, s ...
, under Andrey Kolmogorov. Martin-Löf is an enthusiastic bird-watcher; his first scientific publication was on the mortality rates of ringed birds..


Randomness and Kolmogorov complexity

In 1964 and 1965, Martin-Löf studied in Moscow under the supervision of
Andrei N. Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
. He wrote a 1966 article ''The definition of random sequences'' that gave the first suitable definition of a random sequence. Earlier researchers such as Richard von Mises had attempted to formalize the notion of a test for randomness in order to define a random sequence as one that passed all tests for randomness; however, the precise notion of a randomness test was left vague. Martin-Löf's key insight was to use the theory of computation to define formally the notion of a test for randomness. This contrasts with the idea of randomness in
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
; in that theory, no particular element of a sample space can be said to be random. Martin-Löf randomness has since been shown to admit many equivalent characterizations — in terms of compression, randomness tests, and
gambling Gambling (also known as betting or gaming) is the wagering of something of Value (economics), value ("the stakes") on a Event (probability theory), random event with the intent of winning something else of value, where instances of strategy (ga ...
— that bear little outward resemblance to the original definition, but each of which satisfies our intuitive notion of properties that random sequences ought to have: random sequences should be incompressible, they should pass statistical tests for randomness, and it should be impossible to make money betting on them. The existence of these multiple definitions of Martin-Löf randomness, and the stability of these definitions under different models of computation, give evidence that Martin-Löf randomness is a fundamental property of mathematics and not an accident of Martin-Löf's particular model. The thesis that the definition of Martin-Löf randomness "correctly" captures the intuitive notion of randomness has been called the "Martin-Löf–
Chaitin Gregory John Chaitin ( ; born 25 June 1947) is an Argentine- American mathematician and computer scientist. Beginning in the late 1960s, Chaitin made contributions to algorithmic information theory and metamathematics, in particular a compute ...
Thesis"; it is somewhat similar to the Church–Turing thesis. Following Martin-Löf's work,
algorithmic information theory Algorithmic information theory (AIT) is a branch of theoretical computer science that concerns itself with the relationship between computation and information of computably generated objects (as opposed to stochastically generated), such as st ...
defines a random string as one that cannot be produced from any computer program that is shorter than the string (
Chaitin–Kolmogorov randomness In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program (in a predetermined programming language) that prod ...
); i.e. a string whose Kolmogorov complexity is at least the length of the string. This is a different meaning from the usage of the term in statistics. Whereas statistical randomness refers to the ''process'' that produces the string (e.g. flipping a coin to produce each bit will randomly produce a string), algorithmic randomness refers to the ''string itself''. Algorithmic information theory separates random from nonrandom strings in a way that is relatively invariant to the
model of computation In computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how an output of a mathematical function is computed given an input. A model describes h ...
being used. An algorithmically random sequence is an ''infinite'' sequence of characters, all of whose prefixes (except possibly a finite number of exceptions) are strings that are "close to" algorithmically random (their length is within a constant of their Kolmogorov complexity).


Mathematical statistics

Per Martin-Löf has done important research in
mathematical statistics Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical ...
, which (in the Swedish tradition) includes
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
and statistics.


Bird-watching and sex determination

Per Martin-Löf began bird watching in his youth and remains an enthusiastic bird-watcher. As a teenager, he published an article on estimating the
mortality rate Mortality rate, or death rate, is a measure of the number of deaths (in general, or due to a specific cause) in a particular population, scaled to the size of that population, per unit of time. Mortality rate is typically expressed in units of d ...
s of birds, using data from bird ringing, in a Swedish zoological journal: This paper was soon cited in leading international journals, and this paper continues to be cited. In the
biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditar ...
and statistics of
birds Birds are a group of warm-blooded vertebrates constituting the class Aves (), characterised by feathers, toothless beaked jaws, the laying of hard-shelled eggs, a high metabolic rate, a four-chambered heart, and a strong yet lightweig ...
, there are several problems of missing data. Martin-Löf's first paper discussed the problem of estimating the mortality rates of the Dunlin species, using capture-recapture methods. The problem of determining the
biological sex Sex is the trait that determines whether a sexually reproducing animal or plant produces male or female gametes. Male plants and animals produce smaller mobile gametes (spermatozoa, sperm, pollen), while females produce larger ones (ova, o ...
of a bird, which is extremely difficult for humans, is one of the first examples in Martin-Löf's lectures on statistical models.


Probability on algebraic structures

Martin-Löf wrote a licenciate thesis on probability on algebraic structures, particularly semigroups, while a student of Ulf Grenander at Stockholm University.


Statistical models

Martin-Löf developed innovative approaches to statistical theory. In his paper "On Tables of Random Numbers", Kolmogorov observed that the frequency probability notion of the limiting properties of infinite sequences failed to provide a foundation for statistics, which considers only finite samples. Much of Martin-Löf's work in statistics was to provide a finite-sample foundation for statistics.


Model selection and hypothesis testing

In the 1970s, Per Martin-Löf made important contributions to statistical theory and inspired further research, especially by Scandinavian statisticians including Rolf Sundberg, Thomas Höglund, and Steffan Lauritzen. In this work, Martin-Löf's previous research on probability measures on semigroups led to a notion of "repetitive structure" and a novel treatment of sufficient statistics, in which one-parameter exponential families were characterized. He provided a category-theoretic approach to nested statistical models, using finite-sample principles. Before (and after) Martin-Löf, such nested models have often been tested using chi-square hypothesis tests, whose justifications are only asymptotic (and so irrelevant to real problems, which always have finite samples).


Expectation maximization method for exponential families

Martin-Löf's student, Rolf Sundberg, developed a detailed analysis of the expectation-maximization (EM) method for estimation using data from exponential families, especially with missing data. Sundberg credits a formula, later known as the Sundberg formula, to previous manuscripts of the Martin-Löf brothers, Per and Anders. Many of these results reached the international scientific community through the 1976 paper on the expectation maximization (EM) method by
Arthur P. Dempster Arthur Pentland Dempster (born 1929) is a Professor Emeritus in the Harvard University Department of Statistics. He was one of four faculty when the department was founded in 1957. Biography Dempster received his B.A. in mathematics and physics ( ...
, Nan Laird, and Donald Rubin, which was published in a leading international journal, sponsored by the
Royal Statistical Society The Royal Statistical Society (RSS) is an established statistical society. It has three main roles: a British learned society for statistics, a professional body for statisticians and a charity which promotes statistics for the public good. ...
.


Logic


Philosophical logic

In
philosophical logic Understood in a narrow sense, philosophical logic is the area of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophical ...
, Per Martin-Löf has published papers on the theory of logical consequence, on judgments, etc. He has been interested in Central-European philosophical traditions, especially of the German-language writings of Franz Brentano,
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic ph ...
, and of
Edmund Husserl , thesis1_title = Beiträge zur Variationsrechnung (Contributions to the Calculus of Variations) , thesis1_url = https://fedora.phaidra.univie.ac.at/fedora/get/o:58535/bdef:Book/view , thesis1_year = 1883 , thesis2_title ...
.


Type theory

Martin-Löf has worked in
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
for many decades. From 1968 to '69 he worked as an assistant professor at the
University of Chicago The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private university, private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park, Chicago, Hyde Park neighborhood. The University of Chic ...
where he met William Alvin Howard with whom he discussed issues related to the Curry–Howard correspondence. Martin-Löf's first draft article on type theory dates back to 1971. This impredicative theory generalized Girard's System F. However, this system turned out to be inconsistent due to
Girard's paradox In mathematical logic, System U and System U− are pure type systems, i.e. special forms of a typed lambda calculus with an arbitrary number of sorts, axioms and rules (or dependencies between the sorts). They were both proved inconsistent by Jea ...
which was discovered by Girard when studying System U, an inconsistent extension of System F. This experience led Per Martin-Löf to develop the philosophical foundations of
type theory In mathematics, logic, and computer science, a type theory is the formal system, formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theor ...
, his ''meaning explanation'', a form of proof-theoretic semantics, which justifies predicative type theory as presented in his 1984 Bibliopolis book, and extended in a number of increasingly philosophical texts, such as his influential ''On the Meanings of the Logical Constants and the Justifications of the Logical Laws''. The 1984 type theory was extensional while the type theory presented in the book by Nordström ''et al.'' in 1990, which was heavily influenced by his later ideas, intensional, and more amenable to being implemented on a computer. Martin-Löf's intuitionistic type theory developed the notion of dependent types and directly influenced the development of the calculus of constructions and the logical framework LF. A number of popular computer-based proof systems are based on type theory, for example NuPRL, LEGO, Coq, ALF, Agda, Twelf,
Epigram An epigram is a brief, interesting, memorable, and sometimes surprising or satirical statement. The word is derived from the Greek "inscription" from "to write on, to inscribe", and the literary device has been employed for over two mille ...
, and
Idris Idris may refer to: People * Idris (name), a list of people and fictional characters with the given name or surname * Idris (prophet), Islamic prophet in the Qur'an, traditionally identified with Enoch, an ancestor of Noah in the Bible * Idris G ...
.


Awards

Martin-Löf is a member of the Royal Swedish Academy of Sciences and of the
Academia Europaea The Academia Europaea is a pan-European Academy of Humanities, Letters, Law, and Sciences. The Academia was founded in 1988 as a functioning Europe-wide Academy that encompasses all fields of scholarly inquiry. It acts as co-ordinator of Europea ...
.


See also

* Franz Brentano * Rudolf Carnap * Michael Dummett *
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic ph ...
* Ulf Grenander * Jaakko Hintikka *
Edmund Husserl , thesis1_title = Beiträge zur Variationsrechnung (Contributions to the Calculus of Variations) , thesis1_url = https://fedora.phaidra.univie.ac.at/fedora/get/o:58535/bdef:Book/view , thesis1_year = 1883 , thesis2_title ...
*
Andrei N. Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
*
Anders Martin-Löf Anders Martin-Löf (born 16 March 1940) is a Swedish physicist and mathematician. He has been a professor at the Department of Mathematics of Stockholm University. Martin-Löf did his undergraduate studies at the KTH Royal Institute of Technolo ...
*
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
* Peter Pagin * Dag Prawitz *
Charles Sanders Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism". Educated as a chemist and employed as a scientist for ...
* Frank P. Ramsey *
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ar ...
* Dana Scott *
Alfred Tarski Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician ...
*
Alan Turing Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical c ...


Notes


References


Bird watching and missing data

* * George A. Barnard
"Gone Birdwatching"
''New Scientist'', 4 December 1999, magazine issue 2215. * *


Probability foundations

* Per Martin-Löf. "The Definition of Random Sequences." ''Information and Control'', 9(6): 602–619, 1966. * Li, Ming and Vitányi, Paul, ''An Introduction to Kolmogorov Complexity and Its Applications'', Springer, 1997


Probability on algebraic structures, following Ulf Grenander

* Grenander, Ulf. ''Probability on Algebraic Structures''. (Dover reprint) * Martin-Löf, P. The continuity theorem on a locally compact group. ''Teor. Verojatnost. i Primenen.'' 10 1965 367–371. * Martin-Löf, Per. Probability theory on discrete semigroups. ''Z. Wahrscheinlichkeitstheorie und Verw. Gebiete'' 4 1965 78—102 * Nitis Mukhopadhyay. "A Conversation with Ulf Grenander". ''Statist. Sci.'' Volume 21, Number 3 (2006), 404–426.


Statistics foundations

*
Anders Martin-Löf Anders Martin-Löf (born 16 March 1940) is a Swedish physicist and mathematician. He has been a professor at the Department of Mathematics of Stockholm University. Martin-Löf did his undergraduate studies at the KTH Royal Institute of Technolo ...
. 1963. "Utvärdering av livslängder i subnanosekundsområdet" ("Evaluation of lifetimes in time-lengths below one nanosecond"). ("Sundberg formula", according to Sundberg 1971) * Per Martin-Löf. 1966. ''Statistics from the point of view of statistical mechanics''. Lecture notes, Mathematical Institute, Aarhus University. ("Sundberg formula" credited to Anders Martin-Löf, according to Sundberg 1971) * Per Martin-Löf. 1970. ''Statistika Modeller (Statistical Models): Anteckningar fran seminarier läsåret 1969–1970 (Notes from seminars in the academic year 1969–1970), with the assistance of Rolf Sundberg.'' Stockholm University. * Martin-Löf, P. "Exact tests, confidence regions and estimates", with a discussion by A. W. F. Edwards,
G. A. Barnard George Alfred Barnard (23 September 1915 – 9 August 2002) was a British statistician known particularly for his work on the foundations of statistics and on quality control. Biography George Barnard was born in Walthamstow, London ...
, D. A. Sprott, O. Barndorff-Nielsen, D. Basu and G. Rasch. ''Proceedings of Conference on Foundational Questions in Statistical Inference'' (Aarhus, 1973), pp. 121–138. Memoirs, No. 1, Dept. Theoret. Statist., Inst. Math., Univ. Aarhus, Aarhus, 1974. * Martin-Löf, P. Repetitive structures and the relation between canonical and microcanonical distributions in statistics and statistical mechanics. With a discussion by
D. R. Cox Sir David Roxbee Cox (15 July 1924 – 18 January 2022) was a British statistician and educator. His wide-ranging contributions to the field of statistics included introducing logistic regression, the proportional hazards model and the Cox pro ...
and G. Rasch and a reply by the author. ''Proceedings of Conference on Foundational Questions in Statistical Inference'' (Aarhus, 1973), pp. 271–294. Memoirs, No. 1, Dept. Theoret. Statist., Inst. Math., Univ. Aarhus, Aarhus, 1974. * Martin-Löf, P. The notion of redundancy and its use as a quantitative measure of the deviation between a statistical hypothesis and a set of observational data. With a discussion by F. Abildgård, A. P. Dempster, D. Basu,
D. R. Cox Sir David Roxbee Cox (15 July 1924 – 18 January 2022) was a British statistician and educator. His wide-ranging contributions to the field of statistics included introducing logistic regression, the proportional hazards model and the Cox pro ...
, A. W. F. Edwards, D. A. Sprott,
G. A. Barnard George Alfred Barnard (23 September 1915 – 9 August 2002) was a British statistician known particularly for his work on the foundations of statistics and on quality control. Biography George Barnard was born in Walthamstow, London ...
, O. Barndorff-Nielsen, J. D. Kalbfleisch and G. Rasch and a reply by the author. ''Proceedings of Conference on Foundational Questions in Statistical Inference'' (Aarhus, 1973), pp. 1–42. Memoirs, No. 1, Dept. Theoret. Statist., Inst. Math., Univ. Aarhus, Aarhus, 1974. * Martin-Löf, Per The notion of redundancy and its use as a quantitative measure of the discrepancy between a statistical hypothesis and a set of observational data. ''Scand. J. Statist.'' 1 (1974), no. 1, 3—18. * Sverdrup, Erling. "Tests without power." ''Scand. J. Statist.'' 2 (1975), no. 3, 158–160. * Martin-Löf, Per Reply to Erling Sverdrup's polemical article: ``Tests without power'' (''Scand. J. Statist.'' 2 (1975), no. 3, 158–160). ''Scand. J. Statist.'' 2 (1975), no. 3, 161–165. * Sverdrup, Erling. A rejoinder to: ``Tests without power'' (''Scand. J. Statist.'' 2 (1975), 161—165) by P. Martin-Löf. ''Scand. J. Statist.'' 4 (1977), no. 3, 136—138. * Martin-Löf, P. Exact tests, confidence regions and estimates. Foundations of probability and statistics. II. ''Synthese'' 36 (1977), no. 2, 195—206. * Rolf Sundberg. 1971. ''Maximum likelihood theory and applications for distributions generated when observing a function of an exponential family variable''. Dissertation, Institute for Mathematical Statistics, Stockholm University. * Sundberg, Rolf. Maximum likelihood theory for incomplete data from an exponential family. ''Scand. J. Statist.'' 1 (1974), no. 2, 49—58. * Sundberg, Rolf An iterative method for solution of the likelihood equations for incomplete data from exponential families. Comm. Statist.—Simulation Comput. B5 (1976), no. 1, 55—64. * Sundberg, Rolf Some results about decomposable (or Markov-type) models for multidimensional contingency tables: distribution of marginals and partitioning of tests. Scand. J. Statist. 2 (1975), no. 2, 71—79. * Höglund, Thomas. The exact estimate — a method of statistical estimation. ''Z. Wahrscheinlichkeitstheorie und Verw. Gebiete'' 29 (1974), 257—271. * Lauritzen, Steffen L. ''Extremal families and systems of sufficient statistics''. Lecture Notes in Statistics, 49. Springer-Verlag, New York, 1988. xvi+268 pp.


Foundations of mathematics, logic, and computer science

* Per Martin-Löf. A theory of types. Preprint, Stockholm University, 1971. * Per Martin-Löf.
An intuitionistic theory of types
'' In G. Sambin and J. Smith, editors, Twenty-Five Years of Constructive Type Theory. Oxford University Press, 1998. Reprinted version of an unpublished report from 1972. * Per Martin-Löf. An intuitionistic theory of types: Predicative part. In H. E. Rose and J. C. Shepherdson, editors, Logic Colloquium ‘73, pages 73–118. North Holland, 1975. * Per Martin-Löf.
Constructive mathematics and computer programming
'' In ''Logic, Methodology and Philosophy of Science VI, 1979''. Eds. Cohen, et al. North-Holland, Amsterdam. pp. 153–175, 1982. * Per Martin-Löf.
Intuitionistic type theory
'' (Notes by Giovanni Sambin of a series of lectures given in Padua, June 1980). Napoli, Bibliopolis, 1984. * Per Martin-Löf. ''Philosophical implications of type theory'', Unpublished notes, 1987? * Per Martin-Löf. ''Substitution calculus'', 1992. Notes from a lecture given in Göteborg. * Bengt Nordström, Kent Petersson, and Jan M. Smith. ''Programming in Martin-Löf's Type Theory''. Oxford University Press, 1990. (The book is out of print, bu
a free version
has been made available.) * Per Martin-Löf.
On the Meanings of the Logical Constants and the Justifications of the Logical Laws
'' '' Nordic Journal of Philosophical Logic'', 1(1): 11–60, 1996. * Per Martin-Löf.
Logic and Ethics
'' In T. Piecha and P. Schroeder-Heister, editors,
Proof-Theoretic Semantics: Assessment and Future Perspectives. Proceedings of the Third Tübingen Conference on Proof-Theoretic Semantics, 27–30 March 2019
', pages 227-235. URI:
http://dx.doi.org/10.15496/publikation-35319
'. University of Tübingen 2019.


External links

* {{DEFAULTSORT:Martin-Lof, Per Members of the Royal Swedish Academy of Sciences Members of Academia Europaea Mathematical logicians Swedish logicians Swedish statisticians 20th-century Swedish philosophers 21st-century Swedish philosophers Swedish information theorists 20th-century Swedish mathematicians 21st-century Swedish mathematicians Stockholm University faculty Birdwatchers Swedish ornithologists 1942 births Living people Tarski lecturers Gödel Lecturers