Oskar Becker
Encyclopedia
Oscar Becker was a German philosopher, logician, mathematician
, and historian
of mathematics
.
. His dissertation under Otto Hölder
and Karl Rohn (1914) was On the Decomposition of Polygons in non-intersecting triangles on the Basis of the Axioms of Connection and Order.
He served in World War I
and returned to study philosophy with Edmund Husserl
, writing his Habilitationsschrift on Investigations of the Phenomenological Foundations of Geometry and their Physical Applications, (1923). Becker was Husserl's assistant, informally, and then official editor of the Yearbook for Phenomenological Research.
's Being and Time
. Becker frequently attended Heidegger's seminar
s during those years.
Becker utilized not only Husserlian phenomenology but, much more controversially, Heideggerian hermeneutics, discussing arithmetic
al counting
as "being toward death". His work was criticized both by neo-Kantians and by more mainstream, rationalist logicians, to whom Becker feistily replied. This work has not had great influence on later debates in the foundations of mathematics
, despite its many interesting analyses of the topic of its title.
Becker debated with David Hilbert
and Paul Bernays
over the role of the potential infinite in Hilbert's formalist metamathematics
. Becker argued that Hilbert could not stick with finitism
, but had to assume the potential infinite. Clearly enough, Hilbert and Bernays do implicitly accept the potential infinite, but they claim that each induction in their proofs is finite. Becker was correct that complete induction was needed for assertions of consistency in the form of universally quantified sentences, as opposed to claiming that a predicate holds for each individual natural number.
. He developed a semantics of intuitionistic logic based on Husserl's phenomenology, and this semantics was used by Arend Heyting
in his own formalization. Becker struggled, somewhat unsuccessfully, with the formulation of the rejection of excluded middle appropriate for intuitionistic logic. Becker failed in the end to correctly distinguish classical and intuitionistic negation, but he made a start. In an appendix to his book on mathematical existence, Becker set the problem of finding a formal calculus for intuitionistic logic. In a series of works in the early 1950s he surveyed modal, intuitionistic, probabilistic, and other philosophical logics.
Becker made contributions to modal logic
(the logic of necessity
and possibility
) and Becker’s postulate, the claim that modal status is necessary (for instance that the possibility of P implies the necessity of the possibility of P, and also the iteration of necessity) is named for him. Becker's Postulate later played a role in the formalization given, by Charles Hartshorne
, the American process theologian, of the Ontological Proof of God's existence, stimulated by conversations with the logical positivist and opponent of the 'proof', Rudolf Carnap
.
. Becker, as did several others, emphasized the "crisis" in Greek mathematics occasioned by the discovery of incommensurability of the side of the pentagon
(or in the later, simpler proofs, the triangle) by Hippasus of Metapontum, and the threat of (literally) "irrational" number
s. To German theorists of the "crisis", the Pythagorean diagonal of the square was similar in its impact to Cantor's diagonalization method of generating higher order infinities, and Gödel's diagonalization method in Gödel
's proof of incompleteness of formalized arithmetic. Becker, like several earlier historians, suggests that the avoidance of arithmetic statement of geometrical magnitude in Euclid
is avoided for ratio
s and proportions, as a consequence of recoil from the shock of incommensurability. Becker also showed that all the theorems of Euclidean proportion theory could be proved using an earlier alternative to the Eudoxus
technique which Becker found stated in Aristotle's Topics, and which Becker attributes to Theatetus. Becker also showed how a constructive logic that denied unrestricted excluded middle could be used to reconstruct most of Euclid's proofs.
More recent revisionist commentators such as Wilbur Knorr
and David Fowler have accused historians of early Greek mathematics writing in the early twentieth century, such as Becker, of reading the crisis of their own times illegitimately into the early Greek period. (This “crisis” may include both the crisis of twentieth century set theory and foundations of mathematics, and the general crisis of WWI, the overthrow of the Kaiser, communist uprisings, and the Weimar Republic.)
. In his Dasein und Dawesen Becker advocated what he called a "mantic" divination. Hermeneutics of the Heideggerian sort is applicable to individual lived existence, but "mantic" decipherment is necessary not only in mathematics, but in aesthetics
, and the investigation of the unconscious
. These realms deal with the eternal and structural, such as the symmetries of nature, and are properly investigated by a mantic phenomenology, not an hermeneutic one. (Becker's emphasis on the timelessness and formal nature of the unconscious has some parallels with the account of Jacques Lacan
.)
, Adolf Fraenkel (later Abraham), Arend Heyting
, David Hilbert
, John von Neumann
, Hermann Weyl
, and Ernst Zermelo
among mathematicians, as well as Hans Reichenbach
and Felix Kaufmann
among philosophers. The letters that Becker received from these major figures of twentieth century mathematics and leading logical positivist philosophers, as well as Becker’s own copies of his letters to them, were destroyed during WWII.
Becker's correspondence with Weyl has been reconstructed (see bibliography), as Weyl's copies of Becker’s letters to him are preserved, and Becker often extensively quotes or paraphrases Weyl’s own letters. Perhaps the same can be done with some other parts of this valuable but lost correspondence. Weyl entered into correspondence with Becker with high hopes and expectations, given their mutual admiration for Husserl’s phenomenology and Husserl’s great admiration for the work of Becker. However, Weyl, whose sympathies were with contructivism and intuitionism, lost patience when he argued with Becker about a purported intuition of the infinite defended by Becker. Weyl concluded, sourly, that Becker would discredit phenomenological approaches to mathematics if he persisted in this position.
Two able philosophers who were students of Becker, Juergen Habermas and Hans Sluga
, later grappled with the issue of the influence of Nazism on German academia. The application of Heidegger's ideas to theoretical science (let alone mathematics) has only recently become widespread, particularly in the English-speaking world
. Furthermore Becker's polemical replies probably alienated his critics still further.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, and historian
Historian
A historian is a person who studies and writes about the past and is regarded as an authority on it. Historians are concerned with the continuous, methodical narrative and research of past events as relating to the human race; as well as the study of all history in time. If the individual is...
of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
.
Early life
Becker studied mathematics at LeipzigLeipzig
Leipzig Leipzig has always been a trade city, situated during the time of the Holy Roman Empire at the intersection of the Via Regia and Via Imperii, two important trade routes. At one time, Leipzig was one of the major European centres of learning and culture in fields such as music and publishing...
. His dissertation under Otto Hölder
Otto Hölder
Otto Ludwig Hölder was a German mathematician born in Stuttgart.Hölder first studied at the Polytechnikum and then in 1877 went to Berlin where he was a student of Leopold Kronecker, Karl Weierstraß, and Ernst Kummer.He is famous for many things including: Hölder's inequality, the Jordan–Hölder...
and Karl Rohn (1914) was On the Decomposition of Polygons in non-intersecting triangles on the Basis of the Axioms of Connection and Order.
He served in World War I
World War I
World War I , which was predominantly called the World War or the Great War from its occurrence until 1939, and the First World War or World War I thereafter, was a major war centred in Europe that began on 28 July 1914 and lasted until 11 November 1918...
and returned to study philosophy with Edmund Husserl
Edmund Husserl
Edmund Gustav Albrecht Husserl was a philosopher and mathematician and the founder of the 20th century philosophical school of phenomenology. He broke with the positivist orientation of the science and philosophy of his day, yet he elaborated critiques of historicism and of psychologism in logic...
, writing his Habilitationsschrift on Investigations of the Phenomenological Foundations of Geometry and their Physical Applications, (1923). Becker was Husserl's assistant, informally, and then official editor of the Yearbook for Phenomenological Research.
Work in phenomenology and mathematical philosophy
He published Mathematical Existence his magnum opus, in the Yearbook in 1927. A famous work that also appeared in the Yearbook that year was Martin HeideggerMartin Heidegger
Martin Heidegger was a German philosopher known for his existential and phenomenological explorations of the "question of Being."...
's Being and Time
Being and Time
Being and Time is a book by the German philosopher Martin Heidegger. Although written quickly, and despite the fact that Heidegger never completed the project outlined in the introduction, it remains his most important work and has profoundly influenced 20th-century philosophy, particularly...
. Becker frequently attended Heidegger's seminar
Seminar
Seminar is, generally, a form of academic instruction, either at an academic institution or offered by a commercial or professional organization. It has the function of bringing together small groups for recurring meetings, focusing each time on some particular subject, in which everyone present is...
s during those years.
Becker utilized not only Husserlian phenomenology but, much more controversially, Heideggerian hermeneutics, discussing arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
al counting
Counting
Counting is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a counter by a unit for every element of the set, in some order, while marking those elements to avoid visiting the same element more than once,...
as "being toward death". His work was criticized both by neo-Kantians and by more mainstream, rationalist logicians, to whom Becker feistily replied. This work has not had great influence on later debates in the foundations of mathematics
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
, despite its many interesting analyses of the topic of its title.
Becker debated with David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
and Paul Bernays
Paul Bernays
Paul Isaac Bernays was a Swiss mathematician, who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant to, and close collaborator of, David Hilbert.-Biography:Bernays spent his childhood in Berlin. Bernays attended the...
over the role of the potential infinite in Hilbert's formalist metamathematics
Metamathematics
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories...
. Becker argued that Hilbert could not stick with finitism
Finitism
In the philosophy of mathematics, one of the varieties of finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps...
, but had to assume the potential infinite. Clearly enough, Hilbert and Bernays do implicitly accept the potential infinite, but they claim that each induction in their proofs is finite. Becker was correct that complete induction was needed for assertions of consistency in the form of universally quantified sentences, as opposed to claiming that a predicate holds for each individual natural number.
Intuitionistic and modal logic
Becker made a start toward the formalization of L. E. J. Brouwer's intuitionistic logicIntuitionistic logic
Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...
. He developed a semantics of intuitionistic logic based on Husserl's phenomenology, and this semantics was used by Arend Heyting
Arend Heyting
Arend Heyting was a Dutch mathematician and logician. He was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, and did much to put intuitionistic logic on a footing where it could become part of mathematical logic...
in his own formalization. Becker struggled, somewhat unsuccessfully, with the formulation of the rejection of excluded middle appropriate for intuitionistic logic. Becker failed in the end to correctly distinguish classical and intuitionistic negation, but he made a start. In an appendix to his book on mathematical existence, Becker set the problem of finding a formal calculus for intuitionistic logic. In a series of works in the early 1950s he surveyed modal, intuitionistic, probabilistic, and other philosophical logics.
Becker made contributions to modal logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
(the logic of necessity
Necessity
In U.S. criminal law, necessity may be either a possible justification or an exculpation for breaking the law. Defendants seeking to rely on this defense argue that they should not be held liable for their actions as a crime because their conduct was necessary to prevent some greater harm and when...
and possibility
Possibility
Possibility is the condition or fact of being possible. The Latin origins of the word hint at ability. Possibility also refers to something that "could happen", that is not precluded by the facts, but usually not probable...
) and Becker’s postulate, the claim that modal status is necessary (for instance that the possibility of P implies the necessity of the possibility of P, and also the iteration of necessity) is named for him. Becker's Postulate later played a role in the formalization given, by Charles Hartshorne
Charles Hartshorne
Charles Hartshorne was a prominent American philosopher who concentrated primarily on the philosophy of religion and metaphysics. He developed the neoclassical idea of God and produced a modal proof of the existence of God that was a development of St. Anselm's Ontological Argument...
, the American process theologian, of the Ontological Proof of God's existence, stimulated by conversations with the logical positivist and opponent of the 'proof', Rudolf Carnap
Rudolf Carnap
Rudolf Carnap was an influential German-born philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle and an advocate of logical positivism....
.
History of mathematics
Becker also made important contributions to the history and interpretation of ancient Greek mathematicsGreek mathematics
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...
. Becker, as did several others, emphasized the "crisis" in Greek mathematics occasioned by the discovery of incommensurability of the side of the pentagon
Pentagon
In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...
(or in the later, simpler proofs, the triangle) by Hippasus of Metapontum, and the threat of (literally) "irrational" number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
s. To German theorists of the "crisis", the Pythagorean diagonal of the square was similar in its impact to Cantor's diagonalization method of generating higher order infinities, and Gödel's diagonalization method in Gödel
Godel
Godel or similar can mean:*Kurt Gödel , an Austrian logician, mathematician and philosopher*Gödel...
's proof of incompleteness of formalized arithmetic. Becker, like several earlier historians, suggests that the avoidance of arithmetic statement of geometrical magnitude in Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
is avoided for ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...
s and proportions, as a consequence of recoil from the shock of incommensurability. Becker also showed that all the theorems of Euclidean proportion theory could be proved using an earlier alternative to the Eudoxus
Eudoxus
Eudoxus or Eudoxos was the name of two ancient Greeks:* Eudoxus of Cnidus , Greek astronomer and mathematician.* Eudoxus of Cyzicus , Greek navigator....
technique which Becker found stated in Aristotle's Topics, and which Becker attributes to Theatetus. Becker also showed how a constructive logic that denied unrestricted excluded middle could be used to reconstruct most of Euclid's proofs.
More recent revisionist commentators such as Wilbur Knorr
Wilbur Knorr
Wilbur Richard Knorr was an American historian of mathematics and a professor in the departments of philosophy and classics at Stanford University. He has been called "one of the most profound and certainly the most provocative historian of Greek mathematics" of the 20th century.-Biography:Knorr...
and David Fowler have accused historians of early Greek mathematics writing in the early twentieth century, such as Becker, of reading the crisis of their own times illegitimately into the early Greek period. (This “crisis” may include both the crisis of twentieth century set theory and foundations of mathematics, and the general crisis of WWI, the overthrow of the Kaiser, communist uprisings, and the Weimar Republic.)
Later thought
At the end of his life Becker re-emphasized the distinction between intuition of the formal and Platonic realm as opposed to the concrete existential realm, moved to the terminology, at least, of divinationDivination
Divination is the attempt to gain insight into a question or situation by way of an occultic standardized process or ritual...
. In his Dasein und Dawesen Becker advocated what he called a "mantic" divination. Hermeneutics of the Heideggerian sort is applicable to individual lived existence, but "mantic" decipherment is necessary not only in mathematics, but in aesthetics
Aesthetics
Aesthetics is a branch of philosophy dealing with the nature of beauty, art, and taste, and with the creation and appreciation of beauty. It is more scientifically defined as the study of sensory or sensori-emotional values, sometimes called judgments of sentiment and taste...
, and the investigation of the unconscious
Unconscious mind
The unconscious mind is a term coined by the 18th century German romantic philosopher Friedrich Schelling and later introduced into English by the poet and essayist Samuel Taylor Coleridge...
. These realms deal with the eternal and structural, such as the symmetries of nature, and are properly investigated by a mantic phenomenology, not an hermeneutic one. (Becker's emphasis on the timelessness and formal nature of the unconscious has some parallels with the account of Jacques Lacan
Jacques Lacan
Jacques Marie Émile Lacan was a French psychoanalyst and psychiatrist who made prominent contributions to psychoanalysis and philosophy, and has been called "the most controversial psycho-analyst since Freud". Giving yearly seminars in Paris from 1953 to 1981, Lacan influenced France's...
.)
Contacts and correspondence
Becker carried on an extensive correspondence with some of the greatest mathematicians and philosophers of the day. These included AckermannWilhelm Ackermann
Wilhelm Friedrich Ackermann was a German mathematician best known for the Ackermann function, an important example in the theory of computation....
, Adolf Fraenkel (later Abraham), Arend Heyting
Arend Heyting
Arend Heyting was a Dutch mathematician and logician. He was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, and did much to put intuitionistic logic on a footing where it could become part of mathematical logic...
, David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
, John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
, Hermann Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
, and Ernst Zermelo
Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo was a German mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem.-Life:He graduated...
among mathematicians, as well as Hans Reichenbach
Hans Reichenbach
Hans Reichenbach was a leading philosopher of science, educator and proponent of logical empiricism...
and Felix Kaufmann
Felix Kaufmann
Felix Kaufmann was an Austrian-American philosopher of law.Kaufmann studied jurisprudence and philosophy in Vienna. From 1922 to 1938 he was a Privatdozent there. During this time Kaufmann was associated with the Vienna Circle...
among philosophers. The letters that Becker received from these major figures of twentieth century mathematics and leading logical positivist philosophers, as well as Becker’s own copies of his letters to them, were destroyed during WWII.
Becker's correspondence with Weyl has been reconstructed (see bibliography), as Weyl's copies of Becker’s letters to him are preserved, and Becker often extensively quotes or paraphrases Weyl’s own letters. Perhaps the same can be done with some other parts of this valuable but lost correspondence. Weyl entered into correspondence with Becker with high hopes and expectations, given their mutual admiration for Husserl’s phenomenology and Husserl’s great admiration for the work of Becker. However, Weyl, whose sympathies were with contructivism and intuitionism, lost patience when he argued with Becker about a purported intuition of the infinite defended by Becker. Weyl concluded, sourly, that Becker would discredit phenomenological approaches to mathematics if he persisted in this position.
Nazism and neglect
It is possible that regard for Becker's earlier work suffered from his later Nazi allegiances, leading to lack of reference or published commentary by émigré logicians and mathematicians who had fled Hitlerism. His lecture on "The Vacuity of Art and the Daring of the Artist," presents a "Nordic Metaphysics" in fairly standard Nazi style.Two able philosophers who were students of Becker, Juergen Habermas and Hans Sluga
Hans Sluga
Hans D. Sluga is a German academic, who is a professor of philosophy at the University of California, Berkeley. He teaches and writes on, among other things, Gottlob Frege, Ludwig Wittgenstein, Martin Heidegger, Michel Foucault, and German philosophy in the Nazi period.He studied at the...
, later grappled with the issue of the influence of Nazism on German academia. The application of Heidegger's ideas to theoretical science (let alone mathematics) has only recently become widespread, particularly in the English-speaking world
English-speaking world
The English-speaking world consists of those countries or regions that use the English language to one degree or another. For more information, please see:Lists:* List of countries by English-speaking population...
. Furthermore Becker's polemical replies probably alienated his critics still further.
Becker's works
- Über die Zerlegung eines Polygons in exclusive Dreiecke auf Grund der ebenen Axiome der Verknuepfung und Anordnung (Leipzig, 1914)
- "Contributions Toward a Phenomenological Foundation of Geometry and Its Physical Applications," from Beiträge zur phänomenologischen Begründung der Geometrie und ihre physikalischen Anwendungen (Jahrbuch für Philosophie und phänomenologische Forschung IV 1923, 493–560). Selections trans. by Theodore Kisiel, in Phenomenology and the Natural Sciences, ed. Joseph Kockelmans and Theordore J. Kisiel, Evanston IL: Northwestern University Press, 1970, 119–143.
- Mathematische Existenz. Untersuchungen zur Logik und Ontologie mathematischer Phänomene (Jahrbuch für Philosophie und phänomenologische Forschung, Vol. VIII, 1927, 440–809.
- "The Philosophy of Edmund Husserl," transl. R. O. Elverton, in The Phenomenology of Husserl, ed. R. O. Elverton, Quadrangle Books, Chicago: 1970, 40–72, originally "Die Philosophie Edmund Husserl's," in Kantstudien vol. 35, 1930, 119–150.
- “Eudoxus-Studien: I: Eine voreudoxische Proportionenlehre und ihre Spuren bei Aristoteles und Euklid,” Quellen und Studien zur Geschichte der Mathematik, Astronomie und Phyik B. II (1933), 311–330. [reprinted in Jean Christianidis, ed. Classics in the history of Greek Mathematics, Boston Studies in the Philosophie of Science, vol. 240, Dordrecht/Boston: 2004, 191–209, with intro. by Ken Saito, 188–9.] “II: Warum haben die Griechen die Existenz der vierten Proportionale angenommen,” 369–387, “III: Spuren eines Stetigkeitsaxioms in der Art des Dedekindschen zur Zeir des Eudoxos,” vol. 3 (1936) 236–244, “IV: Das Prinzip des ausgeschlossenen Dritten in der griechischen Mathematik,” 370–388, “V: Die eudoxische Lehre von den Ideen und den Farben, 3 (1936) 389–410.
- "Zur Logik der Modalitäten", in: Jahrbuch für Philosophie und phänomenologische Forschung, Bd. XI (1930), pp. 497–548
- Grundlagen der Mathematik in geschichtlicher Entwicklung, Freiburg/München: Alber, 1954 (2. Aufl. 1964; diese Aufl. ist auch text- und seitenidentisch erschienen als Suhrkamp Taschenbuch Wissenschaft 114. Frankfurt a. M. : Suhrkamp, 1975)
- Dasein und Dawesen (1964)
- Letters to Hermann Weyl, in Paolo Mancosu and T. A. Ryckman, “Mathematics and Phenomenology: The Correspondence between O. Becker and H. Weyl,” Philosophia Mathematica, 3d Series, vol. 10 (2002) 174–194.
Secondary sources
- Annemarie Gethmann-SiefertAnnemarie Gethmann-SiefertAnnemarie Gethmann-Siefert is a professor of philosophy at the University of Hagen, Germany.-Biography:Gethmann-Siefert was born in 1945, studied philosophy, art history and theology in Münster, Bonn, Innsbruck and Bochum. She earned her PhD in philosophy 1973 from the Ruhr University Bochum...
, Jürgen MittelstraßJürgen MittelstraßJürgen Mittelstraß is a German philosopher especially interested in the philosophy of science.He was born in Düsseldorf in 1936 and studied philosophy, history and protestant theology at Bonn, Erlangen, Hamburg and Oxford from 1956 till 1961.He received his Ph.D...
(eds): Die Philosophie und die Wissenschaften. Zum Werk Oskar Beckers (Philosophy and the Sciences: On the Work of Oskar Becker), Munich, Fink, 2002 http://www.springerlink.com/content/b4dwda35ab6km23g/. - Wilbur R. Knorr, “Transcript of a Lecture Delivered at the Annual Convention of the History of Science Society, Atlanta, Dec. 28, 1975” in Jean Christianidis, ed. Classics in the history of Greek Mathematics, Boston Studies in the Philosophie of Science, vol. 240, Dordrecht/Boston: 2004, 245–253, esp. 249–252.
- Joseph Kockelmans and Theordore J. Kisiel, intro. to transl. of Becker, in Phenomenology and the Natural Sciences, Evanston IL: Northwestern University Press, 1970, 117–118.
- Paolo Mancosu and T. A. Ryckman, “Mathematics and Phenomenology: The Correspondence between O. Becker and H. Weyl,” Philosophia Mathematica, 3d Series, vol. 10 (2002) 130–173, bibliography 195–202.
- Paolo Mancosu, ed. From Brouwer to Hilbert,Oxford University Press, 1998, 165–167 (on Hilbert's formalism), 277–282 (on intuitionistic logic).
- Zimny, L., “Oskar Becker Bibliographie,” Kantstudien 60 319–330.