Person: Finsler, Paul
Paul Finsler was a German mathematician who worked on set theory, differential geometry, number theory, probability theory and the foundations of mathematics.
Mathematical Profile (Excerpt):
 Paul attended a grammar school in Urach, then, between 1908 and 1912, he attended a secondary school, the Real Gymnasium in Cannstatt, which emphasised science in its teaching.
 Kutta had been appointed as an ordinary professor of analysis and analytical geometry at the Technische Hochschule in Stuttgart in 1911, the year before Finsler began his studies there.
 Finsler also attended lectures at Stuttgart by Rudolf Mehmke (18571944).
 Leaving Stuttgart in 1913, Finsler entered Göttingen University to undertake graduate studies.
 Finsler' doctoral dissertation was supervised by Carathéodory on curves and surfaces in general spaces.
 This thesis, entitled Über Kurven und Flächen in allgemeinen Räumen Ⓣ(On curves and surfaces in general spaces) (1918), secured Finsler a name for himself as a differential geometer.
 A large number of treatises and also some books have since been written about "Finslerian geometry" inaugurated in Finsler's dissertation, and it has already become clear that this groundbreaking work is worthy in adding to the Swiss geometric tradition that Steiner and Schläfli began.
 Since Finsler's dissertation has never been published by the book trade, it has been enjoyed by only a few mathematicians who were Finsler's friends.
 The reviewer concludes from his own sad experience that there must be many instances where apparently new results were found to be already in Finsler's thesis after the latter finally became available through interlibrary loan.
 In addition to the thesis the present edition contains a comprehensive list of books and papers concerning Finsler spaces (until 1949) compiled by H Schubert.
 It has a wide scope and comprises references to tensor calculus, ordinary and Riemannian geometry, the geometry of paths, the calculus of variations, spaces of infinite dimension, which are ordinarily not considered as contributions to Finsler spaces.
 "Finsler spaces" and "Finsler manifolds" became standard terminology after the publication of Elie Cartan's book Les espaces de Finsler Ⓣ(Finsler spaces) in 1934.
 A Finsler space is a generalisation of a Riemannian space where the length function is defined differently and Minkowski's geometry holds locally.
 Despite the importance of this work, differential geometry was not Finsler's research topic for long since he moved to take up set theory.
 Finsler's habilitation thesis was submitted to the University of Cologne in 1922 and the following year he gave his inaugural lecture Gibt es Widersprüche in der Mathematik?
 Russell's paradox, Finsler points out that one needs to distinguish between satisfiable and unsatisfiable circular definitions.
 Now Finsler had another passion in addition to mathematics, namely astronomy.
 On 15 September 1924 Finsler was in Bonn when, with a pair of binoculars, he discovered a comet which is now named Comet C / 1924 R1 (Finsler).
 After Finsler's report it was observed at several observatories on 19 September.
 For his discovery, Finsler received the Donohoe Comet Medal from the Astronomical Society of the Pacific.
 In 1927 Finsler was appointed to the University of Zürich, where the professor was Andreas Speiser.
 At Zürich, in addition to his work on set theory Finsler also worked on differential geometry, number theory, probability theory and the foundations of mathematics.
 Soon after this Finsler began to teach the course on the introduction to the calculus.
 Finsler's set theory was in the spirit of Georg Cantor.
 In 1926 Finsler published the paper Formale Beweise und Entscheidbarkeit Ⓣ(Formal Proof and Decidability) and also the first part of a major work on set theory Über die Grundlegung der Mengenlehre.
 Finsler develops his approach to the paradoxes, his attitude towards formalised theories and his defence of Platonism in mathematics.
 From the foundational point of view, Finsler's et theory contains a strengthened criterion for set identity and a coinductive specification of the universe of sets.
 Combinatorially, Finsler considers sets as generalised numbers to which one may apply arithmetical techniques.
 Of course, as mentioned above, the set paradoxes were of particular significance to Finsler.
 Now Gödel had not been aware of Finsler's work when he wrote his 1931 paper but, after receiving Finsler's letter, he read the 1926 paper and immediately saw that it had major problems.
 However, he didn't want to offend Finsler so, in his reply, he merely pointed out the difference in their approach but said the Finsler's system was "not really defined at all".
 Finsler did not react well to criticisms that came from Gödel and other logicians.
 Finsler continued to develop his ideas despite the criticisms.

 It seems to the reviewer that Finsler's solution of the antinomy of the liar is not of the required kind: it does not provide a test for proposed proofs, but rather explains away the antinomy after the fact.
 However, Finsler omits exactly the main point which makes a proof possible, namely restriction to some welldefined formal system in which the proposition is undecidable.
 If Finsler had confined himself to some welldefined formal system S, his proof ...
 The reader will see from the list of Finsler's publications that he undertook research on topics other than differential geometry and the foundations of mathematics.
 On 4 July 1937, while in Zürich, Finsler discovered another comet, now named Comet 1937 f (Finsler), and immediately informed Copenhagen.
 This is Professor Finsler's second comet discovery; the first was in 1924.
 On 31 July 1937 Finsler's discovery of the comet made the front page of the Chicago Daily Tribune under the headline "Heads Up! You May See New Comet Tonight".
 His comet discoveries certainly gained Finsler far wider recognition than his mathematical discoveries.
 There was another publication by Finsler which caused something of a sensation, namely his article Vom Leben nach dem Tode Ⓣ(On Life after Death) which he published in 1958.
 Many who read this article probably misunderstood what Finsler was trying to say.
 The most valuable thing Finsler has given to many of his students, is a great confidence in the ability of human reason and a distrust of any mathematical formalism, which pretends to be more than a shorthand notation, compared to that which can be expressed in mathematical thoughts and ideas.
 In 1959 Finsler retired from his professorship in Zürich and was made an honorary professor.
Born 11 April 1894, Heilbronn, Neckar, Germany. Died 29 April 1970, Zürich, Switzerland.
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Astronomy, Origin Germany
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References
Adapted from other CC BYSA 4.0 Sources:
 O’Connor, John J; Robertson, Edmund F: MacTutor History of Mathematics Archive