MATH 4400, History of Mathematics Turn of the 19th to 20th centuries, The Computer Age Peter Gibson January 14, 2020 Leading figures of 1900 (and their successors) Henri Poincar´e(1854-1912) Professor at the Sorbonne P. Gibson Math 4400 2 / 43 David Hilbert (1862-1943) Professor at G¨ottingen P. Gibson Math 4400 3 / 43 Hermann Minkowskii (1864-1909) Professor at ETH Z¨urich P. Gibson Math 4400 4 / 43 Jacques Hadamard (1865-1963) Professor at Coll`egede France P. Gibson Math 4400 5 / 43 Charles Jean de la Valle-Poussin (1866-1962) Professor at Catholic University of Leuven P. Gibson Math 4400 6 / 43 Tullio Levi-Civita (1873-1941) Professor at University of Rome P. Gibson Math 4400 7 / 43 Albert Einstein (1879-1955) Professor at Institute for Advanced Study, Princeton P. Gibson Math 4400 8 / 43 Hermann Weyl (1885-1955) Professor at Institute for Advanced Study, Princeton P. Gibson Math 4400 9 / 43 Minkowski 1872 (aged 8) moved to K¨onigsberg from Russian kingdom 1883 prize of the French Academy of Sciences friendship with David Hilbert, Adoph Hurwitz 1885 doctorate under Ferdinand von Lindemann appointments at Bonn, K¨onigsberg, Z¨urich,G¨ottingen geometry of numbers Minkowski space time P. Gibson Math 4400 10 / 43 Hadamard List of things named after Jacques Hadamard - Wikipedia https://en.wikipedia.org/wiki/List_of_things_named_after_Jacques... List of things named after Jacques Hadamard From Wikipedia, the free encyclopedia These are things named after Jacques Hadamard (1865–1963), a French mathematician. (For references, see the respective articles.) Cartan–Hadamard theorem Cauchy–Hadamard theorem Hadamard product: entry-wise matrix multiplication an infinite product expansion for the Riemann zeta function Hadamard code Hadamard's dynamical system Hadamard's inequality Hadamard's method of descent Hadamard finite part integral Hadamard's lemma Hadamard manifold Hadamard matrix Hadamard's maximal determinant problem Hadamard space Hadamard three-circle theorem Hadamard Transform and Hadamard gate Hadamard–Rybczynski equation Ostrowski–Hadamard gap theorem Retrieved from "https://en.wikipedia.org /w/index.php?title=List_of_things_named_after_Jacques_Hadamard&oldid=679491174" P. GibsonCategories: Lists of things named after mathematiciansMath 4400 11 / 43 This page was last modified on 4 September 2015, at 21:37. Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. 1 of 1 17-02-14 4:47 PM Poincar´e Born in 1854 in Nancy, to a prominent family Top prizes in the concours g´en´eral Graduated from the Ecole Polytechnique, then the Ecole des Mines and worked as a mining engineer 1879 doctorate in mathematics from University of Paris under Charles H´ermite 1881 Professor at the Sorbonne (University of Paris) worked in many different areas, including on the three body problem pioneering work in geometry and topology carried out early work on relativity was active in philosophy, and wrote several widely-read popular works P. Gibson Math 4400 12 / 43 Hilbert 1885 doctorate under Ferdinand von Lindemann 1886-1895 lecturer at K¨onigsberg 1895 professor at G¨ottingen 1900 Paris address 1910 Bolyai prize pre-eminent mathematician of his day P. Gibson Math 4400 13 / 43 One sometimes reads of rivalry and dispute between Hilbert and Poincar´e, the leading mathematicians of 1900. This tends to be overstated. Hilbert contributed to many fields, including mathematical physics|his ideas on the foundations of mathematics are sometimes emphasized at the expense of his many other fundamental contributions. Poincar´e'srejection of Cantor's ideas have not been born out by history. P. Gibson Math 4400 14 / 43 The Institute for Advanced Study in Princeton, New Jersey (established 1930) P. Gibson Math 4400 15 / 43 \spaghetti and Levi-civita" P. Gibson Math 4400 16 / 43 P. Gibson Math 4400 17 / 43 According to Einstein, the theory of relativity relies on the work of: Bernhard Riemann (1826-1866) Hermann Minkowski Tullio Levi-civita Hermann Weyl P. Gibson Math 4400 18 / 43 To a certain extent, Hermann Weyl brought Hilbert's legacy and tradition to the US. P. Gibson Math 4400 19 / 43 The Mathematical Origina of Computers Mathematics is sometimes popularly conceived as ethereal and impractical. In certain instances this is true. P. Gibson Math 4400 20 / 43 Radio waves, microwaves Time dilation (Digital electronic) computers Nuclear fission Wireless communication networks Antiparticles Quantum computers The Higgs boson Yet mathematics has a profound power to make predictions about the world that has been manifest again and again. The existence of each of the following was predicted using mathematics (long) before it was confirmed experimentally. P. Gibson Math 4400 21 / 43 Yet mathematics has a profound power to make predictions about the world that has been manifest again and again. The existence of each of the following was predicted using mathematics (long) before it was confirmed experimentally. Radio waves, microwaves Time dilation (Digital electronic) computers Nuclear fission Wireless communication networks Antiparticles Quantum computers The Higgs boson P. Gibson Math 4400 21 / 43 What is a computer? The word computer has a revealing etymology (as illustrated in the historical uses cited by the Oxford English Dictionary, for example). It's meaning has evolved substantially in the last century. P. Gibson Math 4400 22 / 43 Of course the idea of building a machine to carry out computation was not new|but Turing's mathematical analysis revealed unexpected possibilities, which would take many years to realize. Computers The computer revolution is one of the most dramatic transformations of the late 20th century. It's origins are mathematical, and were largely effected by a single publication, A.M. Turing. On comutable numbers with an application to the Entscheidungsproblem. Proceedings of the London Philosophical Society, S2-42(1):230-265, 1937. P. Gibson Math 4400 23 / 43 Computers The computer revolution is one of the most dramatic transformations of the late 20th century. It's origins are mathematical, and were largely effected by a single publication, A.M. Turing. On comutable numbers with an application to the Entscheidungsproblem. Proceedings of the London Philosophical Society, S2-42(1):230-265, 1937. Of course the idea of building a machine to carry out computation was not new|but Turing's mathematical analysis revealed unexpected possibilities, which would take many years to realize. P. Gibson Math 4400 23 / 43 Major scientific developments of the first part of the 20th century include: special and general relativity quantum mechanics the theory of computing Each of these is deeply mathematical. Aside What were some of the internationally important events of the first half of the twentieth century? P. Gibson Math 4400 24 / 43 Aside What were some of the internationally important events of the first half of the twentieth century? Major scientific developments of the first part of the 20th century include: special and general relativity quantum mechanics the theory of computing Each of these is deeply mathematical. P. Gibson Math 4400 24 / 43 Back to Turing Let's look at Turing's paper directly in some detail. Alan Turing (1912-1954) P. Gibson Math 4400 25 / 43 Exercise Question How many Turing tables are there corresponding to a machine having m states and an alphabet of n symbols? P. Gibson Math 4400 26 / 43 Strangely, the ghost of Cantor emerges to tell us: Theorem (Turing) Almost all real numbers are not computable. Recap Theorem (Turing) There exists a universal computing machine. P. Gibson Math 4400 27 / 43 Recap Theorem (Turing) There exists a universal computing machine. Strangely, the ghost of Cantor emerges to tell us: Theorem (Turing) Almost all real numbers are not computable. P. Gibson Math 4400 27 / 43 A Turing complete electronic computer called the ENIAC (=Electronic NumericalIntegratorAndComputer) was completed in 1946. Other, roughly contemporaneous projects include the Electronic Discrete Variable Automatic Computer (EDVAC) (1951) the Colossus (1944) the Z3, developed by Konrad Zuse (∼1943) Physical devices corresponding to Turing machines can be built out of electronic circuits, using vacuum tubes or transistors. A computing machine is said to be Turing complete if, apart from the restriction of finite memory, it is universal. P. Gibson Math 4400 28 / 43 the Electronic Discrete Variable Automatic Computer (EDVAC) (1951) the Colossus (1944) the Z3, developed by Konrad Zuse (∼1943) Physical devices corresponding to Turing machines can be built out of electronic circuits, using vacuum tubes or transistors. A computing machine is said to be Turing complete if, apart from the restriction of finite memory, it is universal. A Turing complete electronic computer called the ENIAC (=Electronic NumericalIntegratorAndComputer) was completed in 1946. Other, roughly contemporaneous projects include P. Gibson Math 4400 28 / 43 Physical devices corresponding to Turing machines can be built out of electronic circuits, using vacuum tubes or transistors. A computing machine is said to be Turing complete if, apart from the restriction of finite memory, it is universal. A Turing complete electronic computer called the ENIAC (=Electronic NumericalIntegratorAndComputer) was completed in 1946. Other, roughly contemporaneous projects include the Electronic Discrete Variable Automatic Computer (EDVAC) (1951) the Colossus (1944) the Z3, developed by Konrad Zuse (∼1943) P. Gibson Math 4400 28 / 43 The ENIAC was used in the development of the hydrogen bomb under the auspices of the Manhatten Project. All the early electronic computers were developed for military purposes. They provided the impetus for computer science as an independent subject. Most early computer science departments at universities grew out of the local mathematics department. Computers are now ubiquitous, and affect daily life in myriad ways. This is just one example of the predictive power of mathematics and its material consequences.