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 (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 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- 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

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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) 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.

P. Gibson Math 4400 29 / 43 The Dawn of the Information Age

The information revolution A Mathematical Theory of Communication, by Claude Shannon Bell Labs

P. Gibson Math 4400 30 / 43 computer revolution −→ information age

Remember 1997?

A look back

P. Gibson Math 4400 31 / 43 Remember 1997?

A look back

computer revolution −→ information age

P. Gibson Math 4400 31 / 43 A look back

computer revolution −→ information age

Remember 1997?

P. Gibson Math 4400 31 / 43 Google me. Text me. How much memory does your phone have? mouse, download, bandwidth, gigabyte, resolution... New vocabulary and a new mode of existence has emerged.

What would this mean in 1997?

P. Gibson Math 4400 32 / 43 New vocabulary and a new mode of existence has emerged.

What would this mean in 1997?

Google me. Text me. How much memory does your phone have? mouse, download, bandwidth, gigabyte, resolution...

P. Gibson Math 4400 32 / 43 What would this mean in 1997?

Google me. Text me. How much memory does your phone have? mouse, download, bandwidth, gigabyte, resolution... New vocabulary and a new mode of existence has emerged.

P. Gibson Math 4400 32 / 43 How did this all happen? Gradual evolution or design?

P. Gibson Math 4400 33 / 43 Information can be given a precise quantitative definition in terms of choice.  0 1 2 3 4 5 6 7 8 9   10 11 12 13 14 15 16 17 18 19     20 21 22 23 24 25 26 27 28 29     30 31 32 33 34 35 36 37 38 39     40 41 42 43 44 45 46 47 48 49     50 51 52 53 54 55 56 57 58 59     60 61 62 63 64 65 66 67 68 69     70 71 72 73 74 75 76 77 78 79     80 81 82 83 84 85 86 87 88 89  90 91 92 93 94 95 96 97 98 99 Choose a message from an array of possible choices. information= const.× log(total no. of choices) 102 choices → 2 units of information per message

How can communication possibly have a mathematical theory?

P. Gibson Math 4400 34 / 43 information= const.× log(total no. of choices) 102 choices → 2 units of information per message

How can communication possibly have a mathematical theory? Information can be given a precise quantitative definition in terms of choice.  0 1 2 3 4 5 6 7 8 9   10 11 12 13 14 15 16 17 18 19     20 21 22 23 24 25 26 27 28 29     30 31 32 33 34 35 36 37 38 39     40 41 42 43 44 45 46 47 48 49     50 51 52 53 54 55 56 57 58 59     60 61 62 63 64 65 66 67 68 69     70 71 72 73 74 75 76 77 78 79     80 81 82 83 84 85 86 87 88 89  90 91 92 93 94 95 96 97 98 99 Choose a message from an array of possible choices.

P. Gibson Math 4400 34 / 43 102 choices → 2 units of information per message

How can communication possibly have a mathematical theory? Information can be given a precise quantitative definition in terms of choice.  0 1 2 3 4 5 6 7 8 9   10 11 12 13 14 15 16 17 18 19     20 21 22 23 24 25 26 27 28 29     30 31 32 33 34 35 36 37 38 39     40 41 42 43 44 45 46 47 48 49     50 51 52 53 54 55 56 57 58 59     60 61 62 63 64 65 66 67 68 69     70 71 72 73 74 75 76 77 78 79     80 81 82 83 84 85 86 87 88 89  90 91 92 93 94 95 96 97 98 99 Choose a message from an array of possible choices. information= const.× log(total no. of choices)

P. Gibson Math 4400 34 / 43 How can communication possibly have a mathematical theory? Information can be given a precise quantitative definition in terms of choice.  0 1 2 3 4 5 6 7 8 9   10 11 12 13 14 15 16 17 18 19     20 21 22 23 24 25 26 27 28 29     30 31 32 33 34 35 36 37 38 39     40 41 42 43 44 45 46 47 48 49     50 51 52 53 54 55 56 57 58 59     60 61 62 63 64 65 66 67 68 69     70 71 72 73 74 75 76 77 78 79     80 81 82 83 84 85 86 87 88 89  90 91 92 93 94 95 96 97 98 99 Choose a message from an array of possible choices. information= const.× log(total no. of choices) 102 choices → 2 units of information per message

P. Gibson Math 4400 34 / 43 600 000 words.

According to this definition

A picture is worth

P. Gibson Math 4400 35 / 43 According to this definition

A picture is worth 600 000 words.

P. Gibson Math 4400 35 / 43 Claude Shannon (1916-2001)

P. Gibson Math 4400 36 / 43 Brief Bio

(1932-1936) Studied mathematics and electrical engineering at the University of Michigan (1936-1940) Graduate studies at MIT 1940 Research Fellow at the Institute for Advanced Study 1941 Joined Bell Labs, working on war-related research 1943 Met Alan Turing, during the latter’s visit to Bell Labs 1948 Published “A Mathematical Theory of Communication” 1956 Became a professor at MIT

P. Gibson Math 4400 37 / 43 When you look at Shannon’s paper...

Some questions: Is this really mathematics? If so, what sort of mathematics is involved? What are the central notions introduced? What are the implications?

P. Gibson Math 4400 38 / 43 The next question we shall consider is: in what institutional context did this work come about?

P. Gibson Math 4400 39 / 43 Bell Labs was established in the 1880s by Alexander Graham Bell

P. Gibson Math 4400 40 / 43 On Bell Labs in the 1990s

Our only obligation was to do good research. I can’t tell you how liberating that was. Everyone I know who was there says it’s the closest thing to paradise that you can imagine.

-Margaret Wright, Head of Scientific Computing at Bell Labs in 1997

P. Gibson Math 4400 41 / 43 Bell Labs has produced numerous fundamental inventions: radio astronomy photovoltaic cell transistor laser charge coupled-device C, C++ WLAN

P. Gibson Math 4400 42 / 43 Shannon’s 1948 established the theoretical framework for information theory. This made it possible, for example, to design communication networks of computers—ultimately leading to the internet. This continues to have a huge impact on day-to-day life.

P. Gibson Math 4400 43 / 43