Alan Turing in America

“Beyond the way they speak, there is only one (no two!) features of American life which I find really tiresome. The impossibility of getting a bath in the ordinary sense and their ideas on room temperature” — Alan Turing (1936)

Pioneering British computer scientist and mathematician Alan Turing first arrived in America on the 28th of September 1936. Born in 1912, Turing was twenty-four years old when he was invited attend graduate school at Princeton University, studying under Alonzo Church. A Master’s student of mathematics at the University of Cambridge, Turing was at the time working on the theory of computability and would in the coming months be publishing his now renowned paper “On Computable Numbers, with an Application to the Entscheidungsproblem” which re-proved his eventual Ph.D. supervisors’ result known as Church’s theorem.

Turing would travel to America twice, first in ’36 and next in ’42. His first visit was for graduate studies in mathematics, his second as a liaison for the UK’s team of cryptoanalysts at Bletchley Park. This essay aims to recount some of his experiences while there. Continue reading

Richard Feynman on the Distinction between Future and Past

“We have a different kind of awareness about what might happen than we have of what probably has happened”

In physicist Richard P. Feynman’s fifth lecture at Cornell University in 1959 he entertained for a moment the question of what distinguishes the future from the past. The lectures starts out with the following primer:

“It is obvious to everyone that the phenomena of the world are evidently irreversible. I mean, things happen that do not happen the other way. You drop a cup and it breaks, and you sit there a long time waiting for the pieces to come together and jumb back into your hand. If you watch the waves of the sea breaking, you can stand there and wait for the great moment when the foam collects together, rises up out of the sea, and falls back farther out from the shore - it would be very pretty!”

Of course, as Feynman proceeds to point out: this would never happen in the real world. In fact, if one played a video in reverse of this happening in the early days of cinema, the spontaneous reaction of the crowd would likely be laughter — an indication of its surreal nature.

“Even without an experiment, our very experiences inside are completely different for past and future”

Why is so that the progression of still water transformed into breaking waves is so perfectly natural, but the transformation of waves that have broken reversed into their previous state of perfect stillness is absurd? Why do we think of the future and past as different? Continue reading

The Great Purge of 1933

How Antisemitism Destroyed Mathematics in Germany

“If the dismissal of Jewish scientists means the annihilation of contemporary German science, then we shall do without science for a few years!” — Adolf Hilter

Prior to World War II, Germany had led the world in science for more than one hundred and fifty years. Its reputation for excellence in chemistry, physics, biology, medicine and mathematics was rivaled, if at all, only by Britain (Medawar & Pyke, 2000). Of the 100 Nobel Prizes awarded between 1901 and 1932 (the year before Hitler came to power) 33 were awarded to Germans or scientists working in Germany. Britain had won 18, and the United States a mere six.

Then, as the result of a series of events following Hitler’s takeover of Germany in 1933 and the passing of the Berufsbeamtengesetz (“Law for the Restoration of the Professsional Civil Service”), in order to “re-establish a national and professional civil service”, members of certain groups of public employees began being dismissed from German universities. That is, civil servants who were not considered to be of “sufficiently Aryan” descent had to leave their jobs. Continue reading

Shannon Ciphers and Perfect Security

Shannon cipher, named after mathematician Claude Shannon (1916–2001) is a simplified cipher mechanism for encrypting a message using a shared secret key. A cipher is generally defined simply as an algorithm for performing encryption or decryption, i.e. “a series of well-defined steps that can be followed as a procedure”.

Example (Boneh & Shoup, 2020)Suppose Claude and Marvin want to use a ciper such that Claude can send an encrypted message that only Marvin can read.Then, Claude and Marvin must in advance agree on a key k ∈ K. Assuming they do, then when Claude wants to send a message m ∈ M to Marvin, he encrypts m under k, obtaining the ciphertext c = E(k,m) ∈ C, and then sends c to Marvin via some communication channel. Upon receiving the encrypted message c, Marvin decrypts c under k. The correctness property ensures that D(k,c) is the same as Claude's original message m. 

Regarded by many as the foundation of modern cryptography, the concept of a Shannon cipher were first introduced in the 1949 paper Communication Theory and Secrecy Systems published by Shannon in a Bell Systems Technical Journal. The results Shannon presented in the paper was based on an earlier version of his research in a classified report entitled A Mathematical Theory of Cryptography, and preceded Shannon’s publication of his well-known A Mathematical Theory of Communicationpublished a year before, in 1948. The following discussion of Shannon ciphers is based on Chapter 2.1 “Shannon ciphers and perfect security” in the book A Graduate Course in Applied Cryptography by Dan Boneh and Victor Shoup.

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Ramanujan’s First Letter to G.H. Hardy

On or about the 31st of January 1913, mathematician G.H. Hardy of Trinity College at Cambridge University received a parcel of papers from Madras, India which included a cover letter from an aspiring young Indian mathematician by the name of Srinivasa Ramanujan (1887–1920). The cover letter discussed three topics:

  1. An introduction of Ramanujan and his precarious situation;

  2. A claim about the domain of the gamma function; and

  3. A claim about the distribution of prime numbers;

This essay provides an overview of the mathematical content of Ramanujan’s first letter, as well as Hardy’s reaction and response. Read more

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