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Legacy of Alan Turing in Mathematics

Sep 25, 2024

Lecture Notes on Alan Turing

Introduction to Mathematics

  • Mathematics is fundamentally the search for truth.
  • This year marks 100 years since Alan Turing's birth.

Who was Alan Turing?

  • Mathematician, cryptographer, and pioneer of computer science.
  • Known for breaking the German Enigma Code during WWII.
  • Life was a mix of secret triumphs and public tragedies.

Early Life

  • Born on June 23, 1912.
  • Childhood spent in Hastings and Sherborne.
  • Demonstrated talent in math and science early on (e.g., condensed Einstein’s theory of relativity at age 15).
  • Received a scholarship to King's College, Cambridge, in 1931.

Turing's Contributions to Mathematics

  • Focused on the Entscheidungsproblem (decision problem).
  • Introduced the concept of a Turing Machine in 1935:
    • A hypothetical machine that reads symbols and acts based on rules.
    • Demonstrated that you cannot determine whether the machine will stop or run indefinitely.
  • Disproved Hilbert's question of decidability showing limits in mathematics.
  • Turing Machine model became a cornerstone of computer science.

Career and Codebreaking During WWII

  • Worked for the Government Code and Cipher School at Bletchley Park.
  • The Enigma Machine:
    • A device used by the Germans to encrypt messages.
    • Had vast settings and complexity due to its rotors.
    • A challenge for Allied codebreakers.
  • Turing's vital contributions:
    • Made initial breakthroughs with naval Enigma codes.
    • Designed a machine called the Bomb with Gordon Welchman to decode Enigma messages.
    • Used common phrases to assist in breaking codes, reducing the time to find Enigma settings to about 15 minutes.

Impact of Turing's Work

  • His work is believed to have shortened WWII by two years, saving countless lives.
  • Received OBE in 1945 for services, though the reasons were classified for many years.

Post-War Contributions

  • Continued work at the University of Manchester, contributing significantly to computing and biology until his death in 1954.
  • Credited with laying the foundations for computer technology and artificial intelligence.

Turing's Personal Life and Legacy

  • The British government issued a public apology in 2009 for Turing's treatment post-war.
  • He was gay during a time when homosexuality was illegal, leading to tragic consequences:
    • Endured hormone treatment and lost his job.
    • Committed suicide in 1954.
  • Turing's focus was always on knowledge, truth, and solving problems, rather than just technology itself.

Conclusion

  • Alan Turing's legacy continues to shape our world dominated by technology, illustrating the importance of his quest for truth and understanding in mathematics.