turing.pdf

Transcript

1. A Brief, Incomplete, and Mostly Wrong Introduction to Alan Turing's Work Phil Calçado – SoundCloud @pcalcado http://philcalcado.com
2. Sets are weird.
3. How many numbers are there between 1 and 3?
4. Georg Cantor
5. How many numbers are there between 4.4 and 4.5?
6. 4.4...0001 4.4...0002 ... 4.49...0001 ...
7. Now that's some crazy shit.
8. Barbie
9. Math is hard, let's go shopping!
10. David Hilbert
11. For the mathematician there is no Ignorabimus.
12. You lazy bastards
13. „Wir müssen wissen, und wir werden wissen”
14. Given a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers. 10. Determination of the Solvability of a Diophantine Equation (Entscheidung der Losbarkeit einer diophantischen Gleichung.)
15. 10. Determination of the Solvability of a Diophantine Equation (Entscheidung der Losbarkeit einer diophantischen Gleichung.) Given a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers.
16. Bertrand Russel
17. Sets can contain sets.
18. Sets can contain themselves.
19. What about the set of all sets that do not contain themselves? Does it contain itself?
22. Types of sets: • Type 1: Sets of stuff that are not sets • Type 2: Sets of Type 1 sets • Type 3: Sets of Type 2 sets
23. It's not technically cheating...
24. Kurt Gödel
25. Yo, Russel, I'm gonna let you finish, but no formal system extending basic arithmetic can be used to prove its own consistency.
27. Whatever.
28. “The end the work was finished, but my intellect never quite recovered from the strain. I have been ever since definitely less capable of dealing with difficult abstractions than I was before. This is part, though by no means the whole, of the reason for the change in the nature of my work.”
29. Alan Turing
30. “It is of fundamental importance for the character of this problem that only mechanical calculations according to given instructions [...] are admitted as tools for the proof.”
32. Imagine a machine.
33. It reads and writes from and to an infinite paper tape.
34. The tape is the machine's memory.
35. It writes to the tape accordingly to some simple rules defined by configuration.
36. Now imagine a machine that can simulate other machines.
37. By reading the desired configuration from the tape.
38. Question: Is it possible to create a machine that examines another machine's configuration and input and verify if it halts?
39. “There's no general process for determining whether the machine might scan a character it's not expecting, or gets into an infinite loop printing blanks, whether it crashes, burns, goes belly up, or ascends to the great bit bucket in the sky.”
40. tl;dr: No, sorry.
41. Now imagine a machine that can simulate other machines. By reading the desired configuration from the tape.
42. i.e. Imagine a general-purpose computer.
43. Howard H. Aiken
44. “If the basic logics of a machine designed for solution of equations coincide with the logics of a machine for a department store, I'd regard this as the most amazing coincidence I've ever encountered.”
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