a brief, incomplete, and mostly wrong introduction to alan turing's work

A Brief, Incomplete, and Mostly Wrong

Introduction to Alan Turing's Work

Phil Calçado – SoundCloud@pcalcado

http://philcalcado.com

Sets are weird.

How many numbers are there between

1 and 3?

Georg Cantor

How many numbers are there between

4.4 and 4.5?

4.4...00014.4...0002

...

4.49...0001...

Now that's some crazy

shit.

Barbie

Math is hard, let's go shopping!

David Hilbert

For the mathematician there is no Ignorabimus.

You lazy bastards

„Wir müssen wissen, und wir werden wissen”

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.)

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.

Bertrand Russel

Sets can contain sets.

Sets can contain

themselves.

What about the set of all sets that do not contain themselves? Does it contain itself?

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

It's not technically cheating...

Kurt Gödel

Yo, Russel, I'm gonna let you finish, but no formal system extending basic arithmetic can be used to prove

its own consistency.

Whatever.

“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.”

Alan Turing

“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.”

Imagine a machine.

It reads and writes from and to an infinite paper

tape.

The tape is the machine's memory.

It writes to the tape accordingly to some

simple rules defined by configuration.

Now imagine a machine that can simulate other

machines.

By reading the desired configuration from the

tape.

Question: Is it possible to create a machine that

examines another machine's configuration and input and

verify if it halts?

“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.”

tl;dr: No, sorry.

Now imagine a machine that can simulate other

machines.

By reading the desired configuration from the

tape.

i.e.

Imagine a general-purpose

computer.

Howard H. Aiken

“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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