Question

EITHER illustrate Turing’s claim that something practical might go wrong if one has an inconsistent deductive system OR defend Wittgenstein’s claim that the problem would not be a logical one.

Answer 1

illustration for :-"Turing’s claim that something practical might go wrong if one has an inconsistent"

The program is a simulated Turing machine, a mathematical model of computation created by Alan Turing In 1936, he showed that the actions of any computer algorithm can a simple machine that reads and writes 0s and 1s on an infinitely long tape by working through a set of states, or instructions. The more complex the algorithm, the more states the machine requires.

The Scott Aaronson and Adam Yedidia of the Massachusetts Institute of Technology have created three Turing machines with behaviour that is entwined in deep questions of mathematics. This includes the proof of the 150-year-old Riemann hypothesis – thought to govern the patterns of prime numbers.

Turing’s machines have long been used to probe such questions. Their origins lie in a series of philosophical revelations that rocked the mathematical world in the 1930s. First, Kurt Gödel proved that some mathematical statements can never be proved true or false – they are undecidable. He essentially created a mathematical version of the sentence “This sentence is false”: a logical brain-twister that contradicts itself.

Practical benefits

practically, the pair have no intention of running their Turing machines indefinitely in an attempt to prove these problems wrong. It’s not a particularly efficient way to attack that problem,

Expressing mathematical problems as Turing machines has a different practical benefit: it helps to work out how complex they are. The Goldbach machine has 4888 states, the Riemann one has 5372, while Z has 7918, suggesting the ZFC problem is the most complex of the three.

Yedidia has placed his code online, and mathematicians may now compete to reduce the size of these Turing machines, pushing them to the limit. Already a commenter on Aaronson’s blog claims to have created a 31-state Goldbach machine, although the pair have yet to verify this.

Fortnow says the actual size of the Turing machines are irrelevant. “The paper tells us that we can have very compressed descriptions that already go beyond the power of ZFC, but even if they were more compressed, it wouldn’t give us much pause about the foundations of math,

But Aaronson says shrinking Z further could say something interesting about the limits of the foundations of maths – something Gödel and Turing are likely to have wanted to know. “They might have said ‘that’s nice, but can you get 800 states? What about 80 states?’” Aaronson says. “I would like to know if there is a 10-state machine whose behaviour is independent of ZFC.”