I bet you didn't know that this thread would have a part 2

stimpson wrote: »

Vulcan Point Island in the Philippines is world’s largest island within a lake that is situated on an island located in a lake on an island.

https://goo.gl/maps/TrqpNy28DSs

What are the next five largest islands within a lake within an island within a lake on an island?

SimonTemplar wrote: »

Stanley Donen, the director of Singing In The Rain, died a few days ago. So I thought I'd post here that the famous singing in the rain scene was filmed by mixing some milk with the water so that the rain stood out more on screen.

One of the biggest lies in movie history. No one knows where the myth came from but the rain is made visible by backlighting the rain in the plate glass windows and the biggest hurdle was to make sure the cameras weren’t reflected in the glass. It has been said that to p1ss off the camera crew who had made film history someone said it was as easy as mixing milk and water. Whoever said it, it has certainly stood the test of time.

https://www.radiotimes.com/news/2016-12-25/is-the-rain-milk-is-that-debbie-reynolds-voice-gene-kellys-widow-busts-the-myths-about-singin-in-the-rain/

New Home wrote: »

If Pinocchio were to say "My nose will grow now.", it would create a paradox: if it didn't grow, he'd have told a lie, so the nose would grow on account of that, but if it did grow Pinocchio would have told the truth, making it impossible for his nose to grow.

The Pinocchio paradox was thought of in February 2001 by an 11-year-old called Veronique Eldridge-Smith – the daughter of Peter Eldridge-Smith, who specializes in the philosophy of logic. The article was first published in the journal Analysis.

I think it's worth saying what Véronique did here considering she was so young at the time and it links into the creation of computers.

Essentially the Pinocchio paradox is a variation of the Liar Paradox dating back to Ancient Greece and independently discovered by Indian and Arabic mathematicians. The paradox is in essence "This sentence is false". If the sentence is true then it is false, if it's false then what it says is true (what New Home explained basically).

A mathematician called Alred Tarski had proposed you can get rid of the Liar paradox by not letting your language freely talk about truth. Language here means a logical language like mathematics, Lojban (an attempt at a spoken language with completely logical grammar) or computer languages.

However the Pinocchio version is special because in all other versions like:
This sentence is false
I always lie
The person is in some way talking about truth. In Véronique's version she's talking about a physical fact. A fact in a fictitious fantasy world, but it shows the paradox doesn't just arise from letting your language talk about "truth". So Tarksi solution seems to be incorrect.

The Liar Paradox had a great effect on the foundations of mathematics. In 1903 Gottlob Frege was ready to publish the second volume of his "The Foundations of Arithmetic". This was an attempt at a completely airtight proof of the validity and soundness of basic addition, subtraction, multiplication and division. Frege was interested in this because another mathematician David Hilbert had recently given a completely logically sound presentation of standard geometry from which one could answer any potential geometric question. Hilbert asked if the same could be done for the rest of mathematics at a conference in 1897 as one of his 23 problems for mathematicians to solve in the next 100 years. Frege left, Hilbert right:



Frege thought he had constructed a logical presentation like Hilbert's for arithmetic, but just before he took his book to the press Bertrand Russell wrote him a letter to say that due to a version of the Liar Paradox his 119 page work was fundamentally flawed.

Frege gave up shortly after that, but Russell decided to attempt it himself. He and another mathemaitician Alfred Whitehead began work on their enormous three volume "Principia Mathematica". Russell left, Whitehead right:



This was often worked on at Whitehead's estate over breakfast. Over the time they wrote it Russell and Whitehead had raging arguments as Russell often wanted to start all over again as he theorised possible flaws that would develop down the line. The graphic novel Logicomix has a good depiction of this (I'd recommend it, it's a really good look at mathematicians of the period and their personal quirks and obsessions):



Working on the book put a great strain on Russel's marriage. To complete it he and his family moved into Whitehead's estate and over time he began to fall in love with Whitehead's much younger wife Evelyn. I've linked before to this work, where it takes them hundreds of pages to prove 1 + 1 = 2:



As the volumes were published nobody found a flaw like Frege's works had. It seemed they had a completely logical derivation of basic arithmetic from which anything you wanted to know could be proved and began to move onto the rest of mathematics.

However in 1931 Kurt Gödel found that although there was nothing wrong with the logic of Russell and Whitehead's work, it wouldn't be able to tell you everything about arithmetic. This is Gödel's Incompleteness Theorem. It's actually quite simple to explain via a board game. Normally you have the basic rules of a board game which then decide how moves etc work and how any gameplay situation is resolved. Gödel's work showed that a sufficiently advanced board game would always have some situation where the rules can't decide what should happen and you'd have to come up with a new rule to cover that situation. However even then there would be another undecidable situation eventually and so on. Similarly all of arithmetic doesn't follow from a bunch of simple rules, you will always reach a statement whose truth can't be decided and you have to pick what's true.




Gödel's original proof of the Incompleteness theorem is quite hard to follow, so Alan Turing came up with a different way of phrasing it. Imagining a machine that read in data from a tape and transformed the data into new data through a series of simple steps. It makes the proof of the Incompleteness theorem go from tens of pages to a single page essentially. In this case the incompleteness theorem shows up as the fact that are some programs that can be loaded into the machine where the machine can't decide what to do next based on its simple rules and gets caught in a loop. Phrased this way it is called the Halting Problem: The fact that there is no way for a computer to definitely detect if a program will make it crash before it starts running it.

These tape reading machines, Turing machines, would be the foundation of the modern computer when Turing went on to construct an approximation of them for the British government. (Earlier machines by Babbage and Zuse although they do compute are not truly like modern computers as they cannot execute any algorithm). John von Neumann used Turing's ideas to construct the first machine that has the architecture of modern computers for use in the Manhattan project.

So there's a clear chain of ideas from the Liar paradox through the foundations of maths to modern computers.

Candie wrote: »

What has to be one of the most incredible surgeries is osteo-odonto-keratoprosthesis, or tooth-in-eye surgery. Put simply, patients with damaged corneas have one of their teeth removed and either the whole or part of the tooth is implanted in the eye. When the tooth is implanted the risk of rejection is low, and a part of the tooths surface is sheared off and an artifical optical lens implanted in the space. People with eyes severely damaged by chemical injury or disease have had their sight fully restored by this surgery, though it's a long way from being widespread just yet.

I've heard the phrase eye teeth but i didnt realise i was meant to take it literally