Tag Archives: Philosophy

Gödel’s Proof

Gödel showed (i) how to construct an arithmetical formula G that represents the meta-mathematical statement: ‘The formula G is not demonstrable’. This formula G thus ostensibly says of itself that it is not demonstrable. Up to a point, G is constructed analogously to the Richard Paradox. In that Paradox, the expression ‘Richardian’ is associated with a certain number n, and the sentence ‘n is Richardian’ is constructed. In Gödel’s argument, the formula G is also associated with a certain number h, and is so constructed that it corresponds to the statement: ‘The formula with the associated number h is not demonstrable’. But (ii) Gödel also showed that G is demonstrable if, and only if, its formal negation ~G is demonstrable. This step in the argument is again analogous to a step in the Richard Paradox, in which it is proved that n is Richardian if, and only if, n is not Richardian. However, if a formula and its own negation are both formally demonstrable, the arithmetical calculus is not consistent. Accordingly, if the calculus is consistent, neither G nor ~ G is formally derivable from the axioms of arithmetic. Therefore, if arithmetic is consistent, G is a formally undecidable formula.  Gödel then proved (iii) that, though G is not formally demonstrable, it nevertheless is a true arithmetical formula. It is true in the sense that it asserts that every integer possesses a certain arithmetical property, which can be exactly defined and is exhibited by whatever integer is examined, (iv) Since G is both true and formally undecidable, the axioms of arithmetic are incomplete. In other words, we cannot deduce all arithmetical truths from the axioms. Moreover, Gödel established that arithmetic is essentially incomplete: even if additional axioms were assumed so that the true formula G could be formally derived from the augmented set, another true but formally undecidable formula could be constructed, (v) Next, Gödel described how to construct an arithmetical formula A that represents the meta-mathematical statement: ‘Arithmetic is consistent’; and he proved that the formula ‘A ⊃ G’ is formally demonstrable. Finally, he showed that the formula A is not demonstrable. From this it follows that the consistency of arithmetic cannot be established by an argument that can be represented in the formal arithmetical calculus.

(From Nagel and Newman‘s “Gödel’s Proof”

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Thus Composed Nietzsche

“Without music, life would be a mistake.”
— Friedrich Nietzsche

 

Manfred Meditation is one of many musical pieces composed by German Philologist and Philosopher Friedrich Nietzsche. Most of his compositions were created at an early age, and were harshly criticized by Wagner (discretely) and Hans von Bulow (openly, saying that his music is a practical joke). It is even said that Wagner left before the end of a performance, and was found lying on the floor in a hysterical fit of laughter. Nevertheless, this was not the reason behind the later rupture between the Philosopher and ‘The Master’.

Nietzsche may have been affected by that verdict, but I doubt he ever knew how Wagner reacted to his music: Nietzsche kept visiting and writing the Wagners even after that laughing fit. And Wagner’s low opinion of Nietzsche as a composer probably had little to do with the philosopher’s later disgust with everything related to Wagnerism.
—Edward Rothstein, The New York Times

In any occurrence, whether or not Nietzsche’s compositions contain any musical value is not an issue. These early pieces are but an honest expression of a young Nietzsche, who, through several stages in life, came to be the Philosopher we all know so well. So don’t be too cruel !

Live a good lif…

Live a good life. If there are gods and they are just, then they will not care how devout you have been, but will welcome you based on the virtues you have lived by. If there are gods, but unjust, then you should not want to worship them. If there are no gods, then you will be gone, but will have lived a noble life that will live on in the memories of your loved ones.

by Marcus Aurelius (26 April 121 AD – 17 March 180 AD), the philosopher-emperor.