Antinomy or the Deceit of Clarity

“In point of fact, Bertrand Russell constructed a contradiction within the framework of elementary logic itself that is precisely analogous to the contradiction first developed in the Cantorian theory of infinite classes. Russell’s antinomy can be stated as follows. Classes seem to be of two kinds: those which do not contain themselves as members, and those which do. A class will be called “normal” if, and only if, it does not contain itself as a member; otherwise it will be called “non-normal”. An example of a normal class is the class of mathematicians, for patently the class itself is not a mathematician and is therefore not a member of itself. An example of a non-normal class is the class of all thinkable things; for the class of all thinkable things is itself thinkable and is therefore a member of itself. Let ‘N’ by definition stand for the class of all normal classes. We ask whether N itself is a normal class. If N is normal, it is a member of itself (for by definition N contains all normal classes); but, in that case, N is non-normal, because by definition a class that contains itself as a member is non-normal. On the other hand, if N is non-normal, it is a member of itself (by definition of non-normal); but, in that case, Nis normal, because by definition the members of N are normal classes. In short, N is normal if, and only if, Nis non-normal. It follows that the statement ‘N is normal’ is both true and false. This fatal contradiction results from an uncritical use of the apparently pellucid notion of class. Other paradoxes were found later, each of them constructed by means of familiar and seemingly cogent modes of reasoning. Mathematicians came to realize that in developing consistent systems familiarity and intuitive clarity are weak reeds to lean on.[1]


[1] Nagel and Newman, in Gödel’s Proof, pp. 23,24 & 25,  University of Florida Libraries.

Scattered Bits

  • Shadows can move faster than light (this does not violate any “law of physics” because shadows transfer neither energy nor matter).
  • Ludwig van Beethoven was not the only composer struck by deafness (and who still composed afterwards) : the Czech composer Bedřich Smetana also lost his hearing between September and October of 1874, ten years prior to his demise. Additionally, in the last movement of his String Quartet No. 1–dubbed “From My Life“–  the first violin sustains a harmonic  E, reminiscent of the ringing in his ear that forebode his deafness (the ringing was really a chord in A) .
  •  “Madam I am Adam. Able was I ere I saw Elba” are two palindromes, so is the third movement from Hayden’s Symphony No. 47,  Alban Berg’s Lulu, and many of Anton Webern’s compositions.
  • Johann Sebastian Bach used to subtly sign his work with the following sequence of notes : Sib-La-Do-Si (French Nomenclature) or B-A-C-H (German Nomenclature). There are also other similar motifs such as : DSCH (Dmitri Schostakovich), SACHER  hexachord (Paul Sacher; it was used  in twelve compositions by various 20th century composers (12 Hommages à Paul Sacher)).

Sexy Prime (Numbers)

They’re sexy, they’re quirky, they  come in pairs and wouldn’t mind a threesome either, they’re Sexy Primes! Now, before you get the impression that I publish some kind of an unorthodox display of coitus-ridden innuendos and inappropriate shenanigans, I’d like to say that “sexy” here derives from the Latin word “sex” meaning the number “six”. And yes, they do come in pairs, triplets, quadruplets and quintuplets. So what are sexy primes? It’s so simple that you’d…  : they are prime numbers that differ from one another by six.

Examples of sexy prime :

  • Pairs : (5,11), (7,13), (11,17), (13,19), (17,23) etc. ;
  • Triplets : (7,13,19), (17,23,29), (31,37,43) etc. ;
  • Quadruplets : (5,11,17,23), (11,17,23,29), (41,47,53,59) etc. ;
  • Quintuplets : there is only one possible quintuplet : (5,11,17,23,29) and that’s it!

There are also “Twin Primes” that differ by 2, “Cousin Primes” that differ by 4.

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 !

"Sanity is not Statistical"