“Mathematics is a game played according to certain simple rules with meaningless marks on paper.” – David Hilbert
Two intertwined musical careers – those of Pyotr Illych Tchaikovsky and Claude Debussy – came to a climactic head in 1885. The towering Russian Romanticist, Tchaikovsky, premiered his Orchestral Suite No. 3 in Saint Petersburg to overwhelming adulation. In the same year, Debussy, the revolutionary French Impressionist, won the prestigious Prix De Rome piano composition contest and began his work under a royal scholarship at the French Academy in Rome.
The mournful, melodic violin solos of Romanticism and the experimental tonalities of Impressionism contrasted drastically. “Not a single idea is expressed fully, the form is terribly shriveled, and it lacks unity,” Tchaikovsky once wrote of one of Debussy’s works. “Do you not remember the… music, able to express every shade of meaning,” Debussy himself reminisced, “which makes our tonic and dominant seem like ghosts?” 
Music, though, is not alone as a discipline of schools and schisms. Decades later, a similar division began to form in mathematics: the Platonists, led by Kurt Gödel, and the Intuitionists, led by L. E. J. Brouwer, began to stretch the very laws of logic themselves. Mathematics – just like music – became a house divided.
In this article, we take a tour of the fascinating and diverse branches of mathematical thought. Continue reading