*This article is part of a series on Intuitive Math Epistemology. See also:
*1. Is Math Discovered or Created? 2. Does Math Have An End? 3. Tchaikovsky and Debussy

*“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.[1] “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?” [2]

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