Tchaikovsky and Debussy

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


Does Math Have An End?

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

Is there a limit to the number of “true facts” contained in the discipline of mathematics? If (for the sake of argument) humans were around indefinitely, would the discipline ever end? Would we, one day, proclaim that we’d “reached the bottom”? Or does the field continue indefinitely into the depth, reaching arbitrary levels of complexity? If this is the case, where do these true facts come from, and why do they exist?


What will be the math of the future? What will math look like, a hundred years from now, or five thousand, or (again hypothesizing that we’ll be around indefinitely) even a hundred thousand?

These questions make math seem like a strange, magical and universal discipline. The pursuit of their answers has long eluded me. Here, I finally make some attempts.

Continue reading

Is Math Discovered or Created?

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

Math is highly creative. Mathematicians forge onward into unknown worlds, artfully shaping and uniting diverse tools in the resolution of their problems. Their progress, however, is regulated by the rigid requirements of logical consistency; unbreakably bound to itself, the discipline progresses forward without risk of retreat or collapse. The logic guiding it, of course, must come from somewhere beyond those who employ it, and the mathematician seems but an agent in the revelation of something much more profound.

Mathematical research is difficult to describe. To call math “discovered” in the Platonic sense – that is, an already-existing “fact of the universe” – is to neglect the role of the mathematician as leader of an expedition: he or she makes very real, and difficult, decisions regarding the path through the unknown which most promisingly portends success. To call it “created”, however – in the sense that math is but a human invention – is to ignore the role of a seemingly supra-human logic in dictating the progress of the field. Continue reading