Here are some useful fundamentals that I found helpful assuring myself that LEAN works: The Foundational Systems Underneath Lean is based on dependent type theory, an alternative foundation as ZFC. (Lean is strictly stronger than ZFC, roughly ZFC + ω inaccessibles.) (“Dependent” means types can depend on values). But note that the continuum hypothesis is independent of Lean too. Proposition as Types Reference: Philip Wadler Proposition as Types (2014). Section 6 and 7. ...

Vittorio Monti was an Italian violinist working in Paris, and in 1904 he wrote the one piece that kept his name alive. Czardas follows the Hungarian dance form: a slow, brooding opening, then a fast section that keeps speeding up. It belongs to the violin, but everyone borrows it. Today it’s Brian Luce on flute and Carrol McLaughlin on harp, the album Allume again. The flute’s tone is woody and a little dusky, which suits the slow opening. In the fast part, the harp plays the cimbalom. ...

Grieg wrote it in 1875 for Ibsen’s Peer Gynt: Peer stumbles into the hall of the Troll King, and the trolls want to eat him. The whole piece is one short tune in B minor, played over and over, faster and louder each time, until the hall collapses. Grieg didn’t think much of it. He wrote that it reeked of cowpats and Norwegian self-satisfaction. It is now probably the most famous thing he wrote, heard in cartoons and car ads. ...

Kreisleriana, Op. 16, is a set of eight piano pieces Schumann wrote in a few days in the spring of 1838. The title refers to Johannes Kreisler, a fictional Kapellmeister in E.T.A. Hoffmann’s stories: brilliant, unstable, and half mad. The pieces alternate between wild, fast music and slow, inward music. These were the two sides Schumann gave names to: Florestan, the impulsive one, and Eusebius, the dreamer. He told Clara Wieck that she was the main idea behind the work. At the time, her father was refusing to let them marry. He dedicated it to Chopin. ...

If I could keep one Firebird, it would be Antal Doráti with the London Symphony, recorded for Mercury in 1959: the complete 1910 ballet, not the suite. Doráti conducted for the Ballets Russes de Monte Carlo in the 1930s, and it shows. The tempos are ones you could dance to, and nothing is lingered over for its own sake. Mercury’s engineers did the rest with a few microphones hung close to the orchestra. Every wind solo lands in your lap, and the bass drum in the Infernal Dance is a physical event. Sixty-odd years on, it still sounds better than most recordings made last year. ...

The story comes not from Goethe but from Nikolaus Lenau’s Faust, and the episode is called “The Dance at the Village Inn.” Liszt printed the plot in the score: a wedding is in full swing, with music and dancing. Mephisto and Faust walk in. Mephisto takes the fiddle from a sleepy village fiddler and starts to play, and the room goes mad. Faust waltzes a dark-eyed girl out the door and into the woods. A nightingale sings. ...

Yesterday I said Perlman likes to push the orchestra. Here’s the best evidence: Tchaikovsky’s Valse-Scherzo, live at the Tchaikovsky Gala. It sounds pretty under-rehearsed. Listen to how the winds struggle. Perlman and the orchestra are rarely in the same place: he’s ahead, then behind, stretching phrases wherever he likes. On paper, a bad collaboration. In practice, a fantastic performance. He dances around the orchestra, and the rubato becomes the waltz. Next to him, the Leningrad Philharmonic, arguably the best orchestra of its day, sounds a little flat-footed. ...

The last movement of Vivaldi’s Autumn: one of my favorite autumn pieces, and I’ve always found it cute. Is it a mazurka? It’s in triple time and it bounces, so I see why it feels like one. But mazurkas didn’t reach the concert hall until Chopin, a century later. Vivaldi titled this movement La caccia, the hunt. The orchestra plays the horn calls and the solo violin runs the chase. ...

Paganini again: the finale of his Second Concerto, played by Alexandru Tomescu. It’s nicknamed La Campanella for the little bell that rings in the orchestra, answered by the violin in harmonics. Most people know it as Liszt’s piano étude, which says something about how often the original gets played. Is it hard? Yes: harmonics, double-stop harmonics, ricochet, leaps across the fingerboard, all fully exposed. But it’s a different kind of hard from Brahms or Tchaikovsky. One tune comes back again and again, each time with a new trick. The orchestra mostly goes oom-pah. There’s no forty-minute structure to hold together. Paganini was friends with Rossini, and you can hear it: this is an aria, and the violin is the soprano. ...

Same album as yesterday’s Godard: Brian Luce on flute and Carrol McLaughlin on harp, this time playing Paganini’s Caprice No. 24. On flute and harp, the surprise is the harp. Wow, I’ve never heard the harp so angry and aggressive before. Listen on Apple Music