Leonard Bernstein gave a famous discussion of Beethoven’s 5th Sympony. All of these comments apply equally as well to the Peano Axioms and to axiomatics in math. The meaning of the first 4 notes of Beethoven’s 5th Symphony according to Bernstein are all the notes that come after them. The same is the true of the Peano Axioms. The meaning of the Peano 5 Axioms are all the definitions, theorems, proofs, examples and exercises that come after them.
Exercise: Write down some of the things that Bernstein says about Beethoven’s 5th Symphony and then compare those to the Peano Axioms.
Exercise: Look at Dedekind’s 1888 book and compare that to Beethoven. How does Dedekind have the ability to put what the right sequence of lemmas together compare to Beethoven’s ability to put the right note after each other?
Exercise: How do the Peano Axioms compare to the theme in Beethoven’s 5th Symphony? The 3 G’s and an E Flat.
Exercise: Compare the “long gestation ” in Wiki’s phrase of Beethoven’s 5th Symphony to the many years of Dedekind’s developing his 1888 book?
Exercise: How does form in music as Bernstein discusses it above compare to form in math? In axiomatics? In Dedekind?
Exercise: Listen to the end of the Bernstein discussion. Was Dedekind inevitable? Are the Peano Axioms even more “something we can trust that will never let us down”?
Exercise: If Bernstein finds in Beethoven’s 5th Symphony something he can trust that will never let us down as a source of inspiration, and if the Peano 5 Axioms are the same, then isn’t part of inspiring students in math in elementary school to teach the Peano 5 Axioms? Do we wait to let children hear classical music until college? When do children start to study classical music? Shouldn’t it be the same for axiomatics in math? Doesn’t that supply the need for trust? Isn’t Leonard Bernstein saying in this 1954 that we need something we can trust that will never let us down talking about a need we develop in child hood? Can’t the Peano Axioms do this as much as Beethoven’s 5th Symphony?
Exercise: Look at the above Bernstein piece and search on romanticism. What if anything in math corresponds to the classicism-romanticism transition in music?
Exercise: Are axiomatics like the classical period of Bach, Haydn, and Mozart? How? How are they unlike romanticism? If so, why did the classical period in math come at the end of the 19th century? Internal movement of math and music? The sequence of composers is a logical self paced development? Math is the same way?
Exercise: How does sketch work in music compare to Grassmann 1861 as sketch work for Dedekind? Dedekind was inspired by Grassmann, but Dedekind is more inspiring than Grassmann for us. Why is this? Because Dedekind has a tighter logical development. This comes from Dedekind including the small step lemmas that each encapsulate a single idea and use a simple idea for its proof.
Exercise: Dedekind builds up to climaxes such as the Natural Numbers, The Recursion Theorem, the definition of addition, and the definition of multiplication. Compare this build up and crescendo to those in Beethoven’s 5th Symphony.
Exercise: Discuss how set theory provides a foundation for Dedekind the way notes and the music staff do for Beethoven.
Exercise: How does Beethoven build on Mozart and Haydn compared to how Dedekind builds on Grassmann?