Bach’s Aria with Diverse Variations

- Popularly known as Goldberg Variations - All of the pieces, except the final one, are based on a single theme - In the last variation is a sort of “post-ending ending” that contains extraneous ideas (two German folk tunes) that are outside the original theme. Variation called a “quolibet” - Written in 1742, when Bach was Cantor in Leipzig

Copper, Silver, Gold themes

- Kupfergödel, Silberescher, Goldbach - Copper, Silver, Gold: an Indestructible Metallic Alloy - Indestructible Metallic Alloy - Eternal Golden Braid - Giraffes, Silver, Gold (G - Gödel) - Copper, Elephants, Gold (E - Escher) - Copper, Silver, Baboons (B - Bach) - Silva and Gould, who are coppers

Goldbach

- In 1742 sent letter to Euler (who was at the court of King Frederick the Great) with the conjecture: “Every even number can be represented as a sum of two odd primes” - Never been proven - Schnirelmann in 1931 proved that any number, even or odd, can be represented as the sum of not more than 300,000 primes

“Vinogradov property”

- Vinogradov in 1937 showed that every “sufficiently large” odd number can be represented as a sum of no more than three odd primes. - It has not been determined what number is “sufficiently large,” only that it must be finite

Predictably terminate

Goldbach property

- Look at all numbers less than 2N to determine property within a fixed amount of time
- Recognizable

Connection with Zen Dualism

- Statement that a number has the “Achilles property” involves infinite amounts of information - Same with “29 is prime”, etc…

Chaos integral part of beauty and harmony

- Order and Chaos by Escher

Does not predictably terminate

“Goldbach variation”

- Contrived by Hofstader?
- Is every even number the difference of two prime numbers?
- Hofstader does not state whether this holds for all numbers or not
- “Tortoise property”: Number is the difference of two odd primes
- “Achilles property”: Number is not the difference of two odd primes

WONDROUS numbers

- If n is odd, make it 3n+1
- If n is even, make it n/2
- If it reaches 1, it is a WONDROUS number, otherwise it is unWONDROUS
- Seems to expand chaotically, and thus does not predictably terminate

Very Asian Gold Box variation

- very gold Asian box
- then becomes very Asian gold box
- finally, Very Asian Gold Box
- What did Hofstader mean by this final variation?

Real ending:

- “Good night”
- “Sufficiently assiduous” reader can spot.
- Misspellings spell out “end”:
- Whatver (e)
- Outstanding (n)
- Golbach Conjecture (d)

False ending

- Introduction of extraneous event and characters which are inconsistent
- Isomorphism with Bach’s Aria with Diverse Variations (“Post-ending ending”)
- Isomorphism with recognizable/predictably terminate theme. “Sufficiently assiduous” search will always terminate, though cannot predict when it will terminate, thus ending is recognizable

Complete List of All Great Mathematicians

- De Morgan - Abel - Boole - Brouwer - Sierpinksi - Weierstrass - Hidden mathematician in diagonal: Dboups (-1) = Cantor (mathematician who invented diagonalization argument) (“Bach in Liepzig”)