the original Bait and Switch Exposed - the Liar of the Lyre and irrational magnitude as the Greek Miracle

So this von Fritz article pdf link is one I am returning to.

What I call the "bait and switch" as the "rotten root" of the Greek Miracle is exposed in the above paragraph. Notice how in the first definition the concept of ratio as rational depends on Number as arithmetic - "two sets of integers" and then the second NEW definition of ratio then removes the number so that what is left is pure geometry. But the geometry must be proved to "always ...go in the same direction."





So here von Fritz is making the same connection that Professor Luigi Borzacchini calls the "negative judgement paradox" that both state origins from music theory and then was applied to geometry.

But what gets left out of this "transfer" from music into geometry is that in music the direction can be changed since the Perfect Fifth is both 2/3 and 3/2 as a noncommutative phase - with 3/2 as C to G (geometry) and 2/3 as C to F (geometry). So that F=3=G at the same time! This originated from Philolaus as the Liar of the Lyre - as PRofessor Richard McKirahan has detailed - Philolaus literally "flipped" his lyre around to justify transforming musical ratios into geometric irrational magnitude. By flipping the lyre around then the "root tonic" or the "one" value could be changed in geometry but still retain the arithmetic value of the same "one." Yet in music theory - we can hear when the "one" changes.

So von Fritz makes the fascinating case that Hippasus who was a Pythagorean of a generation before Archtyas, was experimenting with sound from discs, as ratios, and since the ratios were still just number despite difference in diameter or thickness, therefore this could also be applied to geometry as the logos of irrational numbers (alogon). And then Archytas further developed this from Philolaus. And then Eudoxus solidified it into the Greek Miracle, as von Fritz details.

The ultimate source of the story seems to be John Burnet's 1892 book Early Greek Philosophy (London, 1892), p. 117:

For the present, the incommensurability of the diagonal and the square remained, as has been said, a "scandalous exception." Our tradition says that Hippasos of Metapontion was drowned at sea for revealing this skeleton in the cupboard.

In a footnote Burnet attributes this story to Iamblichos, Life of Pythagoras 247 (§34).

Unfortunately, and perhaps unsurprisingly, this is absolutely not what the text of Iamblichos says. That is to say: the story is a complete fabrication.

Iamblichos does record a legend that Hippasos (known from other sources to be an important Pythagorean in the 5th century BCE) divulged the secret dodecahedron, the last of the supposedly perfect solids to be discovered, and as a result a "divinity took vengeance upon him...on the grounds of committing an impiety" by ending his life in the sea -- i.e. he drowned in a storm or shipwreck, and some people with vivid imaginations attributed it to divine vengeance.

Here's a translation of Iamblichos -- the relevant bit is at the bottom of p. 47 and top of p. 48 -- but unfortunately the translation's slightly inaccurate, and leaves out the bit about the divinity. For the sake of completeness, here's the Greek text:

οἳ δέ φασι καὶ τὸ δαιμόνιον νεμεσῆσαι τοῖς ἐξώφορα τὰ Πυθαγόρου ποιησαμένοις. φθαρῆναι γὰρ [sc. Hippasos] ὡς ἀσεβήσαντα ἐν θαλάσσῃ τὸν δηλώσαντα τὴν τοῦ εἰκοσαγώνου σύστασιν· τοῦτο δ’ ἦν δωδεκάεδρον, ἓν τῶν πέντε λεγομένων στερεῶν σχημάτων, εἰς σφαῖραν ἐκτείνεσθαι.

A much better account of the discovery of the irrationality of root two is Kurt von Fritz, "The discovery of incommensurability by Hippasus of Metapontum", Annals of Mathematics 46(2) (1945): 242-264. Von Fritz correctly observes that the discovery was one of the greatest achievements of Pythagoreanism, and not a horrid skeleton in the closet as Burnet claims.

So for example the Pythagoreans taught that "one is not a number" and recent scholarship has argued that Parmenides, close to the Pythagoreans, considered the one to be light as consciousness or being. So when we consider the frequency of light based on wavelength, inverse to number, then the value of the "one" changes according to the color (frequency) of the light. So each color is like an octave as the root tonic of the energy - which is literally how Louis de Broglie realized that frequency is momentum - as a "standing wave" as geometry. So the wavelength is inverse to the momentum as a particle but it is moving only in relation to the speed of light that does not change (in a vacuum).

So this is another way of expressing the noncommutative phase source of the "one" of sound as well - that de Broglie had rediscovered with his Law of Phase Harmony - there has to be a SECOND time from the future going in the opposite direction! This is what the Greek Miracle had covered up when it created irrational magnitude as only going in the "same direction." The Orthodox Pythagorean philosophy was based on the secret of time going in both directions as the same time!

So obviously this music role vanished - as Borzacchini emphasizes - and states he is SHOCKED and STUNNED by this vanishing of the music origin. But then Fields Medal math professor Alain Connes has picked up the music again as the basis for noncommutative phase logic to create a new unified field science.