Popper, Plato, McClain, Square roots of 2

Karl Popper uses as an example how Plato based his Cosmology on the Greek's overthrow of the Pythagoreans arithmetical system with the geometrical one caused by the discovery of the square root of two being an irrational number. If this the case then McClain's idea of Plato implying Equal temperment might have been influenced by this shift in cosmological view point that happened at the time. It is interesting to me that this same battle continues to this day.
--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

Hi Kraig,

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> Karl Popper uses as an example how Plato based his
> Cosmology on the Greek's overthrow of the Pythagoreans
> arithmetical system with the geometrical one caused by
> the discovery of the square root of two being an
> irrational number. If this the case then McClain's
> idea of Plato implying Equal temperment might have
> been influenced by this shift in cosmological view point
> that happened at the time. It is interesting to
> me that this same battle continues to this day.

The only problem with that is that the Greeks didn't
discover it -- the so-called "Pythagorean theorem"
was common knowledge to the Babylonians of c.1800-1600 BC.
The Greeks got it from them, centuries later.

The Babylonians knew that the ratio of a diagonal to a
side was exactly the square-root of 2, they knew that that
was an irrational number, and they also knew how
to find a remarkably good approximation of it.
See this analysis of the famous tablet YBC-7289:

http://www.math.ubc.ca/%7Ecass/Euclid/ybc/analysis.html

The base-60 version of 2^(1/2) is 1,24,51,10. In
decimal form that's 1.41421,296 (the 296 decimal part
repeats infinitely).

-monz
http://tonalsoft.com
Tonescape microtonal music software

Joe Monzo --

The origin of the theorem is irrelevant to Popper's idea, which concerns a rivalry between two schools within the Greek complex. It is not at all clear that it is necessary to read Popper's interpretation as plausible history but rather as a metaphor for a change of regimes in more contemporary science.

All of which, of course, is independent of any claims to authenticity in McClain's theses. (For what it's worth, I read McClain's work as unsupportable on the evidence of surviving classical music theory (harmonics); however his intuition does strike me as correct that a small group of proportions are utilized in classical philosophical texts to illustrate notions of balance, immensity, moderation, etc.. The problem is to determine what reference would have been understood by a study of philosophy when Plato throws out a number like 729 instead of a round quantity like a thousand or "many": is this proportion meant to be understood as aesthetic, musical, or abstractly mathematical, or is it just mathematical enough to indicate that one has reach the limit of the wieldly in everyday calculation?)

Kraig --

I don't think we need to treat matters like this as conflicts or battles in the fashion of Popper. For what its worth, I think that Feyerabend's anarchic view of innovation in science is the more realistic one, but then again, you know my anarchic preferences.

Best regards,

Daniel

>
> Posted by: "monz"
>
> Hi Kraig,
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, Kraig > Grady <kraiggrady@
> ...> wrote:
> >
> > Karl Popper uses as an example how Plato based his
> > Cosmology on the Greek's overthrow of the Pythagoreans
> > arithmetical system with the geometrical one caused by
> > the discovery of the square root of two being an
> > irrational number. If this the case then McClain's
> > idea of Plato implying Equal temperment might have
> > been influenced by this shift in cosmological view point
> > that happened at the time. It is interesting to
> > me that this same battle continues to this day.
>
> The only problem with that is that the Greeks didn't
> discover it -- the so-called "Pythagorean theorem"
> was common knowledge to the Babylonians of c.1800-1600 BC.
> The Greeks got it from them, centuries later.

correct that others knew about irrationals before the Greeks, and Popper mentions this, but it was what bringing down the current "atomism" hence the reliance on whole numbers at the time. This would be instrumental in Plato's proposal.
--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:

> The only problem with that is that the Greeks didn't
> discover it -- the so-called "Pythagorean theorem"
> was common knowledge to the Babylonians of c.1800-1600 BC.
> The Greeks got it from them, centuries later.

The Greeks proved it, which makes it a theorem. Yhere's no reason to
think the Babylonians did or even could have.

> The Babylonians knew that the ratio of a diagonal to a
> side was exactly the square-root of 2, they knew that that
> was an irrational number, and they also knew how
> to find a remarkably good approximation of it.

Where is the evidence they knew it was irrational?

hank you Dan.
Popper actually brings thus up not so much as a battle as to illustrate that how recent innovation will cause new philosophical problems and the origin of them are more commonly outside of philosophy. I will check out Feyerabend
--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

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Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:

> > The Babylonians knew that the ratio of a diagonal to a
> > side was exactly the square-root of 2, they knew that
> > that was an irrational number, and they also knew how
> > to find a remarkably good approximation of it.
>
> Where is the evidence they knew it was irrational?

I'd have to do some serious hunting to find real evidence,
but here's something you might like to ponder:

http://it.stlawu.edu/%7Edmelvill/mesomath/Rectangular.html

The Babylonian method for solving a quadratic equation
always involved a procedure which i think is akin to
"completing the square".

I think any further discussion on this should migrate
to tuning-math. I'm interested to see what comments you
might have on this.

-monz
http://tonalsoft.com
Tonescape microtonal music software

Hi Daniel,

--- In tuning@yahoogroups.com, Daniel Wolf <djwolf@...> wrote:
>
> Joe Monzo --
>
> The origin of the theorem is irrelevant to Popper's idea,
> which concerns a rivalry between two schools within the
> Greek complex. It is not at all clear that it is necessary
> to read Popper's interpretation as plausible history but
> rather as a metaphor for a change of regimes in more
> contemporary science.

Thanks for that clarification -- i haven't read Popper,
and was simply commenting on the phrase Kraig used:
"the discovery of the square root of two being an
irrational number", since i know that this discovery
was made in Mesopotamia long before the Greeks found it.

>
> All of which, of course, is independent of any claims to
> authenticity in McClain's theses. (For what it's worth,
> I read McClain's work as unsupportable on the evidence
> of surviving classical music theory (harmonics); however
> his intuition does strike me as correct that a small group
> of proportions are utilized in classical philosophical texts
> to illustrate notions of balance, immensity, moderation,
> etc..

(sorry to break into the middle of your parenthetical
comment)

In this regard, it's interesting to me to find that
ratios which are familiar to those working in 5-limit JI
as important small intervals, such as 81:80 and 25:24,
pop up as discrepancies between various ancient measuring
systems. See:

http://www.metrum.org/measures/appendix.htm

> The problem is to determine what reference would
> have been understood by a study of philosophy when Plato
> throws out a number like 729 instead of a round quantity
> like a thousand or "many": is this proportion meant to
> be understood as aesthetic, musical, or abstractly
> mathematical, or is it just mathematical enough to
> indicate that one has reach the limit of the wieldly
> in everyday calculation?)

Well, for sure the ancient Greeks knew about the
"pythagorean comma" (which we can describe as 3^12),
which in constructing diatonic scales is normally found
between what we today call the the two versions of the
"tritone", which in pythagorean tuning can be described
as 3^-6 and 3^6. 729 = 3^6.

So i agree with McClain that there is some significance
to Plato's use of 729. Whether that significance has
anything to do with music is another matter, since as
i just pointed out above these kinds of discrepancies
can be found in other measuring systems too. But i think
the fact that Plato actually invokes descriptions of
music and tuning argues in favor of at least some of
McClain's interpretation.

-monz
http://tonalsoft.com
Tonescape microtonal music software