So I did an earlier blogpost on how Plato relied on defining "twoness" as the original "unit" of reality - before any actual number. Then yesterday as I reread the classic Hugly and Sayward philosophy professor article, "Did the Greeks Discover the Irrational?" I realized the logical error they expose is that of the concept of the "Unit" as a circular tautology that does not exist mathematically.
“A popular view is that the great discovery of Pythagoras was that there are irrational numbers, e.g., the positive square root of two. Against this it is argued that mathematics and geometry, together with their applications, do not show that there are irrational numbers or compel assent to that proposition.”
Philip Hugly & Charles Sayward Philosophy 74 (2):169-176 (1999). (my emphasis).
So obviously people fall for this logical lie in high school (secondary education) and all sorts of mass mind control b.s. ensues (Western genocide based on repressed homosexual patriarchy as ejaculation addiction, etc.) That's the "initiation" that males get in the West - a logical lie as a "deep pre-established disharmony" to quote correspondent math professor Luigi Borzacchini.
In The Cult of Pythagoras Math and Myths, Alberto Martínez presents a historically accurate discussion of the many great stories in mathematics. As the title suggests Martínez discusses Pythagoras of Samos and the discovery or proof of the hypotenuse theorem. It is surprising that even though grade school students call the hypotenuse theorem the Pythagorean Theorem there is little historical evidence that Pythagoras has anything to do with its proof or discovery.One of the examples I gave of this logical lie was from Julyan Cartwright He appears to have taken his paper "down" from researchgate after I did a blog post on it. But I didn't mention him by name - so must have been a "coincidence."
OK I was pointing out his argument here is the same old "bait and switch" tactic of ignoring the noncommutative phase of pitch with 2/3 as C to F subharmonic as 3/4 as C to F doubled based on changing the value of the 1, the original root tonic! Instead he just "left out" those pesky actual music pitch harmonics. haha.
But instead of researchgate, his "google scholar" has a lot of pitch nonlinear perception - and a "three frequency" foundation
I'll take a look in a second (no pun intended as the "second" is defined as symmetric visual time!)
| Aesthetics, dynamics, and musical scales: a golden connection
JHE Cartwright, DL González, O Piro, D Stanzial
Journal of New Music Research 31 (1), 51-58
| 14 |
There it is. The plot thickens......
At the 2007 Helmholtz Workshop in Berlin, two seemingly disparate papers were presented. One of these, by Julyan Cartwright, Diego González, and Oreste Piro, dealt with a nonlinear dynamical model for pitch perception based on frequency ratios and forced oscillators, while the other, by Jack Douthett and Richard Krantz, focused on musical scales, maximally even (ME) sets, and their relationship to the one-dimensional antiferromagnetic Ising model. Both these approaches lead to a fractal structure involving Farey series known as a devil's staircase. Why is this? What is the connection between them? The ME sets approach is related to the Ising model of statistical physics; on the other hand, the forced oscillator model relates to the circle map of dynamical systems. Thus we find ourselves facing a deeper question: what are the links between these two paradigms, the Ising model and the circle map, that are …
Sounds like the dreaded Platonic Unit again as even "twoness." http://ecoechoinvasives.blogspot.com/2018/02/philolaus-as-liar-of-lyre-as-test-of.html So then this Unit as twoness is really the "X" as the unit of the geometric mean, that Julyan Cartwright also promotes....
But in my previous blogpost I exposed this all as a lie - as noncommutative math has figured out the truth of reality is Male-Female numbers "all the way" down - as infinite resonance of a geometric zero dimension (as the 5th dimension as time).....
OK now let's go back to Julyan Cartwright to see the can of worms he constructs....but keep in mind Alain Connes is relying on music theory as the "formal logic" of noncommutative geometry! So it is precisely the truth of the musical pitch that exposes the lie of Platonic "twoness" as a "unit" before Pythagorean Number and Harmony.
So here is the pdf link for Julyan Cartwright's 2002 paper arguing that the Golden Ratio solves music harmonics
OK back to this original pdf image I posted - Peralta, 2008, on Pythagorean continued fractions
Of course these are NOT really Pythgorean - they are from Archytas and Plato! So fascinatingly the "diagonal" of the triangle is assumed to ALSO be the number 1 - thereby creating a divide and average process based on the circular logic of "A minus B" (or N minus 1) instead of A plus B. I pointed out that this is the logical error of the Golden Ratio "closed form" solution that Kepler was against. Kepler instead embraced the real Pythagorean "male and female" numbers - and so disagreed with this closed form solution using Zero. Aristotle was ALSO against the use of zero as a materialistic negative infinity.
............
So what we are now showing is that the "continued fraction" solution is actually debunked by math professor Louis Kauffman who has demonstrated how the complex numbers are derived from an time iteration of the number 1 as plus 1 then negative 1. In other words, instead of using "zero" as the divide and average solution of the "unit" of 1 that is symmetric between 1 and negative 1 - there is no "unit" as zero but rather an infinite male and female positive 1 and negative 1.
So for the Pythagoreans the number 1 was not a number but instead, as philosophy professors have shown, studying Parmenides (as I blogged) that the number 1 is instead LIGHT as spiritual consciousness that is alchemical resonance. So the light as the one is turned back upon itself thereby resonating with noncommutative spacetime itself, generating relativistic mass that creates matter as physical form (the Yin Matter that is superluminal).
In other words as Alain Connes points out, in music theory, the "quotient" is noncommutative! And so the continued fraction of 1 with 2 and 1 with 1 is also noncommutative. Kepler understood that like in India, as Stephen Hawking points out in his book God Created the Integers - geometry was "transcendental" with the arithmetic being male and female numbers. There was no "irrational" number.
So what is Julyan Cartwright doing - simply RECREATING the original lie that Plato established using the "Greater Perfect System" of Philolaus - doubling the octave and then flipping the Lyre around! Meanwhile not discussing how this changes the pitch relations to the original tonic.
So Julyan even uses the term "palindromic" when Alain Connes emphasizes that the palindrome IS noncommutative! Instead Julyan tries to claim that because it is palindromic as a "hidden" solution therefore it can just be covered up into the symmetric geometric mean solution:
So in this 2010 pdf Cartwright et. al. have a fascinating proposal that pitch perception and hearing in general is based on a "three frequency response" of rational numbers that create a nonlinear subharmonic oscillation. This, of course, is 2/3 as noncommutative phase with the first two numbers as 1 and 1. But they assume an "equal distance" physical solution which, of course, is NOT the empirically true musical solution, as Alain Connes points out.
This more recent analysis, 2018, on Just Pythagorean harmonics as neural consonance (therefore pleasant perception sensation) gives more details.
So they are doing exactly what I did in my master's thesis - utilizing "2nd order beats" - as combination tones:
It is possible to neglect the explicit consideration of time and visualize tone relationships within the octave by computing the ratio of two simultaneous frequencies and then plotting the interval ratio against the amplitude. This is achieved by forming a linear combination of two pure tone waves, a glissando from the unison (f1) to the octave (2f1) and a firm wave at frequency f1. Similar stimuli were previously adopted by Helmholtz (1954) and Kameoka and Kuriyagawa (1969a,b).
phase-locking, as in the mammalian auditory system, in which neural activity in areas including the cochlear nucleus, inferior colliculus, and primary auditory cortex is phase-locked to the stimulus waveform (Large and Tretakis, 2005). The mode-locking model was proposed by Lots and Stone (2008) as the basis for musical consonance, leading to the development of a dynamical system model based on stylized neural oscillators producing both synchronization and mode-locking. These results support the idea that both parts of the communication system (the sender and the receiver of sounds) are similarly “wired.”http://euro-math-soc.eu/review/pythagorean-world Notice this puff-piece review of the 2017 book - has no mention of the ecological crisis and social justice crisis! But hey, "math works"! Symmetry is awesome right! haha.
The Pythagorean World
Jane McDonnell has a PhD in theoretical physics (U. of Cambridge, 1987) and a PhD in Philosophy (Monash U., 2015). In this book she places the philosophy of science and mathematics in a broader metaphysical framework. She has three sources that triggered her ideas: Wigner's well known quote asking about the `unreasonable effectiveness of mathematics in physics' (1960), Plato's Timaeus (360 BC), in which the nature of the physical world is explained, and Leibniz's Monadology (1714) where he explains his philosophy. First she provides a speculative framework that is inspired by set theory, then combines this with elements from the monadology and from quantum theory to arrive at her synthesis that she calls quantum monadology. I am sure this requires some explanation.
Pythagoras' view was that everything is number, which resounds in the contemporary view of Max Tegmark who claims that our world is just a mathematical structure. In McDonnell's view, the Pythagorean vision should be combined with 'mind' to explain our world. In the first chapter she examines Wigner's paper, and the quest of physicists for a theory of everything (TOE). It is the intention to arrive at a consistent Pythagorean metaphysical framework which will be the proper explanation to Wigner's problem. This leads to claims such as 'there is nothing special about the applicability of mathematics in general, but there is certainly something curious about the applicability of mathematics to cutting-edge fundamental physics'.
This is explained by the growing role of symmetry in theoretical physics. A short survey follows how symmetry was introduced and how it gave rise to quantum chromodynamics (QCD). It is clear that here mathematics took the leading role and that particles are proposed that are not observable anymore. QCD is almost a Pythagorean theory, although string theory or M-theory and multiverses pose new challenges. There is a layered hierarchy as more symmetry is required in the theories to explain physics at growing energy levels and this symmetry leads to simplification of the theory. Either there is a 'fundamental layer' which will be the TOE, or there is a nesting: the theory explaining next energy level contains the theory at a previous level as a special case. Thus also in the latter case there is some 'limit theory' that should be fully symmetric and hence extremely simple to explain everything. This is an explanation of Wigner's mystery. Mathematics and fundamental physics both rely on the same principles that can be experienced in the physical world and in our mind.
If mathematical principles underly a physical reality, should we then accept a Pythagorean view of mathematics, i.e., accept the existence of a unique mathematical truth. Here pluralists and universalists have different views. Clearly McDonnell is in favour of the latter. According to Gödel's theorems there are statements that cannot be proven to be true or false. Second order logic may resolve this, but it will have its own unresolved statements, etc. So, each time one can accept one of two possibilities, true or false, and this may lead to the pluralist view of (infinitely) many different mathematics that may exist independently. All this is explained in a very technical chapter discussing set theory, the Zemelo-Fraenkel theory completed with the axiom of choice (ZFC) and the axiom of infinity. Besides other other independence results that are, or are not, compatible with the axiom of choice, the hope for those adhering the Pythagorean idea is to extend the ZF(C) set theory with additional axioms that will eventually lead to a unique kind of mathematics. Several possibilities are described. McDonnell seems to hope that the the winner in this pluralist-universalist controversy in set theory will come from Hugh Woodin and his Ω-logic.
In the next chapter we learn about McDonnell's own speculative metaphysics that lean towards a realistic structuralism. There is the One (in math the empty set, in physics the Big Bang physics) and there is the Many (in math the universe of sets, in physics the physics of the cooled universe) and in between is Being (the mind) and Becoming. The One and the Many are not reachable, and thus not real, but everything happens in between these. So, if mathematics is the structure of being, there must be one, true mathematics, and the physical reality is its mental interpretation. This universal mathematics is not the mathematics that we are used to. There is indeed some human mathematics that can be fictional, but there may exist rational beings that have a deeper insight in this true universal mathematics where no absolute undecidable propositions exist.
A more concrete proposal if this idea is given in the penultimate chapter. Two paradigms are discussed: Leibniz' monad theory (rooted in classical physics and following the structure of set theory) and the consistent histories quantum theory (CHQT) of Griffiths, Gell-Mann, Hartle and Omnès. This CHQT is a generalization of the classical Copenhagen interpretation which allows an almost natural interpretation of quantum cosmology because probabilities of the histories are assume additive. Eventually the two paradigms are melted into what McDonnell calls quantum monadology. In her proposal a monad is described by a consistent history that exists in the Being. It starts from One: 'I exist' and builds up a history by projecting quantum collapses onto Being, thus learning concepts (a monad has a mind) while building up its history towards a universal theory that will explain its whole world. Depending on stochastic outcomes, monads may develop different histories with different theories. This raises a question whether an optimal evolution exists and whether or not this is chosen by some transcendental entity. Are we living in the best of all possible worlds? Her answer: yes we do.
The concluding chapter gives and overview of the theory with some additional remarks and gives links with other philosophers.
This Pythagorean view is certainly not mainstream and therefore, this book is filling a gap in the literature. Of course, on this subject, it is difficult to propose something that is absolutely true and everything is necessarily speculative. Much will depend on what the next advances in the foundations of mathematics as well as physics will bring. One might expect that, given the background of McDonnell as a theoretical physicist, the quantum theoretical part is the hardest part of the book, but it turns out that the most difficult technical part is the one on set theory. Some of the technical stuff has been moved to an appendix, together with another one about problems with plenitude. When you are an ordinary mathematician, not specializing in the philosophical foundations, I do not think this book will influence in any way what you are doing every day. You will probably loose solid ground after set theory gets its metaphysical interpretation. If you are indeed involved in the philosophy of mathematics or more general philosophy, it is certainly a source that will raise controversy. I can imagine you feel the urge to put your heels in the sand or even to dig in and take a stand preparing for an intense polemic. But that is not exceptional. Most of these philosophical explanations provoke controversy.
yes you raise interesting connections. So people keep asking about how light is connected to gravity - and it was Dirac studying Heisenberg who realized that the noncommutative phase math of Heisenberg was actually the same as what Whittaker used in his treatise on electromagnetics - and that Whittaker himself got this math from Poisson. So in quantum mechanics it's called the "Poisson Bracket" which converts the noncommutative time-frequency into the symmetric math of geometry. This is based on a new property of quantum physics that Wolfgang Pauli discovered - "spin."
So now the Pythagorean Theorem - in the essay I linked yesterday - it is "precise" but it is not "accurate." So you make the analogy of a rifle shot. This is what Gregory Bateson describes as the logical paradox - the "error of logical types." So Bateson, in his book, "Mind and Nature: A Necessary Unity" compares a rifle to a shotgun.
These analogies are actually historically proven. So the Pythagorean Theorem was first used to "square the circle" as an approximation for centering the axles on chariot wheels - so we are talking the first Solar Dynasty Tantric alchemy empires. math professor Abraham SEidenberg dug this up - tracing it back to around 1200 BCE or earlier to 3000 BCE. The chariot was first invented on the plains north of the Caucus. Now DNA science proves these chariot warrior priests then spread into India - and to Iran. And even the "spoke" wheel analog in Daoism is from this chariot culture spreading into Western China.
But Daoism itself based on Dragon and Tiger as yin and yang is much older. Music perception is what is truly universal in human culture and is now proven to be biologically hard-wired in mammals - I'm talking about the Yang as 3/2 and Yin as 3/4 and the Emptiness as 1/2 - as infinite Taiji resonance. In India this was the "three gunas" - the oldest philosophy of India.
So then the chariot in terms of math as a religion of mass ritual sacrifice was then tied to turning the circular altars based on the Lunar calendar and female silver energy - into the same area as the square altars based on the solar calendar. Try doing that math of subtracting the lunar calendar from the solar - using ideograms. It's very difficult to do the "divide and average" convergence of time as a closed infinite geometry! So the Chinese adopted the rod-based 10 number system as a potential better means to do the math.
But if you study ancient Chinese astronomy it is based on Daoist harmonics - so the first "note" is actually the lunar calendar energy - and this is biologically accurate with the pineal gland of only human females being precisely synchronized with the lunar calendar of 29.5 days - as females living in Nature. Chimpanzee females are not even close compared to human females.
OK - so moving along with your analog of the rifle shot - in terms of the math then in ancient Greece - to avoid the plagues of Wheat monocultural farming that spread with intensive animal husbandry - God is etymologically from the Indoeuropean word meaning Bull (from go) and Brahman also means Bull - so this plow-based patriarchy then spread disease. The chariots were to evade disease by spreading empire and the first "white people" (from Wheat lack of vitamin D in the diet) spread into Europe since the lime ash fire cement - to waterproof the adobe housing - caused massive deforestation and drought - so around 8000 BCE the first white people were ecological crisis refugees spreading into Europe (where non-white hunter-gatherers lived)....
OK so then the catapult was next developed based on the need to "precisely" double the cube - of the mass ritual sacrifice altar. This is then in terms of music harmonics - the cube root of 2 as 5/4, the Major Third music interval. This was spread via Marsilio Ficino (after the Islamic-Platonic culture spread it back into Europe in 9th century via John Scotus Erigena, the Benedictine Monk)... So then the Modern Science revolution was based on Simon Stevin in the 1500s developing logarithms with the Alogon of Platonic philosophy from Archytas and Euduxus and ARchimedes - as the early form of the calculus. And so this was directly tied to music theory - the idea being that natural number ratios (as found in Daoism) are not needed at all since reality is defined inherently as a closed infinite geometry (the irrational magnitude).
So as I pointed out yesterday in that link - it's a logical error relying on "unit" as a tautology of closed circular meaning - literally the Absurdity of the Surd. Surd meaning "unwhisperable" or not heard - the Unheard origin of the square root of two - it's secret musical origin as Alogon, passed on in Islamic culture as the Surd or irrational number. So this is from Plato - to define the "one" as the hypotenuse unit as well as the length of the triangle sides. And so the order of infinity can be reversed without encountering the inherent male and female value of number as one and negative one - rather time is turned into a static geometric image as "unit" before number as geometry (analytic geometry). So then Plato infamously stated, "Time is the Image of Eternity."
So then with the Modern Science Platonic revolution in Europe - this was funded by the aristocrats with Galileo bragging he could cut down on labor costs - and Newton being in charge of the gold treasury mint - so viciously pursuing any fake currencies with the death penalty. Yes now it was the Cannon trajectories to be best modeled as the inverse square law that very blatantly originated from the music harmonics origin of the Pythagorean Theorem from Archtyas. This is to say Newton argued that a weight stretches a string in tension by squaring the weight and so doubling the string to its octave. This, of course, was NOT the real Pythagorean harmonics but again from Archytas and Plato.
So you state it is much harder for a rifle to be precise in aiming - and so the question here though is not precision it is ACCURACY. In other words if Western science started based on a logical error - and then just kept amplifying that error to greater degrees of precision - then what happens is this is actually a left-brain projection as a repression of the N/om kundalini right brain energy (from logical inference of time-frequency listening). So there is then an external oppression based on patriarchy as ejaculation addiction - called "Missile Envy" by Dr. Helen Caldicott - or called "entropy" by Western science defining randomness based on logarithmic math. So the math and science continues to be more precise as an increased TENSION of mass on Earth - with an acceleration of time to a zero point of geometry (apocalypse). This is all due to defining the math based on the solar calendar from patriarchal plow-based farming as a "symbolic revolution" - the archaeological term. So at first the "Alpha and Omega" of religion (latin for "to bind") was from the upside down bull horns - the spirals as the letter A and Omega of time, now "contained" as geometry that is rectilinear using triangles and squares. So Science is literally a religion based on Platonic geometry - as I pointed out called the "music logarithmic spiral" that then spread from the Platonic Archytas math - Archytas was a military engineer who doubled the catapult size precisely by doubling the cube precisely...
OK so now let's take this to its logical conclusion. The math "works" - right? It's magical! Or is it?
The Pythagorean World: Why mathematics is unreasonably effective in physics (Palgrave Macmillan). 2017 Physics philosopher professor book.
So she expands the wrong Platonic Archytas math all the way into the unified field theory, etc.
I blogged on this -
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