http://archives.math.utk.edu/hypermail/historia/
We need to dig into an esoteric archive to find our evidence. http://archives.math.utk.edu/hypermail/historia/apr99/0134.html
The "quadrivium" is based on a double opposition: the first betweenMath Professor Luigi Borzacchini
<multiplicity> ("poson", arithmetic+music) and <magnitude>
("peelikotes", geometry+astronomy), the second inside both pairs between
rest and motion, but even maybe between theory and model. This last
aspect can be another "divination", but in the above Archytas' fragment
(and in other his arithmetical-musical fragments) the approach to the
music/arithmetic connection is model-driven, whereas in Philolaus' ones
the approach is in 'intrinsic' and 'numerologic' (I can try, if needed,
to give some supports to these statements).
But first the mainstream symmetric math science claim about music theory - or is it the other way around!
http://www.math.wustl.edu/~wright/Math109/M&MCh05 pdf link
So that's the STANDARD math analysis of music theory. But we now know from math professor Luigi Borzacchini that the actual truth is "really shocking" and "astonishing" - the math originated from the wrong music theory itself!!
http://archives.math.utk.edu/hypermail/historia/aug99/0147.html
Let's review what Professor Borzacchini states:
Continuumoops!
The answer is: for an ancient prejudice, concerning the presumedly
'empirical', 'natural', 'phenomenological' character of the idea of
"continuum" and of the opposition discrete/continuous: it is a common
prejudice not only for historians and philologists, but even for most
mathematicians.
It is evident the connection between incommensurability, infinite and
geometrical continuum (in the Aristotelean ambiguous sense: 'denseness', the
potentially infinite divisibility of a finite magnitude, or 'separability',
the common limit between the parts). It is explicitly stated in Proclus and
in a scholium to Euclid, it is even clear in Aristotle, Euclid, and in
Philip of Opus, if my translation of Epinomis' passage is correct. This is
the positive face of the incommensurability, and then, if the idea of
continuum were something 'natural', it had to be also natural to consider
the geometrical continuum the right embedding for the discovery of
incommensurability, and natural as well the organization of mathematics in
the discrete/continuous opposition of the Quadrivium.
I call it a prejudice because it is hard to believe it from a
'cognitivist' point of view:
(i) in cognitive psychology (Piaget) the developed idea of continuity is a
very late one and its appearance seems connected more to a general process
of cognitive development than to empirical experiences,
(ii) Continuity and its opposition to Discreteness is substantially
completely absent in Chinese mathematics, which is a mathematical tradition
till XVII century largely comparable with the European one, even according
to European standards. In particular in the late Mohist logic, which is the
nearest to our standards among the Chinese schools, the same pair of terms
renders pairs for us sharply different: unit/total, member/class,
part/whole, undermining any discrete/continuous distinction; in addition the
Mohist idea of point does not overcome paradoxes analogous to Zeno's and
Sophists' antinomies.
(iii) after Dedekind, Cantor, Hilbert, Zermelo, Goedel, Cohen we know that
the Aristotelean and Euclidean continuum admits numerable models, that we
can not give to its modern versions a first order categorical
axiomatization, that the geometrical continuum can not be proved coincident
with the numerical one, that it can not be empirically verified, that the
place of the numerical continuum in the transfinite hierarchy is one of the
greatest so far open questions, that it is linked to the most disputed axiom
of set theory, etc.
If the continuum has never got empirical nor logical evidence, derives
from complex processes of cognitive development and does not appear outside
our civilization, we can suppose that its evolution in Greek mathematics was
quite complex.
Without writing now the history of this evolution, it is sufficient to
underline that:
(i) in Homer "continuous" has got a temporal meaning: "with no interruption"
(ii) in Pythagoreanism the distinction between monad and point is just
"having position", reflecting probably only the employment of points as
pebbles on abacus or in polygonal shapes. This ancient distinction among
quantities appears still in Aristotle's Categoriae,6, together with the
'modern' discrete/continuous dichotomy.
(iii) in Eleatism continuity is just 'homogeneity' of the being, and is
completely inside the being/not-being and one/many quandaries, and among the
Atomists, to solve those problems, continuity substantially disappears or is
reduced to 'contact'.
(iv) In Anaxagoras the 'absence' of a minimum is necessary just to allow the
change, by mutual intermixtion of everything in everything, in homogeneous
bodies: it is a sort of 'physical' continuity. Still for Aristotle physical
indivisibility is easier to be accepted than mathematical indivisibility.
(De Caelo 306 a28-30)
(v) in Plato the continuum preserves Eleatic features (for example compare
the idea of 'instant' in Plato's Parmenides and in Aristotle's Metaphysics)
and is not accepted in reality and mathematics as well. It is credible he
always considered, as the Mohists, a 'point' existing just as the 'beginning
of the line'
(vi) Even Plato and the Platonists sometimes did not distinguish between
point and monad, for example putting the monad, instead of the point, at the
beginning of the hierarchy line-surface-solid (according to Aristotle's
fragm.28 and Metaph. 1085 a8).
Those Aristotelean and Euclidean characters of continuity which became
the right embedding of the theory of incommensurability credibly did not
appear before Eudoxus and probably were fostered by the discovery of
incommensurability, and the Quadrivium in its earlier Pythagorean version
(if any) did not know any discrete/continuous opposition.
In other words when music theory paved the road toward the discovery of
incommensurability the idea of geometric magnitude was too clumsy to develop
and even to understand such discovery, and it was exactly the possibility of
the geometric drawing of a not-existent music interval to foster the
development of the Aristotelean continuity.
More precisely such not-expressible existence was a breach in the rigid
Parmenidean isomorphism between being, thought and language: "you could not
know what is not nor indicate it�for the same thing is there both to be
thought of and to be�it is not to be said nor thought that it is not".
Parmenides and Zeno in Aristotle are not so awful as they appear in Plato.
And this breach rid the continuum of the being-not being paradoxes where it
dwelt before (and where it remained in Chinese mathematics).
Incommensurability in Aristotle is the paradigmatic example of a "being
as true", a kind of being which is warranted on a purely 'theoretical' way,
and the Aristotelean potential infinite is something that cannot become
actual, because it can happen just in 'thinking'. Autonomy of thinking is
the crucial step for overcoming the Parmenidean deadlock.
In the first letter I wrote that the refusal of speaking of "what is
not", a crucial 'topos' in the Sophists-Platonic times, was the reason why
the musical incommensurability fell into oblivion (and the absence of later
references is probably the strongest element against the thesis of a crucial
role of music theory in the discovery).
History of Mathematics and Cognitive Science.
Are these cognitive hypotheses stranger to historian methods? I don't
think so, and I think impossible to interpret a fragment without a cognitive
hypothesis, because it has no meaning outside a general architecture of
knowledge in which to embed the involved mathematical concepts. Who claims
it useless, simply chooses the cognitive hypotheses inherited from his
scholastic training.
For example the passage (fragm.4) where Archytas claims the superiority
of logistic on all the other sciences, geometry included, is rejected by
some authors (W. Burkert included) on the base of a cognitive prejudice: how
could a great mathematician consider a practical art of computation superior
to the great Greek geometry? If however we recognize the possibility that
the very idea of divisible continuous magnitude is far from being 'natural'
and that it was before Eudoxus nothing more than a soup of paradoxes,
geometry had consequently to be little more than its Egyptian and Babylonian
models, i.e. simple similitude properties, superposition techniques to
compute areas, figurate numbers and simple properties of algebraic geometry,
some connections between geometric figures and Gods, plus the first results
about squaring the circle and doubling the cube.
On the other side the 'theory of the logoi', under the impulse of the
music theory, could have been the main stream of mathematical research,
producing the results of the VIII book of the Elements, the first negative
proof of incommensurability and more advanced geometric applications, as in
Archytas' algorithm to find two mean proportionals.
Here different cognitive hypotheses give completely different meanings to
the same fragment. At the same time I think that history of cognition (in
particular of mathematics) must become an essential ingredient of cognitive
science.
Best wishes.
Luigi Borzacchini
http://archives.math.utk.edu/hypermail/historia/jul99/0063.html
see also...
And so in my 2012 free pdf book I quote this:
So standard science can not consider causality to be noncommutative - hence the "cognitive bias" that math professor Luigi Borzacchini refers to - hiding the music origins of Western science and creating instead a "deep pre-established disharmony" as the "guiding evoltive principle" of symmetric-based science.
and
Here different cognitive hypotheses give completely different meanings toSo then we can see in standard music math analysis it is assumed that the arithmetic number can be "converted" as the exponential into the geometric logarithms: (pdf)
the same fragment. At the same time I think that history of cognition (in
particular of mathematics) must become an essential ingredient of cognitive
science.
Best wishes.
Luigi Borzacchini
But as Fields Medal math professor Alain Connes emphasizes - in FACT the origin of this inverse exponential (arithmetic) as logarithmic (geometric continuum) dynamic is from non-local noncommutative phase logic!!
https://elixirfield.blogspot.com/2018/10/julyan-cartwright-pythagorean-male-and.html
So I did this earlier blog spot on Julyan Cartwright claiming the square root of two is the symmetric resolving the Perfect Fifth and Perfect Fourth asymmetry of the octave. But then he calls it a Palindrome and I note how Alain Connes points out that Palindromes are asymmetric and noncommutative! oops!!
So why is Alain Connes correct? Because again of the Bait and Switch! The music ratios are given without the pitches to the octave.
Cartwright, J. H. E., Gonz�lez, D. L., Piro, O., & Stanzial, D. (2002). Aesthetics, Dynamics, and Musical Scales: A Golden Connection. Journal of New Music Research, 31(1), 51–58. doi:10.1076/jnmr.31.1.51.8099
url to share this paper:
sci-hub.tw/10.1076/jnmr.31.1.51.8099
So notice that Cartwright makes the same wrong assumption about the geometric ratio of the octave as being equivalent in relation to the Perfect Fifth as 2/3 and 3/2. So as I pointed out - the Kaplans made this same "inductive math" logic error about Pythagorean harmonics as I wrote to them:
So as Charles Sayward and Philip Hugly point out:
Did the Greeks Discover the Irrationals? Philip Hugly and Charles Sayward
So this is what Math Professor Luigi Borzacchini calls the "Negative Judgment Paradox" about the music theory proof of incommensurability.
So the subtle yet crucial point of Hugly and Sayward is that for an irrational number to be created then the "units" have to be converted to a standard common denominator:
In other words we can't just say it's the "square root of two" but rather the "inches" HAS to be used to convert the geometric length into arithmetic units.
So for music theory the unit of measurement is the pitch. For standard symmetric Western math then the unit of measurement is Hertz.
So we can agree that Hertz is an approximate measurement... with as much precision as anyone would like... but is it accurate?
So this is NOT the same as the irrational number as the continuum and Connes' big point is that in fact the origination of the logarithmic and the exponential inverse IS the noncommutative phase logic from music theory!!
So in other words you can approximate with as much precision as you want but it is not accurate to give a non-approximate answer!
Hilarious - I found someone trying to rehash my analysis in their book!
Oops - they misspelled noncommutative as noncommunicative mathematics. hahaha.
Wow - there is another math error here - with C to G as 2:3 should be C to F as 2:3.
Since they use "complimentary opposites" that means they are quoting me from around 2006 or so....
OH WAIT - yes they are correct - that is how I stated it and it does make sense! haha. I had to ponder this again. I really appreciate this person's summarizing in their own words - rewriting what I wrote.
When was this book written? 2015 I'm pretty sure this person contacted me on "radiomisterioso"
Still it's a pretty good summary of my research and captures the concept well! Awesome.
The axiom of power set appears in most axiomatizations of set theory. It is generally considered uncontroversial, although constructive set theory prefers a weaker version to resolve concerns about predicativity.so.... on that link:
In mathematics, logic and philosophy of mathematics, something that is impredicative is a self-referencing definition. Roughly speaking, a definition is impredicative if it invokes (mentions or quantifies over) the set being defined, or (more commonly) another set that contains the thing being defined. There is no generally accepted precise definition of what it means to be predicative or impredicative. Authors have given different but related definitions.https://en.wikipedia.org/wiki/Impredicativity
The rejection of impredicatively defined mathematical objects (while accepting the natural numbers as classically understood) leads to the position in the philosophy of mathematics known as predicativism, advocated by Henri Poincaré and Hermann Weyl in his Das Kontinuum. Poincaré and Weyl argued that impredicative definitions are problematic only when one or more underlying sets are infinite.
He gives two examples of impredicative definitions – (i) the notion of Dedekind chains and (ii) "in analysis wherever the maximum or minimum of a previously defined "completed" set of numbers Z is used for further inferences. This happens, for example, in the well-known Cauchy proof of the fundamental theorem of algebra, and up to now it has not occurred to anyone to regard this as something illogical".[12] He ends his section with the following observation: "A definition may very well rely upon notions that are equivalent to the one being defined; indeed, in every definition definiens and definiendum are equivalent notions, and the strict observance of Poincaré's demand would make every definition, hence all of science, impossible".[13]
And so now back to Borzacchini:
No comments:
Post a Comment