First and foremost, I want to thank the HM-members who helped me in these
summer reflections about the music and the discovery of incommensurability
with their remarks, suggestions and criticisms.
I do not try to 'answer' those remarks, I just propose further reflections
that they were induced, written under the Mediterranean Midsummer Sun (!).
About music and arithmetic in ancient civilizations.
Could the discovery of the consonant arithmetic ratios have been made
in Egypt or Babylonia? The question is intriguing, but I know nothing about
archaeological evidence on this subject. There are many interesting
ethnomusicology books about Chinese and Indian or primitive civilizations,
which show how deep was the religious, anthropological and cosmological role
of music (for example in the Upanisad), and even about ancient Egypt and
Babylonia, but I do not know anything about the mathematical aspects.
I can remark just that
(i) It is quite evident the connection between superparticular ratios and
unitary fractions typical of Egyptian mathematics, and (ii) Iamblichus tells
us that the musical sequence 6, 8, 9, 12 was taken by Pythagoras in Babylon;
in addition I could add (iii) a reference to the problem of 'cutting the
tone' in Plutarchus' De Iside et Osiride. As a matter of fact Plutarchus in
that passage ascribed it to the Pythagoreans, but there he is speaking about
Osiris and the unlucky 17; and last a simple problem whose answer however I
did not find in the literature: (iv) the discovery of the mathematical
ratios of the basic consonances is quite easy on a lute, where the strings
can be shortened by the finger and the tuning is made 'by division' on the
string, virtually impossible on a lyre where there is no neck and the tuning
is made 'by ear'. Well the lyre and not the lute was common in Greek music,
whereas lutes were known in Egypt and in Babylon since the III millennium
B.C.
About the sources
Boethius' De Musica is a technical text (a likely translation of
Nichomachus) with very precise and detailed references to the Pythagoreans
(Hyppasus, Philolaus and Archytas) largely coherent with each other, and
with other fragments of these authors, including in the III book long and
detailed computations which show the development of the ancient Pythagorean
music theory since the first 'naive' numerological efforts, continuing with
the negative 'cutting' results on the different ratios involved in the
theory, before giving Archytas' main proof. Even according to Burkert's
standards he seems trustworthy.
The fact that Boethius never mentions incommensurability makes fake
reconstructions or forgeries unlikely. All the book concerns the Pythagorean
theory most of all against Aristoxenus and is oriented toward the main
results about the 'cutting the superparticularis ratio'. I think it reports
exactly the ancient Pythagorean mathematical music theory since its
beginning, and its conclusion is the negative face of the
incommensurability, with a proof whose rigor can not be ascertained. In
order to give a perfectly rigorous proof the techniques of the VIII book of
the Elements are sufficient, but we do not know how near to these was
Archytas' logistic (even if van der Waerden gives a positive answer).
Boethius' text has been discussed by many authors (Szabo, Knorr, Burkert,
etc.) and none of them rejected completely a musical phase in the discovery
of incommensurability (Knorr is the most sceptic, and I discussed his
objections in the first letter); with the words of Burkert (Lore and
Science, 463): "Music theory advances as far as the problems of the
irrational, but stops there and declares them nonexistent. The irrational
belongs to the domain, not of arithmetic, but of geometry". Hence I think
the crucial problem is the translation of the musical 'advances' in the
geometric statement of the problem that we find in Aristotle.
Actually we can read in Boethius the 'musical' road with all its stages,
steps and reasons, but the final 'geometric' step appears nowhere, and the
mathematical music theory is "one of the few fixed points in the
reconstruction of Pythagoreanism before Plato" (Burkert, Lore and Science,
371). On the other side we know nothing about the steps and the reasons of
the geometric tradition: we have just later simple assertions, for example
in Aristotle, of the incommensurability of side and diagonal of the square
with a short reference to an arithmetical proof.
Hence the problem is: why is this traditional and vague hypothesis of a
geometrical origin so largely and acritically accepted, as if the musical
phase, even if so well documented, were a historical accident, without any
role to play in a completely 'geometric' problem?
Continuum
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