Art and Mathematics in Matila Ghyka’s Philosophical Aesthetics – Cornel-Florin MORARU

Art and Mathematics in Matila Ghyka’s Philosophical Aesthetics.
A Pythagorean Approach on Contemporary Aesthetics

Writing his main works on aesthetics and philosophy of art in the first half of the twentieth century, Matila Ghyka had a great influence on the artists and thinkers of his age, but failed to grant himself a place in the canonical history of philosophy because his works did not to meet the modern criteria of the philosophical canon. Although these works changed the way artists like Salvador Dali (Lomas 2006, 11) and André Lhote (Lhote 1969, 68) made and theorized art, the sources of Ghyka’s thinking, namely  the Pythagorean tradition, are often viewed as pseudo-philosophical because of the so-called “mystical” elements which are present in it and the “secrecy” in which Pythagoras’ teachings are covered. This is one of the reasons why many philosophers and scientists are reluctant in approaching Ghyka’s work, his field of influence being restricted at large to artists and art theorists.
The problem with this attitude towards Ghyka is, as I will try to argue,
that the interpretation to the Pythagorean philosophy as “mystical” and its teachings as “secret” is based on a preconception and is somewhat misleading.
The preconception is that the early pre-Socratic philosophy as a whole and the Pythagorean tradition in particular do not constitute models of “pure philosophy”, but some kind of proto-philosophical endeavours belonging to the “pre-history” of rigorous philosophy. The philosophical tradition starts, from this point of view, with Plato and Aristotle, which are viewed as thinkers who purified the conceptions of their predecessors of mythological and mystical elements. This is misleading because it neglects the ancient meaning of the words “mystical” and “secrecy” and, at the same
time, contradicts the attitude of admiration and respect with which “canonical
philosophers” – such as Plato, Aristotle or Plotinus – view the Pythagorean tradition of thinking as a valuable and respectable one. This is why, in order to reveal the preconceptions and cultural biases of the common view on this problem, we must turn back to the pre-Socratic and classical Greek thinking and show the original meaning in which Pythagorean doctrines were “secret” and “mystical”. For this, we must sketch a new way of interpreting the early Ancient Greek philosophy that is able to relieve these
concepts of the negative meanings which were imposed by Christianity in the Late Antiquity and in the Middle Ages. By means of this “hermeneutical detour” we will gain a better understanding of both Pythagorean philosophical tradition and of the relevance that Ghyka’s philosophy has for the contemporary research in domains such as aesthetics and philosophy of art.
Along with the above-mentioned preconception which affects the
general attitude towards Matila Ghyka’s thinking, we can observe another
preconception that is also widespread in the contemporary philosophical
community: the idea that philosophy of art and aesthetics are not suited for
a rigorous mathematical or scientific approach. The origin of this preconception
lays in a modern view of the world, according to which judgements
about scientific facts are constituted in a different way than judgements
about artistic facts. This led to the opinion that science and art triggers two
different modes of knowledge – “conceptual” and “sensitive” –, which have
different logical structures (Baumgarten 1750, §14) and cannot be intertwined
at the level of philosophical discourse or in practice.
In time, these differences between the two domains of human knowledge
established the idea that science in general and mathematics in particular
have a reductionist character, while all phenomena concerning art are
essentially non-reductionist. This preconception can manifest itself more
clearly nowadays, if we look at the fact that, although there are many philosophers
which activate in the fields of human cognition, philosophy of
mind and neuro-phenomenology which pay close attention to the new scientific
approaches to cognition and human thinking, most art theorists and
aestheticians ignore any kind of scientific implications of the aesthetical experience.
There are, indeed, some notable exceptions, but these approaches
Art and Mathematics in Matila Ghyka’s Philosophical Aesthetics…
come very often from scientists with a good philosophical background (e.g.
Thomas Metzinger, Antonio Damasio, Semir Zeki) and not form philosophers
and art theorists as such. At the same time, although there are lots of
philosophers pertaining to both continental and analytical traditions of philosophy
that reflect upon the nature of numbers, there are quite few thinkers
that try a mathematical approach to problems of artwork’s ontology and
aesthetical experience. In this case, many philosophers view the mathematical
approach to art in a reductionist manner, although mathematicians themselves
left this reductionist view on mathematics a few decades ago (Ian
Stewart 2015, 484-487), along with most of the contemporary scientists that
work in the field of neurosciences (Damasio 2016, 127-130). As I will argue,
this reluctance towards a mathematical approach to art should apply neither
to Matila Ghyka’s thinking nor to the Pythagorean tradition, because in
both cases we can observe a non-reductionist view on mathematics.
For this to become clear, we should start by reflecting on the hermeneutic
nature of the process through which the corpus of the philosophical
tradition is formed and the eventual alternative modes of interpreting this
process. By rethinking the criteria of philosophical historiography, we have
a chance of rediscovering a “vein of thought”, which was ignored by the
modern tradition but could prove itself valuable to postmodern thinking.
1. Towards a meontological history of philosophy
The reflection upon Matila Ghyka’s works puts the researcher in a
situation in which he must redefine the criteria and the essential traits by
which a certain thinker’s studies can be accepted in the canonical “history of
Western philosophy”. There is a manifest discrepancy between Matila
Ghyka’s reception into the non-academic world of artists, art theorists and
philosophers and its academic reception. But this is not the only case in
which this state of affairs becomes manifest. In many other cases, it is not
the influence which a certain work had upon its age or its proven utility in a
certain field of study, but rather a kind of “intrinsic value” attributed to the
text that grants it a place in the history of philosophy. Unfortunately, this
“intrinsic” is often founded on a bunch of preconceptions which the
researcher himself is not aware of, among which, in our case, the above
mentioned two ones are central.
This is why we need to define a perspective from which Matila Ghyka’s
Pythagorean approach to art gets encompassed into meontological a tradition
of thought. To achieve this goal, we should focus not on the ontological
interpretation of the history of philosophy, which started with Parmenides,
but was developed mostly in Plato’s γιγαντομαχία περὶ τῆς οὐσίας (Plato
1900b, 246a-248d) and in Aristotle’s sketch of a “history of Being” in the
Metaphysics (Aristotle 1970, 983b-988a). In return, we will focus on non-
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being and nothingness, as the main phenomenon that guides the historical
philosophical efforts. This will grant the so-called “pre-ontologic” philosophers
(i.e. the pre-parmenidian tradition, among which we can also count
the Pythagoreans) a central place in the history of philosophy.
Meontology is usually defined as the philosophical discourse on non-being
or, yet better, nothingness, and is quite well represented in both Romanian
(Cernica 2002; Cernica 2005; Cornea 2010) and European contemporary
philosophy (Heidegger 1988; Sartre 1943; Merleau-Ponty 1964). However,
there are yet no attempts of a systematic meontological approach on the
history of philosophy, as far as my knowledge goes. This is quite strange
because the beginning of the Western philosophy in the pre-Parmenidian
period had a very strong meontological approach and there are several
arguments that can legitimate this view. First of all, in the very first of the
philosophical fragments conserved, Anaximandros stated that the ἄπειρον,
or the limitless, is the principle (ἀρχή) of all things. But ἄπειρον is essentially a
“meontological entity”, which can be defined only by negation and is closer
to non-being than it is to being.
In addition to this argument, I could add another one which I developed
at large elsewhere (Moraru 2017, 107-112), namely that the word which
signifies “Being” (τὸ ἐόν) appears for the first time, as far as our textual
accounts show, in Parmenides’ poem. Although this institution of Being is
viewed by Plato as one of Parmenides’ most important contributions to
philosophy (Plato 1900b, 241c), he also felt the need to surpass the idea of
Being in order to coherently think the interdependence of the five “supreme
ideas” – being, non-being, rest, change, sameness and alterity (Plato 1900b,
241c). Given this context, it would take a lot of presuppositions to assume
that the thinkers which preceded Parmenides were practicing ontology,
because there was no such concept as “Being” that they could study.
Instead, it would be much less risky to assume that they studied nature
(φύσις), as the whole Ancient doxographical tradition states, and that nature
is, in itself, different from Being just like Nothingness is different from
Being. This latter claim, although not present in the doxographical tradition,
can be argued for in a quite simple manner.
First of all, nature is a process, not a substance, as is the case of Being. It
implies change and becoming and this is why the early Greek philosophers
sought it’s “beginning” (ἀρχή), which is, at the same time, the principle of
change in the case of nature. The beginning of a process is that τόπος which
governs its whole horizon of progress along with its possibility of
realization. This is why, the beginning cannot be thought in a “substantial
manner”, but in a temporal one. It is not something, but a certain “rift” in
the structure of temporality which marks the “starting point” of a process.
Only by means of its retrospective re-thinking and re-interpretation can the
beginning gain some substance and be instituted as something which exists
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or existed in a substantial manner. It is because of Plato’s efforts to
reconcile the φύσις with the τὸ ἐόν that one might think otherwise, although
Plato himself was aware that the principle of all thing, the supreme idea,
should be non-substantial and, in a certain sense, approach non-being (Plato
1902, 509b).
This paradoxical status of the ἀρχή of all things as a non-substantial entity
that offers the possibility of any substance or being can also be observed
as a constant problem of ancient philosophy or even of philosophy
as a whole. From Aristotle’s apories of the first principles from the Metaphysics
(Aristotle 1970, 993a-995a), to the problem of Oneness in Neoplatonism,
form the apophatic theology to Nietzsche’s nihilism, there are many texts in
the history of philosophy that could be considered as “meontological”. The
main problem is that the “onto-centric” interpretation of philosophy is so
well established that it governs most of the hermeneutical assumptions of
philosophical historiography. To shake these foundations, one must systematically
follow the meontological history of philosophy – roughly sketched
here – and see the way Being is intertwined with non-Being throughout the
history. Just like two veins that start from the same place, but provide blood
to different parts of the body, ontology and meontology are two veins of
thought which animate two different domains of philosophy.
3. Nothingness as the primary non-substantial entity and its hypostases
The first difficulty we encounter when talking about nothingness is a
logical one. Whatever we say about this phenomenon violates the rules of
traditional logic. As soon as we utter “Nothingness is the limitless”, for
example, we fall upon a contradiction, just because nothingness, by
definition, does not exist or have being, so, it cannot be something. This
shows that, in the field of meontology, classical logic reaches its limits and
can no longer provide us with an instrument of rigorous analysis. However,
this is not a sign that meontology is, in itself, an inaccessible path of
thinking, as Parmenides thought, but should be linked to the fact that our
language as a whole, along with its logic, is designed from an ontological
standpoint and is generally used in order to express substantial entities, not
meontological ones. Through logic we create a synthesis between the
linguistic and ontic realms (Cernica 2013, 78-88) and the “categories” of
Being are also concepts which structure our language and understanding of
the world (Aristotle 1949, 1a-1b; Aristotle 1970, 1028b). In other words,
language itself is designed to designate beings or states of affairs and has no
means to properly express “nothingness”.
This is why Aristotle stays that “one speaks of nothingness, according to
its cases, in the same number of ways as there are categories and, in
addition, one speaks of nothingness as false and as potentiality” (Aristotle
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1970, 1089a)1
. So, our language “ontologizes” things and imprints the
categories of being onto nothingness, donating its substance even before we
can properly grasp it. This is, in fact, the fundamental paradox of
meontology: Although we consent to the thought that “nothingness is non-substantial”,
we give it somewhat of a substance in the very first moment our mind tries to aim it
The fact that Aristotle uses in the cited place the same word as the
grammarians from Alexandria will use to describe the “cases” of a noun –
πτώσεις – is relevant for the strong relation between logic, linguistics and
ontology. We cannot properly address the nothingness because our
language and our logic transform it into “something” and give it some
degree of Being. Caught in language, the nothingness gets mixed or
intertwined with Being and, thus, becomes something. This transformation
of nothingness into being through our discursive thinking by means of
language is what we call hypostasis. It is a “stabilization” of the sense of
nothingness itself in a certain conceptual and linguistic context and
following a certain principle.
This is why nothingness has many hypostases and shows itself to the
human consciousness in different ways. In an ontological context, nothingness
is the non-being; in a linguistic or rhetorical context, nothingness is the
ineffable; in an epistemological context, nothingness is the unknowable; in
the context of psychology, nothingness is the unconscious etc. All these
ways of talking about the nothingness show us that every meontological
discourse has a conceptual genealogy which gives meaning to a certain hypostasis,
usually through negation. The so-called “negative prefixes” (a-, non- or un-)
each give a certain “meontological flavour” to a certain “kinds” of nothingness
and paradoxically set up its foundation in absentia.
But, in each of the mentioned cases, there is another thing we should pay
attention to, namely that every conceptual genealogy has its own structural
principle. In other words, every conceptual context has a “central idea” in
accordance to which all other concepts are organized. For example, the
nothingness is, for a linguist, the ineffable in accordance with the rules of
language, for the epistemologist, the nothingness is the unknowable according
to the rules of human knowledge etc. This is what we can call an
meontologic archaeology and is the active principle which structures the conceptual
genealogy as individual conceptual context. In this way, we obtain a
roughly sketched view on the process that enables us to speak about nothingness
in many different ways. Each of these ways, however, corresponds
to a particular use of human λόγος or “reason” in a very broad sense.
This broad sense of λόγος refers to the wide sphere of human
discursivity, as it is designed in the history of philosophy by the Heraclitean
κοσμικὸς λόγος or the Christian θεῖος λόγος from the Gospel of John. Some
aspects this meaning of λόγος were also observed by contemporary
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phenomenologists such as Martin Heidegger (1999, §48) or Maurice
Merleau-Ponty (1960, 105-110) However, the philosophy of language, be it
phenomenological or analytical, failed many times to notice the essential
role of the λόγος in the hypostatization of the nothingness. From this point
of view, language is a set of hypostatic functions because it works by means
of sedimentation of certain meanings in our mind (Merleau-Ponty 1960, 111-
115) and, thus, it transforms something that is fluid and ungraspable intro a
stand-alone entity, which we can aim intentionally. In other words, through
language, we “give substance” to the world, that otherwise would be a series
of transient intuitions, hunches and forebodings.
But our language has its limits and so does its possibilities of expression.
This is why, we might expect some phenomena to be harder or even impossible
to express by means of the everyday language. This is what the last of
the Neoplatonic philosophers, Damascius, referred to as a “the retorsion”
or περιτροπή (Damascius 1889, I, 7) of the λόγος, a phenomenon which
occurs when we try to talk or think about the first principle of all things that
lays beyond being and even beyond the One itself (Damascius 1889, I, 6),
namely the nothingness (τὸ οὐδέν). About this “ultimate phenomenon” we
cannot have a logically coherent cataphatic speech, but only an apophatic
one, although, rigorously speaking, even this latter kind of speech is not
fully adequate (Damascius 1889, I, 6) to express the vague consciousness or
consenting of the ineffable – εἰς τὴν ἄρρητον συναίσθησις (Damascius
1889, I, 5) – that we feel at the most profound affective and dispositional
level of our being. The first principle cannot be spoken of per se,
but it can be hypostatized into language according to a meontologic
genealogy and archaeology. We cannot refer to the nothingness as nothingness,
but we can refer to nothingness as ineffable, as unknowable, as
non-being etc.
So, if we would like to make a meontological interpretation of the history
of philosophy, we should trace the ways and means by which the nothingness
was hypostatized into philosophical discourse in each and every
historical period. In this way, we can better understand that which stands
beyond a philosophical text, considered as a positum, namely the ineffable
motifs and intuitions that give reason, force and aim to every philosophical
endeavour. The consenting of the nothingness is that which drives our
curiosity and will to understand that which cannot be understood. Thus, we
are able to reach down to the hidden root of the original philosophical
discourse, namely the wonder towards the ineffable principle of all things,
which is a kind of pure an ungraspable nothingness for the human
The two biases, which simultaneously prevent the Pythagorean thinking
and Matila Ghyka’s philosophy of art to enter the official curriculum of
Western philosophy, are manifest due to the ignorance of the original
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meontologic and hypostatic character of the philosophical discourse and of
thinking in general. This is why a reinterpretation of Pythagorism in the
light of the above-mentioned observations could shed a light on the
authentic philosophical character of Matila Ghyka’s aesthetics and on the
possibility of a non-reductionist “mathematical” approach to art.
4. Pythagorean Mathematics and Hypostatic Character of Numbers
In order to show that de Pythagorean concept of “number” is, in fact, a
hypostasis of the nothingness viewed as ἀρχή of all things, we must
determine its meontologic archaeology and genealogy. The archaeology is
that of the human μάθησις (Aristotle 1970, 985b), namely the process of
human learned knowledge (διδασκαλία) which is deposited into information
in a conceptual and propositional sense (μαθήματα). The information we
refer to here can be constructed in various modes, depending on the
domain of knowledge we refer to. For example, the “scientific knowledge”
differs from the “historical one” and the “practical” one. So, there can be
various types of μαθήματα, each of them having the main trait that is
constructed through a process of learning, which implies some effort of
memory (μνήμη) and re-collection (ἀνάμνησις).
This “mathematic” kind of knowledge is somehow opposed by the early
Greek philosophers to σοφία, which means rather “clarity of sight”
(Aristotle 1962, 1141a) or, yet better, a clear insight into the nature of things
provided by the human direct intellectual intuition (νοῦς), which is, as
Aristotle himself points out, a non-discursive or ineffable grasping of the
first principles of things (Aristotle 1962, 1141a). This is why, “the multitude
of information doesn’t teach one how to have insight” (Heraclitus 1951, B
. In other words, knowledge won’t necessary provide wisdom.
This doesn’t mean, however, that learned knowledge is completely
useless. In fact, in the Pythagorean tradition, the μάθησις is some kind of
“bringing to stability and grasping” – ἐπιστήμην καὶ κατάληψιν – of the truth
by means of philosophy (Nicomachos 1866, I, 1), conceived as “appetence
for wisdom” (σοφίας ὄρεξις ). So, the essential character of μάθησις is that it
grasps that which can be grasped from an ineffable intellectual intuition
provided by the νοῦς and deposits it in a well-defined concept with a standalone
meaning. In fact, this bringing to stability and concept of our ineffable
intuitions by means of “syllogism” or “deduction” is the main character of
science (ἐπιστήμη) in the Ancient Greek sense of the word (Nicomachus,
1866, I, 1; Aristotle, 1964, 71b; Beekes and Van Beek 2010, 445).
The relation between σοφία and μάθησις becomes clearer when we try to
think about the nature of Pythagorean concept of “number”, which is the
first way in which the ἀρχή affects the human soul. For example, the idea
that the infinite number existent of things can be reduced to one principle is
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an ineffable intuition (νόησις) or a non-judicative experience, which cannot
be founded on rational arguments alone because it is that which gives possibility
to any reasoning. This is the reason why, in the Pythagorean tradition,
the different numbers are perceived by the human consciousness, primarily
and before any rational determination, as some kind of affection (Aristotle
1970, 985b), not as concepts per se. “Reason” and language in general would
be impossible without the firm belief that we can refer to different objects
of a class by the same word and that this word corresponds in some way
with the ontic entities we perceive. Behind the Pythagorean doctrine
(μάθησις) of numbers lays a “hidden meaning” which cannot be fully
grasped by reason and which pertains to σοφία.
This means that the intention of the Pythagorean thinkers is to gain
wisdom by studying the way in which our non-judicative experiences of the
world could be grasped by means of the study of the doctrine of numbers.
The reason is that “it [the doctrine of numbers n.n.] is, by nature, the vision
(θεωρία) through which the most simple and original things can be
elaborated [by reason]” (Iamblichus 1984, I, 1) and that “the discourse
about it precedes any other doctrine” (Iamblichus 1984, I, 1). In other
words, understanding the hypostatic character of numbers that makes the
transformation of non-judicative experiences into “scientific” concepts and
theories possible is the main aim of the Pythagorean philosophy and must
be achieved before any other “scientific theory”.
This shows that the μαθήματα specific to the Pythagorean philosophical
endeavour have an epistemological genealogy. In other words, the conceptual
context in which the nothingness is hypostasized is formed by concepts
pertaining to science in the Greek sense of the word and are different from
other hypostases of μάθησις. However, that which lays beyond these
numbers, conceived as first affections of the consciousness that spring from
the first principle, is “secret doctrine” of the Pythagoreans in a peculiar
broad sense of ἀπόρρητος μάθησις.
5. The “Secret Doctrines” and The Ineffability of The Nothingness
For a modern person, a secret is something that should be “kept” and
that, in principle, could also be “revealed” by propositional means. Basically,
a secret is some piece of knowledge which is “covered in silence” but could
be uncovered by anyone who knows it. So, the secret is something someone
may not speak and “keeping the secrecy” is an act of individual will. None of
these meanings hold for the ancient ἀπόρρητον, whose main signification is
“that which cannot be spoken”, the ineffable. For Greek philosophers, the
secrecy is not something one may not reveal, but something one cannot reveal.
However, this is the sense in which Plato refers to the “secret doctrines”
(Plato 1900, 62b) and the sense that results from his disclaimer from the
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Seventh Letter concerning the accusations that he revealed the secret
doctrines of philosophy to Dionysius (Plato 1907, VII, 341). He did not
reveal any “secret doctrine” because the secret doctrines cannot be expressed
propositionally. The “hidden sense” of every philosophy and especially
of the Pythagorean doctrine of numbers is not a μάθησις that can be taught
and learned, because all μαθήματα need a set of non-judicative experiences
that are grasped in a non-discursive way by the νοῦς. In fact, this is the main
argument by which Plato denies writing’s role as an aid for knowledge and a
medicine (φάρμακον) for forgetfulness (Plate 1901, 275a-277b) – writing
only helps the ones that have the proper non-judicative experiences or the
proper insights to “remember”, but it doesn’t “teach” anything stricto sensu.
The question we have to raise is: how could one “gain” insight, if not
through reading and learning in general? Plato answers this when he states
that the “shared substantiality” (συνουσία) and the “shared living” (συζῆν)
with philosophy kindles the intuition of the ineffable just like rubbing two
sticks together kindles the fire (Plato 1907, VII, 341). In consequence, the
non-judicative experience is gained by means of an existential, not a
cognitive effort. This is why this kind of experience is not “propositional”
knowledge, but rather a “mystical” one.
As is the case of the “secrecy”, the mystical character of the authentic
philosophical knowledge should not be understood in the terms of modernday
conceptions. For the Greeks, “mystical” meant “silent knowledge” or
“intellectual intuition of the ineffable”. This is confirmed by an inscription,
dated around the second century A.D., which speaks about mystical
knowledge as “the ineffable knowledge of the initiation in mysteries”
(Dittenberger 1883, 873.9)3
. This “silent” and inexpressible knowledge is
what is “imprinted” into our consciousness by συνουσία and συζῆν and is
somehow expressed through μάθησις, but just for those who already have a
certain intuition of the ineffable.
The reflection on the two forms of community with the subject matter
that cause the insight about the ineffable brings us about some kind of
“existential learning”, made by an effort to interiorize and live according to
one’s philosophical occupations. Just like an actor which enters into a
community of substance and of lived time with his character, the philosopher
enters into a community of substance and time with the nothingness
itself. As Plato puts it, philosophy is a form of exercise for death (Plato
1900, 81a) because death is the ἀπόρρητον of life. In some sense, death
leads to the “secret” side of life, that which cannot be spoken of and cannot
be conceived rationally, but of which we all have some insight through our
deepest anxieties and fears. So, the authentic philosopher “imitates” death
by his way of living – he neglects the body and the material things and tries
to “unfasten” his soul from its knotting with the body. Just like an actor, the
philosopher tries to act as if he lives among the incorporeal entities. From a
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Greek point of view, the philosopher is an actor that imitates the Gods, hoping that
someday he himself will transcend his human condition and become Devine.
As we perceive the Pythagorean tradition from this point of view, it
becomes clear that the first preconception we have analysed is no longer
sustainable. The Pythagorean philosophy is not a form of “proto-philosophy”
because of the “mystical” elements and the ritualistic and religious
character of the “secret doctrines”. On the contrary, judging by the aim of
this endeavour, it is rather close to contemporary philosophical projects.
However, this common aim, namely the indication towards some kind of
insight of the ineffable or non-judicative experience, is pursued in each case
by different methods, among which the Pythagorean one is the most
undetermined and elusive. For it to become manifest, it is needed a rigorous
reconstruction of the Pythagorean thinking in its “systematic” form in the
late pythagoreic philosophers.
Nevertheless, what becomes manifest from the concept of “existential
learning” carried out through συνουσία and συζῆν is the connection between
the Pythagorean doctrine of numbers and art. The two domains are
essentially linked from the point of view of the process of imitation
understood in the above-described sense, which we may call a “scenic
sense”. From the perspective of the one who lives a philosophical life, art
and arithmetic are essentially linked as two ways of hypostatization of
nothingness into the vast domain of the human λόγος. As hypostatic
μαθήματα, art (as a skill) and the philosophy of numbers have the same
meontological archaeology, which means that they both are modes in which the
insights of the ineffable are imperfectly expressed through learned skills and
information. However, their meontological genealogies are different, which
makes them different modes of expressing the same ungraspable phenomenon
that is the nothingness.
6. Art and arithmetic as two forms of mathematical knowledge
In Nicomachos’ Introduction to Arithmetic there is a distinction between
two ways or “methods” (μέθοδοι) to deal with numbers as hypostatic
concepts. First of all, the essential trait of numbers is that they can express
the quantity and size of the “magnitude” and “multitude” of things
(Nicomachus 1866, I, 3). But both “magnitude and multitude are, by their
own nature and with necessity, indeterminate (ἄπειρον)” (Nicomachus 1866,
I, 3.5). Magnitude can virtually stretch out to infinity and multitude can be
divided into an infinite number of parts (Nicomachus 1866, I, 3.5). We can
think about an infinity of numbers and about a number with an infinite
number of decimals. So, the concept of number is used to approximate (i.e.
create a hypostasis) of the primordial ἄπειρον of the κόσμος, the same
ἄπειρον Anaximandros designated as ἀρχή.
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In this context “wisdom” simply means the “scientific” account on these
two “forms” of indeterminateness by means of numbers (Nicomachus
1866, I, 3.5)4
. From this point of view, philosophy naturally aims to grasp
the two main meontological features of the world – the absence of borders
and absence of limits, i. e. ἄπειρον and ἀόριστον – into apparently determinate
concepts or products of the human mind in general. This aim can be
reached in two ways – through music in a broad sense and through the
doctrine of numbers or arithmetic (Nicomachus 1866, I, 3.1).
The first method was analysed in the previous pages. It deals with
“quantity in itself” (τὸ περὶ τοῦ καθ’ ἑαυτό), which means that it deals with
de nature of every number, considered as a stand-alone entity and with the
affections (πάθη) through which that number manifests in the domain of
human “mathematical” λόγος. But numbers are, on the other hand, interrelated
in what we call proportions. In fact, this “relative” aspect of
numbers is manifested in nature in general and in human’s artistic products.
From this discipline springs one of the oldest theories of art, namely the
Harmonic Theory of Beauty, which states that the κόσμος as a whole is a
harmonic and proportional system of beings that manifests these harmonic
qualities through what we call “Beauty”. The work of art is some kind of
“microcosm”, harmonic and proportional in itself, which also takes part in
the cosmic harmonic whole.
In consequence, there must also exist a domain that studies the concept
of number in a “relative” sense (περὶ τοῦ πρὸς ἄλλο), and this domain is
exactly what we could call “mathematical” or “arithmetical aesthetics”. As
we can see, it is not an “reductive” discipline in the modern sense of
mathematical sciences, but a discipline that aims to adequately integrate the
indeterminateness and infinity of nature itself in the domain of human
“reason” (λόγος) and consciousness in general. This is no mystical or
esoteric endeavour in the modern sense of the world, just a natural impulse
of the human mind. The Pythagorean arithmetic and philosophy of art are
“mystical” only in the sense that they operate with an ineffable insight into
the principle of all things, with a “ineffable” knowledge that cannot be
expressed propositionally. But, at a closer view, thus are all kind of human
effort to understand the profound nature of our world and our own being.
Having these in mind, the two preconceptions about Pythagorean philosophy
(in general) and Matila Ghyka’s “mathematical aesthetics” (in particular)
collapse. They are not pseudo-philosophical and mystical conceptions about
art and the world that aim the reduce the complexity of the artistic and
creative process to some obscure mathematical proportions and formulas,
but an effort to grasp that which cannot be grasped, namely the first
principles of things thought from a meontological point of view. Starting
from this understanding of the Pythagorean philosophy, Matila Ghyka
Art and Mathematics in Matila Ghyka’s Philosophical Aesthetics…
builds a philosophy of art which has the concepts of “number”, “proportion”
and “harmony” at its centre.
7. Matila Ghyka’s Pythagorean Philosophy of Art and Contemporary
Ghyka’s main concern is to establish a correspondence between the
microcosmos of the human consciousness and art, a correspondence that
could reveal why and how art manages to impress us and help us express our
deepest insights of the world (Ghyka 1938, 13-25). The means by which
such a task could be accomplishes is of the Pythagorean philosophy of
numbers and proportions, especially, by the understanding of the so-called
“golden” or “divine ratio” (Φ = 1,6180339887…).
Although the number Φ has a long history in the Western philosophical
and scientific tradition, one fact about it usually escapes the modern-day
thinkers, namely that it was called by the ancients a “secret” or “ineffable”
number. Those numbers we call “irrational numbers” were called ἄρρητος
(Plato 1902, 546c) or ἄλογος, which means they were “ineffable” or “secret”
in the Pythagorean sense. They could not be grasped by the human reason
and they fully express the paradoxical relation between nothingness and its
hypostases. “Irrational numbers” encompass the indefinite and ineffable
nature of the world into a “arithmetic” determination supressing its
meontologic character.
Another strange thing about Φ and the irrational numbers in general is
that they can be expressed by means of proportions (Ghyka 2016, 58)
between two “rational” numbers. So, the “irrationality” of the cosmos and
of art lays in the ratio or analogy between two rational entities, by putting
together two “rational numbers” or even “rational arguments” we can
obtain an “irrational result”. This shows that the human λόγος has, in itself,
encompassed some degree of irrationality which becomes manifest through
analogy (or, in our terms, hypostatization). So, the irrationality is announced
by the study of numbers in an “analogical” or “relative sense” (περὶ τοῦ
πρὸς ἄλλο), which is essentially the study of “music” as “arts governed by
the Muses”. This is why art was viewed throughout the history of
philosophy as ineffable and impossible to reduce to a scientific formula.
This view was especially promoted in modern philosophy and still
predetermines our attitude towards art and science. At the same time, this is
the place where the ineffability becomes apparent as the foundation of
every “mathematical” experience we might have, and also of the aesthetic
In meontological terms, we gain insight into the “irrationality” of our
conceptions when we confront with art and notice that there is something
more to it than what we actually perceive and/or imagine. Art acts on our
Hermeneia – Nr. 20/2018 Cornel-Florin Moraru
consciousness as a charm or incantation (Ghyka 2016, 102-110) because it
makes us perceive something that is not there and, generally, doesn’t exist at all.
When we look at a piece of painted cloth and we say this represents Napoleon,
we basically create an image based on the ineffable sense of proportionality,
harmony and rhythm that springs from between the various elements of the
painting. In some sense, looking at a work of art is an “error of perception”
because we see there what our minds construct aesthetically, not what
“really is” there for the everyday consciousness – namely stains of colour on
a cloth. The proportions and analogies between the elements of the work of
art make us create a hypostasis of the “subject” so art basically manifests
the same hypostatic character as the “arithmetic” theory of numbers.
But what we call “ratio” and the Greeks called ἀνάλογον is the principle
from which we can construct, by multiplication, addition or other mathematical
operation, an infinite number of equivalent proportions (Ghyka
1998, 49-53) or, as we might say, hypostases. These strings of proportions
or strings of “analogic numbers” can be viewed as another aesthetical mode
to express the ineffable principle of all things, encompassed in an arithmetical
progression. At an intuitive level, we perceive these progressions as
rhythm (Ghyka 2016, 198-208), which is another fundamental element of
any work of art, not just of music. Architectural works, for example, have
their own rhythm, which is constituted by the repetition of certain element
(Ghyka 2016, 74-75) that could be expressed through a string of numbers
or proportions, just like any other (musical) rhythm. So, if an art moves us,
it’s because these subtle proportions and rhythms which govern the form
and constitution of any work of art.
This does not mean that the artist must necessarily be aware of these
proportions and harmonic rhythms. What we call “beauty” is, in Matila
Ghyka’s view, characterised by the so-called Golden Ratio, a “secret” and
“ungraspable number”, so the explicit proportions need not be manifest for
a certain person’s view. These proportions are hidden in the work and
silently guide our perception, so that we need not make an effort to view a
certain work of art as a work of art. The ineffability of the message of art
reaches us because, as part of the κόσμος, we are by nature capable of
observing its harmony and rhythms. The numbers and rhythms do not
simply present themselves to us, but they evocate images and associations
of ideas and comparations in our souls (Ghyka 2016, 134). In other words,
they are hypostatic.
Because our mind has a “metaphoric” or “hypostatic” nature, we transform
perceived proportions and rhythms into affections, insights and ideas in
natural manner. This is possible because the artistic experience has a
different meontological genealogy and archeology as that of arithmetic theory
and “musical” theory, which we analysed earlier. When we perceive a work
of art, we don’t have a “theoretical attitude”, by which employ in a willingly
Art and Mathematics in Matila Ghyka’s Philosophical Aesthetics…
manner the different μαθήματα we gained throughout our lives, but rather a
“aesthetic attitude”, which automatically creates a hypostasis of the ineffable
in the domain of our “affective rationality”. When the average person looks
at a painting, he does not usually search for compositional elements or
other theoretical items, but for some “understanding” of its own feelings.
We “learn” what’s love by reading Romeo and Juliet and what’s nostalgy by
reading The Odyssey. This means that some kind of “interpretation” of our
own non-judicative experiences (in a scenic sense rather than a cognitive
one) takes place in the authentic aesthetic experience. We could call this
“affective hermeneutics”.
This kind of hermeneutics is the reason why Pythagoreans, along with
Matila Ghyka, thought that numbers produce affections of the soul and
that, to understand these affections, we must understand the true hypostatic
nature of numbers. This means translating “affective rationality” into
μαθήματα or transforming what the moderns called “intuitive thinking” into
“conceptual thinking”, without suppressing its ineffability. Although we
should take these conclusions cum grano salis, until the nature of “intuitive”
and “conceptual” thinking is fully understood into a meontological manner,
they open up a new domain of philosophy by which we can build some
connections between science and art – a very fruitful research horizon for
contemporary philosophy.
Having these in mind, we think that the arguments presented in this
study are sufficient for the inclusion of Matila Ghyka’s Pythagorean
aesthetics and philosophy of art into account as a domain worthy to be
studied and developed by contemporary philosophers. This effort may lead
to another paradigm in the philosophy of art, which combines the
mathematical approach to art with the more “intuitional” one. In this way,
the history and theory of art may reshape their current structure and open
up the path to a meontological theory of art, which pays attention not to what is
expressed by art, but to that which in not expressed, but just consented at
the deep and affective level of our consciousness. This endeavour, as we
may already conceive it, would be a hermeneutical one, whose main aim is
to develop the instruments and mechanisms of what we called affective
hermeneutics in order to grasp in a more accurate way the ineffable consent of
the nothingness, which is hypostasiated into art.
1 ἀλλ’ ἐπειδὴ ηὸ μὲν καηὰ ηὰρ πηώζειρ μὴ ὂν ἰζασῶρ ηαῖρ καηηγοπίαιρ λέγεηαι, παπὰ
ηοῦηο δὲ ηὸ ὡρ τεῦδορ λέγεηαι [ηὸ] μὴ ὂν καὶ ηὸ καηὰ δύναμιν.
2 πολςμαθίη νόον ἔσειν οὐ διδάζκει· Ἡζίοδον γὰπ ἂν ἐδίδαξε καὶ Πςθαγόπην αὖηίρ ηε
Ξενοθάνεά ηε καὶ Ἑκαηαῖον
3 ηὰ ἀπόππηηα ηῆρ καηὰ ηὰ μςζηήπια ηελεηῆρ
4 ηῶν ἄπα δύο εἰδῶν ηούηυν ἐπιζηήμην νομιζηέον ηὴν ζοθίαν
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