THINKING A LOT
ABOUT ONE
One possible aim when teaching History of Philosophy is that the pupils
grasp the birth of philosophy as the collective construction of a kind of discourse with
its notions, relations and rhetorical uses. As originally the philosophic language was not
separated from the mythical-religious, the origin of philosophical terms can be followed
up parting from ritual or technical experiences, from mythical-poetical stories or
political dispositions, from medical or judicial uses. It is interesting that the pupils
verify the relative unity and generality of the philosophical knowledge of that time in
comparison with the plurality and specialisation of present sciences, because I think that
we all agree that interdisciplinarity is one of the aims and advantages of the History of
Science.
The historical knowledge, apart from explaining the past, is useful for
giving an account of the present, making us conscious of the historicity of our own times.
And vice-versa: in the tuition of the History of Science, it cannot be avoided that the
pupils have certain knowledge of the sciences, and therefore, an implicit ideology about
them. They feed beliefs and expectations, they extract values and procedures; in definite,
science is today one of the sources of the meaning of life. That is why I do not think it
is adequate to reduce this discipline to a formal system of statements about objectives
and events, or to present science as a neutral axiomatic system in an aseptically ideal
world. It must be proven that in past times and at present, the construction of knowledge
is work and chance, accumulation and crisis, transmission and controversy, that happens in
a real society, with a determined kind of politics, economy, and legality, and diverse
ideologies.
1. Language and
Science
Some time ago I came across one of Heisenberg’s texts (1955),
where he exposed the possibility that the ultimate accessible matter to the human
knowledge was a wide group of elementary particles, and not only a few, as most of the
scientists seemed to expect. This made me reflect upon the presence in the scientific
language of certain concepts and relations, that have been of decisive importance in the
history of thought; beginning and end, the one and the multiple, cause and consequence,
body, shape, limit, space and vacuum, chance and necessity, to be and to become, truth...
We use them as the main beams in the scaffolding of thought, and because of their
frequency, they have generated automatism of thought, such as the implicit belief that the
fundamental is not only simpler than what derives from it, but it tends towards absolute
simplicity. This preference for the simple is not specific of science; we also come across
it equally in mythology and philosophy. In most of the mythological cosmogonies, the known
universe originates in a sole or double divinity, even in polytheist religions. The first
cosmologies of the Greek philosophers are unitary and we have to wait until a century
later for the pluralist physical theories to appear.
The physics of the twentieth century presents, fortunately, quite a
problematic appearance. The theories of relativity and quantum mechanics, as well as the
impossibility to reconcile them, has produced outstanding changes in the perception of the
universe, that has moved on to other fields of culture. This is why it is not strange that
in Nature and the Greeks, Schrödinger states that he had studied the pre-Socratic
philosophers in order to understand current science better. That Popper calls Einstein
"a four dimensional Parmenides". That the scientists rediscover
old philosophical themes, such as Bohm with his distinction between explained order and
implied order, or Wheeler when he asserts that in quantum mechanics no phenomenon is
real until it is an observed phenomenon. The advances in microscopic physics even seem
to have diluted the concept of reality: But the atoms and particles are not so real;
they constitute a world of potential more so than things or actions (Heisenberg). In
the macroscopic field relativity wrecks the idea of absolute simultaneity and the actual
concept of time evaporates: past, present and future are an illusion (Einstein), the
world simply is, it does not happen (H. Weyl). Some scientists maintain a realist
epistemology and they assert that the theories of physics and mathematical formalisms
describe the world exactly how it is; others, more cautious, adopt an instrumentalist
position: it is wrong to think that the job of physics is to discover how Nature is.
Physics deals with what we can say about Nature (Bohr).
It is normal that with this problematic panorama many scientists set
their hopes on a future Theory of All, or in a unified field, or the superstrings. For the
majority of physicists the Big Bang theory is probably very true and with time it will be
perfected. Some critics disagree because of physical or epistemological reasons -like
Hoyle or Feyerabend- and they think that this theory is a mythical-scientific story
devised upon the base of local laws of the matter in our area of the universe, plus some
observations, multiple extrapolations, and a few predictions. But the big style
speculative physics does not stop because of scepticism. In consequence, the university
students of Science believe in it almost like the theorem of Pythagoras, despite its
predictive mistakes in important matters: such as the density of matter in the universe,
the concentration of each atomic element or the temperature of background radiation,
despite it does not explain the existence of X and gamma isotope radiation, despite that
the distribution of the galaxies contradict the postulated isotropy of the universe or
despite of the doubts of the range of validity of Hubble’s law that emerged from the
observations of certain galaxies. I believe there is an interesting parallelism between
the Big Bang and the Milesian cosmology, whose intellectual dignity compares favourably to
other mythical or scientific cosmogonies, like the cosmic Egg of the Orphics or the
Cartesian vortices. In effect, as with Tales the mythical-philosophical cosmology starts,
with the Big Bang the mythical–scientific cosmology begins.
Quantum mechanics have been a fertile field for plural and challenging
interpretations: Those of Copenhagen, of Einstein and Schrödinger, Bohm’s hidden
variables...Certain basic questions are still in the air. ) Should quantization be
interpreted as a physic discontinuity or as a mathematical discontinuity? ) Is it an
experimental or a statistical question? ) What does it mean to be a wave-particle duality?
) Are the properties of matter mere statistical potentials? ) What meaning has causality
in the quantum sphere? ) Does the principle of locality work there or is there action at a
distance? ) Is the vacuum a field full of energy or pure geometrical space? ) Does the
uncertainty principle imply the irruption of the observer’s subjectivity upon physics
objectivity or not? These subjects have made certain paradoxes and mental experiments
popular, such as the simultaneity and the trains, Langevin’s twins and the elasticity
of time, Schrödinger’s cat, the electron and the two slots... Concepts with a
long-standing tradition in our culture have been affected: space, time, reality, matter,
continuity and vacuum, cause and determinism, subject and object...
The extension of science is a factor that contributes to the
proliferation of alternative models, while the increase in communication amongst
scientists favours the reduction and concentration of alternatives. It is usually said
that the majority of physicists are only interested in their work and they are insensitive
to these debates. Although, for those who do appreciate theoretical speculation, this
situation of ambiguity allows to think up hypothesis and suggestive and extravagant
models, such as the many worlds of Everett or Wigner’s reinterpretation of the
macroscopic /microscopic difference like matter/ mind, meanwhile other physicists reclaim
the urgency of a linguistic-conceptual change. This is the case of Bohm, with his
proposition to substitute the linguistic atomism of occidental languages for a new
language that describes reality in terms of a flowing totality, the reomode. In more
modest terms Bell has written: It well could be that a real synthesis of the quantum
and relativity theories does not require a technical development, but a radical conceptual
renovation.
The development of theoretical physics has favoured the development
of novel mathematical techniques, like matrix algebra or tensor calculus, as well as
theories where physics and mathematics almost completely overlap, such as the superstrings
and the reference symmetries. At the same time the physicists are accused of frequently
turning to mathematical artifices ad hoc without theoretical meaning in order to resolve
anomalies. It has been doubted, for example, the logical pertinence of the re-
normalisation techniques in the quantum theory of fields to avoid that certain variables
take infinite values.
This panorama of ambiguity and stable crisis is not exclusive of
present day physics; in each science it has its own nuances. In mathematics there is a
less delicate situation. There we have the self-referential paradoxes of the set theory or
the theorem of incompleteness of Gödel. The intuitionist mathematicians turned down the tertio
excluso principle and the concept of infinite, declaring themselves supporters of a
purification of the activity and the language of mathematics. The theorems obtained by
computer have questioned the proof rule.
Some historians of mathematics have tried to connect these with the
structure of language. In the same way that the occidental philosophy of Being has been
related to the predicative structure of Greek, or the philosophy of universal concepts
with the importance of the nominal function in the Indo-European languages, Chinese
algebra has been related with its ideograph writing and the peculiar character of its
mathematics with the importance of the verbal function in the Chinese language, and the
relation between the algebraic character of Hindu mathematics and the linguistic writing
of Sanskrit.
The need to adapt the language to the new results in science has
nothing to do, in my opinion, with the illusion of creating a new perfect language,
neither in algebraic terms (Lulio, Leibniz) nor of rational morphology (Esperanto,
Volapuk). Language is a social construction and therefore, it is historically ambiguous,
as it implies the experiential subjectivity of many people and its transmission in time.
So, the process of adaptation between language and science should be a permanent job whose
effects will only be perceived in the long run.
2. Our ancestors
and the numbers
Several ethologic studies have proven that animal species much older
than mankind had the capacity of distinguishing amongst small amounts. We can suppose that
this was a very useful ability, for example, to control the litters of the species that
had a limited number of offspring in each breeding season. The studies of our closest
relatives, the primates, confirm their capacity of recognising small amounts and simple
geometrical shapes when they are given the appropriate stimulus. We can then start from
the hypothesis that the hominids of two million years ago, who it seems lived as
hunters-collectors in stable groups of 30 to 50 members, were capable of recognising each
other as members of the same community without needing to postulate that they had an idea
of the quantity of persons that formed it. In effect, even today our capacity to perceive
exact quantities of similar objects is very limited. At a glance it is difficult to
distinguish if there are six or seven donkeys in a field, four or five flies buzzing over
our heads. This reduced quantifying perceptive has left its mark, even in recent times, in
languages where the words for quantity were limited to one, two, three and many. I believe
that we could reasonably embrace the conviction that the advances in the distinction of
quantity could not have happened before the development of language. A process which the
present day palaeontologists, when studying the evolution of the phonic apparatus of the
hominids, calculate that it began 400,000 years ago maximum and 100,000 years ago minimum.
All origin is mythical; to seek for the origin is a hazy dream in which
you have to be satisfied with approximations. It is almost sure that we will never know
the origins of articulated language or numbers, but I find hard to believe that the humans
who carried out ritual burials 70,000 years ago, had not already developed certain notions
of quantity more complex than those of today’s primates, although there is no proof
of this. The oldest fossils that show probable numerical uses are the bones of Lebembo and
Czechoslovakia, of more than 30,000 years ago, they have distinctive groupings of notches
and they seem to have been used as quantity files. Some historians refuse that this
graphic system of quantity representation is strictly numerical, as there are no names of
quantity, but I think that that is an example of logocentrism. As recent investigations
point out that the stockbreeding activity could have began in Africa 35,000 years ago, it
has been postulated that the first development of numbers was linked to the need of
counting the herds, and the later emergence of agriculture, 15,000 years ago only
reinforces this.
The hypothesis is plausible, even if it looks like an extrapolation, of
what we know about the appearance of writing in Mesopotamia. The oldest boards with
writing, of cuneiform type, come from the Sumerian civilisation and have been dated in the
XXXIII century. They only contain numerical marks and signs that represent sheep, goats,
sacks of cereal... It is believed that writing appeared halfway through the fourth
millennium, both in Sumeria and in Egypt, so that the state could register quantities of
agriculture possessions and heads of stock, for administrative purposes. In this way, the
invention of writing would be linked to the constitution of the ancient empires,
characterised by the demographic growth, urban development and the centralisation of
power.
The operation of counting by accumulation of inscriptions upon an
object can be considered as a base 1 system. It later evolved with the appearance of base
two and base five. It is unquestionable that the relief of number 2 is linked to sexual
division. Sex as a metaphor was very powerful in the ancient civilisations; for example
the historians have collected many myths of sexualisation of metal work, where they
conceive the earth giving birth to the metals from its entrails; amongst the Pythagoreans,
the letter delta symbolised the arche geneoseos, generating principle, triangle-vulva. Of
the process of counting by pairs many examples have been found in Central African,
Polynesian, Amazonian and Patagonian civilisations. Their languages conserve in the name
of the numbers the mark of their construction as a base two system.
Nobody has denied that the use of base 5, very extended, and its later
amplifications to base 10 and 20, has its origin in the fact that the calculus was manual.
Several hypotheses have tried to explain the Sumerian choice of such a high base as 60 for
its numerical system. One of them considers it was the result of the fusion of two
civilisations, with numerals in base 5 and base 12. This base 12, that is still used by
Europeans in restricted areas, has a darker origin, because there is no evidence of the
previous existence of a base 6, and it could come from the relation between the lunar year
and the solar year.
The extension and speed of its diffusion could measure the success of
an invention; it is sure the developments concerning the measurement of quantity, spread
quickly because of their utility. There are signs that already during the fourth
millennium a system of measurement had spread through different civilisations of the
Fertile Crescent elaborated by some Indo-European civilisation and it was applied to
different techniques, like the manufacturing of bricks with proportional measures 1:2:4.
In the same way, the success of the cuneiform writing, adopted by the Sumerian’s
neighbouring villages, reflects the importance of being able to register quantities in an
undeletable way.
The psychologists tell us that the first separation a child establishes
-and it takes months to do so- is between me and not me, between their body and the
exterior world. In the older civilisations, the main separation is established between us
-the group- and the rest. To count is to join: implying recognition that different things
are also the same; together with otherness union appears, a game of difference and
repetition. To count is also to separate. To separate what is different, to unify what is
the same. ) How would the primitive shepherd see the same and the different when counting?
) To count a herd, would he first see the same- that they were goats- or the different
– a group of individuals? This method of separation and union after millenniums of
perfectionism would lead to Plato’s method par excellence: generalisation and
division.
Poincaré and Brouwer thought that the conception of whole numbers was
a result of perception, of the intuition of sensitive things; Russell, on the other hand,
considered that it was innate to the human mind, part of its implicit logic. Studying
empirically the intellectual development of the child, Piaget reached an intermediate
conclusion: From the experience of operating with precise things the child manages to
develop the formal scheme of the number, beginning of the conservation of quantity.
3. A world of gods
The nineteenth century anthropologists wanted to see in totemism the
human answer to the polymorphism of the world, a way of selecting human beings and natural
phenomena, elaborated depending on their specific interest for the human group, upon which
to articulate the corresponding rites to symbolise mutual relations; rites to which the
myths give the verbal sense, a way of conservation and transmission. To that point of
view, the twentieth century anthropologists have superimposed a vision of totemism as a
symbolic system to structure features of social interest, such as the relations of
kinship. Confronted to the vast plurality of the natural beings and the social
relationships, mankind responds by developing a taxonomic spirit that tries to organise
reality schematising it.
Maybe fear, as the prime emotion of man before a world that he barely
controls, justifies the deification of those beings and natural phenomena with which they
admit having a relationship of dependency. Therefore, when converting necessities into
rituals, man sets out to magically dominate those relationships by means of repetition and
keeping a historical memory of it. The animism, the most ancient phase of religious
thinking, is characterised for attributing consciousness and will power to all that
exists. This projection of the human over the world indicates the value that man gives to
his own perceptive capacity and his capacity of acting as basic ways of relating with the
other. This recognition of the difference human / non human, that at the same time
postulates a community of feeling and wanting, must have been accompanied by a sensation
of inferiority of man with regards to Nature. It must have been the invention of
stockbreeding and agriculture, as well as the development of metalwork techniques, that
meant a change of direction in the symbolic relations of man with Nature. The
domestication of animals, plants and metals, reduced the dependence of man on his habitat,
encouraging his narcissism. The societies with the oldest history that we know, the
Mesopotamian and the Egyptian, at the end of the fourth millennium, were in an advanced
phase of transition from animism to polytheism. Their Gods were not yet totally human,
their bodies were still partially zoomorphic; when the process is finished the animal and
vegetal features will have transformed into conventional symbols, which accompany the
iconographic representation of totally human Gods. It is the step that goes from the
serpent Tiamat, the falcon Horus, the monkey Hanuman, or the winged angels, to Dionysus
with his goblet and his grapes or to Hefesto with his anvil. Polytheism means an increase
of abstraction with regards to animism: The event has unfolded into agent and action. The
God is no longer the ibis, the palm tree or the river; now there is a god who gives
presents with the arrival of the ibis, another who makes the palm tree give fruits and
another who makes the river grow. The technical manipulation is projected upon nature and
changes the symbolic interpretation of the world.
The first known attempt to reduce polytheism to monotheism,
establishing the cult to Atón as the only God was attributed to the pharaoh Amenhotep IV
–XIV BC-. Although this attempt failed it was an early sign of an intellectual
process that the Greek philosophers and the Jewish priests would culminate in the VI
century BC. The appearance of monotheism is usually linked to regimes of dynastic
monarchy, where the own deification of the monarch operates. This symbolic separation
between the king and his subjects contributes to justifying that the power with which he
rules over them is the same power, alone and total, with which the sole God leads the
course of events. We read in Homer: The command of many is not good, the command of one
is enough.
4. Mythology,
poetry, philosophy
Plutarch says that religion is the contemplation of mysteries and
philosophy is the contemplation of the eternal. The kind of discourse baptised as
philosophy is born within the mythical-religious discourse and will slowly mark the
distances; that is why there are abundant religious elements in the pre-Socratic thinkers.
The religious experience can be expressed in the form of mythological stories, oracular
sentences, and chanted or recited poems or in staging like the tragedies. The poetic
expression will remain in some of the first philosophers, such as Parmenides and
Empedocles; on the other hand the Ionian physicists wrote in prose. Between the poetry of
Homer and the prose of Tales there is an enormous distance that goes from the oral world
to a world with a written memory; because of this it has been said that the philosophic
discipline was born as a new form of writing.
In Greece, since the VIII century the religion of the Olympic gods has
a didactic monument: the Homeric poems. When Homer invokes the Muses I could not even
name the throng of heroes who went to Troy he is pleading for divine inspiration to do
his job, reeling off the tale of a past time whose actions should live on as a lesson in
the memory of the Hellenic civilisation. When Hesiod, towards 700 BC, also reclaims the
help of the Muses to compose his Theogony he is questioned by them like this: We know
how to say many false things that sound like truths, but we also know how to tell the
truth when we want to. In this way, Hesiod is keeping his distance from Homer: his
intention is not to outline an emotional and constructing tale with gods and heroes, but
to give a logical representation of the big realities of Nature symbolised by the gods:
Earth, Eros, Heavens, Night, Day... This connection of truth with the divine stays as a
constant of the pre-Socratic philosophy, with rare exceptions –such as the Sophists
and Democritus- and it will live on in Plato’s Academy.
The cosmogony of Hesiod drinks from Babylonian, Hittite and Hurrita
sources. With his Theogony he tries to explain the universe as a Whole, whose becoming
escapes the power of man, who has not got another choice than to submit. Hesiod situates
the origin in the Chaos; although there is no unanimity about its meaning, today it is
interpreted as the empty space, underlying abyss that separates Earth from the firmament,
or, the act of the separation; the version of Chaos as the prime disorder was a stoical
addition not in accord with the spirit of the work. When the action of Eros, dynamical
principle, unifies the earth with the heavens (Gea and Uranus) the cycle of divine
generations begins. In this work a monotheist spirit can be felt whose purpose is to show
the power of Zeus as king of gods and lord of Nature. These three theogonic elements, the
search for origin, the dynamism of Eros and the monotheist tendency, will continue to be
present in the cosmologies of the pre-Socratic thinkers, who in general, have certain
common elements.
-The analogy between the Cosmos – celestial order – and the
polity, organised by laws. The word cosmos comes from the verb Kosmein, to put the army in
line for combat, which indicates a projection of the political order upon the heavens.
-The hilozoism, the belief in that all what is natural is alive; no
differences are established between animate world and inanimate world. The physis is the
set of the living, field of reproduction and generation, live totality.
-The consideration of the natural events as intentional actions, that
show at the same time an animist survival and a projection of human psyches upon Nature.
THE
PRESOCRATIC PHILOSOPHERS
Thales of Miletus
(approx. 625-545).
It is not by chance that philosophy emerged to the east of Greece,
in Ionia, area that was in permanent contact with oriental cultures and with Egypt. Thales
most famous thesis, that the world is made of water, has been related to Babylonian myths
(Marduk and Tiamat), Egyptians (the god Nun) and Hittites, and with archaic Greek myths
about the god Ocean, that still echo in Homer and Hesiod. But in Thales it is not simply
about that the Earth and the heavens have emerged from the original waters. His
rationalising stamp consists in understanding the physis as a dynamical process, starting
from an original stadium. His geniality lies in reducing the multiplicity of what exists,
with its wide variety of differences, to the union of water. Here it is not about going
from what is similar to the one, as when we give names, but from what is different from
the one. A unit that is not hidden, but one that shows its importance among the rest of
things. The water turns into a patron of change, that of which all comes and to which all
returns. Thales could not have explained the exact mechanism of the transformation of
water into the rest of things, because there would have been some traces in later authors,
and it is possible that he just pointed out the dynamism of water, that can be observed in
the rain and snowfalls, or in the tides, which he attributed to be the cause of
earthquakes. When Thales states that all is full of gods, he is making the physis
independent of the tutelage of the Olympian gods, when he says that each thing has within
its self –and not outside-the principles of its nature, of its dynamism; with that he
reinforces the union and autonomy of the physical beings.
Some authors present Thales as a practical traveller who during his
visits for commercial reasons to cities of Mesopotamia, Egypt and Phoenicia, learnt
several techniques that he introduced to the Ionians, such as to predict a sun eclipse
with the Babylonian astronomical tables, to measure the height of a pyramid comparing the
length of its shadow with the shadow of an object of known height, to measure the distance
of a boat at high sea from the harbour and to sail following the stars. But for the
majority Thales is, above all the inventor of the science of geometry. Being too
eurocentric, the historians place Thales at the beginning of the science of mathematics,
because they accept as its definition the axiomatic-deductive model, that began to take
shape in the Hellenic culture from Enopides to Euclid. The demand of abstraction,
demonstrations, general rules and exactitude, supposed the devaluation of pre-Greek
mathematics, practical, algorithmic, particular and approximate. However with Thales
mathematics begins its transition from the empirical to the theoretical, it goes from its
technical use in the polity to be a speculation about the world; going beyond its mere
magical and technical manipulation it begins to elaborate a new image of nature.
The theorems traditionally attributed to Thales, seem to indicate that
his demonstrations were based upon the symmetry or superimposition of figures. I will only
stop in the theorem of the division of a circle by its diameter in two equal halves. In it
the one and the multiple is presented in two ways:
-The centre is a unique point with regards to the multiplicity of
points that can be determined in the circle; it is the point that defines, precisely,
which of the possible strings are diameters. Thanks to the centre all the diameters are
the diameter.
-All the circles are the circle; the plurality of the size of the
circles is reduced to the union of its figure by the definition of its properties.
Two necessary conditions to be able to consider as a demonstration that
jump from the many empirical circles to the idea of a single circle.
Thales advised the Ionians to have only one seat for the political
assembly and for it to be in Teos, polity situated in the centre of Ionia. We can
appreciate in this a return trip from the theory to the polity, an applied example of the
virtues of theory.
Anaximander of
Miletus (approx. 610-545).
In Anaximander’s cosmology the one is presented as a Whole,
that he calls Apeiron, term that Homer applied to the endless sea and land that one never
completely covers. It seems that with this term he wanted to signify both the unlimited
and the undetermined. Unlimited means absence of spatial limits, of external shape.
Undetermined means the absence of internal limits, of separations. That is why Apeiron
cannot have qualities that only belong to concrete things. In this sense, the Apeiron at
the same time as a Whole is a Nothing, being a theoretical concept constructed by the
negation of the sensitive qualities. Maybe it is the first pre-figuration of what
Aristotle would call raw material, with the exception that this only had a virtual
existence.
The main difference with regards to Thales lies in not having chosen an
empirical element as a principle of the physis. The Apeiron is the hidden surroundings, an
inexhaustible reservoir from where all the present comes; it is immortal and
indestructible, so it is divine; it has not been generated but it generates everything.
The cosmic process begins through the separation of Fire and Mist (damp air). As
Anaximander asserts that the Apeiron embraces all the things, we must understand that the
separation takes place internally. Some authors have believed that that separation was
produced from a rotation movement of the Apeiron, although there are no textual
indications of this. The hypothesis is plausible, because the majority of the
pre-Socratics considered that movement was a fundamental principle, something that had
always existed and there was no need to explain its origins because it did not have any.
Another possibility is that Anaximander conceived that separation as the beginning of
movement, which would approximate the Apeiron to the Chaos of Hesiod.
Ancient commentators attributed to him the doctrine that the Apeiron
generates many worlds, although there are not any textual references either. It is
compatible and coherent with the notion of Apeiron, the idea of multiple successive
worlds; less reasonable is the idea of multiple coextensive worlds, defended by other
commentators.
While Thales thought that the world was flat and the universe
hemispherical, Anaximander seems to have been the first Greek that speaks of a spherical
universe. Despite that he conserved the archaic idea that the earth was cylindrical. His
interest in mathematics can be seen in his celestial map, pioneer in Greece, where he
proportionally ordered the distances to the Earth from the rings of the sun –27
terrestrial diameters-, from the moon –18-and from the stars –9-.
I would like to focus the attention on an idea that I consider of
mathematical origin and extrapolated to the physis. According to Anaximander the Earth
lies fixed in the centre of the universe because it is in equilibrium. Here appears the
idea of the centre as one different to the rest of multiple places, as we already saw in
the geometry of Thales. It will be Archimedes, three centuries later, who will develop
this fertile idea linking geometry to the statics of solids, anticipating with the notion
of centre of weight, the idea of centre of gravity.
Anaximenes of
Miletus (approx. 585-520).
The cosmology of Anaximenes situates the Air as the main principle of
all that exists. As in the case of Thales in water, there also precedents in mythology,
for example the Theogony of the Phoenician Sanconiaton, that took as origin the unlimited
eerie and windy air, and with which some want to relate to the Chaos of Hesiod. Anaximenes
seems to have taken certain ideas from his Milesian predecessors; his Air is, like the
Water of Thales, an empirical element, abundant and necessary for life; on the other hand,
he states that it is unlimited, like the Apeiron of Anaximander, and that it is invisible
when it is very close, although it manifests itself by heat, by dampness and by its
movement. The importance of Anaximenes lies in having given an explication of how the one
–the Air- converts into the multiple, the things. He sustained that through a
reversible process of rarefaction and condensation, through which Air rarefies into fire
or it condenses progressively into wind, vapour, clouds, water, earth and stone. This
dynamical process is eternal, all is born from the Air and can return to the Air. But, he
does not seem to have imagined alternative or successive worlds.
But, he made it obvious the need of Air for everything alive. He
considered that the world breathed Air, like natural beings while they are alive. Already
in Homer we find that breath (pneuma) is the vital soul of man. The Air, as it is the
original and eternal, penetrates everything, it is the Soul of the World, as the dampness
was for Thales. It keeps the cosmos together the same as the soul confers union to the
body. Taking up Anaximander’s idea, he places the Earth in the centre of the cosmos,
which he conceives flat and not very thick, floating in the Air. The geometrical-physical
idea of equilibrium has disappeared, probably substituted by the empirical observation
that extensive, flat and thin bodies, as they offer less resistance to the wind, are more
stable.
Pythagoras (approx.
570-505) and the Pythagoreans.
It is believed that he was the first to call himself a philosopher;
although he was an Ionian, because of political reasons he emigrated to Croton, in the
south of Italy, where he founded his brotherhood towards 525 BC. There is not unanimity
about the dates of the different Pythagorean doctrines, because subsequent commentators
speak, in general, of the Pythagoreans. Here we are referring to the doctrines considered
the oldest, in any case before the expulsion of the brotherhood from Croton towards 450
BC.
The Pythagorean cosmology is based on two principles: the bodies and
the vacuums, this is, the limited and the unlimited. At this time they still did not
distinguish clearly between air and vacuum, therefore, the Milesian tendency of the
unlimited is clear; on the other hand, to consider the bodies as what has limits, and
therefore, a determined figure, takes us immediately to geometry. For the Pythagoreans the
Earth is round and has two movements: translation and rotation.
In my opinion, it was the discovery that mathematical truths were
eternal truths, what led them to consider mathematics as a sacred knowledge that reveals
the essence of the world. From there his famous thesis is born: The things are numbers.
The mysticism of the Pythagorean brotherhood was the mathematical knowledge. It seems that
its origin was with Pythagoras’ discovery that when monotonic strings were plucked
they emitted harmonious sounds if the lengths of the strings kept between them proportions
expressible in numbers, like 1:2, 2:3, 3:4. This caused an interest in mathematics that
materialised in the development of arythmo-geometry, stage in which we could ascribe the
demonstration of the Pythagorean theorem, the polygonal numbers and the study of odd and
even numbers.
The Pythagoreans were the first atomists; if all things were numbers
they had to be formed by units. They called them hňros and they were at the same
time arithmetical, geometrical and physical units .The properties of each thing depended
on their quantity and spatial disposition. There were no physical differences, among the
hňros, they were all equal. The figurative numbers constitute an attempt to express a
geometrical reality in arithmetical terms. When, in the first half of the V century BC,
the Pythagorean Hispaso of Metapontum discovers the irrational magnitudes, this ruins the
arithmetic cosmology, although you could operate in geometry with such magnitudes, you
could not express them in numbers. Maybe this is the starting point of the transformation
of the mathematical atomism of the Pythagoreans into the physical atomism of Leucippus.
If we observe the Pythagorean table we do not only see the prominent
places that are occupied by the one and the multiple, but we can connect them with other
pairs of opposites.
Limited-Unlimited
Odd-Even
One-Multiple
Right-Left
Masculine-Feminine
At rest-In movement
Straight-Curved
Light-Dark
Good-Bad
Square-Oblong
The superiority of the limited over the unlimited is that of the Earth
over the Air, shape over the shapeless; every natural being has a union of shape, the air
has no figure.
The number is made up of two elements: the odd and the even. The
primacy of the odd derives because the even, by splitting, can be reduced to the odd, but
not vice versa, so the odd is simpler.
For the Pythagoreans, the One is not a number, but the beginning-arjé-
of the numbers, their generator. It is neither even nor odd. Numbers are born from the
One, as in Anaximander things are born from Apeiron. Thus, the union is superior to the
multiplicity formed by units.
Rest conserves the union of the thing, but movements can be multiple.
Straight things have only one shape, curved things many. The square and the quadrate
numbers have only one shape, the rectangle and the oblong numbers, many.
There are certain indications that allow us to suspect that the
Pythagoreans adored a sole deity, under the name of Apollo, the most adequate to their
mathematical cosmovision. It is known that the brotherhood believed in reincarnation and
it is considered that Pythagoras was the first Greek to postulate the immortality of the
individual soul, an idea that appears in other cultures at the same time, in Persia with
Zoroaster and in India with Buda, without being able to prove a means of diffusion in one
way or the other. The mysterious cults of that era, like that of Eleusis, are linked to
the traditional cycle of death and renaissance, as can be seen in the myth of Dionysus,
torn to bits and scattered, from whose remains life sprouts again. The rationalisation of
myth in the Pythagoreans results as that life is born from death and death is born from
life, in the same way as opposites generate each other. According to the oldest Hellenic
notion, the individual soul is the portion of the Soul of the World present in each living
being. For the Pythagoreans the individual soul is the harmony of the body. The Soul of
the World is to the macrocosms the same as what the individual soul is to the microcosms
of the body. There is an advance towards the differentiation of the individual souls,
necessary condition to articulate an ethical discourse in which the guilt and the
responsibility are no longer only collective. The idea of personal immortality is the bait
that gets the emotions moving in pursuit of the road to salvation taught by the
brotherhood. To reach excellence as a man means to liberate the soul from its destiny of
successive reincarnations and obtain the fusion with deity. Therefore, Pythagoreanism
implies a double change with regards to traditional religion: It progresses from the
multiple gods towards the one deity and the Soul of the World is multiplied in the
immortal individual souls.
Xenophanes of
Colophon (approx. 570-477)
Of this Ionian poet, who emigrated when young to Sicily, are very
well known his fragments of philosophic theology where he refutes the anthropomorphic
ideas about the gods. God is One, motionless, and all of him sees, thinks and hears
without need of eyes, mind or ears, and tireless he shakes everything with the strength of
his thoughts. It seems that he defended, related to Anaximander’s idea, that the
multiple worlds possible were the equal between themselves.
A peculiar feature in his cosmology is the assertion that the sun is
new every day; it is born every dawn in the East because of the accumulation of igneous
vapours and it advances towards the West in a straight line until it disappears; the
distance makes its apparent movement circular. In this way, Xenophanes separates the union
of the sun in the multiplicity of its daily appearances.
Heraclitus of
Ephesus (approx. 540-480)
Heraclitus represents a new kind of philosopher; his prime interest is
ethics and his physics doctrines are subordinated to his moral attitude. With him, the
lone thinker appears, who disdains most of this fellow human beings because they prefer to
live according to pleasure instead of according to knowledge; the sole teacher opposite
the many ignorant. His method legitimates this stance: in order to find the truth about
the world, Heraclitus does not scrutinise the sky: he investigates himself. His main
thesis is that there is only one rule that governs the world, the Logos, which acts
according to measure and proportion. Union of natural law that contrasts with the
multiplicity of human laws. Supposedly the thought of all men is able to know the Logos,
but it is not easy: nature likes to conceal itself, the one reality hides behind its
multiple manifestations and behind the multiple names that are given to these. Each man
believes to have his own knowledge without realising that the reason is common: the truth
is only one, everyone’s, the mistakes are multiple, and those of each person are
their own.
The good and the bad, the just and the unjust, the beautiful and the
ugly, the day and the night, are opposites only in the appearance of speech; that is how
every pair of opponents conceals their identity. The real, the diké, the way
things are, always manifest themselves as opposition, discord, war, that is how Logos
rules the evolution of nature. Real union and evident duality.
For Heraclitus there is not an original stadium of the cosmos different
from the present one. Fire is the Soul of the World, the Logos incarnated in the Physis,
its strength that transforms everything. It is eternal and unlimited. The human soul is
made of fire. Under the plurality of the things and the bodies lies the union of
Fire-Logos.
Alcmaeon of Croton
(approx. 530-470).
This doctor, although it is believed that he was not a member of
the brotherhood, was greatly influenced by the Pythagoreans. His doctrine is clearly
dualistic: Most of the human things go by pairs. His fame comes from being the
first to distinguish between perception and thought, and for his theory of health as the
equilibrium between multiple pairs of opposites: heat/cold, wet/dry, sweet/bitter... In
three of his fragments we find arguments related to the one and the multiple:
-The truth is only at the reach of the gods; men must be satisfied with
their opinions and speculations.
-To possess the capacity of comprehension distinguishes the human
species from the rest, that are only provided with perception.
-Men die because they are not capable of putting together the beginning
with the end. According to Alcmaeon the heavenly bodies are divine because they have a
perpetual circular movement; man is mortal because his soul does not manage to perpetuate
its movement. Here we see once more the use of the circle as union, which has neither
beginning nor end.
Parmenides of Elea
(approx.520-445).
Parmenides was educated within the Pythagorean tradition, although his
famous Poem constitutes a direct attack to the fundamentals of that philosophy. He
will reject dualism and he will elaborate a monist system; of the two Pythagorean
principles –the bodies and the vacuum- he denies the existence of the second and will
give a new shape to the first. In a way similar to Heraclitus, Parmenides is a wise man
illuminated by the deity that warns men that they live in mistake. For the first time he
sets out with full rigour the problem of reality and appearance. The reality is only one
and to discover it demands being in possession of a method of knowledge; the appearances
are multiple and they are constantly present before us. The reality is only accessible to
rational thought, whilst the world of appearances can be grasped by means of our senses.
In his Poem he explains first the Way of the Truth, the only one possible for knowledge,
and then the Way of the Mistake, a justification of the appearance of the cosmos.
In opposition to tradition and perception, the method of Parmenides is
the logical deduction, in particular, the so-called indirect demonstration or reductio ad
absurdum. It is a question still debated if he took this procedure from the Pythagorean
mathematicians or vice versa. His doctrine begins with a dogmatic assertion: what is is
and what is not is not; in other words, the real is asserted and the vacuum, the nothing
is denied. But, not all that exists is; we must distinguish between what is –the
Being- and what seems to be, but is not Being: appearances are misleading. From this
thesis, by reductio ad absurdum, he will prove the properties of what is: it has not had
genesis, it will not have end in time, it is unique, indivisible, homogeneous, continuos,
immutable and immobile, similar to a sphere. Some of these properties belong to the upper
half of the Pythagorean table of values; others like the homogeneity and continuity seem
to be directly aimed against Anaximenes and his conception of the physical change as
condensation and rarefaction. But what is important is to know what Parmenides is
referring to with his expression what is, or, the One.
The answer is found in his insistence that being and thought are
equivalent. The One is the reality-thought that underlies the world of appearances. As it
is thought this reality is only known through reason, as it is physical this thought is
the fundamental of nature. And although it does not appear in the conserved fragments, all
leads us to believe that this thought rules nature, it is a rational version of the Soul
of the World.
Only a few fragments are conserved of the Way of the Mistake. For
Parmenides the world of appearances is also ruled by need, it has its own sense. To
explain its credibility he thought up a dualistic cosmology: nature is made up of two
principles, Light-Fire and Darkness-Night. From there he retakes elements of the Ionians
and Pythagorean cosmologies.
In summary, in the philosophy of Parmenides we find the Union of Being
opposite the multiplicity of the appearances, the union of reason opposite the union of
perceptions, the union of truth opposite the diversity of opinions, the union of eternity
opposite the plurality of the future.
Zeno of Elea (approx.
490-430).
Zeno, disciple of Parmenides, in consonance with the indirect method of
his master, preferred to defend Parmenides’ doctrine by attacking the doctrine of his
rivals, specially, Pythagoreanism. Instead of claiming that the Being is one and immobile
he prefers to expose the contradictions that are deduced from asserting that the Being is
plural and mobile; when he proves the second he proves the first. Plurality and movement
belong to the world of appearances, to the sensitive experience and not to logic thought,
to reason. His arguments are called paradoxes because they culminate in contradictions
related to the unlimited.
The paradox of the grain –a multitude of grains make a noise when
they fall, a single grain does not,) is a grain sonorous or not? – Using the
opposition between union and plurality to attack the sensitive knowledge, of which we
cannot rely on, because it is contradictory.
The paradox of space –space does not exist because if it existed
it should be in a space and this at the same time should be in another space, and like
this successively– rejecting the Pythagorean principle of vacuum indicating that if
there were a space there should be unlimited spaces, which is absurd.
The paradoxes against plurality are aimed against the Pythagorean
doctrine that says that equal and indivisible units (hóros) form all things. Zeno
demonstrates skilfully that when considering at the same time the hóros as indivisible
physical units -with dimension-, arithmetic –as a numerical quantity- and geometric
-as dots- results in insurmountable contradictions. It is very probable that the logic of
Zeno has forced the Pythagoreans to substitute their mathematical horism for a physical
horism, where atomism comes from.
The paradoxes against movement try to demonstrate that this cannot be
thought about logically, and therefore it is not reality, but appearance, because they
produce contradictions both if you think that space is something continuous –the
first two- or discontinuous –the last two-. The paradox of the stadium states that it
is impossible to cover any given distance, because first you have to cover half the
distance, then half of the half left, and so on successively. As we can see Zeno uses the
dichotomy to convert a unit –any distance- in an unlimited multiplicity of parts. In
effect, who considered space as continuous thought of it as unlimitedly indivisible. Zeno
pointed out that from this unending process immediately resulted in contradictions. Zeno
denies admitting that a limited distance could be conceived as the sum of an unlimited
quantity of parts. The paradox of Achilles and the tortoise reiterates the same argument
as the paradox of the stadium, but related to two moving bodies, or what is the same, a
relative movement.
The paradoxes of the arrow and the parade is addressed against who
considered that space was formed by a multiplicity of juxtaposed indivisible units, this
is, a discontinuous space. Zeno states that during the flight of the arrow –in
apparent movement- it is really at rest. To demonstrate it he splits the distance covered
into the multiplicity of its intermediate positions between the initial position and the
final position. This group of positions is limited, because the number of units that
compose any distance is limited. Then Zeno makes us see that in every position the arrow
is at rest. Translated into modern terms, twenty odd centuries before Draguerre and the
Lumiere brothers, Zeno had decomposed in the black camera of his mind, the illusion of
continuous movement into the multiplicity of its stills.
The paradox of the parade is aimed against those who argued, not that
movement was to cover successively a series of positions, as many as indivisible units has
the journey, but that it consisted in the passing from each position or indivisible unit
to the next. The paradox of the arrow deconstructs an absolute movement and the paradox of
the parade deconstructs a relative movement, that of two mobile bodies between each other
with regards to a third fixed body. We will obviate his explanation here because it would
take us too long.
Melissus of Samos
(approx. 485-420)
He was the commander of the Samian fleet that defeated the
Athenians led by Pericles;
Even though he was an Ionian, he adopted the philosophy of Parmenides,
introducing an important change. For Melissus, what is, the One Being, is an unlimited
Whole, because if it was not, it would not be One, but three things: The One Being, the
limit and the limiting. In the same way, from the fact that the Being is unlimited we
deduce its eternity, because it cannot have beginning or end. He stated that the One is
non-physical, because if it had volume, it would have parts and it would not be One. He
maintained that every so-called pluralist philosophy would have to allot to its
fundamental principles the same characteristics that defined the One, what in fact
happened with Anaxagoras and Leucippus.
The
pluralist philosophers
Towards the middle of the V century BC, some philosophers
dissatisfied with Parmenides’ monism produce pluralist theories of physics. Although
they reject that the Being is One, the powerful logic of the Parmenides, will induce them
to attribute to their respective fundamental principles most of the characteristics of the
Parmenidean Being: eternity, indivisibility, immutability, and homogeneity. In effect,
after Parmenides it cannot be said that generation and corruption form things, therefore,
the elemental principles cannot have beginning or end. In the pluralist philosophers the
changes in the natural beings are explained as changes of the proportion in which the
constituent principles are combined. The Pythagorean mathematicians had profusely
developed the concept of proportion. Precisely the pluralist philosophies share the fact
of synthesising aspects of the Pythagorean and Eleatic tradition and of the Ionian
physics.
Empedocles of Acragas
(approx. 485-425).
This Sicilian doctor and politician, probably educated in the
Pythagorean tradition, seems to have written –as well as a treatise of medicine- two
philosophical books: Purifications, of deep Orphic resonance, and On Nature.
Both expose the same model: the transition from the One towards the multiple by means of
Discord and the return from the multiple to the One by the action of Love. In the first
book this model is applied to the vicissitudes of the soul, in the second to the Nature as
a whole.
The souls of the natural beings have been separated from the prime Soul
by the action of Discord; its terrestrial life has to be an ascetic road to perfection
until managing to reintegrate into the union of the immortal original Soul.
In the Physics of Empedocles the primitive stadium was a Sphere
composed by a perfectly homogeneous combination of four elemental principles, that he
calls roots: Air, Fire, Earth and Water. There are also two non-physical vital principles
in continuous movement: Love and Discord. The penetration of Discord into the Sphere
unified by Love is going to produce the separation of the roots until the total disunion
is reached: the Sphere formed by four concentric of subspheres of Earth, Water, Air, and
Fire. Then, by the action of Love the return to the original union begins.
The doctrine of the four roots is probably of Pythagorean ascendance
and its marks are already found in Xenophanes and Heraclitus. Each natural being is
constituted by a combination of the four elements in a determined proportion and it is
constantly emitting essences, that allow them to be perceived by other natural beings,
grasped by their sensorial organs. Here we see the union of Being as an unending source of
a multiplicity of essences.
A curious aspect is the biological theory of Empedocles about the
origin of the living beings. In a first stage multiple members and odd organs are born; in
the second, they combine by chance, creating monsters. In the third the successful
combinations are predominant, those whose harmony allow them to reproduce, in a non sexual
way; in the fourth, the natural sexed beings appear. In this doctrine we see the advance
from the multiplicity, by means of random combinations to the union as harmony.
Anaxagoras of Clazomene
(approx. 500-428).
His cosmology has a clear connection with Anaximander’s. He
conceives the original stadium of the cosmos as a Whole, a heterogeneous combination
without qualities, indefinite composed of fire or ether, air, earth, water and seeds.
These are the fundamental constituent principles of the natural beings, the difference
between these are due to the variations in the proportions of their combination.
Anaxagoras clearly differentiates the microcosm -the elements and the seeds- from the
macrocosms -the natural beings-. The novelty of his microcosm is the seeds: there are
unlimited in quantity, size and kind. They are infinitesimal, and therefore, imperceptible
for our senses. A force of rotation, induced by the Nous upon an area of the initial
combination, extends all over it, separating hot from cold, dry from wet, dense from
sparse, luminous from dark. The first that separates is fire-ether and the air, that
because of their unlimited extension they will be the most abundant components of the
cosmos.
In his conception of macrocosm Anaxagoras accepts some points of the
philosophy of Parmenides: the vacuum does not exist, so there is no generation or
destruction, but composition and separation. The cosmos is full and continuous, the
physical bodies are unlimitedly divisible. On the other hand, he rejects that the Being is
One and he admits a plurality of eternal principles. He defended the plurality in
opposition to the paradoxes of Zeno, only valid, according to him, against those who
thought of the cosmos as discontinuous and composed of a plurality of indivisible units,
but not valid against an unlimited divisible continuous cosmos.
In the cosmos, as well as the seeds, the Nous exists, the only thing
that never combines with the rest, the subtlest and purest substance, perfectly homogenous
that spreads everywhere and its autonomy allows it to rule over all things, being
omniscient and omnipotent, initiator of the movement of the cosmos and present in
different ways in each species of living beings. Therefore the Nous is the Soul of the
World and the soul of the particular beings, the Logos that rules all, the Mind that
embraces and surrounds the cosmos. That is why the commentator Simplicio, said
exaggerating that in Anaxagoras there was only really two principles: the Nous and the
unlimited combination of seeds.
In each natural being –stone, plant, animal- there is a portion of
each kind of seed, in different proportion; its own nature comes given by the seed whose
portion is the biggest. This makes us see that Anaxagoras’ appeal to the unlimited
plurality was his answer to the question about how could things transform into others.
When answering that by means of the variation of the relative proportion of the seeds, he
offers an anti-economical, but ingenious theory, that saves a lot of speculative effort to
explain the exuberance of nature.
Leuccipus of Elea or Miletus
(approx. 480-425) and Democritus of Abdera (approx. 470-380).
Leuccipus elaborated the main lines of atomism, as a
reinterpretation of the Pythagorean horism after the attacks of Parmenides and Zeno.
Democritus perfected the theory diffusing it in many books. The scarce fragments that had
been conserved of both do not allow separating with certainty the contributions of each of
them. For the atomists there are two principles: the vacuum, unlimited in extension, and
the plenitude, unlimited in quantity, the atoms. These are compact, homogeneous,
indivisible, impenetrable and impassive; they have an unlimited number of shapes, although
they all possess the same nature. They have an unlimited number of sizes, although they
are all imperceptible because of their smallness; their weight depends on their size.
The vacuum and the atoms are eternal and these have always been in
constant movement. Through the times innumerable worlds have formed and been wiped out in
the vacuum. When great quantities of atoms converge in a same region of the vacuum a world
is formed. That conglomeration produces a whirl, called need, that keeps the heavy atoms
in the centre and projects the lighter ones outwards, these form a membrane, the
firmament, that in its rotation captures bodies of the exterior vacuum. The bodies are
produced by aggregation of atoms, that when they collide against each other they link up
or bounce, but they do not fuse. The fundamental characteristics of bodies are the shape
of its atoms, their internal order and their configuration. The sensitive qualities are
not real, that is to say, they do not belong to the individual atoms; they are
conventional, they only characterise the macroscopic bodies.
The movement of the atoms obeys to mechanical causes, with the
exception of the spherical atoms of soul and fire, which have capacity of self-propulsion.
Thanks to this the bodies can know and rule. The perception is produced when the senses
grasp the essences emitted constantly by the bodies.
In summary, the relation between union and plurality is presented in
the atomists under several aspects:
-One is the vacuum and innumerable the worlds that form in it.
-They convert the One Being of Parmenides in a multiplicity of atoms
that have many of their same characteristics.
-The atoms have union of nature and plurality of shapes, sizes and
movements.
-Provisional union of body and multiplicity of emitted essences, mocks
of themselves
Diogenes of Apollonia
(approx. 475-415).
By the fragments that have been conserved it is believed he was a
doctor and usually it is attributed to his technical profession the empiricism manifested
in his philosophy, that agrees with the tradition of the Hippocratic school. Rejecting
both the consequences of Eleatics and the pluralist physics, Diogenes returns to the sole
cosmological principle: Air, or according to other sources, the intermediate substance
between Air and Fire, the Ether that would generate both. This Air or Ether is eternal and
unlimited and in its interior innumerable worlds emerge. He justified this return to
monism arguing that the interaction that we perceive between natural beings is only
possible because they have the same underlying nature; if it was not like that there would
only be interrelation between beings of the same species.
The cosmos is ordered in the best possible way, this proves that Air or
Ether is intelligent. As it is the subtlest substance it is present in all things,
although in a different way in each one, what allows it to rule them. The soul of the
living beings is formed by hot air. Their capacity of knowledge is due to different forms
of interaction between the soul or internal air with the external air through the organs
of the body, following the doctrine which states that the similar recognises the similar.
With regards to the relation between union and multiplicity, Diogenes
returns to the starting point of the Ionian physics; the same as Thales of Miletus he
explains the immense diversity of sensitive things by means of a sole principle that
confers them union of nature.
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