Everything is in everything.

Therefore, it is not possible to isolate, for example, pure water itself or pure air as such, for everything is in everything else.
Anaxagoras further held that the greatest (largest) or maximum and the smallest or minimum do not exist, for the large or the small are mere relative concepts and that anything can be large or small. This is very important and has been held to be true even until today.
Against Empedocles who considered the two principles of motion, love for mixture and strife for separation, Anaxagoras held only Reason (ho nous) as the principle of separation.
If everything is contained in everything, why and how could the differences among things be explained? To this question, Anaxagoras answered it by saying that multiplicity of things in nature are due to the quantitative differences of its elements (everything) contained in them (everything).
In order to explain that the ordered universe (ho cosmos‹ the cosmos as the universe implied an order in the Ancient Greece) came into being out of chaos, Anaxagoras used the principle of Reason (ho nous) to explain the differentiation and the ordering of the universe out of the chaos, this principle of motion as Reason functioned to cause a vortex (hé perichórésis) which separates and differentiates all in an order. Here, the concept of Reason is, for the first time, associated with the concept of discrimination and ordering. This motion is, according to Anaxagoras, even today still retained in the motion of the celestial bodies.
From chaos (in which everything is in everything equally, so there is no separation), the separation of matter took place through the motion of vortex. First, the air, surrounding around the chaos, then the dry from wet, rare from dense,..., finally, the heaven is separated from the earth.
The speed of this vortex can not be comprehensible by the cognitive faculty of the mortal human.
Being a rationalist, Anaxagoras held also that, since this universe, i.e., this sun with its planets, has no reason to be the sole one, there are many universes.
All these changes including those which we observe everyday are caused by Reason.
According to Anaxagoras,

Nous or Reason is,
1) the principle of motion, of the vortex (perichórésis)
2) the principle of living organism
3) the gnósis, knowing everything which is mixed and separated,
therefore, knowing the universe in the past, in the present and in the future
4) not only knowing the future, but possessing the power to have the future actualize in that way, to diakosmein.

While Empedocles' philotés (love) and neikos (strife) as the principle are blind controls (without cognitive faculty, knowledge), Anaxagoras' Nous knows and controls.
Aristotle in his Metaphysics, Book B said, that Anaxagoras made Nous the principle of the universe, "man as a rational animal, against other animals and plants, appears as if he were the only one awaken among those who were asleep..."
By Anaxagoras, the strength and power of knowledge, Nous' knowledge, was clearly recognized and emphasized.
Nous alone exists, according to Anaxagoras, in its pure form and infinite, self-sufficient and being by itself.
Therefore, it possesses the power of separation and motion. Anaxagoras' world is thus dualistic in that there are everything (matter) and Nous. This dualism is quite different from Parmenidean dualism of reality (knowledge) and appearance (illusion or opinion). Anaxagoras' philosophy of nature is a radicalization of Ionian monism.

The Atomists: Leucippus (430 B.C.) and Democritus (420 B.C.)

Leucippus was supposed to be born in Abdera, Elea or Miletus, none of which is certain. Besides Epicuros, there are some scholars who doubt the existence of Leucippus, but Aristotle spoke of him as an independent philosopher (independent of and distinct from Democritus), which substantiates his real existence.
Democritus was born in Abdera, that is relatively certain. According to another story, Democritus was born in Miletus.
The generally established, most acceptable theory purports that Leucippus was born in Miletus and Democritus was born in Abdera and both of them studied philosophy in Elea.
When Anaxagoras was old, Democritus was a very young man and the difference may be 40 years. This is the only reference to Democritus' year, so we take this as the basis of his acmé, i.e., 460 - 40 = 420 B.C. (we know for sure that Anaxagoras was 500 - 460 B.C.)
Democritus was said to have lived quite long, 90 or even 109 years old. If therefore, Democritus died at the age of 100 for instance, then he lived form 460- 360 B.C. This date puts Democritus as a contemporary to those Athenian philosophers as:
Socrates (470 - 399 B.C.)
Plato (427 - 347 B.C.)
Aristotle (387 - 322 B.C.)
Protagoras, the most famous sophist who long lived in Athens, was senior to Democritus.

Thus, the Atomists have been considered as preceding to the Athenian philosophies (i.e., Socrates, Plato, Protagoras, Epicuros, Aristotle, etc.), but in reality, it may be more appropriate to consider the Atomists as the contemporary opposition to the Athenian philosophies. Those socalled Athenian philosophers lead by Socrates focused their philosophical attention to the human matters, education, and socio-political problems. Against those philosophical interests, the atomists tried to revive the strong interest in the philosophy (science) of nature. Probably, to the Atomists, the human affairs seemed very trivial in comparison to the question of nature and of the universe.
It has been recorded that Epicuros (341 - 270 B.C.) said that Leucippus did not exist. Nevertheless, Aristotle referred to Leucippus either alone or in conjunction with Democritus, that indicates Aristotle's distinction between the thought of Leucippus alone and that of Leucippus and Democritus together. This seems clearly to indicate that Aristotle was aware that Leucippus was the founder of the Atomist philosophy.
Epicuros often ignored his predecessors and it is probably because of this tendency which influenced him to say that there did not exist Leucippus.
Leucippus should be considered as the founder of the Atomist philosophy. In a sense, the Atomist philosophy may be said to be a synthesis of the Eleatic Melissus' and the Miletian Natural Philosophy.
Atomists' influence is discernible in Aristophanés' famous comedy Clouds (423 B.C). Therefore, Leucippus' acmé is 430 B.C.
The writings of the Atomist philosophers have been hardly kept. The differences between Leucippus' and Democritus' philosophy amy be determined by the following reasons:

1) Some fragments which were identified as Democritus'
2) Those which were ascertained by the later scholars as Democritus'
3) those which Aristotle referred to both Leucippus' and Democritus' are Leucippus' and not Democritus'
4) Those which Aristotle did not name Leucippus' are Democritus'

Thus, the original Atomist philosophy was conceived by Leucippus, while Democritus developed Leucipppus' thoughts and organized them more systematically.

Leucippus (430 B.C.)

The starting point of Atomist philosophy is the same as that of Anaxagoras. Being challenged by the Eleatic philosophy, it became increasingly difficult to comprehend nature by means of its one element with the change of generation and corruption.
As Leucippus life reveals, being born in Miletus or Abdera, studying Parmenides in Elea, returning to Abdera, Leucippus attempted to comprehend nature as "many" and "changing" as well. What Leucippus learned from the Eleatic school was that there exists absolute distinction between Being and Non-being. Further, Leucippus had to face the dilemma of Melissus' thesis of the non-existence of void (to kenon).

Leucippus' problem is how to resolve
1) that Being exists whereas void does not,
2) that reality is many and changing.

Leucippus' solution is a quite bald one, namely that there "is" void in a certain sense, i.e., in a different sense from that of Being's existing.
Leucippus's solution is that in order for "many"and "change" to exist as real, there must be "void." This solution requires the distinction of the meaning of "to be" between the case where it is applied to "Being" (matter) and the case where it is applied to "void."
In other words, Leucipppus articulated two meanings of existence and contended that void too "exists." This was possible by conceiving the opposite of what the others considered as the most basic contradiction.

Being and
Void exist at the same time!

Now, thanks to this distinction of two meanings of "to be", the universe, in stead of being One, is indeed "many" and "divisible" due to this "existence" of to kenon (void), according to Leucippus of Miletus.

However, as Zeno had clearly demonstrated it, this divisibility of Being as (material) substance, if allowed to infinity, involves a contradiction or paradox (e.g. infinitely large or infinitely small).
Therefore, abandoning the premiss of the infinite divisibility of matter, Leucippus contended that matter, or a material substance, is only finitely divisible.
Thus, the ultimately indivisible unit of material substance is called a-tom (to atomon = no longer indivisible last element).

Atom = the Absolute Being, which has equal in quality, unchanging, i.e., neither generating, nor corrupting.
The universe consists of an infinite number of atoms.

The dualism of atom (to atomon) and
void (to kenon).

However, this void was conceived by Leucippus as the absence, i.e., the Non-being of the atom. In this sense, this is not dualism in the pregnant sense.
Since an atom is qualitatively equal to one another, Leucippus and Democritus were burdened with the task of explaining the multiplicity and variety of natural phenomena which seem to imply qualitative differences.

1) Leucippus attempted to explain the multiplicity by the structural differences of the atom.
a) differences in shape (rhydmos, rhysmos)
b) differences in arrangement (diathigé)
c) differences in direction (of movement) (tropé)
2) Democritus tried to explain what could not be explained by 1),
namely by means of our sense organs.
Among 1), b) and c) are rather derivative and secondary to a), for they were needed to explain motion.

Thanks to the "existence" of kenon, the possibility of motion is demonstrated but not its actuality.
Anaxagoras attempted to explain it by Nous.
Empedocles attempted to explain it by philotés and neikos.
According to the Atomists,
"motion" belongs to the essence of the atom and
the atom (as a substance) does not need any external (efficient) cause for motion.

In order to comprehend the "rest" (in opposition to "motion"),
to kenon (void) obtained significant meanings.
1) Void means distance (hé diastéma) , the space between two atoms (das Zwischenraum)
2) Void signifies the place (ho topos) in which an atom occupies and also moves

To the Atomists, the question of motion (locomotion, mixture and separation) are extremely complex and a "substance" is formed as followed from a bunch of atoms and their motions. And further the origin of the universe was explained.
1) Motion by nature does not have any particular direction (i.e., at random)
2) By such motion, a crash takes place. By crashing with each other, two atoms go to two different directions.
3) Because of shapes and different arrangements of atoms, an entanglement may take place.
4) When some atoms are concentrated in a certain place, a group of atoms may be formed. For example, not only a matter (water or fire), but a living organism and even human soul were considered to be formed of atoms.
5) Such a group of atoms as a whole may be in rest or in motion
6) Further within the group, atoms may be in motion or in rest, etc.
7) However, according to the Atomists, there is no necessary reason for this universe to come into existence as it actually is.
8) In fact, an infinite number of atoms are in an infinite space. There exists an unbalance which may cause a kind of motion like "vortex."
9) The atoms of heavier weight with same "size", shape get together around the center of the "vortex", while the atoms with lighter, "smaller" shape were spanned out. Thus the Heaven and Earth came into Being.
10) While in the inside of the vortex the speed of the motion is slower, the outside of vortex it is faster.
It is clear that the Atomists attempted, unlike Anaxagoras' vous (reason), to explain the nature by means of atoms and void (and accident) alone.

Democritus of Abdera (450 B.C.)

Democritus was born in Abdera, Ionia and was supposed to live for one hundred years (It is interesting to note that perhaps the greatest sophist Protagoras, who said, "Man is measure of all things,..." was also born in Abdera).
Democritus' nickname was "Sophia" (wisdom).
As mentioned above, Democritus, if he died at the age of 100 years, was no doubt a contemporary to those sophists such as Protagoras, to Socrates and even to Plato. We may, therefore, very well say that philosophically speaking, apart from what Democritus systematically reorganized Leucippus' Atomistic philosophy of nature, his philosophy was a response to and the alternative to the Athenian philosophy. His theory of knowledge as well as ethics appear as if Democritus knew the sophists and their relativistic theories both in epistemology and moral philosophy.
As we have seen that Empedocles considered our sensation as the effect of material substances affecting our sense organs, according to Democritus, those effluences are nothing but atoms, images (hé deikela, hé eidola). Those images come through the organs of sense and are the imprints on the soul, just passages (hoi poloi).
Democritus also held that our soul was also consisting of atoms. Sight is explained by the image passing through the air, whereby it is distorted due to its travelling a long distance. Colors were explained by differences of smoothness or roughness in the images, and hearing was explained as the stream of atoms flowing from the sounding body causing motion in the air. Likewise were explained taste, small and touch.
Since Democritus like Leucippus did not recognize anything else but the ultimate indivisible units (atoms) and void, none of which are known by our senses, all the sensations of the special senses do not and cannot portray or convey the reality.

"In nomos, there is sweet, in nomos there is bitter; in nomos, there is war and in nomos there is cold; in nomos there is color. But eteré, there are atoms and void." (Frag. 9)

Namely Democritus held that all our sensations are subjective.

"By senses we in truth know nothing sure, but only something that changes according to the disposition of the body and of the things that enter int it or resist it." (Frag. 9)

"There are two forms of knowledge (gnómé), the true born (gnésié) and the bastard (skotié).
"To the bastard belong all these: sight, hearing, smell, taste, touch.
"The true born is quite apart from these." (Frag. 11)

And yet Democritus did not claim this the knowledge of the true born is the so-called rational knowledge, independent of our sense experience.

Democritus' ethical theory does not seem consistent with his Atomist theory of Being and knowledge, as far as we see from his Fragments.
His ethics is an Eudaimonism (the criterion of the moral right consists in pursuing or being conducive to pursing happiness -- eudaimonia = the well-being of spirit (daimonion)). According to Democritus, this happiness consists in pleasure and fulfillment of cheerfulness (euthymié or euesto) or avoidance of pain or suffering.
Democritus wrote a treatise on cheerfulness (Peri euthymiés), which was later used by Seneca and Plutarch.

"Happiness dwelleth not in herds nor in gold; the soul is the dwelling-place of the 'daimon'." (Frag. 171)

"The best thing for a man is to pass his life so as to have as much joy and as little trouble as may be." (Frag. 189)

In order to attain well being (euthymia) or cheerfulness (euesto), it is required to weighing, deliberating and judging and distinguishing various pleasure. Democritus said that we should be guided, therefore, by the principle of symmetry or harmony. thus we shall be able to attain health = the calmness for body and the cheerfulness = the calmness of soul.