Xenophanes and Parmenides

The search for the “One” led the philosophers of the School of Miletus and the Pythagoreans to various principles and basic statements. The specific answers they gave are not so important to us today. It is much more interesting that they departed one step from mythological thinking and began to formulate rationales for their statements. Authorities didn’t have to be respected, doubts were allowed. What is even more interesting, however, is that in this effort they have already come across a number of problem areas that should prove to be fundamental for future sciences. These fields can be described by the concepts of “infinity” and “motion”.

The “Eleates”, those philosophers of the 6th and 5th centuries BC who lived in a Greek settlement Elea on the west coast of southern Italy, were to raise awareness of a third problem area of such importance. It was the question: How can we obtain reliable knowledge?

Thus, three fundamental questions are already under discussion in such an early period of our intellectual history. And each of these questions or problem areas stood for an area of philosophy at that time: “infinity” for mathematics, “motion” for physics and the question of reliable knowledge for what we call today epistemology. We know it by now: When you tackle big tasks, other big tasks also come into focus. And even if the original task cannot be mastered, there is usually fruitful insight in the treatment of the subsequent problems. In the technological field today, this is called “spin-off”. I will always keep an eye on these fundamental questions in the course of the later blog posts.

Who were the Eleates and how did they answer the question of reliable knowledge? Their most important representatives were Xenophanes (about -570 to about -470), Parmenides (about -515 to about -455) and Zenon of Elea (-490 to about -430).  Let us take a look, which possibilities of knowledge they saw and how the concept of motion came to the fore again, and in the wake of it the concept of infinity. 


Xenophanes was at first probably a Rhapsode, someone who recited Epen Hesiods and Homers, but also appreciated and recited elegies or mocking poems. He only settled in Elea after a long life in which he had got to know many a foreign culture.

Xenophanes takes a completely new view on the question of the possibilities of knowledge.  In a fragment (Mansfeld & Primavesi, 2011, pp. 229, No.32) it says:

The gods did not reveal everything to man at the very beginning, but in the course of time they seek and find what is better.

That is now strikingly topical. Here, already 2,500 years before our time, the knowledge of modern science theory is formulated that knowledge grows in a kind of evolution. We see this particularly clearly in the development of modern physics: the theories are adapted to the ever-increasing number of observations and experimental results. The range of validity of a theory thus becomes ever larger; it becomes “better”. A theory that does not work well enough in such a process of adaptation will at most become a topic in the history of physics. Using the example of motion, we will soon see clearly how such better theories will look in the course of the development of physics.

 Xenophanes thus formulates a path to knowledge in which people search and find something “better”, quite in contrast to mythological thinking, which “seeks nothing, always speaks apodictically and claims to be simply true” (Schupp, 2003a, p. 87).

But there is no mention of the possibility of once being in a definitive possession of the truth. On the contrary – according to Sextus Empiricus (160 to 210) Xenophanes (Mansfeld & Primavesi, 2011, pp. 231, No.39) says it:

Of course, no man has seen the clear, and there will be no man who has seen it with regard to the gods and all the things I declare. For even if one had succeeded to an extraordinary degree in saying perfection, he would still not be aware of it: In all things, there is only assumption.

The sentence “In all things there is only assumption” even presupposes that all our reliable knowledge of the world of all things, even if it exists in the form of physical theories, is ultimately based on certain basic thoughts, principles or equations.

Xenophanes therefore only calls our findings “opinions”, whereby this term does not mean arbitrariness, but the idea that our findings can still be outbid by a better opinion. The opinion could, however, be regarded as “similar to truth”, since it was as well founded as possible in each case. Xenophanes does not believe in the possibility of a revelation of a final truth, as Parmenides later accepts it. 

One can thus call Xenophanes the first decided metaphysical agnostic. There is nothing definitive to say about a God and the world as a whole. The theses of the Ionian philosophers may have seemed too ambitious to him, too far removed from empiricism, which he regarded as more important than the embellishment of myths.

Thus, he contrasted the idea of the Greeks that the goddess Iris appears in a rainbow with a natural explanation: “What they call Iris is also a cloud, one that looks purple, light red and yellow-green (Mansfeld & Primavesi, 2011, pp. 223, No. 21). He also relied more on common sense: “For all is of earth, and all ends as earth” (ibid. No. 4) and “The sea is the source of water, the source of wind” (ibid. No. 23a). 

Xenophanes also discovered that every culture formed its gods in its image: “The Ethiopians claim that their gods are blunt-nosed and black, the Thracians blue-eyed and redheaded” (Mansfeld & Primavesi, 2011, pp. 227, No. 28)
“But if the oxen and horses and lions had hands or could paint with their hands and works, as men do, the horses would paint like horses, the oxen like oxen like gods” (ibid. No. 30).

 He, on the other hand, sets a “single God, the greatest among gods and men, neither in form similar to mortals nor in thought” (ibid. No. 35).

Here we also see in ancient Greece the idea of a single god appearing. Whether we have a separate root of monotheism in front of us here is probably difficult to decide. In other fragments, however, we read how he also speaks of “gods”. The development of monotheism in the early religions is a highly exciting topic, but we do not want to address it here. We are more interested here in how Xenophanes imagines the only God (Mansfeld & Primavesi, 2011, pp. 231, No. 37):

He always stays in the same place, without any motion.

So, the only God distinguishes a place and represents absolute rest. Also, Anaximander had already spoken of an absolute state of rest, but it was the Earth that stood in perfect rest in the centre of the world.  So here it is the only God.


Parmenides from Elea is regarded by some as a pupil of Xenophane, but in any case he is said to have known his works. He is probably regarded as the most prominent philosopher from the time before Socrates. Relatively many fragments of his writing ABOUT NATURE (Περι φύσεωζ = Peri Physeos) have been preserved. But his speeches are considered “dark” by all. This has earned him a special prominence among the pre-Socratics, for his work invites a wide variety of interpretations. There has also been much discussion about the right interpretation of the fragments.

I don’t want to get involved. On the one hand one should have penetrated deeply into the language of that time, on the other hand it does not seem so important to me to know the thoughts of a thinker as exactly as possible. It is much more interesting to know which new thoughts a thinker has brought into the discussion with which motives and how important these are for the treatment of important questions. Parmenides is also primarily concerned with the question of how to acquire knowledge, and his merit is at least that he makes an alternative quite clear with regard to this question. He speaks of two very different ways of knowledge.

The first way is like a revelation. In this way the knowledge of the being is given to one. For Parmenides, it is a goddess who leads a human being to an absolute certainty about what exists by means of a “destiny”. He dresses these thoughts in the story of a journey to the goddess Dike. In a fragment we read (Mansfeld & Primavesi, 2011, pp. 321, No.4):

So, the goddess, […] received me confidently and spoke the following words: “Young man, […]. It is not an evil destiny that has led you away through this path to reach your goal […], but divine providence and justice. So, it is appropriate that you experience everything: on the one hand the unshakable heart of the well-rounded truth, ….

To the well-rounded truth belong then the statements about the “being”: This is timeless, there is no origin and decay. It is “because unborn also imperishable”, it is “present together in the present as a whole, one, connected”. It is also an indivisible continuous, because “… it is not divisible, because it is quite similar. And there is not a stronger being here or there that could hinder its connection, nor a lesser one. […] That’s why it’s quite coherent, because being is close to being.” And:

As the same and persisting in the same and on itself it is, and in this manner remains firmly in the same place. (Mansfeld & Primavesi, 2011, pp. 327, No.11).

Here again we encounter the topos “absolute rest”, i.e. no motion, no change of place. The extraordinary, the imperishable and the comprehensive had to have a motion which is exceptional compared to other motion. According to the view of the world at that time, such a motion could only be rest. Even today it seems to us humans that this is often the case. However, we are not always aware that motion is not a property, but a two-digit relation, i.e. a motion in relation to another object, in everyday life. This unconscious and erroneous assumption that there is absolute rest should allow Zenon of Elea his so-called movement paradoxes, with which we want to deal in the next blog post.

But let us first consider this first path of the Parmenides to true knowledge. While according to Xenophanes the knowledge about the “perfect”, about the “gods and all things” is denied to us, according to Parmenides man can only attain this knowledge if it is revealed to him.

This is now water to the mills of mythical thought. Here an unshakable truth is promised. There is no uncertainty from possible criticism and constant changes in knowledge, as was the case with the Milesians and the Pythagoreans.

Now, every human being will probably never get along completely without a mythical component in his world view, also he may stand firmly to his convictions of such a kind. But a person who believes that the deepest secrets of the world are revealed to him personally as reliable knowledge easily risks seeing this knowledge as obligatory for all other people. It does not always have to be a goddess or a god, one can also feel a strong sense of the evidence of one’s ideas, which one may regard as a profound unshakable truth.

But now we know that different people can also experience very different revelations and feelings of evidence. When power and mission are added to an unshakeable conviction, even from a single source of revelation, different faith communities can gradually develop that divide society. Intolerance, contempt, oppression and annihilation of dissenters was indeed often the result. Violence has been perpetrated in every form.

So, what is the second way to knowledge for Parmenides? According to the above fragment, the young man, on the other hand, should also experience: the opinions of mortals who do not possess true reliability.

The second way is thus the way of the usual mortals, who can only arrive at “opinions”. According to Parmenides, however, this knowledge is at most truth-like and uncertain when measured against the abundance of the existing. Here the influence of Xenophanes shines through: We mortals have insights that can only be regarded as “opinions” in view of our knowledge of what exists. We only find “names” for what we explore:

Therefore, all the name that mortals have set is in trust that it is true. Emergence and decay, being and non-being, changing place and changing the bright colour. (Mansfeld & Primavesi, 2011, pp. 329, No. 11).

To give something a “name” does not yet mean to recognize it: Thus, according to Parmenides, the existing is in reality motionless. But we see motion. They’re just “names,” they just seem to exist. On our own, we can only develop delusions that cannot be reliable. This thesis has of course attracted a lot of attention, as it has the charm of the extraordinary, the anti-intuitive.

Both paths to knowledge were taken in the following millennia. In an effort to understand the world in which we live, the thinkers have, more or less consciously, adopted one side or the other, with sometimes dramatic consequences for the living together of people and the further development of humanity.

With concern to the reliability of the acquired knowledge, the thoughts of Parmenides about the two ways of knowledge are in stark contrast to what has been shown in the course of time. The “opinions” about nature proved to be highly reliable, because their fundamentals were directly inquired by nature – through observations and experiments. The fact that we can use this knowledge in the form of the development of technical devices is evidence of this. But the situation with revelations is quite different. There can be no question of reliability here.

The Presocratics: The Mathematics of the Pythagoreans

At the time when Anaximenes in Miletus represented the tradition of the Ionic school, Pythagoras was active in another part of the Greek culture (ca. -570 to -490).  Born and raised on the island of Samos, he is said to have spent his youth in Babylon and Egypt. There he got to know the religious beliefs and also the mathematical knowledge of these cultures. Around -530 he settled in Kroton, a town rich through trade in a southern Italian Greek settlement, and he founded a school there. This became a conspired community; he himself became an unquestioned authority for his pupils. He must also have been a charismatic speaker, and he must have impressed them greatly with the ideas he took up in Babylon and Egypt. With the sentence “He said it himself”, the Pythagoreans are said not to have even had any discussions (Schupp I, p.63). So today we see strong religious traits in this school.

Much has been written about Pythagoras, much attributed to him. None of this is considered certain today. There are no writings by him, not even by his pupils, the Pythagoreans. From the ancient sources such as those of Heraclitus, Empedocles, Aristotle and from the late antiquity of Diogenes Laertios or Jamblichos we learn contradictions and highly different assessments. 

Two themes were at the forefront of Pythagoras and his pupils, the Pythagoreans, but in any case: the transmigration of souls and mathematics. Pythagoras was probably inspired by his travels on both topics. 

I will not go into the ideas that the Pythagoreans had of a “soul” and a “transmigration of souls”. The Pythagorean theory of the soul had a great effect and influenced later philosophers like Plato in particular. It was the dominant theme of Neuplatonism in the first centuries AD and has also flowed into Christianity through this philosophical current. However, this doctrine did not contribute to the strengthening of the Logos in people’s thinking.

The situation was quite different in this respect, however, with the study of mathematical problems. A wealth of mathematical insights was obtained. A completely new field for philosophical considerations was discovered. Aristotle reports in his Metaphysics:

“At the same time as those philosophers [Parmenides, Empedocles, Anaxagoras, Leukipp, Democritus] were active, but even before that, those called “Pythagoreans” began to pursue mathematical studies. They were the first to advance these studies; and when they were educated in them, they were convinced that the constitutive principles of mathematics were also the constitutive principles of things being. (Mansfeld & Primavesi, 2011, pp. 147, No.31).

The Pythagoreans were, after Aristotle, the first to increase mathematical knowledge, and Pythagoras had given the impetus. This was the second great impetus in the history of ideas. The first impetus was given by the Ionic philosophers when they began to explain the world from within, namely with arguments based on regularity in nature. Here now, in the second step, a language was discovered in which one can note regularities. But not only this; in this language one could also provide arguments that are undeniable.  And for the relationship between the “constitutive principles of mathematics” and the “constitutive principles of things in existence”, the Pythagoreans, according to Aristotle, provided a first answer. We will see in the next blog post how they felt about it. Later, the question of such a relationship will become very topical.

But let us first have a look at the studies that the Pythagoreans and other philosophers, infected by them, have carried out:


Most people associate the name Pythagoras with the SATZ OF PYTHAGORAS via the relationship between the squares of the sides of a right-angled triangle: If one designates the cathets of such a triangle with a and b, the hypotenuse with c, then a2 + b2 = c2 applies. The Babylonians already knew this relationship and Egyptians and Pythagoras, like Thales, had brought such knowledge from his travels. Legend has it that Pythagoras was inspired to prove this theorem by looking at an Egyptian tile (Fig.1).

Fig. 1: Left: Illustration of the theorem of Pythagoras: right: The tile could have looked like this (but with the same strokes everywhere), from which Pythagoras was inspired, according to the legend, to prove the proposition. The squares above the hypotenuse are marked by dash dots. The squares above a, b, c contain two or four triangles of equal size. (from Honerkamp, 2018).

Here we are dealing with a geometric figure from which we can derive a statement about the relation of three numbers, i.e. an arithmetic relation. This is probably the first connection between geometry and arithmetic. About 2,000 years later Descartes was to strengthen this connection with “Analytical Geometry” to such an extent that arithmetic became the focus of mathematics from then on.

The Greeks represented their numbers at that time still by letters, i.e. there were prominent numbers, i.e. those, which we today 1, 2, …, 9, 10, 100,… any other number was seen as the sum of these celebrities, and the summands were then noted in size in the form of letters. We still know such an addition system from the Roman numbers.

In the context of such a number writing, calculating is very difficult. This may have led Pythagoras to invent a geometric number representation in which each number is seen as a sum of ones. For each one a point is noted, whereby the points are arranged to a certain geometrical figure (fig.2). For example, there are squares, triangles, cubes, etc.

Fig.2: A square number (16=42), a triangle number (10 = 1+2+3+4) and a cube number (8= 23).

From such figures one can already read many other relationships between different numbers, e.g. the statement that the sum of successive odd numbers, if it begins with 1, is always a square number (Fig.3).

Fig. 3: A deduction from square numbers (from Schupp,2000a, p.74)

If you still write the last line on the right side of this illustration as :

(1 + 3 + 5 + 7) + 9 = 25, so 16 + 9 = 25, or 42 + 32 = 52,

one obtains a so-called Pythagorean number triple, i.e. three numbers, where the square of the largest can be represented as the sum of the squares of the two smaller numbers. In this way one discovers how easy it is to form such number triplets:  Each sum of such odd numbers ending with a square number yields a triple. So take only an odd square number, say 25 = 52, and form

1 + 3 + … + 23 + 25 = 169, i.e. 144 + 25 = 169, or 122 + 52 = 132.

The Babylonians already knew such triplets of numbers, and since a large number of such triples were found on their cuneiform tablets, methods for the formation of such triples must have been known for much longer.  Even in the megalithic culture of southern England the knowledge of such triples is said to have been used for constructions (van der Waerden, 1983, p. 9).

Inspired by such experiences, the Pythagoreans discovered many relationships between two, three or four numbers, e.g. the arithmetic and geometric mean, including the harmonic mean. Probably they already knew the “golden ratio”. Today one would book such studies under the title “number theory”.


Geometry has not been neglected in all these studies of ratios of numbers. Especially in Egypt, Thales and Pythagoras learned a lot about geometry. After Nile floods the Egyptians often had to reconstruct their fields. It was very useful to know how to construct rectangles and, above all, how to calculate the area of a field.

Not only right angles and rectangles were important. Also circles, triangles and other polygons wanted to be constructed, in practice – and with the Greeks then also on the papyrus with compass and ruler, completely without regard to a use. And one also wanted to be able to calculate the area contents of such areas.

A notorious problem for the Babylonians and Egyptians had been the calculation of the area F of a circle. It was well known that F must be proportional to the square of the radius, but the proportional constant, which we call today π, could not be determined exactly. The Babylonians had used the value 3 for this constant, the Egyptians (16/9)2 = 3.1604… . The Pythagorean antiphon of the Sophist (5th century B.C.) took advantage of the fact that one already knew how to determine the area of regular polygons, approached the circle by such polygons, and thus obtained an approximate value, which is the better the more corners this polygon has. This method was to become the norm. Bryson of Herakleia (-450 to -390) looked not only at regular polygons inscribed in the circle, but also at polygons circumscribing the circle. He thus received two approximations, one smaller and one larger than the value he was looking for. Archimedes was then to drive the method to a perfection that can only be admired today (Honerkamp, 2018, p. 82ff).

Not only surfaces, but also three-dimensional bodies in space were studied, besides the sphere the polyhedrons (polyhedron, after ἔδρα gr. = seat, καδ-έδρα gr. = catheter) were of particular interest. These are bodies that are bounded by several regular surfaces. There is a tetrahedron (quadhedron), the cube is a hexahedron, there is an octahedron (octahedron), a dodecahedron (dodecahedron) and an ikosahedron (twentyhedron) (see Fig. 4).

Fig.4: The five platonic bodies: tetrader, hexahedron, octahedron, dodecahedron, ikosahedron.

As the mathematician Theaitetos (-415 to -369) was able to show, they all have a certain point inside, which is equally far away from all corners, so that all these polyhedrons can be inscribed in a sphere. He also proved that there can be no other polyhedra of this kind. The regularity of these bodies and their uniqueness caused Plato later to assign tetrahedra, hexahedron, octahedron and ikosahedron to the elements fire, earth, air and water, in the sense of the Pythagoreans, who see everything in nature dominated by numbers. Plato thus gave these bodies a mythical shine; this led to the fact that they were later called “Platonic bodies”.

The French historian of science André Pichot lists in his work THE BIRTH OF SCIENCE twenty-one Greek mathematicians of the 6th and 5th centuries BC (Pichot, 2000, p. 371ff). Many mathematical theorems in geometry and arithmetic have been found by these. Also textbooks, called “elements”, must have existed in pre-Socratic times to introduce newcomers to mathematics. None of them has been handed down to us. Only from the time around -300 such a textbook is available to us. But it is probably the most complete and mature work from which we can take the state of mathematics at that time. They are the “elements” of Euclid of Alexandria. This work consists of three chapters (“books”) on arithmetic and 10 chapters on geometry. It has had a hardly surpassable influence in the intellectual history of the western world, above all because it could represent geometry in the form of an axiomatic deductive system and thus made the logical order of the knowledge represented transparent. I will come back to this at the appropriate time.


There was another, even more concrete, field where the Pythagoreans discovered that relations in space can be mapped to relations between numbers. This field was the music they played on the lyre, and here you could even experience relationships between nature and numbers with your senses.

The “tetrachord” had four strings, which could be plucked, struck or beaten to produce a certain tone, depending on length and tension. If one plucked two strings at the same time, one could hear a sound which one felt to be harmonious if the lengths of the two strings were in an integer ratio such as 1:2, 2:3, 3:4. Especially with a ratio of 2:1, the sound sounded almost like a single note. Let us consider here the simplest form of a tetrachord (Fig.5, see also Wikipedia: Tetrachord).

Fig. 5: The strings of a tetrachord and the ratios of the lengths of adjacent strings (after (Pichot, 2000)). Attention: Here the tones get lower from left to right!

Be the first string on the far left the shortest. The second, third and fourth string may then be 4/3, 3/2 or 2/1 times the length of the first. We then call the notes C’, G, F and C, the frequency of the notes G, F, C are 3/4, 2/3 and 1/2 times the frequency of the note C’ of the left string. The Pythagorean Philolaos gave the intervals certain names, later they were called fourth (from C’ to G), fifth (from C to F) and octave (from C’ to C) (fig.6)

Fig.6: The strings of an octochord and the ratios of the frequencies of neighbouring strings in the Pythagorean tuning (according to Pichot,200, p.385).

For an “octochord” with eight strings a finer subdivision was needed. Philolaos first introduced a “second” as the distance between fourth and fifth (from G to F, from Mese to Paramese). For this, the ratio of the frequencies is (3/4):(2/3) = 9/8. With two consecutive seconds (whole tones), the lowest note C (nete) can be used to reach the note F (paramese), but not yet. The remaining “distance” is (4/3):(9/8)2= 256/243. This interval is called a “semitone”, the Greeks called it Diёsis (from δί-ειμι, gr. = pass through).

The interval from G to C’, from the mese to the hypate, is also a fourth, and the result is the same as from C to F.

So, such a “semitone” is an interval consisting of two tones with a frequency ratio of 256/243. But it is not really something like half the interval of a whole tone. If you put two such intervals on top of each other, you get an interval with a frequency ratio of (256/243)2, which is not equal to 9/8. A very small interval with frequency ratio is still missing.

(9/8) : (256/243)2 = 531441/524288.

This interval was today called the Pythagorean comma (from gr. κόπτειν->κόμμα = something like “section”). One would have to call the “semitone” more exactly a small semitone, and then there is also a large semitone, an interval that is larger by the Pythagorean comma. 

As long as you play on the octochord, and everyone who plays at the same time uses the same fundamental C, this Pythagorean comma does not matter. 

This determination of a stock of six tones between the two tones of an octave for a given fundamental leads to the so-called Pythagorean tuning. It is therefore realized with such an octochord.

The six tones (now starting from “below”: D, E, F, G, A, H) can also be determined in this way: From a given tone you can find the pure fifth (frequency ratio 3/2). If this results in a tone above the interval under consideration, set the tone an octave lower, i.e. halve the frequency. So if you start from C, you get one tone at a time: G, D’->D, A, E’->E, H. Now the F is still missing. Then you should start from F, so now you get: F, C, G, D, A, E, H.

If you now want to increase the stock of notes, you can continue to use the fifth rule for this, so you get notes that we could call F#, C#, G#, D#, A#, because their frequencies are one Diёsis higher than F, C, … in each case. The next note would then be ice, and that would have to be F again. But that’s not true, because there are two small semitones (H-C and E-F) between H and the F sharp of this construction, instead of just one like the other such intervals. So the interval H-F sharp is too small, not a pure fifth, a “wolf’s fifth” (because it “howls like a wolf”). So, if you want to strictly follow the rule, you have to increase the frequency of the F sharp by the Pythagorean comma and so do all following notes. The frequency of the ice finally reached would thus be higher by this decimal point than F. This can also be seen if this construction is computed. It is

(3/2)12: 27 = 531441/524288.

If one were to construct the notes B, E flat and A flat from F “downwards” to the same extent, one would obtain the circle of fifths in the usual form. Then the notes G#/As show that the “circle” does not close completely. The tone G sharp is higher by the Pythagorean comma than the As.

The Pythagorean tuning system was fashionable until the 16th century. After a long period of experimentation, the equal tuning was introduced in the 16th/17th century. In this the octave is divided into twelve intervals of equal frequency ratios f. So it must be f12 = 2, hence f = 1.05946…. > 1,05349… = (256/243).

The Pre-Socratics: The school of Miletus

The city of Miletus, situated on the coast of a Greek settlement of the landscape of Ionia (Asia Minor), gained economic importance in the 6th century B.C. through extensive trade relations. A social class developed for which “just about everything was available for comfort and a higher standard of living”.

The cultural milieu was marked by Orphic or Dionysian cults, stories of Homer and epics of Hesiod (born before -700); influences from Mesopotamia and Egypt, Phoenicians and Chaldeans must have been noticeable as well. But time and leisure also brought with them the attempt “to escape ignorance without expecting any benefit beyond knowledge” (Aristotle, after (Schupp, 2003a, p. 43)). Philosophy can be born in such an atmosphere.


Thales lived in this town from about -625 to -547. At first he was a merchant and must have been around a lot in the world at that time. He is said to have brought geometrical and astronomical knowledge from Egypt. In his later years he also interfered in politics. But we only know of all this from later sources, whose credibility is not always given.

What is certain, however, is that he must have lived around 585 BC. He had predicted an eclipse of the sun, which according to modern astronomical calculations took place just in 585 BC. Babylonian astronomers already knew the cycles with which solar and lunar eclipses were repeated. Sun eclipses are not visible from every place equally well; but one knew thus when one could count on such at all. Thales had probably benefited from this knowledge, either through his earlier travels or through more recent cultural contacts. But he was also someone himself who studied the stars. Diogenes Laertius knew that Thales was said to have written two writings about solstices and equinoxes. Because of his knowledge he was also admired by later philosophers (Laertius, 2015, p. 13).

Many of us can surely remember that in school they learned the SATZ OF THALES in mathematics lessons. We do not know whether Thales himself discovered this sentence. Diogenes Laertius writes only 800 years later: “In geometry, a pupil of the Egyptians, he first entered the right-angled triangle into the semicircle, as Pamphile reports. (Laertius, 2015, p. 13). There it is also mentioned that Thales in Egypt should have measured the height of the pyramids, “giving their shadow, which he measured exactly at the time when our shadow and our body have the same length” (Laertius, 2015, p. 13). (Laertius, 2015, p. 15). Thales must have been highly intelligent in any case and he knew how to use his new knowledge independently.

The philosophers regard Thales of Miletus as the first of their guild. We do not know how Thales came to such thoughts, which today are called philosophical thoughts. The processes “emergence and decay” were in the foreground in the mythical narratives and on his travels Thales will not only have taken up geometrical and astronomical knowledge, but also become acquainted with the most diverse of such narratives. Since one constantly observes changes in nature, including emergence and decay, the question may arise as to whether there must not be something lasting and eternal “behind” all this. This must then be something causal and “underlying”, something like a “primordial principle” or a “primordial substance”, that is, something that could also be called “the One”.

Aristotle later wrote about the ideas Thales had developed for this:

“Thales, the first representative of this direction of philosophical investigation, describes water as such a principle. He therefore taught that the land, too, rests on water. The reason for this view was probably the observation that the food of all beings is moist, that the heat itself arises from it and lives from it. But what everything becomes of, that is the principle of everything. If this was one of the reasons for his opinion, then another was probably the fact that the seeds of all beings are of moist nature, but that water constitutes the principle for the nature of wetness.

It was not far-fetched that water should be the primordial cause of all things, for the Sumerians and Egyptians also spoke of an “primordial sea”, and in Homer’s case the inhabited world was surrounded by the Okeanos, a mighty river, which was also considered the father of the gods and the origin of the world.

It is important, however, that Thales argues this way here and elsewhere: Water causes moisture, food needs water, otherwise it dries up and becomes inedible. Water can form fog and clouds, and in the last instance even air (water vapour) and fire were formed from it. Sedimentation on the coast shows how land can be removed from the water.


Thales brought a whole new way of thinking into the world. Anaximander (ca. -610 to -546), who also lived in Miletus and is regarded by many as a pupil of Thales, took up this way of thinking, but immediately showed that one can come to a different conclusion in it. For him, water was perhaps too concrete as primordial matter, and he concentrated more on one particular aspect: the One which should be a primordial reason had to be unlimited, an apeiron (from Greek ἀπέραντος=aperantos, unlimited, infinite). The Greeks had the same word for “unlimited” and “infinite”, but could not yet imagine a curved space. Today we know spaces of different dimensions; a vivid example for an unlimited, but not infinite space is the surface of a sphere, a two-dimensional curved space.

This is where the term “infinity” came into play, a term which subsequently occupied almost all philosophers and mathematicians and which was only learned to be dealt with consistently and formally at the end of the 19th century within the framework of the Cantor set theory. According to Anaximander, what people do not know  nor what they can imagine, should apply to the Apeiron: “The Apeiron is without age” and the “Apeiron is without death and decay” (according to Schupp, 2003a, p. 53).

These were qualities that were granted only to the gods. The One was thus placed on the same level with the gods. Thus Anaximander came suspiciously close to the world of myth, but he answered the question of how the One would create multiplicity, more like a physicist: He spoke of opposing elemental forces, dry hot fire and humid cold steam, that could lie in battle with each other.

In this way he had an idea of what the sun, moon and stars were made of, and even knew how to explain how solar or lunar eclipses occur. According to him, the earth was at the centre of the circles on which the celestial bodies were moving, and it was in perfect equilibrium with all the stars.  In the shape of a cylinder it was similar to a “stone column segment”. Humans lived on the top of the cylinder; according to the geographer Agathemeros, Anaximander even “dared to be the first to draw the map of the inhabited world” (Mansfeld & Primavesi, 2011, pp. 65, No.2). The distances of the celestial bodies from the earth were in certain proportions to each other. Thus, Hyppolyt of Rome reports in the 3rd century that according to Anaximander the circle of the moon and the sun is 19 and 27 times the diameter of the earth cylinder, respectively (Mansfeld & Primavesi, 2011, pp. 75, No.20).

Even a weakening of the struggle between the hot and the humid a passing of the “existing” as well as a new creation from the Apeiron was planned. For Anaximander the Apeiron is therefore not only infinite; it can also create infinite many worlds and let them pass away again. One inevitably thinks of the quantum vacuum in Stephen Hawking’s M-theory and of the constant emergence of new universes from it through quantum fluctuations.

For the interaction between the opposing pairs he had an answer, which again fits more to the myth and the heaven of gods: “From which things the things in existence have their origin, in these also their decay takes place, as it must be, because they do each other justice and punishment for the injustice, according to the temporal order” (Mansfeld & Primavesi, 2011, pp. 71,Nr.15).

The whole picture already contains astonishingly many modern aspects such as the description of spatial relations in quantitative form and the concept of infinity.


At Anaximenes (approx. -586 to -527) the primary substance is now something you know from everyday life: the air. Instead of an abstract, it is now something concrete again. From the Doxographen Aёtios from the 1. century we experience: “Anaximenes set as principle of the being things the air, because from this everything develops and into this everything dissolves again. Just as our soul, which is air, holds us together through its power, so also the whole cosmos includes “breath and air”. (Mansfeld & Primavesi, 2011, pp. 87,Nr.3).

Anaximenes presumably took the air as the primary substance because he observed opposite properties in the air; it could be warm or cold, compressed or diluted. From the observation of a breath he believed he could deduce a relationship between these pairs of opposites: “For when the breath is compressed and solidified by the lips, it becomes cold, while when the mouth is open it escapes, it becomes warm through dilution”. (Mansfeld & Primavesi, 2011, pp. 89, No.5).

So he saw compression or dilution as the basic principle for the difference in things: Clouds consist of weakly compressed air, rain of more compressed air and ice like earth of even more compressed air. The solidification of matter generally has its origin in the cold air, the thinness and looseness in the warm air. Wind was moving air. The primordial matter was now something that could also produce multiplicity. Thus, he had an idea of how the One could become the Many. The air was a primordial substance which, as a breath of life, absorbed all living beings into the unity of all things.

So here we find the connection between breath and life, a thought that one encounters again and again in the intellectual history – e.g. also in the idea of Christianity that the Holy Spirit originated from the Father and Son through breathing (see Wikipedia: Hauchung).

In this earliest school of the pre-Socratics, the Ionian school or the school of Miletus, the concept of “being” or “One” as a primordial substance or principle is in focus. In later schools also the “existing”  or the “Many” will become more strongly in the view and the question about which possibilities of the knowledge we can have about the being like the existing.

Paradigm shift

With the Ionic School, a world view was created that was derived from observations of nature and not from stories about interventions from a “supernatural”. This was a big step for mankind. But it was only a first step in a new direction. Of course, there were no clear answers. The reasons for such statements as “Everything is water” or “Everything is air” were only more or less plausible, necessarily not at all. Later Empedocles (-490 to -430) should still claim that “everything is fire” applies. Generally one spoke of the four basic elements water, air, earth and fire.

It was to take more than 2,000 years before a second step was taken in natural science, from which followed what we now call modern physics. Two new thoughts had to emerge:

First: One does not immediately try to understand the world as a whole, but one has to “begin it in the small”, thus with a simple and most clear phenomenon.

Secondly, one should try to understand the phenomenon not only qualitatively, but also quantitatively, so that an as exact as possible examination of the reasoning becomes possible. This requires sufficiently developed mathematics.

Galileo Galilei was the one to whom these thoughts came in the early 17th century and who was fully aware that he had founded a “new science” with a demonstration of the fertility of such thoughts. If one loves the term “paradigm shift”, which was so unnecessarily strained by Thomas Kuhn (Kuhn, 1973), then one can say that Galileo caused a paradigm shift. It would be the second – after the first paradigm shift by Thales and the Ionian school. Perhaps today we are experiencing a third paradigm shift in artificial intelligence with data-driven machine learning.

I will come back to all this in later blog posts. First of all, we must follow the path that has created all the conditions for this second and perhaps third paradigm shift. This includes, in particular, mathematics in which one learns quantitative and truth-preserving reasoning. Pythagoras (ca. -570 to after -510) is at the beginning of such a mathematics. In the next two blog posts we have to deal with him and his students.

The Pre-Socratics: A Brief Overview

“The greatness of the pre-Socratics […] lies not only in the fact that philosophy began with them. For this can be argued about and has actually been argued about. Rather, it lies in the fact that many essential questions, themes and conditions of science and philosophy are to be found for the first time in the statements of these pioneers that we have received.” (Mansfeld & Primavesi, 2011, p. 9).

With these sentences, Mansfeld and Primavesi introduce their reworking of a collection of texts handed down to us by the Pre-Socratics themselves and the most important secondary testimonies of their doctrines and work. And in a somewhat later sentence, they characterize these questions and themes as follows:

“If there is an exemplariness of the pre-Socratics, it is above all founded in a critical and rational attitude, which should not be a mere cultural-historical fact, but which today hardly has to be achieved less than at that time”.

The pre-Socratics showed this new attitude in dealing with questions about the origin and nature of the world. Instead of inventing or embellishing stories with supernatural actors, they used observations of nature to find answers to their questions. In doing so, they relied on regularities of nature and used analogies and generalizations. To a certain extent, they invented rational reasoning, discovered logos as a tool of thought in the search for the truth about the constitution and order of nature. Equipped with today’s knowledge, we recognize here the first beginnings of our scientific age.

Things in common

One will first ask oneself why it is precisely those philosophers who worked before Socrates that are grouped together. In particular, this question arises when one notices that Zenon (-490 to -430) and Democritus (-440 to -370), for example, can already be regarded as contemporaries of Socrates (-469 to -399).

It is probably decisive that with Socrates a completely new topic arose in this so young philosophy. The philosophers before Socrates were natural philosophers, they were concerned about the order in the world and its beginning, about the “physis” – they were, so to speak, the first physicists. Socrates, on the other hand, “was the first to call philosophy from heaven to earth”, as Cicero (-106 to -34) said in his conversations in Tusculum (Cicero, 2008, pp. V, 10-11), and Diogenes of Laertius learned through several intermediaries that Socrates had recognized that natural philosophy was no good for “us” [by which he probably meant himself and his discussion partners]. So, he turned to the moral doctrine (Laertius, 2015, p. 77). Ethics became the new topic, questions about the best rules for the togetherness of people and for a “good” and happy life. One speaks of a Socratic turning point: Away from physics – towards ethics. This marked the beginning of a new era in philosophy.

Legacy and life data

Only a few fragments of the works of most pre-Socratics have survived. Our knowledge of their doctrines is often based on equally fragmentary works by Plato, Aristotle, Theophrastos and many later doxographers.

Much has been written and puzzled about the pre-Socratics. I found the books by Schupp (Schupp, 2003a) and Pichot (Pichot, 2000) particularly illuminating, in addition to the Mansfeld and Primavesi collections. The work “Lives and opinions of famous philosophers” by Diogenes Laertius, written around 220, is also worth reading, alone because of the many anecdotes. One gets an impression of how much fantasy must have been involved in such reports.

Fig.1 shows the life data of prominent pre-Socratics. With the ellipses the philosophers are grouped together who are assigned to a certain school.

Fig. 1: Chronology of prominent Greek philosophers from the 7th to the 2nd century BC (© J. Honerkamp)

Thales of Miletus created the earliest of these schools, the School of Miletus. The Pythagorean school around Pythagoras and the Pythagoreans played as influential a role as the school of Elea, whose most important representatives are Xenophanes and Parmenides. Elea was a coastal city of an Greek settlement in southern Italy. The philosopher Heraclitus, who is also listed here, cannot actually be assigned to any school. The philosophers Anaxagoras, Zeno of Elea, Empedocles and Democritus are the most prominent representatives of the pluralists and atomists. These were mainly active in Athens.

Areas of activity

As can be seen in Fig.2, most of the important philosophers and mathematicians of Greek antiquity did not live and worked in the Greek motherland, but in the coastal cities of Greek settlements, whether in southern Italy, Sicily, present-day Turkey or Egypt. It was not until Anaxagoras that Athens became the center of Greek philosophy.

Fig.2: Areas of activity of ancient Greek philosophers (after (Symonyi, 1990, p. 59)).

You can understand that. The basis of every civilization was already at that time trade, administration and building activity. Trade, in particular, attracted people who were courageous enough to take on the dangers of a journey and who could also get used to the conventions and customs of other countries. From the coastal towns there must have also been a lively cultural exchange with the respective hinterland, e.g. with Babylonia or Egypt. All this may cause a certain alertness and open-mindedness and favor unconventional thoughts. Moreover, the coastal cities had become rich precisely because of trade and allowed lifestyle that even nonconformists could endure or even appreciate.

This was an atmosphere in which philosophical thoughts could flourish and which attracted philosophers. Thus, we know from some pre-Socratics that in their young years they had travelled the world a lot, whether as traders, as refugees from political persecution or in search of a teacher. Here you can also see that cultural exchange can bear fruit in the long run. The ascent of an overseas settlement of Europeans in the 16/17th century comes to mind at this point: The United States of America soon attracted many intellectuals from Europe and has been culturally and economically dominant since the early 20th century to the present day.

The cultural situation of the time

The pre-Socratics could not know that the questions of the origin and nature of the world as a whole were also being discussed in other important cultural areas of the world at that time. There had probably always been a lively exchange of ideas about gods between all the cultures around the Mediterranean. But independently of this, Confucius (ca. -551 to -479) appeared in China and taught that the world had an order and that man’s highest goal was to live in harmony with this order. In India, between -800 and -600, a collection of philosophical writings, the Upanishads, was created. In our context the Vaisheshika is particularly interesting, a system of a natural philosophy in which five basic elements are mentioned; besides the four, which were also named by the Greeks, there was also the ether here. In Persia, Zoroastrianism spread from -800 to -300, and in Palestine the biblical prophets worked.  The philosopher Karl Jaspers (1883-1969) therefore coined the term “Achsenzeit” (axis time) for the period from -800 to -200. In the meantime, however, many historians see this as not particularly meaningful.

There was a parallel development not only in the field of myth and a beginning natural philosophy. People learned to carry out elementary calculations with numbers and to find solutions for simple mathematical tasks independently of practical problems. The Dutch mathematician B.L. van der Waerden studied the early mathematics of the Chinese and Babylonians and found astonishing parallels in problems and proposed solutions. Likewise, strong similarities with the mathematics of the Hindus were noticeable. Since the knowledge about Pythagorean triangles was also used in the construction of the megalithic monuments in southern England (Stonehedge) (cf. a later blog post), van der Warden sees the source of all this knowledge in a megalithic culture of the period from -3,000 to -2,500, and thus assumes a very early cultural exchange in the field of mathematics (van der Waerden, 1983, p. XI).

But I rather believe that humans were always roughly at the same evolutionary stage and therefore had to solve the same problems in their world. Trade, administration and construction require planning and therefore some skills in arithmetic and geometry. This phenomenon of parallel development shows at most the universality of mathematical thinking.

Be that as it may. The fact is that the ancient Greeks decisively developed the mathematics they received from the Babylonians and Egyptians and made it into a first science. In a blog post about the Pythagorean school I will address this. With this step, which was initiated by this school, the Logos has thus become “alive”, for mathematics acquires its rigor and infallibility by using only logical rules of inference in its deductions (cf. a later blog post). The Greeks discovered the mathematical proof. This is the uniqueness of ancient Greek culture.

With the rediscovery of this culture at the time of the Renaissance in Western Europe, it was possible to build on it and thus set in motion a development that for a certain time led to the cultural and economic dominance of the Western world. However, this soon seems to be a matter of the past.

Cicero, 2008. Tusculanae disputationes/Gespräche in Tusculum. Lateinisch/Deutsch. Stuttgart: Reclam.

Laertius, D., 2015. Leben und Meinungen berühmter Philosophen. Hamburg: Felix Meiner.

Mansfeld, J. & Primavesi, O., 2011. Die Vorsokratiker. Stuttgart: Philipp Reclam jun..

Pichot, A., 2000. Die Geburt der Wissenschaft – Vone den Babyloniern zu den frühen Griechen. Wissenschaftliche Buchgesellschaft Hrsg. Frankfurt, New York: Campus.

Schupp, F., 2003a. Geschichte der Philosophie im Überblick – Bd.1 Antike. Hamburg: Felix Meiner.

Symonyi, K., 1990. Kulturgeschichte der Physik. Thun: Harri Deutsch.

van der Waerden, B., 1983. Geometry and Algebra in Ancient Civilizations. Berlin: Springer.

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