Zenon of Elea, the Motion or how to find seeking the better

A disciple of Parmenides, Zenon of Elea (around -490 to around -430) still attracts special attention today because of the paradoxes with which he annoyed his philosophically interested contemporaries. We now understand paradoxes to mean argumentations that lead to contradictions because an unclear or incorrect idea of a concept is at play. That was the case for both movement and infinity at the time. Today we have clear ideas about these terms and can resolve the paradoxes that have so unsettled his discussion partners. 

Zenon wanted to support the theses of his teacher Parmenides with his considerations. At least this is how we read it in Plato’s dialogue PARMENIDES, in which he lets Zenon say: (after (Mansfeld & Primavesi, 2011, pp. 365, No. 5):

In reality, my writing is something of a support for the Parmenides’ thesis, which is aimed at those who try to make him ridiculous.

His four movement paradoxes are particularly famous. We want to deal here with the third paradox, because it is based on an error, the enlightenment of which stood at the beginning of modern physics. This is about the apparent contradiction between the observation of a flying arrow and the Parmenides assertion that this movement of the arrow is only apparently present, since the being remains in absolute silence.

A flying arrow, Zenon argues, is at a given time in a certain place, which is always as big as the arrow itself. Since he was there in the “now”, he could not be in motion then. So, he is at rest in every moment, the arrow actually stands still. The motion we’re watching is a fake.

I don’t find that argument convincing. After all, Zenon already shows a scepticism towards our everyday perceptions, albeit in the most extreme form. In any case, Zenon seems to me to have the idea that in every “now” the state of an arrow is determined by a location alone, and he assumes that there is no motion. Aristotle says in contradiction: “In the “now” neither rest nor movement can take place” (Mansfeld & Primavesi, 2011, pp. 383, No. 23). So he already sees that Zenon only thinks of the most obvious when describing the state of a moving body. Zeno’s argument does not convince him either. But his counterargument does not go any further as well.

The arrow paradox dissolves when one knows that the state of a body in space is determined by a location and (!) by a velocity (or impulse) at any time, in any “now”.

This was Galilei’s discovery in the early 17th century. He studied the motion of a small ball as it rolls down an inclined long wooden channel. Not only did he find that the distance she travels on the wooden channel increases with the square of time. He also extended the channel beyond the inclined part and observed that the ball on the horizontal channel continues to run the longer the less its movement is affected by unevenness of the ground. He concluded that, with ideal ground, it would then have to continue to run forever. The motion remains as it is if there are no external influences on what is moving.

So movement is a state. In medieval “impetus theory” the movement was still a process: an “impetus” had to work constantly. This would be given to the body at the beginning, kept the movement upright, but was also slowly used up, so that it gradually came to a standstill. Galilei, on the other hand, attributed a slowing of a motion to external influences, e.g. friction. By refraining from external circumstances, he was able to discover a principle of nature that would prove extraordinarily fruitful for the further development of physics. I’ll come back to that soon. But first we have to look at how a velocity in the “now” can be grasped in concrete terms.

The instantaneous velocity

Galilei did not yet have the possibility to calculate an instantaneous velocity. In mathematics he was still at the level of the ancient Greeks, where geometry as a description of nature stood in the foreground. But his contemporary, the French philosopher and mathematician René Descartes, discovered how geometric problems could be converted into arithmetic problems. An “Analytical Geometry” was created, which represented a great advance compared to ancient mathematics and which for the first time enabled one to go beyond the status of the ancient Greeks.

One now learned to describe the location of a point in a coordinate system and to see such points as locations of material bodies, if one abstracted from their expansion. One could also represent the place x(t) of the body in dependence on the time t in a coordinate system.

An average velocity in a time period dt was easy to calculate by forming the ratio dx/dt, where dx may be the distance covered in the time dt. But it became difficult if you wanted to extrapolate to the instantaneous velocity, i.e. if you had to determine the velocity in the “Now”. The time span dt should actually be zero, the distance dx thus also, and the ratio 0 to 0 makes no sense. One had to choose somehow a very small time span dt, which should be however arbitrarily small, but still unequal 0. Somehow these quantities had to have something to do with the “infinitely small”. They called them infinitesimal. It was not a clear idea, but they succeeded in consistently calculating the ratio dx/dt in the “now”, the “differential quotient”. This “infinitesimal calculus” was developed independently by two great thinkers of the time for general functions f(x): Isaac Newton needed this knowledge for his reflections on motion. Gottfried Wilhelm Leibniz regarded it as a purely mathematical problem which had to be solved if one wanted to determine the tangent in a point of the curve of a function in a diagram.

For some time such calculations were very popular; they inspired many new ideas and questions. At the end of the 18th century, mathematicians were no longer satisfied with the justification of such calculations with infinitesimals. The Italian mathematician Lagrange found a method for calculating the differential quotient without having to use the term infinitesimal. In the 1960s, a new type of number could finally be defined using hyperreal numbers in a so-called non-standard analysis. A clear definition of the infinitesimal was now possible: they were certain hyperreal numbers.

Mathematics is sometimes like physics and actually every science: New concepts are not always clearly defined at first. But you can already use them and when you notice that they are “good for something”, you start to take care of the conceptual basics at some point. But it often takes some time for satisfactory clarity to be achieved.

The Evolution of Motion Theory: One Finds “Seeking the Better”

Knowledge of how to calculate the instantaneous velocity from a time-dependent spatial coordinate was a prerequisite for a theory of motion in the language of mathematics. While Galilei had discovered a relationship between distance and time in free fall, one theory was now concerned with describing the position and velocity of a body as a function of time.

The physicists and mathematicians of that time knew their ancient models very well. Especially the ELEMENTS of Euclid of Alexandria, in which he brought the then known laws of geometry into a “logical order”. Euclid has thus set a benchmark for what a mathematical or physical theory should look like. At the beginning there are definitions, conventions and axioms. According to this, all statements of the theory must be logically deducible from the axioms according to mathematical rules.

Newton formulated his theory according to this model. Galileo’s idea that motion can be a state was the first axiom he incorporated into his theory of motion, which is now known as “Newtonian mechanics”: “A body remains at rest or in straight-line, uniform motion if no forces act on it.

Here, of course, something must have been said about space and time in the definitions beforehand, so that one knows what uniform motion means. So you have to know what a straight line is, and you have to say something about the course of time before you can speak of a uniform velocity, one that is constant in direction and size.  Only then can one speak of this particular motion in the axiom and postulate that this motion remains if there is no external influence on the moving body.

In a second axiom, Newton then logically describes a procedure for formulating a mathematical equation for the case that an external force now acts on the body. With a suitable mathematical expression for the force, one can then calculate all motions in the sky and on earth from such an equation of motion, taking into account existing circumstances.

This Newtonian theory of motion, briefly outlined here, was regarded as the only ideal of a scientific theory for over 200 years from the end of the 17th century, and its structure as an axiomatic-deductive system represented a model for future sciences.

Here is an opportunity to speak of two other theories of movement: on the one hand a theory that Aristotle had formulated about 2,000 years earlier, and on the other a theory that Albert Einstein developed some 300 years later and was soon called “special relativity theory”. One can very beautifully demonstrate with these three theories how indeed in the course of time people found “seeking the better”. This story doesn’t describe a special case. One can find many such examples. But let us first characterize the other two theories:

Aristotle was a great systematist, and so he first of all distinguished the motions in the sky from the motions on earth. He divided the earthly motions again into movements of living beings, into natural and finally into forced motions. He gave a different reason for each type of movement. The motions in the sky showed the eternal order. In natural motions the “disturbed order” was restored, e.g. smoke rises to heaven and a stone falls to earth, because light has its place above and heavy has its place below. In a forced motion, a force must always act, otherwise the motion would come to a standstill.

This Aristotelian motion theory had more than 2,000 years of existence. It was still taught in the academies during Galilei’s lifetime, and Galilei had studied it intensively himself until he finally overcame it.

Newton’s theory finally completely replaced it, for it is obviously “better”. From the equations of motion of Newton’s theory, it was possible, with a certain expression for the force which a body exerts on another body due to its mass, to derive the three Kepler laws for the motion of the planets around the sun, even to accurately predict the return of a comet. So with fewer assumptions one can explain more phenomena. The theory also makes predictions that can be tested and indeed have been confirmed. To this day, it is indispensable for calculations in everyday motions.

The theory of motion developed by Albert Einstein, the special theory of relativity, is in turn better than Newton’s theory. The motivation for the development were problems with the idea that the whole universe is filled with a subtle substance, an “ether”, which has prevailed since ancient times. It should also be the carrier of the electromagnetic waves, which at that time had only been known for about two decades. Ether should also mark absolute rest and one wanted to measure motion of earth versus the ether. Whenever and however you did it, you couldn’t see such motion.

Einstein took the bull by the horns, so to speak. He made this negative result the principle of his new theory: “The speed of light in any inertial system is independent of the speed of the light source”. So, this means that no matter how I move relative to the light source, I always measure the same speed for the light.

This theory is also structured as an axiomatic-deductive system. As such, it is even particularly “elegant”, because it is based only on this principle, as on a “principle of relativity” already known from Maxwell’s theory of electromagnetic phenomena. A wealth of phenomena could then be predicted; some of them differ strikingly from our everyday experiences, but in the mean time they have all been proven experimentally.

The hypothesis that there should be something like an ether was dropped. He was no longer needed. There is no absolute rest, but instead an absolute speed: the speed of light as measured in a vacuum. It is an upper limit for the transmission of effects.

When comparing these two theories, one finds that the special theory of relativity is an extension of Newton’s mechanics, in the sense that the smaller the speeds to be considered are in relation to the speed of light, the better the agreement between the statements of both theories. For velocities close to the speed of light, however, the astonishing phenomena already mentioned above are predicted, which have all been confirmed in the meantime.

A useful yardstick for the “goodness” of a theory is its scope of validity. Newton’s theory already had a very large validity range, because with it one can explain all motions which are “non-relativistic”, i.e. sufficiently small compared to the speed of light of approx. 300,000 km/sec. Here one could also try the theory of relativity. That wouldn’t be necessary, it’d just be harder. If one now considers increasingly higher speeds, the statements of the two theories will differ more and more. One leaves the scope of Newton’s theory but remains within the scope of the special theory of relativity. In this sense it is therefore an extension of Newton’s mechanics and thus the better theory.

If we look at the image of an evolution of theory, then we can say that the Aristotelian theory survived 2,000 years because there was no other theory that could become dangerous to it. But Newton’s theory was such a dominant competitor that Aristotle’s theory became extinct. The special theory of relativity is then a further development of Newton’s theory, so that there are now two theories that both have their own habitats. Where these overlap any theory can come into its own.  

Aristotle – Newton – Einstein: Aristotle has submitted, Newton and Einstein “found seeking the better”. Who knows when someone will come and find something better, and what further insights we will gain about motion and thus about space and time. Only one thing seems clear to me after 2,500 years: The way of Xenophanes to find “seeking the better” is also the better way to knowledge.

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 Pythagorean Crisis

After having studied the mathematics of the Pythagoreans and other Greek philosophers of the time in the last blog post, we now have to deal with the conclusions the Pythagoreans drew from the results of their studies. For this we must first have a look at the Pythagorean community in order to understand the fate of Pythagorean philosophy.

The Pythagorean Community

The group around Pythagoras consisted by no means of predominantly unworldly philosophers who enjoyed mathematical problems alone. From later ancient sources (Iamblichos, 3./4. century) one learns that the Pythagoreans later had two groups: “’Mathematicians’ were called those who had been taught more in the special teaching of his science, cultivated with regard to accuracy, ‘acousmatics’, those who had heard only the brief regulations without exact justification”. (Mansfeld & Primavesi, 2011, pp. 137, No.20). 

This has already shown how social divisions arise:  For the acousmatics (ἄκουσμα = Akousma = auditory perception) the mathematicians were not real Pythagoreans at all. On the other hand, the mathematicians saw themselves as a kind of “higher” Pythagoreans. Franz Schupp mentions in this context that later there had been a similar distinction between Gnostics and early Christians: There were “Pistiker”, the simple believers, and “Gnostics”, who have a higher insight or only claim such (Schupp, 2003a, p. 67). In our time they are in the religions the “common people” and on the other side the group of priests and theologians.

But the comparison does not fit in all aspects. For one could not call it unreasonable if Pistikers or believers were to cast doubt on the teachings of the Gnostics or fall away completely from the faith. They then only turn away from the decisions of certain authorities. It would, however, be unreasonable, i.e. acting against reason, if acousmatics experts would not recognize mathematical proofs in principle. They could be informed only to the extent that they would be able to verify the accuracy of the evidence.  

Both ways of thinking, the ones close to the myth and the one close to the Logos, existed in the community and of course also in individual minds. And for later purposes it is advisable to make another distinction in the myth, namely in a “purely philosophical” direction, which saw a similarity between structures of nature and the structure of mathematical concepts, that is, between the “principles of mathematics and the principles of things in existence”, and in a strongly religious direction, which interpreted this view of the world to such an extent that one believed one could deduce from it strict rules for life and coexistence. To this religious direction I also want to count the doctrine of the soul, the assignment of numbers to any virtues as well as all the rules for life, which one finds e.g. in the list of “Akusmata” of Iamblichos (Mansfeld & Primavesi, 2011, pp. 193, No.102).

What I have said about the doctrine of the soul should, of course, apply generally to the religious direction of Pythagorean thought. This is not meant to play a role here.  

The Principles of Mathematics and the Principles of Being Things

If one wants to get an overview of the mathematics of the early Greeks, then one must take the trouble to study the “Elements” of Euclid of Alexandria. Euclid must have written this textbook in the years around -300, and according to Proklos Diadochos (412 to 485) he “used much of Eudoxos, brought to a close much of Theaitetos’ treated, and what was represented by earlier only superficially, he supported by incontestable evidence”. We don’t know if that’s a fair judgement.

In any case, we must note that among the mathematicians who later appear in any lists of early Greek mathematicians, there have not been too many Pythagoreans. From the Pythagorean communities, however, we know two groups whose leaders Philolaos (-470 to -399) and Architas of Taranto (-428 to -347) were famous mathematicians, and whose mathematical results are still remarkable today.

From Philolaos we know about the collector of ancient philosophical writings of Stobaios (5th century), which world view the Pythagoreans derived from their mathematical studies. In one of the fragments that we have of him:

 “And it is truly all that can be recognized, number, for it is not possible to understand or recognize anything without it” (Mansfeld & Primavesi, 2011, pp. p. 147, No.28).

The concept of number is therefore the basis of every knowledge. If you want to recognize something, it must be possible to formulate it quantitatively. And in the more quantitative knowledge one discovers the order and harmony of the world. Another fragment says:

“However, it would be impossible that one of the things recognized by those who exist and by us humans would have come into being if it had not already existed the essence of the things from which the world is composed: the essence of the limiting and unlimited. But since these principles […] are not equal, […] it must necessarily be united by such harmony if it is to be contained in the world order.” (Mansfeld & Primavesi, 2011, pp. 145, 27).

With the concept of the number also the unlimited comes into view. From the fact that there are being things, one must therefore conclude that this Unlimited exists in harmony with the Limited. It is obvious that it was especially the discovery of the rational frequency ratios of the strings, e.g. of the tetrachord, which suggested the connection of numbers with a harmony. (Aristotle, no date) says:

“Since they became aware of the fact that the relationships and laws of musical harmony can be represented in numbers, and since all other phenomena also showed a natural relationship to numbers, but numbers are the first in all nature, they came to the conclusion that the elements of numbers are the elements of everything that exists and the entire universe is a harmony and a number.

Once one has conceived such a thought, one also associates the beauty of the regular polyhedra with the numerical proportions readable there. 

But the Pythagoreans went further. Aristotle mocks it: “What could only somehow be found in agreement between the numbers and harmonies on the one hand and the processes and parts of the vault of heaven and the entire structure of the world on the other hand, they collected and sought to establish a connection; but where they missed the opportunity to do so, they were not afraid of artificial assumptions, only to make their systematic procedure appear as strictly uniformly carried out.

He gave an example of this:

“Since they consider the ten to be the perfect number, and believe that these is concerned with the entire nature of numbers within itself, they make the assertion that even the bodies which turn in the sky are ten in number, and since we know only nine in real experience, they invent a tenth body in the form of the opposing earth.

Philolaos had invented this opposing earth. One also had an argument ready for the fact that one could never see these: It should always be right behind the sun when seen from Earth.

One remembers involuntarily some hypotheses of physics, e.g. the assumption of Wolfgang Pauli that there must be a certain particle, that a certain energy carries with it at the decay of a neutron, so that the preservation of the energy could also be confirmed here. But the difference is quite substantial: In Pauli’s day, the hypothesis was a mandate to examine, in Philolaos it was unthinkable to verify. So, it was pure metaphysics.  

The crisis: Incommensurable variables instead of numbers

But not from these and other unverifiable statements did the Pythagoreans threaten disaster with their philosophy. This came with a discovery that at its core shook her philosophy. To understand that, you need to make a little extra effort:

The Pythagoreans knew the natural numbers {1,2,3,…} and the positive rational numbers, i.e. ratios of natural numbers like 3/4 or 2/5. Since these numbers should reflect the condition of the world, also all things had to have a measure which can be expressed by these numbers. The length of a route, for example, had to be expressed by a multiple of a unit length and for two routes there had to always be a common unit length. Expressed in numbers: there had to be a common divisor g for two numbers, so that the two numbers m and n could be represented as integer multiples of g. The largest of these divisors is then called the “largest common divisor”. For the Pythagoreans, therefore, the length of all routes had to be “commensurable” in this sense, yes, all things in the world had to be commensurable, i.e. have a common measure. For numbers rule the nature of the world.

The ancient Greeks had even found an algorithm with which they could easily calculate the greatest common divisor of two natural numbers. This is demonstrated in Fig. 1:

Fig. 1: The algorithm of alternating removal to determine the largest common divisor of 44 and 12: One subtracts from the larger number, here 44, so often the smaller number, (here 12), until the remainder is smaller than 12. Then 12 is the larger number and you repeat the game – until you get zero. The last number to be subtracted for this, here 4, is the largest common divisor or common measure. https://de.wikipedia.org/wiki/Datei:Euklidischer_Algorithm.png

This procedure is found in Euclid’s “Elements”; but already the Pythagoreans are said to have known this algorithm.

But now the Greeks, and even the Pythagoreans, knew the so-called Pythagorean theorem. In a square of side length 1, the diagonal has a length whose square equals 12 + 12 = 2 according to this theorem. But they didn’t know a number with a square equal to 2. Side length and diagonal can therefore not be commensurable.

If someone is still looking for any way out, one can convince him with a strict proof:

Let us call the quantity whose square equals 2 already √2, as we do today, and make the assertion that this quantity can be represented as a number in the sense of the ancient Greeks, i.e. as a ratio of natural numbers. So then

√2 = m/n.

The numbers m and n can be chosen without loss of generality in such a way that the statement

A:= “m and n are relatively prime”

is true. Squaring the equation results in

2 = m2/n2, therefore also m2 = 2∙n2

It follows that m is an even number, so that m = 2∙k can be written, so m2 = 4∙k2. Thus, with the help of the previous equation 4∙k2 = 2∙n2 is also valid, i.e. n2 = 2∙k2, which finally means that n like n2 is divisible by 2. So: m and n are divisible by 2.

Altogether we conclude that m and n are not relatively prime, i.e. statement A is false, although it was assumed to be true. Then A can’t be true. Because one can never logically conclude from a true statement that it is wrong. That’s plausible. In a later blog post I will show this logical conclusion strictly formal within the framework of a so-called propositional logic.

The ratio √2:1 can therefore not be represented as a ratio of natural numbers m/n.  So, a natural number and a quantity like √2 have no common measure, they are incommensurable.

The quantity √2 was not accepted as a number until modern times. Such non-rational, irrational numbers were seen at most as “impossible” or “imaginary” numbers. You could count with them, their square was equal to 2, but you could never write down these quantities completely, as it was “appropriate for a number”. Only at the end of the 19th century did people learn to define the concept of numbers in such a way that even irrational quantities could be accepted as a special class of numbers.

This discovery that in nature there can be distances whose length cannot be represented as a rational number has greatly shaken the Pythagoreans. It was believed that this knowledge must remain secret. Iamblichos (245 to 325), in his book On Pythagorean Life, tells the story that someone is said to have divulged this discovery. He was then excluded from the common circle and later even perished in the sea (Mansfeld & Primavesi, 2011, pp. 171, No.61,62). Like all anecdotes from this time, you can sprinkle them in for entertainment. You don’t have to believe them.

The legacy of the Pythagoreans

The ancient Greeks were the first to turn the mathematical knowledge, handed down to them by earlier peoples, into a science. They not only found interesting relationships between several numbers and between numbers and geometric figures, bodies and sounds. Even more significant is the fact that they discovered an argumentation that is incontestable, that is, what we call mathematical proof today.

Euclid of Alexandria collected this knowledge and brought it into a logical order. In this order definitions and axioms are placed at the beginning, and all knowledge is derived from them in the form of mathematical proofs. An “axiomatic deductive system” was thus created. Thus, the idea of a strict science was not only born in the time around the year -300, but was also already realized once. This idea still inspires everyone who thinks about what a science actually is. In Die Idee einer Wissenschaft – Ihr Schicksal in Physik, Rechtwissenschaft und Theologie (The Idea of a Science – Your Fate in Physics, Jurisprudence and Theology) I have elaborated on this further (Honerkamp, 2017).

Also, in the years after Euclid the mathematics of the Greeks progressed further. With Archimedes (ca. -287 to -212) it has reached a temporary climax. Historians of science, however, speak of the first signs of the disappearance of creative forces in the coming centuries (Russo, 2005).

Pythagoras and the Pythagoreans gave in ancient Greece the impulse to this first bloom of a science. Though, with their doctrine of harmony they exaggerated it, their religious zeal including their rules for the way of life today seems to us highly strange, sometimes bizarre. They failed with their idea of numbers as a basic pattern for nature. 

Yet they were on the right track. It was not until the second attempt, 2,000 years later, that a combination of mathematics and natural science was to emerge, which then led to an understanding of nature from which people could develop machines rich in both blessings and horrors.

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).

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