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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 CommunityThe 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 &#38; 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 &#38; 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 ThingsIf 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 &#38; 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 &#38; 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 numbersBut 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 statementA:= "m and n are relatively prime"is true. Squaring the equation results in2 = m2/n2, therefore also m2 = 2∙n2It 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 &#38; 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 PythagoreansThe 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.