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:

Arithmetic

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

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.

Music

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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