Galileo and the New Science: Experiment and Mathematics

The notion of a concluding rule was the dominant theme of the second part of this blog. Aristotle had already recognized such rules as decisive tool of thought for dialogues and discourses. A conclusion, according to him, is “a discourse in which some things are presupposed and then something different […] results from it”. He also realised that the rules themselves and the nature of the assumptions were important. Thus, he distinguished between the logical conclusion and the dialectical conclusion.

The logical conclusion was at the centre of his teaching on such tools of thought. Here one could already give clear concluding rules at that time. Even today, any introduction to logic begins with an examination of these concluding rules. The further development of Aristotelian logic in the form of propositional logic is the basis for all further studies of human cognitive abilities. 

The realization that in a sea of mysticism and dialectic there is the possibility at all to transfer the truth of statements to another statement, has driven me very much in my youth, when I had become so properly aware of this.

What’s the use of all this? One could establish such a kind of logical order between statements, in which it becomes clear which true statements follow from which other true statements.  One could start from true statements and build a whole thought building on them, which consists only of true statements.

But – what statements can you start with? That was the big question.

The mathematicians and logicians of antiquity had already demonstrated how this question about a beginning of true knowledge can be answered. Aristotle had shown, as already mentioned in an earlier chapter, that from the syllogisms of the 1st form all other syllogisms can be derived . He had thus solved the problem of how to arrive at true statements at all in such a way that he regarded the syllogisms of the 1st form as true propositions. These were immediately evident for him.

A few decades later, Euclid of Alexandria had then logically ordered the knowledge of geometric areas and bodies and thus created the first larger axiomatic-deductive thought structure. Here, too, he had to regard a few sentences as true at the beginning. They seemed evident from intuition.

So, at term-logic and at geometry already was demonstrated, how knowledge of secure transport of truth can be extended to an axiomatic-deductive system. Throughout the centuries, mathematics has remained an unsurpassed model for such an organization of secure knowledge.

There had been attempts to introduce a similar rigour of argumentation in philosophy and ethics. Such approaches, however, all ran into the sand (see Wikipedia: Mathesis universalis). Had they been the wrong areas for a rigor of thought according to the mode of mathematization?

Perhaps axioms did not necessarily have to be immediately obvious, but it was more important to find a source of true knowledge at all. Just as Euclid could refer to a large number of mathematical proofs and arranged this material according to logical points of view and, if necessary, supplemented it, a “small” axiomatic deductive construct of thought may also emerge after knowledge of some true statements by clarifying the logical relationship between them. Gradually one could then combine these “small buildings” into larger ones. 

Galileo Galilei was the first to recognise that nature was the source of true knowledge, as well as the importance of mathematics for the formulation of such knowledge. He was the first to describe a result of a physical experiment in the language of mathematics.

He certainly saw the implications of this combination of mathematics and experiment. It was immediately clear to him what a revolution a mathematisation represented for the understanding of science at that time. Thus, he spoke of a “new science”, which he had founded. His sentence “The Book of Nature is written in the language of mathematics” bears witness to this, as does the passage of his letter to the Tuscan Secretary of State Vinta in 1610: “Therefore, I take the liberty of calling this a new science discovered by me from its foundations”.

Galileo thus took up the Pythagorean idea again, but in a completely new way. He probably also saw that there is an order, that is, regularities in nature, which can be expressed in mathematical relations, and he had also become acquainted with the rigour of mathematical conclusions through his study of Euclidean geometry. But he also recognized that one must “question” nature through experiments in order to discover this order of nature, to make true statements out of it in mathematical language and to bring these into a logical order. Not empiricism alone, not mathematics alone, but experiment and mathematics are the pillars of his new, strict science.

We all know the consequences of this discovery, without which our world today would be a completely different one. At some point, however, this “new science” had to be discovered; nature and mathematics – or rather nature and logic – are too close to each other.

When is an implication true?

Why does empiricism play such an important role, why do ” inquiries ” of nature in the form of experiments play such an important role, if one wishes a theory after the model of Euclidean geometry, thus as axiomatic-deductive system?  So let’s look again at the modus ponens as a prototype of a logical conclusion:

A, A → B ⊨ B.

In order to deduce a statement that is incontestably true, premises A and A → B must be true. There is one statement, namely A, which occurs in both premises.  The implication forms the bridge to a new statement, namely B, which is then deduced. There must be such “bridges” in every concluding rule, because nothing can be inferred from statements that are completely independent of each other. Also, the syllogisms each have a middle term, which occurs in both premises.

A true implication A → B means that A is sufficient for B: Always if A, then B. Where is that the case?

We can find true implications by questioning nature. We then receive the following answers: “If I throw a ball into the air, it falls to the earth” or “If an electric current flows in a wire, there is a magnetic field in its environment”. The experimental physicists are therefore suppliers of true implications, which we then also call laws of nature.

True implications can also be found if we transform the statement “All Greeks are human beings”, for example, into “If x is Greek, then x is human.”

Here we have formed the terms “Greeks” and “humans” in such a way that the implication is true. The statement thus becomes true by the fact that we form the concepts accordingly.

Then we’re already at the end of our rope. For all other implications the dialectical conclusion is probably responsible, i.e. here an implication belongs to the category of sentences about which Aristotle said:

Sentences are credible if they are recognized by all, or by most, or by wise men, the latter by all, or by most, or by the most experienced and credible.

We can add: And what is recognized by “wise men” also depends on time. Let us only think of the laws of legal science, e.g. the law §1356 of the German Civil Code (BGB), which until 1977 still read: “The woman manages the household on her own responsibility. It shall be entitled to be economically active to the extent compatible with its matrimonial and family responsibilities.”

When it comes to regulations for human coexistence, morals, customs and traditions, yes, everything that nature does not tell us, there can be no generally acceptable true implications. We are referred to the dialectical conclusion and thus to a negotiation about which implications are to be set as true. So here we can only “set” truth, not find it.

The consequence of this is that the statements of the natural sciences are universally valid, but there are countless religions and legal systems. In the natural sciences there is also a change over time. However, as we will see in the next chapters, this is a kind of evolution, a “finding of the ever better” basic assumptions based on ever new discoveries about nature’s behaviour.    

For some time, it was believed that rules for human coexistence could also be read from human nature. Such a doctrine of natural law can be used for the most diverse ideologies. Ultimately, it is always the “wise men” who generally decree the sentences, which actually only seem credible to some, to be true. The Catholic Church still adheres to this doctrine today. For centuries, however, one has been talking of a “naturalistic fallacy” when one infers “ought” from “is”. An implication that links statements about “is” with statements about “ought” be cannot be read from nature.  We owe the first explicit formulation of this insight to the philosopher David Hume (1711 to 1776).

The new science of Galileo Galilei

The “hot” topic of nature research at the time of Galileo was motion. In his work “Discorsi” he says: “Nothing is older than motion, and about it there are neither few nor few writings of philosophers. Nevertheless, I have experienced their peculiarities in great quantity, and among them very worth knowing”.  The motion had already been an issue for the pre-Socratics. Aristotle had distinguished different classes of motions and had found a special explanation for each. Motion is the phenomenon that we encounter most immediately, but which can also be observed in the sky as the course of the stars. If you wanted to learn anything at all from nature, you first had to “understand” the motion.

What was the experiment Galileo used to study motion, and what form of mathematics did he use to describe the results? How Galileo approached the problem is remarkable and symptomatic of the course of modern science. He did not focus on “the whole” as the pre-Socratics did, nor did he try to create a general overview like Aristotle. Instead of this he started it “on a small scale”. He let a small, smoothly polished ball roll down an inclined plane, i.e. an inclined narrow wooden board into which he had a channel buried – a child’s play in modern times. 

This turn of the view alone demonstrates the independence of his thinking, as it is characteristic of a genius. Even in Goethe’s day, philosophers had to think about “what holds the world together at its innermost”, and Faust has only mockery for Mephistopheles when he fights for people: “You can do nothing on a large scale, and now you can begin it on a small scale.  Religions only know this question about “the whole”.

Actually, Galileo has taken up the trail of Xenophanes again. If one trusts that it will be possible to “search for the better”, one appreciates also “small successes” in the search for knowledge; one looks for a template on which one can build. This is how modern science, modern technology works. That is why there is research and also development.

Galileo now had to measure times and distances for each roll of the sphere. How he could determine in particular a time unit in which he used his feeling for an even measure in a song is described in detail in (Fölsing, 1983, p. 177ff). In his notes, he reports: “… with probably a hundred times repetition, we always found that the distances behaved like the squares of time, and this for every inclination of the plane, that is, the channel in which the sphere ran. (Discorsi, after (Fölsing, 1983, p. 174)).

Galileo formulated the result in the form of proportions, ratios, as was customary at the time and as had not yet been learned in any other way. Time periods and distances were variables of different physical dimensions, and one had not yet understood how such variables could be directly related. Therefore, he wrote down his result not in the form in which the distance proportional to the square of the time required is given, but as equality of the ratios of two distances and two squares of corresponding times. In a graph, in which the times are plotted against the distances, this presents itself as a semiparabola, as is indeed found in the Dialogo Quarto of Discorsi Galileis in the discussion of thrown bodies (Fig. 1).

Fig. 1: Semi-parabola as drawn by Galileo during the discussion of tossed bodies ( (Galileo, 2015, p. 276) after (Simonyi, 1990, p. 200).

Here one must say something about the state of mathematical knowledge of Galileo’s time. This could not have been higher than what one knew from late antiquity and how it was probably also taught at the universities of the time in the faculties of the artists, the faculties of the “artes liberales”, the “free arts”. Thus, in mathematics one thought predominantly in geometrical terms, since geometry had always been dominant in antiquity. It was only about a generation after Galileo that René Descartes (1596 to 1650) was to develop an “Analytical Geometry” in which geometric relations could be expressed as arithmetic relationships. Geometric problems could thus be analysed within the framework of arithmetic. Afterwards mathematics became essentially arithmetic and algebra, the doctrine of transforming arithmetic relations. But the fact that the relationship between times and distances in the case on the inclined plane could now be represented by a parable fitted well into the world in which mathematics consisted for the most part of geometry.

Galileo had also been initiated into the beauty and stringency of Euclid’s geometry by an engineer and geometer Ostilio Ricci. He was already “infected” by the idea of having to logically arrange his experimental statements. He was therefore also looking for a principle from which all these statements could be derived. However, he was caught on the wrong track. Four years later he was able to correct this error (Fölsing, 1983, p. 175ff). Such a “theory” for a falling movement would soon have been obsolete anyway. He could not have imagined that at the end of his century a theory would emerge that could explain all motions in the sky and on earth from a few axioms. His falling motion became a small special case in it.

The English physicist and mathematician Isaac Newton stood on Galileo’s shoulders during the development of this theory. The first axiom in this theory was based on Galileo’s hypothesis, on which he had been guided in his falling experiments.  It was the hypothesis that, on a horizontal plane, the motion of the rolling sphere “would continue forever at a uniform speed” if it were not affected by unevenness of the ground (Galilei, 1982, p. 30). 

For Aristotle, a motion that gradually comes to rest through friction is the natural, actual motion. So this is a process for him, only with “force” the motion can be maintained. The rest is then a very special state, “essentially” different from a motion.

With Galilei, on the other hand, the uniform motion is the natural one, and this is a state. Through external circumstances such as friction it can come to rest, but this is only a special state of this kind. This insight stands at the beginning of modern physics.

With which statements can one begin with the formulation of an axiomatic-deductive system for a theory of motion? The answer to this question was obvious for Newton: Galileo’s insight, which was later formulated as the law of inertia, must have been at the beginning of a theory of motion.

Let us take a closer look at which statements have been put at the start in this theory, but also in other physical theories. We will see that this happened in very different ways. But let us first get an overview of these theories in the next chapter.

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.

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.