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.

Socrates, Plato and Aristotle

If I wanted to write a treatise on the pre-Socratics, I would now have to deal in this blog post with the so-called pluralists and atomists such as Anaxagoras, Empedocles, Leukipp and Democritus. But I do not want to give an outline of the philosophy of the pre-Socratics here. This has happened often enough elsewhere, and certainly after a much more detailed study of the literature of the past millennia than I can show. I am essentially interested in the thoughts of the pre-Socratics, in which one can discover the precursors of the way of thinking of modern physics, mathematics and logic.
There it is without doubt the three basic ideas which have been expressed so aptly and so early in the history of ideas by the school of Miletus, by the Pythagoreans and by Xenophanes: In nature there is such a thing as a causal connection, the regularities in nature can be formulated in the language of mathematics, and knowledge can be attained by “finding the better by searching”. But there can be no ultimate knowledge.
The first tentative steps of the pre-Socratics towards basic concepts such as ” motion ” are also remarkable. The long history of the clarification of this concept over the millennia clearly shows how much effort it takes to gradually move from the darkness of the first reflections to a clear idea with which one can argue reliably.
If I now follow the development of the “logos” in the history of philosophy, I will be even more selective. I will concentrate on the highlights, i.e. the works where substantial progress has been made. Of course, one often recognizes these only by looking back from today’s standpoint. The focus will be on such developments in logic, mathematics and physics that have become significant for the present state of the art in terms of both methods and the formation of concepts.
So, if one begins to look at the development in the time after the pre-Socratics, one must first speak of the three greats of Greek philosophy: Socrates (-469 to -399), Plato (-428 to -348) and Aristotle (-384 to -322). As already mentioned in the first blog post, Socrates introduced a new topic into philosophy: Ethics. But it was not only this new topic that led to a new era in philosophy. Anaxagoras from Kleizomenai ( Ionia ) had moved to Athens in the year -462 and had made the thoughts of the Milesians and other pre-Socratics known there.
Where more people can come together, it is also more likely that people with the same interests will meet and talk about them. In addition to teacher-pupil relationships, communities can now also emerge in which discussions can take place on an equal footing. Dialectics came into fashion. Already Zenon of Elea had been her great friend, Aristotle had later even praised him as their inventor (Mansfeld & Primavesi, 2011, pp. 361, No.4). Attic democracy flourished, it was an early type of democracy in which the “people of state” ruled, where the people of state consisted of the full male citizens of the city of Athens who had reached the age of 30. These full citizens could also turn to matters which were not directly necessary to daily life; for such work there were enough slaves, women or immigrants. Especially in political meetings and in court, eloquence and the art of dialectics were in demand, i.e. to find the right reasons for opinions and to spread those that could be applauded by listeners. Sophists taught for money to master this art, even to be able to skilfully involve the opponent in a contradiction in discussions.
About Socrates we know some about Diogenes Laertius (Laertius, 2015, pp. 67-90), in particular he is supposed to have said: ” I know nothing except that I know nothing” (DL 83). And “the good is nothing small, but start with small things” (DL 83). But we don’t know any of his writings. Plato, however, puts a lot into his mouth in his works. These works always represent dialogues that Socrates conducts with a wide variety of dialogue partners and in which it is always a matter of clarifying a concept or a question.

Plato

After the death of Socrates, Plato, in his years of teaching and travel, first stayed with a follower of the philosophy of Parmenides, “travelled to Cyrene to the mathematician Theodoros and from there to Italy to the Pythagoreans Philolaos and Eurytos; from there to Egypt to the prophets” (Laertius, 2015, p. 141). Thus he collected various impressions and combined them to a grandiose mythical thought building, which is still effective today and continues to inspire many people.
He must have been a great storyteller. In his works he described in the form of a dialogue the effort to find concepts and answers to the question of correct behaviour or a good life. Who first chose this literary form is controversial. “It is said that Zenon, the Eleate, was the first to write dialogues, but Aristotle names as such […] Alexamos from Styra […]. I, on the other hand, think that Plato, through his strict treatment and training […], has secured himself the right to first place […]”. Dialogue is “a question-and-answer execution of a philosophical or political theme”, but dialectics “is the art of persuasion, through which we prove something void or right on the basis of the question-and-answer procedure of the undermentioners” (all quotations: (Laertius, 2015, p. 159)).
Diogenes Laertius says of Plato’s works: “TIMAIOS belongs to the field of physics, to the logic belongs POLITIKOS, KRATYLOS, PARMENIDES and SOPHISTES” (Laertius, 2015, p. 160).
When he, as in TIMAIOS, came to talk about the ideas of the pre-Socratics about the cosmos, he did so in mythical form or tied to Pythagoras and saw the world built up from geometric forms. At that time, logic was still predominantly understood to mean the work on terms, i.e. the uncovering of the relationships between related terms, e.g. their classification as superordinate or subordinate terms. A clearer definition was expected from such a division of terms, called “dihairesis”. One could already see, however, that a definition often required definitions of the determinants, thus leading to an infinite regress.
Aristotle grew up in the Academy with this method. Thus, in his logic, which he was soon to develop, the concepts were also at the centre. This thus became a so-called term logic. Modern logic, on the other hand, is a propositional logic. In the next blog post I will work out the difference exactly.
Plato mainly used induction, i.e. the conclusion from special to general, for a demonstration of evidence, “which through some true cases opens up the same truth for other cases in an appropriate way”, as Laertius (Laertius, 2015, p. 161) says. For a long time this conclusion was considered “appropriate”, only David Hume stressed that this conclusion is not mandatory, thus not always “appropriate”. In modern logic there is no place for this conclusion either.
Plato’s central theme, however, was the human soul and how it can express itself in language, ethics, art and politics. He adorns existing myths, redesigns them or invents completely new stories.

The Platonic Academy

For the development of the “logos” all this does not give much. It is interesting in this context, however, that he also promoted training in analytical subjects, probably motivated by the impression that Pythagorean mathematicians made on him. He was not only a great narrator, but also a good organizer and designer. He founded a school where young people were taught philosophy and science. For this he acquired, probably in -387, a plot of land near a grove called Akademeia, because it had been dedicated to the Heros Akademos. In addition to metaphysics, ethics, dialectics and the doctrine of the soul, physics and especially mathematics were studied there, which soon became part of the basic education of every student of a philosopher. The academy remained, with interruptions, over many centuries, it was closed only around 530.
The Platonic Academy became a model: since Augustine (354 to 430) and Martianus Capella, a Roman encyclopedist from the 5th or 6th century, a canon of seven subjects had become established for the schools of late antiquity. This was divided into a trivium (grammar, rhetoric and dialectic or logic) and a quadrivium (arithmetic, geometry, astronomy, music theory). These subjects were called the seven liberal arts because they were “worthy of a free man,” and “free” was a man when he was free from the necessity of earning a living.
In the Middle Ages this study of the liberal arts was regarded as preparation for scientific studies in theology, jurisprudence and medicine. In the universities, the liberal arts were soon taught within the framework of a separate faculty, the Facultas Artium. The “Liberal Arts”, which we know from the USA e.g. as “undergraduate studies” and which are introduced nowadays also e.g. in Germany, try to continue this tradition. However, dialectics is more in the foreground here than any of the mathematical disciplines.

Aristotle

Aristotle, a pupil of Plato’s, had particularly fertile ground for his rigorous training at the Platonic Academy. Born into an educated and wealthy family in Stageira (Chaldidike) in the year -384, he was sent to Plato’s Academy in Athens in the year -367 at the age of 17. The mathematician Eudoxos von Knidos (ca. -395 to ca. -350) played an important role there.
Aristotle, on the one hand, loved to argue and discuss with dissenters. On the other hand, he showed – as with people who have not passed a mathematical education without consequences – a tendency to work systematically and to collect the doctrines of former philosophers such as Pythagoras or Democritus (Schupp, I 256).
Aristotle became such a great systematist, attentive to the methodical in a thought process and vigilant for connections. He was the first of this kind, and he was also to establish a completely new kind of philosophy, which is no longer a poiesis, a “production” of doctrines in a poetic manner. Instead, the focus is on analysis and methods, reasons are required and the results are checked for conclusiveness.
It is probably plausible that one feels the urge for such things when one has a large corpus of doctrines before one’s mind’s eye and at some point wonders why the respective philosophers can so firmly hold their views. But you also have to live in a stimulating environment and, above all, have the talent to make such a new start. So Aristotle soon went his own way in the academy. According to Diogenes Laertius, Plato is said to have said: “Aristotle beat me like young fillings do against their own mother” (Laertius, 2015, p. 225).
In the list of Aristotle’s writings, Laertius lists 146 titles and speaks of a total of 445 270 lines. According to him, the work on logic is “very clearly marked as a tool for all sub-areas.” These writings were later combined into a collection of six books entitled “Organon” (gr. ὄργανον = tool).

The Organon

The titles of the six books in the Organon Collection are as follows:
1. the categories,
2. on the interpretation (peri hermeneias),
3. the doctrine of the logical conclusion (analytika protera, first analysis),
4. the study of evidence (analytika hystera, second analysis),
5. the topics and
6. the Sophist Refutations
The books 3,4 and 5 are particularly important for the development of logical thinking, i.e. the doctrines of the logical conclusion and the proof as well as the topics. Here the so-called Aristotelian logic is developed. This is therefore the fruit of a time in which dialectics occupied a high position in society. Even in discussions about more fundamental questions such as those about a good life or “reasonable” morals, it was soon more a question of the power of the arguments than of authoritarian settlements, and finally it could not fail to be thought about directly which forms of argumentation possess such power that they are incontestable, i.e. must be accepted by every sensible person.
The new era of philosophy, with its emphasis on dialectics, has therefore by no means allowed the idea of the logos to recede into the background or hindered its dissemination. On the contrary, dialectics was precisely the field in which this idea could prove to be a particularly useful and valuable tool for thinking. The idea took shape in Aristotelian logic, which was to become a model for a science that today plays a significant role within the framework of artificial intelligence. In the next blog post I will deal explicitly with Aristotelian logic.

Aristotle and the System of Sciences

Aristotle’s talent for analytical thinking and systematic work, as well as his passion for collecting the doctrines of earlier philosophers, showed other fruits. He establishes a systematics of all sciences known at that time. It divides the sciences into practical and ethical sciences, into poetic (manufacturing) sciences such as medicine or craftsmanship, and finally into theoretical sciences. In the latter he distinguished mathematics, natural science and “first philosophy”, to which he added theology, ontology and logic. Theology at that time consisted essentially in a study of the unchangeable “divine” stars.
This division of Aristotle is, so to speak, the first “cashing down” of knowledge of a time and subsumes everything that creates knowledge under science. Probably he also counted the theoretical sciences of mathematics and natural science among philosophy, albeit not among the “first”. About 500 years later Diogenes writes of Laertius:
As for the parts of philosophy, there are three: physics, ethics, and dialectics.(Laertius, 2015, p. 10),
whereby dialectic here also stands for the doctrine of thought principles, i.e. for logic and more generally for epistemology. Ethics had thus immigrated into philosophy and mathematics had been eliminated. There could no longer be a thinker like Pythagoras, who practiced mathematics but also represented a certain world view with religious fervour. Important ancient mathematicians, such as Euclid of Alexandria or Archimedes of Syracuse, do not appear in the work of Diogenes Laertius on the lives and opinions of famous philosophers – except for Eudoxos of Knidos, who was a member of Plato’s Academy for a time.
But physics, too, was to emigrate from philosophy, albeit only about 2,000 years later, when Galileo Galilei overcame the Aristotelian theory of motion. By demonstrating how to describe regularities in nature in the language of mathematics, he discovered a “new science”. Natural science became a modern physics. The consequences of this discovery will be discussed in detail.
Finally, at the end of the 19th century, logic in the form of mathematical logic emigrated from philosophy. It is now a branch of mathematics and computer science. Nowadays it is even observed how epistemology is becoming an area of cognitive science that not only deals with our ability to think, but also with all conscious and unconscious processes in our brain such as perceiving, learning or remembering.
Cosmology, the theme of the pre-Socratics and the first theme of philosophy ever, is today an area of modern physics. A history of cosmology from the pre-Socratics to Hawking would be highly interesting: the “question of the whole” has arisen in mythical thinking and in this form is still present in all religions, and the pre-Socratics already began to look for physical reasons for their ideas. But only for about 100 years has there been a physical cosmology in which the logos rule strictly. This cosmology is a reconstruction of the history of the universe with consistent consideration of physical theories. The development of the cosmos could only be convincingly told after the phenomena of nature had been understood on the grounds of reliable theories. Because this knowledge was just the necessary guidelines for a history of the universe, which no longer wants to be a mythical but a logical one.