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 Syllogisms of Aristotle

In the last blog post, Aristotle presented the three types of conclusions: the logical conclusion, the dialectical conclusion and the false conclusion. Now we will have to deal with the logical conclusion, which derives a true conclusion from two true premises.

This is, of course, the most important conclusion, which ensures a safe “transport” of the truth of statements. That such a possibility exists is the good news, and you can’t overestimate it. The “bad news” would then be that truth is not gained in any way, it is only passed on. The premises must be true. The question as to how and where one can begin with true statements in practice will be of great concern to us later on.

With the logical conclusion one can distinguish now again several kinds. The individual proofs, as he also called them, differ in the nature of the premises and the conclusion, which then results from a “summation” of the premises. In the Greek “summation” means “συν-λογισμός” (syllogismos); thus, one also speaks of a syllogism, the doctrine of syllogisms is called syllogistics. 

Let us first look at an example of syllogism:

All humans are mortal.
All Greeks are humans.
Therefore: All Greeks are mortal.

So, there are three statements, two propositions from which one proceeds, and a conclusion in which the propositions are “added together”.

The form of sentences and their representation

In this example, all sentences are of the form “All A are B”.  The sentence “All A are B” can also be formulated as follows: “B applies to all A.”  This formulation suggests that B is a predicate of all A: B is attributed to all A (predicated), e.g.: To be mortal applies to all human beings.

The formulation is also the one that comes closest to the Greek text, so it is the more original form. In scholasticism however, this was rewritten then as “All A are B”. All A has the predicate B: All humans are mortal. 

The above syllogism is then in its original form:

To be mortal belongs to all human beings,
To be a human being belongs to all Greeks.
Therefore: To be mortal belongs to all Greeks.

Moreover, this formulation also suggests a method that was discovered much later to illustrate the relationship between two terms. One takes advantage of the fact that each term applies to a set of entities. “Human beings” can be Greeks but also Egyptians, Thracians or Asians or Europeans. If one represents the set of human beings by a circle or any closed curve (B), the set of Greeks as well (A), then the circle for the Greeks lies within the circle for the human beings.

Fig.1: The set of Greeks (A) is a subset of the set human beings (B). To be a human being (B) applies to all Greeks (A).

Such pictures are called Venn diagrams, after the mathematician John Venn (1834 to 1923), who introduced them following Leonard Euler (1707 to 1783). Actually, the philosopher Gottfried Wilhelm Leibniz (1646 to 1716) already used them. These Venn diagrams thus illustrate relationships between two sets in general, no matter of what kind the elements are. Within the framework of set theory, we would also write: A ⊂ B, i.e. A is a subset of B.

But this form of a sentence is not the only one at syllogisms. Aristotle obtained an overview of all possible forms of sentences and statements. This resulted in: (Prior Analytics, Book I, 1,  see
https://ebooks.adelaide.edu.au/a/aristotle/a8pra/book1.html):

A premiss then is a sentence affirming or denying one thing of another. This is either universal or particular or indefinite.
By universal I mean the statement that something belongs to all or none of something else;
by particular that it belongs to some or not to some or not to all;

by indefinite that it does or does not belong.

Thus, beside the general sentences like “All Greeks are human beings” or “To be human belongs to all Greeks” there is also the negation (no-compliance) and the particular statement. Altogether you get the following types of sentences:

B belongs to all A, (all A are B),
B doesn’t belong to any A, (no A is B),
B belongs to some A, (Some A are B),
B doesn’t belong to some A, (some A are not B).

We can illustrate the other types of statements as follows:

 Fig. 2: Left: B doesn’t belong to any A. Middle: B belongs to some A. Right: B doesn’t belong to some A.

In scholastic circles these forms are abbreviated as (A a B), (A e B), (A i B) and (A o B). The letters “a” and “i” should remind us of “affirmo”, “e” and “o” of “nego”.

The structure of syllogisms

Two such forms of statement then form the premises, one such the conclusion. Each conclusion can thus be characterized by three letters from the set {a,e,i,o} and by the position of the three letters, which stand for the concepts in the respective sentence.

One of these terms, the so-called middle term, must occur in both conditions, it can be in first or second place, or in one condition in first, in the other in second. This results in four different shapes or figures. The first figure is the following (see example above)

A – B

B – C

A – C.

Now Aristotle selects in all forms that combination of two forms of statement that necessarily lead to a conclusion. He simply sorts out the conclusions for which he finds a counterexample.

The valid conclusions can then each be represented by the form and a specific combination of the letters a,e,i,o. And in order to be able to remember such combinations better, one has integrated them into corresponding peculiar words, e.g. one remembers the combination a a a with the word “Barbara”, and knows in addition that here the 1st figure is present. This conclusion corresponds exactly to the above example.

Fig.3: The Barbara syllogism

Another important conclusion, also of the 1st figure, is called “Celarent” in this way:

Fig.4: The syllogism Celarent

Aristotle demonstrated in this way that one could systematically formulate a system of conclusions in which from the truth of the premises necessarily follows the truth of the conclusion.

With the systematics of Aristotle, one now has a complete overview of all possible conclusions. Previously however, in syllogistics, “the whole art of syllogistics had consisted in searching around with great effort of time and effort”. as Aristotle wrote in his work On Sophistical Refutations (after Schupp, I 275).

But here one already has a system of statements that is reminiscent of the formal predicate logic that will be introduced later. Like there, you can “quantify” using the predicates, i.e. you can operate with quantities such as “all”, “none” and “some” for the predicates.

Most importantly, Aristotle must select the valid conclusions “by hand” from the set of all possible combinations, simply by discarding those that he recognizes as invalid with the help of an example. This “recognition” is an intuitive one, one with “common sense”. One does not doubt the correctness of the conclusion, but for a strict science in today’s sense this kind of knowledge is not sufficient.

Even if one describes the relations of terms mentioned above with the help of set theory, and thus justifies the conclusions within the framework of set theory, the insight is a mathematical one. Thus, the conclusions would be only indirectly logically justified, because mathematics only uses, as we know today, the rules of inference that are ultimately gained in modern predicate logic. Only in this predicate logic can the conclusions be strictly justified by deriving them from tautologies. We’ll see about that in a later blog post. Only in this logic is the bottom reached on which one can incontestably win a true sentence again from true sentences.

We still would have to comment on Aristotle’s remark on the case that the conditions are only likely to be true or credible. Then the conclusion could also only probably be true. Saying more here was not possible at that time.  It was not until the beginning of the 20th century that a theory of probability was developed with which one can become more precise in this case. This will be explained in more detail in a later blog post on the topic “How to deal with insecure knowledge”.

Syllogisms in an axiomatic-deductive system

But Aristotle’s logic has not only shown us how safe knowledge can be passed on safely. He has also shown that “It is possible also to reduce all syllogisms to the universal syllogisms in the first figure.”, as he writes in the Prior Analytics, 1st book, 7th chapter (see https://ebooks.adelaide.edu.au/a/aristotle/a8pra/book1.html).

Thus, we already have a concrete axiomatic-deductive system. This logical order of the statements in a field of knowledge represented for him the ideal of a science. In the Posterior Analytics, 1st book, 3rd chapter he writes:

On the other hand, I maintain that any science must be based on proofs, but that the knowledge of the unmediated principles is not provable. And it is clear that this must be necessary. For since a knowledge of the earlier propositions from which the proof is made is necessary, but one stops once at unmediated propositions, these must necessarily be unprovable. This is my view and I maintain that there are not only sciences, but also supreme principles of them through which we learn the concepts of conclusion. (Here I prefer the translation of a german version in http://www.zeno.org/Philosophie/M/Aristoteles/Organon/Zweite+Analytiken+oder+Lehre+vom+Erkennen/1.+Buch/3.+Kapitel).

The mathematician Euclid of Alexandria logically arranged the geometric knowledge of that time in this way in the time around -300. This organization of a scientific theory as an axiomatic-deductive system is still a model for any rigorous science today. One tried in all centuries to imitate this organization of a thought building, thus “more geometrico”, after kind of the Euclidean geometry, to arrange the knowledge of his science. In “Die Idee der Wissenschaft – Ihr Schicksal in Physik, Rechtswissenschaft und Theologie” I described how successful it has been to this day to realize this idea (Honerkamp, 2017).

Who first had this idea is not clear. The mathematical proof was already known to the Pythagoreans. If there are enough statements in an area, secured by proofs, one will probably at some point consider which statements could be regarded as axioms. This poses the question: Which statements do I need as a basis in order to be able to deduce all the others from them? Or also: How do I create a logical order?

Such specifications will then always be special statements, have special properties. They can be immediately insightful, i.e. “certain by themselves”, as in theories of mathematics, but also highly abstract and far from our idea, as in theories of physics.