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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 realizationthat in a sea of mysticism and dialectic there is the possibility at all totransfer the truth of statements to another statement, has driven me very muchin my youth, when I had become so properly aware of this.What's the useof all this? One could establish such a kind of logical order betweenstatements, in which it becomes clear which true statements follow from whichother true statements.&#160; One could startfrom true statements and build a whole thought building on them, which consistsonly of true statements. But - whatstatements 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, atterm-logic and at geometry already was demonstrated, how knowledge of securetransport of truth can be extended to an axiomatic-deductive system. Throughoutthe centuries, mathematics has remained an unsurpassed model for such anorganization of secure knowledge. There had beenattempts to introduce a similar rigour of argumentation in philosophy andethics. Such approaches, however, all ran into the sand (see Wikipedia:Mathesis universalis). Had they been the wrong areas for a rigor of thoughtaccording to the mode of mathematization? Perhaps axiomsdid not necessarily have to be immediately obvious, but it was more importantto find a source of true knowledge at all. Just as Euclid could refer to alarge number of mathematical proofs and arranged this material according tological points of view and, if necessary, supplemented it, a "small"axiomatic deductive construct of thought may also emerge after knowledge ofsome true statements by clarifying the logical relationship between them.Gradually one could then combine these "small buildings" into largerones.&#160; Galileo Galileiwas the first to recognise that nature was the source of true knowledge, aswell as the importance of mathematics for the formulation of such knowledge. Hewas the first to describe a result of a physical experiment in the language ofmathematics. He certainly sawthe implications of this combination of mathematics and experiment. It wasimmediately clear to him what a revolution a mathematisation represented forthe understanding of science at that time. Thus, he spoke of a "newscience", which he had founded. His sentence "The Book of Nature iswritten in the language of mathematics" bears witness to this, as does thepassage of his letter to the Tuscan Secretary of State Vinta in 1610:"Therefore, I take the liberty of calling this a new science discovered byme 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 theconsequences of this discovery, without which our world today would be acompletely different one. At some point, however, this "new science"had to be discovered; nature and mathematics - or rather nature and logic - aretoo close to each other. When is an implication true?Why doesempiricism play such an important role, why do " inquiries " ofnature in the form of experiments play such an important role, if one wishes atheory after the model of Euclidean geometry, thus as axiomatic-deductivesystem?&#160; So let's look again at the modusponens as a prototype of a logical conclusion: A, A → B ⊨ B.In order todeduce a statement that is incontestably true, premises A and A → B must betrue. There is one statement, namely A, which occurs in both premises. &#160;The implication forms the bridge to a newstatement, namely B, which is then deduced. There must be such"bridges" in every concluding rule, because nothing can be inferredfrom statements that are completely independent of each other. Also, thesyllogisms each have a middle term, which occurs in both premises. A trueimplication A → B means that A is sufficient for B: Always if A, then B. Whereis that the case? We can find trueimplications by questioning nature. We then receive the following answers:"If I throw a ball into the air, it falls to the earth" or "Ifan electric current flows in a wire, there is a magnetic field in itsenvironment". The experimental physicists are therefore suppliers of trueimplications, which we then also call laws of nature.Trueimplications can also be found if we transform the statement "All Greeks arehuman beings", for example, into "If x is Greek, then x ishuman."Here we haveformed the terms "Greeks" and "humans" in such a way thatthe implication is true. The statement thus becomes true by the fact that weform the concepts accordingly. Then we'realready at the end of our rope. For all other implications the dialecticalconclusion is probably responsible, i.e. here an implication belongs to thecategory of sentences about which Aristotle said:Sentences arecredible if they are recognized by all, or by most, or by wise men, the latterby all, or by most, or by the most experienced and credible. We can add: Andwhat is recognized by "wise men" also depends on time. Let us onlythink 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 onher own responsibility. It shall be entitled to be economically active to theextent compatible with its matrimonial and family responsibilities." When it comes toregulations for human coexistence, morals, customs and traditions, yes,everything that nature does not tell us, there can be no generally acceptabletrue implications. We are referred to the dialectical conclusion and thus to anegotiation about which implications are to be set as true. So here we can only"set" truth, not find it. The consequenceof this is that the statements of the natural sciences are universally valid,but there are countless religions and legal systems. In the natural sciencesthere 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" basicassumptions based on ever new discoveries about nature's behaviour.&#160;&#160;&#160;&#160; 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 GalileiThe "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 theview alone demonstrates the independence of his thinking, as it ischaracteristic of a genius. Even in Goethe's day, philosophers had to thinkabout "what holds the world together at its innermost", and Faust hasonly mockery for Mephistopheles when he fights for people: "You can donothing on a large scale, and now you can begin it on a small scale.&#160; 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 Englishphysicist and mathematician Isaac Newton stood on Galileo's shoulders duringthe development of this theory. The first axiom in this theory was based onGalileo's hypothesis, on which he had been guided in his falling experiments.&#160; It was the hypothesis that, on a horizontalplane, the motion of the rolling sphere "would continue forever at auniform speed" if it were not affected by unevenness of the ground(Galilei, 1982, p. 30).&#160; 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 whichstatements can one begin with the formulation of an axiomatic-deductive systemfor 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 havebeen at the beginning of a theory of motion. Let us take acloser look at which statements have been put at the start in this theory, butalso in other physical theories. We will see that this happened in verydifferent ways. But let us first get an overview of these theories in the nextchapter.