I am grateful to Prof. David Braithwaite for having revised my poor English and to an anonymous referee for some important suggestions.

1 Introduction: What are the foundations of science? The three dialectics

2I suggest the following definition of the foundations of science that I obtained on the basis of four decades of historical works on the main theories of logic, mathematics and physics; they are constituted by two more dialectics of a theoretical nature; one between the two kinds of infinity (either potential, or actual) and a dialectic between the two types of organization of a theory (either deductive form principles-axioms, as Aristotle theorized in ancient times, AO, or aimed at the solution of a crucial problem, PO). The infinity dialectic was formalized, by, on one hand, classical mathematics relying on the actual infinity (AI) (e.g., Zermelo’s axiom) and on the other hand by constructive mathematics relying on (almost) only potential infinity (PI) [Markov 1962], [Bishop 1967]. The organization dialectic was formalized by means of the kind of logic managing it; on the one hand classical logic, managing AO (e.g., in Euclid’s Elements) and, on the other hand, non-classical logic, in particular intuitionist logic [Dummett 1977] managing PO [Drago 1990, 2012b]. When a scientist builds a theory, each theoretical dialectic appears under its formal aspect; a formal alternative is opposite to and incompatible with the other alternative; the scientist has to choose one. In sum, both dialectics constitute two scientific dichotomies. Once the choices are made, each dialectic is dissolved and the scientist proceeds within the scientific realm to establish the notions and principles of his theory. However, when we consider all the theories of a scientific discipline—e.g., physics—we again recognize these theoretical dialectics in the different couples of choices on which the various theories rely. In the light of the above illustrated foundations of science, the question of the birth of modern science receives a first, partial answer: modern science was born when not only its method for producing accurate results was recognized, but also its foundations, i.e., the two theoretical dialectics. Since the alternative choices of these dialectics were not formalized and recognized as relevant for science before the second half of the last century, we obtain a surprising result: the birth of modern science is a historical process spanning several centuries, at least half a millennium. In the following sections, we will detail this long process in relation to Galileo. Notice that the three dialectics of the foundations of science are mutually independent. Thus, according to the above definition, the question about the birth of modern science has to be disentangled into the three questions about when each of these dialectics emerged and was then established, at first in intuitive and then in formal terms.

3In what follows, I will deal with them by considering each dialectic at a time.

2 The dialectic experiment/ mathematical hypothesis. Its historical birth and its problematic definition

4Certainly, the historical start of modern science was marked above all by the birth of the experimental method through its innumerable applications giving new scientific results. Even a first, rough definition of this method was enough to mark an unprecedented novelty with respect to the philosophical world of the ancient Greeks. Rightly, historians of science attributed a crucial importance to this method. However, regarding this dialectic it is no longer possible to assume a positivist viewpoint, which attributes an eminent role to Galileo because he, more than any other, set hard experimental facts against the idealistic philosophical tenets of Aristotelian philosophers. If the experimental method was what positivist historians claim, that is, dealing above all with hard, experimental facts, it would be easy to determine a date for the births of both scientific activity and the method governing it. Instead, Grosseteste, Francis Bacon, Cusanus, Lavoisier, etc., suggested different aspects of this dialectic. Moreover, Galileo is a complex figure; in contradiction to the above appraisal, he also declared that he could obtain scientific results without experiments [Galilei 1638, II Day]. In addition, the question of whether Galileo’s activity conformed to a well-defined experimental method is largely undecided. For instance, about Galileo’s capital achievement—the discovery of the law of the accelerated motion—Galluzzi wrote:

[...] it is not still made irrefutably clear which element of the Galilean investigation was decisive in such an enterprise: either the natural deduction or the observation, either the geometrico-mathematical analogy or the experiment. [Galluzzi 1979, 158]

5Notwithstanding having pondered for three centuries on this dialectic, philosophers of science did not suggest any common agreement on its main features. We know that not only positivist, but also neo-positivist philosophy, although supported by excellent scholars, failed in this task. No surprise if this unsatisfactory situation is presented by Feyerabend as no method at all in science [Feyerabend 1975]. Recently, two scholars wrote an even more discouraging appraisal:

[...] More recent debate has questioned whether there is anything like a fixed toolkit of methods which is common across science and only science. [Andersen & Hepburn 2015]

6The plain conclusion to be derived from these unsuccesses is that this dialectic is philosophy-laden. Indeed, both Koyré and Lindberg stressed that the establishment of the experimental method was the result and at the same time the cause of a profound change not only in the methodology aimed at obtaining answers from nature, but also in the metaphysical conception of reality [Koyré 1957], [Lindberg 1992].

3 The dialectic experiment/ theoretical hypothesis. Its historical birth and Cusanus’ contributions

And thus, by means of experiments done with weight-scales he [the physician] would draw nearer, through a more precise surmise, unto all that is known. [Cusanus 1450, 608, 164]

[...] Experimental knowledge requires extensive written records. For the more written records there are, the more infallibly we can arrive, based on experiments, at the art elicited from the experiments. [Cusanus 1450, 615, 178]

In the mid-fifteenth century appeared what should have been the crowning work of this genre, the Idiota de staticis experimentis, from the pen of Cardinal Nicholas of Cusa. This incidental piece from one of the best-known philosophers and churchmen of the time does not appear to have attracted much attention. It advocates the use of balance and the comparison of weights for the solution of a wide range of phenomena, ranging from mechanics to medicine, and including those of chemical. He advocates differences in weights as a guide to the evaluation of natural waters, the condition of blood and urine in sickness and health, the evaluation of the efficacy of drugs and the identification of metals and alloys. It would be difficult to find a more specific prescription for what scientists were actually doing two centuries later. [Multhauf 1978, 386]

4 The dialectic experiment/ mathematical hypothesis. Its historical birth and Galileo’s determinant achievements

11A century and a half after Cusanus, Galileo extensively applied the experiment/hypothesis dialectic to several fields of physical phenomena. However, Galileo had a particular conception of this dialectic, as his celebrated words emphasize:

Philosophy [read: science] is written in that great book which ever lies before our eyes—I mean the universe—but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth. [Galilei 1890, vi, 232]

13Mathematics indeed may be joined with experiment also according to an instrumental conception of it, as is the case in both chemistry and thermodynamics [Drago 2009b, 2012a]. In fact, a century and a half before Galileo, Cusanus had suggested an alternative philosophical conception of mathematics, which was summarized by McTighe [McTighe 1970]. According to him, ideal mathematical entities are built from reality (and not, as Galileo claimed, recognized as the ideal elements of reality), i.e., they are constructs of our minds. Whereas ordinary knowledge of reality is always approximate and hence it is not subject to the principle of non-contradiction (through a sharp and exhaustive division into true and false), instead, mathematics is perfect knowledge because it enjoys a specific feature; it strictly obeys this principle. Owing to this logical property, mathematics is considered by Cusanus the most certain knowledge we have; for this reason, we apply it to understanding reality. Moreover Cusanus conceived numbers as the explicit forms of unity, i.e., a number enjoys the property of being both as great and as small as it can be; thus it is through numbers that all things are best understood, i.e., referred to unity. For these reasons, the creation of our minds, mathematics, can direct our interpretation of reality. Indeed, by combining experimental data with mathematics the intellectus conjectures the mathematical formula of a physical law. This Cusanus’ way of describing the relationship between mathematics and experiment more adequately than Galileo’s previous quotation describes the activity of an experimental physicist (e.g., in such a way—by means of a conjecture—the law of falling bodies actually originated in Galileo [Wisan 1974, 207-222]). Furthermore, Galileo knew only the mathematics of geometry and proportions and did not share in the effort—elicited, e.g., by his disciples Cavalieri and Torricelli—to extend the realm of mathematics [Drago 2003]. Also for this reason, the theory of mechanics was born later, through Newton, since it required the invention of a very different mathematics, calculus (just as at present time the advances in theoretical physics require the new mathematical technique of symmetries). Before Galileo, Cusanus had, on the other hand, been able to expand the mathematics of his time (geometry and proportions) to new achievements (we will see them in the next section).

14I conclude that Cusanus offered to theoretical physics a more adequate metaphysics of the scientific methodology than Galileo’s, owing to latter’s Platonist view of mathematics. Yet, most historians disregarded the entire metaphysical and mathematical change brought about by the introduction of the scientific method. For an instance, Cohen gave a totally objectivist definition [Cohen 1984, 85-86]. The experimental method developed by Galileo enjoys the following two features:

• it is mathematical in that the relationships between the parameters are quantitative, in that the proofs are geometrical, and above all the properties of falling and projecting bodies are logically derived from a set of a priori postulates;

• it is experimental in that not [only] daily experience but nature subjected to artificial manipulation provides both the starting point and the final empirical check of the axiomatic system.

5 The second dialectic: potential infinity/ actual infinity. Its historical birth

6 The third dialectic: axiomatic organization/ problem-based organization. Its historical birth

7 The third dialectic: classical logic/ non-classical logic. Its historical birth

19The dialectic regarding the kind of organization is formalized by means of two different kinds of logic, respectively classical logic—governing the deductions of a AO theory—and non-classical logic—governing the inductions of a PO theory. Some scholars have remarked that the logical ways of arguing of both Cusanus and Galileo are different from those of classical logic. These differences were attributed to their attendance at the Padua school of logic, where Zabarella made the greatest effort to theorize a different way of reasoning from that of Aristotle. Actually, after the Padua period, Cusanus wanted to found a new kind of logic and he was successful in this aim. He believed that Aristotle’s logic represented a specific activity of the ratio, a particular way of arguing of the mens; yet, according to him there exists another activity of the mind, i.e., the intellectus, which generates coniecturae according to a new kind of logic. He wanted to characterize the specific laws of this new logic. He suggested a celebrated “coincidence of opposites”, which he later abandoned as ineffective. Rather he implicitly changed logic. An accurate inspection of his texts shows that he made use of the characteristic propositions of non-classical logic, those for which the double negation law fails; i.e., the doubly negated propositions whose corresponding affirmative propositions lack evidence (DNPs). Through them he solved his main problem—the best to name God—, by suggesting names pertaining to non-classical logic (Not-Other, Posse=est). Moreover, he was capable of developing the logical arguments, which are specific to the ideal model of a PO theory, i.e., ad absurdum arguments [Drago 2012a].

1. The parts of his text written in Latin and concerning formal theorems do not include DNPs;

2. The other parts—written in vulgar Italian—include more than 100 DNPs;

3. However, some of them are uncertain because some are dubious;

4. Moreover, there is no DNP in the part illustrating rectilinear uniform motion;

5. Instead there are around 90 DNPs in the 25 pages dedicated to solving the problem of naturally accelerated motion;

6. The latter DPNs compose 8 cycles of reasoning, which are illustrated by table 1;

7. Three of these cycles are ad absurdum proofs;

8. His reasoning eventually obtains a DNP which is a universal predicate [Galilei 1638, 190-191]; it represents an hypothesis solving the problem;

9. Which then the author translates—by merely dropping the two negations of this DPN—into the corresponding affirmative proposition in order to deductively derive from it, now considered as a postulate subject to classical logic, all possible consequences to be tested by experiment. Galileo is the only scientist apart from Einstein to have illustrated this logical step through admirably precise words [Einstein 1905a, 891]:

• 16 This author’s translation is justified by appealing to the principle of sufficient reason, which Ga (...)

Let us then, for the present, take this as a postulate, the absolute truth of which will be established when we find that the inferences from this hypothesis correspond to agree perfectly with experiment [Galilei 1638], [Galilei 1638, 183, period before Theorem 1, Proposition I of the Naturally accelerated motion; emphasis added]16

Table 1: The eight cycles of reasoning in De motu locali

1°	2°	3°	4°(3 →⊥)	5° (the case v=0)	6° (v=ks→⊥)	7°?	8
Hp of a law v=kt	Hp of infinite degrees of velocity	Objection: Is time infinite?	Refutation: As many degrees of velocity as the degrees of time	Refutation of the objection: Even in the case v=0?	Falsity of the other law v=ks	Laws on falling bodies on inclines

Legenda: Hp = hypothesis; ⊥ = absurd; k = constant; s = distance; t = time; v = velocity

21However, Galileo ignored the alternative logical nature of both the DNPs and the ad absurdum arguments, maybe because his way of reasoning by means of DNPs was sometimes invalid (actually, it was invalid, according to recent studies, also in the deductive reasoning) [De Luise & Drago 2009, fn. 15]. I conclude that, his genius was capable of achieving exceptional, but irregular results in logic.

8 Conclusion: What is Galileo’s role in this centuries long historical process?

22First, one has to take into account that Galileo’s methodological change and his several scientific results were not enough to give rise to modern science because he did not achieve any complete physical theory. Rather both Cusanus and Galileo anticipated scientific theories. On this point a comparison is not easy because their advances anticipated two very different scientific theories, respectively chemistry [Drago 2009b], which was born three centuries after Cusanus; and mechanics, which was born a few decades after Galileo’s death. Strangely enough, neither anticipated geometrical optics; it was instead anticipated by Grosseteste four centuries before its birth, which occurred in the last period of Galileo’s life. Hence, an appraisal on Galileo’s role cannot refer to a complete scientific object, but only to the elements of physical theories. We will consider their most important elements, i.e., the foundational choices.

23Let us compare Cusanus and Galileo with regard to the two theoretical dialectics. Cusanus was the first to conceive mathematics non-Platonically and he improved the mathematics of the time by first introducing infinity; and regarding infinity, he formulated, albeit in verbal terms, the notion of limit in PI mathematics and a basic notion of AI mathematics. First Cusanus made them manifest through the invention of new mathematical notions and new (theological) theories which relied on the alternative choices—respectively, AI and PO—to the dominant ones, PI and AO. He introduced a new logic, which he qualified as non-Aristotelian, at present recognized as intuitionist. Moreover, he intensively discussed the two dialectics regarding infinity and logic. Already Cassirer stressed that no one more than Cusanus, after the first of his main books, De docta ignorantia of 1444, accomplished such a metaphysical change [Cassirer 1950, 277].

24One century and a half after Cusanus came Galileo, who discussed the relevance of the two kinds of infinity in theoretical physics, any way he did not decide whether and how this dialectic had to be solved by a scientist; but he did not want to appeal to AI. In addition, he knew well both the organization of a theory and the logic suggested by Aristotle. He looked for an alternative logic, yet his Paduan period did not lead him to suggest a specific novelty; although as a matter of fact he adapted some of his theoretical works to the characteristic features of the model of a PO theory, e.g., the DNPs, he considered an alternative organization of a theory to be no more than a Platonic dialogue.

25I conclude that regarding the two theoretical dialectics Cusanus was more advanced than Galileo.

27After Galileo, without any discussion about the said dialectics, Newton decided the theoretical dichotomies with a pair of choices, which became a paradigm throughout the two subsequent centuries of development of theoretical physics. Due to this fact, a winter set in thinking about the two theoretical dialectics; in this winter only two scientists, i.e., L. Carnot [Carnot 1803, xiii-vxii, 3] and independently Einstein [Einstein 1905], made manifest the four choices, without receiving attention on this subject [Drago 2013a]. Hence, one more merit has to be attributed to Galileo, i.e., at the very beginning of modern science to have presented through his discussions almost all the foundational issues of the scientific enterprise which have been re-discovered only after a very long period of time.