II. THE NATURE OF SCIENTIFIC KNOWLEDGE

1 SCIENCE IN THE CLASSROOM, FOR EARLY CHILDHOOD AND PRIMARY EDUCATION TEACHERS II. THE NATURE OF SCIENTIFIC KNOWLEDGE María José Gómez Díaz El CSIC en la Escuela, Vicepresidencia Adjunta de Cultura Científica y Ciencia Ciudadana Consejo Superior de Investigaciones Científicas (CSIC) José M. López Sancho Instituto de Física Fundamental Consejo Superior de Investigaciones Científicas (CSIC) [email protected] ABSTRACT This is the second article in a series dedicated to the teaching of science in Early Childhood and Primary Education. It deals with scientific knowledge and the specific differences and characteristics that distinguish it from ordinary knowledge—differences that must be understood in order to design a pedagogical approach to science in the early stages of schooling. To this end, science is framed within the history of epistemology, as yet another attempt to satisfy the curiosity of our species: to achieve true knowledge of the world in which we live. To understand the historical evolution of knowledge, we distinguish three paths that have intertwined over time. The philosophical thread, which begins with the existence of a real world external to us and seeks to determine what we can know of it. The mathematical thread, whose world is composed solely of ideas and concepts; in Galileo’s words, its characters are triangles, circles, and other geometric figures, and it also encompasses real numbers, complex numbers, symmetries, invariances, and a long etcetera. These ideas inhabit a sort of cloud, both in the computational and Platonic sense, which is accessible to all human beings. Their truth is established through logical demonstration within an axiomatic system, using a symbolic language of its own that guarantees their validity. Finally, the Galilean thread, which inaugurates modern science, begins by limiting the scope of scientific study to the world of magnitudes. It adopts Bacon’s scientific method as a procedure for discovering the laws that govern nature and uses these laws to replace the axioms of the deductive structure of mathematics, thereby fulfilling its own criterion of truth. The interaction among these three threads marked the beginning of what we now know as science. 2 Since Galileo restricted scientific knowledge to that which can be expressed through the language of mathematics, all threads have evolved together. At first, it encompassed little more than the world of Euclid, with his Geometry and Elements, but today—thanks to the expansion of mathematics into the field of structures—it has extended into areas of knowledge far removed from its origins. Linguistics, economics, mathematical biology, computer sciences, and even logic itself, among many other fields of scientific inquiry, have been described in the form of symbolic algorithms expressible by mathematics. By thus meeting Galileo’s criterion, they have become part of scientific knowledge. Keywords: science education, epistemology, scientific language, symbolic thinking, science as a second language, early childhood education. 1. INTRODUCTION: WHY IS SCIENCE POSSIBLE? The history of humankind has many aspects and many levels, but one of the most important—and, of course, the one that most concerns us—is what is known as the search for true knowledge. In this context, science is one of the most successful attempts at that search, as well as the object of our work as teachers and scientists. For this reason, the first question we must ask is whether it is possible to know the behavior of nature—in other words, whether science is possible. Although it may seem overly abstract, this debate can be presented to our students, always at a level we consider appropriate: how can we know whether a law of nature is true? If the question is posed as a riddle whose answer is that it can only be resolved by asking nature itself, curiosity and interest are awakened. Not surprisingly, this has been a recurring problem since the emergence of the human species. For the natural sciences to be possible, three requirements must be met: First: that the same causes always produce the same effects, that is, that there exist general laws of nature. Second: that human beings are capable of discovering and formulating them (such as Newton’s famous principles) and, moreover, of understanding them. Third: that we possess a deductive procedure—consistent with human logic—that allows us to derive from general laws expressions applicable to particular cases, while maintaining the criterion of truth. An example might be the application of the laws of Newton’s Principia to the specific case of the motion of Mars, in order to determine its trajectory. The first two conditions were addressed by Leucippus in the 5th century BC. According to him, we can know the behavior of nature because it always acts according to the same laws, which do not change arbitrarily with time or place. He is credited with the statement that nothing happens by chance: everything occurs through a cause that 3 necessarily determines what happens, in such a way that it cannot occur otherwise. This idea is far deeper than it might appear at first glance. That regularity is what makes possible the formulation of scientific laws that describe and predict the behavior of natural phenomena, as well as their application to the design of devices or buildings— that is, what we now call engineering. If nature were chaotic or unpredictable, we would not be able to construct theoretical models or make reliable predictions (Barnes, 1987). The third condition for the existence of science is expressed in Einstein’s famous statement: “The most incomprehensible thing about the universe is that it is comprehensible,” formulated in his essay Physics and Reality (1936). This idea points to the fundamental question of the epistemology of science: the mysterious congruence between the logic of human thought and the behavior of the universe. For Kant, this correspondence was not a mystery but a necessary consequence: we do not know the world as it is in itself (noumenon), but rather as it appears to us filtered through the a priori structures of our mind—space, time, and the categories of understanding. The fit between thought and reality is guaranteed because it is the mind that imposes its way of knowing reality onto what we observe in nature. For Einstein, on the other hand, such congruence was surprising and even enigmatic. From his perspective of scientific realism, the laws of the universe exist independently of us. What is truly astonishing is that, through thought experiments subject exclusively to human logic, we can reach conclusions that coincide with reality. Contemporary neuroscience offers a third perspective. Researchers such as Rodrigo Quian Quiroga (2017) have shown that the human brain possesses remarkable plasticity, allowing it to generate increasingly precise representations of the regularities of the environment. From this standpoint, the congruence between thought and the universe is understood as an adaptive process: our cognitive capacity has been evolutionarily configured to detect patterns, construct symbolic models, and anticipate natural phenomena—because the survival of the species depended on it. The most profound solution to this enigma may be approached from the point of view of the geometry of the space in which we live. Through the diagrams introduced by John Venn in 1880, concepts are represented as sets subject to the basic operations of modern algebra (Venn, 1880). With this resource, it is possible to demonstrate the validity of any syllogism of Aristotelian logic, as well as its general principles—for instance, that two negations lead to an affirmation, as shown in the figures. 4 The shaded part is the region NOT (A) If we now wish to represent what something is NOT [NOT(A)], we once again obtain the original set A. The shaded part is the region NOT [NOT (A)] = A 5 This formal equivalence between logical operations and geometric configurations leads us to think that logic is not independent of the space in which it develops, but rather conditioned by its geometry. If the geometry of space were different, so too would be our logic. The fact that the universe—and we, as observers—share the same geometric structure, which seems to determine both physical phenomena and the possible forms of thought, could be the deep reason why the world is comprehensible to us. Universe and mind share geometry and, therefore, logic. In 1936 Garrett Birkhoff and John von Neumann published The Logic of Quantum Mechanics. In that work, they proposed replacing classical logic—based on Euclidean geometry—with a different logic, in which the operations correspond to the mathematical structure of Hilbert space, the formal framework within which quantum mechanics is described. In this way, logic ceases to be associated with the geometric intuition of our everyday space and comes to be supported by an abstract geometry, specific to the quantum world. But this subject lies beyond both the scope of this paper and our knowledge. In this direction, we find the research on positive geometry by Claudia Fevola and AnnaLaura Sattelberger (Fevola & Sattelberger, 2025). These authors work on a mathematical approach that attempts to describe, within a single structure, both the phenomena of the microscopic world and those of the early universe on a large scale. The idea is that many processes of nature can be represented not only with equations but also with multidimensional geometric figures that more clearly reveal their internal relations. Thus, it becomes evident that nature seems to be organized according to a common geometric logic, reinforcing the hypothesis that the universe and our mind share the same mathematical structure. In other words, we understand the world because, in some way, we speak the same “geometric language” as it does. Although these new fields of geometric logic fall outside our objectives, it is interesting to note that the path of logic meets that of the nature of space—the geometry—in which we live. In sum, whatever the reason, science is possible because the universe operates according to causal laws from which consequences can be deduced through symbolic manipulations that, fortunately, coincide with the way human beings reason. 2. IN SEARCH OF TRUE KNOWLEDGE Once convinced that we can discover and understand the behavior of nature, and that from one truth we can deduce others by applying human logic, we face the problem of how to discover the laws of nature with guarantees that they faithfully describe reality. And thus begins our adventure—an adventure of thought—in search of a recipe that will allow us to attain true knowledge, which in time will be called the scientific method. We have chosen physics as our case study because it is the simplest of the sciences and the 6 one in which the historical connection between science and mathematics is most clearly seen. In this search for truth, two paths were initially followed by Western thought: the philosophical path and the mathematical path. As we shall see, these two paths follow zigzagging trajectories that intertwine and influence each other until they converge in a general theory of knowledge with Kant’s Copernican turn, which we addressed in the previous article. Having reached that point, we shall advance only along the path of science. As will become clear—and it is entirely logical—each of these approaches is characterized by its way of conceiving the relationship between reality and the observer, a relationship that determines a specific and concrete criterion of truth. The philosophical path Throughout the history of philosophy, the criteria for distinguishing true knowledge from that which is not have been diverse, diffuse, and strongly conditioned by cultural, religious, and philosophical contexts. The question has been present in philosophy since its very origins. In modern times, Immanuel Kant (1724–1804) proposed the so-called Kantian constructivism that we studied in the previous article, according to which there are two domains: the real world and its mental representation. In this scenario, the criterion of truth for a representation or theory would be defined by the degree of correspondence between these two domains. However, the problem arises of how to verify that the mental representation actually corresponds to the real world, so Kant’s ideas did not displace naïve realism, which remained a reference point well into the twentieth century. This naïve realism—reaching its height with Auguste Comte— assumed that scientific theories directly reflect reality (Chalmers, 2013; Haack, 2003). The first thread thus begins with naïve realism, which holds that the external world is just as we perceive it, without cognitive or linguistic constraints. The senses provide immediate, truthful, and reliable access. Within this framework, truth is understood as correspondence between what is asserted and what actually exists or occurs (Ferrater Mora, 2004; Alston, 1996). Thus, when Christopher Columbus claimed that he could reach the Indies by sailing westward, the statement was true if the facts confirmed it and false otherwise. Seeing is believing. In developmental psychology, Jean Piaget showed that naïve realism characterizes the earliest cognitive stages. In the preoperational stage (ages 2–7), children believe that things are exactly as they perceive them, without distinguishing between perception and reality. This is observed in conservation tasks: if the same amount of water is poured from a short, wide glass into a tall, narrow one, children assert that the second contains more, because they focus on height and do not yet integrate two dimensions (Piaget, 1952; Flavell, 1963). 7 These findings suggest a parallel between children’s cognitive development and the historical evolution of thought in human communities—a kind of ontogeny of societies, an especially suggestive idea. The mathematical path The second thread is articulated around geometry and arithmetic, from their Platonic roots to their formalization by Euclid (325–265 BC). If we ask spectators at a Roman circus chariot race like the one described in Ben-Hur, or fans filling a twenty-first-century football stadium, or students in any classroom to perform the same task—to imagine an equilateral triangle or a square—we observe a striking phenomenon: all, with minimal variation, conjure up the same mental figure. This is not the case if we ask them to imagine a horse or a mountain; in those cases, each mind constructs a different image. This fact reveals something profound about the nature of certain concepts: geometric figures seem to have a universal existence, shared by all. They are, in the truest sense, Platonic Ideas (Plato, Meno, 81d–86b). According to Plato, Ideas are not acquired through experience but already exist in our minds, as flight is innate to birds. They remain dormant in a corner of thought and can only be recovered through anamnesis, the method exemplified by Socrates in his dialogue with Meno’s slave. For Plato, the world of Ideas is as real as the paradise described in some religions: in it, geometric figures are perfect, immutable, and eternal; triangles are sharp, circles never tremble, and squares never warp. It is an absolute realm where relations are exact, inhabited by the gods who, as Russell remarked, “are always doing geometry” (Russell, 1945, p. 57). In that ideal universe there is no need for measurement. There are no graduated rulers with markings, nor any need to estimate lengths. Theorems, such as Pythagoras’, are not verified empirically but demonstrated with the straightedge—without numbers—and the compass, symbolic instruments that allow us to deduce universal truths without recourse to examples and observations in the real world. In this domain, truths are attained through pure demonstration, neither by induction nor by approximation (Heath, 1956). To clarify ideas, we may imagine Euclid surrounded by his disciples in a geometry class, with nothing but his words and his students’ imagination. “Let us imagine,” he would say, “an equilateral triangle.” All see it in their minds, with its three equal sides and three identical angles. “Now,” Euclid continues, “let us mentally draw a perpendicular from one vertex to the opposite side. What do we obtain?” The students, without need of compass or ruler, reply almost in unison: “Two right triangles, equal to each other and each with half the area of the original triangle.” 8 And so Euclid would continue his class, without blackboard or sand on the floor. That was the way to do geometry: not with measurements, but with pure reason. It was an exercise of the mind, not the body. Demonstration required nothing but the clarity of ideas and the precision of language. Whether or not Ideas exist in a world of their own, what is certain is that geometry developed in that shared mental space. By carefully analyzing the figures he and his disciples held in their minds, Euclid began his systematic work: he investigated the simplest elements into which those forms could be decomposed—a kind of atoms of geometry—identified them, and gave them names. Thus were born points, lines, planes, angles, and other fundamental concepts, defined in a very particular way. If we reflect, his definitions seem designed to identify something that already exists in the mind. And he did so with a precision that we still admire (Euclid, ca. 300 BC/1956). • A point is that which has no parts. • A line is a length without breadth. • The ends of a line are points. • A straight line is that which lies evenly with the points on itself. • A surface is that which has only length and breadth. • The edges of a surface are lines. With this sort of periodic table of the world of geometric figures, Euclid investigated the laws governing their behavior and how they combined with one another, discovering a series of universal and self-evident truths, which he called axioms, and which constitute the foundation of his Elements (Heath, 1956). They are five, and we present them below, both for their importance and their beauty: • Given two points, a straight line can be drawn joining them. • A straight line can be extended indefinitely in both directions. • Given a point and a distance, a circle can be drawn with that point as center. • All right angles are equal to one another. • If a straight line intersects two other lines so that the interior angles on one side sum to less than 180°, then those two lines, if extended far enough, will meet (the parallel postulate). From these elements or atoms of figures and from the rules combining them—as if it were a kind of chemistry—Euclid employed Aristotle’s logic to demonstrate the whole of Euclidean geometry, using only a straightedge without markings and a compass, without making a single measurement. Possibly for this reason, in the intellectual tradition of ancient Greece, geometry occupied a privileged place as the model of true knowledge (Nets, 1999). 9 3. GALILEO’S EPISTEMOLOGICAL REVOLUTION: THE BRIDGE BETWEEN THE WORLD OF IDEAS AND THE REAL WORLD Galileo, familiar with all these ideas, framed his research program within that paradigm. For Plato, the ideas of geometric figures had a reality as full as it is possible for an idea to have. They existed before human beings did, and independently of them, like Mount Olympus for the Greeks, the Capitol for the Romans, or Svarga for the Hindus. Aristotle, although denying the real existence of universals, admitted that such perfect geometric figures could be recognized in the forms of windows and doors, in roof drawings, or in the appearance of the Moon and Sun in the sky. They were what gave form to things. From these observations, Aristotle argued, universals are elaborated through a process of abstraction carried out by human beings—a position we may define as moderate realism (Cornford, 1952): universals manifest themselves in things (not apart from them, as in Plato), but they have full existence only in our minds, when we create them through abstraction. But Galileo was not concerned with whether universals existed in reality or virtually. What really mattered to him was that, whatever their nature, the geometry built upon them provided true knowledge. He set as his goal to find a connection between this geometry of the world of true knowledge and the reality in which we live. It was necessary to locate that bridge that would allow reasoning about the phenomena of the world without losing the rigor and validity of mathematical truths. One had to discover that connection in order to apply the laws of geometric thought to the study of the universe, preserving their characteristics as true knowledge. The conviction that such a bridge existed is evident in his major works, where he affirms, as we shall see, that the book of nature is written in mathematical language, and without geometry it is impossible to understand it (Galilei, 1623/1957; Koyré, 1973). Galileo found the answer in astronomy, in architecture, and in the engineering of bridges and tunnels. Eratosthenes (c. 276–194 BC), librarian of the Library of Alexandria, had employed geometry and the theory of proportions to measure the radius of the Earth, the size of the Sun and the Moon, and the distances among these three celestial bodies (Evans, 1998). For centuries, Euclidean geometry served as a quantitative tool in the construction of cathedrals, bridges, aqueducts, cloisters, and fortifications, determining the radius of arches and the weight of walls. And it had all been accomplished through the simple act of measuring things in this world and transferring those measurements to the triangles, circles, and spheres of the world of geometry. Water flowed through aqueducts thanks to careful measurements of descent angles, and cathedrals reached precisely the height the master mason had calculated they would. The act of measuring was the method that, by turning lengths, areas, and angles into numbers, allowed geometry—and all its true knowledge—to be used in the real world, with the reliability of its predictions. 16 donkeys) from the world of ideas to the real world, where, although they may seem paradoxical, they prove to be strikingly correct. Aristotle, for his part, would offer a slightly different explanation: it is we who abstract the nature of the mathematical objects associated with magnitudes (the universals), creating them, studying them, and determining their behavior. As we have pointed out elsewhere, throughout the history of philosophy the interpretations have not changed appreciably. Kant, straddling both positions, would say that it is not things-in-themselves that exhibit mathematical behavior, such as vectorial behavior, but rather that these mathematical objects are innate categories of human understanding used to analyze and organize reality. Being, for Kant, a priori forms like space and time, his interpretation would not be too far removed from Plato’s. Likewise, Piaget’s interpretation—that human beings actively construct the Kantian categories in childhood from observing and manipulating objects in the real world— would not be very far removed from Aristotle’s. Finally, in modern science, the interpretation of this extraordinary fact continues to present a mysterious nature and remains a subject of debate, as we shall see below. But the same does not hold true in quantum mechanics. When we associate the mathematical object “wave”—more precisely, a wave function represented by a complex number—with a material particle, the interpretation of what that physically means is no longer immediate. In this case, the mathematical formalism does not translate directly into an intuitive image of the world. This is evident in Schrödinger’s equation (1926), which describes how the wave function evolves over time. Since its formulation, its physical meaning has been a matter of debate. The standard interpretation (Copenhagen, defended by Bohr and Heisenberg) affirms that the wave function does not represent reality itself but the probability that a measurement will yield a certain result. Schrödinger sought to highlight the implausibility of this outcome by translating it into the macroscopic world, leading to the peculiar idea of his cat in a superposition of states (a live cat and a dead cat simultaneously), where it would remain in that unsettling state until the box was opened and the animal observed. Over time, various interpretations have been proposed to explain the meaning of the wave function. Among the most prominent are: • The Copenhagen interpretation, proposed by Niels Bohr and Werner Heisenberg in the 1920s, which holds that the wave function does not describe a physical reality but our knowledge of the system; before measurement, it is in a superposition of possible states, and measurement provokes its collapse into a single outcome. • The many-worlds interpretation, formulated by Hugh Everett III in 1957, eliminates the notion of collapse: all possible results of a measurement occur, 17 each in a different universe. Universes multiply from the original, in a way similar to the trajectories generated in a Monte Carlo method, all existing simultaneously, although without our being able—at least for now—to move from one to another. • The hidden variables theory, defended by David Bohm in 1952, posits that the wave function is real and guides the evolution of particles with well-defined properties at all times; the apparent indeterminacy is due to our ignorance of certain hidden variables that we should eventually uncover. However much we try, part of the wave function seems to remain in Plato’s world of Ideas: it has characteristics not directly observable, but which determine what happens in the real world—such as phase, which determines interference phenomena, or quantum coherence (Dirac, 1930; Feynman, Leighton & Sands, 1965; Penrose, 2004). And perhaps for that reason the wave function so closely resembles an Idea from the myth of the cave: we cannot directly access it; we can only observe the shadows it projects. We could also venture into the strange world of tachyons, with their imaginary mass and velocity greater than light, which—if they exist—could connect the future with the past and explain to Bohr the mysterious collapse of the wave function. From the perspective of the philosophy of science, this scenario leads us to a fundamental question, one already posed by Kant: to what extent do our theories describe the world as it is, and to what extent do they construct it from our own categories of thought? All these interpretations lead us into a territory reminiscent of the magical realism of Fernández Flórez’s The Enchanted Forest, García Márquez’s One Hundred Years of Solitude, or Carroll’s Alice in Wonderland, in which the boundaries of reality blur and take on multiple, shifting forms. It is in this world, at the very limits of knowledge, that our students will develop their professional lives. And they deserve to be prepared as best as possible. 6. NEW CONSEQUENCES OF THE GALILEAN REVOLUTION: THE EXPANSION OF THE WORLD OF SCIENCE THROUGH MATHEMATICS By linking the domain of science to what can be described through the mathematical language, science became, at once, somewhat enslaved to mathematics and simultaneously its beneficiary. Yet this also brought significant advantages. Starting in the seventeenth century, mathematics broadened its domain: it ceased to deal exclusively with numbers, equations, and geometric figures, and began to work with symbols, relations, and formal rules. The mathematical object was no longer limited to being a quantity but came to be conceived as an abstract structure. This shift also transformed the very scope of science, as defined by Galileo (1564-1642): if science is that which can be formulated mathematically, and mathematics now includes structures 18 and relations among them, then science expands into domains that had previously been considered extra-mathematical. We therefore continue along the path of history, pointing out the most significant milestones for our account. The first of these milestones appears with symbolic algebra in the seventeenth century. Descartes (1596-1650), in his Géométrie (1637), replaced numbers with letters. This strategy—apparently as simple as writing y = x + 3 to encompass infinite specific equalities, such as 7 = 4 + 3; 8 = 5 + 3, etc.—allowed him to represent the symbolic equality y = x + 3 in his Cartesian coordinates (also his invention), revealing that this equation corresponds to a straight line. In this way, he unified geometry and algebra into a new discipline: analytic geometry, which uses the language of algebra to study geometric figures through coordinates and equations. From the introduction of analytic geometry by Descartes, the representation of figures by equations became a fundamental principle of modern mathematics. Within this framework, any curve or geometric figure can be described as the set of points that satisfy a given algebraic or analytic relation. Thus, a circle can be expressed by the equation x² + y² = r², a parabola by y = ax² + bx + c, or more complex curves through parametric formulations. The development of the latter has made it possible to approximate virtually any shape, including recognizable silhouettes such as that of a horse—not from “natural” equations, but through specific algebraic constructions that translate contours into mathematical expressions. In fact, sets of such equations circulate on the internet today for drawing hearts, cats, or even Homer Simpson. 19 In this way, analytic geometry demonstrates not only its capacity to unify algebra and geometry but also to extend mathematics toward the symbolic representation of any conceivable form, consolidating the idea that everything imaginable can, in principle, have a mathematical formulation. Thanks to this unification, Galileo was able to describe the trajectories of projectiles and the fall of bodies by means of parabolas, and Newton (1642-1727) formulated his laws of motion and universal gravitation through equations, describing natural phenomena mathematically. The second milestone—impossible without the first—was the formulation of infinitesimal calculus by Newton and Leibniz, introducing the concept of the derivative as the change in the value of a variable, that is, the change in the measurement of a magnitude. With it, physical laws—such as those of gravitation or electromagnetism— were expressed by means of differential equations, relations among variations in measured values. Another important figure in the history of science is Gottfried Wilhelm Leibniz (1646– 1716). In addition to developing differential and integral calculus independently of Newton, using the notation still employed today, he also invented the binary number system, fundamental in numerous fields such as data representation, computing and information technology, digital electronics, cryptography, and cybersecurity, as well as in advanced areas such as artificial intelligence and machine learning. He complemented this with another invention in the world of ideas: his characteristica universalis, a 20 symbolic system or universal language to represent human thought. This allowed him to connect logic with algebra, introducing logical connectors to operate with propositions as if they were numerical expressions, and to construct the famous truth tables. Another fundamental figure is Évariste Galois (1811–1832), who invented—or perhaps discovered—group theory, revealing new algebraic structures such as the permutation group associated with the roots of an equation. What matters, he showed, is not the elements of a set but the structure that relates them—a new concept that broadened the field of mathematics. On the eve of his death in a duel, at barely twenty years of age, Galois, foreseeing his destiny, feverishly wrote a letter-memoir to his friend Auguste Chevalier, in which he condensed his fundamental discoveries. In it, he declared: “I have no time,” and begged mathematicians to analyze and publish his ideas. That night he drafted what we now know as Galois theory, the foundation of modern algebra. His contribution was so revolutionary that it was not understood until some time later—yet another example of Kuhn’s structure of scientific revolutions. The twentieth century consolidated this shift. With Hilbert (1862-1943) and his formalist program, mathematics was redefined as a system of logical rules applied to symbols, without the need for immediate semantic interpretation. Mathematical logic, advanced by Frege, Russell, and Gödel, established the formal foundations of the entire mathematical edifice. In this way, mathematics ceased to be only what could be treated quantitatively (the mathematics known to Galileo) and extended to everything that possesses a formally recognizable and definable structure. This expansion of the mathematical domain had direct consequences for science. Recall Galileo’s famous criterion: what can be expressed in mathematical language is scientific (to satisfy the criterion of truth). Without modifying that criterion, what can now be formalized within a logical-mathematical structure is considered scientific. This transformation has redefined not only the boundaries of science but also those of the representation of knowledge. Thanks to this broadening of mathematics, science has penetrated fields traditionally beyond quantification. In linguistics, for example, Noam Chomsky introduced a generative grammar based on formal rules that allowed natural language to be described as a computational system. In economics, game theory, optimization models, and econometrics have made it possible to formalize human decisions and social interactions within algebraic and statistical structures. In biology, the development of formal genetics from the work of Mendel and Watson-Crick has led to mathematical representations of DNA, and the theory of neural networks has provided mathematical models simulating the functioning of the nervous system, linking artificial intelligence with computational neuroscience. In short, the growth of mathematics constantly redefines what may be considered scientific knowledge. 21 7. THE MATHEMATIZATION OF THE WORLD AND THE TRANSFORMATION OF THE SCIENCES From Hilbert and the formalist turn of the twentieth century onward, the relationship between mathematics and science was radically transformed. Until then, mathematics had been conceived as a descriptive instrument for physics, useful for expressing laws or making predictions. However, the axiomatic formulation of general relativity, Hilbert space as the foundation of quantum mechanics, and the central role of group symmetries in particle physics reveal a profound change: physics is formulated directly as a mathematical theory (Hilbert, 1915/2009; von Neumann, 1932/1996). This shift implies that the physical world is increasingly understood as a mathematical world. The objects of the theory are defined by the mathematical structures that represent them. The notions of field, particle, or interaction cannot be conceived without the algebraic, geometric, or topological apparatus that underlies them (Weyl, 1952; Cartan, 1966). From this arises a broader thesis: the classical sciences (physics, chemistry, biology, economics, linguistics) can be seen as local applications of mathematics to the empirical world. Each discipline selects certain aspects of reality—population dynamics, molecular structures, economic exchanges, grammatical rules—and models them with specific mathematical tools. From that moment on, the world becomes mathematical in a double sense: ontological, because the structure of the world is conceived as accessible only through mathematical categories; and epistemological, because the validity of our knowledge is grounded in the possibility of constructing formal models that can be tested against experience (Wigner, 1960; Tegmark, 2014). 22 Thus, progressive mathematization is not a mere technical process but a true paradigm shift in the conception of scientific knowledge: mathematics ceases to be an auxiliary tool and becomes a kind of virtual reality in which all the sciences live. We are certain that this is what Plato is explaining to Aristotle in the moment captured by Raphael in his painting The School of Athens. 8. THE BUILDING OF SCIENTIFIC KNOWLEDGE In the context of science education, it is essential to understand the internal structure of scientific knowledge. For this, it is necessary to introduce some key concepts that, in addition to offering a rigorous epistemological framework, facilitate pedagogical practice. Just as buildings are composed of basic elements—pillars, doors, windows, columns, rooms, floors, etc.—scientific knowledge is articulated around fundamental components that can be identified and analyzed: magnitudes, laws, theories, models, and paradigms. The first two—magnitudes and laws—have already been discussed, since it was necessary to address them before introducing the criterion of truth and the mathematical language that, in the form of equations, expresses these laws. However, we will now return to them from a different perspective. 23 Theories We have seen that the primary elements of scientific knowledge—the “atoms” of knowledge—are the concepts and magnitudes that constitute the ontology of the domain in question. Defining these elements rigorously is not an easy task, and to avoid falling into circular explanations like those in dictionaries, we will turn to historical examples that illustrate their functional role in the development of science. As a suitable case, we will analyze Isaac Newton’s Philosophiæ Naturalis Principia Mathematica (1687), a complete theory of classical mechanics, still today a model of theoretical structuring and one that coincides with the intuitive framework that human beings, and therefore our students, naturally adopt. In this example, we can observe how scientific knowledge is organized at its different levels. Ontology and epistemology of Newtonian classical mechanics Recall that all knowledge is defined as the mental representation we elaborate of the real world we wish to understand. We can imagine this representation as unfolding within a scenario limited to that part of reality on which we focus our interest. In this context, we call ontology the set of concepts and magnitudes present in that scenario, within which the phenomena under study occur. In Newtonian classical mechanics, this ontology includes: • Absolute space: a fixed, immutable framework in which coordinate axes can be set to define positions and trajectories. These positions and trajectories are the same for any observer, regardless of their location or speed. • Absolute and universal time: it flows uniformly and is the same throughout space for every observer, regardless of their state of motion. It is independent of both space and the observer’s condition. • Mass: quantity of matter, an invariable property of bodies, independent of their velocity or position. • Forces: entities that propagate instantaneously and produce changes in the state of motion of bodies (acceleration). Associated with this ontology is an underlying epistemology that defines what can be known through measurements and with what degree of precision. In Newton’s mechanics, it is as follows: • Determinism: in Newton’s theory, it is possible to know exactly the state of a system and predict its future state. • Objective measurement: physical magnitudes possess defined values that are independent of the observer, the location, and the moment of measurement. 24 Summarizing: we may say, in agreement with Chomsky’s idea, that the ontology and epistemology of a problem define the nouns and verbs of the scientific language necessary to describe and treat it. This ontological and epistemological framework defines the context in which Newton’s laws are formulated, telling us how bodies behave when subjected to a given force, regardless of its nature. As we have noted, these empirically obtained laws replace the axioms of mathematics and come to be called principles. 1. First principle or law of inertia: a body remains at rest or in uniform rectilinear motion if no external force acts on it. 2. Second principle or fundamental law of dynamics: the acceleration of a body is directly proportional to the net force acting on it and inversely proportional to its mass, expressed with the equation F = m·a. 3. Third principle or law of action and reaction: when body A exerts a force on body B (action), another force of equal magnitude and opposite direction appears that body B exerts on A (reaction). In addition to these laws, Newton formulated a causal law, the law of universal gravitation, which defines the force of attraction between two bodies possessing mass, depending on their masses and the distance separating them. This force propagates instantaneously through absolute space, with infinite velocity, which is one of the defining characteristics of classical mechanics. The set formed by ontology (space, time, mass, force), epistemology (determinism, objectivity), and a precise definition of force (as action at a distance) constitutes a theory—in this case, Newton’s gravitational mechanics. Models Returning again to Kantian knowledge, let us recall that its essence is the mental representation of the external real world, and that this representation necessarily has to be simplified. Each of these representations of an aspect of the world is what we call in science a model. We can speak of models of family, of society, of the atom, of the solar system, or of the universe, and for each we find diverse examples depending on the aspect we wish to emphasize. Once the model is determined, we can analyze it within the theoretical framework we choose—that is, within the theory that provides the appropriate conceptual and mathematical tools for its study. To illustrate what a model is, a classic joke in didactics and philosophy of science is often told. A group of horse breeders ask a physicist for help in improving the performance of 25 their racehorses. After a few days of reflection, the physicist presents his theoretical model beginning with the phrase: “Let us suppose a perfectly spherical horse without friction….” The joke illustrates how physicists resort to extremely simplified models in order to solve overly complicated problems. Although in reality spherical horses without friction do not exist, such idealization is fundamental in science: • It allows the isolation of relevant variables and the elimination of irrelevant ones. • It facilitates the mathematical formulation of the problem. • It serves as a starting point for gradually introducing more realistic corrections. It is analogous to Galileo’s case when studying the fall of bodies and assuming no air resistance, or to the use of the ideal gas in thermodynamics. As a more serious example, and continuing along the line we have been developing in these works, let us illustrate the modeling process of the solar system, long equivalent to the universe. The heliocentric versus the geocentric model: from Aristarchus to Newton The history of astronomy up to the eighteenth century can be read as the struggle between two models of the cosmos, both born, unsurprisingly, in classical Greece. The geocentric model, heir to Aristotle and Ptolemy, and the heliocentric model, sketched in antiquity by Aristarchus of Samos and recovered many centuries later by Copernicus. It is remarkable that already in the third century BC Aristarchus proposed a universe centered on the Sun, with Earth and the planets revolving around it. We know of its existence thanks to Archimedes, who mentions it in The Sand Reckoner (Archimedes, trans. 1990), and Plutarch, in Natural Questions, who confirms both the bold proposal and its lack of acceptance in antiquity (Plutarch, trans. 2006). In 1543, Nicolaus Copernicus resumed and developed this hypothesis in De revolutionibus orbium coelestium, proposing a system in which the planets moved in circular orbits around the Sun. Although it retained elements of the Aristotelian tradition, such as supra-lunar circular motion, his work posed a direct challenge to over a thousand years of hegemony of the Ptolemaic geocentric system and marked the beginning of the scientific revolution. The progress of the heliocentric model was gradual. Between 1580 and 1597, Tycho Brahe collected astronomical data on the orbit of Mars with unprecedented precision, which Johannes Kepler later used to formulate his three laws of planetary motion (1609– 32 • Matthews, M. R. (2015). Science teaching: The contribution of history and philosophy of science (2nd ed.). New York: Routledge. • Netz, R. (1999). The shaping of deduction in Greek mathematics: A study in cognitive history. Cambridge: Cambridge University Press. • Newton, I. (1687/1999). Philosophiæ Naturalis Principia Mathematica. Berkeley: University of California Press. • Penrose, R. (2004). The road to reality: A complete guide to the laws of the universe. London: Jonathan Cape. • Piaget, J. (1926/2007). La representación del mundo en el niño. Madrid: Morata. • Piaget, J. (1952). The origins of intelligence in children. New York: International Universities Press. • Piaget, J., & García, R. (1981). Psicogénesis e historia de la ciencia. México: Siglo XXI. • Plutarco. (2006). Cuestiones naturales (Trad.). Madrid: Gredos. • Popper, K. (1972/1994). Conjectures and refutations. London: Routledge. • Putnam, H. (1981). Reason, truth and history. Cambridge: Cambridge University Press. • Quian Quiroga, R. (2017). Borges and memory: Encounters with the human brain. Cambridge, MA: MIT Press. • Russell, B. (1945). A history of Western philosophy. New York: Simon & Schuster. • Schrödinger, E. (1926). Quantisierung als Eigenwertproblem. Annalen der Physik, 79, 361–376. • Tegmark, M. (2014). Our mathematical universe: My quest for the ultimate nature of reality. Vintage. • Venn, J. (1880). On the diagrammatic and mechanical representation of propositions and reasonings. Philosophical Magazine, 9(59), 1–18. • Van Fraassen, B. C. (1980). The scientific image. Oxford: Clarendon Press. • Vygotsky, L. S. (1934/1995). Pensamiento y lenguaje. Madrid: Paidós. • von Neumann, J. (1996). Mathematical foundations of quantum mechanics (R. T. Beyer, Trad.). Princeton University Press. (Trabajo original publicado en 1932). • Weyl, H. (1952). Symmetry. Princeton University Press. • Wigner, E. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications on Pure and Applied Mathematics, 13(1), 1–14.