Dipartimento di Scienze Pure e Applicate Corso di Dottorato di Ricerca in Scienze di Base e Applicazioni Ph.D. Thesis THE POINT MASS AS A MODEL FOR EPISTEMIC REPRESENTATION. A HISTORICAL & EPISTEMOLOGICAL APPROACH Tutor: Candidate: Chiar.mo Prof. Vincenzo Fano Dott.ssa Antonella Foligno Curriculum Scienze della Complessità Ciclo XXX – A. A. 2016/2017 Settore Scientifico Disciplinare: M-FIL/02 To Martina and Giada If our highly pointed Triangles of the Soldier class are formidable, it may be readily inferred that far more formidable are our Women. For, if a Soldier is a wedge, a Woman is a needle; being, so to speak, all point, at least at the two extremities. Add to this the power of making herself practically invisible at will, and you will perceive that a Female, in Flatland, is a creature by no means to be trifled with. But here, perhaps, some of my younger Readers may ask how a woman in Flatland can make herself invisible. This ought, I think, to be apparent without any explanation. However, a few words will make it clear to the most unreflecting. Place a needle on the table. Then, with your eye on the level of the table, look at it side-ways, and you see the whole length of it; but look at it end-ways, and you see nothing but a point, it has become practically invisible. Just so is it with one of our Women. When her side is turned towards us, we see her as a straight line; when the end containing her eye or mouth – for with us these two organs are identical – is the part that meets our eye, then we see nothing but a highly lustrous point; but when the back is presented to our view, then – being only sublustrous, and, indeed, almost as dim as an inanimate object – her hinder extremity serves her as a kind of Invisible Cap. [E. A. Abbott, Flatland. A Romance in Many Dimensions, 1882] Mathematical truth, unlike a mathematical construction, is not something I can hope to find by introspection. It does not exist in my mind. A mathematical theory, like any other scientific theory is a social product. It is created and developed by the dialectical interplay of many minds, not just one mind. When we study the history of mathematics we do not find a mere accumulation of new definitions, new techniques, and new theorems. Instead, we find a repeated refinement and sharpening of old concepts and old formulations, gradually rising standard of rigor, and an impressive secular increase in generality and depth. Each generation of mathematicians rethinks the mathematics of the previous generation, discarding what was faddish or superficial or false and recasting what is still fertile into new and sharper form. [N. D. Goodman 1979, p. 545] Table of Contents Introduction 1 Chapter 1 Models for Scientific Representation 11 1. 1 The Structure of Scientific Theories 18 1. 2 Models and Model-Building Practice 31 1. 3 One Model for many purposes 41 Chapter 2 Ancient Mathematics 60 2. 1 A General Introduction to Ancient Mathematics: the methods and tools of Greek mathematicians 60 2. 2 The standard Euclidean geometry 65 2. 3 Towards a Heuristic tradition 78 2. 4 Archimedes and the Centres of Gravity 82 2. 5 The Method of Mechanical Theorems 90 2. 6 Floating Centres of Gravity 104 Chapter 3 The Renaissance Turns 109 3. 1 The ‘Restoration’ 113 3. 2 The rediscovery of Archimedes in the ‘School of Urbino’ 129 3. 3 Mathematical Humanism in the Renaissance 138 3. 4 Federico Commandino, Head of the Urbino School 140 3. 4. 1 The new Centre of Gravity 145 3. 5 Francesco Maurolico 151 3. 6 The Scientia de Ponderibus 155 3. 7 Guidobaldo del Monte: the new Archimedes? 162 3. 7. 1 Guidobaldo the Natural Philosopher 174 3. 8 Luca Valerio: the turn towards the mathematical analysis of physical states of affairs 187 Chapter 4 From Galileo to the Modern Turns 196 4. 1 Galileo Galilei: a Bridge to the Modern Mathematical Physics 203 4. 2 The Modern Turns 232 4. 2. 1 Hobbes: Abstraction and Idealization as Acts of Mind 234 4. 2. 2 Newton 242 4. 2. 3 Euler, Rational Mechanics and the Algebraic Representation of Physics 252 Conclusions 264 References 270 Acknowledgements 305 Introduction Any casual philosopher of science writing in the twentieth century cannot help but showcase the recurrent presence of Newton’s science of mechanics in introducing her diverging views on the nature of (physical) science. Newton’s Principia embodies a powerful and coherent research program for linking mathematical representations with real-world structures. A crucial passage where Newton himself expresses clearly what he is up to in his Principia occurs in a scholium to section 11 of the first book: In mathematics we are to investigate the quantities of forces with their proportions consequent upon any conditions supposed; then, when we enter upon physics [my emphasis], we compare those proportions with the phenomena of Nature, that we may know what conditions [or laws] of those forces answer to the several kinds of attractive bodies. And, this preparation being made, we argue more safely, concerning the physical species, causes and proportions of the forces.1 The first two books of his treatise deal with forces treated abstractly; they contain purely mathematical exercises in determining the implications of the laws of motion under different conditions. The investigation of all sorts of systematic relations which occur in physical phenomena is constructed by the means of mathematical models,2 which function as instruments of theoretical measurement, and which in turn allow the determination of parameters that characterise forces from parameters that characterise motion. In a later stage, or what Newton describes as “enter[ing] upon physics”, these models are put to use in the third book to measure the characteristics of the forces that can be found in our solar system. 1 Newton (1687) 1726, p. 218. In mathesi investigandae sunt virium quantitates et rationes illae, quae ex conditionibus quibuscunque positis consequentur: deinde, ubi in physicam descenditur conferendae sunt hac rationes cum phaenomenis; ut innotescat quaenam virium conditiones singulis corporum attractivorum generis competant. Et tum demum de virium speciebus, causis et rationibus physicistutius disputare licebit. 2 Hereafter, the terms ‘mathematical model’ and ‘mathematical object’ are used interchangeably. 1 This powerful methodology has the scope to function as an example for emphasising how the world could be formally defined with mathematical models, which can additionally be used for deductive reasoning, simplifying computations and mathematical proofs. Moreover, these models can also be used in physics for representational, explicative and predictive purposes. Examples of them include numbers, equations, functions, relations, and so on. Geometry too has its own mathematical objects, such as points, lines, planes, and solid figures. The philosophy of science deals with the ontological commitments relating to the nature of those objects. Are they mind-dependent objects arbitrarily produced by the scientific community for practical and speculative purposes? Or are they rather mind- independent objects, which have their own autonomy and existence? The fact that mathematics represents the best way we have today of representing the world in which we are embedded, allows us to use those objects – no matter what is the nature of their existence – as a model for representing physical phenomena in the natural world. They are also considered to be useful for mathematical manipulation and clarification, but they represent only idealized states which, strictly speaking, do not exist ontologically. To clarify this point, I suggest to the reader an example, which will also be the case study throughout the research presented in this dissertation. Let us look at the point mass3 as a mathematical entity which is used as a model for epistemic representational purposes of the phenomena under observation. First, let me define this entity with reference to its current definition taken from modern rational mechanics.4 3 The term ‘point mass’ can be found in the literature under several different expressions, including ‘material point’, ‘mass point’, ‘point particle’ and ‘point charge’. Hereafter the terms ‘material point’ and ‘point mass’ will be used synonymously in order to indicate either small bodies or their mathematical idealizations with the geometrically unextended point, or a point of a continuum model of matter. Of course the smallness in question refers to the extension of the space considered; for instance at the astronomical scale, the planets could be considered as punctiform. Moreover, I think that the expression ‘material point’ is not anachronistic, because it helps to refer to the materiality of the object, whilst at the same time the term ‘point mass’ is appropriate because of its explicit reference to mass. 4 The term ‘rational mechanics’ is used here in distinction from the mere mechanics of the past, in order to ensure that when we speak about modern mechanics, we are aware that statics, dynamics and kinematics 2 The material point – regarding its kinematic aspects (i.e. position, trajectory, velocity, acceleration, etc.) – could be considered as a geometrical point. However, under the action of a force it still should be considered as a natural body. Using this schematic representation it might be possible, on the one hand, to obtain the fundamental dynamic laws which underlie its behaviour; but on the other hand the material point’s dynamics will give us the fundamentals of mechanics. Therefore the laws of motion of any kind of body could be established just considering that body as an aggregate of material points.5 In this respect, the point mass is a theoretical and abstract entity which has dimensions that can be considered negligible when compared to a problem under examination.