The Point Mass as a Model for Epistemic Representation. A Historical and Epistemological Approach

The aim of this dissertation is twofold. Firstly, I will aim at showing the role that models play in the philosophy of science as epistemic instruments for representing the variegated aspects of the phenomena in the physical world. Secondly, I will intend to support this first aim by providing extensive analysis of a single case study, namely the point mass – which is one of the fundamental models of today’s mathematical physics – by pursuing a hypothesis concerning its possible development. In this respect my thesis aims to show that some mathematical models – in this particular case, the point mass – derive from a process involving the objectification of scientific practice, instead of being considered as abstractions from natural properties or objects. The outcome of this thesis will be to emphasise the idea that the point mass is a useful idealization used in mathematical physics as a model for epistemic representation. First, let me define what an epistemic representation actually is. Following Contessa (2007) and Swoyer (1991), we can maintain that a scientific model is an epistemic representation insofar as it gives a representation of any particular set of physical phenomena under observation, even though the knowledge drawn about it does not correspond to what we directly observe in nature. There is, in fact, an epistemic gap between the conclusions that we infer from the model and what we directly observe in nature. In this respect, the model of a certain system is an epistemic representation of reality, insofar as it allows the user – by following a certain set of rules – to infer some conclusions and acquire some knowledge concerning that particular system of reality. Therefore, those conclusions need to be “translated”, before one can judge them in a certain respect. The model is first built in such a way that it is easier to study than the target-system, and this factor therefore allows us to derive results. Second, it is assumed to represent its target system, in which representation stands for something like a licence to draw inferences. This approach aims at emphasising the cognitive role played by the scientific community: every user takes the model to stand for the target system, and, moreover, the user is also able to perform valid surrogative inferences from the model to the target. This requires that the user interpret the surrogative inferences in order to acquire knowledge about the phenomenon under observation. Every user is a cognitive agent, who is able to develop representations of the world and make judgements about both the world and their representations of it. In an attempt to understand what it takes to develop a mathematical representation of nature, the historiographical purpose of this dissertation – to which Chapters 2-4 are dedicated – is invoked to support the idea of considering the point mass as part of the phenomenon that Enrico Giusti has labelled the ‘objectification of procedure’. His hypothesis states that mathematical objects are obtained from idealizations of procedures through three stages: i) ‘as’ investigative tools ‘within’ demonstrative procedures; ii) by becoming common elements that are used for solving problems, for which reason they also became objects of study, depending on a specific practical context; and iii) by becoming new and abstract mathematical objects (e.g. as mediators between the world and the scientific theories or models of intelligibility) which deserve to be studied on their own. The claim is not to establish who was able to identify this mathematical object, but rather to find an answer to the question: ‘When and in which demonstrations does the idea of the point mass first become conceptualised as such?’. This dissertation is divided into four chapters. The first chapter focuses on the role of models in science, which are seen not solely as tools that provide a means for interpreting a formal system, but also as tools for representing the world. After this, each of the three subsequent chapters – Chapters 2, 3 and 4 – is devoted to one of the stages of the objectification of procedure. However, none of them pretends to deliver an exhaustive introduction to the historical period under observation, but each respectively sketches out only a historiographical perspective. For example, my discussion of Archimedean mechanics in Chapter 2 does not follow a straightforward chronological approach, because I wish specifically to pursue the claim that I made use of a portion of treatises that have been recently discovered (e.g. The Method, which was rediscovered only at the end of the nineteenth century). However, this aspect should not be seen as a limitation and defect of my reconstruction, because, although Renaissance mathematicians did not have direct access to this last Archimedean treatise, it can be maintained – and I will prove this assertion throughout my dissertation – that there exists a direct methodological affiliation between the two lines of thought (i.e. those in the Archimedean and the Renaissance eras). The only point of dissonance between the Archimedean outcomes and the Renaissance works was the purpose that they had in mind when they were carrying out their research. The first chapter begins by aiming to examine the various perspectives relating to the structure of scientific theories. It will, however, only give an overview of the philosophical context in which the notion of the model assumes a decisive role in science. We shall focus on the different kinds of theoretical models (e.g. abstraction, idealizations and analogies) commonly used in scientific practice in order to represent physical reality and simplify computations. Only in the last section of the first chapter will we emphasise the surrogative role fulfilled by a model; in fact, models are more often used as surrogates of states of affairs which are taken into account for a certain purpose, i.e. as a source from which every user can infer conclusions specifically relating to that model. These conclusions in turn need to be translated from the users in advance, in order to reach some scientific conclusions about that part of reality in itself. In Chapter 2, we shall see that the first stage of the objectification of procedure relates to ancient Greek geometry, where its main impact lay in those demonstrations in which the notion of the centre of gravity – which is of central importance – shares some essential properties with the modern notion of the point mass. The point mass, as a mathematical entity having an algebraic meaning, is not yet present in Greek geometry; rather it is the centre of gravity that functions as a demonstrative tool detached from some of its geometrical features. This stage corresponds to the enquiry made within the field of Archimedean equilibrium concerning the application of the law of the lever, according to which geometrical objects stand in equilibrium at distances that are inversely proportional to their extensions. This constitutes one of the foundational principles of statics imported into a geometrical context. My emphasis focuses on the fact that Archimedes should be held among the first authors to give importance to the heuristic and mechanical meaning of geometrical demonstrations. His heuristic practice can be read as a ‘physicization’ of mechanics, which allows us to consider only the quantitative relations between bodies, without considering some of their attributes (e.g. mass and spatial dimension). Chapter 3 introduces the restoration phase of the ancient Greek mathematical tradition and the legacy of Archimedes between the Middle Ages and the Renaissance period. In particular, this Chapter is dedicated to the School of Urbino and its rediscovery of the ancient Archimedean treatises, not only from a philological but also – and mainly – from both a mathematical and a speculative point of view. In order to show the different approach of the Renaissance towards mechanics, and the use that scholars made of the notion of the centre of gravity, our focus will lie with the practical and theoretical contributions that Renaissance scientific humanism provided: i) in consolidating the new way of doing and conceiving mechanics; and ii) in the second stage of the objectification of procedure, namely that, as can be seen in several treatises published between the fifteenth and sixteenth centuries by Urbino School mathematicians – such as Federico Commandino and Guidobaldo dal Monte – it was possible to switch attention onto the practical use we can make of geometrical elements, such as the centre of gravity. This notion became fundamental in various practical contexts linked to the scienza de ponderibus (science of weights), the aim of which was to solve and formalize static problems concerning heavy bodies, with particular reference to those hanging from a balance. Moreover, the concept of the centre of gravity is also useful to solve the socalled “Equilibrium Controversy”, which addresses the question whether or not a deflected balance will return to its horizontal position, a controversy which, though it had already seen its birth during the Medieval period, was only during the Renaissance, and following the rediscovery of all the Archimedean corpus on statics and hydrostatics, applied to the science of machines and to other purely practical contexts. It is only at the end of the Renaissance, at the turn of the seventeenth century, that the work of Luca Valerio promoted the introduction of a philosophical, almost epistemological, debate over the meaning of physical properties applied in a formalized mathematical context. The Renaissance mathematicians worked towards the foundational programme of constructing an epistemological debate about the analysis of the conditions under which mathematical principles can and should be considered to be true of physical things. A purely mathematical treatment of physical reality was achievable due to the awareness that, in contrast, a solely empirical and mechanical approach is insufficient to understanding the ways in which the physical world behaves. Chapter 4 is devoted to modern rational mechanics. In pursuing the aim of understanding what it takes to develop a mathematical representation of nature by means of an idealized and abstract model, we will see that seventeenth-century mathematical physics represents the framework for the completion of the third stage of the objectification of procedure, insofar as it was at this time that the centre of gravity became an independent object of study as the point mass , i.e. as an idealized entity used to represent physical objects and to which we can ascribe natural properties, such as volumetric extension, mass and forces. The point mass is still a representational geometrical point whose features are idealized, but it now assumes the role of being an independent object of research, which is useful for building the foundational principles of rational mechanics. We shall see how Galileo was able to shift the investigative methodology used in the field of mechanics to a more theoretical and physical perspective, by showing how the purely practical operations carried out through machinery could be used to interpret the working principles behind nature. Or, in other words, within Galileo’s research we can observe that he uses machines as aids in a bid to confer a discursive structure onto the phenomenal world. Within the second part of this chapter, I will examine the more metaphysical contributions of Thomas Hobbes concerning the rule of human imagination in building our representational model of physical phenomena and the point mass. Finally, in the last part of this chapter, I will examine the way in which the model of the point mass began to be used from a purely mathematical point of view, in order to represent, with purely algebraic language, a series of states of affairs which were not only made up of simple rigid bodies, but which were also typified by a higher level of complexity. This represents the climax of the procedure of objectification of our centre of gravity. Now it is no longer a geometrical tool shifted along the arms of a real mechanical balance, but instead a mathematical entity, investigative tool and a model of intelligibility – which is explicitly introduced for the first time in Newton’s and Euler’s research – that has two main characteristics: i) it is the centre of mass of any rigid body, no matter how it is shaped; and ii) it is the point of applicability of all the forces – gravity, works, pressure and so on – acting on a stationary or a moving body, be it travelling in uniform motion, in a state of acceleration or in parabolic motion. The analysis of this scientific practice allows us to reach the conclusion that models serve to understand how the world works and not to help us argue about the ontological claims of what we can observe or think about the nature of the physical world itself. Models have an epistemic value that is strongly dependent on the interpretation that we – as users – attribute them with respect to their specific and (crucially) context-dependent rules.