Brentanian continua and their boundaries Arthur Heller Britto Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2020 c 2020 Arthur Heller Britto All Rights Reserved Abstract Brentanian continua and their boundaries Arthur Heller Britto This dissertation focuses on how a specific conceptual thread of the history of mathematics unfolded throughout the centuries from its original account in Ancient Greece to its demise in the Modern era due to new mathematical developments and, finally, to its revival in the work of Brentano. In particular, we shall discuss how the notion of continuity and the connected notion of continua and boundaries developed through the ages until Brentano’s revival of the original Aristotelian account against the by then established mathematical ortodoxy. Thus, this monograph hopes to fill in a gap in the present state of Brentanian scholarship as well as to present a thorough account of this specific historical thread. Contents Introduction 1 Chapter 1. Aristotle and the origins of the notion of continuity5 1. Aristotle’s theory of continuity5 2. The connection with motion 22 3. Sorabji’s piecewise solution 28 4. Antiquity after Aristotle 31 5. The Middle Ages 34 Chapter 2. The manifold-theoretic conception of continua 43 1. Medieval dissidents and the new mathematical methods of the 17th century 45 2. The “barren” 18th century 65 3. From continua to the continuum of real numbers 67 4. Conclusion 112 Chapter 3. Brentanian continua and their boundaries 114 1. Brentano’s criticisms 118 2. Actual infinities 122 3. Boundaries 124 4. Plerosis and coincidence of boundaries 126 5. Are continua gunky? 143 6. Tension between gunky spaces and measure theory 150 7. Teleosis 152 8. Conclusion 155 Bibliography 157 i Acknowledgments I would like to express my gratitude, first of all, to my advisor and committee members: Achille Varzi, Wolfgang Mann, Justin Clarke-Doane, Tamar Lando and Simona Aimar. They not only reviewed this work, but also offered guidance throughout its writing and the whole program. I couldn’t fail to mention my ex-advisor and friend Mario Porta, who was the one who taught me most of what I currently know about philosophy and who definitely opened the doors for me to be able to enter the realm of philosophical questions that permeate this monograph. I have also learned a lot from friends and colleagues, in particular those who were part of the Informal Formal Philosophy Group at Columbia: Rush Stewart, Robby Finley, Michael Nielsen, Yang Liu, and Ignacio Ojea Quintana. That group developed into perhaps the most significant source of learning in my graduate education. In particular, I’d like to mention Robby’s help in dealing with the intricacies of the Aristotelian and later Medieval notions of infinity and continuity. Finally, I would like to express my deepest gratitude to my family. They were always there to help me with whatever I needed in this journey. ii Introduction Imagine we have a light that shines with a blue colour. At one point, though, say at noon today, this colour immediately changes and the light starts to shine a red light. Then, one could ask: what is the last instant in which the light’s colour was blue? And, analogously, one could also ask: what is the first instant that the light started to shine red? An obvious answer to both questions seems to be precisely noon. However, if we take a closer look at these answers, we will realize, first that they cannot be both true together, if we assume that there is only one instant happening at noon; for if this instant is both the last instant in which the light was blue and the first instant that the light was red, we will necessarily have to say that the light was both fully blue and fully red at this instant, which seems to be a contradiction. Maybe, then, one might reply, we should say that noon is the first instant the light was red and we can take an instant that is one millisecond before noon and claim that that instant is the last instant in which the light was blue. A first obvious objection to this, however, would be: but what in the world allows one to pick the later colour as the one actually present at noon and not the earlier one? Indeed, this attempted solution is guilty of some apparent inherent air of arbitrariness; one cannot present a principled reason to choose red instead of blue as the colour of the light at noon. However, there is a further problem hiding in this simple alleged solution. A natural assumption about time is that between any two instants there is at least another instant. Now, if this is true, then, looking back at our example, we must conclude that there is at least another instant between that last instant in which the light was blue and noon; and, hence, we can ask: what was the colour of the light in that instant. If we answer “blue”, then we would contradict the claim that the instant one millisecond before noon was the last instant in which the light was blue; and, if we answer “red”, then we would contradict the claim that the instant at noon was indeed the first instant in which the light was red. Thus, we must conclude that in this particular instant between noon and a millisecond before noon, the light was neither blue not red. But we assumed that the change was 1 immediate, that is, that no instant passed between the light being blue and then red. We definitely seem to have sunk deeply into a murky logical swamp. Indeed, this kind of problems have a very old history, dating back at least to Ancient Greece, but they are nonetheless also relevant today, with discussions as to how to solve them coming from all directions of research. In particular, we can mention the 20th century attempts, pioneered by Whitehead, de Laguna, Broad and other authors, at constructing a “point-free” version of topology. However, the main point to keep in mind here is that all these discussions have as common ground the discussion of the topological notion of continuity. This notion is, however, not an intrinsically clear one. In fact, different circumstances call for slightly different ideas regarding how to cash out this notion of continuity. Nowadays, in mathemat- ics, the most common usage of this notion of continuity is perhaps in connection to functions. In this sense, a function — more specifically, let us think here of a simple function from real numbers to real numbers — is said to be continuous if one can draw its graph without taking the pen off the paper, i.e., if the line that we recognize as its graph does not contain any “jumps”. The usual formal characterization of this property is the classical -δ property, which shall be studied in a little more depth further on in this dissertation, but which can be intuitively described here as the property according to which a “small” change of the argument of the function only produces a “small” change in its value at this argument — or, in other words, a change to a “nearby” argument point moves the value of the function also to a “nearby” value. We can see in this sense how this notion is also connected to the notion of “closeness”. Nonetheless, this transition into a rigorous definition does away with the original intuition of gaplessness in the sense that we are now always considering a “close” argument point, but one that is indeed some distance away from the original point and that cannot be considered the “next” point following the original argument in the sense that there is a whole interval of points strictly between the two points we are considering. The main goal of this dissertation, thus, is to unravel a specific thread in the history of math- ematics. More specifically, we shall discuss the particular development of this notion of continuity, that started in Ancient Greece and eventually evolved into a key notion of the modern mathematical theory of point-set topology. However, we shall look at this historical thread from the perspective of Brentano’s ideas con- cerning these matters as they were presented in his late texts from the first decades of the 20th 2 century. In doing so, our goal is to fill out a gap in Brentano’s scholarship, as well as to offer a fairly self-contained presentation of this particular historical thread — giving special emphasis on details that share connections with Brentano’s ideas on the matter. It is not only my personal opinion that problems — and, ultimately, their solutions — are much better understood in historical context, but also, Brentano himself was a fairly historically oriented thinker and, although he presented many radically new ideas, he always relied on the historical tradition of every problem he dealt with. Thus, the first chapter of the dissertation deals with the beginning of the history of this concept of continuity, as it was presented by Aristotle and further developed both by later Ancient and by Medieval thinkers. Special attention will be given to a particular interpretation of these topics by a specific Medieval tradition that became known to contemporary writers as quasi-Aristotelianism. This is done since this tradition contains ideas that are very much in tune with the later Brentanian account of the subject, so that the study of this earlier tradition shall provide us not only with the necessary historical underpinning of Brentano’s position, but also with an early presentation of the logical structure of the problems of the Aristotelian position, and of the structure of a solution that will eventually turn out to be analogous to the later Brentanian solution.