Patrick J. Ryan

Since its formulation in the nineteenth century, Bertrand’s paradox has played a prominent role in discussions of probability theory and has received a great deal of attention from philosophers, mathematicians, and physicists alike. This is because the paradox is often interpreted as refuting the Principle of Indifference, a central pillar of classical probability theory. In this paper, I articulate a novel interpretation of Bertrand’s paradox according to which it is ultimately a paradox of infinity produced by a conflict between our intuitions and techniques for measuring infinite sets and the properties we wish to preserve when doing so. I proceed by first analyzing Bertrand’s original formulation of the paradox and show that one can make sense of this discussion entirely in terms how we deal with the infinite sets in question. I then discuss two possible approaches for dealing with this construal and argue that the paradox can be avoided by jettisoning our Cantorian intuitions for measuring infinite sets. Next, I analyze recent mathematical frameworks developed to resolve the paradox and argue that disagreement between these accounts indicates that a "meta-level" Bertrand’s paradox is lurking in the wings. This recurrence of the paradox leads us into deep questions about the possible indeterminacy latent in all attempts to measure infinite sets. Finally, I take stock of what we have learned from my analysis, delineate possible approaches for dealing with the meta-level paradox, and suggest directions for future research.

The great mathematician, physicist, and philosopher, Hermann Weyl, once called mathematics the “science of the infinite.” This is a fitting title: contemporary mathematics—especially Cantorian set theory—provides us with marvelous ways of taming and clarifying the infinite. Nonetheless, I believe that the epistemic significance of mathematical infinity remains poorly understood. This dissertation investigates the role of the infinite in three diverse areas of study: number theory, cosmology, and probability theory. A discovery that emerges from my work is that the epistemic role of the infinite varies, often in surprising ways, across different domains of knowledge. My first chapter examines the role of mathematical infinity in number theory. It is reasonable to think that theorems concerning finite patterns and structures in the natural numbers are particularly “simple” or “elementary.” Indeed, such statements are comprehensible to a wide range of investigators, regardless of their mathematical training. One might then expect proofs of these theorems to be similarly comprehensible. However, many proofs, especially those that utilize only finitary methods, are exceedingly difficult to understand. Consequently, one finds that finitary theorems are often re-proved using infinitary techniques. My claim is that this is because infinitary proofs are often explanatory, while finitary proofs are not. This chapter analyzes why this is the case. Along the way, I investigate other questions of long-standing interest in the philosophy of mathematics, e.g., the role of purity/impurity ascriptions and the nature of the content of a theorem. In particular, I diagnose the explanatory potential of the infinite by articulating a new construal of content. This new construal both saves intuitive epistemic ascriptions made in mathematical practice and explains the unexpected role of the infinite in providing explanatory proofs of finitary statements. Thus, in number theory, my claim is that the infinite often plays an explanatory role. My second chapter turns to the role of the infinite in cosmology. It investigates a question much discussed by philosophers and physicists alike: is the spatial extent of the universe finite or infinite? Contemporary cosmological research has indicated that one of the essential determinants of the extent of the universe is the topology we ascribe to space. Topology is a global property, which may suggest that it is not testable through local observation. Nonetheless, some cosmologists have indicated that it may be empirically detected, thereby providing an answer to the question of spatial extent. I argue that, in fact, the epistemic status of the topology of space is extremely subtle and not well captured by any of the categories commonly employed by philosophers of science. In particular, I argue that topological properties are neither empirical nor a priori (even in suitably weakened senses). Furthermore, I claim that we should prefer topological properties that generate finite universe models (consistent with our best data) in order to avoid extremely thorny issues concerning the physics of an infinite universe. I argue for such a preference on the grounds of the simplicity and explanatory power of finite universe models. Thus, in cosmology, my claim is that the finite often plays an explanatory and simplifying role. My third chapter investigates several paradoxes that arise in the foundations of infinitary probability theory: the Label Invariance Paradox, God’s Lottery, and Bertrand’s Paradox. I argue that these have been poorly understood because they do not expressly concern probability theory, but rather our intuitions about—and formal techniques for dealing with— infinite sets. The paradoxes in question are, in fact, symptoms of our complete reliance upon Cantorian cardinality and its associated criterion of sameness of “size.” That is, two sets have the same cardinality if and only if the elements of the sets can be placed in 1-1 correspondence. When applied to infinite sets, this criterion produces counterintuitive verdicts. For instance, given a fair lottery on the natural numbers, we expect that the probability of drawing an even number is 1/2, and likewise for drawing an odd number. However, one can construct “relabellings” of the naturals such that the probability of drawing an even number remains 1/2, while the probability for drawing an odd number becomes 1/4. I argue that, ultimately, it is the coarseness of Cantorian cardinality that generates the probabilistic paradoxes. I then propose that finer-grained measures of infinite sets from mathematical logic and number theory can help to dissolve the paradoxes in question. Thus, in probability theory, we find that particular kinds of infinitary techniques effectively systematize our theory, while others lead to paradox.

It is well-known that the global structure of every space-time model for relativistic cosmology is observationally underdetermined. In order to alleviate the severity of this underdetermination, it has been proposed that we adopt the Cosmological Principle because the Principle restricts our attention to a distinguished class of space-time models (spatially homogeneous and isotropic models). I argue that, even assuming the Cosmological Principle, the topology of space remains observationally underdetermined. Nonetheless, I argue that we can muster reasons to prefer various topological properties over others. In particular, I favor the adoption of multiply connected universe models on grounds of (i) simplicity, (ii) Machian considerations, and (iii) explanatory power. We are able to appeal to such grounds because multiply connected topologies open up the possibility of finite universe models (consistent with our best data), which in turn avoid thorny issues concerning the postulation of an actually infinite universe.

In this paper I argue for an association between impurity and explanatory power in contemporary mathematics. This proposal is defended against the ancient and influential idea that purity and explanation go hand-in-hand (Aristotle, Bolzano) and recent suggestions that purity/impurity ascriptions and explanatory power are more or less distinct (Section 1). This is done by analyzing a central and deep result of additive number theory, Szemerédi’s theorem, and various of its proofs (Section 2). In particular, I focus upon the radically impure (ergodic) proof due to Furstenberg (Section 3). Furstenberg’s ergodic proof is striking because it utilizes intuitively foreign and infinitary resources to prove a finitary combinatorial result and does so in a perspicuous fashion. I claim that Furstenberg’s proof is explanatory in light of its clear expression of a crucial structural result, which provides the “reason why” Szemerédi’s theorem is true. This is, however, rather surprising: how can such intuitively different conceptual resources “get a grip on” the theorem to be proved? I account for this phenomenon by articulating a new construal of the content of a mathematical statement, which I call structural content (Section 4). I argue that the availability of structural content saves intuitive epistemic distinctions made in mathematical practice and simultaneously explicates the intervention of surprising and explanatorily rich conceptual resources. Structural content also disarms general arguments for thinking that impurity and explanatory power might come apart. Finally, I sketch a proposal that, once structural content is in hand, impure resources lead to explanatory proofs via suitably understood varieties of simplification and unification (Section 5).