My dissertation investigates the relation between Immanuel Kant’s Metaphysical Foundations of Natural Science (MAN) and the Critique of Pure Reason. While it is clear that Kant somehow draws on the first Critique in MAN, the nature of the connection is mysterious. I solve the puzzle by arguing that Kant’s central philosophical objective in MAN is Leibniz-Wolffian: to express apodictically certain knowledge (in regards to empirical physics) in a systematic philosophy; rather than to extend the distinct kind of “a priori” knowledge that the first Critique proposes. Even though Kant’s core philosophical objective is Leibniz-Wolffian, his methods are not. The dissertation explores three crucial dimensions of this theme. First, I distinguish Kant’s conception of a priori knowledge in MAN from that in the first Critique. Second, I clarify the role of the “construction” of the concept matter in MAN through a comparison with his views on construction in geometry. Third, I examine Kant’s view on true judgment in MAN, i.e., the analytic/synthetic distinction.

My first step towards establishing the main thesis consists in showing that Kant’s conception of a priori knowledge in MAN differs from that in the first Critique (Chapter 1). To see this, I begin by articulating Kant’s hierarchy of logical perfection, which is his proposal for how different cognitive faculties unify previously indistinct representations. This hierarchy serves as Kant’s overall epistemic order on which the first Critique (i.e., the Analytic) precedes empirical concept formation (e.g., the empirical concept matter), which, in turn, precedes Kant’s “pure” natural science in MAN. The latter approach reflects Kant’s distinct conception of a “particular” logic, a “logic” of empirical physics. I argue that while a priori knowledge in the first Critique functions to unify indistinct perceptions, “a priori” knowledge as developed in MAN isolates the necessary elements of the empirical concept matter, in particular the trajectories of objects in pure intuition, and (Kantian) absolute space as a first ground.

The second chapter clarifies Kant’s suggestion that he imitates the “mathematical method” and proceeds according to a “construction” of the concept matter in pure intuition in MAN. I develop Kant’s position through comparison with his conception of Euclidean geometry, in particular his view on mathematical definitions and proofs. My thesis is that Kant’s geometrical construction serves as his model for isolating constructable component concepts through which the concept matter can be securely cognized. Kant modifies Leibniz-Wolffian philosophy by proposing that a proper “reflection” on empirical science draws not only on the pure representations from the understanding but also on those from the sensibility. Kantian construction in MAN can hence be thought of as a distinct form of “analysis” of an empirical concept, one that leads to “spatially” encoded (and thereby apodictically certain) determinations of the concept matter.

The third chapter investigates Kant’s view of true judgment in MAN. The analytic/synthetic distinction Kant proposed in the first Critique manifests his departure from Leibniz-Wolff’s approach that all truths can be derived through an application of the principle of contradiction by showing that the predicate is “contained” in the subject concept. I argue that the three criteria Kant gives for the distinction in the first Critique (containment, identity, and differing epistemic uses) express a unified view when understood in regards to the judgments Kant defends in MAN. A key move is to interpret his fundamental Explications of the concept matter as paradigmatic instances of his critical conception of analytic judgment. On this view, the containment criterion reflects containment in the constructed concept, not the “mere” concept.

Overall, my approach to MAN differs from the literature in the following respects. Kant’s essential philosophical moves can, I suggest, be interpreted in a way that MAN comes out as a consistent and interesting project, at least by Kant’s own standards, not as a partial or even a complete failure. In essence, Kant’s programmatic suggestions to derive a priori knowledge and to construct the concept matter in pure intuition have a consistent interpretation. The reading entails that the relation between the first Critique and MAN is rather loose. In the literature, one typically finds the view that Kant’s core philosophical objectives and conceptions in those two works align closely. My interpretation of MAN centers on a fine-grained comparison with Leibniz-Wolffian philosophy, in particular the joint goal of logically perfected cognitions. Despite the clear evidence for such a connection—Kant’s appeal to “metaphysical foundations” and the mathematical method—this angle is underappreciated in the literature.

This dissertation analyzes the ways in which scholars talk about the relation between religion and science. In the late 1980s, the physicist and theologian Ian Barbour proposed that we approach this massive scholarship through the lens of a fourfold typology: scholars tend to conceive of the religion-science relationship (RSR) as one of Conflict, Independence, Dialogue, or Integration. This model, though acknowledged as problematic, still dominates the field of religion-and-science---an interdisciplinary field with hundreds of specialists drawn from philosophy, history, and the natural and social sciences. Extant work which analyzes the discipline as a whole either extends or slightly modifies Barbour's four original categories. In my dissertation, I propose an entirely new way of approaching the religion-and-science literature, by focusing on the methods that scholars employ to reach their conclusions about the RSR rather than focusing on the conclusions themselves. Doing so, I argue, will help to resolve the current widespread feeling that scholars are talking past one another and also help public readerships of the literature clarify what is actually going on in the literature by highlighting the modes of reasoning being used.

I identify four main methods that scholars tend to use when characterizing the RSR: conceptual analysis, (historical) case studies, deconstruction, and fieldwork. Conceptual analysis focuses on the definitions of `religion' and `science,' and seeks to derive their relation logically from those definitions. The method of case studies instead proceeds by first surveying a variety of of historical encounters between religion and science and then arguing, via induction, for some general characterization of the RSR. Deconstruction, on the other hand, emphasizes the contingency of the concepts ``religion" and ``science", either historically or cross-culturally, and explains the emergence of the current RSR on the basis of that contingency. Finally, scholars employing fieldwork extract their characterization of the RSR from empirical data gathered from scientists and religious folk themselves. Although these different methods often draw from particular disciplinary backgrounds, they can be---and are---used by scholars in any discipline.

Each of these methods faces unique issues and challenges which I discuss and further develop, proposing recommendations for those who use these methods in light of the critiques. I argue that no method is better ``on the whole" than any other, for such a determination will depend essentially on the aims, goals, and values scholars and other readers may have in trying to understand the RSR. Thus, I also explain what kinds of audiences may find the different methods relevant, with an especial focus on non-academic audiences.

Throughout the dissertation, I pay especial attention to scholarship in public-facing contexts. Hence, the main sources I consider are academic, book-length tracts written by scholars with public-facing aims. The various critiques I discuss also focus on the public-facing nature of the works examined. An issue all of the current scholarship faces, which has so far gone unrecognized in the literature, revolves around the question, ``Whose `science,' whose `religion'?" Scholars almost always focus on religion and (especially) science as practiced among elites. Standard treatments of science, for instance, draw on the large-scale theories produced by famous scientists, or examine the personal beliefs of scientists employed at prestigious research universities. Left out are the vast majority of practicing scientists which members of the public may interact with (or be), many of whom work in non-research, non-theory-oriented spaces. This leads to a sense in which prevailing accounts of the RSR work with notions of science which fail to accurately reflect the nature of science as practiced in the world. I thus show how the religion-and-science scholarship can be improved by taking these non-research, non-theory-oriented sciences seriously---not only will it make the work more relevant to the publics scholars often wish to reach, but it will also open up new avenues of research in understanding how religion and science are related by real-world actors, not just in the minds of academics.

Overall, my dissertation provides a novel approach to the field of religion-and-science by providing a high-level, overview analysis of the methods used in the literature on the religion-science relationship.

In his Metaphysiche Anfangsgründe der Naturwissenschaft(1786), Immanuel Kant claims that chemistry is an improper, but rational, science. In this dissertation, I explain Kant's conception of chemistry by situating his discussions of the science with respect to his theoretical philosophy and his scientific context.

In the first chapter, I explain why Kant believes chemistry to be an improper science. In the Metaphysische Anfangsgründe der Naturwissenschaft, Kant maintains that the a priori application of mathematics in proper science distinguishes it from improper science. Because of his opposition to mechanical philosophy, which reduces natural phenomena to mathematically expressible qualities, Kant took the application of mathematics to be a nontrivial problem. He contends that there must be a priori, metaphysical principles that validate the application of mathematics to a proper science. Ultimately, Kant argues that the forces of chemistry, unlike those of physics, are incapable of such a priori validation, making chemistry a merely improper science.

The second chapter concerns chemistry's status as a rational science. I contend that rational sciences, unlike mere sciences, are capable of genuine, causal laws for Kant. I argue that there are different kinds of causal laws in different sciences: whereas the laws of physics are conditions for the possibility of experience of external objects, the laws of chemistry are quite different. Kant believes that the cognitive faculty of reason postulates chemical elements as the absolute, fundamental bearers of chemical powers, and that chemical laws are possible only insofar as they follow from the nature of these postulated entities.

In the last chapter, I argue that Kant continues to believe chemistry to be an improper, though rational, science in his unfinished Opus Postumum (ca. 1795-1803). In this work, after his exposure to Lavoisier's chemical revolution, Kant claims that the existence of the caloric can be deduced a priori and that the elements can be enumerated a priori. Nevertheless, I contend that the newly added a priori components neither belong to chemistry nor validate the mathematization of the science. Rather, they are parts of the transition (Übergang), which explains the systematicity of natural science.

This dissertation investigates the origins of the completeness theorem for first-order predicate logic in the algebraic logic work of L\"{o}wenheim and Skolem. When G\"{o}del proved the completeness theorem in 1929, he was unaware that all the components of a completeness proof were already contained in earlier papers by L\"{o}wenheim and Skolem in which they prove the model-theoretic result known as the L\"{o}wenheim-Skolem theorem. This is not, however, a question of G\"{o}del’s completeness proof having been preempted. For neither were L\"{o}wenheim or Skolem show recognition of the result that G\"{o}del would later make explicit.

When the similarity between the proofs was noticed in the 1950s, the fact that Skolem in particular had not put the pieces together to prove completeness before G\"{o}del seemed a puzzling oversight. G\"{o}del offered his own answer to the puzzle, appealing to alleged prejudices Skolem had against transfinite methods of reasoning.

Chapter One shows how the puzzle emerged and how G\"{o}del purported to explain it. Chapters Two and Three give reconstructions of the original proofs of L\"{o}wenheim and G\"{o}del. I analyze the meaning of G\"{o}del’s claim that ``finitary prejudices’’ were at the heart of the failure to recognize completeness, and assess the evidence for this claim in L\"{o}wenheim and Skolem. Chapter Four reconstructs Skolem’s proof of L\"{o}wenheim’s theorem and establishes the technical background to understand the relation it bears to the completeness theorem. In Chapter 5, I argue that G\"{o}del’s own answer to the puzzle rests on a false premise. When certain contextual features are accounted for, Skolem’s failure to recognize a completeness theorem in his own work is not the oversight it now seems. I investigate Skolem’s search for a decidability proof for first-order logic and the role this played in leading Skolem away from the discovery of completeness.

In this dissertation, I investigate how humans represent space and other geometric entities. The topics of my three chapters are delimited by three kinds of spatial representation: perception, conception, and propositional knowledge. In chapter 1, argue against the widely-held philosophical view that the content of visual perception includes a “geometry,” or more precisely, a metric space. I appeal to behavioral and neural evidence to argue that the spatial contents of visual perception exhibit a disunity that most philosophers since Kant would find surprising. In chapter 2, I evaluate a theory of geometric concept acquisition known as “core geometric cognition,” and I use behavioral and neural evidence to argue that the most general assumptions of this approach are likely false. In chapter 3, I defend the view that, despite the geometric disunity exhibited by perception, it is still plausible to believe that some propositional knowledge of advanced mathematical theorems is grounded in visual imagination.

Among many things, Frank Ramsey is famous for his invention of a logical device called the Ramsey sentence. The Ramsey sentence of a scientific theory is a conjunction of the theory’s propositions where the theoretical vocabulary is existentially quantified out. The sentence appears in an obscure paper titled “Theories”. This paper is extremely difficult because Ramsey likely did not finish it before his death and it mostly consists of a mathematical model of scientific theories and a labyrinthine toy example. Consequently, philosophers have proposed many radically different and incoherent interpretations of the Ramsey sentence and Ramsey’s philosophy of science. An important fact interpreters have ignored is that Ramsey had intended “Theories” to be a chapter in a book centered on a revised version of his formal theory for decision-making under uncertainty. The principal accomplishment of my dissertation is to leverage this fact to provide the first complete and philosophically satisfying interpretation of “Theories”, including Ramsey’s toy example and the Ramsey sentence.

Ramsey’s theory for decision-making is incompatible with his philosophical commitments in “Theories” and contemporary papers. In my dissertation, I revise that theory to make it consistent with those commitments. I combine this with the formal model he describes in “Theories” to provide a unified reconstruction of Ramsey’s intended model for scientific theories. I use the reconstructed model to explain Ramsey’s toy example, and I furnish a complete interpretation of Ramsey’s arguments in “Theories”. With these parts, I develop a detailed account of the laws and chances derived from scientific theories and a novel interpretation of the Ramsey sentence. This has several important philosophical consequences.

First, Ramsey did not use the Ramsey sentence in the ways philosophers have previously proposed. Philosophers view the Ramsey sentence either as a tool for eliminating theoretical terms or for asserting the existence of theoretical properties. Neither of these claims is correct. Instead, I argue that the Ramsey sentence is a communication and deliberation device aimed at representing the propositions an individual believes with her behavioral dispositions. Informally, it expresses as new propositions the mixture of rules a person uses to guide her behavior so that she may deliberate and communicate those rules with herself and other people. This suggests the Ramsey sentence can be used to read off the content of an agent’s belief from their behavior.

Second, Ramsey formulated a novel and interesting type of non-reductionistic scientific anti-realism. That is, for Ramsey theoretical propositions are fictitious and yet not eliminable for observational propositions. Theoretical propositions are fictitious because their meaning depends on theoretical laws, which, being universally quantified, are also fictitious. But theoretical propositions cannot be eliminated because they entail conceptual possibilities that are richer than those given by the observation language.

Third, Ramsey held laws and chances to be rules of deference for degrees of belief. His account of laws and chances foreshadows pragmatic, subjectivist accounts of laws and chances from later in the twentieth century. Degrees of belief are representations of preferences over gambles involving observable propositions. A chance then is a rule for preferring some gambles over others, and a law is a chance that assigns unity to its degrees of beliefs. This prefigures the views that chances and subjunctive conditionals are reducible to properties of degrees of belief.

In this dissertation, I analyze the distinction between different types of infinity that Leibniz identifies throughout his philosophical and mathematical works. By attending to these differences, I show how Leibniz’s rejection of infinite number as a contradictory notion does not entail the impossibility of infinitely small lines. Chapter 1 explains the differences between three grades of infinity that Leibniz identifies in a 1676 taxonomy on the infinite and how this taxonomy was an attempt to avoid paradoxes of the infinite. It also contains a description of a separate taxonomy of the infinite that Leibniz gave in 1706, in which Leibniz explicitly bans any type of infinity that is a whole composed of infinity many distinct parts. Chapter 2 treats five places where infinity lines arise in Leibniz’s mathematics: infinite number, the composition of the continuum, infinite series, infinitesimals and their bounded infinite counterparts, and unbounded infinite lines. Looking at the different ways Leibniz evaluates each of these concepts, we see that infinitesimals and bounded infinite lines stand on firmer conceptual footing then the others from Leibniz’s point of view. Chapter 3 argues that Leibniz’s claims that infinitesimals are “fictions” or “impossible,” this is in reference to a specific type of impossibility that he calls the impossible per accidens. Unlike the absolutely impossible, this type of impossibility is not rooted in contradiction, but conflict with metaphysical principles that bar their existence in the order of created things. Hence, Leibniz’s stance towards infinitesimals allows them to be perfectly coherent entities for the purposes of geometric reasoning, despite his ban on their existence within the physical world.