The traditional generality problem for process reliabilism concerns the difficulty in identifying each belief forming process with a particular kind of process. Thatidentification is necessary since individual belief forming processes are typically of many kinds, and those kinds may vary in reliability. I raise a new kind of generality problem, one which turns on the difficulty of identifying beliefs with processes by which they were formed. This problem arises because individual beliefs may be the culmination of overlapping processes of distinct (...) lengths, and these processes may differ in reliability. I illustrate the force of this problem with a discussion of recent work on the bootstrapping problem. (shrink)

Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis with a PRA consistency proof, proposed in around 1995 by Patrick Suppes and Richard Sommer. It is based on an earlier system developed by Rolando Chuaqui and Patrick Suppes. Here, we discuss the inherent problems and limitations of the classical nonstandard framework and propose a much-needed refinement of ERNA, called , in the spirit of Karel Hrbacek’s stratified set theory. We study the metamathematics of and its (...) extensions. In particular, we consider several transfer principles, both classical and ‘stratified’, which turn out to be related. Finally, we show that the resulting theory allows for a truly general, elegant and elementary treatment of basic analysis. (shrink)

The conditions involved in the applicability of mathematics in science are the subject of ongoing debates. One of the best‐received approaches is the inferential account, which involves structural mappings and pragmatic considerations in a three‐step model. According to the inferential account, these pragmatic considerations happen in the immersion and interpretation stages, but not during derivation (symbol‐pushing in a mathematical formalism). In this work, I draw inspiration from the later Wittgenstein and make the case that the applicability of mathematics also rests (...) on pragmatic considerations at the mathematical derivation level. I make the further case that pragmatic considerations at the mathematical derivation level may substitute pragmatic considerations in the other two stages, thus showing how they are holistically interrelated. I illustrate these two points with the solution of a simple geometrical problem. (shrink)

In this work we argue that there is no strong demarcation between pure and applied mathematics. We show this first by stressing non-deductive components within pure mathematics, like axiomatization and theory-building in general. We also stress the “purer” components of applied mathematics, like the theory of the models that are concerned with practical purposes. We further show that some mathematical theories can be viewed through either a pure or applied lens. These different lenses are tied to different communities, which endorse (...) different evaluative standards for theories. We evaluate the distinction between pure and applied mathematics from a late Wittgensteinian perspective. We note that the classical exegesis of the later Wittgenstein’s philosophy of mathematics, due to Maddy, leads to a clear-cut but misguided demarcation. We then turn our attention to a more niche interpretation of Wittgenstein by Dawson, which captures aspects of the aforementioned distinction more accurately. Building on this newer, maverick interpretation of the later Wittgenstein’s philosophy of mathematics, and endorsing an extended notion of meaning as use which includes social, mundane uses, we elaborate a fuzzy, but more realistic, demarcation. This demarcation, relying on family resemblance, is based on how direct and intended technical applications are, the kind of evaluative standards featured, and the range of rhetorical purposes at stake. (shrink)

This paper argues that scientific studies distinguish themselves from other studies by a combination of their processes, their (knowledge) elements and the roles of these elements. This is supported by constructing a process model. An illustrative example based on Newtonian mechanics shows how scientific knowledge is structured according to the process model. To distinguish scientific studies from research and scientific research, two additional process models are built for such processes. We apply these process models: (1) to argue that scientific progress (...) should emphasize both the process of change and the content of change; (2) to chart the major stages of scientific study development; and (3) to define “science”. (shrink)

The present paper examines the role of exact results in the theory of many‐body physics, and specifically the example of the Mermin‐Wagner theorem, a rigorous result concerning the absence of phase transitions in low‐dimensional systems. While the theorem has been shown to hold for a wide range of many‐body models, it is frequently ‘violated’ by results derived from the same models using numerical techniques. This raises the question of how scientists regulate their theoretical commitments in such cases, given that the (...) models, too, are often described as approximations to the underlying ‘full’ many‐body problem. (shrink)

Quantum field theories are notoriously difficult to understand, physically as well as philosophically. The aim of this paper is to contribute to a better conceptual understanding of gauge quantum field theories, such as quantum chromodynamics, by discussing a famous physical limit, the ’t Hooft limit, in which the theory concerned often simplifies. The idea of the limit is that the number N of colours goes to infinity. The simplifications that can happen in this limit, and that we will consider, are: (...) the theory’s Feynman diagrams can be drawn on a plane without lines intersecting ; and the theory, or a sector of it, becomes integrable, and indeed corresponds to a well-studied system, viz. a spin chain. Planarity is important because it shows how a quantum field theory can exhibit extended, in particular string-like, structures; in some cases, this gives a connection with string theory, and so with its representation of gravity. Previous philosophical literature about how one theory might be emergent from, and-or reduced to, another one has tended to emphasize cases, such as occur in statistical mechanics, where the system before the limit has finitely many degrees of freedom. But here, our quantum field theories, including those on the way to the ’t Hooft limit, will have infinitely many degrees of freedom. Nevertheless, we will show how a recent schema by Butterfield and taxonomy by Norton apply to the quantum field theories we consider; and we will classify three physical properties of our theories in these terms. These properties are planarity and integrability, as in and above; and the behaviour of the beta-function reflecting, for example, asymptotic freedom. Our discussion of these properties, especially the beta-function, will also relate to recent philosophical debate about the propriety of assessing quantum field theories, whose rigorous existence is not yet proven. (shrink)

This paper analyses the practice of model-building “beyond the Standard Model” in contemporary high-energy physics and argues that its epistemic function can be grasped by regarding models as mediating between the phenomenology of the Standard Model and a number of “theoretical cores” of hybrid character, in which mathematical structures are combined with verbal narratives and analogies referring back to empirical results in other fields . Borrowing a metaphor from a physics research paper, model-building is likened to the search for a (...) Rosetta stone, whose significance does not lie in its immediate content, but rather in the chance it offers to glimpse at and manipulate the components of hybrid theoretical constructs. I shall argue that the rise of hybrid theoretical constructs was prompted by the increasing use of nonrigorous mathematical heuristics in high-energy physics. Support for my theses will be offered in form of a historical–philosophical analysis of the emergence and development of the theoretical core centring on the notion that the Higgs boson is a composite particle. I will follow the heterogeneous elements which would eventually come to form this core from their individual emergence in the 1960s and 1970s, through their collective life as a theoretical core from 1979 until the present day. (shrink)

The spacetime manifold, the stage on which physics is played, is constructed ab initio in a formal program that resembles the logicist reconstruction of mathematics. Zermelo’s set theory extended by urelemente serves as a framework, to which physically interpretable proper axioms are added. From this basis, a topology and subsequently a Hausdorff manifold are readily constructed which bear the properties of the known spacetime manifold. The present approach takes worldlines rather than spacetime points to be primitive, having them represented by (...) urelemente. Thereby it is demonstrated that an important part of physics is formally reducible to set theory. (shrink)

Scientific models need to be investigated if they are to provide valuable information about the systems they represent. Surprisingly, the epistemological question of what enables this investigation has hardly been investigated. Even authors who consider the inferential role of models as central, like Hughes or Bueno and Colyvan, content themselves with claiming that models contain mathematical resources that provide inferential power. We claim that these notions require further analysis and argue that mathematical formalisms contribute to this inferential role. We characterize (...) formalisms, illustrate how they extend our mathematical resources, and highlight how distinct formalisms offer various inferential affordances. (shrink)