Why study notations, diagrams, or more broadly the variety of nonverbal “representations” or “signs” that are used in mathematical practice? This chapter maps out recent work on the topic by distinguishing three main philosophical motivations for doing so. First, some work (like that on diagrammatic reasoning) studies signs to recover norms of informal or historical mathematical practices that would get lost if the particular signs that these practices rely on were translated away; work in this vein has the potential to (...) broaden our view of the specific normativity of mathematics beyond logical proof. Second, some work focuses on signs to highlight the open-ended and “nonrigorous” aspects of mathematical practice, and the role of these aspects in the historical development of mathematics. The third motivation is based on the following observation: the reason differences in signs matter in mathematics is that humans are finite agents, with cognitive and computational limitations. Accordingly, studying signs can be a way to study how human computational constraints shape the mathematics that we do, and to tackle the topic of mathematical understanding. (shrink)

According to standard accounts of mathematical representations of physical phenomena, positing structure-preserving mappings between a physical target system and the structure picked out by a mathematical theory is essential to such representations. In this paper, I argue that these accounts fail to give a satisfactory explanation of scientific representations that make use of inconsistent mathematical theories and present an alternative, robustly inferential account of mathematical representation that provides not just a better explanation of applications of inconsistent mathematics, but also a (...) compelling explanation of mathematical representations of physical phenomena in general. 1Inconsistent Mathematics and the Problem of Representation2The Early Calculus3Mapping Accounts and the Early Calculus3.1Partial structures3.2Inconsistent structures3.3Related total consistent structures4A Robustly Inferential Account of the Early Calculus in Applications 4.1The robustly inferential conception of mathematical representation4.2The robustly inferential conception and inconsistent mathematics4.3The robustly inferential conception and mapping accounts5Beyond Inconsistent Mathematics. (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)

Between 1886 and 1889, the British scientific and engineering communities witnessed several controversies regarding the principles underlying certain components of electrical circuits. The purpose of this paper is to show that these controversies should be regarded as reflecting a stage in the emergence of basic industrial research as a mediating agency between practical engineering and pure science. It will be suggested that the resolution of these controversies required careful formulation of approximations guided by a practical familiarity with the details of (...) the relevant systems. This underscored an approach to basic engineering problems within which practical experience and scientific theory were regarded as equally fundamental. (shrink)

A prized property of theories of all kinds is that of generality, of applicability or least relevance to a wide range of circumstances and situations. The purpose of this article is to present a pair of distinctions that suggest that three kinds of generality are to be found in mathematics and logics, not only at some particular period but especially in developments that take place over time: ‘omnipresent’ and ‘multipresent’ theories, and ‘ubiquitous’ notions that form dependent parts, or moments, of (...) theories. The category of ‘facets’ is also introduced, primarily to assess the roles of diagrams and notations in these two disciplines. Various consequences are explored, starting with means of developing applied mathematics, and then reconsidering several established ways of elaborating or appraising theories, such as analogising, revolutions, abstraction, unification, reduction and axiomatisation. The influence of theories already in place upon theory-building is emphasised. The roles in both mathematics and logics of set theory, abstract algebras, metamathematics, and model theory are assessed, along with the different relationships between the two disciplines adopted in algebraic logic and in mathematical logic. Finally, the issue of monism versus pluralism in these two disciplines is rehearsed, and some suggestions are made about the special character of mathematical and logical knowledge, and also the differences between them. Since the article is basically an exercise in historiography, historical examples and case studies are described or noted throughout. (shrink)