Axioms in Mathematical Practice

Schlimm, Dirk

doi:10.1093/philmat/nks036

Cited by 36 publications

(16 citation statements)

References 44 publications

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“…Use algebraic abstraction. The modern algebraic method involves characterizing classes of structures of interest by specifying the axioms they satisfy and then reasoning abstractly about these structures via this axiomatic interface [58,59]. The fact that the algebraic method incorporates all three previous strategies helps explain its importance.…”

Section: Specific Solutionsmentioning

confidence: 99%

Reliability of mathematical inference

Avigad

2020

Synthese

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Of all the demands that mathematics imposes on its practitioners, one of the most fundamental is that proofs ought to be correct. This is also a demand that is especially hard to fulfill, given the fragility and complexity of mathematical proof. This essay considers some of ways that mathematics supports reliable assessment, which is necessary to maintain the coherence and stability of the practice. * I am grateful to Silvia De Toffoli and Yacin Hamami for helpful comments and suggestions.

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Section: Specific Solutionsmentioning

confidence: 99%

Reliability of mathematical inference

Avigad

2020

Synthese

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“…Another important difference concerns the epistemic status of axioms. Mathematicians tend to vacillate between two extreme approaches: one casts axioms as not just true but self-evidently and indubitably so; the other sees axioms as neither true nor false but as arbitrary stipulations we are only interested in to know what follows from them (see Blanché 1955; Schlimm 2013). In the philosophical project of total axiomatics, in contrast, we do commit to the truth of the statements in our ‘scrutability base’—they are fundamental truths , after all—but it is not built into the project that we must treat them as self-evident or indubitable.…”

Section: Serious Metaphysics Scrutability and ‘Total Axiomatics’mentioning

confidence: 99%

Philosophy as Total Axiomatics: Serious Metaphysics, Scrutability Bases, and Aesthetic Evaluation

Kriegel

2016

J. of the Am. Philos. Assoc.

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ABSTRACT:What is the aim of philosophy? There may be too many philosophical branches, traditions, practices, and programs to admit of a single overarching aim. Here, I focus on a fairly traditional philosophical project that has recently received increasingly sophisticated articulation, especially by Frank Jackson (1998) and David Chalmers (2012). In Section 1, I present the project and suggest that it is usefully thought of as ‘total axiomatics’: the project of attempting to axiomatize the total theory of the world. In Section 2, I raise a problem for the project that I call the ‘problem of multiple axiomatizations’. I consider some initially alluring but ultimately unpromising approaches to this problem in Section 3. In Section 4, I defend a surprising approach to the problem, according to which competing axiomatizations of the total theory of the world are effectively evaluated for their aesthetic virtues.

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“…26 Quoted in Corry (2004: 44); see also Torretti (1978), comp. Schlimm (2003). 27 By contrast, his a priori commitments were the concepts of electron and ether; see Goldberg (1984).…”

Section: Robb's Italian and American Sourcesmentioning

confidence: 99%

Las imágenes y la lógica del cono de luz: rastreando el giro postulacional de Robb en la física geométrica

Cat

2016

RHV

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Previous discussions of Robb's work on space and time have offered a philosophical focus on causal interpretations of relativity theory or a historical focus on his use of non-Euclidean geometry, or else ignored altogether in discussions of relativity at Cambridge. In this paper I focus on how Robb's work made contact with those same foundational developments in mathematics and with their applications. This contact with applications of new mathematical logic at Göttingen and Cambridge explains the transition from his electron research to his treatment of relativity in 1911 and finally to the axiomatic presentation in 1914 in terms of postulates. At the heart of Robb's physical optics was the model of the light cone. The model underwent a transition from a working mechanical model in the Maxwellian Cambridge sense of a pedagogical and research tool to the semantic model, in the logical, model-theoretic sense. Robb tracked this transition from the 19th-to the 20th-century conception with the earliest use of the term 'model' in the new sense. I place his cone models in a genealogy of similar models and use their evolution to track how Robb's physical researches were informed by his interest in geometry, logic and the foundations of mathematics.

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Axioms in Mathematical Practice

Cited by 36 publications

References 44 publications

Reliability of mathematical inference

Reliability of mathematical inference

Philosophy as Total Axiomatics: Serious Metaphysics, Scrutability Bases, and Aesthetic Evaluation

Las imágenes y la lógica del cono de luz: rastreando el giro postulacional de Robb en la física geométrica

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