What Are Structural Properties?†

Korbmacher, Johannes; Schiemer, Georg

doi:10.1093/philmat/nkx011

Cited by 15 publications

(7 citation statements)

References 22 publications

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“…A property is purely structural if it can be defined wholly in terms of the concepts same and different, and part and whole (along with purely logical concepts). (Franklin 2014a, 57; another attempt in Korbmacher and Schiemer 2018) In short, a purely structural property is one definable in logic and mereology. For example, to be symmetrical with the simplest sort of symmetry is to consist of a part and another part which are the same in some respect.…”

Section: The Science Of Structurementioning

confidence: 99%

Mathematics as a Science of Non-abstract Reality: Aristotelian Realist Philosophies of Mathematics

Franklin

2021

Found Sci

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There is a wide range of realist but non-Platonist philosophies of mathematicsnaturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays out and compares these options, including their accounts of what X is, the examples supporting each theory, and the reasons for identifying the science of X with (most or all of) mathematics. Some comparison of the options is undertaken, but the main aim is to display the spectrum of viable alternatives to Platonism and nominalism. It is explained how these views answer Frege's widely accepted argument that arithmetic cannot be about real features of the physical world, and arguments that such mathematical objects as large infinities and perfect geometrical figures cannot be physically realized.

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Section: The Science Of Structurementioning

confidence: 99%

Mathematics as a Science of Non-abstract Reality: Aristotelian Realist Philosophies of Mathematics

Franklin

2021

Found Sci

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“…Following a recent proposal made in Korbmacher and Schiemer (2018), one can capture the mathematical properties or "concepts" (in Hilbert's sense) expressed by the primitive terms of a theory more formally in the following way: 38…”

Section: Structural Definitions and Model Theorymentioning

confidence: 99%

What are Implicit Definitions?

Giovannini

Schiemer

2019

Erkenn

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The paper surveys different notions of implicit definition. In particular, we offer an examination of a kind of definition commonly used in formal axiomatics, which in general terms is understood as providing a definition of the primitive terminology of an axiomatic theory. We argue that such "structural definitions" can be semantically understood in two different ways, namely (1) as specifications of the meaning of the primitive terms of a theory and (2) as definitions of higher-order mathematical concepts or structures. We analyze these two conceptions of structural definition both in the history of modern axiomatics and in contemporary philosophical debates. Based on that, we give a systematic assessment of the underlying semantics of these two ways of understanding the definiens of such definitions, by considering alternative model-theoretic and inferential accounts of meaning.

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“…In terms of the nature of structural properties themselves, there are at least two possible ways in which to identify a structural property in the literature, one in terms of direct definability and another via a particular notion of invariance across structures. SeeKorbmacher & Schiemer (2017) for a formal comparison between these two options andJohnson (2015) for an application of the latter to linguistic theory.…”

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confidence: 99%