Studying mathematical practices: the dilemma of case studies

Rittberg, Colin Jakob; Kerkhove, Bart Van

doi:10.1007/s11858-019-01038-8

Cited by 11 publications

(4 citation statements)

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“…The axioms of ZFC include: (1) the standard first order logical axioms; (2) two axioms for equality: x = y → (x ∈ z → y ∈ z) and x = y → (z ∈ x → z ∈ y); (3) the domain-specific axioms [8].…”

Section: Situating the Continuum Hypothesis Within Zermelo-fraenkel S...mentioning

confidence: 99%

Delving into the continuum hypothesis: A thorough examination of set theory and the nuances of mathematical infinity

Wang

2023

TNS

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This essay delves deep into one of the most intriguing mathematical puzzles of all time: the Continuum Hypothesis. Beginning with a robust foundational exploration, it sheds light on the key concepts of cardinality and power sets, which are pivotal to the realm of set theory. These foundational ideas set the stage for a deeper investigation into the relationship that the Continuum Hypothesis shares with real numbers and natural numbers. Historically, the Continuum Hypothesis has tantalized mathematicians. This paper takes a journey through time, highlighting the various endeavors to either prove or refute this hypothesis. Some of the most brilliant minds have grappled with its complexities, leaving behind a rich tapestry of mathematical thought. Furthermore, a significant portion of our discussion is centered on situating the Continuum Hypothesis within the context of Zermelo-Fraenkel Set Theory (ZFC). The intricate interplay between the hypothesis and ZFC offers profound insights and raises thought-provoking questions about the very nature of mathematical truth.

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Section: Situating the Continuum Hypothesis Within Zermelo-fraenkel S...mentioning

confidence: 99%

Delving into the continuum hypothesis: A thorough examination of set theory and the nuances of mathematical infinity

Wang

2023

TNS

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“…Therefore, the first side of Distinction 1 (solving philosophical issues) is not compatible with the second side of Distinction 2 (finding the entity of analysis in the empirical investigation), and we obtain the following three types of empirically informed philosophy: 1) Apostate speculator: Addressing philosophical questions via empirical information framed by pre-established epistemologies and ontologies/discursive entities 2) Informed analyst: Describing, studying or analytically investigating practices via a philosophical epistemic lens that includes pre-established ontologies/ discursive entities 3) Freeway explorer: Describing, studying or analytically investigating practices mainly drawing from epistemologies and ontologies/discursive entities established in the empirical research Since any analysis depends on previous theoretical baggage however minimal it is (see f.i. Chang et al 2011;Pitt and Pitt 2011;Rittberg and van Kerkhove 2019), the freeway explorer will feature onto-epistemic frames as well. But there is a difference of degree between the informed analyst and the freeway explorer: the latter works with minimal notions and is easily willing to accept the notions and objects of the practitioners that become apparent or emerge in the empirical analysis, while the former usually works with more philosophically sophisticated notions and ontologies, which they tend to cherish and therefore favour over alternatives.…”

Section: Uses Of Empirical Information: a New Categorizationmentioning

confidence: 99%

Three Roles of Empirical Information in Philosophy: Intuitions on Mathematics do Not Come for Free

Kant

Pérez-Escobar

Sarikaya

2021

KRITERION – Journal of Philosophy

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This work gives a new argument for ‘Empirical Philosophy of Mathematical Practice’. It analyses different modalities on how empirical information can influence philosophical endeavours. We evoke the classical dichotomy between “armchair” philosophy and empirical/experimental philosophy, and claim that the latter should in turn be subdivided in three distinct styles: Apostate speculator, Informed analyst, and Freeway explorer. This is a shift of focus from the source of the information towards its use by philosophers. We present several examples from philosophy of mind/science and ethics on one side and a case study from philosophy of mathematics on the other. We argue that empirically informed philosophy of mathematics is different from the rest in a way that encourages a Freeway explorer approach, because intuitions about mathematical objects are often unavailable for non-mathematicians (since they are sometimes hard to grasp even for mathematicians). This consideration is supported by a case study in set theory.

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“…16 This list has been compiled in joint work with Henrik Kragh Sørensen. It was originally published in Rittberg & Van Kerkhove (2019). little discussion about methods, especially in print.…”

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confidence: 99%

“…For answers to this dilemma seeBurian (2001;, Schickore (2011), Chang (2011),Scholl & Räz (2016),Rittberg & Van Kerkhove (2019).…”

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On the Contemporary Practice of Philosophy of Mathematics

Rittberg¹

2019

ABHPS

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In this paper I propose to reflect on contemporary philosophy of mathematics. i argue that the impartiality to methodology that is present in the current practice of philosophy of mathematics is an intellectual vice of our research community because it leaves our philosophical methods to engage with mathematics implicit. This hides difficulties our research community faces, difficulties our students can reasonably expect to learn about and which we need to engage with to overcome them. to be virtuously open-minded about methodology is not to be impartial about methodology but rather to critically assess the proposals on offer in a public debate.

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Studying mathematical practices: the dilemma of case studies

Cited by 11 publications

References 44 publications

Delving into the continuum hypothesis: A thorough examination of set theory and the nuances of mathematical infinity

Delving into the continuum hypothesis: A thorough examination of set theory and the nuances of mathematical infinity

Three Roles of Empirical Information in Philosophy: Intuitions on Mathematics do Not Come for Free

On the Contemporary Practice of Philosophy of Mathematics

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