Conceptual engineering for mathematical concepts

Abstract

In this paper I investigate how conceptual engineering applies to mathematical concepts in particular. I begin with a discussion of Waismann's notion of open texture, and compare it to Shapiro's modern usage of the term. Next I set out the position taken by Lakatos which sees mathematical concepts as dynamic and open to improvement and development, arguing that Waismann's open texture applies to mathematical concepts too. With the perspective of mathematics as open-textured, I make the case that this allows us to deploy the tools of conceptual engineering in mathematics. I will examine Cappelen's recent argument that there are no conceptual safe spaces and consider whether mathematics constitutes a counterexample. I argue that it does not, drawing on Haslanger's distinction between manifest and operative concepts, and applying this in a novel way to set-theoretic foundations. I then set out some of the questions that need to be engaged with to establish mathematics as involving a kind of conceptual engineering. I finish with a case study of how the tools of conceptual engineering will give us a way to progress in the debate between advocates of the Universe view and the Multiverse view in set theory.

What Should Conceptual Engineering Be All About?

“…The first alternative is a general one that consists of two opposite options: On the one hand, the principled views , which take conceptual engineering to be about some specified representational devices (viz., concepts, linguistic meanings, lexical items, conceptions, speaker-meanings, etc. ); and on the other hand, the unprincipled views which, building on anti-foundationalism (be it implicitly or not), take conceptual engineering to be about any kind of representational device more or less indistinctly, that is, with no need for further specification (e.g., Burgess 2020; Burgess and Plunkett 2020;Cantalamessa 2019;Nado 2019;Prinzing 2018;Sterken 2020;Tanswell 2018).…”

Open texture clarified

“…The best currently available definition of is probably due to Tanswell (2018). In order to capture the notion of open texture as sharply as possible, he presents two definitions of it, one of which he ascribes to Waismann: 5 (OT Tanswell) Open texture, Tanswell's definition.…”

“…Throughout this essay, I do not systematically distinguish between concepts and terms (following e.g Prinzing 2018;Tanswell 2018)…”

Cognitive Structuralism: Explaining the Regularity of the Natural Numbers Progression

“…This method is a back-and-forth approach consisting in both: (i) studying the contexts of use of the concept of natural number, and -after analysing the conceptual structure in those contexts -detecting where the partial meanings originate; (ii) studying which conceptual content of symbols can be traced back to the representations generated by core cognitive systems. "Conceptual engineering" is a method used to conform a given intuitive term to serve in a specific scientific, philosophical, or social context(Cappelen 2018); Tanswell writes specifically about the use of conceptual engineering in mathematics(Tanswell 2017). Just as "conceptual engineering" builds upon engineering design where the process of the final production is preceded by a successful proof of concept, "reverse conceptual engineering" builds upon reverse engineering where the object is deconstructed to reveal its designs and architecture.13 The idea of relying directly on analytical work of philosophers can be found inRips et al (2008) for structuralism and the Peano-Dedekind's axioms, and inBuijsman (2019) for axioms of Frege's arithmetic.…”