Far from being unwelcome or impossible in a mathematical setting, indeterminacy in various forms can be seen as playing an important role in driving mathematical research forward by providing “sources of newness” in the sense of Hutter and Farías :434–449, 2017). I argue here that mathematical coincidences, phenomena recently under discussion in the philosophy of mathematics, are usefully seen as inducers of indeterminacy and as put to work in guiding mathematical research. I suggest that to call a pair of mathematical (...) facts a coincidence is roughly to suggest that the investigation of connections between these facts isn’t worthwhile. To say of this pair, “That’s no coincidence!” is to suggest just the opposite. I further argue that this perspective on mathematical coincidence, which pays special attention to what mathematical coincidences do, may provide us with a better view of what mathematical coincidences are than extant accounts. I close by reflecting on how understanding mathematical coincidences as generating indeterminacy accords with a conception of mathematical research as ultimately aiming to reduce indeterminacy and complexity to triviality as proposed in Rota Indiscrete thoughts, Birkhäuser, 1997). (shrink)

Open texture is a kind of semantic indeterminacy first systematically studied by Waismann. In this paper, extant definitions of open texture will be compared and contrasted, with a view towards the consequences of open-textured concepts in mathematics. It has been suggested that these would threaten the traditional virtues of proof, primarily the certainty bestowed by proof-possession, and this suggestion will be critically investigated using recent work on informal proof. It will be argued that informal proofs have virtues that mitigate the (...) danger posed by open texture. Moreover, it will be argued that while rigor in the guise of formalisation and axiomatisation might banish open texture from mathematical theories through implicit definition, it can do so only at the cost of restricting the tamed concepts in certain ways. (shrink)

One of the distinguishing features of mathematics is the exceptional level of consensus among mathematicians. However, an analysis of what mathematicians agree on, how they achieve this agreement, and the relevant historical conditions is lacking. This paper is a programmatic intervention providing a preliminary analysis and outlining a research program in this direction.First, I review the process of ‘negotiation’ that yields agreement about the validity of proofs. This process most often does generate consensus, however, it may give rise to another (...) kind of disagreement: whether the original and new proof are effectively the same, leading to potential disagreement about the validity of the original proof.Second, I historicize the phenomenon of consensus. I show that in earlier European mathematics and other mathematical cultures, consensus about the validity of arguments was substantially weaker or conceived differently than it is today. This means that contemporary consensus about the validity of mathematical proofs should be explained by historical changes in mathematical practice.Finally, I conjecture what brought about the contemporary form of mathematical consensus. Since a sharp rise in consensus occurs around the turn of the 20th century, it makes sense to explain this consensus by the concurrent formalization of mathematics. However, this explanation has a major flaw: it explains a really existing phenomenon (consensus) by something that hardly ever happens (writing proofs in formal languages). I will therefore explain the ways in which formalization does enter mathematical practice so as to account for contemporary forms of mathematical consensus. (shrink)

Who gets to contribute to knowledge production of an epistemic community? Scholarship has focussed on unjustified forms of exclusion. Here I study justified forms of exclusion by investigating the phenomenon of so-called ‘cranks’ in mathematics. I argue that workload-management concerns justify the exclusion of these outsiders from mathematical knowledge-making practices. My discussion reveals three insights. There are reasons other than incorrect mathematical argument that justify exclusions from mathematical practices. There are instances in which mathematicians are justified in rejecting even correct (...) mathematical arguments. Finally, the way mathematicians spot mathematical crankery does not support the pejorative connotations of the ‘crank’ terminology. (shrink)

According to one of the most powerful paradigms explaining the meaning of the concept of natural number, natural numbers get a large part of their conceptual content from core cognitive abilities. Carey’s bootstrapping provides a model of the role of core cognition in the creation of mature mathematical concepts. In this paper, I conduct conceptual analyses of various theories within this paradigm, concluding that the theories based on the ability to subitize (i.e., to assess anexactquantity of the elements in a (...) collection without counting them), or on the ability to approximate quantities (i.e., to assess anapproximatequantity of the elements in a collection without counting them), or both, fail to provide a conceptual basis for bootstrapping the concept of an exact natural number. In particular, I argue that none of the existing theories explains one of the key characteristics of the natural number structure: the equidistances between successive elements of the natural numbers progression. I suggest that this regularity could be based on another innate cognitive ability, namely sensitivity to the regularity of rhythm. In the final section, I propose a new position within the core cognition paradigm, inspired by structuralist positions in philosophy of mathematics. (shrink)

Conceptual engineering is commonly characterized as the method for assessing and improving our representational devices. Little has been said, however, on how best to construe these representational devices—in other words, on what conceptual engineering should be all about. This paper tackles this problem with a basic strategy: First, by presenting a taxonomy of the different possible subject matters for conceptual engineering; then, by comparatively assessing them and selecting the most conducive one with a view to making conceptual engineering an actionable (...) method, that is, a method that can be applied effectively and consistently to specific case studies. The outcome is that conceptual engineering should be all about concepts on pain of pragmatic inconsistencies otherwise. (shrink)

Conceptual engineering means to provide a method to assess and improve our concepts working as cognitive devices. But conceptual engineering still lacks an account of what concepts are (as cognitive devices) and of what engineering is (in the case of cognition). And without such prior understanding of its subject matter, or so it is claimed here, conceptual engineering is bound to remain useless, merely operating as a piecemeal approach, with no overall grip on its target domain. The purpose of this (...) programmatic paper is to overcome this knowledge gap by providing some guidelines for developing the theories of concepts and of cognition that will ground the systematic unified framework needed to effectively implement conceptual engineering as a widely applicable method for the cognitive optimization of our conceptual devices. (shrink)

Ratio, Volume 34, Issue 4, Page 286-302, December 2021.

Conceptual engineering is the method for assessing and improving our representational devices. On its ‘broad‐spectrum’ version, it is expected to be appropriately applicable to any of our representation‐involving cognitive activities, with major consequences for our whole cognitive life. This paper is about the theoretical foundations of conceptual engineering thus characterised. With a view to ensuring the actionability of conceptual engineering as a broad‐spectrum method, it addresses the issue of how best to construe the subject matter of conceptual engineering and successively (...) defends the theses that conceptual engineering should be: (i) About concepts, (ii) psychologically theorised, (iii) as multiply realised functional kinds. Thereby, I claim to theoretically secure and justify the maximum scope, flexibility, and impact for the method of conceptual engineering on our representational devices in our whole cognitive life—in other words, a broad‐spectrum version of conceptual engineering. (shrink)

ABSTRACT What do we disagree about when we disagree about logic? On the face of it, classical and nonclassical logicians disagree about the laws of logic and the nature of logical properties. Yet, sometimes the parties are accused of talking past each other. The worry is that if the parties to the dispute do not mean the same thing with ‘if’, ‘or’, and ‘not’, they fail to have genuine disagreement about the laws in question. After the work of Quine, this (...) objection against genuine disagreement about logic has been called the meaning-variance thesis. We argue that the meaning-variance thesis can be endorsed without blocking genuine disagreement. In fact, even the type of revisionism and nonapriorism championed by Quine turns out to be compatible with meaning-variance. (shrink)

Conceptual engineers have made hay over the differences of their metaphilosophy from those of conceptual analysts. In this article, I argue that the differences are not as great as conceptual engineers have, perhaps rhetorically, made them seem. That is, conceptual analysts asking ‘What is X?’ questions can do much the same work that conceptual engineers can do with ‘What is X for?’ questions, at least if conceptual analysts self-understand their activity as a revisionary enterprise. I show this with a study (...) of Russell's metaphilosophy, which was just such a revisionary conception of conceptual analysis. (shrink)

In this paper, I highlight the significance of practices of _refutation_ in philosophical inquiry, that is, practices of showing that a claim, person or theory is wrong. I present and contrast two prominent approaches to philosophical refutation: refutation in ancient Greek dialectic (_elenchus_), in its Socratic variant as described in Plato’s dialogues, and as described in Aristotle’s logical texts; and the practice of providing counterexamples to putative definitions familiar from twentieth century analytic philosophy, focusing on the so-called Gettier problem. Moreover, (...) I discuss Lakatos’ method of proofs and refutations, as it offers insightful observations on the dynamics between arguments, refutations, and counterexamples. Overall, I argue that dialectic, in particular in its Socratic variant, is especially suitable for the philosophical purpose of questioning the obvious, as it invites reflection on one’s own doxastic commitments and on the tensions and inconsistencies within one’s set of beliefs. By contrast, the counterexample-based approach to philosophical refutation can give rise to philosophical theorizing that is overly focused on hairsplitting disputes, thus becoming alienated from the relevant human experiences. Insofar as philosophical inquiry treads the fine line between questioning the obvious while still seeking to say something significant about human experiences, perhaps a certain amount of what Lakatos describes as ‘monster-barring’—a rejection of overly fanciful, artificial putative counterexamples—has its place in philosophical argumentation. (shrink)

In recent analytic philosophy, there is a growing interest in the project of conceptual engineering. This paper examines two ways this project might be applied to scientific research, specifically anthropological research. It argues that both of them are harmful to this research. Specifically, it argues that a reliance on the axiological standards of analytic philosophy conflicts with the goal of empirical adequacy. Section one proffers two forms that the engineering project might take when applied to the science. Section two proffers (...) a case study drawn from anthropology. Specifically, it lays out Evans-Pritchard’s Witchcraft, Oracles, and Magic Among the Azande, paying particular attention to the inconsistency he discovered in Azande witchcraft. Sections three and four argue that attempting to remove this inconsistency via the two forms of engineering disable critical features of emic research and discard empirical findings. Finally, I close by noting how the conflict between axiological standards and empirical adequacy might generalize beyond anthropology. (shrink)

Paradigmatic cases of ontological disputes are taken to concern whether or not certain objects exist. Some theorists, however, prefer to view ontologists as really debating about what we should mean with the term “exist” (or other cognate terms). This implies interpreting ontological disputes as metalinguistic negotiations, in keeping with a recent trend to interpret other philosophical disputes along these lines (Plunkett and Sundell. Philosopher's Imprint; 2013;13:1–37). A number of issues arise from such proposal. The first is what counts as evidence (...) that an object‐level dispute is actually a metalinguistic negotiation. The second regards the dialectical relevance of considering metalinguistic negotiation as even just a possible interpretative option. The third issue concerns whether, and why, certain object‐level disputes (and especially ontological ones) should be understood as metalinguistic negotiations. (shrink)

Conceptual engineering has been linked by Herman Cappelen to a position called “representational skepticism”, described as one’s refusal to uncritically take over the conceptual representations one is handed. This position is contrasted with an uncritical attitude, called “representational complacency”. Arguably, neither position, or a hybrid of the two, is rationally sustainable. This paper therefore proposes an alternative option, called “critical concept conservatism”, stating that having a concept makes it rational (in a suitable sense of “rational”) for one to retain it, (...) unless there are grounds to question it. Critical concept conservatism avoids the drawbacks of skepticism and complacency; plus, it is independently supported by both positive and negative considerations. Furthermore, it complies with the demanding attitude towards conceptual representations that a conceptual engineer would be expected to have. (shrink)

Logic is about reasoning, or so the story goes. This thesis looks at the concept of logic, what it is, and what claims of correctness of logics amount to. The concept of logic is not a settled matter, and has not been throughout the history of it as a notion. Tools from conceptual analysis aid in this historical venture. Once the unsettledness of logic is established we see the repercussions in current debates in the philosophy of logic. Much of the (...) battle over the ‘one true logic’ is conceptually talking past each other. The theory of logics-as-formalizations is presented as a conceptually open theory of logic which is Carnapian in flavour and grounding. Rudolf Carnap’s notions surrounding ‘external’ and ‘pseudo-questions’ about linguistic frameworks apply to formalizations, thus logics, as well. An account of what formalizations are, a more structured sub-set of modelling, is given to ground the claim that logics are formalizations. Finally, a novel account of correctness, the COFE framework, is developed which allows the notions of logical monism, pluralism and nihilism to be more precisely formulated than they currently are in the discourse. (shrink)

Conceptual engineering is a philosophical method that aims to design and spread conceptual and linguistic devices to cause meaningful changes in the world. So far, however, conceptual engineers have struggled to successfully spread the conceptual and linguistic entities they have designed to their target communities. This paper argues that conceptual engineering is far more likely to succeed if it incorporates empirical data and empirical methods. Because the causal factors influencing successful propagation of linguistic or conceptual devices are as complicated and (...) interwoven as they are, proper empirical research will greatly boost the likelihood that propagation is successful. In arguing for the superiority of empirical conceptual engineering over armchair-based conceptual engineering, this paper proposes a framework for understanding the causal forces at play in propagation. This is a three-part framework between the label of a lexical item, the psychological states associated with the lexical item, and the worldly things associated with the lexical item. By understanding the way causal forces affecting propagation play out at these three levels, conceptual engineers can better conceptualize, study, and harness the different causal forces affecting the success of their conceptual engineering projects. (shrink)

Informal rigour is the process by which we come to understand particular mathematical structures and then manifest this rigour through axiomatisations. Structural relativity is the idea that the kinds of structures we isolate are dependent upon the logic we employ. We bring together these ideas by considering the level of informal rigour exhibited by our set-theoretic discourse, and argue that different foundational programmes should countenance different underlying logics (intermediate between first- and second-order) for formulating set theory. By bringing considerations of (...) perturbations in modal space to bear on the debate, we will suggest that a promising option for representing current set-theoretic thought is given by formulating set theory using quasi-weak second-order logic. These observations indicate that the usual division of structures into \particular (e.g. the natural number structure) and general (e.g. the group structure) is perhaps too coarse grained; we should also make a distinction between intentionally and unintentionally general structures. (shrink)

This thesis focuses on models of conceptual change in science and philosophy. In particular, I developed a new bootstrapping methodology for studying conceptual change, centered around the formalization of several popular models of conceptual change and the collective assessment of their improved formal versions via nine evaluative dimensions. Among the models of conceptual change treated in the thesis are Carnap’s explication, Lakatos’ concept-stretching, Toulmin’s conceptual populations, Waismann’s open texture, Mark Wilson’s patches and facades, Sneed’s structuralism, and Paul Thagard’s conceptual revolutions. (...) -/- In order to analyze and compare the conception of conceptual change provided by these different models, I rely on several historical reconstructions of episodes of scientific conceptual change. The historical episodes of scientific change that figure in this work include the emergence of the morphological concept of fish in biological taxonomies, the development of scientific conceptions of temperature, the Church-Turing thesis and related axiomatizations of effective calculability, the history of the concept of polyhedron in 17th and 18th century mathematics, Hamilton’s invention of the quaternions, the history of the pre-abstract group concepts in 18th and 19th century mathematics, the expansion of Newtonian mechanics to viscous fluids forces phenomena, and the chemical revolution. I will also present five different formal and informal improvements of four specific models of conceptual change. I will first present two different improvements of Carnapian explication, a formal and an informal one. My informal improvement of Carnapian explication will consist of a more fine-grained version of the procedure that adds an intermediate, third step to the two steps of Carnapian explication. I will show how this novel three-step version of explication is more suitable than its traditional two-step relative to handle complex cases of explications. My second, formal improvement of Carnapian explication will be a full explication of the concept of explication itself within the theory of conceptual spaces. By virtue of this formal improvement, the whole procedure of explication together with its application procedures and its pragmatic desiderata will be reconceptualized as a precise procedure involving topological and geometrical constraints inside the theory of conceptual spaces. My third improved model of conceptual change will consist of a formal explication of Darwinian models of conceptual change that will make vast use of Godfrey-Smith’s population-based Darwinism for targeting explicitly mathematical conceptual change. My fourth improvement will be dedicated instead to Wilson’s indeterminate model of conceptual change. I will show how Wilson’s very informal framework can be explicated within a modified version of the structuralist model-theoretic reconstructions of scientific theories. Finally, the fifth improved model of conceptual change will be a belief-revision-like logical framework that reconstructs Thagard’s model of conceptual revolution as specific revision and contraction operations that work on conceptual structures. -/- At the end of this work, a general conception of conceptual change in science and philosophy emerges, thanks to the combined action of the three layers of my methodology. This conception takes conceptual change to be a multi-faceted phenomenon centered around the dynamics of groups of concepts. According to this conception, concepts are best reconstructed as plastic and inter-subjective entities equipped with a non-trivial internal structure and subject to a certain degree of localized holism. Furthermore, conceptual dynamics can be judged from a weakly normative perspective, bound to be dependent on shared values and goals. Conceptual change is then best understood, according to this conception, as a ubiquitous phenomenon underlying all of our intellectual activities, from science to ordinary linguistic practices. As such, conceptual change does not pose any particular problem to value-laden notions of scientific progress, objectivity, and realism. At the same time, this conception prompts all our concept-driven intellectual activities, including philosophical and metaphilosophical reflections, to take into serious consideration the phenomenon of conceptual change. An important consequence of this conception, and of the analysis that generated it, is in fact that an adequate understanding of the dynamics of philosophical concepts is a prerequisite for analytic philosophy to develop a realistic and non-idealized depiction of itself and its activities. (shrink)

Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of 'multiversism', and then I proceed to analyse Maddy's concerns. Among other things, I take into account salient aspects of multiverse-related mathematics , in particular, research programmes in set theory for which the use of the multiverse seems to (...) be crucial, and show how one may provide responses to Maddy's concerns based on a careful analysis of 'multiverse practice'. (shrink)