Paula Quinon

Conceptual engineering is a philosophical activity that involves reframing or refining concepts to suit specific scientific, philosophical, or social contexts. Originally applied to ethical concepts, it has expanded to formal ones. Not all concepts can be engineered. Some are fixed points that preserve the intended conceptual structures of their frameworks; some are natural fixed points playing the same role across frameworks. Conceptual fixed points have been studied in formal contexts. Concepts ‘truth’, ‘existence’, ‘quantifier’ resist engineering. This paper explores the concept of computation as explained by the Church–Turing thesis and argues that it constitutes a natural conceptual fixed point.

This paper investigates intensional differences between programming languages—understood as differences in how computational processes are expressed, structured, and specified rather than merely in what they compute. While such differences have been studied in classical models of computation, they remain underexplored in the context of programming languages. Yet, programming languages undeniably compute, and any account of what “computing” means must include the ways in which they do so. The paper first clarifies the extensional/intensional distinction and introduces a methodological framework to study this distinction based on Carnapian explication. It then follows an idealized programming workflow, which I structure according to the Carnapian framework, to identify where and how intensional differences arise—including during problem specification, algorithm design, language choice, data representation, and physical implementation. The final part situates intensionality within the broader epistemology of programming practice, examining how it is shaped by type-theoretic assumptions, social and historical context, and the implications of bounded outcomes. Throughout, the paper examines both the “nature” (inherent features of computable functions) and “nurture” (human factors influencing programming language design and use) of intensional differences.

According to popular belief, big data and machine learning provide a wholly novel approach to science that has the potential to revolutionise scientific progress and will ultimately lead to the ‘end of theory’. Proponents of this view argue that advanced algorithms are able to mine vast amounts of data relating to a given problem without any prior knowledge and that we do not need to concern ourselves with causality, as correlation is sufficient for handling complex issues. Consequently, the human contribution to scientific progress is deemed to be non-essential and replaceable. We, however, following the position most commonly represented in the philosophy of science, argue that the need for human expertise remains. Based on an analysis of big data and machine learning methods in two case studies—skin cancer detection and protein folding—we show that expert knowledge is essential and inherent in the application of these methods. Drawing on this analysis, we establish a classification of the different kinds of expert knowledge that are involved in the application of big data and machine learning in scientific contexts. We address the ramifications of a human-driven expert knowledge approach to big data and machine learning for scientific practice and the discussion about the role of theory. Finally, we show that the ways in which big data and machine learning both influence and are influenced by scientific methodology involve continuous conceptual shifts rather than a rigid paradigm change.

We present a model of how counting is learned based on the ability to perform a series of specific steps. The steps require conceptual knowledge of three components: numerosity as a property of collections; numerals; and one-to-one mappings between numerals and collections. We argue that establishing one-to-one mappings is the central feature of counting. In the literature, the so-called cardinality principle has been in focus when studying the development of counting. We submit that identifying the procedural ability to count with the cardinality principle is not sufficient, but only one of the several steps in the counting process. Moreover, we suggest that some of these steps may be facilitated by the external organization of the counting situation. Using the methods of situated cognition, we analyze how the balance between external and internal representations will imply different loads on the working memory and attention of the counting individual. This analysis will show that even if the counter can competently use the cardinality principle, counting will vary in difficulty depending on the physical properties of the elements of collection and on their special arrangement. The upshot is that situated factors will influence counting performance.

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.

This paper reassesses the criticism of the Lucas-Penrose anti-mechanist argument, based on Gödel’s incompleteness theorems, as formulated by Krajewski : this argument only works with the additional extra-formal assumption that “the human mind is consistent”. Krajewski argues that this assumption cannot be formalized, and therefore that the anti-mechanist argument – which requires the formalization of the whole reasoning process – fails to establish that the human mind is not mechanistic. A similar situation occurs with a corollary to the argument, that the human mind allegedly outperforms machines, because although there is no exhaustive formal definition of natural numbers, mathematicians can successfully work with natural numbers. Again, the corollary requires an extra-formal assumption: “PA is complete” or “the set of all natural numbers exists”. I agree that extra-formal assumptions are necessary in order to validate the anti-mechanist argument and its corollary, and that those assumptions are problematic. However, I argue that formalization is possible and the problem is instead the circularity of reasoning that they cause. The human mind does not prove its own consistency, and outperforms the machine, simply by making the assumption “I am consistent”. Starting from the analysis of circularity, I propose a way of thinking about the interplay between informal and formal in mathematics.

The core of the problem discussed in this paper is the following: the Church-Turing Thesis states that Turing Machines formally explicate the intuitive concept of computability. The description of Turing Machines requires description of the notation used for the input and for the output. Providing a general definition of notations acceptable in the process of computations causes problems. This is because a notation, or an encoding suitable for a computation, has to be computable. Yet, using the concept of computation, in a definition of a notation, which will be further used in a definition of the concept of computation yields an obvious vicious circle. The circularity of this definition causes trouble in distinguishing on the theoretical level, what is an acceptable notation from what is not an acceptable notation, or as it is usually referred to in the literature, “deviant encodings”. Deviant encodings appear explicitly in discussions about what is an adequate or correct conceptual analysis of the concept of computation. In this paper, I focus on philosophical examples where the phenomenon appears implicitly, in a “disguised” version. In particular, I present its use in the analysis of the concept of natural number. I also point at additional phenomena related to deviant encodings: conceptual fixed points and apparent “computability” of uncomputable functions. In parallel, I develop the idea that Carnapian explications provide a much more adequate framework for understanding the concept of computation, than the classical philosophical analysis.

In this paper, we propose a conceptual-spaces model of numerical cognition, and more precisely, of representations generated by Approximate Number System. The model is an extended and improved version of our earlier result (Gemel A, Quinon P: The approximate numbers system and the treatment of vagueness in conceptual spaces. In: Lukowski L, Gemel A, Zukowski B (eds) Cognition, meaning and action. Jagiellonian-Lodz University Press, Kraków, pp 87–108, 2015), where only purely quantitative information was accounted for. We focused on the idea that ANS evolved to detect numerosity in the input, and to abstract this information from all possible magnitude-related cues, such as size of compared objects, aggregate area of those objects, or density of their location. The idea is that when one sees a pile of apples, ANS acts as a “number sense”, informing one of the approximate quantity of apples. Consequently, the original model was very simplified, accounting only for one, uniform perceptual discrete visual input. With inspiration from computational models (ex. Dehaene S, Changeux JP: J Cogn Neurosci 5:390–407, 1993;, Lourenco SF, Longo MR: Psychol Sci 21(6):873–881, 2010) that can process more complex stimuli, we propose in this paper a conceptual-spaces model for non-uniform input. The improved version of the model accounts additionally for magnitude-related cues related to both “number sense” and “magnitude sense”.

Turing and Church formulated two different formal accounts of computability that turned out to be extensionally equivalent. Since the accounts refer to different properties they cannot both be adequate conceptual analyses of the concept of computability. This insight has led to a discussion concerning which account is adequate. Some authors have suggested that this philosophical debate—which shows few signs of converging on one view—can be circumvented by regarding Church’s and Turing’s theses as explications. This move opens up the possibility that both accounts could be adequate, albeit in their own different ways. In this paper, I focus on the question of whether Church’s thesis can be seen as an explication in the precise Carnapian sense. Most importantly, I address an additional constraint that Carnap puts on the explicative power of axiomatic systems—an axiomatisation explicates when it is clear which mathematical entities form the theory’s intended model—and that implicitly applies to axiomatisations of recursion theory used in Church’s account of computability. To overcome this difficulty, I propose two possible clarifications of the pre-systematic concept of “computability” that can both be captured in recursion theory, and I show how both clarifications avoid an objection arising from Carnap’s constraint.