Paula Quinon - Academia.edu (original) (raw)
Papers by Paula Quinon
Sailing Routes in the World of Computation
Review of Philosophy and Psychology
We present a model of how counting is learned based on the ability to perform a series of specifi... more 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 ...
Journal of Numerical Cognition
As the existence of this very journal attests, there has been a great change in the range of disc... more As the existence of this very journal attests, there has been a great change in the range of disciplines that take part in explaining what natural numbers (0, 1, 2, 3, …) are. For millennia, questions concerning the modalities of existence and knowledge of numbers used to belong almost exclusively to the domain of philosophy. In philosophy, one tradition in particular was dominant for a long time. Starting from at least the Pythagoreans and Plato, it was generally accepted well into modern times that numbers have an objective existence and the way human beings can find out facts about them is by reason, rather than the senses. In the 19th century, this tradition was challenged by empiricist philosophers like John Stuart Mill (1843), but also by mathematicians who were open to psychological influences, such as Ernst Schröder (1873). However, the momentum that the empiricist and psychologist explanations of numbers could gather was quickly stopped when Gottlob Frege's Grundlagen der Arithmetik (1884) gained importance. In that book, Frege (1884/1980) made a compelling case that the epistemology of natural numbers should be detached from any psychological considerations and rooted firmly in logical conceptual analysis. This approach of Frege, further developed by Bertrand Russell (1903), set the tone for much of 20th century discussion on philosophy of mathematics, and that tone was generally resistant to psychological explanations of mathematical objects, including natural numbers. Even empiricist philosophers of mathematics, such as Mill and more recently Philip Kitcher (1983) in The Nature of Mathematical Knowledge, focused on a priori type of argumentation, engaging almost exclusively in theoretical investigations. While this was the prevalent paradigm in the philosophy of mathematics until the late 20th century, in recent times there has been a visible change in Journal of Numerical Cognition jnc.psychopen.eu | 2363-8761 that trend. Currently, increasingly many philosophers are open to including empirical data in their argumentation. This paradigm change in the philosophy of mathematics is understandable. What used to be the exclusive domain of philosophy now involve many domains of research, including neuroscience, psychology, sociology, anthropology and linguistics. With this new abundance of empirical data, philosophers have also had to reconsider the cognitive foundations of mathematics. This is particularly important because the data overwhelmingly favor some kind of constructivist position when it comes to natural numbers. Rather than the independently existing abstract objects of Plato-let alone the divinities of Pythagoras-the consensus in different empirical disciplines seems to be that numbers are something human beings have invented, rather than discovered. Consequently, for researchers dealing with empirical studies, there may appear to be limited value in the kind of theoretical a priori pursuit that philosophy of mathematics used to be. Yet, while the multitude of empirical results indicates that logical and conceptual analysis of natural numbers is not all there is to the epistemology of natural numbers, philosophical methodology can be useful in interpreting those results, as well as in forming hypotheses and theories. The value of philosophical considerations is already apparent in clarifying terminology, which is necessary for constructing coherent theories. Results concerning infant and animal ability with small numerosities, for example, are too often presented in terms of "numbers", and the abilities are referred to as "infant arithmetic" or "animal arithmetic". Presumably, few researchers would be ready to postulate actual arithmetical thinking to infants, yet they see little problem in using arithmetical terminology to describe the infant ability. In general, it is imperative that the conceptual basis of theories is clarified and there is no conflation concerning the concepts the empirical data concerns. The numerosity concepts infants may have, for example, must be distinguished from the exact notion of natural number. Indeed, at every stage of the development, it must be clear what kind of quantity-concepts are being discussed. Unfortunately, this is too often not the case, and it is our contention that several fine works in the field of numerical cognition would have benefitted from a more careful logical and conceptual analysis of the empirical data and the hypotheses presented to explain them. This is also the case with the otherwise highly insightful new book Numbers and the Making of Us: Counting and the Course of Human Cultures, by the University of Miami anthropologist Caleb Everett (2017). The main aim of Everett's book is to construct a theory about the cultural conditions that have played a key role in the development of numbers. Aside from his own field of anthropology, Everett draws from a wide variety of sources dealing with early human and non-human numerical cognition, and the use of numerals in different cultures. In this way, his work lines up with other accounts aimed at explaining the origins of numbers, counting and arithmetic for the larger public, such as The Number Sense by Stanislas Dehaene (2011), What Counts:
Synthese
Turing and Church formulated two different formal accounts of computability that turned out to be... more 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.
History and Philosophy of Logic, 2011
The book under review is the sixth volume of a series devoted to the work of the Polish philosoph... more The book under review is the sixth volume of a series devoted to the work of the Polish philosopher and logician Stanislaw Leśniewski, published by the University of Neuchâtel. Its author, Denis Miéville, analyses the dynamic character of Leśniewskian inscriptional proof theory. ...
History and Philosophy of Logic, 2011
This paper is devoted to the idea that an artist can benefit from shaping her or his thinking wit... more This paper is devoted to the idea that an artist can benefit from shaping her or his thinking with categories from model-theory, including the conscious choice of the language of expression and then study of resulting relations within a model. I claim that this way of thinking - even if far from being the only possible - can be very fruitful. I also claim that that in the era of digital turn it is, or it might become, one of the most natural ways of developing creative reflection.
Sailing Routes in the World of Computation
Review of Philosophy and Psychology
We present a model of how counting is learned based on the ability to perform a series of specifi... more 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 ...
Journal of Numerical Cognition
As the existence of this very journal attests, there has been a great change in the range of disc... more As the existence of this very journal attests, there has been a great change in the range of disciplines that take part in explaining what natural numbers (0, 1, 2, 3, …) are. For millennia, questions concerning the modalities of existence and knowledge of numbers used to belong almost exclusively to the domain of philosophy. In philosophy, one tradition in particular was dominant for a long time. Starting from at least the Pythagoreans and Plato, it was generally accepted well into modern times that numbers have an objective existence and the way human beings can find out facts about them is by reason, rather than the senses. In the 19th century, this tradition was challenged by empiricist philosophers like John Stuart Mill (1843), but also by mathematicians who were open to psychological influences, such as Ernst Schröder (1873). However, the momentum that the empiricist and psychologist explanations of numbers could gather was quickly stopped when Gottlob Frege's Grundlagen der Arithmetik (1884) gained importance. In that book, Frege (1884/1980) made a compelling case that the epistemology of natural numbers should be detached from any psychological considerations and rooted firmly in logical conceptual analysis. This approach of Frege, further developed by Bertrand Russell (1903), set the tone for much of 20th century discussion on philosophy of mathematics, and that tone was generally resistant to psychological explanations of mathematical objects, including natural numbers. Even empiricist philosophers of mathematics, such as Mill and more recently Philip Kitcher (1983) in The Nature of Mathematical Knowledge, focused on a priori type of argumentation, engaging almost exclusively in theoretical investigations. While this was the prevalent paradigm in the philosophy of mathematics until the late 20th century, in recent times there has been a visible change in Journal of Numerical Cognition jnc.psychopen.eu | 2363-8761 that trend. Currently, increasingly many philosophers are open to including empirical data in their argumentation. This paradigm change in the philosophy of mathematics is understandable. What used to be the exclusive domain of philosophy now involve many domains of research, including neuroscience, psychology, sociology, anthropology and linguistics. With this new abundance of empirical data, philosophers have also had to reconsider the cognitive foundations of mathematics. This is particularly important because the data overwhelmingly favor some kind of constructivist position when it comes to natural numbers. Rather than the independently existing abstract objects of Plato-let alone the divinities of Pythagoras-the consensus in different empirical disciplines seems to be that numbers are something human beings have invented, rather than discovered. Consequently, for researchers dealing with empirical studies, there may appear to be limited value in the kind of theoretical a priori pursuit that philosophy of mathematics used to be. Yet, while the multitude of empirical results indicates that logical and conceptual analysis of natural numbers is not all there is to the epistemology of natural numbers, philosophical methodology can be useful in interpreting those results, as well as in forming hypotheses and theories. The value of philosophical considerations is already apparent in clarifying terminology, which is necessary for constructing coherent theories. Results concerning infant and animal ability with small numerosities, for example, are too often presented in terms of "numbers", and the abilities are referred to as "infant arithmetic" or "animal arithmetic". Presumably, few researchers would be ready to postulate actual arithmetical thinking to infants, yet they see little problem in using arithmetical terminology to describe the infant ability. In general, it is imperative that the conceptual basis of theories is clarified and there is no conflation concerning the concepts the empirical data concerns. The numerosity concepts infants may have, for example, must be distinguished from the exact notion of natural number. Indeed, at every stage of the development, it must be clear what kind of quantity-concepts are being discussed. Unfortunately, this is too often not the case, and it is our contention that several fine works in the field of numerical cognition would have benefitted from a more careful logical and conceptual analysis of the empirical data and the hypotheses presented to explain them. This is also the case with the otherwise highly insightful new book Numbers and the Making of Us: Counting and the Course of Human Cultures, by the University of Miami anthropologist Caleb Everett (2017). The main aim of Everett's book is to construct a theory about the cultural conditions that have played a key role in the development of numbers. Aside from his own field of anthropology, Everett draws from a wide variety of sources dealing with early human and non-human numerical cognition, and the use of numerals in different cultures. In this way, his work lines up with other accounts aimed at explaining the origins of numbers, counting and arithmetic for the larger public, such as The Number Sense by Stanislas Dehaene (2011), What Counts:
Synthese
Turing and Church formulated two different formal accounts of computability that turned out to be... more 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.
History and Philosophy of Logic, 2011
The book under review is the sixth volume of a series devoted to the work of the Polish philosoph... more The book under review is the sixth volume of a series devoted to the work of the Polish philosopher and logician Stanislaw Leśniewski, published by the University of Neuchâtel. Its author, Denis Miéville, analyses the dynamic character of Leśniewskian inscriptional proof theory. ...
History and Philosophy of Logic, 2011
This paper is devoted to the idea that an artist can benefit from shaping her or his thinking wit... more This paper is devoted to the idea that an artist can benefit from shaping her or his thinking with categories from model-theory, including the conscious choice of the language of expression and then study of resulting relations within a model. I claim that this way of thinking - even if far from being the only possible - can be very fruitful. I also claim that that in the era of digital turn it is, or it might become, one of the most natural ways of developing creative reflection.