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Number Cognition and Philosophy of Arithmetic

Final Report Summary - NUMBERS (Number Cognition and Philosophy of Arithmetic)

The project “NUMBERS” ran in the Philosophy Department of Lund University from September 1st, 2012, to October 15th, 2014. The project aimed to contribute to human understanding of the concept of “natural numbers”. The problem was approached from three different directions:
(1) [Task 1] Naturalised Epistemology Revisited: An in-depth study of relations, interactions, and the border between the philosophy of
 mathematics (related to the study of natural numbers) and the cognitive sciences, oriented 
towards number cognition. This aspect contributed to the understanding of naturalised epistemology in the context of natural numbers.
(2) [Task 2] Toy Examples: A case study, focusing on a fruitful collaboration between the two fields. The aim was to provide clear examples of successful collaboration between philosophy of mathematics and the cognitive sciences.
(3) [Task 3] Make Minds Meet: An elaboration of a common conceptual field, the common language of communication, and the common curriculum for philosophers of mathematics and cognitive scientists investigating number cognition.

The background assumption of the project was that there exists a common object of study and that both disciplines can improve their achievements by taking inspiration from the other’s field results and methodology. This assumption was modulated as follows:
- there are various aspects of numerical concepts, especially in the early stage of individual development;
- different branches of the philosophy of mathematics highlight different, but not necessarily incompatible aspects of the concept of number.

(1) [Task 1] Naturalised Epistemology Revisited
The scientific goal of this task was to determine the character of interactions between the two areas of study of natural numbers: the philosophy of mathematics and cognitive studies. The main motivation for the project was the observation that the use of results from cognitive science in philosophy of mathematics, and vice versa, suffers from methodological issues and should be revisited. The working hypothesis of this part of the project was that a fruitful collaboration between the two fields was possible. The main objective of this task was to formulate a new version of naturalized epistemology project for the philosophical treatment of natural numbers.

The objective has been successfully attained and we came to the following conclusions:
1. There is a need to reconsider how the concept of natural number and related concepts (systems of numerical notation, quantifiers, number representations, the Approximate Number System) are understood by various disciplines. We investigated these concepts in the philosophy of mathematics and developmental cognitive psychology. Quinon has produced a journal paper describing the problem and proposing methodological constraints, which would be helpful for making make the collaboration fruitful.
2. One major criticism of use of psychological methods in the philosophy of mathematics comes from Frege, who in Grundlagen (1884) blamed psychology for the lack of cooperation between philosophy and mathematics, based on the fact that psychological methods give raise to results that are “fluctuating and indefinite”, while mathematics (along with the philosophy of mathematics) searches for “definiteness and fixity of the concepts and objects of mathematics”. In our opinion, reconsidering Frege’s objection was crucial for the formulation of a reliable description of naturalized epistemology, following the principles of the Second Philosophy program proposed by Maddy. A journal paper “Philosophy of Arithmetic and Number Cognition: Re-assessing the Basis of Interdisciplinarity” is now in preparation for peer-review; Quinon has been invited to present this paper at Workshop of the Association for Philosophy of Mathematical Practice at Congres of Logic Methodology and Philosophy of Science, Helsinki 2015.
3. One non-formal method used in formal arithmetic is the Carnapian method of explications. An explication, in Carnapian terms, consists in providing an intuitive and prescientific concept with more specific scientific meaning. This study examined an example, namely the concept of recursive function as formalized by the Church-Turing thesis. Different versions of the paper “Is the Church-Turing Thesis a Carnapian Expication?” have been presented to audiences of specialists, including groups at Lund University, University of Copenhagen, Univeristy of Leuven, Ohio State University and University of Southern California. The joint paper of Olsson and Quinon is now proofread for peer-reviewing.

(2) [Task 2] Toy Examples
Quinon studied several examples of promising interaction between philosophy and cognitive science.
1. Quinon (in collaboration with A. Gemel) proposed a cognitive spaces model of ANS. A paper “ANS and the Treatment of Vagueness in the Conceptual Spaces”, presenting their results, will appear in P. Lukowski et al. (eds.), Cognition, Language, Society, a volume to be published by Lodz University Press devoted to work of Peter Gärdenfors.
2. Two papers contribute to a better understanding of the cognitive and psychological background for computational structuralism. The first, devoted to discussion of Frege’s Constraint, has been submitted to Philosophia Mathematica. The second, discussing the use of Tennenbaum's Theorem and the Church-Turing thesis in justifying the correctness of using human arithmetic abilities to single out the standard model of arithmetic, is currently under revision and will be shortly resubmitted to Philosophia Mathematica. Both papers have been presented at some of the world’s top universities (e.g. Lund University, University of Southern Denmark, University of California-Irvine).
3. Preliminary research on the computable features of numbers has been extended to the field of real numbers in collaboration with L. Horsten (Bristol).
4. Quinon initiated an extensive collaboration with P. Blackburn (Roskilde). They prepare a paper devoted to computational aspect of natural numbers from the perspective of naturalized epistemology.

(3) [Task 3] Making Minds Meet
Understanding the differences between the conceptual toolkits and methodologies of the two fields was the main motivation behind the third task of the project. We assumed that the experience of collaboration between the main actors in the two fields would be very fruitful. This task has been realized in the following ways:
1. L. Horsten (Bristol) visited the Philosophy Department of Lund University in December 2012 as part of the NUMBERS project. The visit resulted not only in extensive networking, valuable for the philosophical community and the Department, but also Quinon and Horsten are drafting a paper devoted to the concept of computability on real numbers, related to the “Computability” objective of the proposal.
2. An international workshop “Intensionality in Mathematics” has been organized in collaboration with M. Antonutti (Bristol) and C. Proietti (Lund). Antonutti visited the Department of Philosophy at Lund as a Visiting PhD Student. MC IEF research funding partially financed her visit. The goal of the visit was to transfer knowledge and develop collaboration, but also to let Quinon to gain experience with supervising and leadership. The visit led to a special issue of Synthese being produced, and also to a paper entitled “Intensional Differences in Models of Computation”, which will appear in this issue.
Additionally, Quinon co-organized an international conference on “Numbers and Truth”, in Gothenburg, October 2012.
3. Quinon realized several visits to UC at Irvine: she visited twice for a short period LPS (January 2013, January 2014), and once for a period of three months (April–June 2014). She was also a short-term Visiting Scholar at Sarnecka Cognitive Lab. In the Lab she received training in the ethics of conducting experimental research.
4. As planned, Quinon presented her ongoing work and preliminary results at over 12 invited talks, workshops and seminars in leading European and American research centers and universities.

Further career
The project allowed Quinon to take promising steps in her career. She is involved with international research groups, and is in the process of writing grant applications for with the position of PI. In particular she is involved in writing a project on self-control with another researcher at Irvine. She has also been awarded a research grant for travel to Irvine (from Hultengrens Foundation) to work on a research proposal to be presented to various Swedish funding agencies, which will lead to close collaboration between Sarnecka Cognitive Lab (UCI) and Lund’s Department of Philosophy.

Quinon received the Senior Visiting Fellowship in the Munich Center for Mathematical Philosophy in November 2014. In Munich she presented some of the results from the NUMBERS project. She also begun a collaboration with K. Krzyzanowska on a paper devoted to the relation between the use of numerals as vague quantifiers and the system of representations related to ANS.

Quinon has been recognized as an expert in the use of cognitive science in the philosophy of mathematics (see invitations to the Workshop at ECAP 8 and CLMPS 1025).

To conclude, we are confident that NUMBERS has achieved its core goals: to advance interdisciplinary research on the concept of natural number. The project resulted in the establishment of a sustainable collaboration between philosophers in Lund and cognitive scientists at Irvine. It therefore strengthened European research potential by promoting European research abroad and opening this field up to worldwide collaboration.