The general objective of this project is to explain the concept of natural number. Studying these mathematical entities is important because of their broad use in many reasoning processes and in the very practice of science.
Different disciplines study “natural numbers” under many different incomparable angles:
- in mathematics as abstract entities,
- in computer sciences, as certain class of inscriptions,
- in linguistics as quantifiers,
- in psychology as mental representations,
- in neuro-psychology as neuron configurations,
- in philosophy as, for example, Platonic abstract objects.
This list is not exhaustive.
Moreover, each of the disciplines can study numbers under many different aspects: ex. in mathematics natural numbers can be understood in set theoretical terms (ex. to be identified with finite von Neumann's ordinals) or, studied from the axiomatic viewpoint, as these abstract entities which are described by the second order Peano’s arithmetic.
Even though one can find exchanges and influences between some of the listed disciplines (ex. certain schools in philosophy of mathematics aim to give an account of actual mathematical practice), more often each of these disciplines develops in isolation from the influence of the others, using exclusively its own formal or experimental tools. The debates are eventually conducted within one field (philosophers with various ontological orientations discuss with each other, developmental psychologists disagree on certain aspects of how exactly the number concept is constructed in infants), but there is little inter- or multi- disciplinary exchange on what natural numbers are, and how do we know what they are.
Therefore a global understanding of the concept of natural number is crucial for many profoundly important academic disciplines. In this proposed research Project we aim to investigate the possible connections between different ways of studying this concept. The long-term research target is the in-depth treatment of the different numerical tools we make use of in reasoning and their possible interactions. Unveiling the nature or natures of natural numbers is important especially in view of the very fast developing of new technologies that support reasoning and communication. The multidisciplinary and global understanding of numbers, to which this Project will contribute, is new and innovative; contrasted with local and polarised account given usually by disconnected disciplines.
The objectives of the current Project are the following:
1. Study the relations, interactions and the boarder line between the philosophy of mathematics (related to study of natural numbers) and the cognitive sciences oriented towards number cognition.
2. Perform several short-term case studies using the results of both of these disciplines.
3. Elaborate a common conceptual field and the common language of communication for the two fields.
The background assumption is that there is a common object of studies and that both disciplines can improve its own achievements by taking inspiration in the other fields’ results and methodology. This assumption is 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, not necessarily incompatible aspects of the number concept.