Mathematics: Objectivity by representation
Final Report Abstract
The main goal of this project was to clarify the role of representations in mathematical reasoning and proofs and the way they contribute to mathematical ontology and understanding. By appealing to different sorts of representations, the project investigated how appropriate domains of abstract mathematical objects are constituted, and how appropriate reasoning on them is licensed. Crucially, our approach placed central importance on the role of mathematical practice and its connection to mathematical ontology and epistemology. The main objective of our project was to argue for a twofold (hypo)thesis: (a) Mathematical representations established by stipulations contribute to the constitution of mathematical ontology and to the shaping of mathematical reasoning. (b) Mathematical object-constitution is analogous to scientific object-constitution. The project developed a novel approach to the foundations of mathematics, contrasting it with classical foundational perspectives, like Platonism and Nominalism. These classical positions share an “existential attitude” to mathematical objects, as they both take as crucial the question of whether these objects exist or not. This project argued for a different approach, according to which mathematical objects are fixed by appropriate systems of stipulations that someone suggested and which appropriate communities of people adopted. These stipulations also pertain to the form of reasoning on these objects. According to this innovative approach, mathematical objects are bona fide objects insofar as the systems of stipulations responsible for their constitution promote certain inferential or reasoning practices. In a sense, this approach represents the limit to which the classical foundational positions converge. Indeed, the project’s three PIs (Heinzmann and Panza on the French side, and Leitgeb on the German side) were independently led to this new foundational perspective from different starting points. The French-German collaboration has thus resulted in a synergy of these different classical positions.
Publications
- (2016): “A Choice Semantical Approach to Theoretical Truth”, Studies in History and Philosophy of Science, 58, pp. 1 – 8
Schiemer, G. and Andreas, H.
(See online at https://doi.org/10.1016/j.shpsa.2016.02.001) - (2017): “Logical Anti-Exceptionalism and Theoretical Equivalence”, Analysis 77, pp. 759 – 767
Wigglesworth, J.
(See online at https://doi.org/10.1093/analys/anx072) - (2017): “Two Types of Indefinites: Hilbert & Russell”, IfColog Journal of Logics and their Applications, 4/2, pp. 333 – 348
Schiemer, G. and Gratzl, N.
(See online at https://doi.org/10.5282/ubm/epub.41343) - (2018): “Grounding in Mathematical Structuralism”, in Bliss, R. and Priest, G. (eds.) Reality and its Structure: Essays in Fundamentality, Oxford University Press, Oxford, 2018, 217 – 236
Wigglesworth, J.
(See online at https://doi.org/10.1093/oso/9780198755630.003.0012) - (2018): “Non-Eliminative Structuralism, Fregean Abstraction, and Non-Rigid Structures”, Erkenntnis
Wigglesworth, J.
(See online at https://doi.org/10.1007/s10670-018-0096-3)
