My research combines the philosophy of mathematics with history and "practice" of mathematics. In the recent article Philosophy of mathematical practice - motivations, themes and prospects I characterise this recent part of philosophy. I have in particular contributed to two areas: visual thinking and structuralism in mathematics.Visual thinking in mathematics
I am interested in how diagrams, and in general representations, contribute to the development of mathematics. It has been proposed that diagrams are particularly fruitful in this regard. Looking at the history of mathematics, however, one can easily find examples where other types of representations, such as algebraic expressions, were needed in order to prove certain results. In the light of this, I wish to understand how different representations contribute to mathematical knowledge. In these investigations I have made case studies of Riemann's introduction of his surfaces (Carter 2013), in free probability theory (Carter 2010) and of graph-algebras (Carter 2018). I have also recently written the bibliography on Visual Thinking in Mathematics for Oxford Bibliographies in Philosophy.Structuralism in mathematics
The fact that (contemporary) mathematicians claim they study structures rather than "objects" has inspired in a number of structuralist positions in the philosophy of mathematics. In order to make such proposals clear one needs a characterisation of what is meant by a structure. (Carter 2008) considers a variety of cases from contemporary mathematics where mathematicians refer to structures and conclude that they mean quite different things when they say that mathematics studies structure.
A recent interest concerns the writings on mathematics of Charles Sanders Peirce. I have used his writings as an inspiration to say something about the ontology of mathematics (Carter 2014) and the pre-history of structuralism in mathematics (Carter forthcoming). My studies on the role of representations for the development of mathematical knowledge is informed by Peirce's semiotics.
Structuralism as a Philosophy of Mathematical Practice, 2008: Synthese 163, nr. 2, s. 119-131.
Diagrams and Proofs in Analysis, 2010: International Studies in the Philosophy of Science 24, 1-13.
Handling Mathematical Objects: Representations and Context, 2013. Synthese 90 (17), 3983-3999.
Mathematical Objects as Hypothetical States of things, 2014. Philosophia Mathematica 22, 209-230.
Philosophy of mathematical practice - motivations, themes and prospects 2019. Philosophia Mathematica 27 (1), 1-32.
Visual Thinking in Mathematics. To appear in Oxford Bibliographies in Philosophy. Ed. Duncan Pritchard. New York: Oxford University Press.
Logic of Relations and Diagrammatic Reasoning: Structuralist elements in the writings of C.S. Peirce (1839-1914), forthcoming. To appear in the book 'The Pre-history of Structuralism' edited by E. Rech and G. Schiemer.
I am one of 9 founding members of the Assciation of the Philosophy of Mathematical Practice. I was secretary of this assion between 2010 and 2015. I am currently membe of its steering committee, see the website of the assotioation: philmathpractice.org