This area of research concerns mainly the specificity of Mathematics with respect to the other scientific disciplines. First of all, the objects studied by mathematics are ontologically different from the objects of natural sciences: they are intangible, abstract and linguistic. This has important consequences at the epistemological level, such as the interrelation of mathematics with philosophy of language, or the arising of the “commencement problem”, that is the need for mathematics to justify its own investigation, not provided naturally by its own objects.

Moreover, the inaccessibility of the mathematical objects through experience makes representations crucially important. The same mathematical object can in fact be represented in different registers, such as the symbolic, the graphical or the verbal register, and others; the questions related to converting between the different registers and controlling the meaning carried by each representation involve the epistemology and the teaching of mathematics:for example, how can we enable the student to distinguish between a mathematical object and its representations, when the object itself can only be accessed through its representations? In mathematics, there is no noesis (knowledge) without semiosis (representation). It is rather the coordination among representations which makes the conditions for knowledge possible. The comprehension of the object under study acquires in this way hermeneutic elements, since it stems from the dialogical integration of all the possible semiotic representations.

Another interesting question, also inherent to the topic of representations, concerns the epistemological statute of diagrammatic reasoning, that is the mathematical thinking based on diagrams or other kinds of iconical representations. Even if it is widely agreed among mathematicians that this approach is useful in the process of exploring a mathematical problem, its validity as a demonstration tool is currently questioned in philosophy of mathematics.

We have worked also on the importance of the “method”, literally “road towards”, a crucial research element in empirical sciences and in mathematics, but with different meaning. The method is a necessary guiding element in the work of a scientist, the main road which guides the researcher from the design of the experiment to the formulation of the theory; the method is also an unavoidable constraint to scientific progress. Since the time of Euclid, mathematics has been characterized by the axiomatic method; Hilbert’s reinterpretation in the twentieth century was an efficient solution to the crisis of the foundations of mathematics and deprived mathematics of empirical content, emphasizing its relational nature. Then, curiously, the method, which guides and, at the same time, sets a constraint to the progress of science, in mathematics it opens towards new avenues and it is a source of freedom and of predictive power, exactly as it represents a clear cut between logical structure and intuitive content.

- A. I. Telloni, La scienza sospesa: le vie della ricerca fra metodo ed euristica, MATEPristem, http://matematica.unibocconi.it/articoli/la-scienza-sospesa-le-vie-della-ricerca-fra-metodo-ed-euristica.
- A. I. Telloni, Chiaroscuri dell’insegnamento-apprendimento. Alcune lezioni di matematica del mondo antico, L’insegnamento della Matematica e delle Scienze Integrate, 39B (4) (2016), 411-436.
- A. I. Telloni, C. Toffalori, La logica delle equazioni, in corso di pubblicazione.
- A. I. Telloni, Educare alla razionalità. La lezione del mondo classico, in corso di pubblicazione.
- A.I. Telloni, Il metodo scientifico e la matematica. Teleologia dell’univoco o equivoco della teleologia?, in corso di pubblicazione (Aracne Ed.)

Dott.ssa Agnese Ilaria Telloni email. agnesetelloni@gmail.com