The terms of a mathematical problem become precise and concise if they are expressed in an appropriate notation, therefore notations are useful to mathematics. But are notations only useful, or also essential? According to prevailing view, they are not essential. Contrary to this view, this paper argues that notations are essential to mathematics, because they may play a crucial role in mathematical discovery. Specifically, since notations may consist of symbolic notations, diagrammatic notations, or a mix of symbolic and diagrammatic notations, (...) notations may play a crucial role in mathematical discovery in different ways. Some examples are given to illustrate these ways. (shrink)

The thesis of the paper is that semiotic processes are intrinsic to computation and computational systems. An explanation of computation that does not take this semiotic dimension into account is incomplete. Semiosis is essential to computation and therefore requires a rigorous definition. To prove this thesis, the author analyzes two concepts of computation: the Turing machine and the mechanistic conception of physical computation. The paper is organized in two parts. The first part (Sects. 2 and 3) develops a re-interpretation of (...) the Turing machine starting from Peirce’s semiotics. The author shows how this reinterpretation allows overcoming the dualism between a purist and a realist version of the Turing machine. The second part of the paper (Sect. 4) shows how a mechanistic explanation of physical computation such as the one developed by Piccinini is incomplete without considering semiotic relations. The paper intends to be a contribution to the philosophical debate on computation. (shrink)

I argue that mathematical representations can have heuristic power since their construction can be ampliative. To this end, I examine how a representation introduces elements and properties into the represented object that it does not contain at the beginning of its construction, and how it guides the manipulations of the represented object in ways that restructure its components by gradually adding new pieces of information to produce a hypothesis in order to solve a problem.In addition, I defend an ‘inferential’ approach (...) to the heuristic power of representations by arguing that these representations draw on ampliative inferences such as analogies and inductions. In effect, in order to construct a representation, we have to ‘assimilate’ diverse things, and this requires identifying similarities between them. These similarities form the basis for ampliative inferences that gradually build hypotheses to solve a problem.To support my thesis, I analyse two examples. The first one is intra-field, that is, the construction of an algebraic representation of 3-manifolds; the second is inter-fields, that is, the construction of a topological representation of DNA supercoiling. (shrink)