Representations, in particular diagrammatic representations, allegedly contribute to new insights... more Representations, in particular diagrammatic representations, allegedly contribute to new insights in mathematics. Here I explore the phenomenon of a “free ride” and to what extent it occurs in mathematics. A free ride, according to Shimojima (Artif Intell Rev 15: 5–27, 2001), is the property of some representations that whenever certain pieces of information have been represented then a new piece of consequential information can be read off for free. I will take Shimojima’s (informal) framework as a tool to analyse the occurrence and properties of them. I consider a number of different examples from mathematical practice that illustrate a variety of uses of free rides in mathematics. Analysing these examples I find that mathematical free rides are sometimes based on syntactic and semantic properties of diagrams.
I discuss the notion of ‘understanding’ in mathematics in relation to a result from current mathe... more I discuss the notion of ‘understanding’ in mathematics in relation to a result from current mathematical practice. The mathematical example concerns finding the value of a complex mathematical expression and I illustrate how this is found by gradually breaking it down into simpler expressions. A major point is to show what roles iconic and indexical representations play in this process. Furthermore it is argued that representations in the sense of Peirce play a key role for understanding as described here, and in particular,that icons play an essential role.
Page 1. Ontology and Mathematical Practicet JESSICA CARTER* ... In his Proofs and Refutations he ... more Page 1. Ontology and Mathematical Practicet JESSICA CARTER* ... In his Proofs and Refutations he presented a study of the history of Euler's theo-rem, challenging the formalist conception of mathematics, and proposing a fallibilist view of mathematics. ...
I consider the role of diagrams in contemporary mathematics. More specifically the role of certai... more I consider the role of diagrams in contemporary mathematics. More specifically the role of certain diagrams-so-called directed graphs-will be investigated. I propose that these graphs act as mediating objects. This means that they link certain objects, that is, a C*-algebra and its associated K-groups, and that this link yields an epistemic gain. I explain that the link is possible because a graph represents as a metaphor in two distinct ways. In addition, the diagrammatic presentation of a directed graph becomes an object that can be manipulated. As such, it becomes what I will denote a "faithful representation." The notion of a faithful representation tries to capture the fruitfulness of the combination of metaphorical representation with the possibility of controlled manipulation.
Representations, in particular diagrammatic representations, allegedly contribute to new insights... more Representations, in particular diagrammatic representations, allegedly contribute to new insights in mathematics. Here I explore the phenomenon of a “free ride” and to what extent it occurs in mathematics. A free ride, according to Shimojima (Artif Intell Rev 15: 5–27, 2001), is the property of some representations that whenever certain pieces of information have been represented then a new piece of consequential information can be read off for free. I will take Shimojima’s (informal) framework as a tool to analyse the occurrence and properties of them. I consider a number of different examples from mathematical practice that illustrate a variety of uses of free rides in mathematics. Analysing these examples I find that mathematical free rides are sometimes based on syntactic and semantic properties of diagrams.
I discuss the notion of ‘understanding’ in mathematics in relation to a result from current mathe... more I discuss the notion of ‘understanding’ in mathematics in relation to a result from current mathematical practice. The mathematical example concerns finding the value of a complex mathematical expression and I illustrate how this is found by gradually breaking it down into simpler expressions. A major point is to show what roles iconic and indexical representations play in this process. Furthermore it is argued that representations in the sense of Peirce play a key role for understanding as described here, and in particular,that icons play an essential role.
Page 1. Ontology and Mathematical Practicet JESSICA CARTER* ... In his Proofs and Refutations he ... more Page 1. Ontology and Mathematical Practicet JESSICA CARTER* ... In his Proofs and Refutations he presented a study of the history of Euler's theo-rem, challenging the formalist conception of mathematics, and proposing a fallibilist view of mathematics. ...
I consider the role of diagrams in contemporary mathematics. More specifically the role of certai... more I consider the role of diagrams in contemporary mathematics. More specifically the role of certain diagrams-so-called directed graphs-will be investigated. I propose that these graphs act as mediating objects. This means that they link certain objects, that is, a C*-algebra and its associated K-groups, and that this link yields an epistemic gain. I explain that the link is possible because a graph represents as a metaphor in two distinct ways. In addition, the diagrammatic presentation of a directed graph becomes an object that can be manipulated. As such, it becomes what I will denote a "faithful representation." The notion of a faithful representation tries to capture the fruitfulness of the combination of metaphorical representation with the possibility of controlled manipulation.
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