Michael H. G. Hoffmann

The paper aims to show how by elaborating the Peircean terms used in the title creativity in learning processes and in scientific discoveries can be explained within a semiotic framework. The essential idea is to emphasize both the role of external representations and of experimenting with those representations , and to describe a process consisting of three steps: First, looking at diagrams "from a novel point of view" offers opportunities to synthesize elements of these diagrams which have never been perceived as connected before. Second, by forming those observed syntheses to "new objects" of thinking, and by signifying these objects through new signs , new means of thinking and acting are created . And finally, by applying these new means – in proofs, for instance – the "intelligibility" of new discoveries and their power to explain problematic facts must be tested

In recent years, semiotics has become an innovative theoretical framework in mathematics education. The purpose of this article is to show that semiotics can be used to explain learning as a process of experimenting with and communicating about one's own representations of mathematical problems. As a paradigmatic example, we apply a Peircean semiotic framework to answer the question of how students learned the concept of "distribution" in a statistics course by "diagrammatic reasoning" and by developing "hypostatic abstractions," that is by forming new mathematical objects which can be used as means for communication and further reasoning. Peirce's semiotic terminology is used as an alternative for notions such as modeling, symbolizing, and reification. We will show that it is a precise instrument of analysis with regard to the complexity of learning and of communication in mathematics classroom.

Different situations — like school and workplace — demand different forms of knowledge. Even more important, in particular for lifelong learning, are forms of knowledge we need for managing movements between those situations. To develop a better understanding of how to ‘navigate’ knowledge boundaries, this paper analyzes, firstly, interviews with scientists interpreting familiar and unfamiliar graphs. Our goal is to identify those forms of knowledge that should receive special attention in education. Secondly, the article elaborates — based on Peirce’s semiotics — an epistemologically reflected semiotic model to describe the role and conditions of knowledge necessary for crossing knowledge boundaries.

Abductive reasoning takes place in forming``hypotheses'''' in order to explain ``facts.'''' Thus, theconcept of abduction promises an understanding ofcreativity in science and learning. It raises,however, also a lot of problems. Some of them will bediscussed in this paper. After analyzing thedifference between induction and abduction (1), Ishall discuss Peirce''s claim that there is a ``logic''''of abduction (2). The thesis is that this claim can beunderstood, if we make a clear distinction between inferential elements and perceptive elements of abductive reasoning. For Peirce, the creative act offorming explanatory hypotheses and the emergence of``new ideas'''' belongs exclusively to the perceptive side of abduction. Thus, it is necessary to study the roleof perception in abductive reasoning (3). A furtherproblem is the question whether there is arelationship between abduction and Peirce''s concept of``theorematic reasoning'''' in mathematics (4). Both forms of reasoning could be connected, because both arebased on perception. The last problem concerns therole of instincts in explaining the success ofabductive reasoning in science, and the question whether the concept of instinct might be replaced bymethods of inquiry (5).

Discussions concerning belief revision, theorydevelopment, and ``creativity'' in philosophy andAI, reveal a growing interest in Peirce'sconcept of abduction. Peirce introducedabduction in an attempt to providetheoretical dignity and clarification to thedifficult problem of knowledge generation. Hewrote that ``An Abduction is Originary inrespect to being the only kind of argumentwhich starts a new idea'' (Peirce, CP 2.26).These discussions, however, led to considerabledebates about the precise way in which Peirce'sabduction can be used to explain knowledgegeneration (cf. Magnani, 1999; Hoffmann, 1999).The crucial question is that of understandinghow we can get the new elements capableof enlarging our theories. Under thesecircumstances, it might be helpful to step outof the entanglement and reconsider the basis ofthe problem that originally triggered Peirce'sinterest in abduction. This will lead us toanother Peircean concept, that of ``diagrammaticreasoning,'' which I discuss here in the contextof his ``pragmatism.'' In this way, I hope toreach a better understanding of thecontribution of ``abduction'' to the knowledgegeneration process.