Abstract
The most elementary feature of mathematical progress is problem solving within an established area of research by the use of traditional notations and traditional methods. But in many cases mathematical progress is achieved by shaping a new notion, inventing a new method or even building a new theory. In these conceptual developments, there is usually a tendency towards a higher level of abstraction. In this paper I am particularly interested in the transition to this higher level. Distinguished mathematicians, particularly during the twentieth century, have pointed out that generalization is not an end in itself; what is to be found is rather the right generalization or the interesting one. This criterion of progress stems from the meta-level. In fact philosophically the most interesting things happen on the meta-level.
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Breger, H. (2000). Tacit Knowledge and Mathematical Progress. In: Grosholz, E., Breger, H. (eds) The Growth of Mathematical Knowledge. Synthese Library, vol 289. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9558-2_15
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DOI: https://doi.org/10.1007/978-94-015-9558-2_15
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