concept of function

Since the publication of Thomas Kuhn’s influential book, The Structure of the Scientific Revolutions, historical and sociological approaches to science have led to a kind of relativism towards the evaluation of theories as well as the belief in the break in the history of science. The main idea put by Kuhn's historiography is incommensurability. From Kuhn's point of view, the scientific concepts refers to completely different entities in different theories. On the other hand, each paradigm changes the perceptual domain before the conceptual process and formal articulation. Finally, unlike the conventional philosophy of science (Positivism and Falsifiability), he mentions there is no explicit rule for justification of theories. In contrast, Cassirer emphasis on elements in theory as the invariant regulative forms through which scientific thought progress continuously. In this paper, Cassirer's responses to Kuhn have been formulated. At the end, we attempt to show how diverse types of historiography made their difference.

Since studying mathematics develops and activates the mental system, it may be claimed that the profound understanding of mathematics can be effective in finding the facts and in true understanding of the phenomena. Thus mathematical understanding can help the individual perform his tasks with better knowledge and insight. In other words, benefitting from mathematical concepts, one can achieve the capability of deducing facts. Having stated the cognitive position of mathematics based on Plato’s and Des cartes’ quotations, the present article introduces a type of mathematical knowledge, which is the outcome of profound understanding of mathematical concepts, as mathematical cognition and insight. Determining the domain of this kind of knowledge based on the concept of the word” algorithm”, it also deals with the quality of the stages of achieving it. The article closes with giving certain examples for the cognitive aspect of the concept of function, and the graph structure in the theory of graphs.