We assume that mathematics is the objective entity or is constructed through mental activity. The latter point of view is that mathematics is a mental activity to pursue the certainty. And mathematical consequences are followed by errors, as they are not destined nor definite, but they are made up from human behavior. This study aims to investigate historical development of mathematics, especially focused on epistemological obstacles. It was explored in the light of Kitcher, Lakatos, Bachelerd and Brusseau. And we discussed development of mathematical concepts according to perspectives on historico-critical approach. Kitcher provided question-answering, question-generation, generalization, rigorization, and systematization in the process of development of mathematical knowledge. And we applied this process to the development of the calculus. Lakatos identified a simple pattern of mathematical discovery and of the growth of informal mathematical theories. And he introduced the proof-generated concepts in the process of proofs and refutations. According to Brousseau, the evolution of mathematical concepts conforms to a pattern, that is protomathematical concepts, paramathematical concepts, mathematical concepts. From historical analyses, we can find discontinuous level-rising and overcoming epistemological obstacles in the process of developments of mathematical concepts.