The rapid development of cognitive sciences offers a new perspective on the process of the formation of concepts. Contrary to the classical approach where their content was given in an a priori manner, by using the prototype semantics the meaning of concepts can be derived from a set of several basic experiences conditioned by the physical circumstances of a cognizing subject. This strategy is particularly useful in the attempt to shed a deeper light on the foundations of mathematics whereby the classical positions on the status of mathematical entities acquire their proper justification. The aim of the presented article is to introduce and evaluate the program of the embodied mathematics with a metaphor as a central tool of conceptualization.