Real life applications enter primary and secondary school education in two ways – for creating interest in subjects which may otherwise be abstract, and for the purpose of making use of the school subjects in day-to-day situations. Here, the prime example is mathematics. A demand for a close connection between mathematics and applications in school may be found in national curricula, and is present in textbooks. On the other hand mathematics is considered and taught to be a deductive, a priori, science with internal truth makers, structured by propositions and proofs. Mathematics is presented both as empirically grounded and as an analytic science, creating a possible conflict for students. The problem of the applicability of mathematics is also discussed within philosophy of mathematics: How is it possible for a priori truths to contribute essentially to our descriptions of the world? From a philosophical point of view, we try to shed light on how this seeming paradox may be explained and handled. Central are our views on mathematical concepts as explications and on concept formation in mathematics.