Popular Discussion of Classical and Finite Mathematics Felix M. Lev Artwork Conversion Software Inc., 509 N. Sepulveda Blvd, Manhattan Beach, CA 90266, USA (Email: felixlev314@gmail.com) Abstract Following our recently published book (F. Lev, Finite mathematics as the foundation of classical mathematics and quantum theory. With application to gravity and particle theory. Springer (2020)), we discuss different aspects of classical and finite mathematics and explain why classical mathematics is a special degenerate case of finite one. Keywords: finite mathematics, classical mathematics, finite quantum theory Mathematical education at physics departments develops a belief that clas- sical mathematics (involving infinitesimals, limits, continuity etc.) is the most fun- damental mathematics, while finite mathematics is something inferior what is used only in special applications. And many mathematicians have a similar belief. Historically it happened so because more than 300 years ago Newton and Leibniz proposed the calculus of infinitesimals, and since that time a titanic work has been done on foundation of classical mathematics. This problem has not been solved till the present time (see below) but for the majority of physicists and many mathematicians the most important thing is not whether a rigorous foundation exists but that in many cases standard mathematical technique works with a very high accuracy. The idea of infinitesimals was in the spirit of existed experience that any macroscopic object can be divided into arbitrarily large number of arbitrarily small parts, and even in the 19th century people did not know about atoms and elementary particles. But now we know that when we reach the level of atoms and elementary particles then standard division loses its usual meaning and in nature there are no arbitrarily small parts and no continuity. For example, typical energies of electrons in modern accelerators are mil- lions of times greater than the electron rest energy, and such electrons experience many collisions with different particles. If it were possible to break the electron into parts, then it would have been noticed long ago. Another example is that if we draw a line on a sheet of paper and look at this line by a microscope then we will see that the line is strongly discontinuous 1