Logarithms Are Not Infinity: A Rational Physics-Related Explanation of the Mysterious Statement by Lev Landau

Abstract

Nobel-prize winning physicist Lev Landau liked to emphasize that logarithms are not infinity – meaning that from the physical viewpoint, logarithms of infinite values are not really infinite. Of course, from a literally mathematical viewpoint, this statement does not make sense: One can easily prove that logarithm of infinity is infinite. However, when a Nobel-prizing physicist makes a statement, you do not want to dismiss it, you want to interpret it. In this paper, we propose a possible physical explanation of this statement. Namely, in physics, nothing is really infinite: According to modern physics, even the Universe is finite in size. From this viewpoint, infinity simply means a very large value. And here lies our explanation: while, e.g., the square of a very large value is still very large, the logarithm of a very large value can be very reasonable – and for very large values from physics, logarithms are indeed very reasonable.