The Thermodynamics of Mathematical Representation

Thermodynamics is the physical science surrounding work, heat, and relationships across fundamental quantities, and situates itself near the center of multiple disciplines through its generality and timelessness. Its laws required no rewriting after the twentieth century revolutions of quantum mechanics, relativity, and solid state physics, just to name three subjects. The nine chapters of this book make appeal to thermodynamic notions and laws to get under the hood of mathematics―the language of the physical sciences―without just echoing things best said and written in math books. It takes a system to learn about another system―we all need thermometers, voltmeters, and other gadgets to get to know objects of interest. But just as critical are the numbers and functions we put to the task, however relegated they are to computers in the modern day for the heavy lifting. To be sure, mathematical representations like x = ½, 5.2, π, e, etc., and f(x) = x2, sin(x), etc., are never in physical contact with the solids, liquids, and gases that draw our attention, but they are as impacted by the same natural laws as the lab apparatus itself.

This book shows how the thermodynamic laws impact our number systems. The laws affirm that we have direct access to a vanishingly small fraction of the real numbers. They further establish that the real numbers present a maximum-evolved system impacting all matters of computation, graphing, differentiation, and integration. For completeness, one of the chapters includes cases where the thermodynamic laws have little, if anything, constructive to say about representations in mathematics.

This book presents a novel perspective to students and teachers in the physical sciences, biology, and mathematics, with the goal of enriching classroom and seminar hours. The chapters are self-contained and written informally, and readers with rudimentary knowledge of energy, numbers, and functions should handle the material well.

The author was born and raised in San Francisco, California. He completed his Bachelor of Science in Chemistry at Stanford University and his PhD in Chemistry at Washington University, St. Louis, under the guidance of Professor Tien-Sung Lin. The author pursued his postdoctoral studies with Professor Richard H. Clarke at Boston University.

The author’s independent career began as a chemistry faculty member at West Virginia University in Morgantown, West Virginia. He moved to Loyola University Chicago four years later, where he enjoyed a thirty-four-year sojourn. He is now Professor Emeritus.

The author during his time in Chicago wrote two books published by Taylor & Francis: Chemical Thermodynamics and Information Theory with Applications (2011) and Invitation to Protein Sequence Analysis Through Probability and Information (2019).

The Thermodynamics of Mathematical Representation is the author’s third book. His hope is that this book will be useful in special topics, classes, and seminars for math/science students and for faculty looking for ideas to incorporate into introductory chemistry, physical chemistry, and calculus classes. The author would have taught the classes and seminars himself, but the pandemic, retirement age, the curse of Zoom©teaching, and the wonderful call to be closer to family in Central New York intervened.