Short description:

The nine chapters of this book make appeal to thermodynamic notions and laws to get under the hood of mathematics without just echoing things best said and written in math books. It presents a novel perspective to students and teachers in the physical sciences, biology, and mathematics, with the goal of enriching classroom and seminar hours.

Long description:

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 = ?, 5.2, ?, 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 that is.

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.

The 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.

Table of Contents:

Preface Acknowledgements Author Biography List of Principal Abbreviations Glossary of Principal Terms Chapter One: Aspects of the Thermodynamics of Mathematical Representation Chapter Two: A Tour of Thermodynamic Systems and Operational Laws Chapter Three: A Tour of the Real Number System and Thermodynamic Intertwining Chapter Four: The Information Bridge Joining Thermodynamics and the Number Systems Chapter Five: Information, Integer Partitions, and Small-System Thermodynamics Chapter Six: Information, Prime Number Representations, and Thermodynamic Overlays Chapter Seven: Information, Limit Operations, and Thermodynamic Overlays Chapter Eight: Special Limit Operations and Maximum Entropy Principles Chapter Nine: Points, Paths, and Horizons Epilogue Appendix A: A Synopsis of Thermodynamic Laws Appendix B: Notes and References Appendix C: A Précis on Point Sets and Operations Appendix D: Answers to Selected Exercises Appendix E: Bibliography Index