Preface to a book in draft, Galileo's Book

Galileo’s Book, The Book of Nature is Written in Language of Mathematics

Martin H. Krieger     11/24/19 7:34 AM

Prolog: The Ising Model and The Stability of Matter—our recurring examples

Preface --Galileo was perhaps correct that Nature speaks in the language of mathematics, but it would seem that there are many such dialects, although it may be possible to say the same thing many of the dialects.

Chapter 1: Learning from Newton  .  Lessons—Mathematical physics is never so isolated from the culture of its time, nor in fact is mathematics.

Chapter 2: The Philosophy of Mathematical Physics .  Lessons—That mathematics is useful for physics and may be the language of Nature is perhaps no more surprising than for most people their legs are useful for walking.

Chapter 3: Legerdemain  .  Lessons—It matters if you are persistent, and you have colleagues and teachers, and sometimes nothing helps.

Chapter 4: Ising Matter  .  Lessons—Mathematical choices have physical meaning, and mathematics that does the same work points to interesting physical “equivalences” and perhaps mathematical ones as well.

Chapter 5: Primes and Particles  .  Lessons—It would seem that only a small number of ways of organizing a complex social or physical system prove suggestive and fruitful.

Epilog  —The mathematics matters, it has physical content, and rigor and formalism are likely to point to important aspects of Nature (and the physics).

Notes 

Preface

Galileo was perhaps correct that Nature speaks in the language of mathematics, but it would seem that there are many such dialects, although it may be possible to say the same thing many of the dialects.

It is nowadays a commonplace to quote Galileo (although the attribution is dubious) to the effect that Natures is written in the language of mathematics. My purpose here is to be quite specific about how that language works, here in the realm of mathematics applied to physical systems:

1.         1. Newton provides the archetypal account of the relationship of mathematics and Nature.

2. Mathematical physics releases a charm that allows mathematical agendas to do Nature’s work

3.     Mathematics would seem to do magical work (as in legerdemain) when we are solving a problem, “by the way” incorporating interesting features of a physical system. Tricks and mathematical techniques are not only such, they are also about the world. This is surely the most challenging chapter, since I need to convey both the technical features and how they do the work. I will draw from the Ising example and from proofs of the stability of matter.

4.     Mathematics, in mathematical physics, provides a model of how we might have different perspectives on the same object and so reveal its many aspects. The model here is the various solutions of the two-dimensional Ising model of a ferromagnet, all of which, of course, come to the same result, but each revealing different aspects of this physical system. The phrase from the phenomenologists, “an identity in a manifold presentation of profiles,” informs my thinking.

5.     Mathematics provides structures through which science displays the organization of Nature. Specifically, as a consequence of the development in 19^th century (algebraic) number theory, we have a hierarchy of articulations of the rational numbers, adding in √p, where p is a prime number. What was once prime may no longer be prime, so, for example, adding in √3, 2=(√3-1)(√3+1), so 2 is now composite. The account of primes and their becoming composite provides a model of the particle physicist’s elementary particles, for as the energy is raised (in analogy to adding in √p) particles that were once elementary may

exhibit compositeness.

In each case, I shall explain both the mathematics and the physics, and then make clear how the mathematical language does physical work. What will make this study difficult and distinctive is the actual mathematics and the physics, specific and not at all dumbed down. I have deliberately allowed for repetition of the two main examples, in part because the lay reader is not likely to magically recall their features.