ON PHILOSOPHY, NATURE, AND THE ROLE OF MATHEMATICS

Since ancient times, our level of understanding has grown along with the level of the mathematics that has been developed and employed. Five hundred years ago, Galileo wrote:

Philosophy [i.e., physics or natural philosophy] is written in this grand book—I mean the universe—which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth.1

Newton, by 1687, had developed a differential calculus that he used to describe the motion of the planets.² And even in a much more modern time it was Schrödinger who indicated the need for yet more math. In his notes of December 27, 1925, on the eve of his development of wave mechanics, he wrote:

At the moment I am struggling with a new atomic theory. If only I knew more mathematics! I am optimistic about this thing and expect that if I can only…solve it, it will be very beautiful.³

Through the mathematics of quantum mechanics we have a view into the workings of the quantum world on a submicroscopic scale. What we see is strange to us, and beautiful with its mathematical patterns of uncertainty and probability yielding accurate descriptions of atoms, the building blocks of the world of our senses. And, as you will see, quantum mechanics even explains aspects of black holes, the centers of our galaxies. But is this quantum world only in our minds (because the view provided by quantum mechanics is mathematical and therefore a construct of the human mind)?

Consider: Mathematics is just a way of using logic with the aid of symbols. If protons and electrons do have electrical charges and attract each other as we have observed, and if the electron can be accurately modeled mathematically as diffuse in form and attracted to the nucleus as Schrödinger has done, then the solutions to his mathematical equations are just logical conclusions.

The quantum results of Schrödinger, Heisenberg, and Dirac so well describe this world that we tend to believe that theirs are functionally correct models of the electron and the atom. But the quantum world, including the galaxies, would still exist even if we were not here to make these mathematical models. So it seems that nature itself must be inherently logical, and if there are charged electrons and protons, then the quantum atomic behavior that we observe occurs in a very natural, logical way. Our mathematics just describes it.

HYPOTHESIS, THEORY, LAW, AND CORRESPONDENCE

This book describes parts of quantum theory and the results of its more formal mathematical structuring as quantum mechanics, most specifically as applied to the atom. Theory is defined as a well-tested and proven set of ideas that accurately explains the physical world. Theory by this definition has otherwise sometimes been called a law. Theory is well proven, as compared to untested postulates and ideas, which are scientifically labeled as mere hypotheses.

A successful theory must be able to predict the occurrence or observation of things or happenings not previously seen. As noted earlier in this book, quantum theory in its broader context, including quantum mechanics, is the most proven, predictive, and successful theory in the history of science.4 Quantum theory has become accepted as a broad theory that is valid over the entire range from the macroscopic (large objects) through the microscopic, the submicroscopic, subatomic, and even subnuclear. And it has far-reaching implications, rendering invalid the previous classical (deterministic)⁵ view of what we can observe of our world.

It is not that scientists have capriciously revised their thinking since classical times. Nor that quantum theory was suddenly adopted as a popular new set of ideas. Rather, it was on the basis of more and more experimental evidence that didn't fit with the classical theory that finally scientists had to give up on the classical and seek out something new in explanation. And even when many leading scientists accepted this new quantum theory, there were many who found it just too radical a departure from classical thinking. So the development of quantum theory (including its mathematical formalization as quantum mechanics in 1925) was truly a scientific revolution in the sense described by Kuhn.⁶

As outlined in the preface and early chapters of this book, the classical physics of Newton, Maxwell, and others (which preceded quantum theory) that seemed to so well describe the physical world, began not to fit with newer discoveries starting around 1900. In particular, much of classical physics simply did not explain the submicroscopic world. And we now know that classical physics is also conceptually wrong even as it applies to large objects—say, objects the size of a grain of sand or the planets.

However, despite its failures in the realm of the submicroscopic, classical physics does provide a very good approximation to the results of quantum mechanics for large objects like grains of sand or the planets.⁷ We call this agreement for large objects correspondence, and, because classical physics is easier to use than quantum mechanics, we continue to apply classical physics theory in those many parts of science and engineering where the approximation is sufficiently accurate. In a similar way, should newer, more broadly applicable theories be developed, quantum mechanics and relativity will not simply be cast aside. They will still apply, in another, newer region of correspondence.

Next, before we get a look (in Part Three) at our quantum and relativistic world from the tiniest of fundamental particles to the galaxies, we visit another exciting example quantum application, one that is just emerging with great potential.