# The Astonishing Power of Mathematics in Natural Sciences

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## Chapter 1: Introduction to the Intersection of Mathematics and Physics

Recently, I began a new position where my background is in mathematics, and I find myself collaborating with colleagues who specialize in physics. During a lunch discussion, we explored the interplay between these two disciplines, which led me to reflect on Eugene Wigner’s renowned essay, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Though I had come across references to this work and grasped its general ideas, I realized I had not taken the time to thoroughly read it myself. To enhance my understanding, I decided to engage deeply with the article and summarize its content, hoping it will also benefit you, the reader.

It is important to clarify that my goal here is to analyze Wigner’s article directly, rather than discuss subsequent critiques or related inquiries, such as Richard Hamming's 1980 response. Wigner’s original piece serves as a fascinating entry point into the broader discussion about how mathematics interrelates with scientific inquiry, as well as philosophical questions regarding the nature of mathematics itself. In this exploration, I aim to unpack Wigner's insights and may address these adjacent topics in future writings.

## Chapter 2: The Life of Eugene Wigner

Eugene Wigner (1902–1995), originally Wigner Jenő, studied at the Kaiser Wilhelm Institute in Berlin, where he worked alongside notable figures such as Karl Weissenberg and Richard Becker, who introduced group theory into the realm of physics. He later served as an assistant to the esteemed David Hilbert at the University of Göttingen. His illustrious career included significant contributions to the Manhattan Project, and he eventually became the Director of Research and Development at what is now known as Oak Ridge National Laboratory. His work extended to various governmental research institutions, including the National Bureau of Standards and the National Science Foundation. In recognition of his groundbreaking contributions to atomic theory and particle physics, Wigner was awarded the Nobel Prize in Physics in 1960.

This video explores Wigner's perspective on the extraordinary effectiveness of mathematics in the natural sciences, providing insights into his influential ideas.

## Chapter 3: The Essence of Wigner's Article

In 1960, Wigner published his seminal article, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” in Communications on Pure and Applied Mathematics. He tackles two interrelated questions: the remarkable utility of mathematics in physics, and the distinctiveness of mathematical theories within the field.

To illustrate the first point, Wigner asserts, “The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and there is no rational explanation for it.” He proceeds to examine the following queries: “What is mathematics?”, “What is physics?”, “What role does mathematics play in physical theories?”, and “Is the success of physical theories truly surprising?”

To delve deeper, let's consider Wigner's views on these inquiries.

### Section 3.1: Defining Mathematics

Wigner characterizes mathematics as “the science of skillful operations with concepts and rules invented just for this purpose.” He emphasizes that these concepts are crucial, as without them, only a limited number of significant theorems could be derived from foundational axioms. The selection of concepts is not merely for simplicity but is aimed at enabling sophisticated logical processes that resonate with our aesthetic sensibilities in both their execution and outcomes.

For example, complex numbers arise not from empirical experiences but from the intrinsic beauty of mathematics itself. The development of complex number theory was driven by its elegance rather than its practical applications. Therefore, mathematicians engage with their field not solely for utility but as an endeavor worthy of exploration in its own right.

### Section 3.2: Understanding Physics

Conversely, physics is fundamentally about discerning the laws of nature. These laws allow predictions of future events based on present circumstances, resembling conditional statements. For instance, Galileo's law of gravitation posits that if two rocks are dropped simultaneously from the same height, they will touch the ground at the same moment. Wigner points out that this law is particularly striking because it holds true regardless of location or time, relying solely on specific variables, and is unaffected by others that might influence outcomes.

### Section 3.3: The Role of Mathematics in Physics

Mathematics serves as a clear instrument within physics, given that physical laws are expressed as conditional statements. However, Wigner argues that mathematics occupies a far more profound role. He states, “The laws of nature must have been formulated in the language of mathematics to be an object for the use of applied mathematics.” He illustrates this with three remarkable instances: the laws governing falling bodies, quantum mechanics, and quantum electrodynamics.

For example, Isaac Newton identified that the trajectories of falling objects and celestial bodies can be understood through the concept of ellipses, leading him to formulate the universal law of gravitation. While initially verified with a margin of error of 4%, later analyses affirmed its accuracy to within a tiny fraction of a percent.

This video reviews Wigner's article and discusses the remarkable efficacy of mathematics in the natural sciences, shedding light on Eugene Wigner’s profound observations.

### Section 3.4: The Uniqueness of Theories in Physics

Regarding the second major issue, Wigner contemplates the uniqueness of physical theories, stating, “We cannot know whether a theory formulated in terms of mathematical concepts is uniquely appropriate.” He suggests that a theory explaining a particular phenomenon may not be the only possible explanation and that alternative theories—possibly using different variables or mathematical frameworks—could yield the same results.

For instance, consider the dissonance between quantum mechanics and the theory of relativity. These theories apply to vastly different realms—quantum mechanics to the microscopic and relativity to the macroscopic—and are built upon incompatible mathematical foundations. This divergence raises questions about whether both theories could be approximations of a more comprehensive framework yet to be discovered.

## References

Hamming, R. W. (1980). “The Unreasonable Effectiveness of Mathematics”. The American Mathematical Monthly. 87 (2): 81–90.

Wigner, E. P. (1960). “The unreasonable effectiveness of mathematics in the natural sciences. Richard Courant lecture in mathematical sciences delivered at New York University, May 11, 1959”. Communications on Pure and Applied Mathematics. 13: 114.

This article is part of a continuing series of essays on mathematical topics published in Cantor’s Paradise, a weekly Medium publication. Thank you for engaging with this exploration!