The Unreasonable Effectiveness of Mathematics in Natural Sciences

Introduction

• Title inspired by Eugene Wigner's paper.
• Anecdote about a statistician explaining Gaussian distribution and pi to a classmate.
• Highlights the disconnect between pure mathematics and its applications in real-world statistics.

Main Goals

1. Explain the appearance of pi in the normal distribution formula.
2. Explore why the function e^(-x^2) is significant in statistics.
3. Connect this discussion to the central limit theorem.

Key Concepts

• Normal Distribution (Gaussian Distribution):

  • Basic function is e^(-x^2).
  • The area under the curve must equal 1 for probability interpretation.
  • Pi arises from the area under the curve being the square root of pi.

• Integral Calculus:

  • Used to find the area under a curve through integration.
  • Specific challenge with e^(-x^2) as it lacks a standard antiderivative.

Classic Proof of Area Under Gaussian Curve

• Volume Approach:

  • Shift from 2D to 3D by considering the volume under a bell-shaped surface.
  • Use cylindrical shells to compute the volume based on circular symmetry.

• Integration Steps:

  • Find the volume of thin cylinders of height e^(-r^2) and circumference 2πr.
  • Integrate from 0 to infinity, leading to the conclusion that the volume = pi.

• Connecting Higher Dimensions:

  • Volume under the surface relates back to the 2D Gaussian curve.
  • Each slice parallel to the x-axis retains the same bell curve shape, leading to a relation with the area under the curve.

Significance of e^(-x^2) in Statistics

• John Herschel's Derivation:

  • Established properties of a probability distribution in 2D space leading to the Gaussian function.
  • Two properties: radially symmetric and independent coordinates.

• Functional Equation:

  • Derived that all functions satisfying Herschel's criteria must take the form of e^(-x^2).

Conclusion

• The connection between geometric interpretations and the Gaussian distribution shows why pi appears in the normal distribution formula.
• Future explorations: Further outline the connection between Gaussian distributions and the central limit theorem.

Additional Note

• Kevin Ega provided a method to derive the volume formulas for higher-dimensional spheres, a delightful mathematical exercise.