The Unreasonable Effectiveness of Mathematics in the Natural Sciences

Introduction

• Reference to the title of a paper by Eugene Wigner
• Wigner opens the paper with an anecdote about a statistician and their friend discussing the Gaussian distribution

Key Concepts

Normal Distribution and Gaussian Distribution

• Central Limit Theorem connects to normal distribution
• Basic function: e to the negative x squared (e^-x^2)
• Requires area under bell curve = 1 for probability interpretations
• Pi shows up due to the area under this curve

The Role of Pi

• Step one: explain the area under e^-x^2
• Step two: connect e^-x^2 to special functions in statistics

Integration & Antiderivatives

• Integration used to find area under curves
• e^-x^2 lacks a straightforward antiderivative
• A clever method involves bumping the problem up to 3D, involving volumes under surfaces

Volumes and Higher Dimensions

• Use of cylindrical shells to understand volume under a 3D bell curve
• Volume under the surface calculated to be pi
• Slicing method to visualize the relationship between 3D and 2D curves

Radical Symmetry and Probability Distributions

• John Herschel’s work on 2D probability distributions
• Properties required: Independence of x and y coordinates, and radial symmetry
• Function f(x, y) dependent on distance from the origin
• Forces e^-r^2 shape for the function

Functional Equations

• Introduction of h(x) function for simplification
• Results in exponential function form: e^(some constant * x^2)
• Shows a deep connection between spatial/geometric properties and probability distributions

Central Limit Theorem & Gaussian Distribution

• Gaussian distribution's relation to central limit theorem
• Normal distribution arises from independent variables
• Connection between spatial properties and addition of variables in CLT

Conclusion

• Next step: deeper dive into central limit theorem and function properties
• Mention of higher-dimensional sphere volume derivation shared by a mathematician

References and Further Reading

• Encourage checking out linked explanations and sources

Acknowledgements

• Thanks to patrons and contributors for feedback and support