An Introduction to Mathematics

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Analysis of An Introduction to Mathematics, a 1911 treatise on the discipline.

Alfred North Whitehead’s An Introduction to Mathematics takes us on a journey of broad strokes, one that traverses the history, notation, and application of mathematical fields. An Introduction to Mathematics is a practical book on theory, great for enticing the study of math at the university level. While covering introductory material, it introduces elegant mathematical insights that will intrigue the reader. It illustrates the principles of generality, form, variable, and abstraction as the elements of mathematics.

After touching on the abstract nature of math, we take a look at variables. The work explores essential concepts of any and some. They are the ideas introduced in algebra which allow for concepts like infinity. Applications of variables, such as Boyle’s law, are brought to light. Illuminations of how variables work in diverse fields present their usefulness. From Archimedes to Newton, this concept proves itself intrinsic.

From there, we approach vectors and their use in mechanics. All along, talking about the originators of these ideas. The pattern of history, application, and theory continues. The history of mechanics, under the title of dynamics tours from the greeks to Kopernicus and beyond. We learn the concept of vector operations and thereby adding velocities together using the parallelogram law. The explanations are simple yet powerful.

After this digression into applications, the author conveys us back to the realm of pure mathematics. The evolution, and importance, of mathematical notation as a function of abstraction becomes apparent. The discussion of notion naturally leads back to generalizations about real numbers. In contrast to real numbers, complex or imaginary numbers are next. To illustrate complex numbers, we step into the realm of cartesian geometry. An eminent explanation of co-ordinate geometry shows that it serves as a bridge between the quantification of algebra and the pure spacial reasoning of euclidean geometry. As an aside, it seems non-euclidean geometries of curved space were unknown in 1911. Thus, neither were the theories of general and special relativity. Hence some outdated ideas are found, such as molecules in ether.

Traveling on the continuum, we come to conic sections. We explore the properties of various curvilinear shapes. The importance of these shapes is made apparent by touching, once again, upon orbital mechanics. Ironically the Greeks, who originally studied conics, had no application for them. The ancient Greek love of pure theory came to fruition in concrete forms many millennia later; this is a demonstration of essential wonder.

We explore the nature of functions in some depth. The ideas of discrete and continuous functions lead to other concepts. Interval, the approximation to a number within a standard, and the neighborhood of a number approach limits. The book does not visit upon the use of multivariate functions. It also does not touch upon the concept of functions in computing. Although the terminology has changed, the elegance of the elements proves itself in future applicability.

Periodicity in nature, the enormous importance of these periods in how we develop discrete measurements of time, and how they exist in mathematics is the next topic. Mathematical methods used to analyze physical phenomena bring us to the discussion of trigonometry. As discussed in Plato’s Timaeus, the whole universe is composed of triangles. We thus take a look at the relationship of these unique rectilinear figures and circles, a special curvilinear conic section, through the lens of the unit circle. The natural occurrence and application of the sine and cosine functions are apparent here.

Earlier on, the treatment of discrete and continuous function approached the idea of limits. Now series are introduced, and limits formally defined in the context of series. Limits and series lead to a discussion on the history and theory of differential calculus. This discussion is complete with a demonstration of how to take the first derivative of a function. Saving the best for last, Euclidean geometry and its nature as the science of dimensional order is next. Lastly, the world of measurement visits the continuum in a discussion of quantity.

At shy of 200 pages, this book covers an immense breadth of ideas. For the most part, it elucidates them readily; I had multiple “Aha” moments during reading. All considered this was $4, I picked up a used copy on (though you can also find it on new on Amazon), and about 20 hours well spent. Rereading this book, after a more in-depth study of the field, would lead to making new connections with what is said. In that line of thought; the bibliography is unique, in that it lays out a plan for self-study through other works. This book may inspire a hunger for mathematical knowledge, read with caution.