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A Treatise on Universal Algebra

Alfred North Whitehead
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Library Collection
[Reviewed by
Tom Schulte
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Over a decade before the first edition of Principia Mathematica was published in three volumes between 1910 and 1913, A Treatise on Universal Algebra was published in 1898. It was also intended to be the first of a series of volumes, but the further volumes which were to cover quaternions, matrices and linear algebra were never published. This large volume that did come forth, which Cambridge University Press has reprinted, covers general principles focusing on Boolean algebra, symbolic logic, algebraic manifolds, and the exterior algebra (Grassmann algebra).

Descriptive geometry is highlighted as an application here, but the theoretical underpinnings of computer science also lie here, in the focus on Boolean algebra, although of course Whitehead could not see that at the close of the nineteenth century. Looking back on this work from the Information Age, we benefit from the state-of-the-art scanning computer technology used for this reprint. The result is very impressive: most of the book looks like one typeset in LaTeX, with only the very rare attenuated serif or withered font element assuring us that it was not. Oddly, there is a marked contrast on this front between Book I: “Principles of Algebraic Symbolism” and the rest. In Book I, there are many more of the scanning-born imperfections that are well-known to readers of Google e-books.

Book I is special for another reason. I feel it is one of the chief reasons to read this book today. While Whitehead’s complete vision for a “universal algebra” may never have been printed, to read him groping toward it is like reading Montaigne grappling with “the force of imagination.” Such sections as “Definition of a Calculus” should be required reading for serious students, for while the notation and terminology have changed over the decades, the marshalling of linkage, presentation, exposition, and erudition toward a holistic vision is enlightening.

Tom Schulte is a guide and traffic director on the unpaved paths of algebra for students at Oakland Community College.

Part I. Principles of Algebraic Symbolism: 1. On the nature of a calculus
2. Manifolds
3. Principles of universal algebra
Part II. The Algebra of Symbolic Logic: 1. The algebra of symbolic logic
2. The algebra of symbolic logic (continued)
3. Existential expressions
4. Application to logic
5. Propositional interpretation
Part III. Positional Manifolds: 1. Fundamental propositions
2. Straight lines and planes
3. Quadrics
4. Intensity
Part IV. Calculus of Extension: 1. Combinatorial multiplication
2. Regressive multiplication
3. Supplements
4. Descriptive geometry
5. Descriptive geometry of conics and cubics
6. Matrices
Part V. Extensive Manifolds of Three Dimensions: 1. Systems of forces
2. Groups of systems of forces
3. In variants of groups
4. Matrices and forces
Part VI. Theory of Metrics: 1. Theory of distance
2. Elliptic geometry
3. Extensive manifolds and elliptic geometry
4. Hyperbolic geometry
5. Hyperbolic geometry (continued)
6. Kinematics in three dimensions
7. Curves and surfaces
8. Transition to parabolic geometry
Part VII. The Calculus of Extension to Geometry: 1. Vectors
2. Vectors (continued)
3. Curves and surfaces
4. Pure vector formulae.