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A History in Sum: 150 Years of Mathematics at Harvard 1825–1975

Steve Nadis and Shing-Tung Yau
Harvard University Press
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
P. N. Ruane
, on

In days of yore, many individuals battled against great odds trying to find time to work on mathematics — and sometimes in total isolation. George Green was a miller who left school at the age of nine. Hermann Grassmann was an overworked schoolteacher, Arthur Cayley was a lawyer and Albert Einstein spent seven years as a clerk in a patent agency. During this period of clerical slavery, Einstein gained his Ph.D. and published innovative papers on the ideas from which his fame emanates.

In modern times (as revealed in this book) it was not uncommon for academics to complain about being overburdened by the prospect of having to do 12 hours teaching per week — even though that may have been only for 24 weeks of the year. But that’s just one of many observations arising from this interesting history of mathematics at Harvard.

Perhaps unwittingly, this book complements two others that focus upon the history of mathematics at particular universities. These are Rouse-Ball’s History of the Study of Mathematics at Cambridge and Constance Reid’s double biography Hilbert — Courant, which is almost a history of mathematics at the University of Götingen. Therefore, the three books provide insight into the growth of mathematics in England (up to the late 19th century), Germany (late 19th century and early 20th century), and the USA (1825–1975).

The book includes photographs of 23 mathematicians who are central to the history of mathematics at Harvard. In chronological order, these include Benjamin Peirce, W. F. Osgood, Maxime Bôcher, George Birkhoff, Marston Morse, Hassler Whitney, Saunders Mac Lane, Lars Ahlfors, Oscar Zariski, Raoul Bott and John Tate. There is also very good description of the work of very many other mathematicians, many of whom fled from Germany in the 1930s, which constitutes an historical thread linking this book to the Hilbert biography.

I found this history of mathematics at Harvard to be engaging reading for three basic reasons. Firstly, it not only provides insight into mathematical achievements of all concerned, but also presents them as very different personalities. Hassler Whitney, for example, is said to have sought solitude when dwelling upon research into algebraic topology (appropriately, his main recreational activity was mountaineering). In contrast, Oscar Zariski, who features most prominently in this book, was much more gregarious, and he collaborated widely with respect to his foundational work on algebraic geometry. When his work on algebraic geometry was moving into increasingly abstract directions, Zariski was once approached by a student seeking career advice (physics? engineering or mathematics?). Zariski swiftly replied ‘Choose mathematics; it’s more useless’.

Some mathematicians mentioned by Nadis and Yau seemed to have resented having to teach. To Mac Lane and Birkhoff, teaching algebra was a chore. To others, such as Raoul Bott, teaching was immensely enjoyable — and it was no seeming impediment to his work on the application of Morse theory to the study of the homotopy groups of Lie groups. However, it may be that those who dislike teaching aren’t very good at it anyway. And not all great mathematicians have been recognised as able lecturers.

The book starts from the mid-19th century, when mathematics came into being as an area of study at Harvard University. It reveals a myriad of personalities who have contributed to its prestige as a centre of mathematical research. It portrays life at Harvard from around 1825 to times of the great depression and the years following the 2nd World War. More importantly, it provides meaningful insight into all sorts of mathematical topics about which I previously knew nothing (geometric invariant theory, Stiefel-Whitney classes, Eilenberg-MacLane spaces etc).

Peter Ruane spent most of his working life in primary and secondary teacher education.

  • Preface
  • Prologue: The Early Days—A “Colledge” Riseth in the Cowyards
  • 1. Benjamin Peirce and the Science of “Necessary Conclusions”
  • 2. Osgood, Bôcher, and the Great Awakening in American Mathematics
  • 3. The Dynamical Presence of George David Birkhoff
  • 4. Analysis and Algebra Meet Topology: Marston Morse, Hassler Whitney, and Saunders Mac Lane
  • 5. Analysis Most Complex: Lars Ahlfors Gives Function Theory a Geometric Spin
  • 6. The War and Its Aftermath: Andrew Gleason, George Mackey, and an Assignation in Hilbert Space
  • 7. The Europeans: Oscar Zariski, Richard Brauer, and Raoul Bott
  • Epilogue: Numbers and Beyond
  • Notes
  • Index