## Notes on “A History of Mathematics”

### Sunday, 24th April, 2016

**Book details**

A History of Mathematics (3rd ed.)

Uta C. Merzbach & Carl B. Boyer

1968, 1989, 1991, 2011

Wiley

**Summary**

There is a lot of maths in this book. If you don’t like maths, you probably won’t like this book. This book is definitely aimed at the lay reader however (or perhaps the beginning maths student), so concepts are explained, worked examples are given, some are even fun to try out for yourself (e.g. in arithmetic and geometry). Having said *that* (negation upon negation), the bulk of this book was written in days when “popular exposition” and “accessibility” were taken far more earnestly than they are today: the reader is not pandered to, and there are no silly jokes or narcissistic digressions.

The first nineteen chapters are geographical and chronological. Chapter 19 is about Gauss (1777-1855) and his immediate influence. After the Gauss chapter there are three topic surveys on geometry, algebra, and analysis, and a chapter on “Twentieth-Century Legacies”. Perhaps the authors as historians made an implicit judgement that Gauss separates past from present. However, these later chapters are still “historical” in the sense that they are about the development (and continuing development) of the discipline. There is a brief final chapter sampling some “Recent Trends”.

A book this size (600 pages before bibliographies and index) doesn’t need to present a single linear “path of progress” and full acknowledgement is given to dead-ends, decadence, re-appearances, simultaneous discoveries, etc. It’s a good “history”.

More than anything else I got an idea of the gradualness and lumpiness of the development of the discipline. For example, the piecemeal move from natural language to symbols for ideas like “the thing” (i.e., “x”), arithmetic operations, exponents, etc. Famously “zero” appeared long after all the other numbers, but using the same base for fractions as for integers was not settled for a long time. The equals sign first appeared in print in 1557.

There is a constant interaction, from the earliest times, between what we might call “applied” and “pure” approaches to (or even “conceptions of”) mathematics. From earliest traces (calculations for temple design, rituals, games, patterns), via figures like Archimedes and Euclid, through to figures like Newton and Gauss who were as involved in practical projects as they were in purely mathematical exploration. The institutionalisation of “pure” and “applied” mathematics in academe and schooling isn’t touched on, but it is clear that it is a very very recent development.

Pedagogy *is* a strong theme in the book. Mathematics is presented as a discipline in which the elite are centrally concerned with defining, presenting, and renewing their discipline. There’s Euclid’s Elements, and several similar projects, but the theme really takes off in the chapter on the French revolution (Chapter 18, “Pre- to Postrevolutionary France”). An interesting 20th century example is Nicolas Bourbaki, a “polycephalic” mathematician producing mathematics textbooks since the 1930s.

I am not a fan of revisionist histories, but it is miserable that only five women were mentioned in the whole book: Hypatia, Sofia Kovalevskaya, Sophie Germain, Mary Winston Newson, Emmy Noether (so that’s no women at all between 415 and 1850).

**Next steps**

On the mathematical side, I am reminded of my interest in Riemann’s non-Euclidian geometry (for Schrödinger‘s wave mechanics). I don’t think reading this history has been a mathematical “preparation” but Schrödinger (“Space-Time Structure”, 1950) or Weyl (“Space-Time-Matter”, 1922) might be a nice next read.

Philosophical or foundational issues were not touched on much, apart from as occasional asides, but there is plenty of historical grist for such considerations. Two introductions to the philosophy of maths I have my eye on are “Thinking About Mathematics” (Stewart Shapiro, 2001) and “Introducing Philosophy of Mathematics” (Michele Friend, 2007).