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# New and Noteworthy Titles on our Bookshelf

November 2024

At the 2024 Joint Mathematics Meetings, I attended a lecture by Terence Tao entitled “Machine Assisted Proof.” I learned about developments in machine assistance in mathematics, including interactive theorem provers. After reading *The Meaning of Proofs*, I wonder how computer assistance in proof writing will change or advance the stories that proofs tell. In this book, Lolli explains how proofs are stories, and if we consider the historical developments of these stories, we can see they have a similar history and form to literature and poetry. In fact, there is an entire chapter devoted to the proof structures of Euclid and the poetic and narrative structures found within.

Lolli’s book is not a set of instructions for writing proofs nor does it offer a full definition of “proof.” Instead, he convinces readers that proofs are part of a rich history of storytelling, including many of the same elements as fairy tales. There is a tradition in both to define the parameters of the world and then see the story “through to its final consequences by following its internal logic.” Mathematicians may feel one way our discipline is unique is that we value writing different proofs of the same statement; however, this is not unlike some of the universal stories and themes in literature that appear again and again in different settings with different characters.

The author is in good company in arguing the poetic side of mathematics; it is clear from quotes that Plato, Cantor, Hilbert, and many others agree. Albert Einstein has been credited with saying, “Pure mathematics is, in its own way, the poetry of logical ideas.” While the text claims to be written for anyone, I found that familiarity with the history of mathematics and some of its well-known players is helpful. The appendix includes more technical mathematics for interested readers.

Data science is a major player when discussing the use of mathematics in industry, business, government, and other applied settings. The ability to analyze data and understand the outcomes in a variety of fields remains a high priority for many industries. Whether you are preparing students for such jobs, considering a move to industry, or wanting to understand where abstract mathematics is used in real-world settings, *The Shape of Data* provides an interesting dive into the topic. Farrelly and Gaba walk through geometric and topological algorithms in nontechnical and intuitive ways. As this is not a textbook, there are almost no formal definitions, proofs, or equations; rather, the authors talk through the concepts hoping to make the tools accessible to those without much technical training. In addition, the reader is supplied with scripts (typically short pieces of code that accomplish a single task) written in R to practice implementing the concepts. The authors also provide a downloadable repository that includes code in R and, when possible, Python.

The book begins with an introduction to geometry in machine learning. It then moves into geometry-based algorithms and network analysis. The remaining chapters include metric geometry, geometry- and topology-based algorithms, and examples of implementations in natural language processing, distributed computing, and quantum computing. The mathematics community might like this book as it could provide intuition for yourself or a student who already uses some of these packages in R or Python, especially if the student is interested in applications of their geometric or topological knowledge. I appreciate that this book offers clarity on and intuition for many frequently applied algorithms in data science.