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A Readable Introduction to Real Mathematics

Daniel Rosenthal, David Rosentahl, and Peter Rosenthal
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
[Reviewed by
Meghan De Witt
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Rosenthal, Rosenthal, and Rosenthal have put together a book they intend as a gentle introduction to theoretical mathematics.  The book is not geared towards a particular class commonly taught in the undergraduate curriculum but claims to be written for students interested in learning theoretical mathematics on their own.  The authors do provide a list of classes they feel the book would be suitable for, but without regard to what the students in these classes would need or be able to handle.  For instance, they recommend the book for a non-mathematics majors to get an appreciation of mathematics (I have found that non-STEM majors are more often confused by theoretical mathematics than intrigued by it) or for prospective teachers (without covering most of the required topics that math classes for preservice teachers are required to cover by their accrediting bodies).  But despite a lack of audience, the book itself is fairly well-written and covers a wide range of topics while providing a good background on proofs and theoretical mathematics. 
The topics chosen for inclusion are similar to those found in other introduction to mathematics texts, though with a decided bend towards topics in logic and geometry.  Indeed, the authors delve into cardinalities further than any non-logic based undergraduate math class I have seen.  Interestingly, they do not provide the typical set theory background usually present in such a text, though they do make use of much of it and assume this knowledge in the reader. 
There is also a strong focus on geometric topics.  The authors cover basics of Euclidean geometry, including constructability, and then delve into topics of higher and infinite-dimensional spaces that again seem beyond the scope of their intended audience, despite being interesting and fairly well presented.  
The authors’ stated criteria for deciding what to include in the book is that “the mathematics is beautiful, it is useful in many mathematical contexts, and it is accessible without much mathematical background.” (Page x) While I generally agree with their choices based on the first two points, much of the mathematics chosen is fairly advanced and requires much more preparation beyond high school algebra and trigonometry (which is all the authors claim is required) and what is provided in the text itself. While there undoubtedly exist students who could read and understood this text with that level of background, they are certainly in the minority. 
Further, while the choice of what to include based in mathematical beauty is sound, the authors could also have used that criteria for their choice of what proofs to display for given results. There are several results (the resolution of the three problems of antiquity and the proof of the Fundamental Theorem of Arithmetic being examples) where the proof is needlessly confusing, wordy, and technical. It is not just the statement of mathematical results that can be beautiful, but the justification as well. 
Generally, the writing style is understandable and pitched to the correct level of audience. The exception being the authors’ tendency to include many mathematical expressions in-line instead of centered on their own line, thus making the paragraphs not only large but difficult to follow in many cases. 
In summary, the book was quite an enjoyable read, though I am skeptical of its suitability to the intended audience and purpose. It is, in fact, very readable (other than the earlier mentioned habit of including too many mathematical expressions in the body of the paragraphs) and covers a wide and interesting set of topics. It would undoubtedly help students just entering the world of theoretical mathematics, though perhaps after more advanced preparatory material than just high school algebra and trigonometry. It would be a good text for an introduction to proofs class, assuming some other supplementary material, particularly on the basics of set theory and structure of proofs, was included.


About the Reviewer: Meghan De Witt ( is a professor at St. Thomas Aquinas College whose primary interests are group theory, number theory, and mathematical outreach programs.