Foundations in mathematics seem to be something of a lost art. Why do people need to gain an understanding of mathematics if the tools they use do it all for them? It’s a bit of irony, though, that we are surrounded by technology that is built through the great use and respect for mathematics (or maths as some say).
I take the stance that having a foundation in mathematics is more critical than ever. If you understand the technology at a deeper level than the user interface, then you will be able to wield the technology better. Where some are afraid to try, you will be enabled to do more since you are more comfortable with the deeper levels.
Set theory is not just a branch of math but a foundation in the application of data. If you do not understand sets, you will not be able to understand data well. Everyone should get a base in set theory in high school education. If not, then this book should help.
The book I will discuss, as you might guess, builds a foundation in set theory. Here are the details:
Naïve Set Theory by Paul R Halmos
Dover Publications, Inc.
Mineola, New York
Dover Edition Published in 2017 (Originally published in 1960)
ISBN 9780486814872 | 0486814874
I found this book on Apple Book Store. Here is the link:
https://books.apple.com/us/book/naive-set-theory/id1230575929
*Note on title: It does use term “naïve” in the title, but I warn you not to think of this as something you can casually read. This, as well as other math studies, requires attention and daily work to fully ingest and internalize.
Here is the outline of the book:
- PREFACE
- 1 THE AXIOM OF EXTENSION
- 2 THE AXIOM OF SPECIFICATION
- 3 UNORDERED PAIRS
- 4 UNIONS AND INTERSECTIONS
- 5 COMPLEMENTS AND POWERS
- 6 ORDERED PAIRS
- 7 RELATIONS
- 8 FUNCTIONS
- 9 FAMILIES
- 10 INVERSES AND COMPOSITES
- 11 NUMBERS
- 12 THE PEANO AXIOMS
- 13 ARITHMETIC
- 14 ORDER
- 15 THE AXIOM OF CHOICE
- 16 ZORN’S LEMMA
- 17 WELL ORDERING
- 18 TRANSFINITE RECURSION
- 19 ORDINAL NUMBERS
- 20 SETS OF ORDINAL NUMBERS
- 21 ORDINAL ARITHMETIC
- 22 THE SCHRODER-BERNSTEIN THEOREM
- 23 COUNTABLE SETS
- 24 CARDINAL ARITHMETIC
- 25 CARDINAL NUMBERS
- INDEX
What I like about the layout of the book is that it is direct. The chapters are well organized and meant to build off each other (more on that later). Further, each chapter is not overdone and should be taken at the reader’s pace of understanding the content.
Beyond the Outline
If one is to dive deep into the world of data, then one must understand set theory. What are fields, records, tables, and schemas but various sets? If you are to master SQL or other data languages, then you better understand how set theory is the foundation.
As an example, there is a chapter on Unions and Intersections. It is critical to understand how a table joined to another table will form intersections. This effects the resultset that is returned from a multi-table query.
There are other examples that come to mind, such as relations and pairs. That is what draws me into recommending this book for data professionals. As I read through the chapters, I think about how the application of these theories is done in the data arena. Theory and practicality come together in real world applications, such as relational databases.
One warning is that some chapters might get a bit deep into words for simple concepts. This happens more in the later chapters. Still, keep with it because there is a final point in every chapter. Taking down some notes on the side will certainly help.
Even though you could skip around the book, be aware that each chapter builds upon the previous ones. What you learn early on is useful in the advanced chapters. You might get lost with some of the conventions and introduced symbols if you do not have the grounding of the early writing. Unless you are informed in the theory, take your time and start at the beginning.
There are also exercises in the book but without an answer key. The reader should give it a go despite that missing feature. At the very least, get some paper and write out a proof while using the book as a guide. In fact, take it as an opportunity for a personal research project.
What the book is not about
This book is theory and not about application. If you are looking for a how-to on understanding SQL and data structures, this is not for you. This is about setting a foundation that can be applied to applications such as SQL and data structures. In other words, it is up to the individual to take the theory and think about the real-world application. For the established data professional, this should come quickly if not later in the day as your brain processes the theory.
Final Thoughts
Mathematics is a topic that some might groan over but it cannot be denied that the world has seen great advances thanks to the study. The data professional must respect that fact. To take one to a higher level of understanding requires deeper skills in a world that is overflowing with data.
Understanding how data are represented in the real-world is a key skill. Alongside are the essential and magical skills of integration and transformation of raw data into usable sets. These skills do not come easily and require a solid foundation in many areas of mathematics.
Do not take the easy way out in knowledge. Spend the extra time and keep at it. These topics are worthwhile for study. In this case, Set Theory is one of those studies that will help in the long term for data professionals.
Thank you
References
[1] Paul R Halmos. Naïve Set Theory. Dover Publications, Inc. 2017.
Andy’s Neural Works 