Introduction to Mathematical Philosophy by Bertrand Russell

Like reading a glossary for another book

Bertrand Russell won the 1950 Nobel Prize in Literature, though he was a philosopher, mathematician, and public intellectual, not a literary writer. (I suppose any nonfiction, however, could be considered essays.) As near as I can tell, Russell explained his mathematical system of philosophy in three works: In The Principles of Mathematics, published in 1903, he spelled out the fundamentals of his system of thought. He then built upon that foundation to create his magnum opus, together with Alfred North Whitehead, entitled Principia Mathematica, which was published in three volumes from 1910 to 1913. Neither of those books is intended for a general audience. Presumably one would need a PhD in mathematics to understand them. Russell did write many books for general readers, however, and in 1919 he released Introduction to Mathematical Philosophy, a book intended for lay readers (like myself).

Most of us, when we do math, take for granted the fact that 1 + 1 = 2 or 2 x 2 = 4. As a philosopher, however, Russell takes nothing for granted. Like Rene Descartes or Baruch Spinoza, he starts with first principles, assuming nothing, and systematically builds his argument from the ground up. It’s not enough to accept that 1 + 1 = 2 , one must ask WHY 1 + 1 = 2. Indeed, what exactly is addition, or for that matter, what is a number? That is what this book is about. Russell essentially defines the basic terms and functions of mathematics from a foundation of pure logical reasoning. Entire chapters are devoted to defining terms like rational numbers, real numbers, cardinal numbers, or differentiating between finite and infinite series. Reading this Introduction is much like reading an appendix or glossary to some other book, and Russell even admits as much. Introduction to Mathematical Philosophy merely provides the basic Rosetta Stone for understanding more advanced Russell books, like The Principles of Mathematics or the Principia.

You may not need a PhD in mathematics to read this Introduction, but you might very well need a master’s degree. Even if you’re good with numbers, however, that may not help you here. Leaning towards the philosopher rather than the mathematician, Russell writes all the mathematical formulae in paragraph prose, often using confusing, serpentine syntax. In order to express these concepts in textual form, Russell has to give new mathematical definitions to everyday words, which requires you to learn and memorize this new complex vocabulary.

Around chapter 14 (out of 18), however, the book finally turns in the direction of what I expected when I chose to read it; that is, how we use all these mathematical foundations to make logical deductions. From this point onward, the book is really quite interesting and can actually be understood by non-mathematicians. I question, however, whether I really needed to know many of those definitions that Russell spent so much time on. Once you get past chapter 13, many of those terms aren’t even used, and the text reads more like common sense.

What’s the point of all this? I suspect that Russell is a modern version of Pythagoras. He thinks mathematics is not merely a part of reality; mathematics is reality. Or rather, the logical rules that govern mathematics are the same that govern the movements of atoms or the thoughts in our heads. Russell sees no separation between mathematics and logic; they both operate on the same principles, whether the propositions are expressed through number or word. I believe Russell would assert that all human thought is fundamentally mathematical. We collect empirical sensory data, group it together in a complicated system of set theory, and then assess and organize that data with logical principles to create knowledge.

I was lost for much of the first half of this book, but I did find the latter half rewarding. Overall, I don’t think this Introduction is very effective in accomplishing what Russell set out to do, which is to provide the general public with an on-ramp to his school of thought. To get a handle on this subject, you’d be better off reading a Logic 101 textbook for college undergrads.