Mathematics: The Loss of CertaintySep 30, 2012
This Summer, I finished “Mathematics: The Loss of Certainty”, an amazing book by Morris Kline. It’s a book that tells the history of mathematics, but with a particular theme. As the author boldly states in the introduction, it’s about the “rise and fall” of mathematics. While no one would argue that the utility of mathematics has in any way diminished, the “downfall” that the author speaks of is about the rise of doubts in the status of mathematics as our most trusted knowledge. It’s about whether the universe has a mathematical design or if “mathematization”, in the words of the mathematician Hermann Weyl, “may well be a creative activity of man, like language or music”.
This “loss of certainty” in mathematics is perhaps best epitomized Godel’s theorem of incompleteness, which poked holes in our axiomatic-deductive system that was thought to guarantee the “truth” of a proposition. Godel proved, surprisingly, that mathematical propositions could be derived that are neither provable nor disprovable by the laws of logic in an axiomatic system. The dream of building mathematics on an all encompassing set of axioms, a dream that so tantalized mathematicians in the late 19th and early 20th century, was shattered.
This book also has a fascinating section on calculus. The use of infinite, and infinitesimal values in the logical foundation of calculus stretched the limits of legitimate mathematical concepts, troubled mathematicians, and presaged Georg Cantor’s investigations into the paradoxes of infinity.
It’s fascinating to read how simple ideas like negative numbers were once controversial. Many European mathematicians up to as late as the 16th and 17th century did not accept negative numbers to be legitimate numbers. At the time, the laws of Euclidean geometry were thought to be inherent in the universe and the basis of all mathematics. Negative numbers were “nonsensical” because there were no geometrical basis for them.
Ultimately, these controversial ideas are accepted because they work, and it is our preconceived notions that have to bend to incorporate them. If you want to understand the state of modern mathematics, this book is immensely valuable (and very readable). It ends with mathematics in an isolated and fragmented state. It's isolated due to the steep rise in abstraction as mathematics pulled away from the sciences in the 20th century. It's fragmented because questions about the foundations of mathematics has resulted in three schools of thought: formalism, intuitionism, and logicism.
The greatness of this book is it’s ability to evoke the thrill of a paradigm changing discovery. It’s filled with quotes from mathematicians as they debate and grapple with new ideas (often ones that they came up with themselves). Some of these ideas might be mind boggingly eery, perhaps enough to drive someone insane (see Godel and Cantor), but I guess that’s the nature of reality.