Solutions

Given two objects, add another two identical objects to them, and the result will be four objects. That's not much harder to say or to grasp than "2+2=4," but imagine trying to express a quadratic equation without being able to use mathematical symbols. My junior high report card shows I had enough trouble trying to do it with the symbols; keeping the different unknowns raised to different powers straight without them has just caused a serious panic attack amongst my neurons.

But math didn't come pre-equipped with symbols. They had to be developed. For that matter, so did numbers. Our remotest ancestors who wrote things down probably just made marks in the amount of whatever number they wanted to represent. We had to develop the idea of writing "5" or some other symbol to represent what we had been expressing with something like |||||.

Joseph Mazur's Enlightening Symbols is a fun romp through the development of these symbols and ideas through history as we gradually collapse complicated ideas into simple symbols. He begins with the development of numerals, including the idea of the zero, and continues with how we came to possess plus signs, equals, minuses, square roots, exponents and so on. Obviously the earlier stages are fuzzier, as they happened much longer ago and are represented in few still existing records.

But as we enter the Renaissance, we see different mathematicians develop individual pieces of the puzzle -- sometimes two versions of the same piece, and Mazur quickly sketches how the eventual winner came to dominate. In some cases, new symbols are probably still appearing as math addresses more and more complex areas and requires new ways to talk about them.

Mazur mostly leaves out the truly head-bonking stuff as he takes his quick trip through math history and writes about his subject with a light and fun tone. He includes enough examples of math statements made without symbols to get his point across and sometimes feels a bit repetitive in doing so. But overall Enlightening Symbols is an excellent look at how essential the development of what we call math was to the advancement of society and technology, even of areas that seem to have little to do with math directly.

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An hoary old philosophical conundrum asks if God could create a rock so heavy even he couldn't lift it. Mathematicians never have to ask if they could come up with problems so tough to solve that they could never figure them out, because they work with and around such problems all the time. Ian Stewart lays out several of them in Visions of Infinity; some were solved after much work and some remained unsolved when he wrote the book in 2013. Some might never be solved.

The way "solve" is being used here is slightly different than the way it might be used in other circumstances. Solving a math problem means finding what happens to the numbers after they've been processed according to the rules represented by the symbols accompanying them: 2+2 "solved" is 4. Solving an equation with a variable in it means processing those numbers in a way that will transform an unknown variable into a known number: Solving for x in "x+2=4" means finding out that x=2.

But solving the problems that Stewart talks about means demonstrating that certain math statements with nothing but unknown variables will always be true, no matter what numbers are plugged into them. Or that they won't be, by finding a set of numbers for which the equation will be false.

Visions is a brief look at several such problems and the story either of how they came to be and why they are still mysterious or how they were solved. It is not a quick read; some of the problems are interconnected and Stewart has a habit of dragging concepts from earlier chapters up without much of a signal or refresher of what they might entail. Some of his explanations of where the equations came from are as head-scratching as the equations themselves and furnish a reader with some seriously dense slogging.

Even though some of the math Stewart talks about may have even less of a "real world" application than algebra does in the eyes of a middle-schooler, he argues that it's still very important. Attempts to solve several of the problems in Visions led to many other mathematical breakthroughs and even failures often helped bring about a clearer understanding of the way the world works. And even if they did not, exploring math's outer reaches is no less a voyage of discovery than those taken the ancients who first ventured out of sight of land. Should human beings who seek to satisfy their curiosity about the physical world stop just because the frontiers are in the minds of the explorers? Stewart's answer  in the different chapters of Visions may be complicated and take a long time to understand, but it boils down to, "No, they shouldn't," which sounded right to me before I read his book and still does afterwards.