Math in Books: The Universe in Zero Words

I recently had the pleasure of reading The Universe in Zero Words: The Story of Mathematics as Told through Equations.  Written by Dr. Dana Mackenzie, the book frames mathematical history in terms of some of the most important equations ever discovered.  While writing about equations for a general audience can be a dangerous game, Dr. Mackenzie tackles mathematical notation head on.  If the sight of an equation causes a chill to run down your spine, fear not; the book eases you in with the very simplest of equations (we’re talking 1 + 1 = 2 here) and guides you gently through a history of mathematics, from antiquity to present day.

Of course, as you move closer to the present, the equations get a little more sophisticated.  Even so, Dr. Mackenzie does his best to ground the equations to something relatable to a wide audience (and by and large, he’s quite successful).  For instance, he uses whales as a way to talk about non-Euclidean geometry: you can read more about this example here, and can download the relevant chapter, too.  While the book won’t turn you into a mathematical genius, it will teach you the history surrounding some of the subject’s most important equations, and will give you a reasonable idea as to what the equations are communicating.  Words can’t always do justice to the economical beauty of an equation (after all, that’s what makes equations so appealing in the first place!), but if you suffer from math anxiety, a book like this may help to alleviate some of your most severe symptoms.

Once I finished the book, Dr. Mackenzie was kind enough to let me pick his brain a little bit.  Here’s a little Q and A to give you a better impression of what the book is about and the thought that went into writing it.  I’ve added some links to more math where appropriate.  Thanks to Dr. Mackenzie for taking the time to chat!

Are there any equations you wanted to include but ended up having to strike?

Absolutely. The first draft of the table of contents had something like 48 equations. On the second draft I pared it down to 30, and as you might expect, the last six cuts were the hardest. Two areas that seem underrepresented in my book are statistics and mathematical biology.

From statistics, I would have really liked to include the formula for the normal distribution and Bayes’ Theorem. However, the 1700s and 1800s, which those formulas came from, were already extremely well-represented, and I don’t know what I would cut to make room for them. Also, Bayes’ Theorem poses a bit of a problem — discovered in the 1700s, but really only appreciated in the second half of the twentieth century. So which century do you put it in?

As for mathematical biology, the problem I faced was a lack of equations that are really central to the entire subject. In mathematical physics, everybody agrees on the importance of Newton’s laws or Maxwell’s equations. But biology is more fragmented, and it seems to deal less in universal laws and more in reasonable models or rough approximations. The closest thing to a universal law in biology is natural selection, but Darwin did not express that in mathematical form.

Having said that, though, I think that a very strong candidate for my book would have been the exponential law of population growth when resources are unlimited, and perhaps the logistic law when resources are limited. Both of these equations have such huge ramifications. It was a tough call leaving them out — the law of exponential growth was on my top 30 list — but again it was a case where there was such strong competition for the six equations from the 19th century.

Do you have any personal favorites among the equations included?

Well, of course to some extent they are all favorites! But yes, there are some that are more closely related to the kind of mathematics I did when I was still actively involved in research. Hamilton’s quaternions are a special favorite, because they came up twice in a VERY unexpected way in my research on minimal surfaces (surfaces of least area). Once may be an accident, but twice really gets you thinking. The quaternions and their cousin, the octonions, are closely related to all sorts of “exceptional” phenomena in algebra and geometry, even including the dimension of space.

In the book I didn’t even mention the connection of quaternions to my own research, because there are too many other interesting things about them. They are closely related to spinors, and as I say in my book, they are the absolute best way to mathematically represent anything that spins or rotates. That includes electrons and really all subatomic particles (except the Higgs particle!), and so it created a direct link between the chapter on quaternions and the chapter on Dirac’s equation for the electron. This makes a nice segue also to your next question…

One of the things I enjoyed about your list of equations is that they are not treated in isolation, but instead weave together a nice narrative on the history of mathematics.  Did working within this narrative framework present any challenges when trying to decide on which equations to include?

This is one of the reasons for the beauty and the incredible power of mathematics. It’s almost impossible to start writing about a sufficiently deep formula or theorem and NOT start finding connections to all of the rest of mathematics. I did not have to go looking for these connections; I also didn’t really plan on them when I worked out the table of contents. They just showed up by themselves without any effort on my part. All I had to do was point them out.

Some of the connections are even meta-mathematical. For instance, when I was working on part 2 of the book, which covers roughly the years from 1500 to 1800, I couldn’t help noticing the conflict between publicity and secrecy that kept playing itself out in different ways in different times. Some mathematicians tried very hard to keep their ideas proprietary. Others understood that the best way to advance the subject and to advance your own reputation is to share your discoveries freely. I think it was Euler, who was perhaps the most prolific mathematician of all time, who really led by example and turned mathematical scholarship decisively toward the mode of sharing. Nevertheless, this is a battle that continues to be fought in every century and every generation. Now we have mathematics done for the military or intelligence agencies, and we have mathematics done for private companies (e.g., every investment firm has its own version of the Black-Scholes equation). Will these discoveries be shared, or secreted away? This isn’t a big theme of my book, but it’s something I noticed and pointed out where appropriate.

As you travel from the past to the present, the equations necessarily get more complicated.  Did you find your approach to writing about an equation vary depending on the mathematical sophistication required to be familiar with the equation?  Was it harder to write about Black-Scholes compared to something like the Pythagorean Theorem?

Yes, this was definitely part of the challenge of writing a mathematics book. One thing I’ve noticed about a lot of what I call “Big Honking Histories” of mathematics is that they stop around the early twentieth century. Modern mathematics seems to be just a bridge too far for these books. I was determined to make this book different and cover mathematical developments right up to the present. I did not want to give the message that math was all completed in the 1800s, or give the message that laymen should not even try to understand anything after 1900. That would be a terrible message. Every other science has its popularizers who are trying to convey the science of TODAY, and mathematics needs to do the same.

But it’s hard. In some cases the material was hard even for me to understand, let alone for the reader. Nevertheless, in every case I think the formula must express some idea that is simple and far-reaching; otherwise it could not be a great formula. In the case of Black-Scholes, the heart of the matter is that the movement of stock prices is a diffusion process, just like the movement of molecules in the atmosphere. If I can uncover that basic idea, then I can write something that is accessible to all readers, even if they don’t understand the mechanics of how you turn the central idea into an equation.

Even so, I might not have been completely successful. I have seen one or two reviews that say the last part of the book is more challenging than the first part, where I was writing about formulas like 1+1 = 2 and approximations to pi. But even if it does require a little bit more effort from the reader to read the last part, I think that the effort is worth making. Dirac’s equation, for instance, is too important for us not to at least try to understand what it says. I may not succeed in making it completely understandable, but I’ll get you closer and I will at least put it on your intellectual radar screen.

If you’d like to know more, pick up a copy of Mackenzie’s book!  I think you’ll enjoy it.  If you’re feeling too lazy to navigate away from this page, you can order the book directly from the widget below, or from the “Shop!” tab up at the top.  Happy reading!

 

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