From Negative Numbers to a World War!

Steven Strogatz’s The Joy of x is a “guided tour” from “an adult perspective” to “math’s most compelling and far-reaching ideas”. Each chapter is short (“bite-size”), and the book can be read in any order.

 

Let’s look at one topic from the book: negative numbers. From early in life, most of us get the “use” of negative numbers. “Minus 1” meaning the basement is self-evident. Or it can mean debt (money owed). But there’s one aspect that doesn’t make intuitive sense to most people:

“The most unsettling thing is that a negative times a negative is a positive.”

And of course, as adults, we tend to ask “if these abstractions have any parallels in the real world”.

 

Strogatz gives a fascinating example. Say, you have 3 entities (people, companies, countries, whatever). A solid line designates the connected entities are friends, a dashed line conveys they are enemies. Take 2 such possible arrangements:

 

Social scientists refer to such triangles that denote relationships as “balanced” or “unbalanced”:

“In a balanced triangle, the sign of the product of any two sides, positive or negative, always agrees with the sign of the third.”

By this definition, the triangle on the left is “balanced”, the one on the right “unbalanced”.

 

Ok, you say, this just seems to be a way of representing things visually. Where’s the practical application in this? Aha, by “balanced”, it means the above relations can stay the way they are. Whereas in the “unbalanced” triangle above, Carol is in an uncomfortable position: while she likes both Alice and Bob, they’re likely to be cursing each other in front of Carol. To avoid this awkward situation, should she stay friends only with one of them? See why is it called “unbalanced”?

 

This may seem mildly interesting, but surely it doesn’t qualify as a real application?! And then Strogatz extends the idea in the context of the relationships over the years between the different powers of Europe leading up to World War I (GB = Britain, AH = Austria-Hungary, Fr = France, Ge = Germany, Ru = Russia, and It = Italy). 

All the arrangements until 1907 have at least one unbalanced triangle. Which makes them unstable, i.e., leading to rearrangement in the relations. Until we get to 1907, and the arrangement is finally fully balanced.

 

No, this isn’t a contrived example:

“Scholars have used these ideas to analyze the run-up to World War I.”

Of course, it needs to be taken only half-seriously. After all:

“It’s too simple to account for all the subtilities of geopolitical dynamics.”

 

And as it turned out, the final arrangement being “balanced” was not such a good thing after all because:

“Europe had split into two implacably opposed blocs – technically balanced, but on the brink of war.”

 

The title of this chapter in his book was perfect: “The Enemy of my Enemy”.