One of the core principles of mathematics is the idea of abstraction, generalizing from various experiences to describe simplified models that enable rigorous reasoning. For example, if you look at a street map of your city, nothing there qualifies as a pure Euclidean triangle: all roads have thickness, varying slopes, squished raccoons, etc. But by reasoning about ideal triangles and lines, we can make powerful deductions about the distances between points that are very useful and accurate for practical purposes. However, there is a dark side to abstraction-- when used too much in your daily life, it can cause you to over-generalize and lead to issues like stereotyping and prejudice.
For example, 20 years ago I had a Scottish roommate named Lloyd. Lloyd was a great guy, but I could not understand a word he said, due to his outrageous Scottish accent. Eventually we started keeping a notepad in the room so he could write down anything important he needed to communicate. After a few months with him, I was on the verge of insanity. Now, whenever I'm being introduced to someone from Scotland, I inwardly cringe, bracing myself for a similar experience. In effect, I have abstracted Lloyd as the general Scotsman in my mind, impacting my further relationships and experiences with his countrymen. It hasn't been that much of an impact in my life, as most residents of Scotland have yet to discover the joys of Hillsboro, Oregon, but it's still a bad habit. Is there something simple I can do to try to cure myself of this way of thinking? One intriguing set of ideas comes from a 20th-century pop philosophy movement known as General Semantics.
General Semantics was first created by Polish count Alfred Korzybski in the 1930s, and detailed in a book called "Science and Sanity". This book describes a wide-ranging philosophy based on evaluating our total "semantic response" to reality, and learning to separate true reality, our observations of reality, and our language that describes the reality. By becoming conscious of our tendency to over-abstract, we can improve our own level of sanity, hence the book title "Science and Sanity". While serious philosophers and linguists generally don't consider Korzybski's ideas very deep, he attracted a devoted cult following, who believe that the General Semantics tools can significantly improve people's lives by reducing the errors that result from over-abstraction. This movement also led to the proposal for "E Prime", the variant English language without the verb "to be", which I described back in podcast 196. Amusingly, Korzybski was also a bit of a math geek: when his Institute for General Semantics in Chicago was assigned the address 1232 East 56th Street, he had the address changed to 1234, in order to create a nice numerical progression.
Among the key tools that General Semantics provides for fixing over-abstractions are the "extensional devices", new ways to think about the world that help you to correct your natural tendencies. Many of these involve attaching numbers to words. The most basic is "indexing", mentally assigning numbers to help emphasize the differences between similar objects. For example, I might think of my friend Lloyd as Scotsman-1. Then, if introduced to another person from Scotland, I can think of him as Scotsman-2, emphasizing that he is a completely different person from Scotsman-1 despite their common origin. If I go out with my new friend for a yummy Haggis dinner, I would think of the waiter as Scotsman-3, again recognizing his essential uniqueness and separating him from the other two. Through this assignment of numbers, I can avoid grouping them all into the single abstraction of Scotsmen, and help force myself to treat them as individuals.
Another important extensional device is called "Dating", similar to indexing but based on time. With this device, you attach dates to objects, indicating when you observed or experienced them. The Lloyd I remember should really be thought of as Lloyd-1993, since that's when I knew him, and I'm really only familiar with his characteristics at that time. If he emails me that he's coming to town, I should now think of him as Lloyd-2015, who may be a different person in many ways. Perhaps he has been working on his accent a bit, or maybe due to my 20+ years of engineering experience, my ears have gotten better at discerning words in unusual accents. I should not over-abstract and assume that his most notable characteristics at one time, and my perception of them, will be the same today as in the past. Like everything in the universe, he and I are constantly changing, and I can use this extensional device to remind myself of that.
There are a number of additional extensional devices in General Semantics, such as the use of Et Cetera, quotes, and hyphens to further qualify your abstracted language. These seem a bit more awkward to me, though some may prefer them. Overall, I think the general concepts behind Korzybski's extensional devices probably can serve as a useful tool, especially if I go to Scotland sometime, though perhaps they are not quite as profound as General Semantics fanatics like to think. Korzybski's movement still seems to be going strong, with active institutes in New York, Australia, and Europe that have a presence on the web and in social media, and a quarterly newsletter still in print since 1943. Naturally, I've grossly oversimplified many of the core ideas for this short podcast, but if this has served to whet your appetite, you can find many other details at the links in the show notes.
And this has been your math mutation for today.