The power of words
“One of the properties of language is its ability to generate sentence that have never been heard before.” — Donald Barthelme, Not Knowing, The Essays and Interviews
Math has been called a language in several books and by many famous thinkers, mathematicians, and YouTubers. But people don’t talk about words in this world — math — because words are the currency of language arts and literature. And this distinction, this piece of tape that splits the bedroom into this half and that, means a lot. Many times, before we can even wipe our nose, we’ve been branded word people or number people.
I am going to make two insane arguments:
1) Words are more important than numbers in math
2) Words are the key to overcoming math fear and anxiety.
Insane argument #1:
Words are more important than numbers in math
I. The high-level reason
(TK — fingerpainting, currently)
The difference between meaningless signs and meaningful statements. The whole crux of math being built on proof. This was David Hilbert’s obsession too, and Cantor, and Frege, and Gödel. This self-consistency of mathematics was a really important thing. Maybe something about Turing.
Bertrand Russell realized something was wrong when he encountered the word “infinitesimal” at Cambridge.
II. The low-level reason
Here’s writer Annie Dillard: “When you write you lay out a line of words. The line of words is a miner’s pick, a woodcarver’s gouge, a surgeon’s probe. You wield it and it digs a path you follow.”
Definitions are sentences — lines of words meant to guide us through a thicket as we grope in the dark and follow. In of my favorite books, How I Wish I’d Taught Maths by Craig Barton, he talks about definitions and teaching:
I used to think that definitions and the subsequent explanation of that definition were the most important part of helping students understand a concept. Hence I would start with the definition, and then follow it up with with examples and practice. The problem was, my students never really seemed to understand or use the definitions all that much. (Barton 222)
We encounter definitions right away when we are learn math, but they are particularly wordy (since we’re not comfortable with all the symbols yet) when we are just starting out:
Vertical angles are two angles such that the sides of one angle are opposite rays to the sides of the other angle. (Jurgensen 2011)
We call a repeating wave of this sort a periodic wave. (Transnational College of LEX, 1995)
The area of a circle is exactly half the product of its radius and circumference. (Lockhart 2012)
A line that touches a sphere only once is called a tangent. (Lockhart 2012)
An ordered pair is a list (x, y) of two things x and y, enclosed in parenthesis and separated by a comma. (Hammack 2018)
In simple terms, a space curve is a set of connected points in the embedding space such that any totally connected subset of it can be twisted into a straight line segment without affecting the neighborhood of any point. (Sochi, 2017)
There are many nouns and adjectives in math that mean things in the real world too, like ring, field, integral, derivative, line, statement, connected, and plane.
All of Euclidean geometry is built on five axioms.
Book 1 starts with 23 definitions. Look at Heath’s explanations of translations.
Maybe a note on the art of translation; I don’t know.
Bertrand Russell, Gödel, the parallel postulate
Insane argument #2:
Where I discuss things like the thrill parallel lines in planes and how there are different sizes of infinity and how these are all high-level concepts you can talk about without a single drip of arithmetic.
Words are the key to overcoming math fear and anxiety.