Why are there so many symbols in math?

“Mathematics is the language of the universe.” — every science popularizer ever

I am a mathematician. In fact, I have a PhD in mathematics. But, I am terrible at arithmetic. Confused? I certainly would have been if a self-proclaimed mathematician told me that 15 years ago.

The answer to this riddle is simple: math is not numbers. Whenever a glimpse of my research is seen by nearly anyone but another mathematician, they ask where the numbers are. It’s just a bunch of gibberish symbols, they say.

IMG_20171212_090428
The whiteboard in my office on 12 December 2017.

Well, they are right. Without speaking the language, it is just gibberish. But why—why all these symbols?

The symbols are necessary because communicating the ideas requires it. A simple analogy is common human language.

Mandarin Chinese, for example, has many more like-sounding syllables than English. This has led to a great number of visual puns, which have become a large part of Chinese culture. For example, the phrase 福到了(“fortune has arrived”) sounds the same as 福倒了(“fortune is upside down”). Often you will see the character 福 (“fortune”, fú, which you pronounce as ‘foo’ with an ascending pitch) upside down. While, like most puns, this has no literal meaning, it denotes fortune has arrived.

Fudao
Fudao, CC BY-SA 3.0, https://en.wikipedia.org/w/index.php?curid=30725479

Not laughing? OK, well, not even English jokes are funny when they have to be explained, but you get the idea. This pun just doesn’t translate to English. (Amusingly, there is also no simple common word for pun in Chinese.)

The point here is that upside down 福, with its intended emotional response, is not something you can even convey in English. The same is true in mathematics. Ideas can be explained in long-winded and confusing English sentences, but it is much easier if symbols are used.

And, there really is a sense in which the symbols are necessary. Much like the example of 福, most mathematicians use symbols in a way that is just impossible to translate to English, or any other language, without losing most of the meaning.

Here is a small example. In the picture above you will see p(x|θ). First, why θ? (theta, the eighth letter of the Greek alphabet, by the way). That’s just convention—mathematicians love Greek letters. So, you could replace all the θ’s by another symbol and the meaning wouldn’t change. It’s like the difference between writing Chinese using characters or pinyin: 拼音 = pīnyīn.

You might think that it is weird to mix symbols, such as Roman and Greek, but it now very common in many languages, particularly in online conversations. For example, Chinese write 三Q to mean “thank you”, because 三 is 3 and, in English, 3Q sounds like ‘thank you”. In English, and probably all languages now, emojis are mixed with the usual characters to great effect. You could easily write, “Have a nice day. By the way, my mood is happy and I am trying to convey warmth while saying this.” But, “Have a nice day :)” is much easier, and actually better at conveying the message.

OK, so we are cool with Greek letters now, how about  p(x|θ)? That turns out to be easy to translate—it means “the probability of x given θ.” Unfortunately, much like any statement, context is everything. In this case, not even a mathematician could tell you exactly what p(x|θ) means since they have not been told what x or θ means. It like saying “Bob went to place to get thing that she asked for.” An English speaker recognises this as a grammatically correct sentence, but who is “she”, what is the “thing”, and what is the “place”? No one can know without context.

What the English speaker knows is that (probably) a man, named Bob, went to store to purchase something for a woman, whose name we don’t know. The amazing thing is that many more sentences could follow this and an English speaker could easily understand without the context. Have you ever read or listened to a story in which the characters are never named or described? You probably filled in your own context to make the story understandable for you. Maybe that invented context is fluid and changes as you hear more of the story.

The important point is that such actions are not taught. They come from experience—from being immersed in the language and a culture built from it. The same is true in mathematics. A mathematician with experience in probability theory could follow most of what is written on that whiteboard, or at least get the gist of it, without knowing the context. This isn’t something innate or magical—it’s just experience.

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