Everyday Math: Abstractions Are Good for Goodness' Sake

"The season of Santa Claus is upon us, whether you believe in him or not. Most children start off believing, if they're taught to, until at some point they realize the story is impossible. But even though no cheerful old man actually flies around giving gifts to the world in one night, there is obviously an idea that motivates people to give gifts in the holiday season: It's the mix of generosity, warmth and sharing that we call "the spirit of Christmas." It is more abstract than Santa's fluffy beard, red outfit and sleigh, but its effects are still tangible. As it happens, this is also how abstract mathematics works.

 

Mathematical objects are abstract concepts that capture ideas about the world. They come from finding analogies among different situations. The number two, for instance, is an abstraction that comes from looking at two apples, two bananas, two people, two chairs and so on, and finding what they have in common. Mathematicians then consider that analogy as an object, albeit an abstract one rather than a concrete one: the number two.

 

Math continues to build up like this, level by level, creating more and more advanced mathematical fields. The similarities among different numerical situations lead to equations involving x's and y's. Analogies among those equations lead to the field of algebraic geometry. In the 20th century, finding things in common among entire fields of math led to the creation of category theory, my own very abstract field of research, which formally studies mathematical analogies themselves.

 

People often ask if math is discovered or created. I think it is both: We discover math in the form of these analogies among situations, and we create ways of expressing and studying those situations. Doing abstract math feels like dreaming things up in our imagination, which is quite different from the decidedly un-dreamy experience many people have of math classes. But it opens us mathematicians up to a different criticism -- how can we just make those things up?

 

Philosophers have considered whether or not abstract mathematical concepts such as numbers are "real." We can't touch them or see them, so what status do they have as objects? Then again, there are plenty of real things we can't touch or see, such as love or hunger. Still, we can experience the effects of love and hunger, and so it is with abstract mathematics.

 

One vivid example is imaginary numbers. The clue is right there in the name "imaginary." Ordinary, or "real" numbers, describe real things in the world -- lengths and other measurable quantities. Imaginary numbers don't. The imaginary number "i" was dreamed up by mathematicians to be the square root of negative one. This can't be an ordinary number, because positive and negative numbers both square to positive numbers, not -1. Instead, mathematicians imagined a different sort of number, and duly called it imaginary.

 

This might seem like just a game played by academics with their heads in the clouds, but it has effects in the real world. Imaginary numbers turn out to be very helpful in physics, which is all about understanding how the real world functions. In fact, recent advances in quantum mechanics seem to indicate that imaginary numbers aren't just helpful but necessary. Quantum mechanics, the study of subatomic interactions, is at the root of most modern electronic devices; it is crucial to understanding the behavior of electrons in semiconductors, essential components of modern computers.

The dreams of pure mathematics end up having a profound effect on our daily lives. Abstract concepts and real life are not so far apart after all -- which is why both the spirit of Christmas and Santa Claus get credit for the delights of the holiday season." [1]

1. REVIEW --- Everyday Math: Abstractions Are Good for Goodness' Sake
Cheng, Eugenia. Wall Street Journal, Eastern edition; New York, N.Y. [New York, N.Y]. 11 Dec 2021: C.17.