When I first started studying mathematics seriously in 2001, I immediately encountered negative numbers. They were just as scary as I remembered them, back in my wretched high school days. If you added them to a positive number, they took away quantity. If you added them to each other, the sum was smaller. (I hadn't been introduced to absolute value yet.) If you subtracted the negative number, it was like adding quantity. I was in a mirror-universe where all the directions were the opposite of what I was (newly) used to.

But if you multiplied two of these mirror-universe numbers, they were redeemed. Their product left the mirror-universe and returned to the world of the positive universe, where everything can be counted upon. But if you had the misfortune to multiply a positive number by a negative number, it was once again plunged into the other world. And fractions, which were annoying enough to begin with, were turned negative if one of those numbers was negative.

If you think this sounds mathematically melodramatic, you're right. I envy those lucky minds who can think about mathematics and scientific data with no pictorial, imaginary or emotional involvement. Numbers are just numbers, data just data, graphs look like nothing but graphs. There are no silly childish stories to be told about them, only what they signify for the experiment. This is one reason why I'm not a scientist. The world of numbers is like a comic book to me.

In a very early posting here from 2004, I discussed synesthesia and numbers. I am synesthetic when it comes to numbers, though not as much for letters. Negative numbers also had a somewhat predictable synesthetic component for me. They were cold, blue, and sad. They sent me toward chilly, depressing realms of the number world where I looked with longing at the bright horizon of the x-axis and the sunny northeastern quadrant of the Cartesian coordinates where everything is positive. On the other side, doomed pairs resided in the opposite realm, the hellish southwestern quadrant where everything was negative. Not only did negative numbers have an emotional element, they even had a moral element. Negative numbers were somehow wrong. After all, negative numbers take money out of your bank account, make your environment colder, and even shorten your lifespan. If you multiplied a positive number by a negative number, it was poisoned. It became afflicted by negativity.

But then, along came mathematical functions. You can, through a miraculous process, take away the negativity and turn the negative into positive. That cheerful self-promoter among functions, f(x) = x², not only helps run the universe, but saves numbers from negativity. Only a sinister mathematical villain with an evil imagination would come up with i, the imaginary number whose square is minus one. And then there is absolute value. In the absolute world, between those staunch uprights, you don't have to worry about negativity. It's just quantity, without its positive or negative value.

But I can't rest on absolute value. Doesn't everyone tell me that there are no absolutes? Not even in mathematics? I am told that there are places where even the holy laws of mathematics break down. What's that saying…? "Black holes are where God divides by zero." There are places in Quantumland, where I may never visit, in which the ordinary mathematics I am currently learning gets blown to bits (or, perhaps, qubits.). I wonder what weird colors, what harrowing emotional landscapes are waiting in that world beyond, where only a few brave souls are able to win their way through.

Posted at 3:53 am | link