Lighting a Candle for Mathematicians

In the sprit of lighting a candle rather than curse the darkness, I offer the following bit of incendiary tallow for our mathematical friends. I have stated before that something is wrong with mathematics as it exists today. One possible genesis for this is that most of our concept of numbers comes from the "number line". This is thinking of numbers like they live on a ruler laying before you. The number one is to the left of the number two; number four is further down to the right, past number three, etc. When the concept of negative numbers and zero came along, places were found for them off to the left more still.

Concepts of adding and subtracting numbers evolved while staring at the number line. Multiplication and division followed in the same vain. When irrational numbers came along, the number line showed its first cracks since π and e were not definable points on the line. They could only be considered to be "between this number and that number, just about here". When they looked for the square root of a negative one, it wasn't there at all, so it was dubbed an "imaginary number". Rather than go back and reconsider the number line metaphor for such inadequacies, they just accepted that the number line was right. It was irrational and imaginary numbers that had problems (hence their names; born of frustration).

What would have happened if the number line had never been conceived? Where would we have gone when negative numbers were invented? (some would say "discovered" but we won't get into that here.) First let's think about this; positive numbers and negative numbers. We have positives and negatives throughout the universe around us. For positive electric charge there is negative electric charge. There are positive reactions, there are negative reactions. Positive numbers - negative numbers. But nature also shows us that besides positive and negative, there is also neutral. Protons are positive, electrons are negative, neutrons are neutral.

Where are the neutral numbers?

The number line never revealed them because it is a flawed metaphor. But describing nature with mathematics demanded more that the number line could accommodate. Maybe it is time to consider the neutral numbers. In fact, in some ways we do. Often when a neutral number is needed from an equation we will instruct the mathematician to take the absolute value of the answer. Basically we are forcing an unsigned or neutral result where our otherwise a flawed equation can only produce positives and negatives.

Let's take a simpler example. If we look at the interaction of positive and negative numbers a curious pattern unfolds:

(+1)×(+1)=(+1)
(+1)×(-1)=(-1)
(-1)×(-1)=(+1)

The series is asymmetrical. More positives shake out than negatives. This shows an incongruity with the real world right there. Look closer at the second equation, here a positive and a negative yield a negative; as if the positive one was actually a neutral one. Now look at the first equation, the two positives yield a positive. Again, the polarity of the number has no effect on the outcome; it is "neutral". So you see, we have endowed positive numbers with the attributes of neutral numbers somewhat haphazardly. Lets see if we can describe a series of equations with positive, negative, and neutral numbers which has more symmetry:

(+1)×(+1)=(-1)
(~1)×(-1)=(-1)
(~1)×(+1)=(+1)
(-1)×(-1)=(+1)
(~1)×(~1)=(~1)
(-1)×(+1)=(~1)

The tilde (~) symbol is used here as the designator for neutral numbers.

Now see how pleasing this looks. Equal quantities of positives, neutrals, and negative products. Now look at the rules that appear: Neutrals never affect the sign of the product, otherwise, two like signs yield the opposite sign, and opposite signs cancel each other. Perfect harmony with our understanding of nature. And the big bonus? Look! The square root of negative one! Right there! That's why simply putting the letter "i" in front of a positive number always worked to show imaginary numbers. But they're not imaginary anymore! This may not be the key to unlocking all of the shackles on modern mathematics, but doesn't it warrant a little consideration and exploration.