Instead of learning math in high school I spent most of my time getting sent out of class. In fact, on one occasion the teacher, Mr Ganpat, was so frustrated with me that he sent me out of the class to chase the cows that were in the school compound. It was not that I was a behavior problem or any such thing; it was just that I asked too many questions. Further, it was not that I was not good at math or that I hated it or any such thing, it was just that I learned differently. I wanted to know the "why" behind the rule or principle. For example when we were being taught the Pythagorean theorem I first wanted to know what Pythagorean meant and what theorem meant. Also, I wanted an explanation of the concept of square root. I knew what a square was, and I knew what a root (I was thinking plants) looked like, so what did those two have to do with a mathematical concept? I wanted to know the "why" behind the order of operations rather than to slavishly follow some PEMDAS rule. Then there was my favorite one - How the multiplication of two negative numbers results in a positive number. My teachers could only give that to me as a rule, but they could not explain why it was so. To this day, I have asked many math teachers to provide the explanation but no one seems to be able to do so. My view is that the individual or individuals who came up with those rules arrived at those principles through thought, and I wanted to be able to follow that thought process to see how they arrived at the rule or principle. Is that too much to ask?
And so the field of Mathematics could not help me, so I turned to Philosophy to answer the question of why two negative numbers when multiplied results in a positive number. The key idea that helped my understanding of this rule is the notion that "every idea implies its opposite" - a basic principle recognized by Hegel to help him to conclude that "the Real is Rational". So the following is how I answered (philosophically) the question I first asked my Math teacher in high school. I have set it up in the structure of a lesson plan. Enjoy or Endure ☺
Purpose:
To understand how the multiplication of two negative numbers results in a positive product
Some Presuppositions:
1. That every idea
implies its opposite
2. That the concept
number is different from
the thing numbered
Clarification of presuppositions:
1. Every idea implies
its opposite is illustrated as follows:
Good implies Evil
Hot implies Cold
Big implies Small
Hard implies Soft
Ugly implies Beautiful
Negative implies Positive
-4 implies +4
Note that neither the idea nor its opposite can exist
without the other. In a sense they exist for each other.
2. The number 4 is
not the same as 4 apples. The essence of apple is not number but something
else; however, the essence of number is quantity. If you remove (eat for example) an apple
nothing remains; if you remove (set aside) a number, its opposite remains.
(Now I understand how Mathematics is a pure science. It
arena is the realm of thought - of essences.)
1. Note that the term
multiply really has in view the addition of sets; so we can say that
multiplication is a
kind of addition. Such as:
4 x 4 or 4 times 4
is also
4 + 4 + 4 + 4
(Ask yourself: how many times does 4 occur in the addition
above? Well, 4 times. Hence 4 x 4)
2. Now let us look at
-4 times -4 or
-4 x -4 or the appearance of -4 negative four times. Now, please note that the number on the right
is called the multiplier. Keep an eye on that number, for in our demonstration
it is critical.
In this particular case of -4 x -4 I suggest that we look at
the multiplier as different from the number multiplied. In other words, let's
see it as removing or taking away the multiplee four times.
This is starting to tire me out! I guess you must be tired out too (assuming you got this far). ☺
[Shadows and Substance. Photographic Art by Ric Couchman]