Yeah, it's a long-winded title. But hey, it's a big subject. So the question is, "Mathematics: Invented or Discovered?"
There are two elements to this; both of which have been touched on in my title. Is the field of mathematics something that we humans have built, in order to better understand our world? To make sense of it and give it structure, meaning, order, organisation, rules? Or is it something that simply is? Is math a force, such as gravity? You cannot say that Sir Isaac Newton invented gravity, that would be absurd. So is math such a force? Well, I find that the best thing to do, when presented with such questions, is to go back to the definition, as I have done in previous blogs. So what's the definition?
From dictionary.com:
The systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically.
This definition, especially considering the word "treatment" suggests the human element of math: that it is something we use in order to understand certain things. That it is something we invented in order to understand these things. Numbers have been invented by humans. I could say that I am now typing on 1 laptop computer. But am I really typing on one object? The laptop is an entity comprised of myriad particles, or atoms, and of course these atoms are made of yet more particles, and for all we know, those particles are made up of even smaller ones and so on. So what are numbers, really? Another popular aspect of numbers is their use in counting money. This is even more theoretical than the laptop! What is money? Paper or metal coins? Not always; it can be represented as a figure on a screen, AND GUESS WHAT?!?! That figure is made up of tiny particles called pixels! But we humans do try to be practical, especially regarding money, and we give this no thought. The fact is, numbers are used to count money. The higher the money, the more the government owes you. All of this is man-made. Numbers, and the adding, subtracting, dividing, or multiplication of those numbers, are just a way to make sense of our lives.
If mathematics is invented, could we give it different rules? Wouldn't that be nice! But because math is something we use to give order to our world, we can't go around changing rules just because it gives us the answer we put on our exam. This inexorability of math doesn't mean that it is some sort of universal force that we have discovered through time, it just means that what our ancestors have invented is solid and dependable, in one word, useful! Math is used everyday, and in this day and age, we would be quite lost without it.
Pythagoras
This mathematician's theorem is still used today, and in his honour we have named it Pythagorean theorem. Of course there are plenty of other important mathematicians that lived hundreds of years ago, and whose discoveries we still put to use today. Notice, I just used the word "discoveries." Why did I use that word? Anyone who knows me should know that I choose my words carefully, and, indeed, use them sparingly. Unless I'm rambling on in a blog ;)I simply had to call them discoveries, because that is what they were. Pythagoras discovered that "The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c)." He did not wake up one morning, and decide that this should be true, because he willed it to be so. It is simply true. It's one of those axioms discussed in class, and Pythagoras merely discovered it. BUT what do these things represent? Ideas that were made by humans! If math, at the very beginning of its conception, had taken a different course, we might have something different. It is impossible to say.
To conclude, and state my opinion in brief, simplified terms, I believe that mathematics is something that humans have invented, as a way of knowing things.
