Time for a New Math?

Mathematics have been with us for a long time, since we started to be interested in counting things and grouping things together. It started out as pretty simple stuff probably, but pretty soon, smart people started to figure out all sorts of things that could be derived from the first mathematical rules. The more new and existing stuff people came up with, the more applications we got for math, and so it has continued through the centuries, until we have ended up with the math we have today.

Math is really useful, it allows us to do all sorts of things, describe the physical reality and help us build awesome buildings and fly into space and many many other things. Endless amounts of math has to be done for me to be able to type and post this article. So, all in all math is doing pretty good for itself. Everything sort of fits together nicely you might say, and math is just fine the way it is. Well I disagree.

Unsound means to be flawed, not perfect, logically invalid. I admit, I don't know much about math, I use it like most people for mostly mundane things, but I also know enough about math to know that the system of numbers and operators we use are not sound. I think it's high time that math is redone in a system that is more precise, less prone to horrible and crippling flaws.

When I was in school, I learned that when you create a vector space, or a mathematical system if you want to call it that, there are 10 criteria that needs to be fulfilled for the system to be sound. I can't say I remember many of them, but I do remember that for a system to be sound, any legal operator in the system when applied to one or more legal numbers have to yield another legal number in the system. Meaning, that if you add two numbers together, the result has to be another valid number. For most operators and numbers this is not a problem in our current system. If you add 2 and 4, you get 6, another perfectly valid number. In the same way, if you take the square root of 25, you get 5, a legal operator applied to a valid number and it yields another valid number. Excellent. So far so good, but complete mathematical doom is lurking right around the corner.

In our mathematical system, negative numbers are valid numbers, meaning, any legal operator applied to a negative number, must yield a valid number as the answer. And this is where everything falls apart. The square root of negative 1 is not a number at all, the answer is mysteriously i. Now, if i a constant representing a valid number, like say pi there would be no problem. But i is not a real number, i is the imaginary number. Imaginary! What in the world possessed people to invent an imaginary number? It's disaster! Our numbers in our mathematical system are one dimensional, meaning they are vectors with just one element. The imaginary number is two dimensional, it has a real and an imaginary part. If we look past the obvious issues of a number having an imaginary part, it's still absolutely outrageous. The legal operator square root, applied to a legal value -1 yields an answer that has more dimensions than the number we started with. A two dimensional answer, a number that is not within the same vector space as the original number! The system is not sound! How can we have a math that breaks down so easily?

Now, some math lovers might try to tell you that the example I gave is not proof that the system is unsound, they might claim that out number system is really two dimensional, and that every number we use is in reality complex numbers with an imaginary part that just happens to be zero. I think this is questionable, but I guess if you want to be technical about it and feel like it's important that the math we have is left alone, you could sort of say that the numbers are all in the complex plane, and all our numbers are two dimensional. It's still a pretty messed up idea to have imaginary numbers, but oh well, let's give the math lovers this one, but with a big fat asterisk for sure . At least in my opinion.

So, is math really sound anyways?

No. Not in the least. Even with the desperate inclusion of imaginary numbers, total mathematical collapse is no further away than division by zero. Zero is a perfectly valid number in our system. The division operator is equally legal, and yet, if you try to apply this operator when the denominator is zero causes complete and utter failure. A massive collapse of math in itself. If you try to divide by zero, the answer is ... undefined. Great. Undefined is most certainly not a valid number. It's not a number at all, it's nothing, it's completely and utterly outside of the vector space we are working in. It is as related to the numbers we started with as a bowl of cereal, or democracy, or a small rock, or any number of things that are totally and completely not a number at all. Math is not sound. Division by zero breaks it to smithereens.

It is my opinion that math is so fundamentally flawed that it needs to be completely reinvented. Start from scratch if necessary, and build a math that is consistent and perfect. After all, how embarrassing would it be if visitors from other galaxies where to make contact, and they realize that our whole math breaks down by something so simple as division by zero? The shame!