The Shaky Foundation of Mathematics

When we watch the Space Shuttle lift itself into the remote extremes of our atmosphere we are conscious of the triumph of technology, but we are also witnessing the triumph of mathematics. For it is mathematics that serves as the foundation and the language of science and technology. But what is "mathematics"?

The math we know today is as different from the math of the Ancient Greeks as is our spoken language. Like our spoken language, our mathematics is a tapestry of concepts and rules accumulated from around the world since prehistory. And while it is common to think that mathematics is a self evident truth of nature which we cleverly discovered and put to use; this is far from reality.

Mathematics has been evolving ever since the first human recognized that two apples is more than one apple. And this evolution has been surprisingly slow. It took many centuries for mathematicians to accept concepts such as zero and infinity. The simple geometry of the Ancient Greeks was inadequate to solve a great many problems. It took the introduction of calculus before many problems could be reduced to a solution. And while our current collection of mathematical tricks allows us to create wonders never dreamed of before, it is far from perfect.

Modern mathematics has some major problems which have long been recognized but never properly addressed. Without venturing into too much detail, let's look at some of the obvious examples and their implications to further advancement.

Irrational Numbers - There is a class of numbers which behave very strangely. So unlike whole numbers are these that they were labeled irrational because when they were first discovered, the primary characteristic was that they could not be represented simply as a ratio of two rational numbers. Irrational came to take its modern meaning for the very reason that mathematicians could not help but feel that something was wrong with irrational numbers. This would not be an issue except that many of these numbers are vitally important, such as π (Pi) which is needed when calculating various attributes of a circle. Unfortunately π has no exact value. While we can calculate the value of π to a degree where there is no practical impact to everyday life, there is currently no hope that an exact value exists. So it is for all irrational numbers.

The most disturbing thing about irrational numbers is the frequency in which they appear in nature. π, e, the Golden Ratio, √2, etc. All of these numbers appear over and over again in nature, and yet our system of mathematics has no exact value for them; only approximations. It is almost as if we purposely created a system of mathematics that could never describe the world around us. The existence of irrational numbers alone should force the abandonment of current mathematics as hopelessly flawed.

The Square Root of a Negative Number - As we were taught in elementary school, any two like-signed numbers, when multiplied together, yield a positive number. So while we know that the square root of nine is three (3 × 3 = 9 therefore √9 = 3 )*, there is no one number which will yield a negative nine (-9) when multiplied with itself (-3 × -3 = 9; 3 × 3 = 9). This is true for all even roots, but not for the odd roots (-3 × -3 × -3 = -27). And since there is an infinite number of negative numbers and even roots, there are an infinite number of "unexplainable" or "missing" roots. While this may seem trivial in the grand scheme of things, in mathematics in has always been a real nuisance.

To simplify the problem, mathematicians used some of the accepted rules of manipulation to reduce the number of missing roots from infinity down to one; no small feat in itself. They did this by factoring out a negative one from every number, or in other words, by changing negative nine (-9) to nine times negative one (9 × -1 = -9) they could take the square root of any negative number and express it as the square root of the number (as a positive) multiplied by the "square root of negative one" (√-1). In our example this would be √-9 = √(9 × -1) = √9 × √-1 = 3 × √-1. So now there was only one "unexplained" number, the square root of negative one. To make things easier to write, the mathematicians called this the "imaginary number" and wrote it as a lower case i or sometimes j. So now they could write √-9 = 3 × √-1 = 3 × i, or by dropping the multiplication sign as is common practice √-9 = 3i.

So a major problem–an infinite set of imaginary numbers–was reduced to a small problem–one imaginary number. Now mathematicians were free to calculate, but they weren't always sure what to do with the imaginary numbers, so they regularly discarded any answer that had an imaginary component. Other mathematicians found that they could not afford this convenience and were forced to hang onto numbers that had both real and imaginary components (now called "complex numbers" because it is not palatable to present "imaginary" answers to your peers). While this "work-around" has survived and allowed mathematics to progress, there still is no answer to the question: what is the square root of negative one.

Zero and Infinity - These "numbers" were late to the party for the simple reason that they don't follow the same rules as the other numbers; and mathematicians live and die by rules. First, while zero and infinity seem similar to whole numbers, all whole numbers are either odd or even; zero and infinity are neither (or perhaps they are both simultaneously). Whole numbers are either positive or negative; zero is neither. Whole numbers divide into themselves one time; zero and infinity do no yield this result.

Zero multiplied by any number yields zero, therefore any number divided by zero can yield any number as a result. This flies in the face of common sense and lead early adopters of zero to state that "division by zero is undefined". In real world experiments however, scientists found that as the denominator of a function approaches zero, the quotient approaches infinity. So it was accepted that allowing the denominator to reach zero would yield an answer of infinity. This forced the development of calculus to help explain how equations behave as certain numbers in the equation move towards limits like zero and infinity.

But these are not firm answers like 2 × 3 = 6 or 9 ÷ 2 = 4½; they are more like convenient "tricks" to yield a useful answer where none would otherwise exist. So mathematicians will say that the answer to 5 divided by zero is infinity (or more precisely they will say that the answer approaches infinity when five is divided by a number that approaches zero; no need to commit too strongly when using a "trick"). But since 5 × 0 = 0 and 19,847 × 0 = 0, then 5 ÷ 0 = 5 and 19847 and 2 and 0 and every other number there is; hardly useful to a serious mathematician.
So as you can see, there are some serious problems with mathematics as we have defined it today. And while mathematicians have created cleaver tricks to work around these problems, we may be facing some serious limitations and outright errors as a result.

The reason that these tricks survive is because they were developed against the problems faced by human beings on planet earth. The scale of our bodies and the things we interact with on a daily basis–baseballs, automobiles, airplanes, etc.–all allow these tricks to yield acceptable answers on a consistent basis. But our scientists have left our familiar scale and are working with galaxies and atoms, and here our tricks are catching up with us.

Our flawed mathematics have our scientist fumbling to understand important fundamental concepts whose explanations are approaching the ludicrous as a result. We can't determine if light is a wave or a particle. Galaxies seem to be missing more matter than they contain. The existence of gravity depends upon the existence of imaginary particles having no size nor mass. We are at a point where the mathematics allow situations to exist where cats are dead and alive simultaneously as long as no one looks at them. Where we all exist in an ever increasing number of parallel universes with that number approaching infinity by adding an infinite number of universes everyday. Where people flying to the edge of the solar system and back at a velocity near the speed of light will take generations to return to earth, while going at a slower rate of speed would result in a quicker trip.

All of these problems stem from our unwillingness to throw away our current system of mathematics and start fresh. And while no one knows what the new mathematics will be, we can state a few of the properties it will have:

1. No imaginary numbers - Absolute truths must exist and be applicable to real situations.
2. No irrational numbers - A structure of fixed dimensions must have absolute values associated with it.
3. No ambiguity between whole numbers, zero and infinity. All must follow the same rules if they are to be considered.
4. No infinite sets of infinite sets - Any set must be less than or equal to infinity, not less than and equal to infinity as today's flawed system allows.
5. No need to discard an answer - Where today a calculation can yield an answer like "the cat weighed 3 kg and -12 kg" leaving it to us to recognize that the former is the only correct answer, a less flawed system would yield only the correct answer.

This is in no way meant to be an exhaustive commentary on this subject, rather it is hoped that you will appreciate that much more needs to be done toward this end.