When I was a child, I remember seeing a Disney short about mathematics called Donald in Mathmagic Land. In it, I learned about the Pythagoreans, a secret math society from Ancient Greece who believed that all the universe consisted of countable numbers.

Pythagorus, the founder of the society, never published any mathematical work. Or at least, nothing survives that can be attributed to him. Pythagorus started the group after discovering that musical harmony at its root was a ratio of whole numbers. Pythagorus wondered if all orderings of the universe were in truth ratios of countable numbers (what today we would call rational numbers) and formed his society with the task of unlocking these secrets.

The story is that Pythagorus was passing a blacksmith when he heard what sounded like musical notes each time the blacksmith hammered on his anvil.  Pythagorus stopped in to investigate and discovered that that the blacksmith had three anvils of different sizes.  One anvil was 1/2 the size of the largest and the smallest was 2/3 the size of the largest. 

On the surface, mathematics seems to be in the open. After all, it's required study in both high school and college. But, deeper down, if you think about it, it is a secret society.

On the surface, mathematics is about problem solving. You are given a formula, then a word problem, and then you must use the formula to solve the world problem. The goal of mathematics seems to be to challenge your problem solving skills and logic.

The way that mathematics is taught, it is perhaps very surprising to learn that the problem solving aspect has little or no interest to the professional mathematician. Indeed, the basic mathematics of school: algebra, geometry, trigonometry, and calculus, are for the most part, completely ignored. They are after all well established and not in need of expansion.

The technique of interest to the mathematician is the mathematical proof. While on the surface, math seems to be about intellectual development and problem solving, it is for the practitioner, a search for underlying patterns and harmonies. Mathematics, the way it is studied by mathematicians, is really about beauty and truth. While it may appear to be about engineering and accounting, mathematics really has more in common with art and science.

It is as if all math papers are encoded in unfathomable mathematical terminology. According to the Disney documentary, each Pythagoreans need to show the symbol of the pentagram in order to gain admittance to Pythagorean gathering. In todays, mathematics, to gauge the underyling logic requires far more substantial knowledge.

The reason that the Pythagoreans chose the pentagram is for me very interesting. The ratio of the sides of the pentragram represent the golden ratio. The details of this ratio can be found here.

Professional mathematicians today seem to feel little or no obligation to share their knowledge with the uninitiated. Indeed, it seems to me that the true wonders of mathematics, today, are only communicated to other mathematicians. If you are not a member of the Modern Pythagoreans, then you are not invited to share in their wisdom and their insights.

There is a coterie of brave souls who try to translate mathematics for the general reader. They try to write on topics that will appeal to the general reader and share some of the amazing ideas which has been part of the last two hundred years of mathematics. Unfortunately, because of the math drillings in school, they have a difficult market. Few people are open to hearing about math voluntarily.

This raises an important question: why can't a math problem be easy and then interesting? Science, when taught by a gifted teacher, can be about the fascinating experiments that challenge our intuitions and reveal the often surprising physical world. Why can't mathematics be presented in the same way?

What if mathematics was taught with the goal of making the underlying ideas easier to understand. Consider this example:

Is it possible to create a three-dimensional one-sided figure?

This is not meant as a trick question. It is really an effort to challenge our intuitions about three-dimensional space. The answer, which should be as obvious as cat is spelled c-a-t, is yes and it's called a Moebius Strip.

Here's another question that I would like to see as a fundamental part of mathematics eduation:

What's wrong with this proof that 1=2?

1. Let a=b
2. a² = ab
3. 2a² = a² + ab
4. 2a² - 2ab = a² - ab
5. 2(a² - ab) = 1(a² - ab)
6. 2 = 1
   

The answer is that we used division by zero. If a=b, then a² - ab=0 and to get from step #5 to step #6, we did something which is not allowed in mathematics. In other words, all we proved was that 2*0 = 1*0.

Here's the last one:

If x² = 2, can we represent x as the ratio of two whole numbers?

The answer is no. If it did, it leads to a contradiction:

1. Let (a/b)² = 2 where a,b are whole numbers
2. Let a,b be in the simplest terms so that a,b do not have any common factors (ie the simplest term for 2/6 is 1/3)
3. So a²/b² = 2 which means a² = 2b²
4. Since 2 divides a², we know that a is even (an odd*odd = odd)
5. So, there exists c such that a=2c.
6. (2c)² = 2b²
7. So, after dividing 2 from both sides, we get 2c² = b²
8. And there is our contradiction. Do you see it?
9. If 2 divides b², then b is also even so 2 divides both a,b but by step #2, we assumed that a,b were in the simplest terms.
   

For mathematicians, this is enough to prove that the square root of 2 is an irrational number (that is, it cannot be represented by a ratio of two whole numbers).

This last proof was especially significant to the Pythagoreans. Remember, they believed that all the world could be represented as the ratio of two whole numbers. In other words, they believed that all numbers are rational. The existence of irrational numbers disproves this hypothesis.

If we wished to, we could teach mathematics the same way as we teach science. We could focus on understanding the details and reasoning behind a proof rather than focusing solely on problem solving.

Now, the secret society nature of mathematics is completely unintentional by mathematicians.  They would love to share their learning and insights.  My main point is that we need to have math appreciation classes in addition to the problem solving classes.