Donald Duck in Mathmagic Land
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.
Modern Mathematics: A New Secret Society
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.
Math Papers: A secret code
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 today's, mathematics, to gauge the underyling logic requires far more substantial knowledge.
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.
Making Math Interesting
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?
- Let a=b
- a2 = ab
- 2a2 = a2 + ab
- 2a2 - 2ab = a2 - ab
- 2(a2 - ab) = 1(a2 - ab)
- 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:
- Let (a/b)2 = 2 where a,b are whole numbers
- 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)
- So a2/b2 = 2 which means a2 = 2b2
- Since 2 divides a2, we know that a is even (an odd*odd = odd)
- So, there exists c such that a=2c.
- (2c)2 = 2b2
- So, after dividing 2 from both sides, we get 2c2 = b2
- And there is our contradiction. Do you see it?
- 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.
A Mathematician's Challenge
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.
cute stuff 69! on March 07, 2013:
THIS HELPED ME A lot THANKS SOOOOO MUCH HUB PAGES!
Ireno Alcala from Bicol, Philippines on April 14, 2012:
It's like Functions during our fourth year in high school. Anything is possible in Mathematics. We can formulate our own equation, just like you did here.
I'm still learning, just like the others.
Manna in the wild from Australia on December 20, 2011:
Voted up and interesting. You explain things well and this generated some interesting comments.
Pathyy on December 19, 2011:
great this information
larryfreeman (author) from Fremont, CA on July 27, 2010:
I'm glad that you enjoyed this hub. I think that when mathematics is presented in its natural form, it becomes more like an architectural structure or a Bach symphony and less like an engineering problem.
Simone Haruko Smith from San Francisco on July 24, 2010:
Wow. As a complete math dunce, it's fascinating to read all this background! I feel like I've been missing out on something really amazing.
larryfreeman (author) from Fremont, CA on January 19, 2010:
I'll be glad to read your hub when it is ready.
HABMATH on January 16, 2010:
I read the comic "Donald in Mathmagic Land" as a first grader. I loved it then ( and still do) and it did get me interested in math.I am presently writing a hub called: " how math really works" I would love your comments on it. I would like a little patience on your part because ( like many mathematicians) I am a terribly slow writer
larryfreeman (author) from Fremont, CA on August 27, 2009:
My father was a computer engineer in the 70's so I got into computers through him.
I consider myself more a computer geek than a mathematician. For me, mathematics is more of a hobby.
I got a bit deeper into mathematics because I was bothered about not understanding why calculus works. In college, I learned the basic rules about derivatives and integration but I had skipped the Fundamental Theorem of Calculus.
Once I started getting to understand the classic mathematical proofs, I was surprised how interesting I found them.
I am very glad that you enjoyed my hubs.
Deborah-Lynn from Los Angeles, California on August 27, 2009:
I liked your Hub and the interesting comments it has generated, was it Disney that inspired you to become a mathmetician? I saw those same cartoons, but I was not inspired, in fact I learned only enough math to get me through college chemistry and organic chemistry, only because I was only motivated to get a high G.P.A. in college. I wish I saw and felt what others do regarding math, not having that love of numbers makes me very aware that I am not a part of the "secret society". Thank you for still including me by allowing me access to your mathematical insights through reading your wonderful Hubs.
i from Earth on June 07, 2009:
"If I have 10 apples and each day I give you x apples, how long before the apples are gone?" Gone where?....to the hell?......do I know you? :) In reality it depends on our choices: If you gave me 10 apples,- I can recycle them 100% or little less, but keep the seeds. If you didn't give me any apples (0 apples),- you don't have many choices. Only recycle them 100% or little less, or keep the seeds?
What do you chose? :)
Jmell from El Paso, Texas, USA on May 19, 2009:
hi Larry. I'm no mathematician, not even close, but I loved your hub and agree whole heartedly that Math should be fun when learning.
Peter from Australia on May 02, 2009:
G'day Larry I'm still working on that apple one.
I'm sure there is an answer hidden away somewhere ?
highway star from Mostly Seattle, Amsterdam and Milan on February 21, 2009:
A really great hub!
larryfreeman (author) from Fremont, CA on February 20, 2009:
Thanks very much for nice comments. I am glad that you enjoyed the hub.
Om Prakash Singh from India, Calcutta on February 20, 2009:
This was a really great Hub. Though I was able to figure out both the problem you had suggested, no not because I am a good mathematician but because I had come across this problem earlier. :)
But more one the serious note, yes you are correct it is very important to have classes that teaches about the passion of mathematics. It should be made a subject that should be learned out to passion and then only students will be able to understand the real power of mathematics. I no math geek but I know how important it is. Thanks for the Hub! It is nice to have you around and nice to be your fan!
Shalini Kagal from India on September 26, 2008:
Great hub! If only we could guide our children through that magical, mystical door and instill in them a love for mathmatics!
larryfreeman (author) from Fremont, CA on September 25, 2008:
Great question. My oldest child is 5 and does not yet understand division but he's getting close. :-) It sounds like your child is very smart!
Here's what I plan to tell my son when he is old enough. In mathematics, there are plenty of mysteries. Most of these mysteries surround very tough math problems. Some, such as division by zero,are more difficult than they appear.
For example, in most division, you can undo it. if 5* 6 = 30, then 30/5 = 6 and 30/6=5. If we have 5x = 30, then we know that x =6.
But with zero, we have a serious ambiguity. 5*0=0 and 10*0=0. If I say that 0x=0 then x can be any number we want. There's no way to recover the value that is multiplied by zero.
In any division problem, there are conditions. If I have 10 apples and each day I give you x apples, how long before the apples are gone? I can't give you 11 apples since I only have 10 and likewise if I don't give you any apples (0 apples), then there will never be a time when the apples run out so in such a situation, there is no answer.
Aya Katz from The Ozarks on September 25, 2008:
Okay, Larry, great hub and I've given you a thumbs up.
But now, supposing your elementary school child doesn't accept that dividing by zero is not allowed. Supposing the child asks: "If zero is just a number, like any other number, and we can add and subtract and even multiply by zero, why can't we divide?" What would you say then?