Incompleteness blog

Mathematical research is considered by many as the art of exploring regularity and trends in a system of axioms.

After defining some premises, mathematicians can use mathematical logic to deduct the rigorous truth of a statement, at that point the statement becomes a theorem and the pursuit for the implied truths starts again.

Devoting ones life to mathematical research isn’t best suited to the minds of most of us (to say the least), but that doesn’t bar us “ordinary people” from enjoying the fruits of centuries of fruitful mathematical research. It’s a given fact that every freshman high school student these days is being taught math way above and beyond the comprehension of the great minds of the Greek and Roman era.

While it’s absolutely reasonable to claim that watching some videos on the interwebz won’t make you a mathematician, there’s nothing wrong with absorbing some intuitive understanding for stretching your mind a bit and inducing inspiration, and is there a better way to do so then watching colorful videos?

So, without further ado, here are some of my favorite “math for dummies” videos.

Different “kinds” of infinity in set theory

One of the most common pet peeves of the curious mind occurs when first stumbling upon the notion that there are different kinds of infinity, and that they can be proved to have different cardinalities (put in simple words, some infinities are bigger then others).

Does that concept sound unreasonable to you? Don’t worry, you are in good company. Georg Cantor, the father of axiomatic set theory and the first mathematician to ever define and explore infinity, was also skeptical. But ruthless, uncompromising research and razor sharp mathematical intuition has led him to prove a series of stunning claims, the most famous of which is that there are more real numbers than rational numbers.

Unfortunately, the mathematical community wasn’t ready for Cantor’s foresight, and the contempt that was expressed towards him broke his spirit. After a series of nervous breakdowns Cantor decided to quit math, and lived his few years in poverty and malnutrition until he finally died in a sanitarium in 1918.

The first video of the day tries to explain Cantor’s proof (also called “Cantor’s Diagonal“) to the different cardinalities of the reals and the rationals in everyday language.

Causality vs. conditionality in probability

Statistics are often considered by many to be the most confusing topics in the list of undergraduate math topics, and to no avail. It’s equation might not seem as cryptic, and you don’t need to pseudo-visualize any n dimensional geometry, but it’s counter-intuitive conclusions and intricate subtleties make the novice statistician very prone to make errors. Statistics are very, very confusing.

Unfortunately for us, statistics are probably also the mathematical field of study whose results have the biggest influence on our everyday life.

We read about statistics in ads and consumer digest on a daily basis, nation-wide companies all around us spread their branches and hire and fire according to statistical data harvest, statistics are even used to calculate our income tax - and yet so little of us have even the most basic tools to audit the statistical claims we hear.

One of the most common (and most commonly abused) fallacies is the misinterpretation of causal statistics. This little video sets out to clarify that misunderstanding.

Turning a ball inside out using topology

In the beginning there was Euclidean geometry, and everything was nice for a while until someone figured out we’re actually living on a ball. Curious minds wondered how that might effect our geometry and started researching the properties of spherical geometry.

The process, however messy, was based upon a very reasonable concept - lets see how we can apply the Euclidean theorems to a sphere instead of a plain. It was soon proved that in this manner you can use all of the axioms but the fifth (which states that parallel lines have no intersection points).

The fact is, that nobody liked that fifth axiom anyway, as it seems far too complex compared to the other four, and at that point people finally understood why. The four basic axioms hold water in any geometry, while the fifth one limits it to a certain type of surface - a plain (it was later recognized that plain wasn’t the only satisfactory type of surface for the 5th axiom, another common example is the class of saddle-like surfaces, also known as hyperbolic surfaces) . It was further argued (and proved) that replacing that axiom with another one, stating that all parallel lines meet at the same point, would force the surface in question to be a sphere.

But math is all about generalization, so the next question was, naturally, what entities could be made using these axioms on any type of surface, on any number of dimensions. In order to tackle that mind bending quandary, concepts which were intuitive in Euclid geometry (such as surface, distance, and angle) had to be redefined using complex algebraic instruments - and thus topology was born.

Even though considered to be arcane and useless outside the doctrine, topology was probably the most common topic to be discussed around water fountains in math faculties around the world during the lite 19th and early 20th centuries. But it was only after Einstein employed it to describe the space-time continuum in his general theory of relativity, that topology rightfully gained it’s place in the hallmark of science.

One of the prettiest demonstrations of topology at work, is the prove that a ball can be turned inside out without being dissected. This video uses that theorem to demonstrate some of the basics of topological thinking. And even though the narration can often be juvenile, I think it’s absolutely charming.

Some loosely relevant links

• A free introductory book on Set Theory
• A detailed biography of Georg Cantor
• Causal inference via causal statistics
• Correlation does not imply causation

Related posts:

1. Chaos Theory for kids
2. Abstract Algebra in a nutshell

Tags: internets, layman, math

This entry was posted on Monday, June 22nd, 2009 at 2:18 AM. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.