Class 1/23/12

Salut mes amis des maths!

Another day, another lesson on the history of math! 

Today's lecture went more into detail about the Pythagoreans and what they're various studies entailed. As stated in my previous blog entry, the Pythagoreans were enfatuated with different types of numbers such as odd, even, triangular and other types of numbers to name a few. For example, the form of pythagorean triples is of the from ( n, (n+1)^2 / 2 , (n-1)^2 / 2) where n is odd and (m, (m/2)^2 - 1, (m/2)^2 + 1) where m is even. Also mentioned previously, the pythagoreans were strong believers that the world in all its entirety could always be expressed with numbers; basically, that is to say, numbers could generate everything. The believe in this concept with an almost occult like faith, so you can imagine their surprise at discovering the concept of commensurable versus incommensurable.  

The Pythagoreans believed everything could be counted, including lengths, but in order to count a length, a measure must be utilized. What the Pythagoreans assumed was that it was always possible to find such a measure that was appropriate for a given problem. If such a unit is found, it becomes the unit and cannot be divided. In particular, it was assumed that it was possible to "find a measure by which both the side of and diagonal of a square could be counted. In other words, there should exist a length such that the side and diagonal of a square are incommensurable; there is no common measure." This is the case in the above picture, portraying two legs of a triangles which measure length 1 and the hypotenuse (diagonal of square) with length square root of two. Since we know the square root of two is irrational, it is obvious to us that the such a value and 1 do not share a common measure, so an appropriate unit cannot be found for this problem. 

In addition to the Pythagoreans, Dr. Daddel also briefly covered Zeno's Paradoxes (one of which is illustrated above) as well as some background information on Plato. Some of the different facts we learned about Plato were that Socrates was his instructor until he was executed in 399 BC for challenging the authorities, he wrote   "The Republic", didn't care for "practical math", studied different aspects of Geometry, and created his own academic which remained influential in Greek Education even after his death. 

Until next time, mes amis! 

Bisous 

-MB