Week 2 - What's an Axiom? (Nature)

What’s an Axiom?

Based on any sort of mathematical knowledge I have, I imagine that an axiom is simply something to believe in relation to mathematics. When something cannot be proved, but seems to always be true, it is assumed to be true. This is what I have believed to be an axiom. Anything I did know essentially came from studying in a course at Grand Valley, and axioms were heavily used in proofs. 

Axioms are actually used universally in logic. They stem from our basic sense of reasoning, and are used in the beginning of logic thoughts or arguments. An axiom is specifically a premise that is accepted as a truth since it is very evident. Axioms came from the Greeks, in part to their development of proofs and logic. Euclid was essential in the development of axioms, and deduced many ideas that developed into axioms that believed to always be true without proof. In Euclidian geometry, there are five basic axioms:

1. Given any two points, they can be joined by exactly one line.
2. Given any finite, non-zero length line segment, it can be extended infinitely into
   exactly one line.
3. Given any line segment, there is exactly one circle with one endpoint of the segment as the center, and with the other endpoint on the circle.
4. All right angles are equivalent modulo translation, rotation, and mirroring.
5. Given a line l and a point p which is not on l, there is exactly one line that passes through
   p but never intersects l.

All other ideas were meant to proven.

Most axioms were used the different kinds of sciences when they were developed, but mathematics has altered that in the last 200 years. Now there are 9 axioms by way the Zermelo-Fraenkel set theory. The axioms are used to develop proofs in modern mathematics. They have been part of other sciences such as physics, and the axiom remains as the expanding set of believed truths that continue to develop the sciences.

Works Cited:

http://www.mrc.uidaho.edu/~rwells/Critical%20Philosophy%20and%20Mind/Chapter%2023.pdf

http://scienceblogs.com/goodmath/2007/03/07/basics-axioms/

What is Math?

What is Math?

Math is the study of numbers, and can be applied to a seemingly infinite number of real-life situations. The creation of numbers, symbols, and systems seemed to be invented in conjunction with various mathematical systems, where these numbers and symbols could be used, in application, to likely make things easier in life. 

The major discoveries, or developments, in mathematics, in my opinion are:

1. Invention of Calculus
2. Development of scientific systems
3. The basic number model
4. Numbers such as pi or e
5. The order of operations

These are massive in the mathematics world, and many industries couldn’t exist without these fields of mathematics. Calculus gives us derivatives, and rates, and the idea of slopes and many other things. The development of financial algorithms come from calculus and differential equations. The scientific systems lead a large part in research in technology and medicine. Our basic number model is the foundation of everything we know today in mathematics. Presumably, the basic number model could be the set of integers, but could be interpreted as any set of numbers, since the original set of numbers could have been any of the sets of numbers. Number such as pi or e as used so much in mathematics. There are entire fields and ideology based on them. Finally, the order of operations allow us to use any number system or set of numbers to function in a progressing society. They simplified life and made many things easier to grasp, as well as became easy for anyone to use. You would be hard pressed to find someone who didn’t understand, at least, part of the order.

Week 1 - History of Zero (History)

History of 0

The history of 0 does not necessarily refer to what we believe zero to always be. We would assume that when we hear “zero”, we would think of something that is actually nothing. “I have zero chance of buying this car”, or “There is zero food left at the buffet....shame”. In history though, dating back to Babylonians before Christ, zero did have its existence, as a measurement. More so, zero represented larger quantities, much like how we have tens and hundreds and thousands and so on. 

It is impossible to tell where zero came from, and cannot be traced to a single source. The concept of zero has been around for centuries, and is thought to be initially used throughout the Middle East, India, and North Africa in the Arabic numerical system, and brought to Europe by Fibonacci in the 13th century. There is evidence that zero was used by the Babylonians but was not written as a single digit or symbol, but interpreted as such. From there, zero became more recognizable, and became more known as how we would typically think of it: as the average between -1 and 1. 

Now, we see zero as applicable as any given single item in mathematics. It became the center of origin in coordinate planes. More familiarly, it represents an empty space. Oddly enough, the Greeks were believed to have put a symbol to zero, as their numbers were not represented by digits, but by lines. Thus, with zero space not be able to represented by any physical measure, they used “0” to represent the empty space in their numerical system. 

To get a rough interpretation of the use of zero by all sorts of history’s cultures and societies, I’d recommend reading: http://www-history.mcs.st-and.ac.uk/HistTopics/Zero.html

Other works cited: 

http://www.scientificamerican.com/article.cfm?id=history-of-zero