Mathematics… (Say What?)

Initially, when people think about the field of mathematics many think about number crunching and intense/confusing calculations involving the use of various algorithms.  However, true this may be, mathematics is much more than this and a large portion of mathematics involves little to no number crunching/calculations!

The core values in the field of mathematics are LOGIC and TRUTH. 

The field of Mathematics is based off of a set of axioms (i.e. statements) from which theorems, postulates, and lemmas can be deduced (i.e. true statements).  For example, a famous axiom set proposed by Hilbert to help develop the field of geometry looks like such:

Undefined Terms: point, line, incidence, betweenness, and congruence.

Incidence Geometry

AXIOM I-1: For every point P and for every point Q not equal to P there exists a unique line l that passes through P and Q.

AXIOM I-2: For every line, l, there exist at least two distinct points incident with l .

AXIOM I-3: There exist three distinct points with the property that no line is incident with all three of them.

One of the theorems deduced from this axiom set is known as the Crossbar Theorem which states:

If ray AD is between rays AC and AB, then ray AD intersects line segment BC.

In simpler words this theorem basically states that if we have a triangle ABC with a line that passes through the one of the vertexes such that the line passes through the interior of the triangle, then the line must intersect another side of the triangle.

It may seem redundant to actually prove this, but a large part of mathematics is proving what is seemingly obvious.  For example in an introductory proving course a person may actually prove 1 + 1 = 2 !

Overall, mathematics is about pattern and structure.  Mathematicians use logical analysis, deduction, and calculations within these patterns and structures.  Which, to state again, mathematics main emphasis is on LOGIC and deriving statements based off statements known to be TRUE (i.e. theorems, propositions, etc).  Mathematics main purpose is to extend on these theorems and propositions to find new patterns/structures and then see if these new findings relate to other fields (i.e. computer programing, biology, physics, chemistry, ecology, oceanography, etc.).  In a way, mathematicians prove true statements/ find patterns before we even know/understand how they relate to the real world or other fields. 

However, once a connection is made between the math and a practical application the ramifications are endless.  This can be seen in many fields such as:

·        Physics (quantum theory)

·        Biology (relations to the way DNA unknots itself before dividing)  

·        Computer science (algorithms behind programming languages and the use of fractals as a practical tool for compressing data on computer disks). 

There are also everyday fun practical applications that abstract mathematics gives us as well!  For example, the mathematics which go behind solving a Rubik’s cube.  Here a very complex series of steps can be reduced to relatively few simple algorithms. 

The link below goes over the math and shows the “everyday person” how to solve any 3 X 3 Rubik’s cube.

http://web.mit.edu/sp.268/www/rubik.pdf

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