Tuesday, July 12, 2011

What Is Your Math Philosophy?

Calvin and Hobbes
Truth is a tricky thing. Some things are true because we say so, others are true independent of us. Lets call these two classes of truths Created Truths and Discovered Truths. Some examples of Created Truths:

  • In chess a bishop moves diagonally.
  • The ninth word in Never Gonna Give You Up is rules.
  • The main character in the above comic strip is named Calvin.
  • Darth Vader is Luke Skywalker's father.
  • Peter Parker is Spiderman.

All of these things are true because someone said so (the creator(s) of chess, Rick Astley, Bill Watterson, George Lucas, and Stan Lee).

Some examples of Discovered Truths:

  • The Earth revolves around the sun.
  • Water and ice have the same molecular structure.
  • Your eye color is determined by your genes.
  • Fire burns wood.
  • Iron is attracted to magnets.

Each of these is true independent of us. No one decided that the Earth revolves around the sun, it just does.

Now consider Mathematical Truths. Some examples:

  • 2 + 2 = 4
  • There is no largest prime number
  • 1729 can be written as the sum of two cubes in two different ways
  • The graph of y = x²-2x+3 has a minimum at (1,2).
  • The area of a circle with radius r is πr²

Are these truths discovered or created?  More on this soon.