Math Mutation 49:  What is Math Anyway?

    For the 50th Math Mutation podcast, I thought it's about time to
discuss a basic question:  what the heck am I talking about?
By the way, for those of you who are confused, I started with podcast
number 0, so this one really is the 50th.  So, what exactly is
mathematics, and why did I create a new podcast about it?
    As for definitions, there are a lot of them floating around on the
net.  Dictionary.com, for example, defines math as " the systematic
treatment of magnitude, relationships between figures and forms, and
relations between quantities expressed symbolically."  The most
popular definition at Urban Dictionary is similar:  "The unified
theorems and theories used in the analysis of order, sequence,
pattern, form, space, figures, and numbers."  But I think these common
definitions are really focusing on the trees and missing the forest.
For example, one exciting subdiscipline I have mentioned in numerous
podcasts is the theory of computation-- this clearly seems to be part
of math, yet one could argue that both these definitions exclude it!
I don't think the right way to define mathematics is to enumerate a
list of subtopics.
    My favorite concise definition comes from the 19th-century
philosopher Benjamin Pierce.  He defined mathematics as "the science
that draws necessary conclusions."  And I think that captures what I
find most interesting:  the idea that from simple premises, we can
discover surprising consequences.  From the basic ideas of length,
area, and angle measurements, we can discover the Pythagorean
Theorem.  From the definition that continuous functions don't skip a
value, we find that there must be opposite points on the globe sharing
the same temperature.  By extending the concepts of solid geometry, we
can derive properties of multidimensional objects we will never see.
Or from precise definitions of what it means to reason in a formal
system, we are led to Godel's theorem that there are fundamental
limitations to our reasoning abilities. 
    This is what I find most interesting about mathematics-- the odd
discoveries and surprising consequences, along with the strange quirks
of the people involved in discovering them.  Of course, applicability
to the real world, and its resulting presence as a requirement in many
areas of our education, are important, but these areas are
well-covered by other podcasts.  If something in the podcast does
happen to apply to the real world or help you out in a college class,
that's great, but that's not the main point.  The main point is to
talk about odd corners or applications that I find interesting,
whether they have some real-world purpose or not!  I'm betting that
there are plenty of people out there as geeky as me, who also enjoy
these kinds of topics.  Perhaps we can help each other to discover new
and fun areas we might have otherwise missed. 
    By the way, I'd like to thank those of you who have reviewed this
podcast on iTunes; I think the user named 'weepleman' did a good job
of capturing the essence, when he wrote that the podcast gives
"useless but interesting ideas and math".  Also, thanks to those of
you who emailed me, at my address erik@mathmutation.com.  I like to
hear from listeners out there, mainly to reassure me that I have some,
and ideas for future podcast topics are always welcome.  
    And this has been your Math Mutation for today.


References:

<a
href="http://dictionary.reference.com/browse/mathematics">Mathematics
at Dictionary.com</a>
<a
href="http://www.urbandictionary.com/define.php?term=mathematics">Mathematics
at Urban Dictionary</a>
<a href="http://en.wikipedia.org/wiki/Mathematics">Mathematics at
Wikipedia</a>