Wednesday, August 25, 2010

Just what is an Epsilon, anyway?

As an introduction to myself, and to my blog, I have written the following:

Just what is an Epsilon?

Definition 1.   Epsilon is a Letter of the Greek Alphabet.

Definition 2.  When Calculus was first discovered by Isaac Newton and Gottfried Leibniz, and refined by Leonhard Euler, the foundation of their discovery was...difficult to describe.  Newton called them fluxions.  Leibniz called them infinitesimals.  I'm not sure what Euler called them, but every time he needed one, he'd ask for a number greater than zero, but smaller than any other number.

As Calculus matured, mathematicians decided that they didn't like this fuzzy, intuitive concept.  They wanted real, down-to-earth numbers, darnit!  So mathematicians formalized this concept:
Give me any number greater than zero--which we'll call our epsilon given--and no matter how small it is, I will find another number, also greater than zero (which we'll call delta) so that, as long as the input of my function is smaller than delta, the output will be smaller than epsilon.
This is a bold claim, and it's been the bane of calculus students for centuries.  But it's really a simple idea:  give me something small, and I can make sure that my function will fit nicely into that itty bitty region.

And it is this tiny idea that has made so many powerful things possible:  travel to the Moon, to Mars, and beyond; rides on roller coasters; tall buildings, long bridges, and high-flying airplanes; business analysis; growth of populations and decay of radioactive materials; understanding of electricity, heat, and water flow; and so much more.

Thus, epsilon is synonimous, not just of the small, but of the great power of the small as well.

What does Epsilon have to do with anything else?

The power of the small isn't just limited to numbers, or to engineering.  The power of the small is an important social principle as well.  Every economy is, of necessity, based on individuals making daily decisions on how they should live.  The individual is the smallest minority.  Any society that empowers the smallest minority--the individual--will thrive.  Any society that hampers the individual will sicken and die.  And the only way to empower individuals is to trust them to make the best decisions for their own lives.

We live in an era where State intervention is the norm:  onerous regulations and fines prevent individuals from exploring new employment opportunities, and even push small businesses out of business--or sometimes just prevent their growth--leaving only big corporations to provide us with goods and services.  Hideous taxes prevent individuals from spending time with their families.  Perverse incentives have been created by our governments to spend rather than to save.  All this drains our motivation to try new things, or to do what's best for ourselves and those we love and care about!

This is not to say I'm against big corporations...but corporations need to start somehow, and the best ones start as a couple of nerds tinkering in a garage somewhere.  Corporations also need competition, so they could continue to provide the best service--and so they could be killed off, when they don't.  Without nerds tinkering in their garages, there is no means to create the competition that ensures the vibrancy of our ways of life.

Hence, this is a place to advocate for the Small, and to highlight the obstacles that they face.

Who am I?

My name is Alpheus, and I am a mathematician.  At one time, I wanted to be a college professor...but after a few semesters of teaching, I realized that I needed to do something different.  I am still trying to figure out how to best leverage my love of mathematics, so that I could provide for my family, and have enough free time to pursue other interests as well.

I have also become increasingly aware of how intrusive our American government, and even our State and Local governments, have become--and how these intrusions interfere with our pursuits of happiness.  Politically, I've come to describe myself as a "conservative libertarian":  I have come to understand that, while we need to seek to learn what is good, and then do it, I have also come to understand that the less that Government interferes with that, the better off--and the better morally--we are.

No comments:

Post a Comment