The Line and the Circle

Okay, so regarding the line of infinite length and the circle of infinite circumference; what do you think? Are they related?

Think about what a circle is, a single line bounding a figure equidistantly from a single point, called the centre. The line itself is, however, unbounded. Do you see a way in which the circle might represent a kind of Little Infinity?

Think about the single straight line. Could this be a way of representing Big Infinity stretching out in both directions for eternity?

Are these two shapes so different though? What would a circle with an infinite circumference look like? Here is a picture of a finite circle:

Note that this picture has no particular size associated with it. The circumference is the distance around the outside of the circle. If this distance were infinite it would be impossible to know exactly where to begin the curve of the circle, precisely because there is an infinite distance the circumference must encompass. Once you begin to curve the circle, the shape would demand that the circle be completed (because all circles are the exact same shape with different radii and circumferences; eg. they are all similar) and therefore finite. The best way to demonstrate this might be to look at a very very very very small part of a circle, which might looks something like this if we zoomed way in on a single point:

This looks an awful lot like a line to me. So now, if a circle is representative of Little Infinity and a line is representative of Big Infinity they seem like they might be very much alike indeed.

Do you buy into this? What do you think? Is this convincing you of something more about infinity? Does it make you think maybe there is something interesting at work here that we need to look at more closely?

For Thought - Your High School Geometry

Remember your High School Geometry?

Remember lines? How they extend in both directions unendingly? They have length, but no width. What sort of thing is a line? What do you suppose an infinitely long line is like? Does it make sense to think or talk about an infinite line?

Now think about circles. What sort of a thing is a circle? What does it look like? If all circles look the same, how big could a circle be and still be considered a circle? What might a circle of infinite circumference look like?

Do you think you see a relationship between these two things? What is the image above a picture of?

Little Infinity

Of the two concepts we are setting out to explore, Big Infinity seems to be the harder of the two concepts to get a hold on, probably because of its lack of visible external boundaries. Perhaps we should start by exploring Little Infinity.

I suggested that one way to model Little Infinity might be to look at all the rational numbers between 0 and 1. How many numbers are there between 0 and 1? We can enumerate them in one fashion as fractions, decimals, or parts of numbers. (0.1, 0.2, 0.3, etc.) How many of these numbers would we say there are between 0 and 1? Does it seem that there is an infinite number of fractional parts between 0 and 1?

Imagine a number line that displays this:

This line is finite in length with distinct boundaries, yet within those boundaries it contains a seemingly endless number of possible divisions. As you approach 0 the numbers seem to get infinitely smaller, and as you approach 1 the numbers seem to get infinitely bigger. With integers like 0 and 1 we can count very easily from one number to the next (0,1,2,3,4, etc.). The next question to ask, then is how we do this with our divisions. What number follows 0? What is the next number after 0 or the last number before 1? What implications might this have for Little Infinity?

You might answer that the number directly following 0 might be something like 0.000...01, such that it is a decimal followed by an infinite number of 0's with a 1 at the end. However, I'm not sure that this means very much at all, since we pretty much arrive at the same problem of being unable to count from 0 to 1. If you believe all of this, it looks like you'll have to accept that there is this thing I have dubbed Little Infinity.

Maybe old Zeno wasn't as wrong as you thought.

If this all seems to be the case, then as you approach 1 from the direction of 0 it seems you could continually come closer and closer without being obliged to count to the number 1. It reminds me of the way some parents might count at their children when scolding them, "One, Two, Two...and a half, Two...and three-quarters..." They rarely reach three and are not obliged to count three because they can continually come closer.

Does all of this make you want to believe in infinity more? What implications does this seem to have for numbers or the number line? What does our investigation seem to say about Big Infinity? Do you think there could be larger and smaller infinities, or is the infinite singular in size?

Big Infinity and Little Infinity

The distinction between what I call "Big Infinity" and "Little Infinity" has been on my mind for a while, so coming back to it now as a starting point for postings seems particularly appropriate. It happens to be pretty central in some of the popular philosophical inquiries these days, which means it's worth getting a clearer understanding of what we mean by "infinity".

What does it mean for something to be infinite? You might answer that it has something to do with endlessness or eternity; perhaps it relates to time or the size of the universe somehow. It seems to me that my very first thought about infinity is "something that goes on forever" or maybe "an amount that cannot be quantified". Either way it seems to have a suggestion of magnitude within it.

"Big Infinity" is the kind of infinity that tries to suggest something that is unbounded on the ends. Infinite space means means space without a boundary at which it no longer continues. If you recall your High School Geometry lesson, you might think of Big Infinity in the definition of a line. Unending in both directions, right? It passes through all of the points that lie on itself.

So it's big. Really big.

In ancient times, the philosopher Zeno is attributed with an interesting paradox, also called the paradox of the arrow. Zeno said that an arrow fired at a target could never actually hit it because it would always cross half of the distance in the approach. By continually halving the distance the arrow would become incrementally closer but never actually hit its target.

This is something like "Little Infinity". Think of all of the numbers between 0 and 1 (0.1, 0.01, etc.) and you have a sense of what Little Infinity represents. The infinite amount of space between two defined boundaries. Tiny differences, minuscule differences, but differences nonetheless.

Start by thinking about bounded and unbounded infinity then. How do you percieve them? Do you believe in them at all? Do you think one makes more sense than the other? What does it mean for something to be infinite?

On Perseverence

Yeah, so I was reminded that I have a blog again. As usual this prompts me to once again examine my desire to write on a regular basis and do a bunch of active, visible thinking while I'm in the process of writing.

I'm just not very good at remembering.

What I need is a blogging alarm clock, something that says, "Hey...that stuff you're thinking about? GO WRITE IT DOWN ON YOUR BLOG!"

Okay, so here we go again with the trying over.