I recognize that I'm in the middle of a difficult subject matter, the generalization of a curve in a topological manifold, but in the interest of tackling a difficult subject matter not quite having the mathematical background (and that I'm a slow learner), I think I should take this section a bit slower. As such, I will interrupt the flow of the presentation and make my way forward by way of a more fundamental level of understanding, eventually getting into the world of differential geometry. In this way, I'm hopeful that the terms being used will become more familiar as I move along.

So, let me begin with what it means to be a real number, which is the foundation of the mathematics being used to describe the so-called space-time continuum. The continuum component turns out to be more continuous than one might imagine, and there are lots of terms floating around that are more or less taken for granted that need to be cleared up.

There are a number of numbers systems, including the integers, the rationals, the reals and the complex numbers. Integers and rational numbers are well understood and they both have the property of being infinite in number, though one is a subset of the other. Interestingly, they also have the property of being countable. Not countable in the sense in which the infinite number of them can be counted, but rather that we can in some sense enumerate them. Even though it appears that the number of rational numbers are greater than the number of integers, they are considered to have the same infinite number of them because we can obtain a one-to-one mapping of the integers and the rationals. I'm not going to prove this but I want you to realize that mappings of one set into another set each of which is infinite is the way in which infinities are brought into some kind of understanding. We can wrap our mind around them, so-to-speak.

Now, it turns out that you can't obtain a one-to-one mapping of the integers or rationals into the set of real numbers. One might say there are too many of them to do that. The infinity of the set of real numbers is larger than the infinity of the set of rationals. In terms of density, the reals are more closely packed, if you like. And it is this "closely packed" idea that caused the concept of continuity, originally understood as the continuation associated with some activity, such as, for example, what Zeno was trying to come to grips with in his drawing attention to the sequence of fractions that never quite reached its goal, somehow proving that motion is impossible, to become transformed from the rationals to the reals, despite that there were irrationals among them (also, what are called transcendental numbers). The naming of these numbers probably carries some significance, but from my experience within the world of mathematics, they are just labels.

In any case, the size of the infinity of the real numbers is a matter of interest to mathematicians, such as in dealing with power sets, where a power set of an infinite set has a greater size associated with its infinity than the set of which it is a power of. A power set is the set of all subsets of the given set. Do the real numbers have the same size as the power set of rationals?

Not to get carried away with the interests of mathematicians here, real numbers have been harnessed by mathematicians by their ability to define them. The way I was taught (and I don't believe this represents the best definition) is derived from what is known as a Dedekind cut. (You can easily look this up on the web.)

The idea behind a cut is that every real number can be expressed as the limit of a potentially infinite sequence (known as a Cauchy sequence) of rational numbers. You can understand this by noting that pi is a real number whose representation is a growing rational number (3.14159.....). In any case, a cut of the rationals is a segmentation of them into two parts typically labelled A and B, with A being numbers less than some given number (say the sqrt(2)) and B, the set that is equal to or greater than this number. Since sqrt(2) is irrational, there is no rational number equal to it, but its definition is well defined by this definition anyway.

And this way of thinking of real numbers leads to the notion of open and closed intervals. And interval of real numbers is open at one end if it does not contain its boundary value -- the cut at which the boundary value is defined. It is a closed interval if it does contain that value. Thus, in the A and B intervals above, the A interval is open (at both ends), while B is closed at the lower value. This is sometime written as (A) and [B), with the ')' sign signaling open, and the ']' signaling closed.

In both cases, the intervals are considered continuous.

Turning to geometry, the reals are represented as continuous lines, where each point of the line "corresponds" to a unique real number. An interval becomes a line segment (or rather the reverse). One might think that the length of the line segment represents the difference of the values of the points at each end of the segment. However, if the line is curved, there is a need to map the points of the curved line, which determines that correspondence (and an one-to-one mapping at that) onto some interval of the real line. A mapping is a function that takes numbers in one domain and "translates" them so as they can be referred to by numbers in another domain. (Note that the domains do not have to be different.) Determining distance, then, requires understanding the structure of the line, identified by the mapping function. And the way they are determined are by taking infinitesimals at each point and summing them over the segment. Each of the infinitesimals are considered line elements.

What, then, is an infinitesimal. This is considered to be an infinitely small length of a line segment surrounding a point and is usually denoted by the prefix symbol

d, prefixing the the segment's length, often referred to by the letter

s (thus the nomenclature

ds), and intends to be the limit of a sequence of lengths of smaller and smaller segments surrounding the point, each term of the sequence usually denoted by the prefix symbol

(thus the nomenclature

. One could sum each of these intervals and reach a better and better approximation of the length, where the summation of the sequence of intervals is denoted by the symbol

, which in its general form becomes

. However, the summation can be made exact by summing the infinitesimals. This is known as an integration over the set of infinitesimals and has the symbol

which in its general form is

. And the determination of the limit of the sequence of shorter and shorter length's is known as differentiation.

In my next post, I'll be moving into the geometry of space, covering ground already covered, but taking care to not let certain terms go by without making sure I've stated them to my satisfaction.

James