Points and Limits

Traditionally, the typical procedure of constructing a metric space S looks like that: let us take a set B ((which will be called a base in the following; her, we don't discuss topology, and there should hardly ever be any confusion), and let us know how, for any two points x and y from B, to find a (real) number ρ, to be called the distance between x and y, provided the following conditions are satisfied:

(1) ρ(x, y) = 0 if, and only if x = y (2) ρ(x, y) = ρ(y, x) (3) ρ(x, y) ≤ ρ(x, z) + ρ(x, z)

In the above, the same quantifier "for any" is implicitly meant (and we do not yet ponder much over its sense and feasibility). The second axiom, to be true, just says that metrics is a function of a subset rather than an ordered pair, as the order of elements does not matter. This makes one think that thus defined distance is basically a kind of measure: it hides the size of what lies between the end points (the length or duration to walk). This too requires a separate treatment. The last property is commonly known as the triangle inequality; it is here, where most alternative theories of distance introduce their specific deviations (for instance, like in ultrametric spaces).

The formal mathematics holds that one is free to define anything in any way. Nobody cares for the reason. In reality, of course, definitions are never spun out of thin air: they are intentionally designed to produce what we need in the end. Since this ultimate goal grows from the common practice, arbitrary solutions are very unlikely to encounter. A layperson, however, may miss the inner sense of what is going on, and this makes it hard to get through the rest of the science, however logically derived from the primary notions. Typically, mathematicians are not inclined to put things plain; even worse, they try hard to disguise the practical motivation. Here, we'll look closer to what is usually left in the shade.

On the next stage of metric theorization, one comes to studying sequences of points from B, which are formally defines as some mappings of the natural numbers into the base. However, in fact, we need an algorithm that would allow us to choose one point of the base after another; it is this practical principle (along with a starting point) that defines the resulting sequence. Devoid of this systemic directedness, the collection of points is not really a sequence, but rather a mere set. One could easily observe that the very construction of the natural numbers is exactly like that: as soon as you reach a certain mark, go on to the next. The common arithmetic operations are introduced later on, formally imposed as an optional structure due to one of the possible ways of identifying different sequences. Well, let the topic wait for a better season.

So far, we are left with the succession of base points, which is traditionally denoted using the subscript notation: x_n, with n = 1, 2, ... (sometimes, starting from zero, or from an arbitrary positive number). Sequences may be of any sort, and the same points may be counted many times. In the trivial case, the whole sequence will contain a single point of the base, stubbornly reproduced at each sampling. In general, a sequence may exhibit multiple self-intersections, numerous loops. The important special case of this generalization is provided by periodic sequences (closed orbits). Eventually, sequences may (appear to) be random; this is yet another reason to avoid considering sequences as instantly given entities, to honestly compute them once needed, without any pretense to get the same result the next time.

Since we are interested in distances, we immediately discover two complementary modes of transforming sequences of base points (which may be complex and poorly tractable, or even not mathematical at all) into sequences of numbers (which seem to be much more familiar). First, we can compute the distances between the elements of the sequence: ρ_in(n; m) = ρ (x_n+m, x_n). Alternatively, one could fix a point in the base and determine the distances of all the elements of the sequence to this reference point: ρ_out(n) = ρ(x_n, x₀). The notation is to stress the different character of these quantities: either inner or outer structure. The former is well known from mathematical statistics, as a variety of autocorrelation. The complete collection of such functions (with all possible m) may be considered as a fait account of the inner organization of the sequence, regardless of its embedding in the incident space. The latter construct puts us in the framework of vector analysis, so that any point of the base could be represented by its radius vector; with a few reference points to start with (the required number depending on the nature of the base and the way of its arithmetization), we can specify the direction as well. When such multiple reference points form an independent sequence, we come to a “synthesis” of the inner and outer structures, sequence comparison: ρ_y,x(n; m) = ρ(y_n+m, x_n), or, conversely: ρ_x,y(n; m) = ρ(x_n+m, y_n).

The outer structure, in general, does not follow from the inner structure, and the other way round. This depends on both the organization of the base and the definition of the distance. Still, in many practical cases, the two structures seem to lead to basically the same view.

One of the most important ideas related to such correspondence is provided by the notion of convergence. Thus, if, for any positive real number ε, there is a natural number N, such that ρ_in(N; m) ≤ ε for all n ≥ N at some fixed m (usually set to unity), the sequence x_n is called a Cauchy sequence. Alternatively, if, for any ε, there is some N, such that ρout(n) ≤ ε for all n ≥ N, we say that the sequence of points x_n converges to the point x₀, or, equivalently, that the point x₀ is the limit of the sequence, which is commonly written as x_n → x₀. Convergence of the points of the base thus gets reduced to convergence of real numbers abstracted from their object area. This may lead to spurious effects, as the properties of numeric sequences do not exactly correspond to the properties of the objects of interest, and an imprudent judgement neglecting the essential features of the object area may lead to logical fallacies.

In metric spaces, the triangle rule makes every converging sequence a Cauchy sequence as well. The converse is not true, as there may be no point of the base infinitely close to the sequence points. On the other hand, the same triangle rule implies that, with x_n → x₀ and y_n → x₀, also ρ_y,x → 0. One is tempted to believe the converse to be true: if ρ_y,x → 0 then x_n and y_n either simultaneously fail to converge or converge to the same base point. If this were so, one could boldly consider all the sequences with ρ_y,x → 0 as equivalent, so that the base could be completed by such equivalence classes taken for the lacking limit points. In elementary mathematics, we simply observe that, if the distance between the limit points of equivalent sequences is non-zero, it is enough to choose ε equal to the half of that distance to make the three convergence conditions (for x_n, y_n, and ρ_y,x) violate the triangle rule; consequently, the distance between the limit points must be zero, and then by the first rule of metrics, the limit point will coincide, which seems to be the desired result.

I dare to make a boring suggestion: let us pierce the inviolable wall of mathematical rigor and look out through the tiny hole into the open of not so elementary world. Logically, the definition of the limit only says that the distance between the limit point and the elements of the sequence can be made smaller than any fixed real number. But this is not an identity, unless we deal with the same point infinitely repeated. Similarly, the parallel convergence to two different point means that the distance between these limits can be made smaller than any number, and not that the distance is zero. In other words, the distance between the limit points of two equivalent sequences is the limit of a numeric sequence rather than a ready-made real number. It tends to zero, but does not equal zero. In the early days of mathematical analysis, its founding fathers spoke of infinitely small quantities, never identifying them with real numbers. Later, the static paradigm has expelled the notion of an infinitely small value from any school courses; in the XX century, the term has been revived in the context of nonstandard analysis (which, however, merely tried to tame the essentially dynamic idea reformulating it in the same static language). Now, the distance between the elements of converging sequences is infinitely small, but it is not zero. Logically, we cannot apply the first rule of metrics to such quantities, except for a few special cases (say, the isolated points of the base).

For the same reasons, the triangle rule does not apply to the convergence process, being initially coined for finite (static) quantities. A similar rule concerning infinitesimal values would look differently: ρ (x, y) is an infinitesimal of the same or higher order compared to ρ (x, z) + ρ (x, z).

Let's dig a little bit deeper. When we compare two sequences, we leave the initial metric space S to arrive at a different space, with the base composed of Cauchy sequences on S. The equivalence of sequences is defined in respect to that new space. So, the zero distance as the measure of the difference of sequences is not the same as the zero distance between the elements of the initial base, and we have no right to compare x_n, y_n and ρ_y,x within the same triangle rule! This is a logical fallacy, term substitution. We have not accounted for the fact that the same number (name) may label qualitatively different entities. Distances in the space of sequences may be computable on the basis of distances between their elements; still, these are different notions of distance, whether their numerical values coincide or not.

To identify a class of equivalent Cauchy sequences in the space S with a point of its base B, one needs a special operation, which does not need to be always feasible, and which may be equivocal. For instance, consider a random identification with the points of some area in B, as described by a probability distribution. It is only in a very special case, when the distribution is represented with the δ-function (which, as we know, is not entirely a function but rather a functional), that the projection would give a kind of a point. In the same lines, if a Cauchy sequence does not converge to a base point, we cannot formally complete the base adding a new point, since such additional points may be logically incompatible with the object area of the theory and need a different theory, with a different base.

And finally, for advanced dummies. The collection of sequences converging to some base point x could be treated as its infinitesimal neighborhood. Each point therefore becomes a center of the cloud of infinitely small deviation from that point, its virtual variations. For every positive real number r, the number of elements in any sequences converging to x that remain outside the ball of the radius ris finite. Provided, in a meaningful theory, the sequences are constructed according to the same generic principle, one could estimate the average number of points –ε(r) outside the r-сферы-sphere; the sign has been chosen to reflect the fact that any sequence is only “shortened” at any level r. Those acquainted with nonstandard analysis may invoke a kind of ultrafilter. The quantity ε(r) serves as a measure of connectedness of the point x to the base B, resembling the binding energies of electrons in an atom or ion. This “energy” is negative due to a special choice of the reference level: we count from the threshold of detaching the element from the set (the analog of the ionization potential in atomic physics). Depending on the structure of the base (the object area of the theory), one will obtain different distributions of such binding energies. Thus, in atoms, we often observe series of discrete levels converging to the ionization threshold.

This can readily serve as a basis of the possible generalizations of the notion of an element's belonging to a set. Normally, an element either belongs to a set or not. Fuzzy set theories admit incomplete (partial) belonging; multiple belonging is an obvious extension. However, there are no indications of the origin and possible forms of the membership functions. Here, we relate membership (the way the element is linked to the set) to the structure of the object area selecting the possible trajectories. Every membership function then becomes a property of the infinitesimal neighborhood of a point and it can be numerically evaluated as an average of some operator acting in this inner space.

That's the point. Any superstructures of the base set produce a higher-level entity. In the same time, their presence means the development of an inner structure within a base point, its inner space. Thus mathematical object become hierarchies.

It is understood that converging sequences are not the only option available. One can consider any trajectories approaching the limit point, including continuous curves (like all kinds of spirals). Alternatively, one could speak of randomly selected approximations. Additionally, there are various non-explicit definitions (like set intersections). One could even associate points with algorithms or physical processes, with their specific symmetries (“spinor” components). The inner space of a point can be extremely complex, while the base retains a simple metric structure, and its points still coincide for zero distances. With all that, infinitely small distances do not imply anything until we indicate the level of discrimination, unfolding the hierarchy in a specific manner. The transition from the inner dynamics of each point to the bas-level properties requires a definite projection procedure (just like quantum mechanics represents the observables with operators).

We encounter inner spaces every time we are to decide on equality (equivalence) of one thing to another. Quantitatively, this means that some measure of difference tends to zero. In a rough overall comparison, the complexity of the objects to compare is not apparent. Still, as soon as we eliminate distinctions at some level, we need to deal with finer variations, penetrate the “inside” of zero. One does not even need quantum mechanics: it is enough to recall that the Solar system, with all its planetary richness, looks like a single point for the inhabitants of the nearest stars, nothing to say about distant galaxies.

For a purely mathematical illustration, take the equality of complex numbers. Formally, two complex numbers are equal when the distance between them (the absolute value of the difference) is zero. However, if a complex number is specified by the absolute value and phase, the points {0, φ₁} and {0, φ₂} are far from being the same. That is, zero distance does not mean perfect coincidence, as we need to approach zero by similar trajectories to ensure the equality of phases. Traditionally, nobody cares for such nuances, and the phase of complex zero is said to be undefined. In other words, the distance in complex plane is defined up to a phase factor, or, alternatively, as a phase average, so that, to be explicit, we have to explain how we compute that average and why.

School mathematics takes the linear algebraic form of the complex number with component-wise equality for primary. In this picture, zero is a single point, like any other. Similarly, a single infinite point is introduced (in projective geometry); however, in real calculations, we have to specify the way we pass these singularities. It is never possible to think of zero and infinity as ordinary numbers; this inclusion is purely conventional. They are not entirely numbers, as they do not exactly behave like numbers.

There is an obvious parallel with vector spaces: a vector as a directed quantity is not entirely the same as its coordinate representation. Where the zero vector is directed?

Can we rationally explain why the linear scheme should be the origin of all? Just admit that rotation might be much more important in the real world than mere translation. Why not? The component-wise representation may be considered as a special case, just how a straight line is a special case of general curve, along which the inner space of each point gets mapped onto the inner space of the next. The standard theory of metric spaces remains true, but only in the zero approximation.

Aug 1988