How Far Can Distance Take Us

I would like to follow up on Alain Connes’ statement in my last blog. The weave of mathematical thought is tight.  The seeds of mathematics are found in early explorations of number relationships and in observations of what we call space. But symbol, stripped of content, has led to heightened powers of thought and discernment.  A simple illustration of what happens is this quick look at the weave of algebraic and geometric ideas.

One of the things that may not be clear to individuals with very little math experience is that the algebraic way of thinking about something and the geometric way of thinking about something are often two ways of thinking about the same thing.  Everyone who has taken an algebra class knows that a function, say defined by  y = 2x  assigns a y value to every x value.  We can draw this relationship using Cartesian coordinates. It is a line passing through the origin (the point 0,0) with a particular positive incline. You can use Pythagoras’ observation about right triangles to calculate the distance between any two points on the line or any two points on the infinitely extended Cartesian plane.  The space here is the plane and the objects in it are the points. We identify a particular collection of points, determined by an algebraic function, when the line is drawn.

But the distance between “points” with any number of coordinates is defined using the Pythagorean idea.  So we can calculate the distance between two points in three dimensions, or four, or infinitely many.  Even more impressive is the extension of this idea of distance, to the more general idea of a metric which is the distance  between any two mathematical objects in a mathematical space that has a metric.  For a any metric to be valid (including the distance formulas in more than three dimensions), just three things about it must be true (inherited from the original):

-the distance must always be positive (or 0 if we measure from one object to itself).

-we get the same result if we measure from say object A to object B or from object B to object A.

-the distance from A to B added to the distance from B to say C is greater than or equal to the distance from A to C.  (mimicking a triangle)

The metric we all know is the Euclidean metric, or the distance formula we started with.  But every non-Euclidean space has a defined metric and, if properly established, we can talk about the distance between two objects in any collection of any number of mathematical objects.  Functions themselves, like the one we started with, can be the objects in a collection. Then the distance between two functions can be defined.  In the theory of fractals, this kind of thinking can describe very complicated shapes as the limit of simpler ones (like the numerical limit in calculus).

The fruits of this kind of interweaving are everywhere in mathematics and, as Alain Connes observes, the weave cannot be separated without threatening the integrity of the whole body of thought.  But my own mind wanders in two directions when I think about this kind of mathematical development.  I wonder first how it is that we successfully identify the essence of an idea or, more specifically, what gives us confidence in our three properties of distance.  This confidence can only be justified mathematically.  It is the very thing that makes research in mathematics so difficult.  When we can no longer see the distance we wish to calculate, the mind has to hold onto something internal to itself.  Somehow the mathematician learns, through purely mathematical experience, the conditions for invention.  Related to this, but looking in the other direction is the miracle that it works.  Physics and engineering are built on these relationships.  And what follows is a book review in Evolutionary Psychology, where the author comments on mathematics applied to observations of evolutionary progression:

……we turn to the kinds of mathematical arenas, invented between the time of Newton and the time of Einstein, used to represent this evolutionary change. By and large these arenas are sets of elements, each element representing or “marking” a possible state of the evolving system. Usually the elements are labeled with numbers, or strings of numbers, that demarcate them quantitatively. The equations of motion specify the rates (or something similar) at which any one state gives way to others accessible from it, and so on through each moment of time in the evolutionary progression. The set of elements often has a metric, or natural measure of distance, associated with it, which allows us to say when a state has changed a little or a lot, and by how much. When suitably posed, the metric can be read as equipping the set of elements with geometric properties.

These properties are intrinsic to the evolutionary change and can be freed from the arbitrary manner in which we might map, or link, the state elements to their numerical indices. It is then natural to think of such a set, equipped with a natural geometry, as a “space” of “points,” each point marking a state, and of the evolutionary change as tracing out a path or trajectory though this space. So, for example, a state element or point represents a population in which the frequency of a gene variant has a specific value and that of a meme variant also has a specific value. Points with slightly different frequency values are nearby in the space. The equations of motion connect the points in an axiomatic game of “join-the-dots,” to predict which point will follow which as the population evolves. A glance in any textbook about mathematical population genetics, ecology, or neural network theory, for instance, will reveal endless content based on this general point of view.