Thursday, December 10, 2009

Naming Functions, Why our love of the closed form is holding us back.

When communicating a function it is necessary to have a name for it. So if we wish to communicate the function which takes as input a real number and returns the product of that number with itself we say f = x^2 or f = x2 or f = x*x. These are all names for the same object just as 1.25 is the same object as 5/4. But more profoundly this "name" for the function gives us a handle by which we are able to hold it in our minds. Because of this x^2 becomes more than just a name for the object that corresponds to that particular function it IS that function. In fact were I to name the function differently say for instance f = x^2*(sin^2(2x) + cos^2(2x)) it is very likely that someone would tell me "but that is really just x squared!".

While the above example may seem rather contrived. I merely invented a function (or rather a "name for the same function") which obviously reduced to the function/representation of x^2. But is generally true that the representation of a function is so important to the way that we think about the function that knowing the representation of a function is considered the same as "knowing" the function itself. Of course if someone would generally be considered to "know" a function then it is not sufficient for them to know a description of the function such as df/dt = 1 and f(2) = 3. Although this is a description of the function that uniquely describes a function object I wouldn't be considered to "know" the function until I had provided the particular representation f = x + 1.

This preference for representing functions in a particular way is overwhelmingly strong. That particular type of representation being what is known as a "closed form". A closed form of a function is a representation which describes the function in terms of a finite combination of special functions and operations. The operations being of course addition subtraction multiplication division and composition. Composition of course being putting the output of one function into another eg. f(g(x)). The special functions change depending on who you ask about what a "closed form" is. The strictest definition would have the only special functions be the constant functions and the function f(x)=x. In which case the set of functions for which we would be able to create closed forms would be the polynomials. Usually though it is understood that certain very common functions should also be included so that ln(x), e^x, sin(x), arcsin(x) should also be included. The functions which can be put into closed form with these few operations and special functions are the stuff with which we almost exclusively deal. When we allow other types of operations such as integration often when we put nice simple closed forms in for the integrand the function we get out cannot be expressed in closed form. For instance a very useful function known as the dilogorithm is the integral from 0 to x of -ln(1-t)/t dt.

Early on in ones mathematical education the mere existence of such functions seems non-sensical. How can there exist functions that you can't write down? But of course you can write down the dilogarithm you write it down as the integral I just described. But it seems so strange to not be able to write that function in our more normal basis that in some sense it and the infinite number of functions like it become somehow not quite really functions. Our inability to name the function in our usual way means that our ability to think about that function is also impededed. We deal with this by simply effectively adding those functions which cannot be written in closed form to our set of special functions by simply assigning it a symbol. In the case of the dilogarithm for instance the symbol is Li2(x). Alternatively of course and more powerfully we can simply add the operations of integration and differentiation to our list of allowed operations. But integration is not as well understood of an operation as addition or multiplication which leads to the fact that it is often difficult or impossible to evaluate a complicated integral.

Here we stumble upon the real criterion which an expression of a function needs to meet in order for it to be an effective means of communicating that function. Any effective representation needs to be as easy to evaluate as possible. Here a "closed form" of a function is not always the best means of communicating a function. For instance in the case of the dilogarithm the integral cannot be computed using the standard methods of integration (obvious considering that the resulting function cannot be represented with the standard methods of representation) and so the numerical evaluation of the dilogarithm at a point using the integral representation is at best awkward. But the taylor series for the dilogarithm is extremely simple and the value of the dilogarithm can be quickly and easily be calculated using it to within the required accuracy (within reason of course).

But if someone were to specify a function by merely listing the values of the first few coefficients of its taylor series they would not be considered to really "know" the function. In order to "know" a function it is necessary to be able to express that function in closed form. Here I use close form in a generalized sense where any well defined operations are allowed and the only special functions are the constant functions and the identity function. For a mathematician in order to really know a function it really is necessary to be able to put it into this more general type of closed form. However in general we should be less picky about how we are allowed to describe functions in order to "know" them.

The situation is similar to the situation with our system of numbers. Early on our systems of numeration was limited to small positive integers. A number like 1000000000000000000 was too large to be able to be given a name and a number like 2.3 too strange. Over time numbering systems became better and stranger and stranger numbers were allowed into the club of numbers that could be named by the system of numeration. One of the biggest leaps in systems of numeration comes from the ability to specify ratios. With the ability to specify a number like a/b where a and b are integers numbers like 3.5 become things that one can use and reason with. But of course there are still numbers that are left out by this scheme. In fact almost every number cannot be specified as a fraction. The existence of numbers like the square root of 2 (which cannot be written as a fraction) is very much like the existence of functions like the dilogorithm. When we first encountered numbers like sqrt(2) they were frightening and mysterious but they are now routine. And while it is important for me to be able to specify a number like the square root of 2 in closed form as sqrt(2) it is at least equally important that I am not paralyzed by the fact that its decimal representation 1.41 etc is not exact. If I can specify a sufficient number of terms of sqrt(2) it should be considered that I "know" the number. In fact this is very much the reality of the way that people think about numbers now. If I specify a number only up to some precision it is understood that the number could vary around the value that I specified by a small amount. The fact that I do not EXACTLY specify the number is unimportant. Similarly it is important that we begin to think about functions in much the same way that we think about numbers. Functions should be something that we specify in the most convenient way with no preference between closed forms and more general forms such as the first few terms of a Fourier expansion. Just as we do not look down on specifying a number as 1.25 instead of 5/4ths. When functions are specified in these more general bases it should become second nature to us (as it has become for numbers) to interpret that as a valid representation of the function. The fact that the actual function could vary somewhat in a L2 ball around the function should not bother us.


Anonymous said...
This comment has been removed by a blog administrator.
Anonymous said...
This comment has been removed by a blog administrator.