Wooden Ploughs

I just found the following in a nifty little book called "Prelude to Mathematics" by W.W. Sawyer [Dover Books, 1955/1982], which uses overviews of several widely separated areas of mathematics to demonstrate the ways in which mathematics is the study of patterns.

The quote is about perceived (and perceiving) limits, and is not far afield from some of the remarks in Rosen's manuscript "The Limits of the Limits of Science", and also seems pertinent, by way of analogy, to the general study of Rosennean complexity:

[Note: The quote occurs at the end of a section discussing the "hypergeometric function", which is a particular common form into which a vast number of mathematical functions can all apparently be translated. p. 63-64]

"Besides the functions that occur in school work, there are many functions used by engineers or physicists - the Legendre polynomial and the Bessel functions, for example - which are particular cases of the hypergeometric function. In fact there must be many universities today where 95 percent, if not 100 percent of the functions studied by physics, engineering, and even mathematics students, are covered by this single symbol F(a, b; c; x).

What does this fact mean? That there are no other functions besides the hypergeometric type? Most certainly not; it is quite easy to write down functions of other types. The explanation lies in a different direction altogether.

Imagine farmers living in a country where no other tool was available except the wooden plough. Of necessity, the farms have to be in those places where the earth is soft enough to be cultivated with a wooden implement. If the population grew sufficiently to occupy every suitable spot, the farms would become a map of the soft earth regions. If anyone ventured beyond this region he would perish and leave no trace.

It is much the same with mathematical research. At any stage of history, mathematicians possess certain resources of knowledge, experience, and imagination. These resources are sufficient to resolve some problems but not others. If a mathematician attacks a problem which is completely beyond the range of the ideas available to him, he publishes no papers and leaves no trace in mathematical history. Other mathematicians, attacking problems within their powers, publish discoveries. Unconsciously, therefore, the map of mathematical knowledge comes to resemble the map of problems soluble by given tools.

But of course the discoveries themselves open the way for the invention of fresh tools. As the coming of the steel plough would change the map of the farmlands, so these new tools open up new regions of profitable research. But the new tools may take centuries to come, and while we wait for them, the frontier remains an impassable barrier.

Something of the sort seems to be the case with the hypergeometric function. It appears to be the limit of the kind of pattern we are able to recognize at present. If one goes just beyond its boundaries, everything seems formless. Beyond doubt, pattern exists there, but it is the kind of pattern we haven't yet learnt to see.

I do not wish to imply that the hypergeometric function is the only function about which mathematicians know anything. That is far from being true. There are other fertile valleys with which the wooden ploughs of the twentieth century can cope; but the valley inhabited by schoolboys, by engineers, by physicists, and by students of elementary mathematics, is the valley of the Hypergeometric Function, and its boundaries are (but for one or two small clefts explored by pioneers) virgin rock."