W. W. Sawyer makes a beautiful analogy regarding the mathematical landscape in his book Prelude to Mathematics.
Imagine farmers living in a country where no other tool was available except the wooden plough. Of necessity, the farms would have to be in those places where the earth was 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. …
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. … 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.
Related post: Easy to guess, hard to prove
2 thoughts on “Mathematical landscape”
On the other hand, there’s occasionally a proof that some regions of the landscape really are non-arable (solid rock? nutrient-free sand?) and will, therefore, never be plowed and certainly never settled.
How apt that Sawyer would use the wooden plough as a metaphor in mathematics. The confluent hypergeometric functions are called the wooden plough of mathematics and were once state of the art. In discrete statistical models, they still occasionally turn up some gems in unploughed soil.