Mathematics is both old and new. Traditionally, research in mathematics is the study of numbers and spaces: numbers such as the natural numbers 1, 2, 3, etc that everyone is familiar with, and spaces such as the 3-dimensional space that we see around us. From a modern viewpoint, one can argue that mathematics is still about numbers and spaces, but then one would have to widen the interpretation of "numbers" and "spaces" to include the great variety of forms that numbers and spaces have recently evolved into. This is what mathematical research has become.

In modern times the distinction between things that resemble numbers and things that resemble spaces is not always clear. Sometimes numbers are interpreted as spaces. Furthermore, the spaces that mathematicians now study often do not resemble what most think of when they consider "space". Consider the following three examples:

You probably know that the list of prime numbers is infinite, but you might be surprised to find out that it is still not fully understood how individual prime numbers are spaced out on the real axis as the list makes its way to infinity. But when considering prime numbers together as a single collection of numbers (i.e. as a space), various things become clear.

The "space" of various economic data is now a subject of mathematical analysis, although one would usually think of data as "numbers".

Some scientists are now proposing that the universe is actually a ten dimensional space, instead of three dimensional as we usually think of it.

So, in modern mathematics, numbers and spaces exist in a wide variety of forms, and included in these forms are mysterious properties and intriguing principles that mathematicians find very compelling.

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