Is it for me?

Well, if you got here in the first place it seems that the answer is yes! Still it may be useful to you to see what the gist of mathematics is as we, folks who do it for a while now, feel it. O.K. then. Did you ever wonder about questions like:

  • What happens if a dam breaks? How will the water flood the valley below the dam? When does the other end of the lake notice the broken dam at the one end? (applied mathematics and differential equations)
  • Can any theorem in plane geometry be discovered by a sophisticated enough computer? Are there bounds to artificial intelligence? (mathematical logic)
  • What does it mean to say two objects are the 'same'? Are they, for example, 'isomorphic', 'homeomorphic', or 'homotopic'? And what does it mean that maps should reflect the true nature of an object? This leads to the notion of a category and of localizing at a class of morphisms.
  • Is there a way to know the shape of our universe by observation? Are the rules of nature the same everywhere? (differential geometry or mathematical physics).
  • How many experiments does one need to justify a theory? (statistics)
  • Does the equation x^n + y^n = z^n have a solution in positive integers when n>2? Which natural numbers N are the area of a right triangle with rational sides? (number theory)
  • How many holes are in the donut x^n + y^n = z^n ? How many curves of a given genus are on a given surface? (algebraic geometry)

Mathematical research, as most of science, starts with an observation. Whereas in the physical sciences, like physics, biology or chemistry, the objects of study are primarily physical entities and observation usually means observation of a physical phenomenon, in mathematics the objects of study do not have physical presence in general: the number three is not a "thing". In fact, often the object of study is the very rule governing a class of phenomena (rules of motions, Hamiltonian systems) or uniting a class of phenomena under a single concept (three: three apples, three cars etc.). Mathematics is thus very much the science of science. This is why often mathematics is named the Queen of the Sciences.

Frequently, a mathematical theory starts with a "what if". It starts with not assuming something that was taken for granted until then. The invention of non-Euclidean geometry starts with "what if there was more than one straight line between two points?" Esoteric as it may seem, those geometries (elliptic, hyperbolic) are at the heart of modern theories concerning the structure of the universe and its future development. Here are some other striking examples:

  • Believe it or not but theoretically it is possible to take a solid ball of unit diameter, divide it into finitely many parts, and then reassemble them into two solid balls, each of unit diameter! This is called "The Banach-Tarski paradox". It conflicts with everything you know. Yet it is true. How the experience reconciles with theory is in this case measure theory.
  • Suppose there is a world where there are three possible answers to the question "is the door open?"
  • Is there a universe where every triangle is isosceles and every point in a ball is a center of the ball? It turns out that there is. It is the (metric space of) p-adic numbers. Not only that, but it replaces in some sense the so useful tool of local analysis that one instinctively does in geometry (by that we mean the study of a function or a manifold at an arbitrary small neighborhood of a point).

Hopefully, the impression that you get is that the bounds of our knowledge are our beliefs. As Alfred North Whitehead once said "It requires a very unusual mind to undertake the analysis of the obvious."

Last edited by on Tue, 06/10/2008 - 10:29