Is Mathematics Limited?
Mathematics, like any human creations, has limitations. But wait! This is not the usual story. The usual story is quite the opposite.
Physicists, the biggest users of mathematics, rave about it's effectiveness. Modern physics would be impossible without mathematics.
If you use a tool, it's important to know its limitations. As an example, in physics the use of continuous functions has been extensive and as a result much of physics is expressed in terms of differential equations. But the big challenge now is to reconcile gravity with quantum mechanics, and continuous functions are almost certainly not up to the task.
You could argue that the use of the mathematical concept of continuity, convenient as it was, has actually set physics back.
But I digress. What about mathematics itself?
The limitation here is simple. In a nutshell, mathematics is a structured language that can make very specific statements. But it turns out that it's possible to make statements in mathematics that the language is not sufficient to prove or disprove. So in that respect it's incomplete. The work of Kurt Godel showed some of these limitations.
The challenge to mathematicians? The next advances in physics may need a whole new generation of mathematics. Perhaps even a rethinking of mathematics itself. Now that's some challenge!
Content written and posted by Ken Abbott firstname.lastname@example.org