Independence Through the Lens of Mathematics

On this Independence Day, let’s think about freedom in a new way—through mathematics.

We know that a function is just a rule that takes one set of objects and matches them to another set (mapping every element of the set to a unique element of the other set). In a way, it’s like how we react to different situations. That is given a scenario(input) we react to that scenario (output) like a function.

A general function has enormous freedom. In one place, it can be smooth and gentle; in another, it can suddenly jump. It can be predictable here, and completely chaotic there. It’s wild and free.

But what happens when we start adding restrictions?
The more rules we give a function, the less freedom it has.

Imagine 2 functions (General Functions)

To say they are identical, they must match at every single point in their domain.
Even if they match at infinitely many points, they could still be totally different everywhere else. That’s how much freedom they have.

What if we require that the 2 functions are continuous
and they identical only on a dense set (a set of points that is everywhere, no matter how small the region), then they must be equal everywhere.
Their freedom to differ get been restricted.

Now consider Analytic functions, these are very special: they are infinitely differentiable and can be written as a power series.
If two analytic functions are the same on just a sequence of numbers that has a limit point in the set, then they are the same everywhere.
They are so tightly constrained and “well-behaved” that a tiny amount of information completely determines them.
They’ve lost almost all their freedom.

What about Polynomial? A polynomial is even more restricted. If two polynomials of degree n are such that such that they match at just n+1 distinct points, they must match everywhere.

And finally Linear Functions (not actually final, becausethe worst condition is of the constant function - they have no free will). Two linear functions that agree only on the basis vectors of a vector space are already the same everywhere.
Just a few points determine their entire identity.

So, the more “well-behaved” we make a function—the more rules and restrictions we give it—the less freedom it has and the more predictable they become.

What This Means for Us?

This reminds me of people. On this Independence Day, I think we should try to be more like general functions—free to take many forms, not born to act in a single fixed way.

We shouldn’t let ourselves become so restricted that a few moments, choices, or labels define our whole life.
We should keep our freedom to be ourselves—in all our messy, unpredictable, and beautiful glory—just like a function with infinite possibilities.

Of course, we all need some structure to function. But let’s not let structure erase individuality.
Just like in mathematics, the more conditions society places on us, the less freedom we have to be who we truly are.
If we try too hard to be “perfect” and well-behaved, we might lose our freedom entirely.

Happy Independence Day! Let’s be free—but stay well-defined. 🎉🇮🇳

(The core idea of equality of different types of functions is inspired by Professor S. Kumaresan’s book The Pathway to Complex Analysis.)

Please read it for fun, I may make some changes in future to refine the analogy. But I wanted to post this thought on Independence Day so I posted it(Freedom).