What makes Math mysterious — some brilliant results in Math
This post is the second in the series “For the Love of Math”. Find the first one here.
This post will be a long one: full of equations, references and anecdotes about famous mathematicians. Similar to my previous post, this too will be updated as and when I get the time to do so.
It’s 4 AM in the morning when I write this. I woke up to the sound of mosquitoes buzzing in my ear and couldn’t sleep so I decided I’d write this. I’ll be listing some well-known but amazing theorems and results in Mathematics that amaze me in their entirety. Here we go:
1. There are more than one forms measuring distances!
Well, in fact, there is an infinite number of forms of distance measurement.
Then the Holder’s Inequality states that:
This result might appear boring to some but if you look closely, this theorem can pour out results you cannot think of! For example, we can get the generalised triangle inequality, called the Minkowski Inequality, by elementary transformations of this inequality.
If you’re still not impressed, many fundamental properties of measures in mathematical theory are developed from this inequality for example: Probability and Lebesgue Measures develop a lot of their properties from these two inequalities.
Fun Fact: The Holder’s Inequality was actually first proved by Leonard James Rogers from Oxford and Holder rediscovered it independently.
Fun Fact [2]: Minkowski actually taught Alber Einstein during his time in Göttingen.
2. The theorem upon which all of Machine Learning is built.
What essentially Machine Learning practitioners do is that they find a map from a given set of numbers (dataset) to another given set of numbers (classes or output). Now have you ever wondered whether such a map exists or not for any sets of numbers? Have you wondered if out of these maps, are there maps that map the two sets 100% accurately? Turns out, and ML practitioners will enthusiastically agree with me here, that as the set of numbers become more and more complex, it becomes harder and harder to find a good map and impossible to find a perfect generalized map.
One of the most common ways to find the said map is to use polynomials as maps from each set to another. Now it’d be nice if we’d have some assurance for the accuracy of the polynomial map before we start finding the polynomial, right?
Weierstrass’s Approximation Theorem states that:
Suppose f is a continuous real-valued function defined on the real interval [a, b]. For every ε > 0, there exists a polynomial p such that for all x in [a, b], we have | f (x) − p(x)| < ε, or equivalently, the supremum norm || f − p|| < ε.
In other words, polynomials can arbitrarily approximate any map of any set of numbers to any other set of numbers. Seems a bit obvious, doesn’t it? The proof of this seemingly basic theorem comes from what is known as the Bernstein Polynomials. This extremely complex looking proof is exactly what it took for Weierstrass to formulate this theorem.
If this property wouldn’t have been true for polynomials, Amazon would not have been making billions through AWS, Deepmind by Google would not be busting the balls of every ML Research organization and China would not have been spying on its people so effectively.
Fun Fact: Weierstrass, “father of modern analysis”, was an autodidact in math and did not have any formal training whatsoever. Well that’s inspiring isn’t it?
3. The Infinite Monkey Theorem
The picture you saw at the beginning of this blog (of a monkey on a typewriter) is what makes this theorem as a more funny theorem in math. In simple language, it states that if a Monkey were to type on a typewriter for an infinitely long time, there will be a section of the typed text which is exactly the same as William Shakespeare’s Macbeth.
Before you start bickering, no this is not specific to Macbeth. Any and all sequences of text you can imagine will occur infinitely many times in the infinitely long text document written by a random operating stupid, albeit cute, monkey. If you’re one of those who scoffs at this result as blatantly obvious and of no practical use in the real world, the next paragraph is for you!
This theorem essentially states the famous quote “only in chaos can there be order”. I’ll give some examples where this theorem becomes really really useful. If you’re familiar with the generative models in Machine Learning, it’s basically teaching a computer how to make new stuff rather than just compute on the preexisting stuff, this theorem is the only thing that makes the generation possible. A computer cannot, latest till today, think on its own and create. It needs randomness to start with. That randomness can be moulded, transformed and made into something meaningful like generating faces of people who do not exist, generating fake videos of people saying stuff they won’t even dream of, fake pornographic content of famous people and everything else you might read in the news by the tag of GANs. Guided randomwalks can help solve problems in optimization that seems unsolvable by any iterative method.
If you walk long enough, you will stumble upon a breakthrough.
At the end of this article I would like to express again the importance of the Infinite Monkey Theorem. This article might also have been written by a monkey, you know? ;)
Footnotes:
More theorems are coming on this blog in the form of updates. Stay tuned!
- Holder’s Inequality — Excellent Wiki Page
- Weierstrass’s Theorem — Article on Neural Networks
