I probably frightened half of my audience already :anguished:.
Maybe you hate equations. Maybe you love them. Maybe they’re irritating pieces of junk. Maybe they make you look insanely cool.
If they’re your worst nightmares, then maybe, just maybe, I got bad news for you. You use them without you even realizing it. As you try to unlock your stylish IPhone X by grinning at it (face recognition), it uses too many equations to calculate the right pixel intensities of patches on the screen, and many more to be confident that it’s you.
Maybe it looks like I’m snubbing you. Okay, go ahead and have an espresso. You might wanna swirl it to mix the contents thoroughly. But beware, the shape of the liquid surface it assumes has a mathematical equation too!
These guys are everywhere. Perhaps they’re the real ‘omnipresent’ entities. In my opinion, these guys are the ones which spin the giant cosmic web, cause day and night on earth, break your bones when you fall from the terrace of a three-storey building, and yes, help you score in a mathematics exam ._.
It so happens that they’re not just omnipresent, but too powerful (and beautiful?). And that’s probably the essence of this post. I will start by sprinkling two so-called equations in mathematics which happen to be ‘mysterious’ and ‘beautiful’ in many ways, in my article. You can stare at them for as long as you like, or flip through if it sounds like nightmare, for there’s a lot more that comes up later!
The Irrational Guys win it after all
If you’ve had a background in science, you would have almost definitely come across the Euler’s identity:
Euler’s Identity (Courtesy: Flickr)
Arguably described as the most beautiful equation in mathematics, it’s interesting to see how three fundamental constants play around with three fundamental arithmetic operations to produce something spectacular.
Look at it this way: 2.7182818… (infinite digits!) raised to a complex number times 3.14159265359… (infinite digits!) gives you a simple number you come across – negative one. Take time to absorb this.
The Million Dollar Baby
Riemann Zeta Function (Courtesy: Wikipedia)
This is an idea that is popularly expressed as the sum of all natural numbers equaling -1/12.
Wait, what?
If you have seen it before, I’m going to spoil the show a little bit – it’s a figurative way of representing the value of the Riemann zeta function at a point where it’s not even defined! But why is it a million dollar baby? A closely connected, famous, unsolved problem is the Riemann hypothesis, and it’s simple – anyone who solves it becomes a million dollars richer.
Whether you have come across this or not, I highly recommend watching 3Blue1Brown’s magnificent video on the Riemann hypothesis. Even if you don’t understand it, watch it for the sake of the visualisations.
The Universe runs on equations
And I’m not even exaggerating!
Let’s say you steal Hermione’s Time Turner, go back in time and enter your school on another boring day. Your physics teacher enters, dabbles around in a monotonous fashion and writes Newton’s law of gravitation on the blackboard finally. Although it’s not completely wrong, it’s not entirely accurate either; the theory of relativity propounded by Albert Einstein is a successor to that, and has survived every experimental test till now.
Let’s stick to Newton’s business for now; it’ll make our life much easier. This equation can move mountains, and not just that; it can literally move planets too.
Every planet’s orbit around the sun is mathematically obtained by using Newton’s law, Kepler’s laws of planetary motion and high school calculus. And so this makes us understand the fact that one year has 365 days much better.
But here’s the fun part; what if we tweak the numbers in those laws a little bit? What will actually happen?
On paper, nothing’s going to happen. Planets don’t give a damn about our existence anyway. But as the laws that explain nature evolve, and we get better and better theories, we see that we can explain a lot of peculiar events in the universe. It is therefore not surprising to say that the universe is bound to function using physical laws expressed in the form of mathematical equations.
Planets move that way because they’re bound to do so by Newton’s laws.
Black holes behave in a bizarre fashion because they’re bound to do so by general relativity.
Suppose we did tweak the numbers in equations. Instead of the inverse square nature of Newton’s law, we had some other number sitting there. Obviously this would mean that the force of gravity on us changes. But science is constantly evolving, and therefore maybe, just maybe, someone decides to alter Newton’s law after tons of experimentation. We are then pulled down by a force of a different strength, because the number in Newton’s law has changed! Imagine if this happens real time: someone is falling from the 12th storey of a building, and suddenly the value of acceleration due to gravity (whose value is obtained from Newton’s law) changes mid-air. You keep toggling the number, and the rate of falling keeps toggling too. That would be wild!
Although something of this sort is not feasible to happen and the universe has always been functioning as it is, imagine if a big debate in science happens: following the same idea, the universe would be confused too, and it wouldn’t be clear on what to do until we are clear of its understanding! This Utopian universe is nice to think about, but something close to this has already been happening: as we understand the universe better and better, we have more and more different ways of explaining how it works.
I’ll return to this viewpoint later. But do you feel exhausted already? I feel like eating Pringles; care to have a bite? But, but, that has a mathematical equation too!
Courtesy: Wikipedia
Math inspired from biology
Yes, that form of science that presumably requires a lot memorization actually has math and mathematical equations in it. I won’t talk about biology as such, but I will state that humans take inspiration from life in unbelievable ways.
Man was inspired by the way birds fly, and so he chose to fly.
However, he was clever enough to not copy-paste nature for the sake of greater returns; birds and insects use wings to lift themselves off the ground as well as propel themselves forward, but aircrafts use wings for lift and engines for propulsion. This is the basis of modern aerodynamics (oops, I have an end semester exam to complete!)
Courtesy: Wikipedia
Man was inspired by the way the human beings think, and so he decided to train machines to think too.
This is the goal of artificial intelligence, i.e. to be able to invent machines that mimic human beings as much as possible. A particular series of algorithms called neural networks is completely inspired by the way information is transmitted from one neuron to another in your body, mimicking the way your brain operates.
Courtesy: Wikipedia
Neural networks are powerful and used in too many real life applications such as face recognition, email spam filtering, medical diagnosis, astronomical data analysis and many more.
I started from talking about equations but sped off to mathematical principles and algorithms. Let’s take a diversion now to try and make sense of something more fundamental: we are intelligent enough to comprehend all of this.
We are intelligent enough to write equations and say the universe revolves around these.
What if it is so that the universe led to our evolution so that it can comprehend itself?
The Anthropic Principle
Any Dark fans over here?
In S01 E07, Jonas travels to the past to see missing Mikkel talking to his own mom, but chooses to not interfere, as it will “question his own existence”. The anthropic principle is not really that, but it has a similar flavor to it.
Simply put, it is the idea that the laws of nature are so cleverly and delicately designed that it gives rise to the evolution of humankind as it is today. Every single theorem in mathematics, every single fundamental constant in science and the universe itself are hardwired to enable mankind to be able to make observations on the universe and extract scientific laws. In short, if life were impossible, it would not have been known.
Unfortunately, like Jonas, you can’t meddle enough to threaten the existence of natural laws. The principle vaguely gives the idea that some random hunter-dweller is going to use the wheel for the first time to move things tomorrow – and that will go on to revolutionize civilization – but something already made everything possible in such a way that the hunter-gatherer will think exactly as he predicts tomorrow. This is because that ‘something’ already ensured that humans walk upright and have a sixth sense called intelligence.
People talk about two variants of the anthropic principle – strong and weak. They talk about the same thing in varying degrees, and it’s actually interesting to note that a couple of scientists predicted a few scientific principles using the anthropic principle.
So, as a plot twist, if you hate mathematical equations, it is not justified at all. The universe is a stage, living organisms are actors and you are one such supporting actor who feels he stands out of the crowd by saying he hates acting. As I said in the beginning, many of you would have been intimidated by seeing the title. But if you made it till here, and if you’re still not intimidated by equations, I’m ready to drop my nuke.
Standard Model Lagrangian (Courtesy: Wikipedia)
Refernces:
- Euler’s Identity: ‘The Most Beautiful Equation.’
- Visualizing the Riemann Hypothesis and Analytic Continuation - YouTube
- Riemann Hypothesis - Clay Mathematics Institute.
- Backpropagation — ML Glossary Documentation
- Hawking, Stephen W. A Brief History of Time.
- Hawking, Stephen, and Leonard Mlodinow. The Grand Design.
- Anthropic Principle, Wikipedia
