A Fresher’s Treatise On Mathematics

As we released the Voyager I space probe out into orbit, it had on it, the engraved identity of our civilisation. It was meant to be nothing more than a unique postcard detailing our scientific evolution and where we stand, along with the pinpointed location of our planet with respect to the massive and endless universe.
The picture below displays the contents of the farthest traversed postcard ever in Earth’s history…or so to speak. Go ahead and have a study of the Voyager’s message.

The most trivial inference one can draw from this photograph, is that it doesn’t involve English anywhere nor any other colloquial languages.
Our belief is that the language of the universe is not in the architecture of arbitrary and ambiguous mix of phrases (that we concocted up over numerous eras for our own communication on Earth). Instead it should centre more on solid deductions and reasoning.
It is the inherent ability of intelligent beings to be forbearers of the scientific method and logical derivation. More specifically in this regard, many would argue that Mathematics is the language of the Universe, because all other sciences are built on top of it; like a tower resting on an immovable foundation.
Math lays out the groundwork for the skyscraper of Science and Scientific Thought in all forms, and that’s what I’ll be diving into, during the rest of this article.

So why is math cool? Why is it by some, considered the base of science? Well, to put it blunt,

Mathematics is just justifying what the Universe gave you, so that you can, in turn, operate back on the Universe.

It started with representation and then moved on to fixed definitions. Shepherds, if you had to take an example, counted their sheep to keep track and traders counted goods to be shipped and so on. The shepherd saw that he had twenty sheep, and simply said “I have 1 times 20 sheep” and stored this representation, to allow him to make his task of taking his flock out to graze much more simple. It’s that effortless. Today, based on the same fundamentals we’re able to engineer supercomputers that can create a time-lapse of our universe. It studies every single star in the visible range of the cosmos and how they would or have behave(d) as time progresses(d).
From an aerial standpoint, all this computer is doing is really just counting too. Just, way way way better. But it’s all the huge blocks of the subject that came in between a shepherd and the supercomputer that contributed to making the latter possible.

Calculus, the subject we created showed us how to extricate the continuous (∫) nature of the universe hidden behind discrete subsistences. It involves the quantitative definitions of the physical world right from velocity calculations (rates of change) to circulation and flow with regard to fluid mechanics
( and even beyond obviously, but none that I would know of at this point of my academia ). This is just one of the many examples where Math literally quantifies our physical world and what we perceive in it. What this means is, I can write an equation on a piece of paper that governs the very motion of whirlpools and celestial bodies.
For millennia, people have been qualitatively understanding Philosophy, Astronomy, Mechanics, etc. Advancement in Math was the most finely carved weapon at our disposal with which we defined our reality time and time again.
I suppose there is no satisfaction in formulas unless you feel their numerical magnitude that embeds the application in our environment.

In this manner,  Mathematics is a way of predicting the future.
Consider a trivial kinematical problem which most of us will be familiar with, say a car starting from rest with an acceleration of x m/s², hence find out the velocity at time t=10s, 20s, and so on. The integration of Mathematical principles with the physical world just provided us a means of accurately predicting 10 to 20 or more seconds into the future
(Obviously neglecting air resistance and other forces, but which can be incorporated nonetheless).
The beauty of Math stems from how far we can go with predicting the future with more complex problems and intuitions branching out into tons of more possible variations.

The subject is also widely considered abstract, because sometimes it’s hard to perfectly visualise what is put down as variables and operations on paper. Take this small write-up, by Reuben Hersh: (Adapted from What is Mathematics, Really?)

“Dieudonne and Cohen among others, have noticed that the typical mathematician is a philosophical split personality.
When he is working at his mathematics, he has no doubt that the objects he is studying have in some sense a real, objective existence. Whether it be ‘N’ the set of all natural numbers, R the uncountable set of real numbers, or perhaps some infinitely smooth infinite dimensional manifold. This is, so to speak, his week-day religion. It is a variety of “Platonism” (also often called “realism“). However, if he should be challenged to explain where, how, in what sense any of these invisible, intangible, infinite entities is real or exist, he is likely to turn tail, and retreat hypocritically into some form of formalism.
He drops any claim that anything in pure math really exists; all we really are doing, he explains, is making logical deductions from meaningless axioms. This is “formalism“, so to speak his Sunday religion. He quotes; This hypocritical schizophrenia is widely practised, and it serves the purpose, of getting the philosophers off our necks and letting us do math as usual.

Going off-piece for a bit, I’m going to talk about the cover image of this article.

First off, there exists a finite difference between the value of π and the value of 22/7. This is because π can never be represented as a ratio of two real numbers.

Secondly, π is irrational, which means it has an infinite string of non repeating real numbers. Now here’s a thought process exercise:
Imagine each of these numbers converted into their binary equivalents.
You now have an infinite string of binary numbers (or bits).
We are already accustomed to binary numbers representing information, in the form of bytes, kilobytes and even exabytes.
So based on the previous statement, what sort of information is conveyed by infinite bits in infinite non-repeated sequencing?
Just something to think about.

The abstract side of Maths allows us to quite symbolically walk on water. We can tread on entities that exist on paper but not in the perceivable world around us.
The simplest examples would be the concept of infinity and how n-dimensional spaces exist on paper. n-dimensions. And here we are in the middle of the three co-ordinate axes, defining ourselves as its origin, standing at the centre of the universe.
However, we are indeed far from ever fathoming the depths of these entities in our lives.

This representation of an n-dimensional manifold was best encapsulated to our understanding by Edwin Abott, in his Flatland Theory. So Here’s another thought experiment:

Take a piece of paper, draw on it, a circle and a square next to each other.
Now imagine if these figures were alive, (and will therefore be referring to them as proper nouns from this point on) how they would discern each other’s existence.
If the Square were to have it’s vision senses on its edge facing the Circle, and the Circle to have the same on the arcline facing the Square, then they would see each other as a finite line. More precisely, as one dimension.
Now take this piece of paper, and just release it into the air. The paper will slowly swivel and float down due to air resistance and finally hit the floor. If we were to neglect the thickness of the paper, we just envisioned a 2 dimensional space ( plane of paper ) floating around in a 3-dimensional space. And the best part? The entities of the 2-dimensional space had no idea they were part of a dimension greater than the ones they perceived.

The argument is that we ourselves may be part of a greater number of dimensions encompassed one on top of the other, but we would have no way of knowing, just like the Square and Circle.

Quite honestly, we don’t need to be intimidated by these Mathematical models all that much.
For one, being human, we’ll be pushing forward trying to solve whatever comes in our face and continuously advancing technology to maybe someday engineer our very own singularity.
Secondly, even at that stage, based on our history it seems like we will be provided with other problems with a higher intricacy at the very instant we make a dramatic breakthrough. That’s how it’s been so far and it seems like that’s how it will be in the future. In this sense, it’s important that we understand that despite human efforts and potential, we are humbled by the gargantuan span of our cosmos and its ramifications. Our greatest strength is also simultaneously our greatest weakness. We will keep solving problems, over various acceptances and disagreements but in the end it’s doubtful we will ever come across a final problem. We will never be done. Never finished. Perhaps when we do, it’s when our purpose ends, …or at least something of the sort.

This write-up was meant to be nothing more than how I personally look at, and feel about mathematics and it’s augmentation with our existence. That said, there’s still a long long way to go. I’m also looking forward to writing “A Sophomore’s Treatise On Mathematics” and perhaps even further on. It’s factually incorrect to end this article in the following way, but nonetheless I will sincerely be looking forward to infinity…

…and beyond.

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