Math Is Never Just Numbers
It’s more
From the very little math I know, and upon reflection, I can conclude that math is never just numbers.
My first introduction to math was numerical.
The books are not ruled, but squared. Squares are unit houses for the numbers. Some numbers can interact with their neighbours, producing new numbers. Interfaces permitting these interactions and the resultant numbers are different. They are not numbers.
Numbers are not enough to explain the interaction among themselves. Numbers cannot deduce the other mathematical symbols. The reverse holds as well. A multiplication sign converts a small number and amplifies it. Zero will take any number and when multiplied with itself, contain it in its empty yet full self — it will convert it to zero.
Beneath a number exists a line — a number line. Without the numbers, it would just be a line. With the numbers, it’s a number line, ordinally spaced equidistant from one integer to the next. The numbers on the left of zero match the ones on its right, except for the negative sign.
These numbers can stretch to insanely large figures, and yet, when multiplied by zero, they are converted to nothing. In the presence of the multiplication sign, zero is the black hole from where no number escapes. Multiplication may be the first mathematical function that introduces the mystery of math, using simple numbers. Multiplication shows us that math is never just numbers.
Addition and subtraction are simpler in comparison. The negative has to be visible, but the positive can be invisible. It’s assumed that an integer lacking the positive sign has it by its side, but the negative has to conspicuously show its state and position on the number line. Math is never just numbers.
You could opt to use the ‘x’ sign or something as simple as a dot. It would have the same meaning and the same numerical products. X marks the spot. Or is it that X spots the mark? Snoop Dogg gets it: Math is never just numbers.
In literature, brackets are an aside, extra information that can be chosen to be ignored by the reader. In mathematics, they are to be given priority. The infamous acronym BODMAS reminds us that brackets are the first house to be visited before the other mathematical instruments.
That’s the word, instruments. Math is never just numbers. It contains instruments synonymous with Harry Potter’s wand — they can convert numbers into other numbers. Without these instruments, numbers remain anaemic, lacking the richness we know about them.
In simple math, brackets can be included or not. In complex math, brackets begin to appear more commonly. The numbers close to the brackets influence the contents of the bracket. Those separated by a function rest until the conflict or resolution is sought among the numbers near the bracket.
Broadly speaking, multiplication increases. Division reduces. Yet, reality shows that you can multiply by dividing. Bacteria and simple cells divide to multiply. Reality reminds us that math is never just numbers.
Reality can be just as puzzling in addition. Two rabbits and two other rabbits added together make four rabbits. But a single drop and another single drop, when added, give you one drop, only bigger. Visually confusing, but once you notice that one of the drops is bigger, a shift from mere addition to volume concentration shows what parameter has been added. It is a real-life manifestation of what the equals sign can do.
These signs make meaning from numbers. The equals sign, for instance, is like the mirror in the universe. It tries to make sense of one side of the universe with the other. 2+2 = 4. Two universes. But some mathematicians note how the equals sign is limiting and consider the alternative — equivalence. I didn’t bother going deeper. I know my limits.
Limits are the other concept in math that we are grateful for understanding. When an iteration is pushed to its limits, we get a better understanding of a concept. Primes, for instance, are closer when close to zero, but become more spaced out when the numbers increase. 2 and 3 are closer than 7 and 11.
Calculus is also a lesson in limits. It’s a limit of the natural count. The extremes of the large and the small. The infinite and the infinitesimal. They inject accuracy into calculations. You either integrate or differentiate. Thanks to the sign introduced by Leibniz, calculus can be simplified.
The world, however, is not as smooth as the sine curves. It is rough. Benoit Mandelbrot made sure we remember that. He took this concept seriously and proved it, advancing the mathematics of fractals. That is, fractured lines, to make an understanding of our rough world. In that regard, he united formal mathematics with the roughness of quantum mechanics. Life is not smooth, and neither is reality, down to the fundamental particles. It’s never just numbers.
Electrons jump from one point to another. Virtual particles appear and disappear. Other particles annihilate each other. It’s never uniform. Our universe is not uniform. And so is math. The number line can give the impression of uniformity, but it’s deceiving.
Consider the two numbers, 0 and 1. Between these numbers lies infinity. An infinite number of fractions is almost consistent with an infinite number of decimals. However, the decimals are far richer than the fractions. The Pi, for instance, is a transcendental number, not ending at the fourth significant number but continuing to infinity. Its fractional equivalent, however, is finite — 22/7. And yet, these infinities are confined to the two finite numbers, 0 and 1. Between any two natural numbers lies infinities.
It’s an interesting segue to infinities. Between two integers lies more than one infinity. When you take one infinity and raise it to the power of infinity, you get more infinities. Yet, infinity is only a symbol, a concept. It’s not ideally a number. Math is never just numbers.
Interestingly, the trope, numbers never lie, shows how insufficient they are after this short discourse. Math is never just numbers. If numbers never lie and yet numbers derive meaning from the symbols they are associated with, it means our ideas of truth are limited.
Indeed, Kurt Gödel showed us how incomplete math is. Some truths are self-sufficient. The very rules used to create meaning in math are not enough to explain the completeness of math. Consistency is not completeness. Math is never just numbers.
What I’m trying to say is…
The introduction to math is the carving of a single corner in a mansion with many rooms, which a sufficiently interested individual can never cover to completion.
The room, which is somehow stable, is unstable because it’s incomplete. The deeper one gets, the more convoluted and labyrinthine it becomes.
Math is never just numbers.
Yet, it’s all mathematics.
This song inspired some of the lines used in this article. Source — YouTube