Uncertainty & Approximation - Physical & Ephemeral
No man has ever drawn a perfect circle, no metal beam ever made has been perfectly straight - no point drawn has ever been zero dimensional, and no sphere ever truly spherical. We strive for predictability and determinism, yet we live in a reality that is inherently uncertain and indeterminate. If there is a message in this post, which there may or may be not, it would be that the only certainty - is that there will always be uncertainty.
99.999...% of all problems ever encountered in the physical sciences (in fact, in any discipline) are not exactly solvable. We are raised in an environment within which infinite precision is impossible, yet we are led to believe it is attainable. Unless you were educated in an accelerated or advanced high school program, you've probably never truly internalized this concept until attending college level quantitative science courses (I know I didn't).
In this current context, I am specifically referring to the inability to solve for a variable(s) in your equation exactly. Take, for example, calculating the necessary curvature for an airplane wing to achieve a certain amount of lift under certain conditions. This calculation inherently relies on the number π, and π itself is an irrational number (in fact, a transcendental number) - that is, it has infinitely non-repeating decimal places. This means that your calculation of the curvature of the wing will only ever be solved to as many significant digits as you have used in your implementation of π in the equation.
And that is only a consideration on a mathematical level. A similar degree of uncertainty arises on a computational level. Hopefully you're using a computer to do these calculations, but you're computer itself has an inherent degree of granularity hard-coded within its ability to perform calculations. This includes the fact that there is a limited set of numbers your computer can use constrained by the number of possible bits allocated for a number type (I may be fudging some terms here - but the message is there). This also includes how your computer traverses a continuous set - say, the number line between 0 and 1. Ultimately, there's a discreteness introduced into what is otherwise a supposed continuous process due to the fact that there is a limited set of numbers a computer has available to itself.
Uncertainty in this form is so prevalent in all fields in the physical sciences that there are whole disciplines and numerous methods devoted specifically to address the problem. Disciplines like Numerical Analysis and methods such as Perturbation Theory are are, by design, meant to expedite approximations so that they can be computed faster and with more certainty (numerical analysis) - and, in fact, to give us at least some approach to solving for something that cannot be exactly solved for (perturbative methods).
I now want to revisit the concept of mathematical uncertainty that I touched upon earlier. If you were to draw a line segment and label one end 0 and the other end 100, and you were then to, say, throw a dart which hits a point on said line. You have a 0% chance of hitting a whole number. In fact you have a 0% chance of hitting even a rational number. MOREOVER, you have a 0% chance of hitting even an algebraic number (algebraic numbers encompass even irrational numbers such as the square root of 2 because it can be written as the solution to an algebraic equation - that is x2 = 2).
The set of whole numbers, rational numbers, and algebraic numbers all have a size of infinity. The set of all real numbers also has a size of infinity. But the size of the set of real numbers is, in a sense, so much more infinite than the size of whole numbers, rational numbers, and even algebraic numbers (see the Continuum Hypothesis). The point is, just as how almost all equations cannot be solved for exactly, the same is true of numbers. 'Almost All' (this is a true mathematical condition, Almost All) numbers are transcendental. They cannot be expressed as the solution to some finite equation. They cannot be computed to infinite precision within finite time. And this is not a physical concept we are referring to. We are referring to the concept of numbers! We are referring to something that exists only in the mind! Yet even that cannot be determined to infinite precision.
The uneasiness about this idea was contemplated as far back as the ancient Greeks (probably even farther). The discovery of irrational numbers is usually attributed to Hippasus, who belonged to a group called the Pythagoreans. The Pythagoreans were a sect who followed the teachings of Pythagoras influenced mostly via a study of astronomy, mathematics, and music. It is said that when Hippasus brought to light his discovery of irrational numbers at sea, his fellow compatriots threw him overboard "...for having produced an element in the universe which denied the...doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios."
If you've made it this far, congratulations, you've basically just had me fart into your ears for the last 5 minutes. I'd like to say more, but this can go on forever (maybe in another blog post). Cherish the moments you have, cherish the ones you love, cherish the time you spend with one another. Their fate is as uncertain as your own. The only thing certain, is uncertainty itself - the only constant, is change itself. And now I go back to coding. As always - stay happy, stay strong, stay passionate - and take care of yourselves.