Meaning, or Lack Thereof

Caution It's another expository post (*I mean it's better than just ranting, but Samyak please stop playing an expert*). The same cautions ahead(I'm no expert; I may be wrong). With that out of the way, let's begin: Foreplay  Introduction So! How do you define natural numbers? As is the case with other topics in intro analysis, you'll think (I definitely thought) that this is so boringly obvious . I mean, it's just how we count right? But many things about $\mathbb{N}$ aren't so obvious. For example, how would you prove that $a+b=b+a$ for $a,b\in \mathbb{N} $? You'd say it's blindingly obvious , but commutativity is not obvious for many other things(try, for example, rotating a Rubik's cube and you'll get it). Note that you can't justify with examples (you'd have to give infinite examples to prove this property) Now, this was a problem that presented itself in the mid-to-late $19^{th}$ century. People always assumed that numbers, in g...

When walking, I know that my aim  Is caused by the ghosts with my name.  And although I don’t see  Where they walk next to me,  I know they’re all there, just the same. -David Morin (Introduction to Classical Mechanics, 2008) Sounds eerie, right? But, in a certain interpretation of the notion of action of a path in physics, this is actually true, and the math behind it, calculus of variations, is truly fascinating in itself. Taylor Series: An overview For those of you who know calculus, it's taught late into differential stuff that we can find minimas and maximas of a function, say $f(x)$ by looking for the points where $f'(x)$ turns zero. Later on, in the same course, we get an intuition for why this is true, in the $\text{Taylor series}$, a powerful way to make sense of any ( differentiable) function, by using simple polynomials. So for any random differentiable function $f(x)$, the Taylor series, of $f(x)$ around the input $x=a$ looks like: $$f(x-a)=\sum_{n=0}^{\i...