I see a lot of posts here questioning the validity of ‘instantaneous rate of change’ and most responses point to the epsilon–delta definition of a limit (or at least some informal version of it). My understanding is that with this definition we calculate the average change of a function across some non-zero interval, and then continue to decrease the interval to no end, at least in principle.
Since this process never ends, the object of study is always a curve, not a point. Albeit, it is an ever shrinking curve but it is a curve nonetheless. In one breath we stress that the interval never actually becomes zero, but then in the next breath we claim that the derivative corresponds to the slope of the tangent line at a point. To me this seems to be a contradictory view.
I have a view on this issue for which I would love to hear your criticisms. I argue that the issue stems from us constructing continua from points when we should be constructing points from continua. To explain my view, consider how we draw graphs. We start with a blank piece of paper (a continuum of sorts) and draw lines on it (i.e. axis lines, grid lines, functions, etc.). Only once these lines intersect can we identify points. With this construction, points are emergent, not fundamental. Since we can continue adding lines to no end, we’ll never completely fill the paper with ink. There will always be continua in between the lines. Might the derivative describe not the points, but the shrinking continua adjacent to the points?
My apologies if I was unclear in the last paragraph, I provided more details on this view in a YouTube video if you are interested.
I believe that continuum-based constructions don’t face the problem of completeness and paradoxes of actual infinity present with point-based constructions. As such, it’s a view that I find hard to let go of but I want to let go of it if it’s wrong. So I would appreciate it if you can point out problems with this view.
Thanks for your time.
If $f$ is differentiable at $a$, then "the tangent line at $a$" or "the tangent line through $a$" refers to a particular line through the point $(a,f(a))$ (specifically, the one with slope $f'(a)$). There's at most one tangent line for each point $(b,f(b))$ on the graph, so we can speak of the tangent line at a point, if it exists. I don't see any problem with this terminology.
I'm not sure, but it seems like you might be objecting to the fact that defining the derivative (the slope of the tangent line) involves secant lines which always pass through two relevant points on the graph. My personal gut reaction is "That's why we don't call it the 'secant' line at the point. The different word 'tangent' is used for that reason."
I'm not certain I understand what you're getting at here, but it seems like your idea might not work well for an example like $f(x)=\begin{cases}x^{2} & \text{if }x\in\mathbb{Q}\\0 & \text{else}\end{cases}$. There are holes in the parabola part of that graph, and it's not anything like a simple contiguous curve near $(0,0)$, but it's still differentiable at $0$ and so most would say it still has a tangent line there.
I'm having trouble understanding what this part of the discussion has to do with the derivatives part, but this depends sensitively on how (or "how fast") we add lines. For example, if I want to fill the unit square with vertical line segments, let's just say I fill $x=0$ through $x=t$ by time $t$, so that by time $0.1$ I've already added a continuum's worth of lines. Then I'll completely fill the square by time $1$.