When considering an infinitesimal distance/interval/in calculus, what is the intuitive interpretation? Is it too small to be measurable but still has some distance on an unattainable scale? Are there different interpretations? If so, what I am considering for the time being is the interpretation in calculus, but I'm still glad to hear of all views.
Note: I may not be talking about "measurable" in the same sense as measure theory. Sorry about that.
The intuitive interpretation (and one construction in non-standard analysis) is a sequence of distances converging to zero. So it's a process rather than one single distance. Every "real" distance $x$ can be thought of as the sequence $$x,x,x,\ldots,$$ whereas infinitesimals are sequences like $$1,\frac{1}{2},\frac{1}{3},\ldots.$$ Getting all this to work is Robinson's non-standard analysis.
First think about these properties of the number zero. Let $r$ be any positive real number, then...
$0 < r$
$0 > -r$
$0 \cdot r = 0$
$0 + r = r$
You can think of infinitesimals as "zero-like numbers". Let $r$ be any positive real number. Then if $\epsilon$ is an infinitesimal, it has these properties:
$\epsilon < r$
$\epsilon > -r$
$\epsilon \cdot r$ is an infinitesimal
$\epsilon + r$ is "infinitely close" to the real number r
They are useful in calculus for many reasons, but the major reason that someone in Calculus 1 will appreciate is that they simplify your derivative. The traditional definition of the derivative is that $f'(x) = lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$, which we must do because simply plugging in $h = 0$ as you would normally do to "solve the limit" yields the undefined algebraic expression $\frac{0}{0}$.
Infinitesimally, we define $f'(x) = ($ "the real number infinitely close to" $\frac{f(x+\epsilon) - f(x)}{\epsilon})$. This definition prevents you from having to do complicated limit arguments and lets you focus on the algebra of the situation.
Another good example where infinitesimals simplify calculus is the definition of continutiy. Usually, you define a function $f$ to be continuous at $a$ if and only if $lim_{x \rightarrow a} f(x) = f(a)$ which unpacks by the definition of limit to $\forall \epsilon > 0 \exists \delta > 0$ such that $|x - a| < \delta$ implies $|f(x) - f(a)| < \epsilon$. That is, the definition requires two universal quantifiers ("for all") and one existential ("there exists").
The infinitesimal definition of continuous is much easier to understand: "if $x$ is infinitely close to $a$, then $f(x)$ is infinitely close to $f(a)$".
The difficulty in using infinitesimals is not necessarily understanding their properties as much as it is difficulty in CONSTRUCTING them. Understanding how "infinitely close to" works is similarly not very difficult for the functions in any first calculus course. Fortunately, we usually don't worry about constructing real numbers in an introductory calculus course, so it should be reasonable to use infinitesimals in a similar way!
I have 2 ideas about how infinitesimals work, but keep in mind that this is just my model, my analisis, my interpretation of infinitesimals, and that I may be wrong, but here we go:
Imagine if the real numbers were quantized (I know they're not, but this model if very useful to understand my interpretation of infinitesimals),in other words, between r and r+dr there are no numbers in that interval: r+dr is the number that comes right after r. dr would be the smaller number you could ever think of, the smaller number you could ever measure, and all the numbers you see are just a huge number times dr. Take for example Planck's theory, he said that energy of oscillators was quantized (E=hf), and no matter what you get as the energy in that oscillator, it must be a number n times hf.
Imagine the classic distance vs time graph typically used to teach derivatives in schools. Speed = change of distance/time taken for the change or s = Δd/Δt . Now what is more interesting is to know the speed at a particular point t , ie; function s(t). Now using the above definition blindly it appears like speed at a point should be infinite, since s=Δd/Δt and Δt=0 at a point. Now that is absurd since you know there is a finite speed at each point if you are driving.
That's where infinitesimal or limits comes into picture, instead of the earlier definition let s(t) be Δd/Δt as Δt -> 0, or it as t becomes infinitesimally small. Think of this being a very small time interval that you cannot really measure the size of it , the moment you measure the size of it you can make it smaller than that (That is a property of real numbers , purposely skipping the math part here). Now here Δt != 0 since it is never the same point. Also the only value of speed which can satisfy the above condition is speed at the point t for if it not the speed at time t and is the average of the speed on the interval then I can make the interval still smaller by definition of infinitesimally small and contradict that. Thus the introduction of 'infinitesimally small interval' concept suddenly gives us a tool to define the value of a function at a point and we can solve for limits to get those values. This also defines fundamentally what a derivative is.