How do we know for sure that we can measure nonlinear curves using linear units? It’s really not that self-evident a truth, yet we treat it as one.

Evidence to the contrary is the proven fact that we can’t depict the surface of a sphere on a flat surface without distortion.

For example, how do we know that once we wrap a real number line along a nonlinear curve, there are no distortions on that real number line that would prohibit us from measuring that curve accurately using linear units? It’s not a simple question.

The same goes for measuring nonlinear areas using linear units. Calculus is fundamentally based off the non-self-evident assumption that it is possible. Yet, no proof nor even informal justification is given. Many Calculus books simply define nonlinear area as the Riemann sum of the areas of rectangles in that area. It’s not that self-evident of a definition.

Actually, I think that one has been covered rather thoroughly.
In the case of a circle, for example, a straight edge ruler can be rolled along the surface. In the case of volumes, the “submerging into water” technique proved the viability of the calculations, which depend entirely on the exact same math as surface areas. There are countless easily provable experiments to validate the mathematics.

I think that we seem to do quite well with the measurement of distances on our curved planet. Prediction of flight paths, parabolic curves of rocket trajectories, space-flight, meteorite/asteroid orbits etc

How do you know the ruler isn’t slipping? How do you know the points on the nonlinear curve CAN be measured using a linear ruler? It seems that in order to measure the nonlinear curve, you MUST use nonlinear units. It seems that linear units, then, are also nonlinear units. That doesn’t seem right…

There must be a way to mathematically prove that nonlinear curves can or cannot be measured using linear units.

Do you NOT find any problem with that fact that the sum of areas of an increasing number of rectangles below the curve–(said areas being definite)–approaches some number that may NOT be definite but is the area below the curve?

What I find curious is that you don’t, apparently, have any problem with the fact that there is no area of anything in the two abstract dimensions.