Well, it’s your (and your students’) loss. I don’t think I’m going to be able to convince you here.

The idea that rulers and coordinate systems are a “foundation” of anything is just complete utter nonsense. Unfortunately, it’s a bit of nonsense which has infected the world, and is dogma among many scientists and engineers, mainly because they haven’t ever really considered the question before.

It’s tragic that our culture, especially our intellectual culture (as compared to e.g. plumbers, carpenters, or mechanics), systematically devalues geometric and spatial reasoning in favor of big tables of abstract numbers. We learn to interact with reality through textbooks and calculators instead of direct physical experience. But oh well.

> the coordinate representation is more fundamental from a physical viewpoint

The two most brilliant engineers I ever met, quite literally scientific geniuses, eschewed standardized measurements wherever possible, and built tools relating objects directly to other objects. Their solutions to problems were built around directly applying one object’s shape (or other attributes) to another object, without ever needing to write down arbitrary numbers.

Want to fit two things together precisely? Trace the shape of one directly onto the other. Want to make sure holes in two wooden surfaces align? Drill them on one board using a rigid metal template, then flip the template over and drill the other board. Want to have a level shelf (with respect to gravity) along a wall? Get a long double-open-ended hose, positioned in a wide U shape, and fill it with water, and then compare the height of the water at one side of the hose to the height of the water at the other side, and mark those two heights on the wall. Want a table to rest on four equal-length legs? Saw one of them and then use it as a template for the other three, and make sure to allow enough flex in the tabletop to keep all four legs stable even when the ground isn’t quite flat.

Describe any electrical or mechanical device you want to one of these guys and he’d build it for you, usually out of $20 of parts bought at a corner hardware store or found in a scrap bin somewhere, with a more effective design than any commercial version you could get for $1000, or take any broken device to him and he’d fix it, all without ever once touching a ruler or constructing a coordinate system.

Measurement has done great things for science, and for society. Being able to write the results of experiments on paper in an unambiguous way and transmit them around the world and down through centuries, so they can be repeated by strangers, is a wonderful thing. Scaling our production processes up to produce millions of nearly-identical objects using mostly machines and unskilled labor, compared to a society where every bit of production requires skilled human intervention, has dramatically raised our standard of living. Nothing quite perfectly interfaces with anything else because neither was made directly with reference to its mate but they get close enough through careful industrial process monitoring.

But treating measurement as fundamental is, as Hestenes says, a kind of disease of the mind, imprisoning our creativity. Rulers are no more fundamental to reality than NAND gates composed of silicon transistors are fundamental to computation. They’re just one type of tool, an arbitrary human choice.