I suspect you still aren’t understanding what Hestenes means about coordinate-free approaches to problems and why he calls coordinates a “math virus” (cf. http://geocalc.clas.asu.edu/pdf/MathViruses.pdf). Again, I’d highly recommend you study his work in detail.
Also note:
> Let there be no mistake about the nature of the coordinate virus. There is nothing wrong with using coordinates when they are appropriate. It is the insidious idea that coordinates are somehow more fundamental or concrete than other mathematical objects that limit conceptual capacity.
Basically, I think you aren’t going to understand what Hestenes is talking about until you engage with his work more directly than just this short summary. Me trying to explain to you all of his ideas about solving physics problems is not going to be a worthwhile use of either of our time. If you want to understand that, you should read his books and papers, try to solve the problems therein, apply those tools and approaches to your own problems, etc. Reading a few paragraphs on an internet forum is not going to make these ideas sink in.
You said:
> it sure sounds like he's talking about teaching vectors as geometric objects with various manipulation rules and avoiding coordinate descriptions. [...] This is actually the sort of angle that [...] is not [possible] for physicists. [... For] physics, it is absolute essential that you are also able to think about vectors in terms of coordinates if you are to connect them to the real physical world, which has rulers and T-squares and so on. Physicists are trying to understand the world, not the abstraction.
The laws of the “real physical world” are absolutely not defined by T-squares or rulers. Those are imperfect measurement devices which we use to help us learn approximate relationships between things in the world and some human-defined standard measurement, but there is nothing fundamental about rulers. The fundamental thing is the structural/spatial relationships between objects which are directly interacting in whatever system we care about, not the tick marks on some stick which were drawn by a machine based on other tick marks drawn on some arbitrary global reference stick.
In any event, it is obvious that Hestenes doesn’t want to abolish measurement devices or ban the use of coordinates. You’re basically arguing with a straw man.
If you really want to get Hestenes’s answer to your comments, I suggest emailing him directly, instead of trying to get some strangers on Hacker News to stand in as proxies. I suspect he’ll be willing to at least point you at other sources to read, even if he doesn’t answer in detail.
> In my teaching experience as a graduate student, students adopt the geometric picture of a vector as soon as they are able to, because it's vastly easier and more pleasant to work with. (That's why the abstraction was originally developed.)
The geometric picture of a vector is the vector, and insofar as it’s “an abstraction” it is because the concept of a vector is itself the abstraction in question. The geometrical properties and relations of vectors are the fundamental nature of vectors. If students “struggle to build the proper abstract machinery in their brains”, what that means is they haven’t actually learned or properly been taught what a vector is yet.
Many many scientists I have met, and especially many engineers, when confronted with a problem, immediately start reaching for measurement devices and coordinate systems. This is an approach to problem solving which is inherently limiting, sometimes severely limiting.
> Basically, I think you aren’t going to understand what Hestenes is talking about until you engage with his work more directly than just this short summary.
Agreed. Unfortunately, I haven't heard anything that makes me think that would be worth the time. I could certainly be mistaken.
> The laws of the “real physical world” are absolutely not defined by T-squares or rulers.
They certainly aren't defined by them. Rather, rulers are how we make contact with the real world.
> The fundamental thing is the structural/spatial relationships between objects which are directly interacting in whatever system we care about
You are referring to the abstraction which humans have constructed, and which we believe approximately describes an external reality. (It's very possible, of course, that space is emergent, in which case neither coordinates or vectors are fundamental.) However, the way we make contact with the world, and the foundation from which we infer the abstraction, is based on coordinate systems.
To discuss this further would require us to dive into significant philosophy of science.
> The geometric picture of a vector is the vector, and insofar as it’s “an abstraction” it is because the concept of a vector is itself the abstraction in question. The geometrical properties and relations of vectors are the fundamental nature of vectors. If students “struggle to build the proper abstract machinery in their brains”, what that means is they haven’t actually learned or properly been taught what a vector is yet.
This is semantics, and I don't think we disagree here. There are multiple ways to build up a vector space from more basic mathematical objects. One is with coordinates, and another is with operators. They turn out to be equivalent, and a student certainly hasn't mastered the subject unless they feel confident with both.
Nonetheless -- and here is where we probably disagree -- the coordinate representation is more fundamental from a physical (through not mathematical) viewpoint, because of the direct connection it has to our empirical observations and tools.
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.
This was a neat summary. I'd add that often, you can use a length of string to measure something, without actually needing to know the numerical length. But then from that perspective, a tape measure is just ~2000 pieces of string neatly rolled up.
Also note:
> Let there be no mistake about the nature of the coordinate virus. There is nothing wrong with using coordinates when they are appropriate. It is the insidious idea that coordinates are somehow more fundamental or concrete than other mathematical objects that limit conceptual capacity.