> I have noted that perhaps a third of the students seem incapable of reasoning with vectors as abstract elements of a linear space. Rather, they insist on conceiving a vector as a list of numbers or coordinates. I have come to regard this concept of vector as a kind of conceptual virus, because it impedes development of a more general and powerful concept of vector. I call it the coordinate virus!
It's hard to tell what the author is proposing, but from the surrounding pages 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 may be possible for mathematicians (although I doubt it's sensible) but is not for physicists. In math, you're studying the logical implications of various axioms, and the abstract ideas are the goal. It's perfectly possible for a mathematician to learn and master an abstraction that was historically inspired by a physical system without personally ever actually connecting the abstraction to the physical system.
But 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. 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 students who don't fail to do so because they struggle to build the proper abstract machinery in their brains, not because they have been "infected" with the coordinate picture. The pictures aren't exclusionary, they are both necessary.
I disagree completely. I am not a physicist but I make video games which has the same kind of constant grounding in applicability. (We are always dealing with running physical systems, it's just that they are simulated.)
The week I learned to treat vectors as abstract objects, rather than arrays of coordinates, I experienced a drastic phase shift in my ability to program geometric operations effectively and clearly. The coordinates are still there, of course, but you have a lot more power over them.
The book "Linear Algebra Done Right" is all about this, and I absolutely recommend reading it if you haven't.
I don't see how we disagree. I wasn't downplaying the importance of the geometric picture, I was just disputing the idea that you could avoid teaching coordinates, or that the coordinate picture was somehow infectious in the sense that it would displace a geometric picture.
I began reading this posting and thinking that I really should acquire material on GA and end up ordering a linear algebra 'refresher" book bases on your comment. Funny considering how difficult it seems for GA to gain mainstream attraction :)
"But 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."
This breaks down the second you encounter, for example: Fourier transforms, wave equations, or hyperbolic geometry. Physicists are trying to understand the world, which in many cases does not comport itself in a way which is considerate of our naive intuition about mechanics and geometry.
The nth dimensional component of a vector still has a specific meaning within the context of the parent model, but it's far easier if you stop trying to force that and learn to deal with vectors as vectors instead. This allows for abstractions which are portable between the very many places where vectors are useful, instead of trying to force each value to fit an ad hoc cognitive model, with whatever flaws and conceptual limits that it introduces.
Considering the author (David Hestenes) has spent his whole career since the 60s as a physicist and physics teacher, I suspect he has a pretty good grasp on what is possible for physicists and helpful for physics students.
If you are a physics teacher too, I highly recommend studying his work on modeling, as well as studying his work on Geometric Algebra. That is, every physicist, especially mechanics teachers at the undergrad level, should try working through New Foundations for Classical Mechanics, and every physics teacher at every level should spend some solid time with http://geocalc.clas.asu.edu/html/Modeling.htmlhttp://modeling.asu.edu/
As for coordinates: there are many many problems in physics which can be solved in a coordinate-free way, saving huge amounts of error-prone symbol-twiddling bookkeeping while improving clarity and intuition. Even if you want to compute a numerical solution to something, solving it in the abstract case in a coordinate free way often lets you plug your numbers into a much nicer formula at the end.
Coordinates are not an essential part of the world, and not even an essential part of most physical models of particular situations. They are imposed by problem-solvers who have learned to always work with coordinates, without thinking, even for problems where the coordinates complicate problem solving. At the point where coordinates become necessary, they can always be added in with little difficulty. However, ripping the coordinates out of a half-solved problem is basically impossible.
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.
Not being a mathematician, I have no idea whether Geometric Algebra is in fact superior to the systems in general use currently, but it sounds exciting.
> he regards the notion that mathematics does not permit division by zero as demonstrating that mathematics is faulty
It’s very oversimplified to say that “mathematics does not permit division by zero”. Mathematics (at least in the modern sense, descending from Euclid) is just the logical extension of whatever set of axioms you want to set up. It’s entirely possible to operate on a concept of numbers which allows division by zero. For example using the extended complex numbers (the complex plane plus a point at infinity). See https://en.wikipedia.org/wiki/Riemann_sphere
Anyhow, what does your friend have to do with Hestenes’s paper?
> what does your friend have to do with Hestenes’s paper?
The comment is clearly totally irrelevant, so I did some googling, and it appears that "trane" is a bot created on Kuro5hin by MichaelCrawford, and that references to "Kuro5hin's trane" are some sort of inside joke.
Which in turn makes it look like the non sequitur above was posted by that bot, even if other posts from that account were by human.
In any case it's just adding noise to the conversation.
His real name is Robert Mitchell. He has a BA in Classical Greek as well as a Master's of some sort from U of Chicago. He worked for several years as a Java coder but for reasons I don't fully understand, decided that paying work is somehow evil and so flatly refuses even to apply for a job.
In part that's because he is very shy and so cannot tolerate office politics, and in part - again for reasons I don't understand - he has a passionate hatred for women and so cannot tolerate being in their presence.
He's heavily into jazz and swing music; he likes to call himself "trane" because that was John Coltrane's nickname.
It's hard to tell what the author is proposing, but from the surrounding pages 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 may be possible for mathematicians (although I doubt it's sensible) but is not for physicists. In math, you're studying the logical implications of various axioms, and the abstract ideas are the goal. It's perfectly possible for a mathematician to learn and master an abstraction that was historically inspired by a physical system without personally ever actually connecting the abstraction to the physical system.
But 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. 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 students who don't fail to do so because they struggle to build the proper abstract machinery in their brains, not because they have been "infected" with the coordinate picture. The pictures aren't exclusionary, they are both necessary.