Reforming the Mathematical Language of Physics (2002) [pdf]

http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

by MaysonL • 11 years ago

41 20 11 years ago

geocalc.clas.asu.edu

jacobolus6• 11 years ago

Previous submissions: https://news.ycombinator.com/item?id=3284160 https://news.ycombinator.com/item?id=9202543

Related conversation currently on HN: https://news.ycombinator.com/item?id=9746051

MichaelCrawford2• 11 years ago

Kuro5hin's trane is a classical linguist (ie. Ancient Greek) who knows enough about physics and math to shoot his own foot off.

For example he regards the notion that mathematics does not permit division by zero as demonstrating that mathematics is faulty.

He's a real smart guy as well as one of my very best friends but it's hard to debate someone who got honors in his Linguistics Masters degree.

jacobolus6• 11 years ago

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?

DougMerritt• 11 years ago

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.

jessriedel6• 11 years ago

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.

jblow• 11 years ago

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.

jessriedel6• 11 years ago

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.

fsloth• 11 years ago

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 :)

cat9• 11 years ago

"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.

jessriedel6• 11 years ago

Like jblow, you are under the mistaken impression that I am downplaying the importance of the geometric interpretation.

jacobolus6• 11 years ago

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.html http://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.

jessriedel6• 11 years ago

No where do I say that the geometric picture is not supremely useful. It's my day job.

sullyj3• 11 years ago

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.

Reforming the Mathematical Language of Physics (2002) [pdf]

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