

Reforming the Mathematical Language of Physics (2002) [pdf] - MaysonL
http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

======
jessriedel
> 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
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.

~~~
jessriedel
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.

------
jacobolus
Previous submissions:
[https://news.ycombinator.com/item?id=3284160](https://news.ycombinator.com/item?id=3284160)
[https://news.ycombinator.com/item?id=9202543](https://news.ycombinator.com/item?id=9202543)

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

------
sullyj3
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.

------
MichaelCrawford
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.

~~~
jacobolus
> _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](https://en.wikipedia.org/wiki/Riemann_sphere)

Anyhow, what does your friend have to do with Hestenes’s paper?

~~~
DougMerritt
> 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.

~~~
MichaelCrawford
His website is [http://www.subbot.org/](http://www.subbot.org/)

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.

