
Even Great Mathematicians Guess Wrong - fogus
http://rjlipton.wordpress.com/2011/04/13/even-great-mathematicians-guess-wrong/
======
xyzzyz
Well, his argument is unconvincing -- it only proves that there is no _normed_
division algebra on R^3. Division algebra on R^3 would be interesting and
maybe useful even without norm. However, its existence is still impossible.
One can prove it for instance by noticing that if it existed, then {1, v, v^2}
for any non-scalar v would span whole R^3 (the hardest part of proof), so v^3
would be a linear combination of them, say v^3 = a 1 + b v + c v^2 for some a,
b, c \in R. Then v is the zero of polynomial f(x) = x^3 - c x^2 - b^x - a. But
deg f = 3, so it has real zero, so for some q, w, p \in R, f(x) = (x-q)(x^2 +
w x + p), and 0 = f(v) = (v-q)(v^2 + w v + p 1), and factors on the right side
are non-zero, so we have non-trivial zero divisors which is imposible in
division algebra.

There is also a theorem due to H. Hopf, stating that the only finite
dimensional, commutative division algebras over R are C and R itself, but the
only proof I know requires pretty heavy topological machinery.

------
ArbitraryLimits
I think it's an even better story how he "guessed wrong" with Hamiltonian
mechanics, also; the only difference is that he never fixed the mistake and it
wasn't really fixed until the early 20th century with the development of
optimal control.

See "The Brachistochrone Problem and Modern Control Theory"
([http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.26....](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.26.825&rep=rep1&type=pdf))

The TL;DR is that classical mechanics can be formulated as finding a time
history of position and velocity which minimizes a certain function of them
both. Hamiltonian mechanics essentially optimizes the function over position
and velocity as if position and velocity were independent, then adds the
constraint that velocity be the time derivative of position via a Lagrange
multiplier. This Lagrange multiplier is the momentum.

Of course, no one in a physics course actually learns it that way;
undergraduates just memorize some equations and graduate students, if they are
lucky, learn if from the Hamilton-Jacobi equation (which is how Hamilton
developed it in the first place) where the momenta become the spatial gradient
of the function to be minimized when evaluated at all positions. (It's exactly
the same as the Bellman return function in dynamic programming; hence the name
"Hamilton-Jacobi-Bellman equation.")

IMHO, neither of these other developments is as intuitive as the Lagrange
multipler one; but because Hamilton's original formulation obscures it, no one
in physics learns it that way.

A fun postscript: when you consider the equations of classical mechanics in
terms of position and momenta (this combination is what physicists call "phase
space") they form a manifold with a special property called "symplectic";
symplectic geometry can be formulated in terms of matrices over the field of
quaternions. According to some people, this idea was a big breakthrough, but I
don't believe Hamilton himself, despite having formulated both, ever noticed.

~~~
starwed
_"IMHO, neither of these other developments is as intuitive as the Lagrange
multipler one; but because Hamilton's original formulation obscures it, no one
in physics learns it that way."_

I'm pretty sure learning a bit about Lagrange multipliers is standard for a
grad classical mechanics class. Granted, I've forgotten all I learned back
then, b/c it has never come up in anything else I do, but it was definitely
part of the course.

~~~
ArbitraryLimits
I believe it used to be standard to use Lagrange multipliers to handle
constraints on the state space in R^n as an alternative to using generalized
coordinates. For example, when you handle the case of motion on the surface of
a sphere you can either use R^3 with a Lagrange multiplier enforcing the
constraint that you stay on the surface of the sphere, or you could use
latitude/longitude as coordinates.

The interpretation of the momenta themselves as Lagrange multipliers is
completely nonstandard, though. Try googling "hamiltonian mechanics lagrange
multipliers"; you'll get either examples like the sphere thing or hits from
optimal control tutorials.

------
yummyfajitas
Another great example (technically a physicist, not a mathematician) is
Einstein and his quest for a local theory of quantum mechanics.

In his original paper deriving EPR (
<http://en.wikipedia.org/wiki/EPR_paradox> ), he believed it was a reducto ad
absurdum which invalidated configuration-space based quantum mechanical
theories:

 _We are thus forced to conclude that the quantum mechanical description of
physical reality given by wavefunctions is not complete [...] No reasonable
definition of reality could be expected to permit this._

<http://prola.aps.org/pdf/PR/v47/i10/p777_1>

Nevertheless, he thought a local and complete theory of QM was possible and
spent many years searching. Bell showed this to be impossible (after
Einstein's death). Eventually experiments showed the predictions of the EPR
paper to hold, thereby implying that the definition of reality is not
"reasonable".

If a scientist never gets things wrong, they aren't doing anything
interesting.

------
happy4crazy
If you enjoy playing around with quaternions, you may also have fun with
geometric algebra.

Geometric Algebra for Physicists is very clear and gently paced.
[http://www.amazon.com/Geometric-Algebra-Physicists-Chris-
Dor...](http://www.amazon.com/Geometric-Algebra-Physicists-Chris-
Doran/dp/0521715954/)

This one also looks good: [http://www.amazon.com/Linear-Geometric-Algebra-
Alan-Macdonal...](http://www.amazon.com/Linear-Geometric-Algebra-Alan-
Macdonald/dp/1453854932/)

------
aksbhat

         Complex numbers are a powerful tool for studying the two-dimensional plane: each point corresponds to a unique complex number . The beauty of this correspondence is that it allows you to add, subtract, and multiply points in the plane
    

I wish someone would have told me this five years ago, it is such a neat way
of thinking!

~~~
happy4crazy
My introductory physics class in college included a problem that investigated
the effects of "2d" gravity (a generic attractive force proportional to 1/r
instead of 1/r^2) on a point mass in the plane.

When the professor mentioned that I could write all the vectors as complex
numbers, and do regular old algebra with them, my brain almost melted.

------
iwwr
Another fascinating idea is to consider the split-complex numbers (and their
split-quaternion cousins) as ways to express hyperbolic geometry (in 2 or 3/4
dimensions respectively).

