Re: [FRIAM] Complex Numbers .. the end of the line?

[Roger Frye]

On Jan 23, 2012, at 5:38 PM, Owen Densmore wrote:
> The obvious question is "what next"?  I.e. if we look at complex numbers at 
> 2-tuples with a peculiar algebra, shouldn't we expect 3-tuples and more that 
> are needed for operations beyond polynomial equations?

The Fundamental Theorem of Algebra states that complex numbers suffice.  But 
that only means if you all you need is to do is express the solutions of 
polynomial equations.  Abel showed that they do not suffice to solve quintics.  
Trigonometric functions allow easy solution of cubic equations with real roots, 
and Ramanujan used theta functions extensively.

Hamilton felt the need for quaternions, which are convenient for 3-D 
transformations.  There are generalizations in many directions: hypergeometric 
functions, Hestenes geometric algebra,  Carl pointed to Baez and octonions, 
which go on to Clifford Algebras.  Penrose has long advocated spinors as 
fundamental.  But conventional mathematical physics chose to generalize in the 
direction of linear operators and functional calculus.

Carl said it nicely as
> Suspect you are about to pop out of algebra and end up someplace else as 
> interesting.

-Roger



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Re: [FRIAM] Complex Numbers .. the end of the line?

[Owen Densmore]

Arlo:

> ...Would it not be better to say, "are there number(data?)-structures that
> provide for interesting algebras not yet considered?"
>

Yes indeed.  I was fumbling for a way to say that but ran out of steam!

Roger Critchlow:

> http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
>
Now that is interesting, and nice to know this is a broader conversation
than I had known.  GA's .. gotta look into them and their unification of
complex numbers and vectors.

Roger/Carl:

> Suspect you are about to pop out of algebra and end up someplace else as
> interesting.

As you say, I think this is the more fruitful approach.

All: The Cayley Dickson generalizations discussed in wikipedia: R C H O did
present an "answer" in that there are successful numeric extensions, that
complex numbers "are not alone".  As much as I wish computer graphics had
used them for their transformations rather than 4-tuples (homogeneous
coordinates) and 4-matrices, I'm not sure just how quaternions differ in
theory from linear algebra, which simply started in on generalized n-tuples.

In other words, simple n-tuple algebras might have put all these
generalizations from R into a single framework.  Why *aren't* complex
numbers simply our first use of 2-tuples, unified with the rest of linear
algebra.  Possibly the answer is that, yes linear algebras uses n-tuples,
but they focus on very different matters such as linear independence,
spanning sets, projections, subspaces, null spaces and so on.

Fun!  So now I hope I can find some interesting problems that ONLY can be
handled with some of these non linear algebraic higher number systems.
 Interestingly enough, I believe all of the extensions mentioned, as well
as all of linear algebra, have the same cardinality .. the continuum, right?

Thanks for the insights,

   -- Owen

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Re: [FRIAM] Complex Numbers .. the end of the line?

[Bruce Sherwood]

This link to an Oersted Medal talk is indeed of great interest. The
author, the theoretical physicist David Hestenes, built on the
foundation laid by mathematicians in the 19th century and in an
important sense completed their work on what is called "Geometric
Algebra", a framework which unifies much of the math done by
physicists, by providing geometric representations in all areas that
complement the algebra.

An analogy: The introduction of vectors by Gibbs made many things
easier to do and to say. Not only were many things easier to do, more
importantly the vector concept provided powerful new ways of thinking.
GA is like that. Some things that are very effortful with vectors
become much easier with GA, but more importantly it opens up new ways
of thinking and, as mentioned above, unifies many maths (plural) that
are usually seen as completely separate. Incidentally, Hestenes feels
that it's unfortunate that Gibbs took a piece out of GA and missed the
full point, but it's only the Gibbs vectors that most physicists know
about. For example, in the Gibbs view there are two kinds of vectors,
the regular kind and "axial" vectors. In GA there's only one kind of
vector; what has been called an "axial" vector is actually a 2D
"bivector" representing a planar element whose magnitude is its area.
An example is the cross product of two vectors.

For me, a striking example of the unifying power of GA is this: The
Pauli spin matrices were taught to me as special 2x2 matrices, special
to quantum mechanics, for describing the spin state of an electron. In
the GA framework, these matrices pop out as just a natural part of
living in a 3D world! Nothing particularly to do with quantum
mechanics! Stunning.

There are additional GA links on my home page, http://www4.ncsu.edu/~basherwo/.

I hasten to say that I am alas not an expert on GA, just a fan
observing from a distance. Also, I've been told that something called
"differential forms" has much of the same flavor and power, and I know
absolutely nothing about that.

Bruce

On Mon, Jan 23, 2012 at 11:43 PM, Roger Critchlow  wrote:
> http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
>
> -- rec --
>


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Re: [FRIAM] Complex Numbers .. the end of the line?

[Frank Wimberly]

Differential forms are (covariant) tensors.  That is they are multi-linear
functionals defined on n-tuples of vectors.  I wonder if tensor analysis
provides a framework for many of the mathematical concepts discussed in this
thread.

Frank
---
Frank C. Wimberly
140 Calle Ojo Feliz
Santa Fe, NM 87505

wimber...@gmail.com   wimbe...@cal.berkeley.edu
505 995-8715 (home)   505 670-9918 (cell)




-Original Message-
From: friam-boun...@redfish.com [mailto:friam-boun...@redfish.com] On Behalf
Of Bruce Sherwood
Sent: Tuesday, January 24, 2012 10:12 AM
To: The Friday Morning Applied Complexity Coffee Group
Subject: Re: [FRIAM] Complex Numbers .. the end of the line?

This link to an Oersted Medal talk is indeed of great interest. The
author, the theoretical physicist David Hestenes, built on the
foundation laid by mathematicians in the 19th century and in an
important sense completed their work on what is called "Geometric
Algebra", a framework which unifies much of the math done by
physicists, by providing geometric representations in all areas that
complement the algebra.

An analogy: The introduction of vectors by Gibbs made many things
easier to do and to say. Not only were many things easier to do, more
importantly the vector concept provided powerful new ways of thinking.
GA is like that. Some things that are very effortful with vectors
become much easier with GA, but more importantly it opens up new ways
of thinking and, as mentioned above, unifies many maths (plural) that
are usually seen as completely separate. Incidentally, Hestenes feels
that it's unfortunate that Gibbs took a piece out of GA and missed the
full point, but it's only the Gibbs vectors that most physicists know
about. For example, in the Gibbs view there are two kinds of vectors,
the regular kind and "axial" vectors. In GA there's only one kind of
vector; what has been called an "axial" vector is actually a 2D
"bivector" representing a planar element whose magnitude is its area.
An example is the cross product of two vectors.

For me, a striking example of the unifying power of GA is this: The
Pauli spin matrices were taught to me as special 2x2 matrices, special
to quantum mechanics, for describing the spin state of an electron. In
the GA framework, these matrices pop out as just a natural part of
living in a 3D world! Nothing particularly to do with quantum
mechanics! Stunning.

There are additional GA links on my home page,
http://www4.ncsu.edu/~basherwo/.

I hasten to say that I am alas not an expert on GA, just a fan
observing from a distance. Also, I've been told that something called
"differential forms" has much of the same flavor and power, and I know
absolutely nothing about that.

Bruce

On Mon, Jan 23, 2012 at 11:43 PM, Roger Critchlow  wrote:
> http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
>
> -- rec --
>


FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org



FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Re: [FRIAM] Complex Numbers .. the end of the line?

[Joshua Thorp]

Thanks Roger, interesting paper.  

I have always been fascinated at the relationship between the language of a 
mathematics and corresponding science that can be described with it.

--joshua

On Jan 23, 2012, at 11:43 PM, Roger Critchlow wrote:

> http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
> 
> -- rec --
> 
> On Mon, Jan 23, 2012 at 5:38 PM, Owen Densmore  wrote:
> Integers, Rationals, Reals .. these scalars seemed to be enough for quite a 
> while.  Addition, subtraction, multiplication, division all seemed to do well 
> in that domain.
> 
> But then came the embarrassing questions that involved the square root of 
> negative quantities and the brilliant "invention" of complex numbers (a + bi) 
> where i = √-1 which allows the fundamental theorem of algebra .. i.e. that a 
> polynomial of degree n has n roots .. but the roots must be allowed to be 
> complex.
> 
> The obvious question is "what next"?  I.e. if we look at complex numbers at 
> 2-tuples with a peculiar algebra, shouldn't we expect 3-tuples and more that 
> are needed for operations beyond polynomial equations?
> 
> This led me to think of linear algebra .. after all, there we are comfortable 
> with n-tuples and we can apply any algebra we'd like to them (likely limiting 
> them to be fields).
> 
> Wikipedia shows this:
> http://en.wikipedia.org/wiki/Complex_numbers#Matrix_representation_of_complex_numbers
> which illustrates an interesting job of integrating complex numbers into 
> matrix form, not surprising 2x2, although the matrices are the primitives in 
> this algebra, not 2-tuples.
> 
> 3D transforms do get us into quaternions which wikipedia 
> http://en.wikipedia.org/wiki/Complex_numbers#Generalizations_and_related_notions
> considers a generalization of complex numbers.
> 
> So the question is: are there higher order numbers beyond complex needed for 
> algebraic operations? Naturally n-tuples show up in linear algebra, over the 
> fields N,I,Q,Z and C.  But are there "primitive" numbers beyond C that linear 
> algebra, for example, might include?
> 
> What's next?  And what does it resolve?
> 
>-- Owen
> 
> 
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at http://www.friam.org
> 
> 
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at http://www.friam.org


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Meets Fridays 9a-11:30 at cafe at St. John's College
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Re: [FRIAM] Complex Numbers .. the end of the line?

[Russ Abbott]

This is *way *outside my area of competence -- to the extent that I still
have one -- but I remember reading about Conway's Surreal
numbers,
which may be of interest.

*-- Russ*


On Tue, Jan 24, 2012 at 10:21 AM, Joshua Thorp  wrote:

> Thanks Roger, interesting paper.
>
> I have always been fascinated at the relationship between the language of
> a mathematics and corresponding science that can be described with it.
>
> --joshua
>
> On Jan 23, 2012, at 11:43 PM, Roger Critchlow wrote:
>
> http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
>
> -- rec --
>
> On Mon, Jan 23, 2012 at 5:38 PM, Owen Densmore wrote:
>
>> Integers, Rationals, Reals .. these scalars seemed to be enough for quite
>> a while.  Addition, subtraction, multiplication, division all seemed to do
>> well in that domain.
>>
>> But then came the embarrassing questions that involved the square root of
>> negative quantities and the brilliant "invention" of complex numbers (a +
>> bi) where i = √-1 which allows the fundamental theorem of algebra .. i.e.
>> that a polynomial of degree n has n roots .. but the roots must be allowed
>> to be complex.
>>
>> The obvious question is "what next"?  I.e. if we look at complex numbers
>> at 2-tuples with a peculiar algebra, shouldn't we expect 3-tuples and more
>> that are needed for operations beyond polynomial equations?
>>
>> This led me to think of linear algebra .. after all, there we are
>> comfortable with n-tuples and we can apply any algebra we'd like to them
>> (likely limiting them to be fields).
>>
>>  Wikipedia shows this:
>>
>>
>> http://en.wikipedia.org/wiki/Complex_numbers#Matrix_representation_of_complex_numbers
>>
>> which illustrates an interesting job of integrating complex numbers into
>> matrix form, not surprising 2x2, although the matrices are the primitives
>> in this algebra, not 2-tuples.
>>
>> 3D transforms do get us into quaternions which wikipedia
>>
>>
>> http://en.wikipedia.org/wiki/Complex_numbers#Generalizations_and_related_notions
>>
>> considers a generalization of complex numbers.
>>
>> So the question is: are there higher order numbers beyond complex needed
>> for algebraic operations? Naturally n-tuples show up in linear algebra,
>> over the fields N,I,Q,Z and C.  But are there "primitive" numbers beyond C
>> that linear algebra, for example, might include?
>>
>> What's next?  And what does it resolve?
>>
>>-- Owen
>>
>> 
>> FRIAM Applied Complexity Group listserv
>> Meets Fridays 9a-11:30 at cafe at St. John's College
>> lectures, archives, unsubscribe, maps at http://www.friam.org
>>
>
> 
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at http://www.friam.org
>
>
>
> 
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at http://www.friam.org
>

FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

[FRIAM] Sustainable Model

[Robert J. Cordingley]

See http://xkcd.com/1007/
Does your model have this problem?  How would you know?

Robert C


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Re: [FRIAM] Sustainable Model

[Greg Sonnenfeld]

Sustainable,

Sustainable sustainable sustainable, sustainable sustainable.


Greg Sonnenfeld

“The scientists of today think deeply instead of clearly. One must be
sane to think clearly, but one can think deeply and be quite insane.”



On Tue, Jan 24, 2012 at 3:25 PM, Robert J. Cordingley
 wrote:
> See http://xkcd.com/1007/
> Does your model have this problem?  How would you know?
>
> Robert C
>
> 
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at http://www.friam.org


FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
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Re: [FRIAM] Complex Numbers .. the end of the line?

[Frank Wimberly]

This is a message from Dean Gerber.  For some reason it didn't reach the
List when he sent it.  I forward it at his request.  I will certainly attend
the lecture he offers.

 

 

Algebras

 

Owen--
  I think what you are looking for is the theory of algebras, generally
known as non-associative algebras. These structures are vector spaces V(F)
defined over a field of scalars F satisfying the usual axioms of a vector
space with respect to the operations of vector addition and scalar
multiplication, and an additional binary operation of vector multiplication
(called a product) that is distributive with respect the vector space
operations. To be specific, let x,y,z be vectors in V(F), let a,b be scalars
in F, and denote the product of x and y by x!y. Then  V(F) and the
product (!) define an algebra if and only if

i) x!y is in V. (x!y is a vector - the very meaning of "binary composition")
ii) a(x!y) = (ax)!y = x!(ay). ( Scalar multiplication distributes with
vector multiplication)
iii) x!(y + z) = x!y + x!z and (y + z)!x = y!x + z!x. (Vector multiplication
distributes with vector addition)

Since a vector space is always equivalent to a set of tuples, this provides
the multiplication of tuples you are looking for. For an n-dimensional
vector space the generic (general) product is completely defined by n-cubed
parameters, known as the structure constants. Specific choices of these
parameters from the field F define specific algebras and the properties of
these algebras vary greatly over the possible choices. For example, for n =
2 there are 8 free parameters and the complex numbers represent a single
point in this 8 dimensional space of structure constants. That particular
choice implies that the algebra of complex numbers is itself field.
Generally, algebras have no properties other than i) to iii) above, i.e.
they are generally not commutative or associative.

 

The Caley-Dickson procedure is a process by which a "normed" algebra can be
extended to a normed algebra of twice the dimension.  The only real one
dimensional normed algebra is the real number field itself. The
Caley-Dickson extension is just the complex numbers as a 2 dimensional
algebra, and it is also a field; in fact the only 2 dimensional field..

 

 The Caley-Dickson extension of the complex number algebra is the 4
dimensional quaternion algebra. But, the quaternions are NOT a field: they
are not commutative even though they are associative and a division algebra.
They are often known as a "skew" field, 

 

The Caley-Dickson extension of the quaternions is the 8 dimensional octonion
algebra, and these are neither commutative or associative, but they are a
division algebra.

 

The next step gives the nonions of dimension 16 at which  point we lose the
last semblance of a field because they are not commutative, not associative,
and not a division algebra. Thus, if we want fields, the complex numbers are
indeed the end point. All real division algebras are of dimensions 1,2,4, or
8! There are many division algebras in the dimensions 2,4,8, but only in n =
2 are all of them classified up to isomorphism.


I could go on, if you could gather up an audience of at least ten for a
(free) three hour blackboard lecture with two breaks. For an audience of
fewer than ten I would have to collect ten hours of Santa Fe minimum wages
for prep and lecture time. Its a beautiful subject with a very colorful
history, and includes the quaternions, octonians, Lie algebras, Jordan
algebras, associative algebras, everything mentioned by the FRIAM
commentariat. 

 

 Regards- Dean Gerber

 





From: friam-boun...@redfish.com [mailto:friam-boun...@redfish.com] On Behalf
Of Owen Densmore
Sent: Tuesday, January 24, 2012 9:34 AM
To: Complexity Coffee Group
Subject: Re: [FRIAM] Complex Numbers .. the end of the line?

 

Arlo:

...Would it not be better to say, "are there number(data?)-structures that
provide for interesting algebras not yet considered?"

 

Yes indeed.  I was fumbling for a way to say that but ran out of steam!

 

Roger Critchlow:

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

Now that is interesting, and nice to know this is a broader conversation
than I had known.  GA's .. gotta look into them and their unification of
complex numbers and vectors.

 

Roger/Carl: 

Suspect you are about to pop out of algebra and end up someplace else as
interesting.

As you say, I think this is the more fruitful approach.

 

All: The Cayley Dickson generalizations discussed in wikipedia: R C H O did
present an "answer" in that there are successful numeric extensions, that
complex numbers "are not alone".  As much as I wish computer graphics had
used them for their transformations rather than 4-tuples (homogeneous
coordinates) and 4-matrices, I'm not sure just how quaternions differ in
theory from linear algebra, which simply started in on generalized n-tuples.

 

In other words, simple n-tuple algebras might have put all these
generalizations fr

Re: [FRIAM] Complex Numbers .. the end of the line?

[Rich Murray]

With only an intuitive, skating on soap bubble films, grasp, I still
enjoyed reading all these posts -- look forward to some kind of
computer interactive game learning process to convey the widest most
comprehensive framework to unify all these partial frameworks -- I
suspect it will have to intimately include a fractal reality in which
every tiny region is in one-to-one correspondence with the whole --
since I experience that each of us is all of single entire unified
creative hyperinfinity -- Rich Murray, Imperial Beach, CA 91932


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