Re: [FRIAM] SOPA URLs

[Owen Densmore]

On Wed, Jan 25, 2012 at 11:10 AM, ERIC P. CHARLES  wrote:

> Owen,
> To shift slightly...  Aren't there conflicting metaphors here? How can
> something be free, and yet be a marketplace? In marketplaces people are
> orderly, follow the rules, pay for things, etc. The idea of a "free
> marketplace of ideas" is inherently contradictory. Either it is an anarchy,
> or it is a marketplace. What am I missing?
>
> Eric
>

Well, possibly "Free as in Beer" and "Free as in Speech" applies here.

My really naive view is that I want to keep the internet protocols neutral
in the sense that the internet simply delivers packets from one end to
another, with no intervention other than routing in between.  That was the
initial design, no application based knowledge within the end-to-end
delivery.

Initially this was an issue of keeping the internet very clean and indeed
has allowed maximal evolution.  Just think how hard it would be to create
new applications if the internet itself had to change for each one!

The closest we get to application specific knowledge is with the port
numbers which are preserved for particular apps.  But even these can easily
use other port numbers such as when we want SMTP to be secure .. we use a
different port number for SSH encoded mail packets.

So I think of the market place as being the endpoints not the transport.

Thus my thoughts on "Free as in Markets" is simply service between
consenting applications on the endpoints of the internet.

   -- 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

Re: [FRIAM] SOPA URLs

[Arlo Barnes]

Dear All:

To shift slightly...  Aren't there conflicting metaphors here? How
>> can something be free, and yet be a marketplace? In marketplaces people
>> are orderly, follow the rules, pay for things, etc. The idea of a
>> "free marketplace of ideas" is inherently contradictory. Either it is an
>> anarchy, or it is a marketplace. What am I missing?
>
> My feeling is that when someone uses the term "free", "equality", (
> in engineering - "robust", "optimal", etc.) It must always be applied with
> respects to something. I assume the congressman means "free" with respects
> to "entry into the marketplace", essentially against rent seeking, barriers
> to entry, etc. I've always found this particularly annoying in political
> cases applying "equality", where capitalist and communist systems apply
> it to different parameters (opportunity vs. wealth distribution ).

Also, the Internet is an anarchy if sorts (there is no overall government
besides technological necessity, only localised governments [site
administration] and citizen action [wikis, comments, and so on]), and a
marketplace in that ideas (including any content, any media) compete via
appeal and/or quality for attention/consumption, just as businesses in a
market, free (in the 'enterprise' sense) or otherwise, compete with prices
for customers.

Sincerely,

Arlo James Barnes

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] SOPA URLs

[Greg Sonnenfeld]

Hello,

I had also called up our congressman about SOPA/PIPA and asked him to
support OPEN instead. Have the senators supporting the bills reversed?

My feeling is that when someone uses the term "free", "equality", ( in
engineering - "robust", "optimal", etc.) It must always be applied
with respects to something. I assume the congressman means "free" with
respects to "entry into the marketplace", essentially against rent
seeking, barriers to entry, etc.

I've always found this particularly annoying in political cases
applying "equality", where capitalist and communist systems apply it
to different parameters (opportunity vs. wealth distribution ).



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 Wed, Jan 25, 2012 at 11:24 AM, Nicholas  Thompson
 wrote:
> Didja get the ms?
>
>
>
> From: friam-boun...@redfish.com [mailto:friam-boun...@redfish.com] On Behalf
> Of ERIC P. CHARLES
> Sent: Wednesday, January 25, 2012 11:11 AM
> To: Owen Densmore
> Cc: The; Group; disc...@sfcomplex.org
> Subject: Re: [FRIAM] SOPA URLs
>
>
>
> Owen,
> To shift slightly...  Aren't there conflicting metaphors here? How can
> something be free, and yet be a marketplace? In marketplaces people are
> orderly, follow the rules, pay for things, etc. The idea of a "free
> marketplace of ideas" is inherently contradictory. Either it is an anarchy,
> or it is a marketplace. What am I missing?
>
> Eric
>
>
> On Wed, Jan 25, 2012 11:35 AM, Owen Densmore  wrote:
>
> Ben Lujan got back to me w/ a form letter.
>
>
>
> I wonder if he'd really like to discuss:
>
> - Off-shore server farms
>
> - How to stop piracy but still be able to lend an e-book
>
> - Non US servers => only can stop within the routers, VERY dangerous.
>
> - Putting onus on ISPs is expensive and very difficult.
>
> - Not clear piracy really is as large a loss as claimed.
>
> - Much piracy occurs because media folks have not caught up yet.
>
>
>
> Example of last one: I torrent old TV shows that are not otherwise available
> .. although Amazon is starting to catch up and a few of my torrents are now
> available as VOD & free for Prime members.
>
>
>
> It is interesting to see his idea of net neutrality:
>
> I have long supported the concept of net neutrality (to ensure unrestricted
> access to the internet without limitation from internet service providers),
> and while I believe we need to address the growing threat of online piracy
> by foreign entities; we must ensure that the internet remains a free
> marketplace of ideas, that individuals' freedoms are protected, and that
> fair and equal access is available to all.
>
>
>
> Interesting way of putting it.  I'd prefer the basic notion that the
> internet protocols are designed to be service independent, depending on only
> the endpoints to provide the services.  Telcos hate this, and never believed
> internet could provide the low latency they need.  Wrong!
>
>
>
> Interesting to see how all this plays out.
>
>
>
>    -- Owen
>
>
>
> -- Forwarded message --
> From: Congressman Ben Ray Lujan 
> Date: Wed, Jan 25, 2012 at 7:30 AM
> Subject: Message from Congressman Ben Ray Lujan
> To: o...@backspaces.net
>
> Dear Mr. Densmore,
>
> Thank you for contacting my office in Washington, D.C. Discussing critical
> issues with my constituents is very important to me.
>
> The Stop Online Piracy Act (SOPA) is designed to address the growing issue
> of online piracy. Online piracy has become a major problem in the digital
> age. Much of this piracy is conducted not by individual users, but by
> world-wide criminal cartels that are frequently involved in other illegal
> ventures. The money from such piracy not only helps to fund criminal
> enterprises, but also reduces U.S. jobs, exports, and the overall economic
> competitiveness of our country. The impact of this is felt not only by large
> corporations, but also by independent artists and musicians, including many
> New Mexicans, who find their intellectual property rights violated online.
> Further, counterfeit medication and contaminated drugs that are sold online
> endanger the health of Americans, particularly seniors.
>
> Over the past few weeks, I have heard from many constituents who agree that
> piracy is an issue that must be addressed, yet have serious concerns with
> provisions in the bill.  After listening to them and hearing further from
> those with expertise in this field, I have taken another hard look at the
> bill. I share many of the concerns that have arisen and have decided that I
> can no longer support this legislation in its current form. I have long
> supported the concept of net neutrality (to ensure unrestricted access to
> the internet without limitation from internet service providers), and while
> I believe we need to address the growing threat of online piracy by foreign
> entities; we must ensure t

Re: [FRIAM] SOPA URLs

[Nicholas Thompson]

Didja get the ms? 

 

From: friam-boun...@redfish.com [mailto:friam-boun...@redfish.com] On Behalf Of 
ERIC P. CHARLES
Sent: Wednesday, January 25, 2012 11:11 AM
To: Owen Densmore
Cc: The; Group; disc...@sfcomplex.org
Subject: Re: [FRIAM] SOPA URLs

 

Owen,
To shift slightly...  Aren't there conflicting metaphors here? How can 
something be free, and yet be a marketplace? In marketplaces people are 
orderly, follow the rules, pay for things, etc. The idea of a "free marketplace 
of ideas" is inherently contradictory. Either it is an anarchy, or it is a 
marketplace. What am I missing?

Eric
 

On Wed, Jan 25, 2012 11:35 AM, Owen Densmore  wrote:



Ben Lujan got back to me w/ a form letter.

 

I wonder if he'd really like to discuss:

- Off-shore server farms

- How to stop piracy but still be able to lend an e-book

- Non US servers => only can stop within the routers, VERY dangerous.

- Putting onus on ISPs is expensive and very difficult.

- Not clear piracy really is as large a loss as claimed.

- Much piracy occurs because media folks have not caught up yet.

 

Example of last one: I torrent old TV shows that are not otherwise available .. 
although Amazon is starting to catch up and a few of my torrents are now 
available as VOD & free for Prime members.

 

It is interesting to see his idea of net neutrality:

I have long supported the concept of net neutrality (to ensure unrestricted 
access to the internet without limitation from internet service providers), and 
while I believe we need to address the growing threat of online piracy by 
foreign entities; we must ensure that the internet remains a free marketplace 
of ideas, that individuals' freedoms are protected, and that fair and equal 
access is available to all.

 

Interesting way of putting it.  I'd prefer the basic notion that the internet 
protocols are designed to be service independent, depending on only the 
endpoints to provide the services.  Telcos hate this, and never believed 
internet could provide the low latency they need.  Wrong!

 

Interesting to see how all this plays out.

 

   -- Owen

 

-- Forwarded message --
From: Congressman Ben Ray Lujan 
Date: Wed, Jan 25, 2012 at 7:30 AM
Subject: Message from Congressman Ben Ray Lujan
To: o...@backspaces.net

Dear Mr. Densmore,

Thank you for contacting my office in Washington, D.C. Discussing critical 
issues with my constituents is very important to me.

The Stop Online Piracy Act (SOPA) is designed to address the growing issue of 
online piracy. Online piracy has become a major problem in the digital age. 
Much of this piracy is conducted not by individual users, but by world-wide 
criminal cartels that are frequently involved in other illegal ventures. The 
money from such piracy not only helps to fund criminal enterprises, but also 
reduces U.S. jobs, exports, and the overall economic competitiveness of our 
country. The impact of this is felt not only by large corporations, but also by 
independent artists and musicians, including many New Mexicans, who find their 
intellectual property rights violated online. Further, counterfeit medication 
and contaminated drugs that are sold online endanger the health of Americans, 
particularly seniors.

Over the past few weeks, I have heard from many constituents who agree that 
piracy is an issue that must be addressed, yet have serious concerns with 
provisions in the bill.  After listening to them and hearing further from those 
with expertise in this field, I have taken another hard look at the bill. I 
share many of the concerns that have arisen and have decided that I can no 
longer support this legislation in its current form. I have long supported the 
concept of net neutrality (to ensure unrestricted access to the internet 
without limitation from internet service providers), and while I believe we 
need to address the growing threat of online piracy by foreign entities; we 
must ensure that the internet remains a free marketplace of ideas, that 
individuals' freedoms are protected, and that fair and equal access is 
available to all.

Once again, thank you for expressing your opinion on this matter. Please feel 
free to contact any of my offices if I can assist you in any way. I am very 
interested in your opinions and your concerns. To find more information 
regarding current legislation, services I can offer, ways my office can be 
helpful to you, or to sign up for my e-newsletter to get regular updates from 
my office, please visit my website at:   
www.lujan.house.gov.

 Sincerely,

Rep. Ben Ray Luján
United States House of Representatives
NM-03

 

 

 

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

Eric Charles

Professional Student and
Assistant Professor of Psychology
Penn State University
Altoona, PA 1

Re: [FRIAM] SOPA URLs

[ERIC P. CHARLES]

Owen,
To shift slightly...  Aren't there conflicting metaphors here? How can
something be free, and yet be a marketplace? In marketplaces people are
orderly, follow the rules, pay for things, etc. The idea of a "free marketplace
of ideas" is inherently contradictory. Either it is an anarchy, or it is a
marketplace. What am I missing?

Eric
 

On Wed, Jan 25, 2012 11:35 AM, Owen Densmore  wrote:
>
>
>Ben Lujan got back to me w/ a form letter.


>>
>
>>I wonder if he'd really like to discuss:
>>- Off-shore server farms
>>- How to stop piracy but still be able to lend an e-book
>>- Non US servers => only can stop within the routers, VERY dangerous.
>>- Putting onus on ISPs is expensive and very difficult.
>>- Not clear piracy really is as large a loss as claimed.
>>- Much piracy occurs because media folks have not caught up yet.
>>
>
>>Example of last one: I torrent old TV shows that are not otherwise available
.. although Amazon is starting to catch up and a few of my torrents are now
available as VOD & free for Prime members.
>>
>
>>It is interesting to see his idea of net neutrality:
>
>>I have long supported the concept of net neutrality (to ensure unrestricted
access to the internet without limitation from internet service providers), and
while I believe we need to address the growing threat of online piracy by
foreign entities; we must ensure that the internet remains a free marketplace
of ideas, that individuals' freedoms are protected, and that fair and equal
access is available to all.
>
>>
>Interesting way of putting it.  I'd prefer the basic notion that the internet
protocols are designed to be service independent, depending on only the
endpoints to provide the services.  Telcos hate this, and never believed
internet could provide the low latency they need.  Wrong!>
>
>
>
>>Interesting to see how all this plays out.
>>
>
>>   -- Owen
>>>
>
>>-- Forwarded message --
>From: Congressman Ben Ray Lujan <<#>>
>
>Date: Wed, Jan 25, 2012 at 7:30 AM
>Subject: Message from Congressman Ben Ray Lujan
>To: <#>
>
>Dear Mr. Densmore,
>>
>
>
>Thank you for contacting my office in Washington, D.C. Discussing critical
issues with my constituents is very important to me.


>
>
>The Stop Online Piracy Act (SOPA) is designed to address the growing issue of
online piracy. Online piracy has become a major problem in the digital age.
Much of this piracy is conducted not by individual users, but by world-wide
criminal cartels that are frequently involved in other illegal ventures. The
money from such piracy not only helps to fund criminal enterprises, but also
reduces U.S. jobs, exports, and the overall economic competitiveness of our
country. The impact of this is felt not only by large corporations, but also by
independent artists and musicians, including many New Mexicans, who find their
intellectual property rights violated online. Further, counterfeit medication
and contaminated drugs that are sold online endanger the health of Americans,
particularly seniors.


>
>
>Over the past few weeks, I have heard from many constituents who agree that
piracy is an issue that must be addressed, yet have serious concerns with
provisions in the bill.  After listening to them and hearing further from those
with expertise in this field, I have taken another hard look at the bill. I
share many of the concerns that have arisen and have decided that I can no
longer support this legislation in its current form. I have long supported the
concept of net neutrality (to ensure unrestricted access to the internet
without limitation from internet service providers), and while I believe we
need to address the growing threat of online piracy by foreign entities; we
must ensure that the internet remains a free marketplace of ideas, that
individuals' freedoms are protected, and that fair and equal access is
available to all.


>
>
>Once again, thank you for expressing your opinion on this matter. Please feel
free to contact any of my offices if I can assist you in any way. I am very
interested in your opinions and your concerns. To find more information
regarding current legislation, services I can offer, ways my office can be
helpful to you, or to sign up for my e-newsletter to get regular updates from
my office, please visit my website at: .


>
>
> Sincerely,


>Rep. Ben Ray Luján
>United States House of Representatives
>
>NM-03
>
> 
>>
>
>
>

>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
>

Eric Charles

Professional Student and
Assistant Professor of Psychology
Penn State University
Altoona, PA 16601



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] SOPA URLs

[Owen Densmore]

Ben Lujan got back to me w/ a form letter.

I wonder if he'd really like to discuss:
- Off-shore server farms
- How to stop piracy but still be able to lend an e-book
- Non US servers => only can stop within the routers, VERY dangerous.
- Putting onus on ISPs is expensive and very difficult.
- Not clear piracy really is as large a loss as claimed.
- Much piracy occurs because media folks have not caught up yet.

Example of last one: I torrent old TV shows that are not otherwise
available .. although Amazon is starting to catch up and a few of my
torrents are now available as VOD & free for Prime members.

It is interesting to see his idea of net neutrality:

I have long supported the concept of net neutrality (to ensure unrestricted
access to the internet without limitation from internet service providers),
and while I believe we need to address the growing threat of online piracy
by foreign entities; we must ensure that the internet remains a free
marketplace of ideas, that individuals' freedoms are protected, and that
fair and equal access is available to all.


Interesting way of putting it.  I'd prefer the basic notion that the
internet protocols are designed to be service independent, depending on
only the endpoints to provide the services.  Telcos hate this, and never
believed internet could provide the low latency they need.  Wrong!

Interesting to see how all this plays out.

   -- Owen

-- Forwarded message --
From: Congressman Ben Ray Lujan 
Date: Wed, Jan 25, 2012 at 7:30 AM
Subject: Message from Congressman Ben Ray Lujan
To: o...@backspaces.net

Dear Mr. Densmore,

Thank you for contacting my office in Washington, D.C. Discussing critical
issues with my constituents is very important to me.

The Stop Online Piracy Act (SOPA) is designed to address the growing issue
of online piracy. Online piracy has become a major problem in the digital
age. Much of this piracy is conducted not by individual users, but by
world-wide criminal cartels that are frequently involved in other illegal
ventures. The money from such piracy not only helps to fund criminal
enterprises, but also reduces U.S. jobs, exports, and the overall economic
competitiveness of our country. The impact of this is felt not only by
large corporations, but also by independent artists and musicians,
including many New Mexicans, who find their intellectual property rights
violated online. Further, counterfeit medication and contaminated drugs
that are sold online endanger the health of Americans, particularly seniors.

Over the past few weeks, I have heard from many constituents who agree that
piracy is an issue that must be addressed, yet have serious concerns with
provisions in the bill.  After listening to them and hearing further from
those with expertise in this field, I have taken another hard look at the
bill. I share many of the concerns that have arisen and have decided that I
can no longer support this legislation in its current form. I have long
supported the concept of net neutrality (to ensure unrestricted access to
the internet without limitation from internet service providers), and while
I believe we need to address the growing threat of online piracy by foreign
entities; we must ensure that the internet remains a free marketplace of
ideas, that individuals' freedoms are protected, and that fair and equal
access is available to all.

Once again, thank you for expressing your opinion on this matter. Please
feel free to contact any of my offices if I can assist you in any way. I am
very interested in your opinions and your concerns. To find more
information regarding current legislation, services I can offer, ways my
office can be helpful to you, or to sign up for my e-newsletter to get
regular updates from my office, please visit my website at:
www.lujan.house.gov.

 Sincerely,
Rep. Ben Ray Luján
United States House of Representatives
NM-03

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?

[Owen Densmore]

+1!  I'd love to attend.  Lets see if we can reach quorum.

Dean may not be on the list (he may be on wedtech, discuss, or sfx_graphics
instead), or he may have multiple email addresses and sent from one not
associated with his friam membership.  That's the typical problem getting
bounces from the list.

   -- Owen

On Tue, Jan 24, 2012 at 8:21 PM, Frank Wimberly wrote:

> 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

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

[Grant Holland]

Dean, Frank, Owen,

That would be 3 hours delightfully spent. Sign me up.

Thanks! -
Grant

On 1/24/12 8:21 PM, Frank Wimberly wrote:


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 quater