George Lekoff?

[Stephen Guerin]

Anyone make George Lekoff's Colloquium and care to give a short review?
        http://www.santafe.edu/sfi/events/abstract/102

or public lecture?
        http://www.santafe.edu/sfi/events/publicLectures.html

-Steve

____________________________________________________
http://www.redfish.com    [hidden email]
624 Agua Fria Street      office: [phone]
Santa Fe, NM 87501        mobile: [phone]

[Roger Frye]

>
>
>Anyone make George Lekoff's Colloquium and care to give a short review?
> http://www.santafe.edu/sfi/events/abstract/102
>
>or public lecture?
> http://www.santafe.edu/sfi/events/publicLectures.html
>  
>
lAkoff's public lecture was well attended despite the rain and had to be
cut a little short for fear that the rain was about to get very heavy.

 From the abstract, I had expected the thesis to be that mathematics is
limited by the range of metaphors that our brains can muster in order to
comprehend and generate mathematical ideas.  And indeed that is a
consequence of Lakoff's cognitive science approach, but it wasn't the
focus or even a concern.  Instead, Lakoff battled the romantic notion
that mathematics is universal -- for example that pi is known to all
sufficiently sentient beings in all galaxies.

I haven't read his book, but I got the impression that the lecture
followed its main developments, touching on Euler's formula, the basic
metaphor for infinity, multiple set theories, and Boolean logic with
lots of examples along the way.  On the way out, I overheard someone
saying that she had never listened so closely to a lecture.  I enjoyed
the lecture and left reconfirmed in my view that the best way to teach
mathematics is to break the notion that I, as the teacher, know the
answers, and instead to lead the students to their own creations with
appreciation for the great creations of the past.  I also concluded that
mathematical expositions would be clearer if they exposed the metaphors
that led to their choices of axioms.

He began with the observation that color is not "out there."  Color is a
creation of the cones in our retinas and the wiring of our brains.  And
women have many more genes for cones than men do, so we probably see
different colors.  This theme, that color, and even mathematical ideas,
are the creations of the sensory-motor system of our brains contradicts
the romantic notion that mathematics can give us the answers to
universal questions.  I suppose he didn't mention Platonic ideals
because people don't learn about them anymore.  He didn't mention 42 either.

Why does -1 * -1 = + 1?  Because of the metaphor of rotating the positve
number line pi radians twice.

How did Cantor prove that there are no more numbers needed to cover the
plane than those needed to cover a line?  By using several metaphors
including matching poker chips to see who wins.  I don't understand
Lakoff's argument that Cantor is wrong.  For some reason, he insists on
saying that Cantor's proof does not come from within mathematics, but
instead comes from the pile of metaphors he uses.  But the same could be
said about multiplying signed numbers.  I think he should say that
Cantor's proof is true within the set of axioms that Cantor uses, but
that it is not universally true for all mathematical systems, and of
course not true for those, like Lakoff, who prefer to think there is a
lot more to a plane than a line because of different metaphors.

We are taught that sets cannot contain themselves because common set
theory is based on the metaphor of a container.  When some
mathematicians needed a set theory that allowed sets to contain
themselves, they changed to the metaphor of a graph, and created a
different set theory.  If there can be multiple set theories in
mathematics, and different theories give different results, then
mathematics cannot give us a single answer to universal questions.  And
since Boole created his logic as a symbolic algebra of classes, it too
is not universal.  And even if human reason were based on Logic, which
Lakoff knows it is not, then Reason also would not be universal.

-Roger Frye

[Stephen Guerin]

Cool review! Thanks, Roger.

> Why does -1 * -1 = + 1?  Because of the metaphor of rotating the positive
> number line pi radians twice.

I don't completely grok the metaphor. I get the 2*pi radians part but not
the connection between rotation and multiplication.

A quick surf turned up this mathworld page:
http://mathworld.wolfram.com/Rotation.html
and this WikiPedia entry:
http://www.wikipedia.org/wiki/Quaternions_and_spatial_rotation

Damn, I knew I should have paid more attention in linear algebra.

-S

____________________________________________________
http://www.redfish.com    [hidden email]
624 Agua Fria Street      office: [phone]
Santa Fe, NM 87501        mobile: [phone]

> -----Original Message-----
> From: [hidden email] [mailto:[hidden email]]On
> Behalf Of Roger Frye
> Sent: Thursday, August 28, 2003 9:32 PM
> To: The Friday Morning Complexity Coffee Group
> Cc: Reuben Hersh
> Subject: Re: [FRIAM] George Lekoff?
>
>
> >
> >
> >Anyone make George Lekoff's Colloquium and care to give a short review?
> > http://www.santafe.edu/sfi/events/abstract/102
> >
> >or public lecture?
> > http://www.santafe.edu/sfi/events/publicLectures.html
> >
> >
> lAkoff's public lecture was well attended despite the rain and had to be
> cut a little short for fear that the rain was about to get very heavy.
>
>  From the abstract, I had expected the thesis to be that mathematics is
> limited by the range of metaphors that our brains can muster in order to
> comprehend and generate mathematical ideas.  And indeed that is a
> consequence of Lakoff's cognitive science approach, but it wasn't the
> focus or even a concern.  Instead, Lakoff battled the romantic notion
> that mathematics is universal -- for example that pi is known to all
> sufficiently sentient beings in all galaxies.
>
> I haven't read his book, but I got the impression that the lecture
> followed its main developments, touching on Euler's formula, the basic
> metaphor for infinity, multiple set theories, and Boolean logic with
> lots of examples along the way.  On the way out, I overheard someone
> saying that she had never listened so closely to a lecture.  I enjoyed
> the lecture and left reconfirmed in my view that the best way to teach
> mathematics is to break the notion that I, as the teacher, know the
> answers, and instead to lead the students to their own creations with
> appreciation for the great creations of the past.  I also concluded that
> mathematical expositions would be clearer if they exposed the metaphors
> that led to their choices of axioms.
>
> He began with the observation that color is not "out there."  Color is a
> creation of the cones in our retinas and the wiring of our brains.  And
> women have many more genes for cones than men do, so we probably see
> different colors.  This theme, that color, and even mathematical ideas,
> are the creations of the sensory-motor system of our brains contradicts
> the romantic notion that mathematics can give us the answers to
> universal questions.  I suppose he didn't mention Platonic ideals
> because people don't learn about them anymore.  He didn't mention
> 42 either.
>
> Why does -1 * -1 = + 1?  Because of the metaphor of rotating the positve
> number line pi radians twice.
>
> How did Cantor prove that there are no more numbers needed to cover the
> plane than those needed to cover a line?  By using several metaphors
> including matching poker chips to see who wins.  I don't understand
> Lakoff's argument that Cantor is wrong.  For some reason, he insists on
> saying that Cantor's proof does not come from within mathematics, but
> instead comes from the pile of metaphors he uses.  But the same could be
> said about multiplying signed numbers.  I think he should say that
> Cantor's proof is true within the set of axioms that Cantor uses, but
> that it is not universally true for all mathematical systems, and of
> course not true for those, like Lakoff, who prefer to think there is a
> lot more to a plane than a line because of different metaphors.
>
> We are taught that sets cannot contain themselves because common set
> theory is based on the metaphor of a container.  When some
> mathematicians needed a set theory that allowed sets to contain
> themselves, they changed to the metaphor of a graph, and created a
> different set theory.  If there can be multiple set theories in
> mathematics, and different theories give different results, then
> mathematics cannot give us a single answer to universal questions.  And
> since Boole created his logic as a symbolic algebra of classes, it too
> is not universal.  And even if human reason were based on Logic, which
> Lakoff knows it is not, then Reason also would not be universal.
>
> -Roger Frye
>
>
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9AM @ Jane's Cafe
> Lecture schedule, archives, unsubscribe, etc.:
> http://www.redfish.com/friam
>
>

[Roger Critchlow-2]

I actually missed some of the talk due to a phone call, but I already
knew this one.

If you take complex numbers as points in the plane, and represent the
point by its radius and angle, then multiplication of complex numbers
works out to be identical to rotation in the plane (add the angles) and
dilation in radius (multiply the magnitudes).  The imaginary unit, i,
rotates by pi/2, -1 rotates by pi, 1 rotates by 0, and every other
complex number of unit magnitude represents some rotation.

A key part of Lakoff's argument, which applies to other fields besides
mathematics, is that all meaning is grounded in having a body which
intrinsically makes sense of a symbolic system which would otherwise be
drifting without anchor.  His career started with an attempt to build a
formal semantics to accompany Chomsky's formal syntactic theory, but he
concluded that it was futile exercise in symbol crunching.  That's where
the throw away comments during his talk about the impossibility of a
pure computer mind come from.  If you think back over what he said,
mathematics is based on metaphors, but all the metaphors are eventually
grounded in our physical, bodily existence, because if they weren't we
wouldn't know what they mean.

Which reminds me of nothing more than Samuel Johnson's refutation of
Bishop Berkeley, http://www.samueljohnson.com/refutati.html, found by
searching google with "refute it thus".

-- rec --

Stephen Guerin wrote:

> Cool review! Thanks, Roger.
>
>
>>Why does -1 * -1 = + 1?  Because of the metaphor of rotating the positive
>>number line pi radians twice.
>
>
> I don't completely grok the metaphor. I get the 2*pi radians part but not
> the connection between rotation and multiplication.
>
> A quick surf turned up this mathworld page:
> http://mathworld.wolfram.com/Rotation.html
> and this WikiPedia entry:
> http://www.wikipedia.org/wiki/Quaternions_and_spatial_rotation
>
> Damn, I knew I should have paid more attention in linear algebra.
>
> -S
>
> ____________________________________________________
> http://www.redfish.com    [hidden email]
> 624 Agua Fria Street      office: [phone]
> Santa Fe, NM 87501        mobile: [phone]
>
>
>>-----Original Message-----
>>From: [hidden email] [mailto:[hidden email]]On
>>Behalf Of Roger Frye
>>Sent: Thursday, August 28, 2003 9:32 PM
>>To: The Friday Morning Complexity Coffee Group
>>Cc: Reuben Hersh
>>Subject: Re: [FRIAM] George Lekoff?
>>
>>
>>
>>>
>>>Anyone make George Lekoff's Colloquium and care to give a short review?
>>> http://www.santafe.edu/sfi/events/abstract/102
>>>
>>>or public lecture?
>>> http://www.santafe.edu/sfi/events/publicLectures.html
>>>
>>>
>>
>>lAkoff's public lecture was well attended despite the rain and had to be
>>cut a little short for fear that the rain was about to get very heavy.
>>
>> From the abstract, I had expected the thesis to be that mathematics is
>>limited by the range of metaphors that our brains can muster in order to
>>comprehend and generate mathematical ideas.  And indeed that is a
>>consequence of Lakoff's cognitive science approach, but it wasn't the
>>focus or even a concern.  Instead, Lakoff battled the romantic notion
>>that mathematics is universal -- for example that pi is known to all
>>sufficiently sentient beings in all galaxies.
>>
>>I haven't read his book, but I got the impression that the lecture
>>followed its main developments, touching on Euler's formula, the basic
>>metaphor for infinity, multiple set theories, and Boolean logic with
>>lots of examples along the way.  On the way out, I overheard someone
>>saying that she had never listened so closely to a lecture.  I enjoyed
>>the lecture and left reconfirmed in my view that the best way to teach
>>mathematics is to break the notion that I, as the teacher, know the
>>answers, and instead to lead the students to their own creations with
>>appreciation for the great creations of the past.  I also concluded that
>>mathematical expositions would be clearer if they exposed the metaphors
>>that led to their choices of axioms.
>>
>>He began with the observation that color is not "out there."  Color is a
>>creation of the cones in our retinas and the wiring of our brains.  And
>>women have many more genes for cones than men do, so we probably see
>>different colors.  This theme, that color, and even mathematical ideas,
>>are the creations of the sensory-motor system of our brains contradicts
>>the romantic notion that mathematics can give us the answers to
>>universal questions.  I suppose he didn't mention Platonic ideals
>>because people don't learn about them anymore.  He didn't mention
>>42 either.
>>
>>Why does -1 * -1 = + 1?  Because of the metaphor of rotating the positve
>>number line pi radians twice.
>>
>>How did Cantor prove that there are no more numbers needed to cover the
>>plane than those needed to cover a line?  By using several metaphors
>>including matching poker chips to see who wins.  I don't understand
>>Lakoff's argument that Cantor is wrong.  For some reason, he insists on
>>saying that Cantor's proof does not come from within mathematics, but
>>instead comes from the pile of metaphors he uses.  But the same could be
>>said about multiplying signed numbers.  I think he should say that
>>Cantor's proof is true within the set of axioms that Cantor uses, but
>>that it is not universally true for all mathematical systems, and of
>>course not true for those, like Lakoff, who prefer to think there is a
>>lot more to a plane than a line because of different metaphors.
>>
>>We are taught that sets cannot contain themselves because common set
>>theory is based on the metaphor of a container.  When some
>>mathematicians needed a set theory that allowed sets to contain
>>themselves, they changed to the metaphor of a graph, and created a
>>different set theory.  If there can be multiple set theories in
>>mathematics, and different theories give different results, then
>>mathematics cannot give us a single answer to universal questions.  And
>>since Boole created his logic as a symbolic algebra of classes, it too
>>is not universal.  And even if human reason were based on Logic, which
>>Lakoff knows it is not, then Reason also would not be universal.
>>
>>-Roger Frye
>>
>>
>>
>>============================================================
>>FRIAM Applied Complexity Group listserv
>>Meets Fridays 9AM @ Jane's Cafe
>>Lecture schedule, archives, unsubscribe, etc.:
>>http://www.redfish.com/friam
>>
>>
>
>
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9AM @ Jane's Cafe
> Lecture schedule, archives, unsubscribe, etc.:
> http://www.redfish.com/friam
>
>

[Roger Frye]

Yes, those rotation links that Stephen found, and Critchlow's
explanation are completely sound, but the idea can be explained to a 3rd
grader.

I've been tutoring a pair of 3rd graders.  When I asked them where the
negative numbers were, they told me that they were in a hole in the
ground.  All of the positve numbers are floating around in the air, and
the negative numbers are in a big hole.  I guess there must have been a
picture like that in their text book.  We began playing with tape
measures and yard sticks in order to introduce the idea of a number
line,  We decided that a million was way off to the right somewhere.  I
showed them how you can use two yard sticks to add numbers.  Put the
left end of the top yardstick on the 3 inch mark of the bottom yard
stick.  Then read off the + 3 table by looking up a number on the top
yard stick and reading the answer on the bottom yardstick.

Then I turned the top yardstick 180 degrees with the smallEndian end
still on the 3.  This is a subtraction slide rule.  3-1 = 2, 3-2 = 1,
3-3 is at the end of the ruler, so that must be where zero is.  They
also knew that 3-4 was -1, and they could see that the answer lay off
the end of the bottom yard stick, so we put a 3rd, turned-around
yardstick next to the other bottom yardstick, so that we could read off
the negative results.

We moved on to the idea that multiplying by -1 is the same thing as
rotating the top yardstick 180 degrees and reading off the same
distance.  And then we rotated another 180 degrees in order to map
multiplying the negative numbers by minus 1 onto the postive yardstick.

Later we invented logarithms by trying to construct a number line that
would let us do multiplication by sliding pieces of paper along each
other, and we downloaded templates for a circular sliderule and cut them
out.

Note how interpeting multiplication by minus 1 as a 180 degree rotation
requires a leap of imagination.  It's because it is just taught to us a
special rule.  "That's just the way it has to be in order to make
arithmetic work out."  The usual metaphors that we use for addition and
multiplication of counting, repeated operations, areas and so forth,
don't require us to use the rotation metaphor.  Once we learn why e^i*pi
= -1 and complex multiplication or linear algebra, we can go back and
relearn why arithmetic works.

-Roger Frye

[Robert Holmes]

Lakoff's metaphor for multiplication by -1 (rotation through 180 degress)
leads into the concept of complex numbers neatly (and comprehensibly) by
posing the question, "OK, but what if I stop halfway round?" In fact, he
devotes a section to this point towards the end of his book. Unfortunately,
this is where the metaphor runs out: it doesn't give any help to (IMHO) the
next logical question "What if I don't rotate in the plane? What if I rotate
in 3D rather than 2D?"

Now I vaguely remember that such a rotation would be A Bad Thing; that
there's a good reason for stopping at x + iy, and not inventing things that
look like x + iy + jz +.... However I can't remember those reasons for the
life of me and the metaphor doesn't shed any light on the goodness/badness
of rotating in this way.

Robert


-----Original Message-----
From: [hidden email] [mailto:[hidden email]] On Behalf
Of Roger Frye
Sent: 29 August 2003 12:09
To: The Friday Morning Complexity Coffee Group
Subject: Re: [FRIAM] George Lekoff?


Yes, those rotation links that Stephen found, and Critchlow's
explanation are completely sound, but the idea can be explained to a 3rd
grader.

I've been tutoring a pair of 3rd graders.  When I asked them where the
negative numbers were, they told me that they were in a hole in the
ground.  All of the positve numbers are floating around in the air, and
the negative numbers are in a big hole.  I guess there must have been a
picture like that in their text book.  We began playing with tape
measures and yard sticks in order to introduce the idea of a number
line,  We decided that a million was way off to the right somewhere.  I
showed them how you can use two yard sticks to add numbers.  Put the
left end of the top yardstick on the 3 inch mark of the bottom yard
stick.  Then read off the + 3 table by looking up a number on the top
yard stick and reading the answer on the bottom yardstick.

Then I turned the top yardstick 180 degrees with the smallEndian end
still on the 3.  This is a subtraction slide rule.  3-1 = 2, 3-2 = 1,
3-3 is at the end of the ruler, so that must be where zero is.  They
also knew that 3-4 was -1, and they could see that the answer lay off
the end of the bottom yard stick, so we put a 3rd, turned-around
yardstick next to the other bottom yardstick, so that we could read off
the negative results.

We moved on to the idea that multiplying by -1 is the same thing as
rotating the top yardstick 180 degrees and reading off the same
distance.  And then we rotated another 180 degrees in order to map
multiplying the negative numbers by minus 1 onto the postive yardstick.

Later we invented logarithms by trying to construct a number line that
would let us do multiplication by sliding pieces of paper along each
other, and we downloaded templates for a circular sliderule and cut them
out.

Note how interpeting multiplication by minus 1 as a 180 degree rotation
requires a leap of imagination.  It's because it is just taught to us a
special rule.  "That's just the way it has to be in order to make
arithmetic work out."  The usual metaphors that we use for addition and
multiplication of counting, repeated operations, areas and so forth,
don't require us to use the rotation metaphor.  Once we learn why e^i*pi
= -1 and complex multiplication or linear algebra, we can go back and
relearn why arithmetic works.

-Roger Frye



============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9AM @ Jane's Cafe
Lecture schedule, archives, unsubscribe, etc.: http://www.redfish.com/friam

[Roger Critchlow-2]

Robert Holmes wrote:

> Lakoff's metaphor for multiplication by -1 (rotation through 180 degress)
> leads into the concept of complex numbers neatly (and comprehensibly) by
> posing the question, "OK, but what if I stop halfway round?" In fact, he
> devotes a section to this point towards the end of his book. Unfortunately,
> this is where the metaphor runs out: it doesn't give any help to (IMHO) the
> next logical question "What if I don't rotate in the plane? What if I rotate
> in 3D rather than 2D?"
>
> Now I vaguely remember that such a rotation would be A Bad Thing; that
> there's a good reason for stopping at x + iy, and not inventing things that
> look like x + iy + jz +.... However I can't remember those reasons for the
> life of me and the metaphor doesn't shed any light on the goodness/badness
> of rotating in this way.

No, Hamilton discovered that the representation for rotations in 3D
requires 4 coordinates and called them Quaternions. They have one real
part and three complex (or hyper-complex) parts.  They are an entirely
Good Way to do rotations.

And Clifford discovered that the general solution in N-dimensions is the
even subalgebra of the Clifford Algebra over Euclidean N space, which
requires 2^(N-1) coordinates.

What breaks down is the identification of the rotation algebra with the
underlying space:  although 2 coordinates describe rotations in 2 space,
you need 4 coordinates for rotations in 3 space, 8 coordinates for
rotations in 4 space, and so on.

I don't have any good sources for making sense of this in terms of
grounded metaphors, but if you weed to understand the geometry (without
the Lakoff metaphors) try one of the intro papers by David Hestenes at
http://modelingnts.la.asu.edu/GC_R&D.html

-- rec --

[Dr. Richard C. Cassin]

        Where can we sign up for Roger's class ?



-----Original Message-----
From: [hidden email] [mailto:[hidden email]]On
Behalf Of Roger Frye
Sent: Viernes, 29 de Agosto de 2003 11:09 a.m.
To: The Friday Morning Complexity Coffee Group
Subject: Re: [FRIAM] George Lekoff?


Yes, those rotation links that Stephen found, and Critchlow's
explanation are completely sound, but the idea can be explained to a 3rd
grader.

I've been tutoring a pair of 3rd graders.  When I asked them where the
negative numbers were, they told me that they were in a hole in the
ground.  All of the positve numbers are floating around in the air, and
the negative numbers are in a big hole.  I guess there must have been a
picture like that in their text book.  We began playing with tape
measures and yard sticks in order to introduce the idea of a number
line,  We decided that a million was way off to the right somewhere.  I
showed them how you can use two yard sticks to add numbers.  Put the
left end of the top yardstick on the 3 inch mark of the bottom yard
stick.  Then read off the + 3 table by looking up a number on the top
yard stick and reading the answer on the bottom yardstick.

Then I turned the top yardstick 180 degrees with the smallEndian end
still on the 3.  This is a subtraction slide rule.  3-1 = 2, 3-2 = 1,
3-3 is at the end of the ruler, so that must be where zero is.  They
also knew that 3-4 was -1, and they could see that the answer lay off
the end of the bottom yard stick, so we put a 3rd, turned-around
yardstick next to the other bottom yardstick, so that we could read off
the negative results.

We moved on to the idea that multiplying by -1 is the same thing as
rotating the top yardstick 180 degrees and reading off the same
distance.  And then we rotated another 180 degrees in order to map
multiplying the negative numbers by minus 1 onto the postive yardstick.

Later we invented logarithms by trying to construct a number line that
would let us do multiplication by sliding pieces of paper along each
other, and we downloaded templates for a circular sliderule and cut them
out.

Note how interpeting multiplication by minus 1 as a 180 degree rotation
requires a leap of imagination.  It's because it is just taught to us a
special rule.  "That's just the way it has to be in order to make
arithmetic work out."  The usual metaphors that we use for addition and
multiplication of counting, repeated operations, areas and so forth,
don't require us to use the rotation metaphor.  Once we learn why e^i*pi
= -1 and complex multiplication or linear algebra, we can go back and
relearn why arithmetic works.

-Roger Frye



============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9AM @ Jane's Cafe
Lecture schedule, archives, unsubscribe, etc.:
http://www.redfish.com/friam

[Carl Tollander-2]

Another link of possible interest for this thread.
After quaternions, we go to octonions. (what else?)
http://math.ucr.edu/home/baez/Octonions/octonions.html
Shows how we get there from Clifford Algebras (at least it might
if I understood it!)

BTW, 3D rotations (non quaternion) while not necessarily "bad" are
potentially confusing, since the 3D rotations don't commute (and they
are subject to the heartbreak of "gimbal lock").  In 3D graphics, its
not a bad idea (at least it used to be) to do the rotational math in
quaternions and convert back to 3D matrices only when you need to actually
draw things (go to the hardware).

-----Original Message-----
From: [hidden email] [mailto:[hidden email]]On
Behalf Of Roger E Critchlow Jr
Sent: Friday, August 29, 2003 4:05 PM
To: The Friday Morning Complexity Coffee Group
Subject: Re: [FRIAM] George Lakoff?




Robert Holmes wrote:
> Lakoff's metaphor for multiplication by -1 (rotation through 180 degress)
> leads into the concept of complex numbers neatly (and comprehensibly) by
> posing the question, "OK, but what if I stop halfway round?" In fact, he
> devotes a section to this point towards the end of his book.
Unfortunately,
> this is where the metaphor runs out: it doesn't give any help to (IMHO)
the
> next logical question "What if I don't rotate in the plane? What if I
rotate
> in 3D rather than 2D?"
>
> Now I vaguely remember that such a rotation would be A Bad Thing; that
> there's a good reason for stopping at x + iy, and not inventing things
that
> look like x + iy + jz +.... However I can't remember those reasons for the
> life of me and the metaphor doesn't shed any light on the goodness/badness
> of rotating in this way.

No, Hamilton discovered that the representation for rotations in 3D
requires 4 coordinates and called them Quaternions. They have one real
part and three complex (or hyper-complex) parts.  They are an entirely
Good Way to do rotations.

And Clifford discovered that the general solution in N-dimensions is the
even subalgebra of the Clifford Algebra over Euclidean N space, which
requires 2^(N-1) coordinates.

What breaks down is the identification of the rotation algebra with the
underlying space:  although 2 coordinates describe rotations in 2 space,
you need 4 coordinates for rotations in 3 space, 8 coordinates for
rotations in 4 space, and so on.

I don't have any good sources for making sense of this in terms of
grounded metaphors, but if you weed to understand the geometry (without
the Lakoff metaphors) try one of the intro papers by David Hestenes at
http://modelingnts.la.asu.edu/GC_R&D.html

-- rec --


============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9AM @ Jane's Cafe
Lecture schedule, archives, unsubscribe, etc.:
http://www.redfish.com/friam