View Full Version : Commutative rings aren't physical spaces (This Week's Finds 205)
Email me at CS not Boole
Apr15-04, 02:57 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>>From: John Baez\n>We get a little\n>dictionary for translating between geometry and algebra, like this:\n>\n> GEOMETRY ALGEBRA\n> spaces commutative rings\n> maps homomorphisms\n\nThis may be well and good for mathematical space on sci.math, but it\'s\nnot how physical space works on sci.physics.* (or shouldn\'t be).\n\nThe difference between a direct product and a tensor product is that\nthe multiplicands of a direct product don\'t talk to each: they can\'t\nexchange information or particles, but simply coexist as\nnoncommunicating parallel worlds each oblivious to the other.\nIntercomponent communication requires tensor product.\n\nAt the level of individual points of a physical geometry, it takes\nnoncommutativity of product to create intercomponent communication.\n\nIf you believe that the algebra of physical space is commutative then\nyour thinking is still in its pre-linear-logic period. A key insight\nof linear logic (not the only one) is that the additive operations of\ndirect product and direct sum (which coincide for vector spaces but not\nfor algebras and most other mathematical objects) juxtapose\nnoninteracting universes, whereas tensor product creates universes\nalong with communication channels between them. (You don\'t have to\nknow linear logic to know this -- it\'s already a basic difference\nbetween tensor product and direct sum of vector spaces -- but if you\nknow that then you\'re in good shape to make a start on linear logic.)\n\nA simple example is the quaternions, which depend on noncommutativity\n(namely ij = -ji) to allow rotation about an arbitrarily oriented\naxis. As Hamilton finally realized in the mid-19th century, a\ncommutative geometry has serious problems trying to swing one axis\naround to another in any physically meaningful way, regardless of\nwhether the angle of swing is circular or hyperbolic (I don\'t know if\nHamilton considered the latter).\n\nThis is why no Clifford algebra whose underlying vector space is of\ndimension greater than two is commutative. If two nonreal axes of an\nassociative algebra (more precisely their unit vectors) commute, it\nmeans that they can\'t communicate with each other. In particular a\nparticle traveling along one such axis has no hope of becoming, or even\ninfluencing, a particle traveling along the other axis\n\nThe Clifford algebras of dimension at most two are commutative because\nthey don\'t *have* two nonreal axes and may as well be treated as\nscalars, as they usually are, but shouldn\'t be when more dimensions are\nbrought in. The "real axis" is ironically named since it is the one\naxis that isn\'t a physical axis, consisting rather of the observable\nvalues as genuine scalars. Complex numbers aren\'t real, they are two\nreal numbers delivered by two concurrent observers separated by quarter\nof a wavelength, though a sixth of a wavelength works better for some\napplications. (Sorry, starting to ramble there... :)\n\nVaughan Pratt\n--\nDon\'t contact me at pratt@boole.stanford.edu, substitute cs for boole instead.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>>From: John Baez
>We get a little
>dictionary for translating between geometry and algebra, like this:
>
> GEOMETRY ALGEBRA
> spaces commutative rings
> maps homomorphisms
This may be well and good for mathematical space on sci.math, but it's
not how physical space works on sci.physics.* (or shouldn't be).
The difference between a direct product and a tensor product is that
the multiplicands of a direct product don't talk to each: they can't
exchange information or particles, but simply coexist as
noncommunicating parallel worlds each oblivious to the other.
Intercomponent communication requires tensor product.
At the level of individual points of a physical geometry, it takes
noncommutativity of product to create intercomponent communication.
If you believe that the algebra of physical space is commutative then
your thinking is still in its pre-linear-logic period. A key insight
of linear logic (not the only one) is that the additive operations of
direct product and direct sum (which coincide for vector spaces but not
for algebras and most other mathematical objects) juxtapose
noninteracting universes, whereas tensor product creates universes
along with communication channels between them. (You don't have to
know linear logic to know this -- it's already a basic difference
between tensor product and direct sum of vector spaces -- but if you
know that then you're in good shape to make a start on linear logic.)
A simple example is the quaternions, which depend on noncommutativity
(namely ij = -ji) to allow rotation about an arbitrarily oriented
axis. As Hamilton finally realized in the mid-19th century, a
commutative geometry has serious problems trying to swing one axis
around to another in any physically meaningful way, regardless of
whether the angle of swing is circular or hyperbolic (I don't know if
Hamilton considered the latter).
This is why no Clifford algebra whose underlying vector space is of
dimension greater than two is commutative. If two nonreal axes of an
associative algebra (more precisely their unit vectors) commute, it
means that they can't communicate with each other. In particular a
particle traveling along one such axis has no hope of becoming, or even
influencing, a particle traveling along the other axis
The Clifford algebras of dimension at most two are commutative because
they don't *have* two nonreal axes and may as well be treated as
scalars, as they usually are, but shouldn't be when more dimensions are
brought in. The "real axis" is ironically named since it is the one
axis that isn't a physical axis, consisting rather of the observable
values as genuine scalars. Complex numbers aren't real, they are two
real numbers delivered by two concurrent observers separated by quarter
of a wavelength, though a sixth of a wavelength works better for some
applications. (Sorry, starting to ramble there... :)
Vaughan Pratt
--
Don't contact me at pratt@boole.stanford.edu, substitute cs for boole instead.
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>pratt@boole.Stanford.EDU (Email me at CS not Boole) wrote in message news:<c5l6tr\\$94a\\$1@news.Stanford.EDU>...\n> >From: John Baez\n> >We get a little\n> >dictionary for translating between geometry and algebra, like this:\n> >\n> > GEOMETRY ALGEBRA\n> > spaces commutative rings\n> > maps homomorphisms\n>\n> This may be well and good for mathematical space on sci.math, but it\'s\n> not how physical space works on sci.physics.* (or shouldn\'t be).\n>\n> The difference between a direct product and a tensor product is that\n> the multiplicands of a direct product don\'t talk to each: they can\'t\n> exchange information or particles, but simply coexist as\n> noncommunicating parallel worlds each oblivious to the other.\n> Intercomponent communication requires tensor product.\n>\n> At the level of individual points of a physical geometry, it takes\n> noncommutativity of product to create intercomponent communication.\n>\n> If you believe that the algebra of physical space is commutative then\n> your thinking is still in its pre-linear-logic period. A key insight\n> of linear logic (not the only one) is that the additive operations of\n> direct product and direct sum (which coincide for vector spaces but not\n> for algebras and most other mathematical objects) juxtapose\n> noninteracting universes, whereas tensor product creates universes\n> along with communication channels between them. (You don\'t have to\n> know linear logic to know this -- it\'s already a basic difference\n> between tensor product and direct sum of vector spaces -- but if you\n> know that then you\'re in good shape to make a start on linear logic.)\n>\n> A simple example is the quaternions, which depend on noncommutativity\n> (namely ij = -ji) to allow rotation about an arbitrarily oriented\n> axis. As Hamilton finally realized in the mid-19th century, a\n> commutative geometry has serious problems trying to swing one axis\n> around to another in any physically meaningful way, regardless of\n> whether the angle of swing is circular or hyperbolic (I don\'t know if\n> Hamilton considered the latter).\n>\n> This is why no Clifford algebra whose underlying vector space is of\n> dimension greater than two is commutative. If two nonreal axes of an\n> associative algebra (more precisely their unit vectors) commute, it\n> means that they can\'t communicate with each other. In particular a\n> particle traveling along one such axis has no hope of becoming, or even\n> influencing, a particle traveling along the other axis\n>\n> The Clifford algebras of dimension at most two are commutative because\n> they don\'t *have* two nonreal axes and may as well be treated as\n> scalars, as they usually are, but shouldn\'t be when more dimensions are\n> brought in. The "real axis" is ironically named since it is the one\n> axis that isn\'t a physical axis, consisting rather of the observable\n> values as genuine scalars. Complex numbers aren\'t real, they are two\n> real numbers delivered by two concurrent observers separated by quarter\n> of a wavelength, though a sixth of a wavelength works better for some\n> applications. (Sorry, starting to ramble there... :)\n>\n> Vaughan Pratt\n\nThis is a rather presumptuous post! Mr. Pratt presumes to know how\nthings "should be" in physics. Everyone knows that the mathematics\nused in physics is just a model for describing the real world. Anyone\nwho says that the real world "is" or "is not" some mathematical\nstructure doesn\'t really get it. Take for example supposedly curved\nspace as in general relativity. Whitehead showed you could do\nrelativity in flat space. It\'s all a device for thinking about things\nand shouldn\'t be confused with reality.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>pratt@boole.Stanford.EDU (Email me at CS not Boole) wrote in message news:<c5l6tr$94a$1@news.Stanford.EDU>...
> >From: John Baez
> >We get a little
> >dictionary for translating between geometry and algebra, like this:
> >
> > GEOMETRY ALGEBRA
> > spaces commutative rings
> > maps homomorphisms
>
> This may be well and good for mathematical space on sci.math, but it's
> not how physical space works on sci.physics.* (or shouldn't be).
>
> The difference between a direct product and a tensor product is that
> the multiplicands of a direct product don't talk to each: they can't
> exchange information or particles, but simply coexist as
> noncommunicating parallel worlds each oblivious to the other.
> Intercomponent communication requires tensor product.
>
> At the level of individual points of a physical geometry, it takes
> noncommutativity of product to create intercomponent communication.
>
> If you believe that the algebra of physical space is commutative then
> your thinking is still in its pre-linear-logic period. A key insight
> of linear logic (not the only one) is that the additive operations of
> direct product and direct sum (which coincide for vector spaces but not
> for algebras and most other mathematical objects) juxtapose
> noninteracting universes, whereas tensor product creates universes
> along with communication channels between them. (You don't have to
> know linear logic to know this -- it's already a basic difference
> between tensor product and direct sum of vector spaces -- but if you
> know that then you're in good shape to make a start on linear logic.)
>
> A simple example is the quaternions, which depend on noncommutativity
> (namely ij = -ji) to allow rotation about an arbitrarily oriented
> axis. As Hamilton finally realized in the mid-19th century, a
> commutative geometry has serious problems trying to swing one axis
> around to another in any physically meaningful way, regardless of
> whether the angle of swing is circular or hyperbolic (I don't know if
> Hamilton considered the latter).
>
> This is why no Clifford algebra whose underlying vector space is of
> dimension greater than two is commutative. If two nonreal axes of an
> associative algebra (more precisely their unit vectors) commute, it
> means that they can't communicate with each other. In particular a
> particle traveling along one such axis has no hope of becoming, or even
> influencing, a particle traveling along the other axis
>
> The Clifford algebras of dimension at most two are commutative because
> they don't *have* two nonreal axes and may as well be treated as
> scalars, as they usually are, but shouldn't be when more dimensions are
> brought in. The "real axis" is ironically named since it is the one
> axis that isn't a physical axis, consisting rather of the observable
> values as genuine scalars. Complex numbers aren't real, they are two
> real numbers delivered by two concurrent observers separated by quarter
> of a wavelength, though a sixth of a wavelength works better for some
> applications. (Sorry, starting to ramble there... :)
>
> Vaughan Pratt
This is a rather presumptuous post! Mr. Pratt presumes to know how
things "should be" in physics. Everyone knows that the mathematics
used in physics is just a model for describing the real world. Anyone
who says that the real world "is" or "is not" some mathematical
structure doesn't really get it. Take for example supposedly curved
space as in general relativity. Whitehead showed you could do
relativity in flat space. It's all a device for thinking about things
and shouldn't be confused with reality.
Brian Quincy Hutchings
Apr17-04, 05:09 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Whitehead\'s cool; Russell\'s kook.\nbut I think his "presumption" was pretty good, although\nI didn\'t follow it too closely. after all,\ncommutativity is just math, two.\n\ngowan4@hotmail.com (gowan) wrote in message news:<fab2a28f.0404150837.56aa9274@posting.google. com>...\n\n> space as in general relativity. Whitehead showed you could do\n> relativity in flat space. It\'s all a device for thinking about things\n\n--Give Earth a Trickier Dick Cheeny -- out of office, after GIGA years.\nhttp://www.benfranklinbooks.com/\nhttp://larouchepub.com/other/2004/book_reviews/3105suskind_oneill.html\nhttp://larouchepub.com/other/2004/3105cheny_trgtted.html\nhttp://larouchepub.com/other/2003/3046chnygte_plmbrs.html\nhttp://larouchepub.com/other/2003/3048iraq_58_const.html\nhttp://www.rand.org/publications/randreview/issues/rr.12.00/\nhttp://members.tripod.com/~american_almanac\nhttp://www.wlym.com/pdf/iclc/howthenation.PDF\nhttp://larouchepub.com/other/2004/3104elect_voting.html\nhttp://larouchepub.com/other/2003/3045dems_dive_soros.html\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Whitehead's cool; Russell's kook.
but I think his "presumption" was pretty good, although
I didn't follow it too closely. after all,
commutativity is just math, two.
gowan4@hotmail.com (gowan) wrote in message news:<fab2a28f.0404150837.56aa9274@posting.google.com>...
> space as in general relativity. Whitehead showed you could do
> relativity in flat space. It's all a device for thinking about things
--Give Earth a Trickier Dick Cheeny -- out of office, after GIGA years.
http://www.benfranklinbooks.com/
http://larouchepub.com/other/2004/book_reviews/3105suskind_oneill.html
http://larouchepub.com/other/2004/3105cheny_trgtted.html
http://larouchepub.com/other/2003/3046chnygte_plmbrs.html
http://larouchepub.com/other/2003/3048iraq_58_const.html
http://www.rand.org/publications/randreview/issues/rr.12.00/
http://members.tripod.com/~american_almanac
http://www.wlym.com/pdf/iclc/howthenation.PDF
http://larouchepub.com/other/2004/3104elect_voting.html
http://larouchepub.com/other/2003/3045dems_dive_soros.html
Esa A E Peuha
Apr21-04, 10:53 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\ngowan4@hotmail.com (gowan) writes:\n\n> Take for example supposedly curved\n> space as in general relativity. Whitehead showed you could do\n> relativity in flat space.\n\nDid he? Not in what is usually known as Whitehead\'s theory of gravity,\nbecause spacetime is not flat in it, just differently curved than in\nEinstein\'s theory.\n\n--\nEsa Peuha\nstudent of mathematics at the University of Helsinki\nhttp://www.helsinki.fi/~peuha/\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>gowan4@hotmail.com (gowan) writes:
> Take for example supposedly curved
> space as in general relativity. Whitehead showed you could do
> relativity in flat space.
Did he? Not in what is usually known as Whitehead's theory of gravity,
because spacetime is not flat in it, just differently curved than in
Einstein's theory.
--
Esa Peuha
student of mathematics at the University of Helsinki
http://www.helsinki.fi/~peuha/
Email me at CS not Boole
Apr22-04, 03:30 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <fab2a28f.0404150837.56aa9274@posting.google.com>, \ngowan <gowan4@hotmail.com> wrote:\n\n>This is a rather presumptuous post! Mr. Pratt presumes to know how\n>things "should be" in physics. Everyone knows that the mathematics\n>used in physics is just a model for describing the real world. Anyone\n>who says that the real world "is" or "is not" some mathematical\n>structure doesn\'t really get it.\n\nWhile there\'s some truth to what you say, there\'s also some pedantry to\nit. To take a simpler example, granted a conic section is only a\n*model* of a physical orbit, and an approximate model at that given the\npresence of other bodies and the finiteness of the involved masses, yet\nthis does not stop people from agreeing with "a captive orbit *is* an\nellipse" and disagreeing with "a captive orbit *is* a hyperbola."\n\nThe truth of the propositions of physics are decided not on the basis\nof whether they contract "is modeled by" to "is" but by whether the\nobviously intended message is factually correct or not.\n\nOn that basis your challenge to me should not be whether I was wrong to\n*identify* physically connected axes (experimentally testable by for\nexample whether a particle traveling in the direction of one axis can\nswitch direction to the other axis) with the mathematical notion of\nnoncommuting axes, but whether either one of commuting or noncommuting\naxes are equally good at modeling that sort of thing. My point was a\nfactual one in that sense, namely objecting to John Baez\'s restriction\nto commutative geometry.\n\nIn a nutshell, what goes wrong with commuting axes in a space is that\nthey permit the algebra of that space to factor as a direct product of\nalgebras. (Or perhaps it should be said the other way round: axes in\nseparate components of a direct product necessarily commute.) A direct\nproduct AxB of algebras (also called direct sum, which is correct for\nthe underlying vector space of the algebra but incorrect for the\nalgebra as a whole including its multiplication such as matrix\nmultiplication) is one in which all the operations of the algebra\nincluding the multiplication are performed within A and B\nindependently, with no opportunity to exchange anything between A and\nB. A and B effectively function in AxB as two worlds completely out of\ncommunication with each other.\n\nNoncommutativity (only in the multiplication, never in the vector space\nhalf of an algebra) is always commutative. For example ij = -ji tells\nyou that i and j used as operators (e.g. i as rotation by 90 degrees, j\nas reflection) interact, in that doing them in the opposite order has\nthe opposite effect, e.g. turning 90 degrees followed by reflecting in\na mirror vs. reflecting followed by turning 90 degrees; these are\nopposite turns. Commuting axes can\'t capture that aspect of the\ninteractivity of the axes of physical space. (Incidentally a\nreflection is a turn through a hyberbolic angle, whose magnitude is\ncorrelated with the fraction of the incident light reflected.)\n\nThat\'s the gist of commutativity and noncommutativity. They\'re only\nmodels, but as such they come with factual content that can checked\nagainst experiment, just as orbits that may be ellipses or hyperbolas\nare only models yet are still experimentally verifiable.\n\n>Take for example supposedly curved\n>space as in general relativity. Whitehead showed you could do\n>relativity in flat space. It\'s all a device for thinking about things\n>and shouldn\'t be confused with reality.\n\nWhether a space is flat depends on the norm chosen for it. The same\nphysical space can support multiple norms serving different physical\npurposes, and be flat with one of them and curved with another. These\nnorms are fruitfully identified with the respective physical purposes\nthey serve. Any confusion here will stem not from identification per\nse but from *incorrect* identification: the wrong norm for the\noccasion.\n\nVaughan Pratt\n--\nDon\'t contact me at pratt@boole.stanford.edu, substitute cs for boole instead.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <fab2a28f.0404150837.56aa9274@posting.google.com>,
gowan <gowan4@hotmail.com> wrote:
>This is a rather presumptuous post! Mr. Pratt presumes to know how
>things "should be" in physics. Everyone knows that the mathematics
>used in physics is just a model for describing the real world. Anyone
>who says that the real world "is" or "is not" some mathematical
>structure doesn't really get it.
While there's some truth to what you say, there's also some pedantry to
it. To take a simpler example, granted a conic section is only a
*model* of a physical orbit, and an approximate model at that given the
presence of other bodies and the finiteness of the involved masses, yet
this does not stop people from agreeing with "a captive orbit *is* an
ellipse" and disagreeing with "a captive orbit *is* a hyperbola."
The truth of the propositions of physics are decided not on the basis
of whether they contract "is modeled by" to "is" but by whether the
obviously intended message is factually correct or not.
On that basis your challenge to me should not be whether I was wrong to
*identify* physically connected axes (experimentally testable by for
example whether a particle traveling in the direction of one axis can
switch direction to the other axis) with the mathematical notion of
noncommuting axes, but whether either one of commuting or noncommuting
axes are equally good at modeling that sort of thing. My point was a
factual one in that sense, namely objecting to John Baez's restriction
to commutative geometry.
In a nutshell, what goes wrong with commuting axes in a space is that
they permit the algebra of that space to factor as a direct product of
algebras. (Or perhaps it should be said the other way round: axes in
separate components of a direct product necessarily commute.) A direct
product AxB of algebras (also called direct sum, which is correct for
the underlying vector space of the algebra but incorrect for the
algebra as a whole including its multiplication such as matrix
multiplication) is one in which all the operations of the algebra
including the multiplication are performed within A and B
independently, with no opportunity to exchange anything between A and
B. A and B effectively function in AxB as two worlds completely out of
communication with each other.
Noncommutativity (only in the multiplication, never in the vector space
half of an algebra) is always commutative. For example ij = -ji tells
you that i and j used as operators (e.g. i as rotation by 90 degrees, j
as reflection) interact, in that doing them in the opposite order has
the opposite effect, e.g. turning 90 degrees followed by reflecting in
a mirror vs. reflecting followed by turning 90 degrees; these are
opposite turns. Commuting axes can't capture that aspect of the
interactivity of the axes of physical space. (Incidentally a
reflection is a turn through a hyberbolic angle, whose magnitude is
correlated with the fraction of the incident light reflected.)
That's the gist of commutativity and noncommutativity. They're only
models, but as such they come with factual content that can checked
against experiment, just as orbits that may be ellipses or hyperbolas
are only models yet are still experimentally verifiable.
>Take for example supposedly curved
>space as in general relativity. Whitehead showed you could do
>relativity in flat space. It's all a device for thinking about things
>and shouldn't be confused with reality.
Whether a space is flat depends on the norm chosen for it. The same
physical space can support multiple norms serving different physical
purposes, and be flat with one of them and curved with another. These
norms are fruitfully identified with the respective physical purposes
they serve. Any confusion here will stem not from identification per
se but from *incorrect* identification: the wrong norm for the
occasion.
Vaughan Pratt
--
Don't contact me at pratt@boole.stanford.edu, substitute cs for boole instead.
Gerry Myerson
Apr23-04, 04:14 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <c61asd\\$h0h\\$1@news.Stanford.EDU>,\npratt@boole .Stanford.EDU (Email me at CS not Boole) wrote:\n\n> Noncommutativity (only in the multiplication, never in the vector space\n> half of an algebra) is always commutative.\n\n?????\n\n--\nGerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <c61asd$h0h$1@news.Stanford.EDU>,
pratt@boole.Stanford.EDU (Email me at CS not Boole) wrote:
> Noncommutativity (only in the multiplication, never in the vector space
> half of an algebra) is always commutative.
?????
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
Brian Quincy Hutchings
Apr24-04, 09:18 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>4th time:\nis there *any* commutative space,\nof more than two dimensions?\nas for ij = -ji, I gues taht you could look at it\nas a right-agnle rotation, followed by a reflection, but\nisn\'t it the usual thing, to view it two rotations,\naround two orthogonal axes?... anyway,\na pair of rotations aren\'t commutative.\n\npratt@boole.Stanford.EDU (Email me at CS not Boole) wrote in message news:<c61asd\\$h0h\\$1@news.Stanford.EDU>...\n\n> Noncommutativity (only in the multiplication, never in the vector space\n> half of an algebra) is always commutative. For example ij = -ji tells\n> you that i and j used as operators (e.g. i as rotation by 90 degrees, j\n> as reflection) interact, in that doing them in the opposite order has\n> the opposite effect, e.g. turning 90 degrees followed by reflecting in\n> a mirror vs. reflecting followed by turning 90 degrees; these are\n> opposite turns. Commuting axes can\'t capture that aspect of the\n> interactivity of the axes of physical space. (Incidentally a\n> reflection is a turn through a hyberbolic angle, whose magnitude is\n> correlated with the fraction of the incident light reflected.)\n\n--ils duces d\'Enron -- conspiracy central?\nhttp://tarpley.net\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>4th time:
is there *any* commutative space,
of more than two dimensions?
as for ij = -ji, I gues taht you could look at it
as a right-agnle rotation, followed by a reflection, but
isn't it the usual thing, to view it two rotations,
around two orthogonal axes?... anyway,
a pair of rotations aren't commutative.
pratt@boole.Stanford.EDU (Email me at CS not Boole) wrote in message news:<c61asd$h0h$1@news.Stanford.EDU>...
> Noncommutativity (only in the multiplication, never in the vector space
> half of an algebra) is always commutative. For example ij = -ji tells
> you that i and j used as operators (e.g. i as rotation by 90 degrees, j
> as reflection) interact, in that doing them in the opposite order has
> the opposite effect, e.g. turning 90 degrees followed by reflecting in
> a mirror vs. reflecting followed by turning 90 degrees; these are
> opposite turns. Commuting axes can't capture that aspect of the
> interactivity of the axes of physical space. (Incidentally a
> reflection is a turn through a hyberbolic angle, whose magnitude is
> correlated with the fraction of the incident light reflected.)
--ils duces d'Enron -- conspiracy central?
http://tarpley.net
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