use of real numbers in mathematics and physics [Archive]

Tobias Fritz

Oct11-04, 03:51 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>\nDear physicists & mathematicians,\n\nas you all know, the real numbers (and probably even more the complex\nnumbers, but I will not distinguish here), are at the very foundation of\nmost part of mathematics and all parts of physics. IMHO, this is partly\ncaused by the imagination we have of a continuum, which itself stems from\nthe spacetime we live in; I do not deny that they have proved to be very\nuseful e.g. in physics: not only QM and GR heavily rely on them, and when\nperforming a physical measurement, what we expect to get is a "real"\nnumber. And, they are somehow natural as the only complete ordered field up\nto iso.\nYet, does this justify that we do not think about alternatives at all (or do\nwe?) ?\n\n\n\nBasically, my questions are:\n\n1) Do you think that there is too little abstraction and generalization in\npure mathematics away from the real numbers?\n\n2) Do you think that the real numbers are the appropriate system for\nformulating a unified physical theory? What about quantization of\nspacetime?\n\n\n\nWhat bothers me most - a few examples to both topics:\n\n1) - abstraction and generalisation as in 1) led to abstract algebra, which\ntoday is one of the most useful and sophisticated branches of\nmathematics.\n\n- analysis: is it possible to formulate analytical concepts in a more\nabstract framework? I have seen so many category-theoretic formulations\nof other familiar notions that I doubt this is not the case. What do\nyou think about nonstandard analysis?\n\n- topology: in homotopy theory, the unit interval is used to study\ngeneral topological spaces. Why not use other Dedekind-complete ordered\nsets? We do not need any arithmetic operations at all (I recently heard\na very interesting talk about a category-theoretic formulation of\nhomotopy theory, so this exists already).\nWhy are we so interested in top. spaces, especially manifolds, only?\nThere are other interesting concepts too, like uniform spaces or\ndiffeological spaces, for example.\n\n2) - We say that the result of a physical measurement is a real number. But\nwhat about errors? Any dense subset, say the rationals Q, would be\nindistinguishable by measurements.\n\n- quantization of spacetime: can we hope to achieve this using a\ncontinuum like R? Or do we need something more discrete? But what about\nisotropy, then? What do you think the mathematics of a unified theory\ncould look like?\nI know that in mathematics, there are always people thinking about the\nabsurdest things, but AFAIK all attempts for unification of QM and GR\nuse algebra, (functional) analysis and topology "over" R; let it be\nnoncommutative geometry, superstrings or whatever. If this is wrong and\nthere are people considering alternatives, please tell me and excuse my\nignorance.\n\n\nPlease do not ask which alternatives I have - of course no serious ones! Nor\ndo I propose to weaken any specific properties R has. But history shows\nthat we should question *everything*, beginning at the very basics.\n\n\n\nHoping for a lively discussion,\nTobias\n\n[deliberately leaving out a famous quotation from Kronecker, because: who\nknows whether the natural numbers have a physical reality? ;-)]\n\n--\neveryone who casts a shadow\nseems to stand in the sun\n\nreverse my forename for mail! - saibot\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>Dear physicists & mathematicians,

as you all know, the real numbers (and probably even more the complex
numbers, but I will not distinguish here), are at the very foundation of
most part of mathematics and all parts of physics. IMHO, this is partly
caused by the imagination we have of a continuum, which itself stems from
the spacetime we live in; I do not deny that they have proved to be very
useful e.g. in physics: not only QM and GR heavily rely on them, and when
performing a physical measurement, what we expect to get is a "real"
number. And, they are somehow natural as the only complete ordered field up
to iso.
Yet, does this justify that we do not think about alternatives at all (or do
we?) ?

Basically, my questions are:

1) Do you think that there is too little abstraction and generalization in
pure mathematics away from the real numbers?

2) Do you think that the real numbers are the appropriate system for
formulating a unified physical theory? What about quantization of
spacetime?

What bothers me most - a few examples to both topics:

1) - abstraction and generalisation as in 1) led to abstract algebra, which
today is one of the most useful and sophisticated branches of
mathematics.

- analysis: is it possible to formulate analytical concepts in a more
abstract framework? I have seen so many category-theoretic formulations
of other familiar notions that I doubt this is not the case. What do
you think about nonstandard analysis?

- topology: in homotopy theory, the unit interval is used to study
general topological spaces. Why not use other Dedekind-complete ordered
sets? We do not need any arithmetic operations at all (I recently heard
a very interesting talk about a category-theoretic formulation of
homotopy theory, so this exists already).
Why are we so interested in top. spaces, especially manifolds, only?
There are other interesting concepts too, like uniform spaces or
diffeological spaces, for example.

2) - We say that the result of a physical measurement is a real number. But
what about errors? Any dense subset, say the rationals Q, would be
indistinguishable by measurements.

- quantization of spacetime: can we hope to achieve this using a
continuum like R? Or do we need something more discrete? But what about
isotropy, then? What do you think the mathematics of a unified theory
could look like?
I know that in mathematics, there are always people thinking about the
absurdest things, but AFAIK all attempts for unification of QM and GR
use algebra, (functional) analysis and topology "over" R; let it be
noncommutative geometry, superstrings or whatever. If this is wrong and
there are people considering alternatives, please tell me and excuse my
ignorance.

Please do not ask which alternatives I have - of course no serious ones! Nor
do I propose to weaken any specific properties R has. But history shows
that we should question *everything*, beginning at the very basics.

Hoping for a lively discussion,
Tobias

[deliberately leaving out a famous quotation from Kronecker, because: who
knows whether the natural numbers have a physical reality? ;-)]

--
everyone who casts a shadow
seems to stand in the sun

reverse my forename for mail! - saibot

robert j. kolker

Oct11-04, 01:06 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>\n\n\nTobias Fritz wrote:\n>\n> 2) Do you think that the real numbers are the appropriate system for\n> formulating a unified physical theory? What about quantization of\n> spacetime?\n>\n\nMy guess is that a purely discrete theory to describe physical reality\nwill be mathematically intractable. We have a dillema. The mathematics\nwe can use, cannot be literally true of reality. The mathematics that\ncan be literally true of reality we cannot use because of its difficulty.\n\nGo figure.\n\nBob Kolker\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>Tobias Fritz wrote:
>
> 2) Do you think that the real numbers are the appropriate system for
> formulating a unified physical theory? What about quantization of
> spacetime?
>

My guess is that a purely discrete theory to describe physical reality
will be mathematically intractable. We have a dillema. The mathematics
we can use, cannot be literally true of reality. The mathematics that
can be literally true of reality we cannot use because of its difficulty.

Go figure.

Bob Kolker

Aage Andersen

Oct11-04, 01:06 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>\n>\n> 2) - We say that the result of a physical measurement is a real number.\n> But\n> what about errors? Any dense subset, say the rationals Q, would be\n> indistinguishable by measurements.\n\nActually a single measurement always gives a rational number.\n\nregards Aage\n\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>2) - We say that the result of a physical measurement is a real number.
> But
> what about errors? Any dense subset, say the rationals Q, would be
> indistinguishable by measurements.

Actually a single measurement always gives a rational number.

regards Aage

Arnold Neumaier

Oct12-04, 10:49 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\nAage Andersen wrote:\n>>2) - We say that the result of a physical measurement is a real number.\n>>But\n>> what about errors? Any dense subset, say the rationals Q, would be\n>> indistinguishable by measurements.\n>\n>\n> Actually a single measurement always gives a rational number.\n\nNo, according to NIST standards, it gives an interval consisting of a\nrational number together with an error bar.\n\nNevertheless there is no contradiction if one assumes that reality is\ngoverned by equations in terms of reals, and only the measurement\nabilities are limited.\n\n\nArnold Neumaier\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>Aage Andersen wrote:
>>2) - We say that the result of a physical measurement is a real number.
>>But
>> what about errors? Any dense subset, say the rationals Q, would be
>> indistinguishable by measurements.
>
>
> Actually a single measurement always gives a rational number.

No, according to NIST standards, it gives an interval consisting of a
rational number together with an error bar.

Nevertheless there is no contradiction if one assumes that reality is
governed by equations in terms of reals, and only the measurement
abilities are limited.

Arnold Neumaier

Aage Andersen

Oct12-04, 01:22 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>\n"Arnold Neumaier" <Arnold.Neumaier@univie.ac.at> skrev i en meddelelse\nnews:416BAE1B.8020109@univie.ac.at...\ n\n>> Actually a single measurement always gives a rational number.\n>\n> No, according to NIST standards, it gives an interval consisting of a\n> rational number together with an error bar.\n>\n> Nevertheless there is no contradiction if one assumes that reality is\n> governed by equations in terms of reals, and only the measurement\n> abilities are limited.\n>\n>\n> Arnold Neumaier\n\nOf course you are right. I just wanted to stress that my voltmeter never\nshows squareroot(2)\nor the like. Perhaps we could say a single reading always gives a rational\nnumber.\n\nregards Aage\n\n\n\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>"Arnold Neumaier" <Arnold.Neumaier@univie.ac.at> skrev i en meddelelse
news:416BAE1B.8020109@univie.ac.at...

>> Actually a single measurement always gives a rational number.
>
> No, according to NIST standards, it gives an interval consisting of a
> rational number together with an error bar.
>
> Nevertheless there is no contradiction if one assumes that reality is
> governed by equations in terms of reals, and only the measurement
> abilities are limited.
>
>
> Arnold Neumaier

Of course you are right. I just wanted to stress that my voltmeter never
shows squareroot(2)
or the like. Perhaps we could say a single reading always gives a rational
number.

regards Aage

Danny Ross Lunsford

Oct13-04, 03:45 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\nTobias Fritz <tobias@mad.scientist.com> wrote in message news:<2sqb28F1n87f0U1@uni-berlin.de>...\n\n> Basically, my questions are:\n>\n> 1) Do you think that there is too little abstraction and generalization in\n> pure mathematics away from the real numbers?\n>\n> 2) Do you think that the real numbers are the appropriate system for\n> formulating a unified physical theory? What about quantization of\n> spacetime?\n\n1) No! Clifford algebras, Grassmann algebras - complex numbers,\nquaternions, and even octonions (strong interactions) all play\nessential roles in physics. In particular, the "quaternion quantum\nmechanics" of Finkelstein and Jauch was a direct precursor to\nWeinberg-Salaam-Glashow.\n\n2) No one is ever going to measure imaginary energy, charge etc. But\nwe live with complex numbers all the time, in the everyday, classical,\n"real" world. Did you know that Euclidean geometry is essentially\ncharacterized by i? What could be more "real"? Yet, it is so. The\nrelation of reals to complexes in physics is often on of *situation*\nvs. *solution*. That is, equations may imply complex configurations be\nthey ever so real. So, for example, if we allow complex numbers, we\ncan state that every line in the (projective) plane intersects any\nnon-degenerate conic exactly twice, with the possibility that the two\nintersection points are coincident. Consider for example a line that\nintersects an ellipse in two real points. Draw the real tangents at\nthose points - these are two real lines. These in turn intersect at\nthe point "polar" to the given line with respect to the ellipse. Now\ntake a line that does not intersect the ellipse in real points - you\ncan still determine imaginary intersection points, and form the\nimaginary tangents there, which now intersect in a unique *real* point\ninterior to the ellipse! The relationship of polarity persists even\nthough the intermediate elements have become imaginary. The *real*\nrelationship of pole and polar implies complex configurations.\n\nI\'ve always thought that QM may be a matter of "phasiness" in the\nsense that real values for physical quantities can only be determined\nat spacetime places that are mutually "in phase" - that the same\nquantity measured in an "out of phase" place will give an imaginary -\nthat is, "unreal" - answer in the direct sense.\n\n-drl\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>Tobias Fritz <tobias@mad.scientist.com> wrote in message news:<2sqb28F1n87f0U1@uni-berlin.de>...

> Basically, my questions are:
>
> 1) Do you think that there is too little abstraction and generalization in
> pure mathematics away from the real numbers?
>
> 2) Do you think that the real numbers are the appropriate system for
> formulating a unified physical theory? What about quantization of
> spacetime?

1) No! Clifford algebras, Grassmann algebras - complex numbers,
quaternions, and even octonions (strong interactions) all play
essential roles in physics. In particular, the "quaternion quantum
mechanics" of Finkelstein and Jauch was a direct precursor to
Weinberg-Salaam-Glashow.

2) No one is ever going to measure imaginary energy, charge etc. But
we live with complex numbers all the time, in the everyday, classical,
"real" world. Did you know that Euclidean geometry is essentially
characterized by i? What could be more "real"? Yet, it is so. The
relation of reals to complexes in physics is often on of *situation*
vs. *solution*. That is, equations may imply complex configurations be
they ever so real. So, for example, if we allow complex numbers, we
can state that every line in the (projective) plane intersects any
non-degenerate conic exactly twice, with the possibility that the two
intersection points are coincident. Consider for example a line that
intersects an ellipse in two real points. Draw the real tangents at
those points - these are two real lines. These in turn intersect at
the point "polar" to the given line with respect to the ellipse. Now
take a line that does not intersect the ellipse in real points - you
can still determine imaginary intersection points, and form the
imaginary tangents there, which now intersect in a unique *real* point
interior to the ellipse! The relationship of polarity persists even
though the intermediate elements have become imaginary. The *real*
relationship of pole and polar implies complex configurations.

I've always thought that QM may be a matter of "phasiness" in the
sense that real values for physical quantities can only be determined
at spacetime places that are mutually "in phase" - that the same
quantity measured in an "out of phase" place will give an imaginary -
that is, "unreal" - answer in the direct sense.

-drl

Frank Hellmann

Oct13-04, 03:45 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"Aage Andersen" <aaa@email.dk (REMOVE)> wrote in message news:<416a8aa0\\$0\\$203\\$edfadb0f@dread12.news.t ele.dk>...\n> >\n> > 2) - We say that the result of a physical measurement is a real number.\n> > But\n> > what about errors? Any dense subset, say the rationals Q, would be\n> > indistinguishable by measurements.\n>\n> Actually a single measurement always gives a rational number.\n>\n> regards Aage\n\nThus if we assume that meassurements can be repeated infinitely, we\nnaturely are led to include all numbers to which the rational numbers\ncan converge -> The real numbers.\n\n--\nf\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>"Aage Andersen" <aaa@email.dk (REMOVE)> wrote in message news:<416a8aa0$0$203$edfadb0f@dread12.news.tele.dk>...
> >
> > 2) - We say that the result of a physical measurement is a real number.
> > But
> > what about errors? Any dense subset, say the rationals Q, would be
> > indistinguishable by measurements.
>
> Actually a single measurement always gives a rational number.
>
> regards Aage

Thus if we assume that meassurements can be repeated infinitely, we
naturely are led to include all numbers to which the rational numbers
can converge -> The real numbers.

--
f

Tobias Fritz

Oct13-04, 03:45 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>\n> We don\'t have to have a dilemma. Both physics and mathematics reveal\n> the fact that the world of integers and continuity can coexist.\n> Differential equations work very well modeling discrete phenomena:\n> nuclear reactors, fluids, electric charge and currents all involve\n> discrete quantities yet differential equations work just as well as\n> would be the case if the atomic theory was false.\n>\nYes (except for nuclear reactors ;-), but IMHO the limitations of classical\nphysics are much more remarkable: e.g. how much electronics would we have\ntoday if we didn\'t understand how transistors work (IIRC, there is some\ntunneling involved)?\n\n\n> There are totally discrete space-time-state systems, such as Cellular\n> Automata on a Cartesian lattice, whose gross behavior is well modeled\n> by differential equations.\n>\nNot surprising, since in numerics CAs are designed to give approximate\nsolutions to PDEs!\n\n\n> Further, it is possible to design such\n> systems so that the representations of many physical quantities, such\n> as momentum, spin or charge are conserved exactly. Noether\'s theorem\n> states that for every such conserved quantity there is a corresponding\n> continuous symmetry. In the case of a CA model, we must assume that\n> there is an apparent continuous symmetry that occurs as one looks at\n> scales far enough above the scale of the lattice.\n>\nSo in current physics, we are looking from some scale far above the\n"lattice". But aren\'t you interested in the lattice itself?\n\n\n> One advantage of up from the bottom totally discrete models of\n> physical systems (if we can find one) is that what occurs at the\n> bottom would be easy to understand exactly!\n>\nI doubt this, because even CA\'s can become extremely complex even on "small"\nscales as we can simulate them today on a computer. If the world is a CA,\nisn\'t it reasonable to assume that it will be extremely complex, just by\nlooking at the huge diversity of things that evolved in it? Hm, maybe not\non the other hand because we have a continuous approximation, at least.\nNote the difference between complex and complicated: a CA, for example, can\nbe complex, even though it is not complicated: its behavior is\nunpredictable in large, but the rules that govern it are very simple (of\ncourse you know the Game of Life ;-)\n\nBut can the bottom model of the universe be a discrete lattice, separating\ntime and space? Maybe, but this would fix specified frames of reference;\nthis is no disproof, but makes it very improbable. What do you think?\n\n\n> Further, it would be\n> possible to derive analytically the differential equations we know and\n> love.\n>\nYeah, we should hope so.\n\n\n> Today, in many areas where we have math that works, it is\n> accompanied by a total lack of understanding as to what is actually\n> going on at the bottom. QM is the best example, the math works but no\n> one knows why and no one understands what is actually going on.\n>\n\n--\neveryone who casts a shadow\nseems to stand in the sun\n\nreverse my forename for mail! - saibot\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>We don't have to have a dilemma. Both physics and mathematics reveal
> the fact that the world of integers and continuity can coexist.
> Differential equations work very well modeling discrete phenomena:
> nuclear reactors, fluids, electric charge and currents all involve
> discrete quantities yet differential equations work just as well as
> would be the case if the atomic theory was false.
>
Yes (except for nuclear reactors ;-), but IMHO the limitations of classical
physics are much more remarkable: e.g. how much electronics would we have
today if we didn't understand how transistors work (IIRC, there is some
tunneling involved)?

> There are totally discrete space-time-state systems, such as Cellular
> Automata on a Cartesian lattice, whose gross behavior is well modeled
> by differential equations.
>
Not surprising, since in numerics CAs are designed to give approximate
solutions to PDEs!

> Further, it is possible to design such
> systems so that the representations of many physical quantities, such
> as momentum, spin or charge are conserved exactly. Noether's theorem
> states that for every such conserved quantity there is a corresponding
> continuous symmetry. In the case of a CA model, we must assume that
> there is an apparent continuous symmetry that occurs as one looks at
> scales far enough above the scale of the lattice.
>
So in current physics, we are looking from some scale far above the
"lattice". But aren't you interested in the lattice itself?

> One advantage of up from the bottom totally discrete models of
> physical systems (if we can find one) is that what occurs at the
> bottom would be easy to understand exactly!
>
I doubt this, because even CA's can become extremely complex even on "small"
scales as we can simulate them today on a computer. If the world is a CA,
isn't it reasonable to assume that it will be extremely complex, just by
looking at the huge diversity of things that evolved in it? Hm, maybe not
on the other hand because we have a continuous approximation, at least.
Note the difference between complex and complicated: a CA, for example, can
be complex, even though it is not complicated: its behavior is
unpredictable in large, but the rules that govern it are very simple (of
course you know the Game of Life ;-)

But can the bottom model of the universe be a discrete lattice, separating
time and space? Maybe, but this would fix specified frames of reference;
this is no disproof, but makes it very improbable. What do you think?

> Further, it would be
> possible to derive analytically the differential equations we know and
> love.
>
Yeah, we should hope so.

> Today, in many areas where we have math that works, it is
> accompanied by a total lack of understanding as to what is actually
> going on at the bottom. QM is the best example, the math works but no
> one knows why and no one understands what is actually going on.
>

--
everyone who casts a shadow
seems to stand in the sun

reverse my forename for mail! - saibot

Arnold Neumaier

Oct13-04, 08:44 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\nAage Andersen wrote:\n> "Arnold Neumaier" <Arnold.Neumaier@univie.ac.at> skrev i en meddelelse\n> news:416BAE1B.8020109@univie.ac.at...\n>\n>\n>>>Ac tually a single measurement always gives a rational number.\n>>\n>>No, according to NIST standards, it gives an interval consisting of a\n>>rational number together with an error bar.\n>>\n>>Nevertheless there is no contradiction if one assumes that reality is\n>>governed by equations in terms of reals, and only the measurement\n>>abilities are limited.\n>>\n>>\n>>Arnold Neumaier\n>\n>\n> Of course you are right. I just wanted to stress that my voltmeter never\n> shows squareroot(2)\n> or the like. Perhaps we could say a single reading always gives a rational\n> number.\n\nNo. Your voltmeter neither shows 1.5 or the like.\n\nInfinitely many rationals (and uncountably many reals) are\ncompatible with any observable state of the voltmeter.\nThat\'s why the error bars are intrinsic to measurement results, even to\nsingle readings. Deleting them and claiming exact measurement results is\njust laziness, acceptable when the resolution of an instrument is known.\n\n\nArnold Neumaier\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>Aage Andersen wrote:
> "Arnold Neumaier" <Arnold.Neumaier@univie.ac.at> skrev i en meddelelse
> news:416BAE1B.8020109@univie.ac.at...
>
>
>>>Actually a single measurement always gives a rational number.
>>
>>No, according to NIST standards, it gives an interval consisting of a
>>rational number together with an error bar.
>>
>>Nevertheless there is no contradiction if one assumes that reality is
>>governed by equations in terms of reals, and only the measurement
>>abilities are limited.
>>
>>
>>Arnold Neumaier
>
>
> Of course you are right. I just wanted to stress that my voltmeter never
> shows squareroot(2)
> or the like. Perhaps we could say a single reading always gives a rational
> number.

No. Your voltmeter neither shows 1.5 or the like.

Infinitely many rationals (and uncountably many reals) are
compatible with any observable state of the voltmeter.
That's why the error bars are intrinsic to measurement results, even to
single readings. Deleting them and claiming exact measurement results is
just laziness, acceptable when the resolution of an instrument is known.

Arnold Neumaier

robert j. kolker

Oct13-04, 02:13 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>\n\nDanny Ross Lunsford wrote:\n>\n> 2) No one is ever going to measure imaginary energy, charge etc. But\n> we live with complex numbers all the time, in the everyday, classical,\n> "real" world. Did you know that Euclidean geometry is essentially\n> characterized by i?\n\nEuclidean geometry is about isometries O(n) and SO(n) for the rigid\nmotions (excluding orientation reversing symmetries). The matrix\nelements of the group representations are real. Where does i come in on\nthis?\n\nBob Kolker\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>Danny Ross Lunsford wrote:
>
> 2) No one is ever going to measure imaginary energy, charge etc. But
> we live with complex numbers all the time, in the everyday, classical,
> "real" world. Did you know that Euclidean geometry is essentially
> characterized by i?

Euclidean geometry is about isometries O(n) and SO(n) for the rigid
motions (excluding orientation reversing symmetries). The matrix
elements of the group representations are real. Where does i come in on
this?

Bob Kolker

Tobias Fritz

Oct14-04, 11:07 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\n> Euclidean geometry is about isometries O(n) and SO(n) for the rigid\n> motions (excluding orientation reversing symmetries). The matrix\n> elements of the group representations are real. Where does i come in on\n> this?\n>\nHm, at least they are useful in the classification of rotations: every\nmatrix in O(n) is normal, so can be diagonalized over C. By transforming to\na real basis again, it is possible to prove that there are invariant 2-dim\nand 1-dim subspaces, which leads to a block structure of the matrix.\n\n--\neveryone who casts a shadow\nseems to stand in the sun\n\nreverse my forename for mail! - saibot\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>Euclidean geometry is about isometries O(n) and SO(n) for the rigid
> motions (excluding orientation reversing symmetries). The matrix
> elements of the group representations are real. Where does i come in on
> this?
>
Hm, at least they are useful in the classification of rotations: every
matrix in O(n) is normal, so can be diagonalized over C. By transforming to
a real basis again, it is possible to prove that there are invariant 2-dim
and 1-dim subspaces, which leads to a block structure of the matrix.

--
everyone who casts a shadow
seems to stand in the sun

reverse my forename for mail! - saibot

J. J. Lodder

Oct14-04, 11:07 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>\nTobias Fritz <tobias@mad.scientist.com> wrote:\n\n> Dear physicists & mathematicians,\n>\n> as you all know, the real numbers (and probably even more the complex\n> numbers, but I will not distinguish here), are at the very foundation of\n> most part of mathematics and all parts of physics.\n\nOn the contrary: real numbers have nothing to do with physics.\nsee below.\n\n> Basically, my questions are:\n>\n> 1) Do you think that there is too little abstraction and generalization in\n> pure mathematics away from the real numbers?\n\nDo you have any idea at all what you are talking about?\n\n> 2) Do you think that the real numbers are the appropriate system for\n> formulating a unified physical theory? What about quantization of\n> spacetime?\n\nReal numbers are irrelevant to experimental physics.\nAny measurement result has a finite precision,\nhence is a rational number.\nIf you insist (suitable units) you can even arrrange\nfor any measurement result to be an integer.\n\nIn theoretical physics some real numbers occur, such as 4pi.\nReplacing these by a rational approximation,\nsay to a hundred decimal places, won\'t change any observable prediction.\n\nIf you realy insist on being as complicated as you need\nthe computable reals might be needed for physics,\nbut this is still only a countable subset of the reals.\n\nBy definition, no theoretical computation can ever\npredict an uncomputable real as the outcome.\n\nBest,\n\nJan\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>Tobias Fritz <tobias@mad.scientist.com> wrote:

> Dear physicists & mathematicians,
>
> as you all know, the real numbers (and probably even more the complex
> numbers, but I will not distinguish here), are at the very foundation of
> most part of mathematics and all parts of physics.

On the contrary: real numbers have nothing to do with physics.
see below.

> Basically, my questions are:
>
> 1) Do you think that there is too little abstraction and generalization in
> pure mathematics away from the real numbers?

Do you have any idea at all what you are talking about?

> 2) Do you think that the real numbers are the appropriate system for
> formulating a unified physical theory? What about quantization of
> spacetime?

Real numbers are irrelevant to experimental physics.
Any measurement result has a finite precision,
hence is a rational number.
If you insist (suitable units) you can even arrrange
for any measurement result to be an integer.

In theoretical physics some real numbers occur, such as 4pi.
Replacing these by a rational approximation,
say to a hundred decimal places, won't change any observable prediction.

If you realy insist on being as complicated as you need
the computable reals might be needed for physics,
but this is still only a countable subset of the reals.

By definition, no theoretical computation can ever
predict an uncomputable real as the outcome.

Best,

Jan

Oz

Oct15-04, 01:54 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>\n\n\nDanny Ross Lunsford <antimatter33@yahoo.com> writes\n\n>I\'ve always thought that QM may be a matter of "phasiness" in the\n>sense that real values for physical quantities can only be determined\n>at spacetime places that are mutually "in phase" - that the same\n>quantity measured in an "out of phase" place will give an imaginary -\n>that is, "unreal" - answer in the direct sense.\n\nYou are not alone.\n\nI\'m not entirely sure, though, that this is \'truly\' imaginary.\nOne can (I think) always map imaginary space onto a 2-D space.\nI presume one can do this analogously for quarternions.\n\nAnyway, call it phase or another dimension the effect is the same.\nThings can pass each other by without interacting because they are not\nin fact co-incident, although that is excessively simplistic.\n\n--\nOz\nThis post is worth absolutely nothing and is probably fallacious.\n\nUse oz@farmeroz.port995.com [ozacoohdb@despammed.com functions].\nBTOPENWORLD address has ceased. DEMON address has ceased.\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>Danny Ross Lunsford <antimatter33@yahoo.com> writes

>I've always thought that QM may be a matter of "phasiness" in the
>sense that real values for physical quantities can only be determined
>at spacetime places that are mutually "in phase" - that the same
>quantity measured in an "out of phase" place will give an imaginary -
>that is, "unreal" - answer in the direct sense.

You are not alone.

I'm not entirely sure, though, that this is 'truly' imaginary.
One can (I think) always map imaginary space onto a 2-D space.
I presume one can do this analogously for quarternions.

Anyway, call it phase or another dimension the effect is the same.
Things can pass each other by without interacting because they are not
in fact co-incident, although that is excessively simplistic.

--
Oz
This post is worth absolutely nothing and is probably fallacious.

Use oz@farmeroz.port995.com [ozacoohdb@despammed.com functions].
BTOPENWORLD address has ceased. DEMON address has ceased.

Danny Ross Lunsford

Oct15-04, 01:54 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>\n\n"robert j. kolker" <nowhere@nowhere.net> wrote in message news:<2t53qeF1rk6crU1@uni-berlin.de>...\n\n> Euclidean geometry is about isometries O(n) and SO(n) for the rigid\n> motions (excluding orientation reversing symmetries). The matrix\n> elements of the group representations are real. Where does i come in on\n> this?\n\n>From the characteristic equation\n\nx^2 + y^2 + ... + z^2 = 0\n\nof the "ideal domain" of Euclidean geometry as it sits inside\nprojective geometry. Amazingly, i is to Euclid as c is to Minkowski!\nIn the latter the ideal domain is simply the light cone.\n\n-drl\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>"robert j. kolker" <nowhere@nowhere.net> wrote in message news:<2t53qeF1rk6crU1@uni-berlin.de>...

> Euclidean geometry is about isometries O(n) and SO(n) for the rigid
> motions (excluding orientation reversing symmetries). The matrix
> elements of the group representations are real. Where does i come in on
> this?

>From the characteristic equation

x^2 + y^2 + .[/itex].[itex]. + z^2 =

of the "ideal domain" of Euclidean geometry as it sits inside
projective geometry. Amazingly, i is to Euclid as c is to Minkowski!
In the latter the ideal domain is simply the light cone.

-drl

John Forkosh

Oct15-04, 01:55 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>\n\nArnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:\n: Aage Andersen wrote:\n: > "Arnold Neumaier" <Arnold.Neumaier@univie.ac.at> wrote:\n: >\n: >> Aage Andersen wrote:\n: >>>Actually a single measurement always gives a rational number.\n: >>\n: >>\n: >>No, according to NIST standards, it gives an interval consisting of a\n: >>rational number together with an error bar.\n: >>\n: >>Nevertheless there is no contradiction if one assumes that reality is\n: >>governed by equations in terms of reals, and only the measurement\n: >>abilities are limited.\n: >>Arnold Neumaier\n: >\n: >\n: > Of course you are right. I just wanted to stress that my voltmeter never\n: > shows squareroot(2)\n: > or the like. Perhaps we could say a single reading always gives a rational\n: > number.\n:\n:\n: No. Your voltmeter neither shows 1.5 or the like.\n:\n: Infinitely many rationals (and uncountably many reals) are\n: compatible with any observable state of the voltmeter.\n: That\'s why the error bars are intrinsic to measurement results, even to\n: single readings. Deleting them and claiming exact measurement results is\n: just laziness, acceptable when the resolution of an instrument is known.\n: Arnold Neumaier\n\nHow does this work foundationally? To begin with, how can you\n(re-)formulate quantum mechanics so that the eigenvalue of a hermitian\noperator is a "value" (real or rational) plus error bars?\nAre you just calculating something like <x>=\\int\\phi*x\\phi,\n<x^2>=\\int\\phi*x^2\\ph i, then calculating standard deviation in the\nusual way and arbitrarily defining "error bars" (or "error\ndistributions") as "six sigma" or something similar?\nIf that\'s the case, then it seems to me like you can\'t\nfoundationally claim that measurements are values plus error bars.\nIn this case error bars are just derived quantities that only\ncome into play at the tail end of the theory.\nAn ensemble interpretation also wouldn\'t seem to affect\nthe interpretation of *individual* measurements.\n\nTo foundationally claim measurements entail error bars, I\'d think\nyou\'d have to initially introduce the concept of a measurement\nprocedure in which error bars are already intrinsic. Then you\'d\nhave to develop a formal model around this idea, i.e., a model\nthat contains elements which can be interpreted as this kind of\nmeasurement (values plus error bars).\n\nOf course, your argument that *individual* measurements indeed\nentail error bars sounds physically reasonable. But how do you\nstart with this idea as foundationally axiomatic, and then build\nup a formalism around it? Thanks,\nJohn Forkosh\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>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:
: Aage Andersen wrote:
: > "Arnold Neumaier" <Arnold.Neumaier@univie.ac.at> wrote:
: >
: >> Aage Andersen wrote:
: >>>Actually a single measurement always gives a rational number.
: >>
: >>
: >>No, according to NIST standards, it gives an interval consisting of a
: >>rational number together with an error bar.
: >>
: >>Nevertheless there is no contradiction if one assumes that reality is
: >>governed by equations in terms of reals, and only the measurement
: >>abilities are limited.
: >>Arnold Neumaier
: >
: >
: > Of course you are right. I just wanted to stress that my voltmeter never
: > shows squareroot(2)
: > or the like. Perhaps we could say a single reading always gives a rational
: > number.
:
:
: No. Your voltmeter neither shows 1.5 or the like.
:
: Infinitely many rationals (and uncountably many reals) are
: compatible with any observable state of the voltmeter.
: That's why the error bars are intrinsic to measurement results, even to
: single readings. Deleting them and claiming exact measurement results is
: just laziness, acceptable when the resolution of an instrument is known.
: Arnold Neumaier

How does this work foundationally? To begin with, how can you
(re-)formulate quantum mechanics so that the eigenvalue of a hermitian
operator is a "value" (real or rational) plus error bars?
Are you just calculating something like <x>=\int\phi*x\phi,<x^2>=\int\phi*x^2\phi, then calculating standard deviation in the
usual way and arbitrarily defining "error bars" (or "error
distributions") as "six \sigma" or something similar?
If that's the case, then it seems to me like you can't
foundationally claim that measurements are values plus error bars.
In this case error bars are just derived quantities that only
come into play at the tail end of the theory.
An ensemble interpretation also wouldn't seem to affect
the interpretation of *individual* measurements.

To foundationally claim measurements entail error bars, I'd think
you'd have to initially introduce the concept of a measurement
procedure in which error bars are already intrinsic. Then you'd
have to develop a formal model around this idea, i.e., a model
that contains elements which can be interpreted as this kind of
measurement (values plus error bars).

Of course, your argument that *individual* measurements indeed
entail error bars sounds physically reasonable. But how do you
start with this idea as foundationally axiomatic, and then build
up a formalism around it? Thanks,
John Forkosh

John Baez

Nov29-04, 02:50 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>In article <416c04a4\\$0\\$199\\$edfadb0f@dread12.news.tele.d k>,\nAage Andersen <aaa@email.dk (REMOVE)> wrote:\n\n>I just wanted to stress that my voltmeter never shows squareroot(2)\n>or the like. Perhaps we could say a single reading always gives a rational\n>number.\n\nThat\'s just because you drew symbols like 1.1 and 2 on the dial of\nyour voltmeter. If you drew symbols like sqrt(2) and pi on it,\nI guess you\'d claim a single reading always gives an irrational number.\n\nThough often repeated, this idea that a single measurement always\ngives a rational number is... irrational! It goes back to an old\nGreek prejudice in favor of rational numbers. When someone proved\nthe square root of two was irrational, they drowned him (according\nto the legend).\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 <416c04a4$0$199$edfadb0f@dread12.news.tele.dk>,
Aage Andersen <aaa@email.dk (REMOVE)> wrote:

>I just wanted to stress that my voltmeter never shows squareroot(2)
>or the like. Perhaps we could say a single reading always gives a rational
>number.

That's just because you drew symbols like 1.1 and 2 on the dial of
your voltmeter. If you drew symbols like \sqrt(2) and \pi on it,
I guess you'd claim a single reading always gives an irrational number.

Though often repeated, this idea that a single measurement always
gives a rational number is... irrational! It goes back to an old
Greek prejudice in favor of rational numbers. When someone proved
the square root of two was irrational, they drowned him (according
to the legend).

John Baez

Nov29-04, 02:50 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>In article <416D1334.7080004@univie.ac.at>,\nArnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:\n\n>Aage Andersen wrote:\n\n>> "Arnold Neumaier" <Arnold.Neumaier@univie.ac.at> skrev i en meddelelse\n>> news:416BAE1B.8020109@univie.ac.at...\n\n>>>>Actua lly a single measurement always gives a rational number.\n\n>>>No, according to NIST standards, it gives an interval consisting of a\n>>>rational number together with an error bar.\n\nThis may be convenient, but it\'s a convention rather than a law of nature:\nwe could use a rational multiple of pi together with an error bar if we\npreferred.\n\n>Infinitely many rationals (and uncountably many reals) are\n>compatible with any observable state of the voltmeter.\n\nTrue!\n\n>That\'s why the error bars are intrinsic to measurement results, even to\n>single readings. Deleting them and claiming exact measurement results is\n>just laziness, acceptable when the resolution of an instrument is known.\n\nIf we fully accepted this principle, we\'d have to admit the error bars\nshould have error bars themselves... and so on ad infinitum... which\ngets a bit tiresome. Everyone gets lazy sooner or later. :-)\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 <416D1334.7080004@univie.ac.at>,
Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:

>Aage Andersen wrote:

>> "Arnold Neumaier" <Arnold.Neumaier@univie.ac.at> skrev i en meddelelse
>> news:416BAE1B.8020109@univie.ac.at...

>>>>Actually a single measurement always gives a rational number.

>>>No, according to NIST standards, it gives an interval consisting of a
>>>rational number together with an error bar.

This may be convenient, but it's a convention rather than a law of nature:
we could use a rational multiple of \pi together with an error bar if we
preferred.

>Infinitely many rationals (and uncountably many reals) are
>compatible with any observable state of the voltmeter.

True!

>That's why the error bars are intrinsic to measurement results, even to
>single readings. Deleting them and claiming exact measurement results is
>just laziness, acceptable when the resolution of an instrument is known.

If we fully accepted this principle, we'd have to admit the error bars
should have error bars themselves... and so on ad infinitum... which
gets a bit tiresome. Everyone gets lazy sooner or later. :-)

Norm Dresner

Nov30-04, 12:48 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>"John Baez" <baez@galaxy.ucr.edu> wrote in message\nnews:codf52\\$cl2\\$1@glue.ucr.edu...\n> In article <416c04a4\\$0\\$199\\$edfadb0f@dread12.news.tele.d k>,\n> Aage Andersen <aaa@email.dk (REMOVE)> wrote:\n>\n> >I just wanted to stress that my voltmeter never shows squareroot(2)\n> >or the like. Perhaps we could say a single reading always gives a\nrational\n> >number.\n>\n> That\'s just because you drew symbols like 1.1 and 2 on the dial of\n> your voltmeter. If you drew symbols like sqrt(2) and pi on it,\n> I guess you\'d claim a single reading always gives an irrational number.\n>\n> Though often repeated, this idea that a single measurement always\n> gives a rational number is... irrational! It goes back to an old\n> Greek prejudice in favor of rational numbers. When someone proved\n> the square root of two was irrational, they drowned him (according\n> to the legend).\n>\nJust because there\'s a specific number on a dial of a voltmeter doesn\'t mean\nthat it represents the *exact* voltage. It\'s no more like to be sqrt(2)\nvolts than it is 1.000000... volts.\n\nNorm\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>"John Baez" <baez@galaxy.ucr.edu> wrote in message
news:codf52$cl2$1@glue.ucr.edu...
> In article <416c04a4$0$199$edfadb0f@dread12.news.tele.dk>,
> Aage Andersen <aaa@email.dk (REMOVE)> wrote:
>
> >I just wanted to stress that my voltmeter never shows squareroot(2)
> >or the like. Perhaps we could say a single reading always gives a
rational
> >number.
>
> That's just because you drew symbols like 1.1 and 2 on the dial of
> your voltmeter. If you drew symbols like \sqrt(2) and \pi on it,
> I guess you'd claim a single reading always gives an irrational number.
>
> Though often repeated, this idea that a single measurement always
> gives a rational number is... irrational! It goes back to an old
> Greek prejudice in favor of rational numbers. When someone proved
> the square root of two was irrational, they drowned him (according
> to the legend).
>
Just because there's a specific number on a dial of a voltmeter doesn't mean
that it represents the *exact* voltage. It's no more like to be \sqrt(2)
volts than it is 1.000000... volts.

Norm

Joe Rongen

Nov30-04, 12:50 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 <416c04a4\\$0\\$199\\$edfadb0f@dread12.news.tele.d k>,\n> Aage Andersen <(REMOVE)> wrote:\n>\n> I just wanted to stress that my voltmeter never shows squareroot(2)\n> or the like. Perhaps we could say a single reading always gives a\n> rational number.\n\nIf you were to measure AC RMS voltages than the\nreadout would actually show the root mean square.\n\n\n---\nOutgoing mail is certified Virus Free.\nChecked by AVG anti-virus system (http://www.grisoft.com).\nVersion: 6.0.802 / Virus Database: 545 - Release Date: 11/26/04\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 <416c04a4$0$199$edfadb0f@dread12.news.tele.dk>,
> Aage Andersen <(REMOVE)> wrote:
>
> I just wanted to stress that my voltmeter never shows squareroot(2)
> or the like. Perhaps we could say a single reading always gives a
> rational number.

If you were to measure AC RMS voltages than the
readout would actually show the root mean square.

---
Outgoing mail is certified Virus Free.
Checked by AVG anti-virus system (http://www.grisoft.com).
Version: 6..802 / Virus Database: 545 - Release Date: 11/26/04

Patrick Powers

Nov30-04, 12:50 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>baez@galaxy.ucr.edu (John Baez) wrote in message news:<codfe3\\$clf\\$1@glue.ucr.edu>...\n> >That\'s why the error bars are intrinsic to measurement results, even to\n> >single readings. Deleting them and claiming exact measurement results is\n> >just laziness, acceptable when the resolution of an instrument is known.\n>\n> If we fully accepted this principle, we\'d have to admit the error bars\n> should have error bars themselves... and so on ad infinitum... which\n> gets a bit tiresome. Everyone gets lazy sooner or later. :-)\n\nError bars for single readings? I see this as having no meaning, as\nbeing at best guesswork.\n\nAs for error bars for error bars, there is no need. This is already\ntaken care of by the mathematics of statistics.\n\nIf you want something to worry about, try systematic bias. Statistics\nand error bars won\'t help you there.\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>baez@galaxy.ucr.edu (John Baez) wrote in message news:<codfe3$clf$1@glue.ucr.edu>...
> >That's why the error bars are intrinsic to measurement results, even to
> >single readings. Deleting them and claiming exact measurement results is
> >just laziness, acceptable when the resolution of an instrument is known.
>
> If we fully accepted this principle, we'd have to admit the error bars
> should have error bars themselves... and so on ad infinitum... which
> gets a bit tiresome. Everyone gets lazy sooner or later. :-)

Error bars for single readings? I see this as having no meaning, as
being at best guesswork.

As for error bars for error bars, there is no need. This is already
taken care of by the mathematics of statistics.

If you want something to worry about, try systematic bias. Statistics
and error bars won't help you there.

jdff

Nov30-04, 12:51 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>baez@galaxy.ucr.edu (John Baez) wrote in message news:<codfe3\\$> >Infinitely many rationals (and uncountably many reals) are\n> >compatible with any observable state of the voltmeter.\n>\n> True!\n>\n> >That\'s why the error bars are intrinsic to measurement results, even to\n> >single readings. Deleting them and claiming exact measurement results is\n> >just laziness, acceptable when the resolution of an instrument is known.\n>\n> If we fully accepted this principle, we\'d have to admit the error bars\n> should have error bars themselves... and so on ad infinitum... which\n> gets a bit tiresome. Everyone gets lazy sooner or later. :-)\n\nCan\'t we invoke the Bekenstein bound, which gives the finite number of\nbits of information extractable from any system, for a given area of\ninterface. If it\'s a finite number of bits, it is not just rational,\nbut not even countably infinite.\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>baez@galaxy.ucr.edu (John Baez) wrote in message news:<codfe3$> >Infinitely many rationals (and uncountably many reals) are
> >compatible with any observable state of the voltmeter.
>
> True!
>
> >That's why the error bars are intrinsic to measurement results, even to
> >single readings. Deleting them and claiming exact measurement results is
> >just laziness, acceptable when the resolution of an instrument is known.
>
> If we fully accepted this principle, we'd have to admit the error bars
> should have error bars themselves... and so on ad infinitum... which
> gets a bit tiresome. Everyone gets lazy sooner or later. :-)

Can't we invoke the Bekenstein bound, which gives the finite number of
bits of information extractable from any system, for a given area of
interface. If it's a finite number of bits, it is not just rational,
but not even countably infinite.

Alfred Einstead

Dec1-04, 11:04 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>baez@galaxy.ucr.edu (John Baez) wrote:\n> >That\'s why the error bars are intrinsic to measurement results, even to\n> >single readings. Deleting them and claiming exact measurement results is\n> >just laziness, acceptable when the resolution of an instrument is known.\n> If we fully accepted this principle, we\'d have to admit the error bars\n> should have error bars themselves... and so on ad infinitum... which\n> gets a bit tiresome. Everyone gets lazy sooner or later. :-)\n\nThe root of the matter, addressing the subject header itself, is that\na real number conveys an infinite amount of information. If, on the\nother hand, we\'re to take limitations seriously such as the Bekenstein\nBound, a finite region of space-time will only possess a finite\ncapacity for information. This puts the real number in the\nspot light as an abstraction that goes further than what nature\nmandates.\n\nBut this and all related issues are already handled and superseded in\nlarge part by the notion of a mixed state. First, the only place\nwhere real numbers make their full impact (i.e., where both the\np\'s and q\'s potentially have unlimited precision together) is\nclassical physics. The pure states then become singular distributions\nover phase space and a gulf of infinite information separates such\na state from the classical mixed states (such as the grand canonical\ndistribution) which may be comprise of it. So, pure states are\nout, except as idealizations and everything is mixed. Therefore,\nthe last of precision is already built into the framework -- for\nthe case of Classical Physics -- at the outset.\n\nIn quantum physics 1/2 of the problem is already gone, since the\np\'s and q\'s in nature no longer contain an infinite amount of\ninformation when taken together. The pure state then, too, have\nan inherent fuzziness associated with them.\n\nThe other 1/2 of the problem still remains: the p\'s or the q\'s\ntaken by themselves can be made arbitrarily precise -- which seems\nto indicate that either, alone, contains an infinity of information.\nThe problem shows up, then, in the question of what q looks like\nafter a "wave function collapse" and what a subsequent measurement\nof, say, p yields after q is measured.\n\nThe root of the problem is then addressed by noting that the p\'s\nand q\'s are singular in the standard interpretation and one is\nforced to either (a) disallow variables with a continuous spectrum\nor (b) bring in a more generalized notion of operator -- the POVM.\n\nLurking underneath this should be some kind of theorem to the effect\nthat there is some kind of equivalence between (a) and (b) so that\neither possibility can be focused on without loss of generality.\n\nTaking the Bekenstein bound seriously seems to point, already, to\n(a). Conversely, allowing either (a) or (b) seems then to provide\na way framework which will allow for the Bekenstein bound or something\nsimilar.\n\nUltimatly, the root of the problem is the continuum: the subalgebra\nof p\'s or of q\'s each by themselves is still classical (i.e. a\ncommutative algebra). But the framework for going beyond the\ncontinuum is already there and has already been 1/2 realized by\nthe preliminary step of limiting p-q\'s together. The last step\nof limiting p\'s or q\'s by themselves (or calling forth the\nBekenstein bound) is what remains.\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>baez@galaxy.ucr.edu (John Baez) wrote:
> >That's why the error bars are intrinsic to measurement results, even to
> >single readings. Deleting them and claiming exact measurement results is
> >just laziness, acceptable when the resolution of an instrument is known.
> If we fully accepted this principle, we'd have to admit the error bars
> should have error bars themselves... and so on ad infinitum... which
> gets a bit tiresome. Everyone gets lazy sooner or later. :-)

The root of the matter, addressing the subject header itself, is that
a real number conveys an infinite amount of information. If, on the
other hand, we're to take limitations seriously such as the Bekenstein
Bound, a finite region of space-time will only possess a finite
capacity for information. This puts the real number in the
spot light as an abstraction that goes further than what nature
mandates.

But this and all related issues are already handled and superseded in
large part by the notion of a mixed state. First, the only place
where real numbers make their full impact (i.e., where both the
p's and q's potentially have unlimited precision together) is
classical physics. The pure states then become singular distributions
over phase space and a gulf of infinite information separates such
a state from the classical mixed states (such as the grand canonical
distribution) which may be comprise of it. So, pure states are
out, except as idealizations and everything is mixed. Therefore,
the last of precision is already built into the framework -- for
the case of Classical Physics -- at the outset.

In quantum physics 1/2 of the problem is already gone, since the
p's and q's in nature no longer contain an infinite amount of
information when taken together. The pure state then, too, have
an inherent fuzziness associated with them.

The other 1/2 of the problem still remains: the p's or the q's
taken by themselves can be made arbitrarily precise -- which seems
to indicate that either, alone, contains an infinity of information.
The problem shows up, then, in the question of what q looks like
after a "wave function collapse" and what a subsequent measurement
of, say, p yields after q is measured.

The root of the problem is then addressed by noting that the p's
and q's are singular in the standard interpretation and one is
forced to either (a) disallow variables with a continuous spectrum
or (b) bring in a more generalized notion of operator -- the POVM.

Lurking underneath this should be some kind of theorem to the effect
that there is some kind of equivalence between (a) and (b) so that
either possibility can be focused on without loss of generality.

Taking the Bekenstein bound seriously seems to point, already, to
(a). Conversely, allowing either (a) or (b) seems then to provide
a way framework which will allow for the Bekenstein bound or something
similar.

Ultimatly, the root of the problem is the continuum: the subalgebra
of p's or of q's each by themselves is still classical (i.e. a
commutative algebra). But the framework for going beyond the
continuum is already there and has already been 1/2 realized by
the preliminary step of limiting p-q's together. The last step
of limiting p's or q's by themselves (or calling forth the
Bekenstein bound) is what remains.

Oz

Dec1-04, 11:06 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>Norm Dresner <ndrez@att.net> writes\n\n>Just because there\'s a specific number on a dial of a voltmeter doesn\'t mean\n>that it represents the *exact* voltage. It\'s no more like to be sqrt(2)\n>volts than it is 1.000000... volts.\n\nGiven its very likely measuring rms values its probably more likely to\nbe reading in sqrt(2) anyway. Don\'t even *think* of rotary dials where\npi is likely to figure as well.\n\nYe gods, we live in an irrational world.....\n\nThat explains everything...\n\n--\nOz\nThis post is worth absolutely nothing and is probably fallacious.\n\nUse oz@farmeroz.port995.com [ozacoohdb@despammed.com functions].\nBTOPENWORLD address has ceased. DEMON address has ceased.\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>Norm Dresner <ndrez@att.net> writes

>Just because there's a specific number on a dial of a voltmeter doesn't mean
>that it represents the *exact* voltage. It's no more like to be \sqrt(2)
>volts than it is 1.000000... volts.

Given its very likely measuring rms values its probably more likely to
be reading in \sqrt(2) anyway. Don't even *think* of rotary dials where
\pi is likely to figure as well.

Ye gods, we live in an irrational world.....

That explains everything...

--
Oz
This post is worth absolutely nothing and is probably fallacious.

Use oz@farmeroz.port995.com [ozacoohdb@despammed.com functions].
BTOPENWORLD address has ceased. DEMON address has ceased.

Arnold Neumaier

Dec2-04, 06:14 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>Alfred Einstead wrote:\n> baez@galaxy.ucr.edu (John Baez) wrote:\n>\n>>>That\'s why the error bars are intrinsic to measurement results, even to\n>>>single readings. Deleting them and claiming exact measurement results is\n>>>just laziness, acceptable when the resolution of an instrument is known.\n>>\n>>If we fully accepted this principle, we\'d have to admit the error bars\n>>should have error bars themselves... and so on ad infinitum... which\n>>gets a bit tiresome. Everyone gets lazy sooner or later. :-)\n>\n> The root of the matter, addressing the subject header itself, is that\n> a real number conveys an infinite amount of information. [...]\n>\n> In quantum physics 1/2 of the problem is already gone, since the\n> p\'s and q\'s in nature no longer contain an infinite amount of\n> information when taken together. The pure state then, too, have\n> an inherent fuzziness associated with them.\n\nBut whether or not a state is pure, all expectations are exact\nreal numbers, with an infinite amount of information.\nAnd this will be so in any reasonable form of physics.\nCast out continuity, and you lose physics.\n\n\n> The root of the problem is then addressed by noting that the p\'s\n> and q\'s are singular in the standard interpretation and one is\n> forced to either (a) disallow variables with a continuous spectrum\n> or (b) bring in a more generalized notion of operator -- the POVM.\n\nBoth quantities with a discrete spectrum, and POVMs have\nreal-valued expectations, and their dynamics is continuous, implying that\none cannot get rid of the continuum that way.\n\n\nArnold Neumaier\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>Alfred Einstead wrote:
> baez@galaxy.ucr.edu (John Baez) wrote:
>
>>>That's why the error bars are intrinsic to measurement results, even to
>>>single readings. Deleting them and claiming exact measurement results is
>>>just laziness, acceptable when the resolution of an instrument is known.
>>
>>If we fully accepted this principle, we'd have to admit the error bars
>>should have error bars themselves... and so on ad infinitum... which
>>gets a bit tiresome. Everyone gets lazy sooner or later. :-)
>
> The root of the matter, addressing the subject header itself, is that
> a real number conveys an infinite amount of information. [...]
>
> In quantum physics 1/2 of the problem is already gone, since the
> p's and q's in nature no longer contain an infinite amount of
> information when taken together. The pure state then, too, have
> an inherent fuzziness associated with them.

But whether or not a state is pure, all expectations are exact
real numbers, with an infinite amount of information.
And this will be so in any reasonable form of physics.
Cast out continuity, and you lose physics.

> The root of the problem is then addressed by noting that the p's
> and q's are singular in the standard interpretation and one is
> forced to either (a) disallow variables with a continuous spectrum
> or (b) bring in a more generalized notion of operator -- the POVM.

Both quantities with a discrete spectrum, and POVMs have
real-valued expectations, and their dynamics is continuous, implying that
one cannot get rid of the continuum that way.

Arnold Neumaier

DickT

Dec2-04, 06:14 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>jdff1001@hotmail.com (jdff) wrote in message news:<3f96fbb1.0411300407.50d3b783@posting.google. com>...\n> baez@galaxy.ucr.edu (John Baez) wrote in message news:<codfe3\\$> >Infinitely many rationals (and uncountably many reals) are\n> > >compatible with any observable state of the voltmeter.\n> >\n> > True!\n> >\n> > >That\'s why the error bars are intrinsic to measurement results, even to\n> > >single readings. Deleting them and claiming exact measurement results is\n> > >just laziness, acceptable when the resolution of an instrument is known.\n> >\n> > If we fully accepted this principle, we\'d have to admit the error bars\n> > should have error bars themselves... and so on ad infinitum... which\n> > gets a bit tiresome. Everyone gets lazy sooner or later. :-)\n>\n> Can\'t we invoke the Bekenstein bound, which gives the finite number of\n> bits of information extractable from any system, for a given area of\n> interface. If it\'s a finite number of bits, it is not just rational,\n> but not even countably infinite.\n\nUmmm, the even numbers have a finite number of bits per number, and\nthey are countably infinite.\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>jdff1001@hotmail.com (jdff) wrote in message news:<3f96fbb1.0411300407.50d3b783@posting.google.com>...
> baez@galaxy.ucr.edu (John Baez) wrote in message news:<codfe3$> >Infinitely many rationals (and uncountably many reals) are
> > >compatible with any observable state of the voltmeter.
> >
> > True!
> >
> > >That's why the error bars are intrinsic to measurement results, even to
> > >single readings. Deleting them and claiming exact measurement results is
> > >just laziness, acceptable when the resolution of an instrument is known.
> >
> > If we fully accepted this principle, we'd have to admit the error bars
> > should have error bars themselves... and so on ad infinitum... which
> > gets a bit tiresome. Everyone gets lazy sooner or later. :-)
>
> Can't we invoke the Bekenstein bound, which gives the finite number of
> bits of information extractable from any system, for a given area of
> interface. If it's a finite number of bits, it is not just rational,
> but not even countably infinite.

Ummm, the even numbers have a finite number of bits per number, and
they are countably infinite.

arivero@posta.unizar.es

Dec2-04, 06:18 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>Alfred Einstead wrote:\n> baez@galaxy.ucr.edu (John Baez) wrote:\n> > >That\'s why the error bars are intrinsic to measurement results, even to\n> > >single readings. Deleting them and claiming exact measurement results is\n> > >just laziness, acceptable when the resolution of an instrument is known.\n> > If we fully accepted this principle, we\'d have to admit the error bars\n> > should have error bars themselves... and so on ad infinitum... which\n> > gets a bit tiresome. Everyone gets lazy sooner or later. :-)\n>\n> The root of the matter, addressing the subject header itself, is that\n> a real number conveys an infinite amount of information.\n\nEr...\n\nSo, I understand that this thread is about the use of non-algebraic\nnumbers in mathematics and physics. Is it? Because as far as I know,\nonly a few of these are used (pi, e...). Any other, as sqrr(2) or the\ncoefficients appearing when composing angular momenta, are always\nalgebraic, ie, they are solutions of some equation with integer\ncoefficients. That is not infinite information.\n\nBy the way, I supposse this is the right method to focus physicist\napproach to numbers. On one side we have measurements, and we can\ndebate if they are rational numbers or... hmm... "statistic numbers?".\nOn the next step we have equations having these numbers as\ncoefficients, and the numbers they appearing can be real or complex, as\nthey are to be interpreted as the roots of these equations. Finally we\nhave some deep equations which express an interactions, as for instance\nthe infinite process of approaching a circle with a polygon, or the\ncomposition of infinitesimal group actions to get a finite move. In\nsuch cases non algebraic numbers are doomed to appear, expressing our\nintention of an indefinitely repeated process.\n\nIn any case, the question of the error bar seems interesting, because\nit is not straightforward how it translates across an equation when\nsome criticality enters play.\n\nAlejandro\n\n\n\nIf, on the\n> other hand, we\'re to take limitations seriously such as the Bekenstein\n> Bound, a finite region of space-time will only possess a finite\n> capacity for information. This puts the real number in the\n> spot light as an abstraction that goes further than what nature\n> mandates.\n>\n> But this and all related issues are already handled and superseded in\n> large part by the notion of a mixed state. First, the only place\n> where real numbers make their full impact (i.e., where both the\n> p\'s and q\'s potentially have unlimited precision together) is\n> classical physics. The pure states then become singular distributions\n> over phase space and a gulf of infinite information separates such\n> a state from the classical mixed states (such as the grand canonical\n> distribution) which may be comprise of it. So, pure states are\n> out, except as idealizations and everything is mixed. Therefore,\n> the last of precision is already built into the framework -- for\n> the case of Classical Physics -- at the outset.\n>\n> In quantum physics 1/2 of the problem is already gone, since the\n> p\'s and q\'s in nature no longer contain an infinite amount of\n> information when taken together. The pure state then, too, have\n> an inherent fuzziness associated with them.\n>\n> The other 1/2 of the problem still remains: the p\'s or the q\'s\n> taken by themselves can be made arbitrarily precise -- which seems\n> to indicate that either, alone, contains an infinity of information.\n> The problem shows up, then, in the question of what q looks like\n> after a "wave function collapse" and what a subsequent measurement\n> of, say, p yields after q is measured.\n>\n> The root of the problem is then addressed by noting that the p\'s\n> and q\'s are singular in the standard interpretation and one is\n> forced to either (a) disallow variables with a continuous spectrum\n> or (b) bring in a more generalized notion of operator -- the POVM.\n>\n> Lurking underneath this should be some kind of theorem to the effect\n> that there is some kind of equivalence between (a) and (b) so that\n> either possibility can be focused on without loss of generality.\n>\n> Taking the Bekenstein bound seriously seems to point, already, to\n> (a). Conversely, allowing either (a) or (b) seems then to provide\n> a way framework which will allow for the Bekenstein bound or something\n> similar.\n>\n> Ultimatly, the root of the problem is the continuum: the subalgebra\n> of p\'s or of q\'s each by themselves is still classical (i.e. a\n> commutative algebra). But the framework for going beyond the\n> continuum is already there and has already been 1/2 realized by\n> the preliminary step of limiting p-q\'s together. The last step\n> of limiting p\'s or q\'s by themselves (or calling forth the\n> Bekenstein bound) is what remains.\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>Alfred Einstead wrote:
> baez@galaxy.ucr.edu (John Baez) wrote:
> > >That's why the error bars are intrinsic to measurement results, even to
> > >single readings. Deleting them and claiming exact measurement results is
> > >just laziness, acceptable when the resolution of an instrument is known.
> > If we fully accepted this principle, we'd have to admit the error bars
> > should have error bars themselves... and so on ad infinitum... which
> > gets a bit tiresome. Everyone gets lazy sooner or later. :-)
>
> The root of the matter, addressing the subject header itself, is that
> a real number conveys an infinite amount of information.

Er...

So, I understand that this thread is about the use of non-algebraic
numbers in mathematics and physics. Is it? Because as far as I know,
only a few of these are used (\pi, e...). Any other, as sqrr(2) or the
coefficients appearing when composing angular momenta, are always
algebraic, ie, they are solutions of some equation with integer
coefficients. That is not infinite information.

By the way, I supposse this is the right method to focus physicist
approach to numbers. On one side we have measurements, and we can
debate if they are rational numbers or... hmm... "statistic numbers?".
On the next step we have equations having these numbers as
coefficients, and the numbers they appearing can be real or complex, as
they are to be interpreted as the roots of these equations. Finally we
have some deep equations which express an interactions, as for instance
the infinite process of approaching a circle with a polygon, or the
composition of infinitesimal group actions to get a finite move. In
such cases non algebraic numbers are doomed to appear, expressing our
intention of an indefinitely repeated process.

In any case, the question of the error bar seems interesting, because
it is not straightforward how it translates across an equation when
some criticality enters play.

Alejandro

If, on the
> other hand, we're to take limitations seriously such as the Bekenstein
> Bound, a finite region of space-time will only possess a finite
> capacity for information. This puts the real number in the
> spot light as an abstraction that goes further than what nature
> mandates.
>
> But this and all related issues are already handled and superseded in
> large part by the notion of a mixed state. First, the only place
> where real numbers make their full impact (i.e., where both the
> p's and q's potentially have unlimited precision together) is
> classical physics. The pure states then become singular distributions
> over phase space and a gulf of infinite information separates such
> a state from the classical mixed states (such as the grand canonical
> distribution) which may be comprise of it. So, pure states are
> out, except as idealizations and everything is mixed. Therefore,
> the last of precision is already built into the framework -- for
> the case of Classical Physics -- at the outset.
>
> In quantum physics 1/2 of the problem is already gone, since the
> p's and q's in nature no longer contain an infinite amount of
> information when taken together. The pure state then, too, have
> an inherent fuzziness associated with them.
>
> The other 1/2 of the problem still remains: the p's or the q's
> taken by themselves can be made arbitrarily precise -- which seems
> to indicate that either, alone, contains an infinity of information.
> The problem shows up, then, in the question of what q looks like
> after a "wave function collapse" and what a subsequent measurement
> of, say, p yields after q is measured.
>
> The root of the problem is then addressed by noting that the p's
> and q's are singular in the standard interpretation and one is
> forced to either (a) disallow variables with a continuous spectrum
> or (b) bring in a more generalized notion of operator -- the POVM.
>
> Lurking underneath this should be some kind of theorem to the effect
> that there is some kind of equivalence between (a) and (b) so that
> either possibility can be focused on without loss of generality.
>
> Taking the Bekenstein bound seriously seems to point, already, to
> (a). Conversely, allowing either (a) or (b) seems then to provide
> a way framework which will allow for the Bekenstein bound or something
> similar.
>
> Ultimatly, the root of the problem is the continuum: the subalgebra
> of p's or of q's each by themselves is still classical (i.e. a
> commutative algebra). But the framework for going beyond the
> continuum is already there and has already been 1/2 realized by
> the preliminary step of limiting p-q's together. The last step
> of limiting p's or q's by themselves (or calling forth the
> Bekenstein bound) is what remains.

Arnold Neumaier

Dec3-04, 04:49 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>arivero@posta.unizar.es wrote:\n> Alfred Einstead wrote:\n>\n>>baez@galaxy.ucr.edu (John Baez) wrote:\n>>\n>>>>That\'s why the error bars are intrinsic to measurement results, even to\n>>>>single readings. Deleting them and claiming exact measurement results is\n>>>>just laziness, acceptable when the resolution of an instrument is known.\n>>>\n>>>If we fully accepted this principle, we\'d have to admit the error bars\n>>>should have error bars themselves... and so on ad infinitum... which\n>>>gets a bit tiresome. Everyone gets lazy sooner or later. :-)\n>>\n>>The root of the matter, addressing the subject header itself, is that\n>>a real number conveys an infinite amount of information.\n>\n>\n> Er...\n>\n> So, I understand that this thread is about the use of non-algebraic\n> numbers in mathematics and physics. Is it? Because as far as I know,\n> only a few of these are used (pi, e...).\n\nDid you realize that the cosine of every rational number is non-algebraic?\nAnd most function values of the exponential, Bessel functions, etc. at\nrational arguments...\n\nNevertheless, any number actually used was written with a finite number of\nsymbols from a finite set, hence contains finite information only once it\nis communicated. Before communication, infinite information is missing,\nhowever, since you don\'t know how long the description is going to be...\n\n\nArnold Neumaier\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>arivero@posta.unizar.es wrote:
> Alfred Einstead wrote:
>
>>baez@galaxy.ucr.edu (John Baez) wrote:
>>
>>>>That's why the error bars are intrinsic to measurement results, even to
>>>>single readings. Deleting them and claiming exact measurement results is
>>>>just laziness, acceptable when the resolution of an instrument is known.
>>>
>>>If we fully accepted this principle, we'd have to admit the error bars
>>>should have error bars themselves... and so on ad infinitum... which
>>>gets a bit tiresome. Everyone gets lazy sooner or later. :-)
>>
>>The root of the matter, addressing the subject header itself, is that
>>a real number conveys an infinite amount of information.
>
>
> Er...
>
> So, I understand that this thread is about the use of non-algebraic
> numbers in mathematics and physics. Is it? Because as far as I know,
> only a few of these are used (\pi, e...).

Did you realize that the cosine of every rational number is non-algebraic?
And most function values of the exponential, Bessel functions, etc. at
rational arguments...

Nevertheless, any number actually used was written with a finite number of
symbols from a finite set, hence contains finite information only once it
is communicated. Before communication, infinite information is missing,
however, since you don't know how long the description is going to be...

Arnold Neumaier

Eckard Blumschein

Dec3-04, 04:51 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>On 11/29/2004 9:50 AM, John Baez wrote:\n> In article <416c04a4\\$0\\$199\\$edfadb0f@dread12.news.tele.d k>,\n> Aage Andersen <aaa@email.dk (REMOVE)> wrote:\n\n> That\'s just because you drew symbols like 1.1 and 2 on the dial of\n> your voltmeter. If you drew symbols like sqrt(2) and pi on it,\n> I guess you\'d claim a single reading always gives an irrational number.\n\nWhile perhaps nobody could imagine an evenly spaced scale using sqrt(2)\nand pi, nonetheless everybody is using a nearly logarithmic cochlear\npartition inside his organ of Corti.\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>On 11/29/2004 9:50 AM, John Baez wrote:
> In article <416c04a4$0$199$edfadb0f@dread12.news.tele.dk>,
> Aage Andersen <aaa@email.dk (REMOVE)> wrote:

> That's just because you drew symbols like 1.1 and 2 on the dial of
> your voltmeter. If you drew symbols like \sqrt(2) and \pi on it,
> I guess you'd claim a single reading always gives an irrational number.

While perhaps nobody could imagine an evenly spaced scale using \sqrt(2)
and \pi, nonetheless everybody is using a nearly logarithmic cochlear
partition inside his organ of Corti.

Oz

Dec3-04, 04:52 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>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> writes\n\n>But whether or not a state is pure, all expectations are exact\n>real numbers, with an infinite amount of information.\n>And this will be so in any reasonable form of physics.\n>Cast out continuity, and you lose physics.\n\nHmmm....\n\n1) How much information is there in the following real number: 1011?\n\n2) How much information is there in an electron?\n\n3) How much information is there in a photon?\n\n4) Is describing 1011 as not (not (1101)) a valid description?\nWell, in some sense (not (1101)) could be considered to be the\ninformation of all numbers except (1101) but then in another sense the\nabsence of (1101) could be considered to have precisely the same\ninformation as (1101).\n\n\n--\nOz\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>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> writes

>But whether or not a state is pure, all expectations are exact
>real numbers, with an infinite amount of information.
>And this will be so in any reasonable form of physics.
>Cast out continuity, and you lose physics.

Hmmm....

1) How much information is there in the following real number: 1011?

2) How much information is there in an electron?

3) How much information is there in a photon?

4) Is describing 1011 as not (not (1101)) a valid description?
Well, in some sense (not (1101)) could be considered to be the
information of all numbers except (1101) but then in another sense the
absence of (1101) could be considered to have precisely the same
information as (1101).

--
Oz

Patrick Powers

Dec4-04, 03:16 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>Oz <oz@farmeroz.port995.com> wrote in message news:<Y6gpmfAld5rBFwX8@port995.com>...\n> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> writes\n>\n> >But whether or not a state is pure, all expectations are exact\n> >real numbers, with an infinite amount of information.\n\nCorrect in theory. In practice no infinity can be observed.\n\n> >And this will be so in any reasonable form of physics.\n> >Cast out continuity, and you lose physics.\n\nTell that to Max Plank. :-)\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>Oz <oz@farmeroz.port995.com> wrote in message news:<Y6gpmfAld5rBFwX8@port995.com>...
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> writes
>
> >But whether or not a state is pure, all expectations are exact
> >real numbers, with an infinite amount of information.

Correct in theory. In practice no infinity can be observed.

> >And this will be so in any reasonable form of physics.
> >Cast out continuity, and you lose physics.

Tell that to Max Plank. :-)

Alejandro

Dec4-04, 03:17 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>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<41AF39A7.8040801@univie.ac.at>...\n> arivero@posta.unizar.es wrote:\n\n> > So, I understand that this thread is about the use of non-algebraic\n> > numbers in mathematics and physics. Is it? Because as far as I know,\n> > only a few of these are used (pi, e...).\n>\n> Did you realize that the cosine of every rational number is non-algebraic?\n\nIndeed I believe that my previous phrase includes such numbers; they\nare combinations of exponentials (e^ix - e^-ix). So they are just\nexpressions\ngot from e. Of course 2 pi, 3 pi, 4 pi, pi^2 &c and also not\nalgebraic.\n\n> And most function values of the exponential, Bessel functions, etc. at\n> rational arguments...\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>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<41AF39A7.8040801@univie.ac.at>...
> arivero@posta.unizar.es wrote:

> > So, I understand that this thread is about the use of non-algebraic
> > numbers in mathematics and physics. Is it? Because as far as I know,
> > only a few of these are used (\pi, e...).
>
> Did you realize that the cosine of every rational number is non-algebraic?

Indeed I believe that my previous phrase includes such numbers; they
are combinations of exponentials (e^{ix} - e^-ix). So they are just
expressions
got from e. Of course 2 \pi, 3 \pi, 4 \pi, \pi^2 &c and also not
algebraic.

> And most function values of the exponential, Bessel functions, etc. at
> rational arguments...

Arnold Neumaier

Dec4-04, 03:18 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>Oz wrote:\n> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> writes\n>\n>\n>>But whether or not a state is pure, all expectations are exact\n>>real numbers, with an infinite amount of information.\n>>And this will be so in any reasonable form of physics.\n>>Cast out continuity, and you lose physics.\n>\n>\n> Hmmm....\n>\n> 1) How much information is there in the following real number: 1011?\n\ndepends on what you knew before. If nothing then the information content\nis infinite. if you knew that it is\n\n\n> 2) How much information is there in an electron?\n\ndepends on what you mean. A pure state of an electron is defined by\nits wave function (up to a phase). Thus knowing all about an electron\nrequires in the traditional interpretation to know all about this\nwave function - an infinite amount of information.\n\n\n> 3) How much information is there in a photon?\n\nAbout the same amount as in an electron. Both have one binary and\nthree continuous degrees of freedom, for spin and position (or momentum).\n\n\n> 4) Is describing 1011 as not (not (1101)) a valid description?\n\nNo, since 1011 is not the same as 1101.\n\nMoreover, logical operations apply only to statements, not to objects.\n\'\'not not x= 1101\'\' is the same as \'\'x = 1101\'\'\n\n\nArnold Neumaier\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>Oz wrote:
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> writes
>
>
>>But whether or not a state is pure, all expectations are exact
>>real numbers, with an infinite amount of information.
>>And this will be so in any reasonable form of physics.
>>Cast out continuity, and you lose physics.
>
>
> Hmmm....
>
> 1) How much information is there in the following real number: 1011?

depends on what you knew before. If nothing then the information content
is infinite. if you knew that it is

> 2) How much information is there in an electron?

depends on what you mean. A pure state of an electron is defined by
its wave function (up to a phase). Thus knowing all about an electron
requires in the traditional interpretation to know all about this
wave function - an infinite amount of information.

> 3) How much information is there in a photon?

About the same amount as in an electron. Both have one binary and
three continuous degrees of freedom, for spin and position (or momentum).

> 4) Is describing 1011 as not (not (1101)) a valid description?

No, since 1011 is not the same as 1101.

Moreover, logical operations apply only to statements, not to objects.
''not not x= 1101'' is the same as ''x = 1101''

Arnold Neumaier

Boo

Dec6-04, 07:16 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>> Did you realize that the cosine of every rational number is non-algebraic?\n\nWell, cos(0) = 1\n\n--\nBoo\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>> Did you realize that the cosine of every rational number is non-algebraic?

Well, cos(0) = 1

--
Boo

Patrick Powers

Dec6-04, 07:17 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>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<41AF39A7.8040801@univie.ac.at>...\n>\n> Did you realize that the cosine of every rational number is non-algebraic?\n\nI\'m behind the times. When did they prove that?\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>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<41AF39A7.8040801@univie.ac.at>...
>
> Did you realize that the cosine of every rational number is non-algebraic?

I'm behind the times. When did they prove that?

Oz

Dec7-04, 08:06 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>\nArnold Neumaier <Arnold.Neumaier@univie.ac.at> writes\n>Oz wrote:\n>> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> writes\n>>\n>>\n>>>But whether or not a state is pure, all expectations are exact\n>>>real numbers, with an infinite amount of information.\n>>>And this will be so in any reasonable form of physics.\n>>>Cast out continuity, and you lose physics.\n>>\n>>\n>> Hmmm....\n>>\n>> 1) How much information is there in the following real number: 1011?\n>\n>depends on what you knew before. If nothing then the information content\n>is infinite. if you knew that it is\n\nEh? I would say that in a sense it was one unit of information.\n\n>> 2) How much information is there in an electron?\n>\n>depends on what you mean. A pure state of an electron is defined by\n>its wave function (up to a phase). Thus knowing all about an electron\n>requires in the traditional interpretation to know all about this\n>wave function - an infinite amount of information.\n\nThat cannot be so. By definition it must contain an electronsworth of\ninformation, and this must be finite. I imagine that the information can\nbe inscribed on the surface of a black hole of one electronmass. Maybe\nadd a bit for charge.\n\n>> 3) How much information is there in a photon?\n>\n>About the same amount as in an electron.\n\nInfinite? Tsk.\n\n>> 4) Is describing 1011 as not (not (1101)) a valid description?\n>\n>No, since 1011 is not the same as 1101.\n\n<damn typos...>\n\n>Moreover, logical operations apply only to statements, not to objects.\n> \'\'not not x= 1101\'\' is the same as \'\'x = 1101\'\'\n\nIndeed. However particles are quantised so removing one electron ought\nto give at most one electronsworth of information, although I must admit\nthis wouldn\'t seem to be precisely correct and needs more thought.\n\n--\nOz\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>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> writes
>Oz wrote:
>> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> writes
>>
>>
>>>But whether or not a state is pure, all expectations are exact
>>>real numbers, with an infinite amount of information.
>>>And this will be so in any reasonable form of physics.
>>>Cast out continuity, and you lose physics.
>>
>>
>> Hmmm....
>>
>> 1) How much information is there in the following real number: 1011?
>
>depends on what you knew before. If nothing then the information content
>is infinite. if you knew that it is

Eh? I would say that in a sense it was one unit of information.

>> 2) How much information is there in an electron?
>
>depends on what you mean. A pure state of an electron is defined by
>its wave function (up to a phase). Thus knowing all about an electron
>requires in the traditional interpretation to know all about this
>wave function - an infinite amount of information.

That cannot be so. By definition it must contain an electronsworth of
information, and this must be finite. I imagine that the information can
be inscribed on the surface of a black hole of one electronmass. Maybe
add a bit for charge.

>> 3) How much information is there in a photon?
>
>About the same amount as in an electron.

Infinite? Tsk.

>> 4) Is describing 1011 as not (not (1101)) a valid description?
>
>No, since 1011 is not the same as 1101.

<damn typos...>

>Moreover, logical operations apply only to statements, not to objects.
> ''not not x= 1101'' is the same as ''x = 1101''

Indeed. However particles are quantised so removing one electron ought
to give at most one electronsworth of information, although I must admit
this wouldn't seem to be precisely correct and needs more thought.

--
Oz

whopkins@csd.uwm.edu

Dec8-04, 06:55 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>The amont of information required to determine a pure state given the\nmixed state is generally finite, not infinite -- in Quantum Theory.\n(The major issue, as pointed out, being when you\'re dealing with\nquantities whose spectra are continuous). That the expectations are\nnumbers is irrelevant.\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>The amont of information required to determine a pure state given the
mixed state is generally finite, not infinite -- in Quantum Theory.
(The major issue, as pointed out, being when you're dealing with
quantities whose spectra are continuous). That the expectations are
numbers is irrelevant.

Daniel Doro Ferrante

Dec9-04, 02:03 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>On Wed, 8 Dec 2004 12:55:20 +0000 (UTC), <whopkins@csd.uwm.edu> wrote:\n\n> The amont of information required to determine a pure state given the\n> mixed state is generally finite, not infinite -- in Quantum Theory.\n> (The major issue, as pointed out, being when you\'re dealing with\n> quantities whose spectra are continuous). That the expectations are\n> numbers is irrelevant.\n>\n\nGiven all that has been said so far, would anyone care to comment on\nG. Chaitin\'s ideas about Real numbers? (Perhaps, even on Wolfram\'s... >;-)\n\n< http://www.cs.auckland.ac.nz/CDMTCS/chaitin/ >\n\n< http://en.wikipedia.org/wiki/Gregory_Chaitin >\n\nHow does "Digital Philosophy" (Digital Physics) fit in all this?\n\nIn fact, in this recent paper,\n<http://front.math.ucdavis.edu/math.HO/0411418>,\nChaitin explicitly tackles this question... but, AFAICT, his ideas are\npretty\nantagonic to what has been said here so far.\n\n--\nUsing M2, Opera\'s revolutionary e-mail client: http://www.opera.com/m2/\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>On Wed, 8 Dec 2004 12:55:20 +0000 (UTC), <whopkins@csd.uwm.edu> wrote:

> The amont of information required to determine a pure state given the
> mixed state is generally finite, not infinite -- in Quantum Theory.
> (The major issue, as pointed out, being when you're dealing with
> quantities whose spectra are continuous). That the expectations are
> numbers is irrelevant.
>

Given all that has been said so far, would anyone care to comment on
G. Chaitin's ideas about Real numbers? (Perhaps, even on Wolfram's... >;-)

< http://www.cs.auckland.ac.nz/CDMTCS/chaitin/ >

< http://en.wikipedia.org/wiki/Gregory_Chaitin >

How does "Digital Philosophy" (Digital Physics) fit in all this?

In fact, in this recent paper,
<http://front.math.ucdavis.edu/math.HO/0411418>,
Chaitin explicitly tackles this question... but, AFAICT, his ideas are
pretty
antagonic to what has been said here so far.

--
Using M2, Opera's revolutionary e-mail client: http://www.opera.com/m2/

frisbieinstein@yahoo.com

Dec9-04, 11:33 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\nrobert j. kolker wrote:\n> Tobias Fritz wrote:\n> >\n> > 2) Do you think that the real numbers are the appropriate system\nfor\n> > formulating a unified physical theory? What about quantization of\n> > spacetime?\n> >\n>\n> My guess is that a purely discrete theory to describe physical\nreality\n> will be mathematically intractable. We have a dillema. The\nmathematics\n> we can use, cannot be literally true of reality. The mathematics that\n\n> can be literally true of reality we cannot use because of its\ndifficulty.\n>\n> Go figure.\n>\n> Bob Kolker\n\nIn my opinion real numbers and other infinities are all useful\napproximations. They are used because they are much more convenient\nand there is no reason to believe that the difference is significant.\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>robert j. kolker wrote:
> Tobias Fritz wrote:
> >
> > 2) Do you think that the real numbers are the appropriate system
for
> > formulating a unified physical theory? What about quantization of
> > spacetime?
> >
>
> My guess is that a purely discrete theory to describe physical
reality
> will be mathematically intractable. We have a dillema. The
mathematics
> we can use, cannot be literally true of reality. The mathematics that

> can be literally true of reality we cannot use because of its
difficulty.
>
> Go figure.
>
> Bob Kolker

In my opinion real numbers and other infinities are all useful
approximations. They are used because they are much more convenient
and there is no reason to believe that the difference is significant.

Arnold Neumaier

Dec10-04, 05:00 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>Boo wrote:\n>>Did you realize that the cosine of every rational number is non-algebraic?\n>\n>\n> Well, cos(0) = 1\n>\nSorry, I forgot to mention trivial exceptions.\n\n\nArnold Neumaier\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>Boo wrote:
>>Did you realize that the cosine of every rational number is non-algebraic?
>
>
> Well, cos(0) = 1
>
Sorry, I forgot to mention trivial exceptions.

Arnold Neumaier

Arnold Neumaier

Dec10-04, 05:00 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>whopkins@csd.uwm.edu wrote:\n> The amont of information required to determine a pure state given the\n> mixed state is generally finite, not infinite -- in Quantum Theory.\n\nOnly if you force it into an eigenstate in a particular experiment.\nBefore measurement, it contained infinitely much information.\nMeasuring a state is killing it, not determining it.\nThis is the reason why quantum computing is so difficult.\n\n\nArnold Neumaier\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>whopkins@csd.uwm.edu wrote:
> The amont of information required to determine a pure state given the
> mixed state is generally finite, not infinite -- in Quantum Theory.

Only if you force it into an eigenstate in a particular experiment.
Before measurement, it contained infinitely much information.
Measuring a state is killing it, not determining it.
This is the reason why quantum computing is so difficult.

Arnold Neumaier

seratend

Dec10-04, 05:01 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>Frank Hellmann Wrote:\n> <div class="vbmenu_control"><a href="jabberwocky:;"\n> onClick="newWindow=window.open(\'\',\'usenetCode\' ,\'toolbar=no,location=no,scrollbars=yes,resizable =yes,status=no,width=650,height=400\');\n> newWindow.document.write(\'<HTML><HEAD><TITLE>Usen et\n> ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0\n> BGCOLOR=#F1F1F1><table border=0 width=625><td\n> bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\\\'s\n> original ASCII form: </font></td></tr><tr><td width=449><br><br><font\n> face=courier><UL><PRE>\\n"Aage Andersen" <aaa@email.dk (REMOVE)> wrote\n> in message\n> news:<416a8aa0\\\\\\$0\\\\\\$203\\\\\\$edfadb0f@dr ead12.news.tele.dk>...\\n> >\\n>\n> > 2) - We say that the result of a physical measurement is a real\n> number.\\n> > But\\n> > what about errors? Any dense subset, say\n> the rationals Q, would be\\n> > indistinguishable by\n> measurements.\\n>\\n> Actually a single measurement always gives a\n> rational number.\\n>\\n> regards Aage\\n\\nThus if we assume that\n> meassurements can be repeated infinitely, we\\nnaturely are led to\n> include all numbers to which the rational numbers\\ncan converge ->\n> The real\n> numbers.\\n\\n--\\nf\\n</UL></PRE></font></td></tr></table></BODY><HTML>\');">\n> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this\n> Usenet post in original ASCII form">**View this Usenet post in\n> original ASCII form </a></div><P>"Aage Andersen" <aaa@email.dk\n> (REMOVE)> wrote in message\n> news:<416a8aa0\\$0\\$203\\$edfadb0f@dread12.news.t ele.dk>...\n> > >\n> > > 2) - We say that the result of a physical measurement is a real\n> number.\n> > > But\n> > > what about errors? Any dense subset, say the rationals Q,\n> would be\n> > > indistinguishable by measurements.\n> >\n> > Actually a single measurement always gives a rational number.\n> >\n> > regards Aage\n>\n> Thus if we assume that meassurements can be repeated infinitely, we\n> naturely are led to include all numbers to which the rational\n> numbers\n> can converge -> The real numbers.\n>\n> --\n> f\n\nIt seems that you are confusing the closure of the rationnal set with\nthe non existence of a bijection between Q (or any countable set) and\n|R sets.\n\nSeratend.\n\n------------------------------------------------------------------------\nThis post submitted through the LaTeX-enabled physicsforums.com\nTo view this post with LaTeX images:\nhttp://www.physicsforums.com/showthread.php?t=47108#post397631\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>Frank Hellmann Wrote:
> <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"Aage Andersen" <aaa@email.dk (REMOVE)> wrote
> in message
> news:<416a8aa0\\$0\\$203\\$edfadb0f@dread12.news.tele.dk>...\n> >\n>
> > 2) - We say that the result of a physical measurement is a real
> number.\n> > But\n> > what about errors? Any dense subset, say
> the rationals Q, would be\n> > indistinguishable by
> measurements.\n>\n> Actually a single measurement always gives a
> rational number.\n>\n> regards Aage\n\nThus if we assume that
> meassurements can be repeated infinitely, we\nnaturely are led to
> include all numbers to which the rational numbers\ncan converge ->
> The real
> numbers.\n\n--\nf\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>"Aage Andersen" <aaa@email.dk
> (REMOVE)> wrote in message
> news:<416a8aa0$0$203$edfadb0f@dread12.news.tele.dk>...
> > >
> > > 2) - We say that the result of a physical measurement is a real
> number.
> > > But
> > > what about errors? Any dense subset, say the rationals Q,
> would be
> > > indistinguishable by measurements.
> >
> > Actually a single measurement always gives a rational number.
> >
> > regards Aage
>
> Thus if we assume that meassurements can be repeated infinitely, we
> naturely are led to include all numbers to which the rational
> numbers
> can converge -> The real numbers.
>
> --
> f

It seems that you are confusing the closure of the rationnal set with
the non existence of a bijection between Q (or any countable set) and
|R sets.

Seratend.

------------------------------------------------------------------------
This post submitted through the LaTeX-enabled physicsforums.com
To view this post with LaTeX images:
http://www.physicsforums.com/showthread.php?t=47108#post397631

Eckard Blumschein

Dec10-04, 05:02 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>On 12/9/2004 6:33 PM, frisbieinstein@yahoo.com wrote:\n> robert j. kolker wrote:\n>> Tobias Fritz wrote:\n>> >\n>> > 2) Do you think that the real numbers are the appropriate system\n>> > for formulating a unified physical theory? What about quantization of\n>> > spacetime?\n>> >\n>> My guess is that a purely discrete theory to describe physical\n>> reality will be mathematically intractable. We have a dillema. The\n>> mathematics we can use, cannot be literally true of reality. The mathematics that\n>> can be literally true of reality we cannot use because of its difficulty.\n>\n> In my opinion real numbers and other infinities are all useful\n> approximations. They are used because they are much more convenient\n> and there is no reason to believe that the difference is significant.\n\nI agree. Both the potentially infinitly large and the potentially\ninfinitely small, constituting the ideal unresolvable continuum, are\nnecessary requisites of a mathematics that optimally fits to all\nbranches of physics. Hilbert was a little bit mislead by the limitation\ntowards the infinitely small set by quantum physics when he in 1925\ncommented on "Das Unendliche". Mathematicians should accept that Hermann\nWeyl was perhaps correct with his somewhat constructivistic \'sauce\', and\nthere is no reason at all to replace the continuum by means of discrete\nnumbers. Spectra of discrete functions are anyway continuous and vice\nversa. Meanwhile, I learned that there are already less brutal and more\nappealing concepts than the usual set theory. In particular, I\nappreciate the constructivistic term apartness, slightly deviating from\nequality.\n\nWhat about the structure of spacetime, I still tend to not prematurely\nexclude the possibility that time might not relate to the observed\nevents according to the traditional notion of time but to the observer\nwho is experiencing it.\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>On 12/9/2004 6:33 PM, frisbieinstein@yahoo.com wrote:
> robert j. kolker wrote:
>> Tobias Fritz wrote:
>> >
>> > 2) Do you think that the real numbers are the appropriate system
>> > for formulating a unified physical theory? What about quantization of
>> > spacetime?
>> >
>> My guess is that a purely discrete theory to describe physical
>> reality will be mathematically intractable. We have a dillema. The
>> mathematics we can use, cannot be literally true of reality. The mathematics that
>> can be literally true of reality we cannot use because of its difficulty.
>
> In my opinion real numbers and other infinities are all useful
> approximations. They are used because they are much more convenient
> and there is no reason to believe that the difference is significant.

I agree. Both the potentially infinitly large and the potentially
infinitely small, constituting the ideal unresolvable continuum, are
necessary requisites of a mathematics that optimally fits to all
branches of physics. Hilbert was a little bit mislead by the limitation
towards the infinitely small set by quantum physics when he in 1925
commented on "Das Unendliche". Mathematicians should accept that Hermann
Weyl was perhaps correct with his somewhat constructivistic 'sauce', and
there is no reason at all to replace the continuum by means of discrete
numbers. Spectra of discrete functions are anyway continuous and vice
versa. Meanwhile, I learned that there are already less brutal and more
appealing concepts than the usual set theory. In particular, I
appreciate the constructivistic term apartness, slightly deviating from
equality.

What about the structure of spacetime, I still tend to not prematurely
exclude the possibility that time might not relate to the observed
events according to the traditional notion of time but to the observer
who is experiencing it.

Arnold Neumaier

Dec10-04, 10:18 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\nOz wrote:\n> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> writes\n>\n>>Oz wrote:\n>>\n>>>1) How much information is there in the following real number: 1011?\n>>\n>>depends on what you knew before. If nothing then the information content\n>>is infinite. if you knew that it is\n>\n> Eh? I would say that in a sense it was one unit of information.\n\nIf you knew before that it has to be a 4-digit decimal number,\nthe information gained is 4*log 10/log2 bits. If you only knew before\nthat it is a number with at most 20 digits, the informaition gained\nis 20*log 10/log2 bits. If you only knew before that it is a number with\nat most 10000 digits, the informaition gained is ...\n\n\n>>2) How much information is there in an electron?\n>>\n>>depends on what you mean. A pure state of an electron is defined by\n>>its wave function (up to a phase). Thus knowing all about an electron\n>>requires in the traditional interpretation to know all about this\n>>wave function - an infinite amount of information.\n>\n> That cannot be so. By definition it must contain an electronsworth of\n> information, and this must be finite.\n\nThis does not follow. Knowing a particular electron intimately is\ninfinitely precious.\n\n\n\n> Indeed. However particles are quantised so removing one electron ought\n> to give at most one electronsworth of information, although I must admit\n> this wouldn\'t seem to be precisely correct and needs more thought.\n\nThe information humans are interested is always finite, since they can\nhardly remember even 20 decimal digits seen only once.\n\nThus they simplify things to the point that all they want to know about\nan electron is its mass and charge. This is only a few bits. But if you\nwant to tell someone else exactly where the electron is that you are\nreferring to, you have an infinitely more difficult task. Of course,\nany human \'else\' will not be patient enough but be staisfied with a\ncrude position and momentum estimate consistent with the uncertainty\nrelation. But his is not the best possible statement about the electron,\nwhich would be telling the complete wave function. You can do it only\nif you forced the electron into a prison where it has to behave in\na dull way, being restricted in its freedom to a few bits of change.\n\n\nArnold Neumaier\n\n\n\n\nArnold Neumaier\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>Oz wrote:
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> writes
>
>>Oz wrote:
>>
>>>1) How much information is there in the following real number: 1011?
>>
>>depends on what you knew before. If nothing then the information content
>>is infinite. if you knew that it is
>
> Eh? I would say that in a sense it was one unit of information.

If you knew before that it has to be a 4-digit decimal number,
the information gained is 4*log 10/log2 bits. If you only knew before
that it is a number with at most 20 digits, the informaition gained
is 20*log 10/log2 bits. If you only knew before that it is a number with
at most 10000 digits, the informaition gained is ...

>>2) How much information is there in an electron?
>>
>>depends on what you mean. A pure state of an electron is defined by
>>its wave function (up to a phase). Thus knowing all about an electron
>>requires in the traditional interpretation to know all about this
>>wave function - an infinite amount of information.
>
> That cannot be so. By definition it must contain an electronsworth of
> information, and this must be finite.

This does not follow. Knowing a particular electron intimately is
infinitely precious.

> Indeed. However particles are quantised so removing one electron ought
> to give at most one electronsworth of information, although I must admit
> this wouldn't seem to be precisely correct and needs more thought.

The information humans are interested is always finite, since they can
hardly remember even 20 decimal digits seen only once.

Thus they simplify things to the point that all they want to know about
an electron is its mass and charge. This is only a few bits. But if you
want to tell someone else exactly where the electron is that you are
referring to, you have an infinitely more difficult task. Of course,
any human 'else' will not be patient enough but be staisfied with a
crude position and momentum estimate consistent with the uncertainty
relation. But his is not the best possible statement about the electron,
which would be telling the complete wave function. You can do it only
if you forced the electron into a prison where it has to behave in
a dull way, being restricted in its freedom to a few bits of change.

Arnold Neumaier

Arnold Neumaier

Arnold Neumaier

Dec10-04, 10:18 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\nPatrick Powers wrote:\n> Oz <oz@farmeroz.port995.com> wrote in message news:<Y6gpmfAld5rBFwX8@port995.com>...\n>\n>>Arnol d Neumaier <Arnold.Neumaier@univie.ac.at> writes\n>>\n>>\n>>>But whether or not a state is pure, all expectations are exact\n>>>real numbers, with an infinite amount of information.\n>\n>\n> Correct in theory. In practice no infinity can be observed.\n\nThis doesn\'t mean much. Only that people are limited. Fleas are even more.\n\n\n>>>And this will be so in any reasonable form of physics.\n>>>Cast out continuity, and you lose physics.\n>\n> Tell that to Max Planck. :-)\n\nWell, he created a puzzle. It was solved by quantum mechanics, using\na continuous dynamics of quantum states! The old, discrete QM soon got\nsterile.\n\n\nArnold Neumaier\n\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>Patrick Powers wrote:
> Oz <oz@farmeroz.port995.com> wrote in message news:<Y6gpmfAld5rBFwX8@port995.com>...
>
>>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> writes
>>
>>
>>>But whether or not a state is pure, all expectations are exact
>>>real numbers, with an infinite amount of information.
>
>
> Correct in theory. In practice no infinity can be observed.

This doesn't mean much. Only that people are limited. Fleas are even more.

>>>And this will be so in any reasonable form of physics.
>>>Cast out continuity, and you lose physics.
>
> Tell that to Max Planck. :-)

Well, he created a puzzle. It was solved by quantum mechanics, using
a continuous dynamics of quantum states! The old, discrete QM soon got
sterile.

Arnold Neumaier

arivero@posta.unizar.es

Dec11-04, 03:09 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>Patrick Powers wrote:\n> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message\nnews:<41AF39A7.8040801@univie.ac.at>...\n > >\n> > Did you realize that the cosine of every rational number is\nnon-algebraic?\n>\n> I\'m behind the times. When did they prove that?\n\nSurely the results on trisection of ancles and constructibility of\nn-poligons can be extended to a claim of the kind Arnold is doing.\nhttp://mathforum.org/library/drmath/view/51698.html\n\nAlejandro\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>Patrick Powers wrote:
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message
news:<41AF39A7.8040801@univie.ac.at>...
> >
> > Did you realize that the cosine of every rational number is
non-algebraic?
>
> I'm behind the times. When did they prove that?

Surely the results on trisection of ancles and constructibility of
n-poligons can be extended to a claim of the kind Arnold is doing.
http://mathforum.org/library/drmath/view/51698.html

Alejandro

Aaron Bergman

Dec12-04, 03:49 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>In article <1102700581.556996.67360@f14g2000cwb.googlegroups. com>,\narivero@posta.unizar.es wrote:\n\n> Patrick Powers wrote:\n> > Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message\n> news:<41AF39A7.8040801@univie.ac.at>...\n> > >\n> > > Did you realize that the cosine of every rational number is\n> non-algebraic?\n> >\n> > I\'m behind the times. When did they prove that?\n>\n> Surely the results on trisection of ancles and constructibility of\n> n-poligons can be extended to a claim of the kind Arnold is doing.\n> http://mathforum.org/library/drmath/view/51698.html\n\nWell, as I\'m sure was pointed out, it\'s not true as cos(0) = 1. I\'d\nguess the correct result (just exclude zero) follows from the\nLindemann-Weierstrass theorem.\n\nThe stuff your talking about has more to do with algebraic numbers and\ntheir Galois groups.\n\nAaron\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 <1102700581.556996.67360@f14g2000cwb.googlegroups.c om>,
arivero@posta.unizar.es wrote:

> Patrick Powers wrote:
> > Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message
> news:<41AF39A7.8040801@univie.ac.at>...
> > >
> > > Did you realize that the cosine of every rational number is
> non-algebraic?
> >
> > I'm behind the times. When did they prove that?
>
> Surely the results on trisection of ancles and constructibility of
> n-poligons can be extended to a claim of the kind Arnold is doing.
> http://mathforum.org/library/drmath/view/51698.html

Well, as I'm sure was pointed out, it's not true as cos(0) = 1. I'd
guess the correct result (just exclude zero) follows from the
Lindemann-Weierstrass theorem.

The stuff your talking about has more to do with algebraic numbers and
their Galois groups.

Aaron

Arnold Neumaier

Dec12-04, 03:50 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>arivero@posta.unizar.es wrote:\n> Patrick Powers wrote:\n>\n>>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message\n> news:<41AF39A7.8040801@univie.ac.at>...\n>\n>>>Did you realize that the cosine of every rational number is non-algebraic?\n>\n>>I\'m behind the times. When did they prove that?\n>\n> Surely the results on trisection of ancles and constructibility of\n> n-poligons can be extended to a claim of the kind Arnold is doing.\n> http://mathforum.org/library/drmath/view/51698.html\n\nNo; these are assertions about the numbers cos(q*pi) with rational q.\nThese are all algebraic numbers, and the constructibilty quest is to\nrepresent them in a field constructed from the rationals by a sequence of\nquadratic extensions.\n\nOn the other hand, though I don\'t have references at hand, I believe it has\nbeen proved that if rational linear combination sum_{k=1:n} a_k exp(x_k)\nwith distinct rational, nonzero x_k and rational, nonzero a_k cannot vanish.\nSince cos(x) is a linear combination of exponentials and products\nof exponentials are exponentials, this implies that if x is rational and\ncos(x) is algebraic then x=0.\n\n\nArnold Neumaier\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>arivero@posta.unizar.es wrote:
> Patrick Powers wrote:
>
>>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message
> news:<41AF39A7.8040801@univie.ac.at>...
>
>>>Did you realize that the cosine of every rational number is non-algebraic?
>
>>I'm behind the times. When did they prove that?
>
> Surely the results on trisection of ancles and constructibility of
> n-poligons can be extended to a claim of the kind Arnold is doing.
> http://mathforum.org/library/drmath/view/51698.html

No; these are assertions about the numbers cos(q*\pi) with rational q.
These are all algebraic numbers, and the constructibilty quest is to
represent them in a field constructed from the rationals by a sequence of
quadratic extensions.

On the other hand, though I don't have references at hand, I believe it has
been proved that if rational linear combination sum_{k=1:n} a_k \exp(x_k)
with distinct rational, nonzero x_k and rational, nonzero a_k cannot vanish.
Since cos(x) is a linear combination of exponentials and products
of exponentials are exponentials, this implies that if x is rational and
cos(x) is algebraic then x=0.

Arnold Neumaier

tttpppggg@yahoo.com

Dec13-04, 09:12 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>The polysigned numbers are relevant to this discussion I think. They\nproduce the reals at sign two, the complex numbers at sign three, and\nhigher dimensional spaces without the use of cross products. A\nmagnitude analysis shows that products in signs higher than three have\ndifferent behavior from the expected |AB| = |A||B|.\nA somewhat thorough description can be found at:\nhttp://bandtechnology.com/PolySigned/PolySigned.html\n\nSo to answer your first question- these are the generalization of both\nthe reals and the complex numbers.\n\n-Tim\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>The polysigned numbers are relevant to this discussion I think. They
produce the reals at sign two, the complex numbers at sign three, and
higher dimensional spaces without the use of cross products. A
magnitude analysis shows that products in signs higher than three have
different behavior from the expected |AB| = |A||B|.
A somewhat thorough description can be found at:
http://bandtechnology.com/PolySigned/PolySigned.html

So to answer your first question- these are the generalization of both
the reals and the complex numbers.

-Tim

Aaron Bergman

Dec13-04, 09:59 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>power-hungry types are\nbetter organized, are more ruthless and Machiavellian and have taken\ncare to build themselves a strong power base.\n\n225. These phenomena appeared clearly in Russia and other countries\nthat were taken over by leftists. Similarly, before the breakdown of\ncommunism in the USSR, leftish types in the West would seldom\ncriticize that country. If prodded they would admit that the USSR did\nmany wrong things, but then they would try to find excuses for the\ncommunists and begin talking about the faults of the West. They always\nopposed Western military resistance to communist aggression. Leftish\ntypes all over the world vigorously protested the U.S. military action\nin Vietnam, but when the USSR invaded Afghanistan they did nothing.\nNot that they approved of the Soviet actions; but because of their\nleftist faith, they just couldn\'t bear to put themselves in opposition\nto communism. Today, in those of our universities where "political\ncorrectness" has become dominant, there are probably many leftish\ntypes who privately disapprove of the suppression of academic freedom,\n\n\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>power-hungry types are
better organized, are more ruthless and Machiavellian and have taken
care to build themselves a strong power base.

225. These phenomena appeared clearly in Russia and other countries
that were taken over by leftists. Similarly, before the breakdown of
communism in the USSR, leftish types in the West would seldom
criticize that country. If prodded they would admit that the USSR did
many wrong things, but then they would try to find excuses for the
communists and begin talking about the faults of the West. They always
opposed Western military resistance to communist aggression. Leftish
types all over the world vigorously protested the U.S. military action
in Vietnam, but when the USSR invaded Afghanistan they did nothing.
Not that they approved of the Soviet actions; but because of their
leftist faith, they just couldn't bear to put themselves in opposition
to communism. Today, in those of our universities where "political
correctness" has become dominant, there are probably many leftish
types who privately disapprove of the suppression of academic freedom,

Arnold Neumaier

Dec13-04, 10:35 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>technology for only ONE purpose: to attack the\ntechnological system.\n\n203. Imagine an alcoholic sitting with a barrel of wine in front of\nhim. Suppose he starts saying to himself, "Wine isn\'t bad for you if\nused in moderation. Why, they say small amounts of wine are even good\nfor you! It won\'t do me any harm if I take just one little drink..."\nWell you know what is going to happen. Never forget that the human\nrace with technology is just like an alcoholic with a barrel of wine.\n\n204. Revolutionaries should have as many children as they can. There\nis strong scientific evidence that social attitudes are to a\nsignificant extent inherited. No one suggests that a social attitude\nis a direct outcome of a person\'s genetic constitution, but it appears\nthat personality traits tend, within the context of our society, to\nmake a person more likely to hold this or that social attitude.\nObjections to these findings have been raised, but objections are\nfeeble and seem to be ideologically motivated. In any event, no one\ndenies that children tend on the average to hold social attitudes\nsimilar to those of their parents. From our point of view it doesn\'t\nmatter all that much whether the attitudes are passed on genetically\nor through childhood training. In either case the ARE passed on.\n\n205. The trouble is that many of the people who are inclined to rebel\nagainst the industrial system are also concerned about the population\nproblems, hence they are apt to have few or no children. In this way\nthey may be handing the world over to the sort of people who support\nor at least accept the industrial system. To insure the strength of\nthe next generation of revolutionaries the present generation must\nreproduce itself abundantly. In doing so they will be worsening the\npopulation problem only slightly. And the most important problem is\n\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>technology for only ONE purpose: to attack the
technological system.

203. Imagine an alcoholic sitting with a barrel of wine in front of
him. Suppose he starts saying to himself, "Wine isn't bad for you if
used in moderation. Why, they say small amounts of wine are even good
for you! It won't do me any harm if I take just one little drink..."
Well you know what is going to happen. Never forget that the human
race with technology is just like an alcoholic with a barrel of wine.

204. Revolutionaries should have as many children as they can. There
is strong scientific evidence that social attitudes are to a
significant extent inherited. No one suggests that a social attitude
is a direct outcome of a person's genetic constitution, but it appears
that personality traits tend, within the context of our society, to
make a person more likely to hold this or that social attitude.
Objections to these findings have been raised, but objections are
feeble and seem to be ideologically motivated. In any event, no one
denies that children tend on the average to hold social attitudes
similar to those of their parents. From our point of view it doesn't
matter all that much whether the attitudes are passed on genetically
or through childhood training. In either case the ARE passed on.

205. The trouble is that many of the people who are inclined to rebel
against the industrial system are also concerned about the population
problems, hence they are apt to have few or no children. In this way
they may be handing the world over to the sort of people who support
or at least accept the industrial system. To insure the strength of
the next generation of revolutionaries the present generation must
reproduce itself abundantly. In doing so they will be worsening the
population problem only slightly. And the most important problem is

Arnold Neumaier

Dec13-04, 10:36 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>damaging parts of the brain or they can be brought to\nthe surface by electrical stimulation. Hallucinations can be induced\nor moods changed by drugs. There may or may not be an immaterial human\nsoul, but if there is one it clearly is less powerful that the\nbiological mechanisms of human behavior. For if that were not the case\nthen researchers would not be able so easily to manipulate human\nfeelings and behavior with drugs and electrical currents.\n\n158. It presumably would be impractical for all people to have\nelectrodes inserted in their heads so that they could be controlled by\nthe authorities. But the fact that human thoughts and feelings are so\nopen to biological intervention shows that the problem of controlling\nhuman behavior is mainly a technical problem; a problem of neurons,\nhormones and complex molecules; the kind of problem that is accessible\nto scientific attack. Given the outstanding record of our society in\nsolving technical problems, it is overwhelmingly probable that great\nadvances will be made in the control of human behavior.\n\n159. Will public resistance prevent the introduction of technological\ncontrol of human behavior? It certainly would if an attempt were made\nto introduce such control all at once. But since technological control\nwill be introduced through a long sequence of small advances, there\nwill be no rational and effective public resistance. (See paragraphs\n127,132, 153.)\n\n160. To those who think that all this sounds like science fiction, we\npoint out that yesterday\'s science fiction is today\'s fact. The\nIndustrial Revolution has radically altered man\'s environment and way\nof life, and it is only to be expected that as technology is\nincreasingly applied to the human body and mind, man himself will be\naltered as radically as his environment and way of life have been.\n\nHUMAN RACE AT A CROSSROADS\n\n\n\n\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>damaging parts of the brain or they can be brought to
the surface by electrical stimulation. Hallucinations can be induced
or moods changed by drugs. There may or may not be an immaterial human
soul, but if there is one it clearly is less powerful that the
biological mechanisms of human behavior. For if that were not the case
then researchers would not be able so easily to manipulate human
feelings and behavior with drugs and electrical currents.

158. It presumably would be impractical for all people to have
electrodes inserted in their heads so that they could be controlled by
the authorities. But the fact that human thoughts and feelings are so
open to biological intervention shows that the problem of controlling
human behavior is mainly a technical problem; a problem of neurons,
hormones and complex molecules; the kind of problem that is accessible
to scientific attack. Given the outstanding record of our society in
solving technical problems, it is overwhelmingly probable that great
advances will be made in the control of human behavior.

159. Will public resistance prevent the introduction of technological
control of human behavior? It certainly would if an attempt were made
to introduce such control all at once. But since technological control
will be introduced through a long sequence of small advances, there
will be no rational and effective public resistance. (See paragraphs
127,132, 153.)

160. To those who think that all this sounds like science fiction, we
point out that yesterday's science fiction is today's fact. The
Industrial Revolution has radically altered man's environment and way
of life, and it is only to be expected that as technology is
increasingly applied to the human body and mind, man himself will be
altered as radically as his environment and way of life have been.

HUMAN RACE AT A CROSSROADS

seratend

Dec13-04, 10:36 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>when their\ndisruptive effects became apparent. Thus, permanent changes in favor\nof freedom could be brought about only by persons prepared to accept\nradical, dangerous and unpredictable alteration of the entire system.\nIn other words, by revolutionaries, not reformers.\n\n112. People anxious to rescue freedom without sacrificing the supposed\nbenefits of technology will suggest naive schemes for some new form of\nsociety that would reconcile freedom with technology. Apart from the\nfact that people who make suggestions seldom propose any practical\nmeans by which the new form of society could be set up in the first\nplace, it follows from the fourth principle that even if the new form\nof society could be once established, it either would collapse or\nwould give results very different from those expected.\n\n113. So even on very general grounds it seems highly improbably that\nany way of changing society could be found that would reconcile\nfreedom with modern technology. In the next few sections we will give\nmore specific reasons for concluding that freedom and technological\nprogress are incompatible.\n\n\n\nRESTRICTION OF FREEDOM IS UNAVOIDABLE IN INDUSTRIAL SOCIETY\n\n\n\n114. As explained in paragraph 65-67, 70-73, modern man is strapped\ndown by a network of rules and regulations, and his fate depends on\nthe actions of persons remote from him whose decisions he cannot\ninfluence. This is not accidental or a result of the arbitrariness of\narrogant bureaucrats. It is necessary and inevitable in any\ntechnologically advanced society. The system HAS TO regulate human\nbehavior closely in order to function. At work, people have to do what\nthey are told to do, otherwise production would be thrown into chaos.\nBureaucracies HAVE TO be run according to rigid rules. To allow any\nsubstantial\n\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>when their
disruptive effects became apparent. Thus, permanent changes in favor
of freedom could be brought about only by persons prepared to accept
radical, dangerous and unpredictable alteration of the entire system.
In other words, by revolutionaries, not reformers.

112. People anxious to rescue freedom without sacrificing the supposed
benefits of technology will suggest naive schemes for some new form of
society that would reconcile freedom with technology. Apart from the
fact that people who make suggestions seldom propose any practical
means by which the new form of society could be set up in the first
place, it follows from the fourth principle that even if the new form
of society could be once established, it either would collapse or
would give results very different from those expected.

113. So even on very general grounds it seems highly improbably that
any way of changing society could be found that would reconcile
freedom with modern technology. In the next few sections we will give
more specific reasons for concluding that freedom and technological
progress are incompatible.

RESTRICTION OF FREEDOM IS UNAVOIDABLE IN INDUSTRIAL SOCIETY

114. As explained in paragraph 65-67, 70-73, modern man is strapped
down by a network of rules and regulations, and his fate depends on
the actions of persons remote from him whose decisions he cannot
influence. This is not accidental or a result of the arbitrariness of
arrogant bureaucrats. It is necessary and inevitable in any
technologically advanced society. The system HAS TO regulate human
behavior closely in order to function. At work, people have to do what
they are told to do, otherwise production would be thrown into chaos.
Bureaucracies HAVE TO be run according to rigid rules. To allow any
substantial

tttpppggg@yahoo.com

Dec13-04, 11:08 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>power mowers, radios, motorcycles,\netc. If the use of these devices is unrestricted, people who want\npeace and quiet are frustrated by the noise. If their use is\nrestricted, people who use the devices are frustrated by the\nregulations... But if these machines had never been invented there\nwould have been no conflict and no frustration generated by them.)\n\n49. For primitive societies the natural world (which usually changes\nonly slowly) provided a stable framework and therefore a sense of\nsecurity. In the modern world it is human society that dominates\nnature rather than the other way around, and modern society changes\nvery rapidly owing to technological change. Thus there is no stable\nframework.\n\n50. The conservatives are fools: They whine about the decay of\ntraditional values, yet they enthusiastically support technological\nprogress and economic growth. Apparently it never occurs to them that\nyou can\'t make rapid, drastic changes in the technology and the\neconomy of a society with out causing rapid changes in all other\naspects of the society as well, and that such rapid changes inevitably\nbreak down traditional values.\n\n51.The breakdown of traditional values to some extent implies the\nbreakdown of the bonds that hold together traditional small-scale\nsocial\n\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>power mowers, radios, motorcycles,
etc. If the use of these devices is unrestricted, people who want
peace and quiet are frustrated by the noise. If their use is
restricted, people who use the devices are frustrated by the
regulations... But if these machines had never been invented there
would have been no conflict and no frustration generated by them.)

49. For primitive societies the natural world (which usually changes
only slowly) provided a stable framework and therefore a sense of
security. In the modern world it is human society that dominates
nature rather than the other way around, and modern society changes
very rapidly owing to technological change. Thus there is no stable
framework.

50. The conservatives are fools: They whine about the decay of
traditional values, yet they enthusiastically support technological
progress and economic growth. Apparently it never occurs to them that
you can't make rapid, drastic changes in the technology and the
economy of a society with out causing rapid changes in all other
aspects of the society as well, and that such rapid changes inevitably
break down traditional values.

51.The breakdown of traditional values to some extent implies the
breakdown of the bonds that hold together traditional small-scale
social

Arnold Neumaier

Dec13-04, 11:20 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>are reasonably confident that the general\noutlines of the picture we have painted here are roughly correct. We\nhave portrayed leftism in its modern form as a phenomenon peculiar to\nour time and as a symptom of the disruption of the power process. But\nwe might possibly be wrong about this. Oversocialized types who try to\nsatisfy their drive for power by imposing their morality on everyone\nhave certainly been around for a long time. But we THINK that the\ndecisive role played by feelings of inferiority, low self-esteem,\npowerlessness, identification with victims by people who are not\nthemselves victims, is a peculiarity of modern leftism. Identification\nwith victims by people not themselves victims can be seen to some\nextent in 19th century leftism and early Christianity but as far as we\ncan make out, symptoms of low self-esteem, etc., were not nearly so\nevident in these movements, or in any other movements, as they are in\nmodern leftism. But we are not in a position to assert confidently\nthat no such movements have existed prior to modern leftism. This is a\nsignificant question to whic\n\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>are reasonably confident that the general
outlines of the picture we have painted here are roughly correct. We
have portrayed leftism in its modern form as a phenomenon peculiar to
our time and as a symptom of the disruption of the power process. But
we might possibly be wrong about this. Oversocialized types who try to
satisfy their drive for power by imposing their morality on everyone
have certainly been around for a long time. But we THINK that the
decisive role played by feelings of inferiority, low self-esteem,
powerlessness, identification with victims by people who are not
themselves victims, is a peculiarity of modern leftism. Identification
with victims by people not themselves victims can be seen to some
extent in 19th century leftism and early Christianity but as far as we
can make out, symptoms of low self-esteem, etc., were not nearly so
evident in these movements, or in any other movements, as they are in
modern leftism. But we are not in a position to assert confidently
that no such movements have existed prior to modern leftism. This is a
significant question to whic

arivero@posta.unizar.es

Dec13-04, 11:25 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>the treatment would be applied to those who might become\ndrunk drivers (they endanger human life too), then perhaps to peel who\nspank their children, then to environmentalists who sabotage logging\nequipment, eventually to anyone whose behavior is inconvenient for the\nsystem.\n\n30. (Paragraph 184) A further advantage of nature as a counter-ideal\nto technology is that, in many people, nature inspires the kind of\nreverence that is associated with religion, so that nature could\nperhaps be idealized on a religious basis. It is true that in many\nsocieties religion has served as a support and justification for the\nestablished order, but it is also true that religion has often\nprovided a basis for rebellion. Thus it may be useful to introduce a\nreligious element into the rebellion against technology, the more so\nbecause Western society today has no strong religious foundation.\n\nReligion, nowadays either is used as cheap and transparent support for\nnarrow, short-sighted selfishness (some conservatives use it this\nway), or even is cynically exploited to make easy money (by many\nevangelists), or has degenerated into crude irrationalism\n(fundamentalist Protestant sects, "cults"), or is simply stagnant\n(Catholicism, main-line Protestantism). The nearest thing to a strong,\nwidespread, dynamic religion that the West has seen in recent times\nhas been the quasi-religion of leftism, but leftism today is\nfragmented and has no clear, unified inspiring goal.\n\nThus there is a religious vaccuum in our society that could perhaps be\nfilled by a religion focused on nature in opposition to technology.\nBut it would be a mistake to try to concoct artificially a religion to\nfill this role. Such an invented religion would probably be a failure.\nTake the "Gaia" religion for example. Do its adherents REALLY believe\nin it or are they just play-acting\n\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>the treatment would be applied to those who might become
drunk drivers (they endanger human life too), then perhaps to peel who
spank their children, then to environmentalists who sabotage logging
equipment, eventually to anyone whose behavior is inconvenient for the
system.

30. (Paragraph 184) A further advantage of nature as a counter-ideal
to technology is that, in many people, nature inspires the kind of
reverence that is associated with religion, so that nature could
perhaps be idealized on a religious basis. It is true that in many
societies religion has served as a support and justification for the
established order, but it is also true that religion has often
provided a basis for rebellion. Thus it may be useful to introduce a
religious element into the rebellion against technology, the more so
because Western society today has no strong religious foundation.

Religion, nowadays either is used as cheap and transparent support for
narrow, short-sighted selfishness (some conservatives use it this
way), or even is cynically exploited to make easy money (by many
evangelists), or has degenerated into crude irrationalism
(fundamentalist Protestant sects, "cults"), or is simply stagnant
(Catholicism, main-line Protestantism). The nearest thing to a strong,
widespread, dynamic religion that the West has seen in recent times
has been the quasi-religion of leftism, but leftism today is
fragmented and has no clear, unified inspiring goal.

Thus there is a religious vaccuum in our society that could perhaps be
filled by a religion focused on nature in opposition to technology.
But it would be a mistake to try to concoct artificially a religion to
fill this role. Such an invented religion would probably be a failure.
Take the "Gaia" religion for example. Do its adherents REALLY believe
in it or are they just play-acting

Arnold Neumaier

Dec14-04, 01:21 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>because the\nliberals and leftists would wish to solve our social problems by\nhaving society guarantee everyone\'s security; but if that could be\ndone it would only bring back the problem of purposelessness. The real\nissue is not whether society provides well or poorly for people\'s\nsecurity; the trouble is that people are dependent on the system for\ntheir security rather than having it in their own hands. This, by the\nway, is part of the reason why some people get worked up about the\nright to bear arms; possession of a gun puts that aspect of their\nsecurity in their own hands.\n\n13. (Paragraph 66) Conservatives\' efforts to decrease the amount of\ngovernment regulation are of little benefit to the average man. For\none thing, only a fraction of the regulations can be eliminated\nbecause most regulations are necessary. For another thing, most of the\nderegulation affects business rather than the average individual, so\nthat its main effect is to take power from the government and give it\nto private corporations. What this means for the average man is that\ngovernment interference in his life is replaced by interference from\nbig corporations, which may be permitted, for example, to dump more\nchemicals that get into his water supply and give him cancer. The\nconservatives are just taking the average man for a sucker, exploiting\nhis resentment of Big Government to promote the power of Big Business.\n\n\n14. (Paragraph 73) When someone approves of the purpose for which\npropaganda is being used in a given case, he generally calls it\n"education" or applies to it some similar euphemism. But propaganda is\npropaganda regardless of the purpose for which it is used.\n\n15. (Paragraph 83) We are not expressing approval or disapproval of\nthe Pa\n\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>because the
liberals and leftists would wish to solve our social problems by
having society guarantee everyone's security; but if that could be
done it would only bring back the problem of purposelessness. The real
issue is not whether society provides well or poorly for people's
security; the trouble is that people are dependent on the system for
their security rather than having it in their own hands. This, by the
way, is part of the reason why some people get worked up about the
right to bear arms; possession of a gun puts that aspect of their
security in their own hands.

13. (Paragraph 66) Conservatives' efforts to decrease the amount of
government regulation are of little benefit to the average man. For
one thing, only a fraction of the regulations can be eliminated
because most regulations are necessary. For another thing, most of the
deregulation affects business rather than the average individual, so
that its main effect is to take power from the government and give it
to private corporations. What this means for the average man is that
government interference in his life is replaced by interference from
big corporations, which may be permitted, for example, to dump more
chemicals that get into his water supply and give him cancer. The
conservatives are just taking the average man for a sucker, exploiting
his resentment of Big Government to promote the power of Big Business.

14. (Paragraph 73) When someone approves of the purpose for which
propaganda is being used in a given case, he generally calls it
"education" or applies to it some similar euphemism. But propaganda is
propaganda regardless of the purpose for which it is used.

15. (Paragraph 83) We are not expressing approval or disapproval of
the Pa

Eckard Blumschein

Apr2-05, 02:41 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>On 12/2/2004 1:14 PM, Arnold Neumaier wrote:\n> Alfred Einstead wrote:\n>> baez@galaxy.ucr.edu (John Baez) wrote:\n\n>> The root of the matter, addressing the subject header itself, is that\n>> a real number conveys an infinite amount of information. [...]\n>>\n>> In quantum physics 1/2 of the problem is already gone, since the\n>> p\'s and q\'s in nature no longer contain an infinite amount of\n>> information when taken together. The pure state then, too, have\n>> an inherent fuzziness associated with them.\n>\n> But whether or not a state is pure, all expectations are exact\n> real numbers, with an infinite amount of information.\n> And this will be so in any reasonable form of physics.\n> Cast out continuity, and you lose physics.\n\nHaving pondered a lot about how my ideas on the subject of real numbers\ncan be agreed with G. Cantor\'s theory, I have to apologize for my\nhopefully not yet too late response supporting you.\n\nPlease find my pertaining reasoning at\nhttp://iesk.et.uni-magdeburg.de/~blumsche/M280.html\nand do not take it an April joke.\n\nEckard Blumschein\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>On 12/2/2004 1:14 PM, Arnold Neumaier wrote:
> Alfred Einstead wrote:
>> baez@galaxy.ucr.edu (John Baez) wrote:

>> The root of the matter, addressing the subject header itself, is that
>> a real number conveys an infinite amount of information. [...]
>>
>> In quantum physics 1/2 of the problem is already gone, since the
>> p's and q's in nature no longer contain an infinite amount of
>> information when taken together. The pure state then, too, have
>> an inherent fuzziness associated with them.
>
> But whether or not a state is pure, all expectations are exact
> real numbers, with an infinite amount of information.
> And this will be so in any reasonable form of physics.
> Cast out continuity, and you lose physics.

Having pondered a lot about how my ideas on the subject of real numbers
can be agreed with G. Cantor's theory, I have to apologize for my
hopefully not yet too late response supporting you.

Please find my pertaining reasoning at
http://iesk.et.uni-magdeburg.de/~blumsche/M280.html
and do not take it an April joke.

Eckard Blumschein

John F

Apr6-05, 12:24 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>Eckard Blumschein <blumschein@et.uni-magdeburg.de> wrote:\n: On 12/2/2004 1:14 PM, Arnold Neumaier wrote:\n: > Alfred Einstead wrote:\n: >> baez@galaxy.ucr.edu (John Baez) wrote:\n: >> The root of the matter, addressing the subject header itself, is that\n: >> a real number conveys an infinite amount of information. [...]\n: >>\n: >> In quantum physics 1/2 of the problem is already gone, since the\n: >> p\'s and q\'s in nature no longer contain an infinite amount of\n: >> information when taken together. The pure state then, too, have\n: >> an inherent fuzziness associated with them.\n: >\n: > But whether or not a state is pure, all expectations are exact\n: > real numbers, with an infinite amount of information.\n: > And this will be so in any reasonable form of physics.\n: > Cast out continuity, and you lose physics.\n:\n: Having pondered a lot about how my ideas on the subject of real numbers\n: can be agreed with G. Cantor\'s theory, I have to apologize for my\n: hopefully not yet too late response supporting you.\n: Please find my pertaining reasoning at\n: http://iesk.et.uni-magdeburg.de/~blumsche/M280.html\n: and do not take it an April joke.\n: Eckard Blumschein\n\nBesides the Cantorian approach, the notions of Kolmogorov\ncomplexity and computable real numbers demonstrate that\nmost (if not all) the real numbers used in physics do _not_\ncontain infinite information.\nThe Kolmogorov complexity of a string (which may be a\nsequence of digits) is, simply, the length of the shortest\ncomputer program which emits that string. So, for example,\npi contains only a little information since a relatively\nshort program can emit it. Of course, it\'ll take quite\na while for that program to finish, and I won\'t go into\nspace (i.e., memory requirements) versus time (number of\ninstructions executed) complexity.\nPhysical behavior is often modelled as differential\nequations or as other mathematical objects that can be\nprogrammed. So all the resulting real numbers are\ncomputable, and hence carry only finite information.\n"Kolmogorov complexity" (also called "algorithmic\ninformation theory") is easily googleable, see, e.g.,\nhttp://en.wikipedia.org/wiki/Algorithmic_information_theory\nThe standard text is\nhttp://homepages.cwi.nl/~paulv/kolmogorov.html\n(which is a bit dumbed down from its more monograph-like\nfirst edition).\n"Computable reals" or "computable real numbers" is\nalso pretty well represented on google. I\'m not sure if\nthere\'s a standard text, but you might try "Computability\nin Analysis and Physics", Marian B. Pour-El and\nJonathan I. Richards, Springer 1989, ISBN 0-387-50035-9.\n--\nJohn Forkosh ( mailto: j@f.com where j=john and f=forkosh )\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>Eckard Blumschein <blumschein@et.uni-magdeburg.de> wrote:
: On 12/2/2004 1:14 PM, Arnold Neumaier wrote:
: > Alfred Einstead wrote:
: >> baez@galaxy.ucr.edu (John Baez) wrote:
: >> The root of the matter, addressing the subject header itself, is that
: >> a real number conveys an infinite amount of information. [...]
: >>
: >> In quantum physics 1/2 of the problem is already gone, since the
: >> p's and q's in nature no longer contain an infinite amount of
: >> information when taken together. The pure state then, too, have
: >> an inherent fuzziness associated with them.
: >
: > But whether or not a state is pure, all expectations are exact
: > real numbers, with an infinite amount of information.
: > And this will be so in any reasonable form of physics.
: > Cast out continuity, and you lose physics.
:
: Having pondered a lot about how my ideas on the subject of real numbers
: can be agreed with G. Cantor's theory, I have to apologize for my
: hopefully not yet too late response supporting you.
: Please find my pertaining reasoning at
: http://iesk.et.uni-magdeburg.de/~blumsche/M280.html
: and do not take it an April joke.
: Eckard Blumschein

Besides the Cantorian approach, the notions of Kolmogorov
complexity and computable real numbers demonstrate that
most (if not all) the real numbers used in physics do _not_
contain infinite information.
The Kolmogorov complexity of a string (which may be a
sequence of digits) is, simply, the length of the shortest
computer program which emits that string. So, for example,
\pi contains only a little information since a relatively
short program can emit it. Of course, it'll take quite
a while for that program to finish, and I won't go into
space (i.e., memory requirements) versus time (number of
instructions executed) complexity.
Physical behavior is often modelled as differential
equations or as other mathematical objects that can be
programmed. So all the resulting real numbers are
computable, and hence carry only finite information.
"Kolmogorov complexity" (also called "algorithmic
information theory") is easily googleable, see, e.g.,
http://en.wikipedia.org/wiki/Algorithmic_information_theory
The standard text is
http://homepages.cwi.nl/~paulv/kolmogorov.html
(which is a bit dumbed down from its more monograph-like
first edition).
"Computable reals" or "computable real numbers" is
also pretty well represented on google. I'm not sure if
there's a standard text, but you might try "Computability
in Analysis and Physics", Marian B. Pour-El and
Jonathan I. Richards, Springer 1989, ISBN 0-387-50035-9.
--
John Forkosh ( mailto: j@f.com where j=john and f=forkosh )

Oz

Apr6-05, 03:25 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>John F <john@PleaseSeeSigForAddress.invalid.com> writes\n> The Kolmogorov complexity of a string (which may be a\n>sequence of digits) is, simply, the length of the shortest\n>computer program which emits that string. So, for example,\n>pi contains only a little information since a relatively\n>short program can emit it. Of course, it\'ll take quite\n>a while for that program to finish, and I won\'t go into\n>space (i.e., memory requirements) versus time (number of\n>instructions executed) complexity.\n\nPerhaps sliding into philosophy as an alternative to mathematically\ndescribing precisely what one is saying (due inability to do same):\n\n1) I don\'t think you can distinguish complexity in a 3+1D world by\ntaking the complexity of the 3D bit (the program) and ignoring the\nlength of the time bit (required to obtain the precise result). I would\nthus say that pi requires infinite information even if its isn\'t\ncomplex. After all the number \'one\' is very simple yet implies and\ninfinite precision. You can\'t even do pi in analogue form since this\nwould require a known precisely flat space over an extended area, which\nitself requires similarly infinite precision.\n\n2) I hadn\'t spotted JB\'s comment that p & q together define the\nfuzziness of space, but I like the idea. Its a similar argument to (1)\nabove.\n\n3) About the only thing one seems to be sure about is that particles\ncome in chunks of one since nobody ever measured (say) half an electron.\nHowever I don\'t even think this is really true as all detectors have a\nfinite detection efficiency and particles do have an ability to be\nelsewhere due tunnelling. It may be here today, but (a low probability)\nof gone tomorrow. So not-measuring an electron doesn\'t mean that there\nisn\'t one there (or even near).\n\n4) Its agreed that there is no global definition for \'the energy of the\nuniverse\' and I would bet similar arguments ought to apply to most other\nconstants. How can one be sure, for example, that the observable\nuniverse is always uncharged?\n\n5) One is thus drawn to the conclusion that any 4-volume of space has a\nmaximum information content. From a distant memory of an aged \'This\nWeeks Finds...\' I think this is the information content on the surface\nof a black hole of equivalent surface (I\'ve probably got this wrong),\nwhich is indeed a rather large number. Its rather irritating that this\nlarge number does not seem to reflect the size of h.\n\n--\nOz\nThis post is worth absolutely nothing and is probably fallacious.\n\nUse oz@farmeroz.port995.com [ozacoohdb@despammed.com functions].\nBTOPENWORLD address has ceased. DEMON address has ceased.\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>John F <john@PleaseSeeSigForAddress.invalid.com> writes
> The Kolmogorov complexity of a string (which may be a
>sequence of digits) is, simply, the length of the shortest
>computer program which emits that string. So, for example,
>\pi contains only a little information since a relatively
>short program can emit it. Of course, it'll take quite
>a while for that program to finish, and I won't go into
>space (i.e., memory requirements) versus time (number of
>instructions executed) complexity.

Perhaps sliding into philosophy as an alternative to mathematically
describing precisely what one is saying (due inability to do same):

1) I don't think you can distinguish complexity in a 3+1D world by
taking the complexity of the 3D bit (the program) and ignoring the
length of the time bit (required to obtain the precise result). I would
thus say that \pi requires infinite information even if its isn't
complex. After all the number 'one' is very simple yet implies and
infinite precision. You can't even do \pi in analogue form since this
would require a known precisely flat space over an extended area, which
itself requires similarly infinite precision.

2) I hadn't spotted JB's comment that p & q together define the
fuzziness of space, but I like the idea. Its a similar argument to (1)
above.

3) About the only thing one seems to be sure about is that particles
come in chunks of one since nobody ever measured (say) half an electron.
However I don't even think this is really true as all detectors have a
finite detection efficiency and particles do have an ability to be
elsewhere due tunnelling. It may be here today, but (a low probability)
of gone tomorrow. So not-measuring an electron doesn't mean that there
isn't one there (or even near).

4) Its agreed that there is no global definition for 'the energy of the
universe' and I would bet similar arguments ought to apply to most other
constants. How can one be sure, for example, that the observable
universe is always uncharged?

5) One is thus drawn to the conclusion that any 4-volume of space has a
maximum information content. From a distant memory of an aged 'This
Weeks Finds...' I think this is the information content on the surface
of a black hole of equivalent surface (I've probably got this wrong),
which is indeed a rather large number. Its rather irritating that this
large number does not seem to reflect the size of h.

--
Oz
This post is worth absolutely nothing and is probably fallacious.

Use oz@farmeroz.port995.com [ozacoohdb@despammed.com functions].
BTOPENWORLD address has ceased. DEMON address has ceased.

John Forkosh

May6-05, 02:36 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>Oz <Oz@farmeroz.port995.com> wrote:\nJohn F <john@PleaseSeeSigForAddress.invalid.com> wrote:\n: Eckard Blumschein <blumschein@et.uni-magdeburg.de> wrote:\n: : On 12/2/2004 1:14 PM, Arnold Neumaier wrote:\n: : > Alfred Einstead wrote:\n: : >> baez@galaxy.ucr.edu (John Baez) wrote:\n: : >> The root of the matter, addressing the subject header itself, is that\n: : >> a real number conveys an infinite amount of information. [...]\n: : >>\n: : >> In quantum physics 1/2 of the problem is already gone, since the\n: : >> p\'s and q\'s in nature no longer contain an infinite amount of\n: : >> information when taken together. The pure state then, too, have\n: : >> an inherent fuzziness associated with them.\n: : >\n: : > But whether or not a state is pure, all expectations are exact\n: : > real numbers, with an infinite amount of information.\n: : > And this will be so in any reasonable form of physics.\n: : > Cast out continuity, and you lose physics.\n: :\n: : Having pondered a lot about how my ideas on the subject of real numbers\n: : can be agreed with G. Cantor\'s theory, I have to apologize for my\n: : hopefully not yet too late response supporting you.\n: : Please find my pertaining reasoning at\n: : http://iesk.et.uni-magdeburg.de/~blumsche/M280.html\n: : and do not take it an April joke.\n: : Eckard Blumschein\n:\n: Besides the Cantorian approach, the notions of Kolmogorov\n: complexity and computable real numbers demonstrate that\n: most (if not all) the real numbers used in physics do _not_\n: contain infinite information.\n: The Kolmogorov complexity of a string (which may be a\n: sequence of digits) is, simply, the length of the shortest\n: computer program which emits that string. So, for example,\n: pi contains only a little information since a relatively\n: short program can emit it. Of course, it\'ll take quite\n: a while for that program to finish, and I won\'t go into\n: space (i.e., memory requirements) versus time (number of\n: instructions executed) complexity.\n: Physical behavior is often modelled as differential\n: equations or as other mathematical objects that can be\n: programmed. So all the resulting real numbers are\n: computable by a finite-length program, and hence carry\n: only finite information.\n: "Kolmogorov complexity" (also called "algorithmic\n: information theory") is easily googleable, see, e.g.,\n: http://en.wikipedia.org/wiki/Algorithmic_information_theory\n: The standard text is\n: http://homepages.cwi.nl/~paulv/kolmogorov.html\n: (which is a bit dumbed down from its more monograph-like\n: first edition).\n: "Computable reals" or "computable real numbers" is\n: also pretty well represented on google. I\'m not sure if\n: there\'s a standard text, but you might try "Computability\n: in Analysis and Physics", Marian B. Pour-El and\n: Jonathan I. Richards, Springer 1989, ISBN 0-387-50035-9.\n: --\n: John Forkosh ( mailto: j@f.com where j=john and f=forkosh )\n:\n:\n: Perhaps sliding into philosophy as an alternative to mathematically\n: describing precisely what one is saying (due inability to do same):\n\nUnfortunately, some mathematics is needed, at least to the extent\nthat you precisely define what you mean by "information." Intuition\nis indispensably essential while you\'re choosing definitions and axioms,\nand while you\'re writing a motivating discussion to defend these choices.\nBut after you complete these essential first tasks (which you might\nargue contain the bulk of the real physics), math is indispensable\nto develop the quantitative consequences of your ideas and to compare\nthem with experiment.\nClassical information is normally formulated in terms of\nShannon entropy, and quantum information in trems of von Neumann\nentropy. Kolmogorov complexity, and its consequences I mentioned,\nis based on the former. I didn\'t explicitly mention this since it\'s\ndiscussed in the literature I cited. An easier introduction is the\nreprint book Information Randomness and Incompleteness, 2nd ed,\nG.J. Chaitin, World Scientific 1990, ISBN 981-02-0171-0 (pbk).\nIf you feel that pi (or even 1.0 from item (1) below) contain\ninfinite information, then I\'d want you to define what you mean by\n"information."\n\n: 1) I don\'t think you can distinguish complexity in a 3+1D world by\n: taking the complexity of the 3D bit (the program) and ignoring the\n: length of the time bit (required to obtain the precise result). I would\n: thus say that pi requires infinite information even if its isn\'t\n: complex. After all the number \'one\' is very simple yet implies and\n: infinite precision. You can\'t even do pi in analogue form since this\n: would require a known precisely flat space over an extended area, which\n: itself requires similarly infinite precision.\n\nTime complexity doesn\'t contribute to the information content (a la\nShannon entropy) of a real number, which is why I only mentioned it\nbriefly. Think of it like this. You\'re familiar with the zip and\nunzip programs that compress and decompress files? Suppose you zip\na file. Does the compressed version contain the same information\ncontent (a la Shannon entropy) as the original? The answer is, "Yes."\nAnd that\'s despite the fact that the unzip program may have to execute\nmany instructions to reconstruct the original file.\n\nYour remaining items, especially (5), seem to deal with\nthe information content characterizing physical systems\nrather than the information content characterizing real\nnumbers. That\'s a whole different question.\n\n: 2) I hadn\'t spotted JB\'s comment that p & q together define the\n: fuzziness of space, but I like the idea. Its a similar argument\n: to (1) above.\n:\n: 3) About the only thing one seems to be sure about is that particles\n: come in chunks of one since nobody ever measured (say) half an electron.\n: However I don\'t even think this is really true as all detectors have a\n: finite detection efficiency and particles do have an ability to be\n: elsewhere due tunnelling. It may be here today, but (a low probability)\n: of gone tomorrow. So not-measuring an electron doesn\'t mean that there\n: isn\'t one there (or even near).\n:\n: 4) Its agreed that there is no global definition for \'the energy of the\n: universe\' and I would bet similar arguments ought to apply to most other\n: constants. How can one be sure, for example, that the observable\n: universe is always uncharged?\n:\n: 5) One is thus drawn to the conclusion that any 4-volume of space has a\n: maximum information content. From a distant memory of an aged \'This\n: Weeks Finds...\' I think this is the information content on the surface\n: of a black hole of equivalent surface (I\'ve probably got this wrong),\n: which is indeed a rather large number. Its rather irritating that this\n: large number does not seem to reflect the size of h.\n: --\n: Oz Use oz@farmeroz.port995.com\n:\n--\nJohn Forkosh ( mailto: j@f.com where j=john and f=forkosh )\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>Oz <Oz@farmeroz.port995.com> wrote:
John F <john@PleaseSeeSigForAddress.invalid.com> wrote:
: Eckard Blumschein <blumschein@et.uni-magdeburg.de> wrote:
: : On 12/2/2004 1:14 PM, Arnold Neumaier wrote:
: : > Alfred Einstead wrote:
: : >> baez@galaxy.ucr.edu (John Baez) wrote:
: : >> The root of the matter, addressing the subject header itself, is that
: : >> a real number conveys an infinite amount of information. [...]
: : >>
: : >> In quantum physics 1/2 of the problem is already gone, since the
: : >> p's and q's in nature no longer contain an infinite amount of
: : >> information when taken together. The pure state then, too, have
: : >> an inherent fuzziness associated with them.
: : >
: : > But whether or not a state is pure, all expectations are exact
: : > real numbers, with an infinite amount of information.
: : > And this will be so in any reasonable form of physics.
: : > Cast out continuity, and you lose physics.
: :
: : Having pondered a lot about how my ideas on the subject of real numbers
: : can be agreed with G. Cantor's theory, I have to apologize for my
: : hopefully not yet too late response supporting you.
: : Please find my pertaining reasoning at
: : http://iesk.et.uni-magdeburg.de/~blumsche/M280.html
: : and do not take it an April joke.
: : Eckard Blumschein
:
: Besides the Cantorian approach, the notions of Kolmogorov
: complexity and computable real numbers demonstrate that
: most (if not all) the real numbers used in physics do _not_
: contain infinite information.
: The Kolmogorov complexity of a string (which may be a
: sequence of digits) is, simply, the length of the shortest
: computer program which emits that string. So, for example,
: \pi contains only a little information since a relatively
: short program can emit it. Of course, it'll take quite
: a while for that program to finish, and I won't go into
: space (i.e., memory requirements) versus time (number of
: instructions executed) complexity.
: Physical behavior is often modelled as differential
: equations or as other mathematical objects that can be
: programmed. So all the resulting real numbers are
: computable by a finite-length program, and hence carry
: only finite information.
: "Kolmogorov complexity" (also called "algorithmic
: information theory") is easily googleable, see, e.g.,
: http://en.wikipedia.org/wiki/Algorithmic_information_theory
: The standard text is
: http://homepages.cwi.nl/~paulv/kolmogorov.html
: (which is a bit dumbed down from its more monograph-like
: first edition).
: "Computable reals" or "computable real numbers" is
: also pretty well represented on google. I'm not sure if
: there's a standard text, but you might try "Computability
: in Analysis and Physics", Marian B. Pour-El and
: Jonathan I. Richards, Springer 1989, ISBN 0-387-50035-9.
: --
: John Forkosh ( mailto: j@f.com where j=john and f=forkosh )
:
:
: Perhaps sliding into philosophy as an alternative to mathematically
: describing precisely what one is saying (due inability to do same):

Unfortunately, some mathematics is needed, at least to the extent
that you precisely define what you mean by "information." Intuition
is indispensably essential while you're choosing definitions and axioms,
and while you're writing a motivating discussion to defend these choices.
But after you complete these essential first tasks (which you might
argue contain the bulk of the real physics), math is indispensable
to develop the quantitative consequences of your ideas and to compare
them with experiment.
Classical information is normally formulated in terms of
Shannon entropy, and quantum information in trems of von Neumann
entropy. Kolmogorov complexity, and its consequences I mentioned,
is based on the former. I didn't explicitly mention this since it's
discussed in the literature I cited. An easier introduction is the
reprint book Information Randomness and Incompleteness, 2nd ed,
G.J. Chaitin, World Scientific 1990, ISBN 981-02-0171-0 (pbk).
If you feel that \pi (or even 1. from item (1) below) contain
infinite information, then I'd want you to define what you mean by
"information."

: 1) I don't think you can distinguish complexity in a 3+1D world by
: taking the complexity of the 3D bit (the program) and ignoring the
: length of the time bit (required to obtain the precise result). I would
: thus say that \pi requires infinite information even if its isn't
: complex. After all the number 'one' is very simple yet implies and
: infinite precision. You can't even do \pi in analogue form since this
: would require a known precisely flat space over an extended area, which
: itself requires similarly infinite precision.

Time complexity doesn't contribute to the information content (a la
Shannon entropy) of a real number, which is why I only mentioned it
briefly. Think of it like this. You're familiar with the zip and
unzip programs that compress and decompress files? Suppose you zip
a file. Does the compressed version contain the same information
content (a la Shannon entropy) as the original? The answer is, "Yes."
And that's despite the fact that the unzip program may have to execute
many instructions to reconstruct the original file.

Your remaining items, especially (5), seem to deal with
the information content characterizing physical systems
rather than the information content characterizing real
numbers. That's a whole different question.

: 2) I hadn't spotted JB's comment that p & q together define the
: fuzziness of space, but I like the idea. Its a similar argument
: to (1) above.
:
: 3) About the only thing one seems to be sure about is that particles
: come in chunks of one since nobody ever measured (say) half an electron.
: However I don't even think this is really true as all detectors have a
: finite detection efficiency and particles do have an ability to be
: elsewhere due tunnelling. It may be here today, but (a low probability)
: of gone tomorrow. So not-measuring an electron doesn't mean that there
: isn't one there (or even near).
:
: 4) Its agreed that there is no global definition for 'the energy of the
: universe' and I would bet similar arguments ought to apply to most other
: constants. How can one be sure, for example, that the observable
: universe is always uncharged?
:
: 5) One is thus drawn to the conclusion that any 4-volume of space has a
: maximum information content. From a distant memory of an aged 'This
: Weeks Finds...' I think this is the information content on the surface
: of a black hole of equivalent surface (I've probably got this wrong),
: which is indeed a rather large number. Its rather irritating that this
: large number does not seem to reflect the size of h.
: --
: Oz Use oz@farmeroz.port995.com
:
--
John Forkosh ( mailto: j@f.com where j=john and f=forkosh )

John Forkosh

Oct11-06, 02:44 PM

Oz <Oz@farmeroz.port995.com> wrote:
John F <john@PleaseSeeSigForAddress.invalid.com> wrote:
: Eckard Blumschein <blumschein@et.uni-magdeburg.de> wrote:
: : On 12/2/2004 1:14 PM, Arnold Neumaier wrote:
: : > Alfred Einstead wrote:
: : >> baez@galaxy.ucr.edu (John Baez) wrote:
: : >> The root of the matter, addressing the subject header itself, is that
: : >> a real number conveys an infinite amount of information. [...]
: : >>
: : >> In quantum physics 1/2 of the problem is already gone, since the
: : >> p's and q's in nature no longer contain an infinite amount of
: : >> information when taken together. The pure state then, too, have
: : >> an inherent fuzziness associated with them.
: : >
: : > But whether or not a state is pure, all expectations are exact
: : > real numbers, with an infinite amount of information.
: : > And this will be so in any reasonable form of physics.
: : > Cast out continuity, and you lose physics.
: :
: : Having pondered a lot about how my ideas on the subject of real numbers
: : can be agreed with G. Cantor's theory, I have to apologize for my
: : hopefully not yet too late response supporting you.
: : Please find my pertaining reasoning at
: : http://iesk.et.uni-magdeburg.de/~blumsche/M280.html
: : and do not take it an April joke.
: : Eckard Blumschein
:
: Besides the Cantorian approach, the notions of Kolmogorov
: complexity and computable real numbers demonstrate that
: most (if not all) the real numbers used in physics do _not_
: contain infinite information.
: The Kolmogorov complexity of a string (which may be a
: sequence of digits) is, simply, the length of the shortest
: computer program which emits that string. So, for example,
: pi contains only a little information since a relatively
: short program can emit it. Of course, it'll take quite
: a while for that program to finish, and I won't go into
: space (i.e., memory requirements) versus time (number of
: instructions executed) complexity.
: Physical behavior is often modelled as differential
: equations or as other mathematical objects that can be
: programmed. So all the resulting real numbers are
: computable by a finite-length program, and hence carry
: only finite information.
: "Kolmogorov complexity" (also called "algorithmic
: information theory") is easily googleable, see, e.g.,
: http://en.wikipedia.org/wiki/Algorithmic_information_theory
: The standard text is
: http://homepages.cwi.nl/~paulv/kolmogorov.html
: (which is a bit dumbed down from its more monograph-like
: first edition).
: "Computable reals" or "computable real numbers" is
: also pretty well represented on google. I'm not sure if
: there's a standard text, but you might try "Computability
: in Analysis and Physics", Marian B. Pour-El and
: Jonathan I. Richards, Springer 1989, ISBN 0-387-50035-9.
: --
: John Forkosh ( mailto: j@f.com where j=john and f=forkosh )
:
:
: Perhaps sliding into philosophy as an alternative to mathematically
: describing precisely what one is saying (due inability to do same):

Unfortunately, some mathematics is needed, at least to the extent
that you precisely define what you mean by "information." Intuition
is indispensably essential while you're choosing definitions and axioms,
and while you're writing a motivating discussion to defend these choices.
But after you complete these essential first tasks (which you might
argue contain the bulk of the real physics), math is indispensable
to develop the quantitative consequences of your ideas and to compare
them with experiment.
Classical information is normally formulated in terms of
Shannon entropy, and quantum information in trems of von Neumann
entropy. Kolmogorov complexity, and its consequences I mentioned,
is based on the former. I didn't explicitly mention this since it's
discussed in the literature I cited. An easier introduction is the
reprint book Information Randomness and Incompleteness, 2nd ed,
G.J. Chaitin, World Scientific 1990, ISBN 981-02-0171-0 (pbk).
If you feel that pi (or even 1.0 from item (1) below) contain
infinite information, then I'd want you to define what you mean by
"information."

: 1) I don't think you can distinguish complexity in a 3+1D world by
: taking the complexity of the 3D bit (the program) and ignoring the
: length of the time bit (required to obtain the precise result). I would
: thus say that pi requires infinite information even if its isn't
: complex. After all the number 'one' is very simple yet implies and
: infinite precision. You can't even do pi in analogue form since this
: would require a known precisely flat space over an extended area, which
: itself requires similarly infinite precision.

Time complexity doesn't contribute to the information content (a la
Shannon entropy) of a real number, which is why I only mentioned it
briefly. Think of it like this. You're familiar with the zip and
unzip programs that compress and decompress files? Suppose you zip
a file. Does the compressed version contain the same information
content (a la Shannon entropy) as the original? The answer is, "Yes."
And that's despite the fact that the unzip program may have to execute
many instructions to reconstruct the original file.

Your remaining items, especially (5), seem to deal with
the information content characterizing physical systems
rather than the information content characterizing real
numbers. That's a whole different question.

: 2) I hadn't spotted JB's comment that p & q together define the
: fuzziness of space, but I like the idea. Its a similar argument
: to (1) above.
:
: 3) About the only thing one seems to be sure about is that particles
: come in chunks of one since nobody ever measured (say) half an electron.
: However I don't even think this is really true as all detectors have a
: finite detection efficiency and particles do have an ability to be
: elsewhere due tunnelling. It may be here today, but (a low probability)
: of gone tomorrow. So not-measuring an electron doesn't mean that there
: isn't one there (or even near).
:
: 4) Its agreed that there is no global definition for 'the energy of the
: universe' and I would bet similar arguments ought to apply to most other
: constants. How can one be sure, for example, that the observable
: universe is always uncharged?
:
: 5) One is thus drawn to the conclusion that any 4-volume of space has a
: maximum information content. From a distant memory of an aged 'This
: Weeks Finds...' I think this is the information content on the surface
: of a black hole of equivalent surface (I've probably got this wrong),
: which is indeed a rather large number. Its rather irritating that this
: large number does not seem to reflect the size of h.
: --
: Oz Use oz@farmeroz.port995.com
:
--
John Forkosh ( mailto: j@f.com where j=john and f=forkosh )