Math questions from a philosopher
Alba Papa-Grimaldi
problems (a brief overview can be seen at
www.timeandreality.co.uk/paradox.htm) and I need some help with some answers
to mathematical questions:
1. Please could you tell me if there is a reason why
square root of two (and of other prime numbers) is an
irrational number?
2. The other question which is related to the first is:
if the hypotenuse of a triangle in which the adjacent
sides measure 1cm and 1cm is an irrational number (ie
square root of 2) why in reality can it be given a
certain value when we measure it?
Thanks in advance for any help you can give me
Alba
G. Frege
wrote:
>
> 1. Please could you tell me if there is a reason why
> square root of two (and of other prime numbers) is an
> irrational number?
>
Because it can't be a rational number. Hence it is a irrational number (if so).
>
> 2. The other question which is related to the first is:
> if the hypotenuse of a triangle in which the adjacent
> sides measure 1cm and 1cm is an irrational number (ie
> square root of 2) why in reality can it be given a
> certain value when we measure it?
>
Because (physical) reality doesn't know about "rational" or "real numbers" etc.
We just measure a certain length. Since any measurement only can be performed
with a finite precision, the (direct) result (of any measurement) actually can
be represented with a rational number.
F.
David C. Ullrich
<al...@ticnet.co.uk> wrote:
>This looks like a group that would be able to help me. I am looking at some
>problems (a brief overview can be seen at
>www.timeandreality.co.uk/paradox.htm) and I need some help with some answers
>to mathematical questions:
>
>1. Please could you tell me if there is a reason why
>square root of two (and of other prime numbers) is an
>irrational number?
The reason for this has been well-known for over 2000
years:
If r is rational then r = n/m, where n and m are integers
and have no common factor. Say n/m is such a representation
of sqrt(2). Then n^2 = 2 m^2 ("n squared is two times m squared"),
so that n^2 is even. Hence n is even, since the square of an
odd number is odd. So n = 2 k for some integer k.
It follows that 4 k^2 = 2 m^2, so 2 k^2 = m^2. Hence m
is even. So n and m are both even, contradicting the
assumption that n and m have no common factor.
>2. The other question which is related to the first is:
>if the hypotenuse of a triangle in which the adjacent
>sides measure 1cm and 1cm is an irrational number (ie
>square root of 2) why in reality can it be given a
>certain value when we measure it?
An actual physical measurement is not exact:
If you measure the hypotenuse you might get 1.4,
which is a rational number but which is also not
_exactly_ the length of the hypotenuse. If someone
else measures it more accurately they might get
1.414, which is also rational, and which is also
not exactly the length of the hypotenuse.
>Thanks in advance for any help you can give me
>
>Alba
>
******************
David C. Ullrich
Alba Papa-Grimaldi
Am I correct in saying then that we cannot actually get a 100% accurate
measurement, either by calculation or measurement, for the hypotenuse of
such a triangle ?
Alba
David C. Ullrich
<al...@ticnet.co.uk> wrote:
>Thank you for the replies so far.
>
>Am I correct in saying then that we cannot actually get a 100% accurate
>measurement, either by calculation or measurement, for the hypotenuse of
>such a triangle ?
You can't get a 100% accurate _measurement_ of _anything_.
If you _knew_ that the sides of a triangle were exactly one unit
long then you'd know that the hypotenuse was exactly sqrt(2),
but if we're talking about an actual physical triangle you never
know that the length of the sides is exactly 1.
In fact there are various reasons one might suggest that it doesn't
even make any sense to say that the length of the side of an
actual physical triangle is exactly 1. The triangle is composed
of atoms, so the length of the side has something to do with
the positions of those atoms, and they don't actually _have_
a precisely defined position.
>Alba
>>
>> >2. The other question which is related to the first is:
>> >if the hypotenuse of a triangle in which the adjacent
>> >sides measure 1cm and 1cm is an irrational number (ie
>> >square root of 2) why in reality can it be given a
>> >certain value when we measure it?
>>
>> An actual physical measurement is not exact:
>> If you measure the hypotenuse you might get 1.4,
>> which is a rational number but which is also not
>> _exactly_ the length of the hypotenuse. If someone
>> else measures it more accurately they might get
>> 1.414, which is also rational, and which is also
>> not exactly the length of the hypotenuse.
>>
>>
>> >Thanks in advance for any help you can give me
>> >
>> >Alba
>> >
>>
>>
>> ******************
>>
>> David C. Ullrich
>
******************
David C. Ullrich
Mechanic
"G. Frege" <g.f...@simple-line.de> wrote in message
news:v9nq8vshdua9n1unn...@4ax.com...
Now, you called me an idiot before. I won't call you any names because
civilized behavior is what distinguished humans from other species.
You could have just stayed silent and read the post by David C. Ullrich.
Maybe you can learn something or at least what is a proper way to answer a
question.
Now, back to the question by Alba. The debate still goes on whether space is
continuous or discrete and whether the Pythagorean theorem is valid. The
answer according to the Pythagoreans is sqrt(2) but we don't know whether
that can resolve or map into actual physical dimensions. If space is
infinitely divisible then sqrt(2) is the correct answer. Whether we can
measure it or not maybe is a problem due to our limited instrumentation
capacity. But just think of an instrument that measures directly sqrt(2) to
infinite precision.
If space is discrete as in Loop Quantum Gravity theories, then Pythagoreans
may be wrong. But nobody has taken the risk so far to replace arithmetic as
we know it to fit with such theory. (Or even pronounce irrationals as
non-existent).
Your question Alba is unsolved to this date as in the times of Socrates. Two
schools of thought and two different answers.
Leonard Blackburn
Yes. OP: Also keep in mind there probably is no such
thing as the "exact length of the hypotenuse" in reality.
A physical object is made of atoms which are flying around
like crazy. Maybe the real number line (which is where
we find irrational numbers) doesn't quite have a place in
physical reality.
Leonard
G. Frege
wrote:
>
> Am I correct in saying then that we cannot actually get a 100% accurate
> measurement [...] of such a triangle?
>
Yes. And it's not even clear what the "true/real length" of one side of that
triangle should be. (We do not even have to refer to quantum theory to
understand the reasons for that fact, at least intuitively.)
Since the side of a real triangle (consisting of "atoms") is rather like:
o o oooo oo o o o o o o o o o oooo oo
o o ooo oo ooo o oo ooo oo o o o o o
oo oo oooooooo ooooooooo ooooo oo oo
??? -------------- L --------------- ???
Now we could say, ok, the length is the MAX distance between atoms that are part
of the side of the triangle. Leading to something like:
| o o oooo oo o o o o o o o o o oooo oo|
|o o ooo oo ooo o oo ooo oo o o o o o |
| oo oo oooooooo ooooooooo ooooo oo oo |
| |
|---------------- L --------------------|
But actually, this also cannot be done with absolute precision. Since an atom
actually is not a solid ball with a definite surface. Actually it consists of a
core and a "cloud" of electrons moving "around" that core (to use a classical
picture).
So if we MAGNIFY the image we have:
| ___ Cloud of electrons
| . : . /
|. . . .
.| . * :
.: . :.
| . . .
|
|------------- ...
So we can't really determine the precise beginning of the side of triangle (that
we want to measure).
Now we could say, well, let's begin at the core of the atom:
___ Cloud of electrons
. : . /
. . . .
. . * :
.: .| :.
. . .
|
|--------- ...
Again, if we would magnify, we would have something like
____ Core of the Atom
/
o.o
o.o.o
o.o
??? --------- ...
Note, that the core is also NOT a static object with a clear boundary, etc. etc.
(Not even if we cool down the triangle to 0 degree Celsius... Ok, at this point
we would have to refer to quantum theory, Heisenberg's uncertainty principle,
etc. etc.)
So finally we have to accept that it's _in principle_ not possible to get a 100%
accurate measurement. Instead we usually use some error margins to deal with
this state of affairs:
L = ... +/- ...
That's THE BEST we can do.
And actually, rational numbers would be good enough to state that result(s).
(But they are not good enough as basis for our physical THEORYs of the
"reality".)
F.
G. Frege
>
> Now, you called me an idiot before. I won't call you any names because
> civilized behavior is what distinguished humans from other species.
>
I appreciate that. :-)
>
> You could have just stayed silent and read the post by David C. Ullrich.
> Maybe you can learn something or at least what is a proper way to answer a
> question.
>
*lol* :-)
Well, if you are unhappy with my answer, what can I do? :-)))
>
> Now, back to the question by Alba.
>
Right.
> The debate still goes on whether space is continuous or discrete and
> whether the Pythagorean theorem is valid. The answer according to the
> Pythagoreans is sqrt(2) but we don't know whether that can resolve or
> map into actual physical dimensions. If space is infinitely divisible
> then sqrt(2) is the correct answer.
>
This are quite interesting but (on the other hand) COMPLETELY "theoretical"
questions - Alba's question actually _may_ be reduced JUST to the measuring
problem.
>
> Whether we can measure it or not maybe is a problem due to our limited
> instrumentation capacity.
>
Imho, it's a FUNDAMENTAL problem, we can't "overcome". This can be understood
even without _directly_ referring to quantum mechanics, I think. See my other
post.
>
> But just think of an instrument that measures directly sqrt(2) to
> infinite precision.
>
Well, this reminds me to the "task": think of a natural number between 0 and 1.
>
> If space is discrete as in Loop Quantum Gravity theories, then Pythagoreans
> may be wrong.
>
Well, since I'm not familiar with Loop Quantum Gravity theories, I don't have an
opinion here. But you may be right. (?)
>
> But nobody has taken the risk so far to replace arithmetic as
> we know it to fit with such theory. (Or even pronounce irrationals as
> non-existent).
>
Again, NO numbers "exist" is the physical reality. On the other hand, they
"exist" as mathematical objects. And NO physical theory will EVER change that!
:-)
F.
Mechanic
Try to refrain from dogmatic stands. Even this view has been the subject of
controversial debate (Platonists vs. constructivist and formalists) You do
not need to shout in caps. Everyone here knows the issues involved. So let
me ask you something: if numbers are not part of physical reality, where do
they belong? Is there another reality? Isn't your brain and whatever is in
there (and insults often) part of physical reality?
if you are interested in these issues try Jackson's knowledge argument. This
is where the debate has been carried in nowadays. Physical vs. mental
knowledge. If all knowledge is physical then numbers is also a part of
physical reality. Someone said, I can't remember now: God made the integers
and then the world.
You see, opinion vary on this important subject and your caps will only
serve to exposing your ignorance.
G. Frege
> >
> > Again, NO numbers "exist" is the physical reality. On the other hand, they
> > "exist" as mathematical objects.
> >
>
> [...] Even this view has been the subject of controversial debate (Platonists vs.
> constructivist and formalists).
>
No. Not really. Even a full-fledged Platonist does not pretend that numbers are
PHYSICAL objects. ;-)
>
> [...] So let me ask you something: if numbers are not part of physical reality,
> where do they belong?
>
I really don't know. But that's for sure: they are NOT PHYSICAL objects (if
objects at all). Or did YOU or ANYONE else ever _see_ or _experience_ a number?
(I don't think so.)
>
> Is there another reality?
>
Now THIS would be the solution of the Platonist. I'm not sure if *I* am a
Platonist in this sense. (BTW: the REAL Frege and Gödel were BOTH Platonists in
this strong sense.)
>
> Isn't your brain and whatever is in there (...) part of physical reality?
>
Sure. But actually numbers are NOT "in a brain". That's for sure. :-)
F.
P.S.
If interested in this questions, you might google for "category error". ;-)
|-|erc
"Alba Papa-Grimaldi" > wrote
The answer is simple and does not have anything to do with physical
measuring capacity. Measuring capacity is well covered by rationals.
The hypotenuse can determine the number, but the number cannot determine the
hypotenuse because it is irrational.
Say we can zoom into the length of the line to infinite increasing scale. I give you
instructions to redraw the line from the expansion 1.66666666666 and I just
keep repeating. You can finish the line. You can deduce I am reciting a rational
and calculate the *exact* point two thirds between 1 and 2. Then consider I
instruct to recreate this lengh, 1.4142684 and I keep adding numbers. You get
more and more precise but can never draw the exact length of sqr(2) when given
the number.
consise representation can derive numeric representation
rational cr <-> nr
irrational ci -> ni
Herc
Poker Joker
I'm still laughing even as you read this.
SO WHAT IF ITS A PHYSICAL MEASUREMENT? I measure
some lengths using rulers that are based on SQRT(2)
inches. Yep using a standard ruler, the best
measurement of the sides of the triangle is 1 inch
and using my SQRT(2) ruler, the hypotenuse is
best measured at SQRT(2) inches. They just so happen
to agree with the EXACT dimensions of the given
triangle. Neither ruler is perfect. The measurements
aren't perfect. But the answer is right on the money.
The reason we can give it a certain value when we
measure it is because we actual do a measurement.
An actual measurement on an actual triangle provides
a real value. The value may not be accurate or it may
be very accurate. It doesn't matter. Its a value
and it may be an integer, a rational, or even irrational.
Its just a number. Just because a number is irrational
does not mean its not REAL.
Accuracy and precision are independent of rational and
irrational numbers. You are dealing with two completely
different concepts.
Virgil
<3e8d92c9$0$80659$7b0f...@reader.news.newnet.co.uk>,
"Alba Papa-Grimaldi" <al...@ticnet.co.uk> wrote:
> Thank you for the replies so far.
>
> Am I correct in saying then that we cannot actually get a 100% accurate
> measurement, either by calculation or measurement, for the hypotenuse of
> such a triangle ?
No physical measurement can ever be 100% accurate.
When you say "by calculation", if you mean something like an
exact representation in, say, decimal notation, of sqrt(2),
it is impossible.
But there are symbolic forms which are exact representations
of lots of things which cannot be given exactly in decimal
form.
Poker Joker
"Virgil" <vmh...@attbi.com> wrote in message
news:vmhjr2-2DE285....@netnews.attbi.com...
Decimal form isn't symbolic?
I will now measure the number of words in the previous
question. 1... 2... 3... 4... The measurement resulted
in four words. The equipment used to measure was very
high-tech yet we all know it isn't accurate. I wonder
how much the actual value differs from the absolutely
positively inaccurate measurement?
By the way, can anybody show the proof that
"no physical measurement can ever be 100% accurate?"
I can't remember who proved that.
Virgil
"Poker Joker" <Po...@wi.rr.com> wrote:
> "Virgil" <vmh...@attbi.com> wrote in message
> news:vmhjr2-2DE285....@netnews.attbi.com...
> > > Am I correct in saying then that we cannot actually
> > > get a 100% accurate measurement, either by
> > > calculation or measurement, for the hypotenuse of
> > > such a triangle ?
> >
> > No physical measurement can ever be 100% accurate.
> >
> > When you say "by calculation", if you mean something
> > like an exact representation in, say, decimal notation,
> > of sqrt(2), it is impossible.
> >
> > But there are symbolic forms which are exact
> > representations of lots of things which cannot be given
> > exactly in decimal form.
>
> Decimal form isn't symbolic?
It is certainly not the only symbolic form, as your question
would seem to imply. Would it have pleased you more if I had
added "other" to "symbolic forms"?
I had presumed that everyone could have supplied that by
themselves, but was apparently too optimistic.
>
> I will now measure the number of words in the previous
> question. 1... 2... 3... 4... The measurement resulted
> in four words. The equipment used to measure was very
> high-tech yet we all know it isn't accurate. I wonder
> how much the actual value differs from the absolutely
> positively inaccurate measurement?
And how do you measure fractional parts of a word.
There is a distinction between making a *count* and making a
*measurement*.
For the former, counting, the result is usually reported
exactly as a positive integer if the counts are not too
large, and then no unit of measure need be attached (though
approximations and units in powers of 10 are common as the
integers get larger).
For the latter, measurements, you must always choose a unit
of measure, which can be scaled to larger or smaller units
when convenient, and usually report your results as decimal
multiples, of limited precision, of that unit.
You seem unaware of the difference.
>
> By the way, can anybody show the proof that
> "no physical measurement can ever be 100% accurate?"
> I can't remember who proved that.
>
As a rule of thumb, one can say that the correspondence
between the world of mathematics and the world of
physicality can never be perfect. But there can be no proof,
in any mathematical sense, about anything outside the
mathematical world, including its interface with other
worlds.
Poker Joker
> In article <OHsja.9967$vb.1...@twister.rdc-kc.rr.com>,
> "Poker Joker" <Po...@wi.rr.com> wrote:
>
> > "Virgil" <vmh...@attbi.com> wrote in message
> > news:vmhjr2-2DE285....@netnews.attbi.com...
>
> > > > Am I correct in saying then that we cannot actually
> > > > get a 100% accurate measurement, either by
> > > > calculation or measurement, for the hypotenuse of
> > > > such a triangle ?
> > >
> > > No physical measurement can ever be 100% accurate.
> > >
> > > When you say "by calculation", if you mean something
> > > like an exact representation in, say, decimal notation,
> > > of sqrt(2), it is impossible.
> > >
> > > But there are symbolic forms which are exact
> > > representations of lots of things which cannot be given
> > > exactly in decimal form.
> >
> > Decimal form isn't symbolic?
>
> It is certainly not the only symbolic form, as your question
> would seem to imply. Would it have pleased you more if I had
> added "other" to "symbolic forms"?
>
> I had presumed that everyone could have supplied that by
> themselves, but was apparently too optimistic.
No, just covering your slip of the keybaord this time around.
> >
> > I will now measure the number of words in the previous
> > question. 1... 2... 3... 4... The measurement resulted
> > in four words. The equipment used to measure was very
> > high-tech yet we all know it isn't accurate. I wonder
> > how much the actual value differs from the absolutely
> > positively inaccurate measurement?
>
> And how do you measure fractional parts of a word.
Who cares? Count the characters. The fractional part
is the number counted divided by the total number.
If you divide characters, some OTHER COUNTING SCHEME
is used.
> There is a distinction between making a *count* and making a
> *measurement*.
>
> For the former, counting, the result is usually reported
> exactly as a positive integer if the counts are not too
> large, and then no unit of measure need be attached (though
> approximations and units in powers of 10 are common as the
> integers get larger).
>
> For the latter, measurements, you must always choose a unit
> of measure, which can be scaled to larger or smaller units
> when convenient, and usually report your results as decimal
> multiples, of limited precision, of that unit.
>
> You seem unaware of the difference.
You seem to to imply that a measurement can NEVER be
an integer count. You think it is always a non integer
rational number ONLY. Open up man. See the world. There
are concepts beyond that.
> >
> > By the way, can anybody show the proof that
> > "no physical measurement can ever be 100% accurate?"
> > I can't remember who proved that.
> >
>
> As a rule of thumb, one can say that the correspondence
> between the world of mathematics and the world of
> physicality can never be perfect. But there can be no proof,
> in any mathematical sense, about anything outside the
> mathematical world, including its interface with other
> worlds.
So what gives you the right to say
"no physical measurement can ever be 100% accurate?"
Measurements CAN be exact and therefore accurate.
Even when they aren't exact, they can certainly be
numbers other than non integer rationals.
Virgil
"Poker Joker" <Po...@wi.rr.com> wrote:
As far as I can recall, every integer has a decimal
representation. Do you know of any integers which do not?
>
> > >
> > > By the way, can anybody show the proof that
> > > "no physical measurement can ever be 100% accurate?"
> > > I can't remember who proved that.
> > >
> >
> > As a rule of thumb, one can say that the correspondence
> > between the world of mathematics and the world of
> > physicality can never be perfect. But there can be no proof,
> > in any mathematical sense, about anything outside the
> > mathematical world, including its interface with other
> > worlds.
>
> So what gives you the right to say
> "no physical measurement can ever be 100% accurate?"
IIRC, there is something about that in the Bill of Rights.
> Measurements CAN be exact and therefore accurate.
Measurements exclude values which are integer multiples of
some essentially indivisible unit, such as words, and thus
are, strictly speaking, counts rather than measurements.
Can you find any measurements that are exact and satisfy the
above exclusion? I thought not.
> Even when they aren't exact, they can certainly be
> numbers other than non integer rationals.
The standard recording of measurements is some (terminating)
decimal multiple of some unit of measure. There is a limit
to how small such units can be in any practical situation
and still be able to measure such units down to the nearest
integer.
Do you know your height to the nearest micrometre?
Do you know is your weight to the neastest microgram?
If you claim to know either of these, describe precisely how
you achieved such unbelievable accuracy.
The unit part of measurements can, I suppose, include
irrationals, like pi, but then how do you MEASURE to get an
irrational number of units of measure?
Jim Burns
Virgil wrote:
>
> In article <hvCja.21548$5j.2...@twister.kc.rr.com>,
> "Poker Joker" <Po...@wi.rr.com> wrote:
> >
[ ... ]
> > Measurements CAN be exact and therefore accurate.
>
> Measurements exclude values which are integer multiples of
> some essentially indivisible unit, such as words, and thus
> are, strictly speaking, counts rather than measurements.
>
> Can you find any measurements that are exact and satisfy the
> above exclusion? I thought not.
The speed of light in vacuum is exactly 299,792,458 meters
per second.
I'm exhibiting a fair bit of wise-ass-ness here, but
beyond that I'm interested in how "measurements" such
as this now-standard value for the speed of light fit
into this discussion of measurements in the usual sense.
Yours,
Jim Burns
G. Frege
>
> The speed of light in vacuum is exactly 299,792,458 meters
> per second.
>
Yeah. But this is NOT a _measured_ value... as you surely know. :-)
F.
G. Frege
>
> Yeah. But this is NOT a _measured_ value... as you surely know. :-)
>
Just googled a little bit to check my claim.
http://www.what-is-the-speed-of-light.com/speed-of-light-defined.html
Read the last paragraph. :-)
F.
Jim Burns
It's a curious value because, even though it's defined
exactly now, that definition was preceded by a great deal
of measurement, as I'm sure _you_ know.
I have a question about the whole process of scientific
measurement, only a small part of which resulted in
this exact value for the speed of light. I doubt I'll
do a very good job of expressing it; it's not really
very clear in my mind, as yet. But, what the hell, I'll
give it a shot.
The usual way I think of a measurement is as an individual
act, for example, weighing a rock: I put the rock in
one pan of the scale and I add or subtract weights on the
other pan until the two sides balance. Done.
However, the more fundamental the thing I want to measure,
the less individual the act becomes, the more dependent
the result reported, the "true"(?) result, becomes on
other measurements, on other measurers, on our
understanding of what it is we're measuring, on a host of
things. If, instead of weighing a particular rock, I
want to measure the density of limestone, I need to think
about what samples are and are not limestone, where
I should collect them from, whether to throw out
particular outlier values, how my results mesh with
others' results, etc.
Along with this burden of finding a wider kind of
agreement for the more fundamental value, I receive
the opportunity to find a greater precision and
a greater accuracy. (Even if there is variation in,
for example, the density of limestone, this variation
can be characterized very precisely.)
I think the speed of light being _defined_ as the
particular value of 299,792,458 m/s is an extreme
example of both a fundamental result (depending
on and depended on by much of the physical sciences)
and a precise result (they don't come much preciser).
It strikes me that these two aspects are opposite sides
of the same coin. I guess my question is: is this
obviously true? obviously false? none of the above?
Yours,
Jim Burns
G. Frege
>
> It's a curious value because, even though it's defined
> exactly now, that definition was preceded by a great deal
> of measurement, as I'm sure _you_ know.
>
Sure. :-)
>
> The usual way I think of a measurement is as an individual
> act, for example, weighing a rock: I put the rock in
> one pan of the scale and I add or subtract weights on the
> other pan until the two sides balance. Done.
>
Yes.
>
> However, the more fundamental the thing I want to measure,
> the less individual the act becomes, the more dependent
> the result reported, the "true"(?) result, becomes on
> other measurements, on other measurers, on our
> understanding of what it is we're measuring, on a host of
> things.
>
Well, yes. Actually we have to eliminate "all" the possible [and/or more
obvious] sources of error; that's not always a simple task.
>
> If, instead of weighing a particular rock, I
> want to measure the density of limestone, I need to think
> about what samples are and are not limestone, where
> I should collect them from, whether to throw out
> particular outlier values, how my results mesh with
> others' results, etc.
>
I see. Yes. Actually this typo of error is called "systematic" error. Of course
we have to try to avoid this type of error (if we can).
In fact, measuring (in this sense) is IN NO WAY a trivial task!
Actually, there's a sub-field of physics which deals especially with this type
of problems: experimental physics.
>
> Along with this burden of finding a wider kind of
> agreement for the more fundamental value, I receive
> the opportunity to find a greater precision and
> a greater accuracy. (Even if there is variation in,
> for example, the density of limestone, this variation
> can be characterized very precisely.)
>
Yes.
>
> I think the speed of light being _defined_ as the
> particular value of 299,792,458 m/s is an extreme
> example of both a fundamental result (depending
> on and depended on by much of the physical sciences)
> and a precise result (they don't come much preciser).
>
Right.
>
> It strikes me that these two aspects are opposite sides
> of the same coin. I guess my question is: is this
> obviously true? obviously false? none of the above?
>
:-)
Hell, It seems that due to my limited knowledge of the English language I just
missed your question. :-)
Well, probably one (more) comment... Take the value of the speed of light:
299,792,458 m/s in "vacuum". Now this is actually a _precise_ but also a IDEAL
value. It's quite likely that almost any beam of light that actually "moves" in
space DOES NOT move with exactly this speed. Since... as you can see, there's a
certain important condition "in vacuum". Now we actually know that "the vacuum"
is a idealization already (---> quantum fluctuations). There is no such thing as
a TOTAL vacuum - at least not considering some space [not zero] over some time.
Moreover if we actually would try to MEASURE that speed our measuring device
[including the scientist who wants to make the measurement] would INEVITABLY
influence the space in which the beam of light is moving in some way... (--->
GRT) which would in turn have an influence on the value of the speed of the beam
of light _we would measure_, etc., etc. (Of course the influence of these [and
probably other] sources of error [of this type] may be VERY small, but the very
point is: we can't eliminate it completely.)
You might also have a look at the other post(s) I wrote in this thread.
F.
George Dance
> Now, you called me an idiot before. I won't call you any names because
> civilized behavior is what distinguished humans from other species.
If you do a google search using "Idiot author:G. author:Frege", you'll
find you're in good company. (And if you substitute his real name,
you'll find a lot more.)
So my advice would be: don't take that personally; it's a reflection
on him, not on you.
> You could have just stayed silent and read the post by David C. Ullrich.
> Maybe you can learn something or at least what is a proper way to answer a
> question.
David Ullrich is another one whom "Frege" (under his real name) has
labelled an 'idiot.' You wouldn't really expect "Frege" to actually
read a so-called idiot's post, can you (or to stay silent, FTM)?
Bill Taylor
> I see. Yes. Actually this typo of error is called "systematic" error.
............................^^^^
A genuine self-referential error!
Bertie Russ would be delighted, I'm sure.
We should send a copy to Smullyan too.
-------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
-------------------------------------------------------------------------------
In quantum mechanics tiny physical systems do things that would be difficult or
impossible to explain using quantum mechanics. - J Baez
Reply: In typo veritas.
-------------------------------------------------------------------------------
G. Frege
wrote:
"This typo of error is called 'systematic' error."
('G. Frege')
>
> A genuine self-referential error!
>
Indeed! :-)
>
> Bertie would be delighted, I'm sure.
> We should send a copy to Smullyan too.
>
*lol*
F.
Jim Burns
> >
> > I think the speed of light being _defined_ as the
> > particular value of 299,792,458 m/s is an extreme
> > example of both a fundamental result (depending
> > on and depended on by much of the physical sciences)
> > and a precise result (they don't come much preciser).
> >
> Right.
>
> >
> > It strikes me that these two aspects are opposite sides
> > of the same coin. I guess my question is: is this
> > obviously true? obviously false? none of the above?
> >
> :-)
>
> Hell, It seems that due to my limited knowledge of the
> English language I just missed your question. :-)
I've been thinking about what I wanted to say, and it doesn't
seem to be any clearer. Nevertheless, I will try again.
I'm interested in the relentless pursuit of precision in
standards laboratories such as http://www.nist.gov/
and I wonder just how far it can go. The earlier discussion
of precision reminded me of this.
It seems to me that in a final Theory of Everything, values
that we certainly consider measurements today, such as the
mass of the electron or the fine structure constant, could
be calculated directly from the theory. In fact, we might
think of this as the very definition of a final theory, one
that is either all right or all wrong, without parameters to
wiggle to make things come out right. (I'm attempting to
paraphrase Brian Greene in _Elegant Universe_ describing
string theory.)
Assuming the perspective of some future in which this final
theory has been completed, it seems to me that the purely
theoretical calculations of, for example, the electron mass
will have precedence over the actual physical measurements of
electron mass. Any disagreement between theory and experiment
would be blamed on errors in experiment and theory would be
considered the "true" value.
An analogy: consider the value for pi. We've had a "final
theory" describing shapes on flat surfaces ever since
Euclid's Elements. If I carry out the measurement of the
diameter and circumference of a circle and find that their
ratio is 3.1418 +/- .0001, everyone, including myself,
would assume my circle was not quite round, the surface not
quite flat, or something else.
I suppose we could then regard the perfected values for the
electron mass, the fine structure canstant and the like as no
longer being measurments, just as we rarely think of our
calculations of pi as true measurements involving the diameter
and circumference of a physical circle.
Perhaps it is just a whim, but I would like to think of these
perfected values as still being measuremnts. However, instead
of being implied by a single action such as weighing a rock or
even by a lot of actions such as the process of finding the true
density of limestone, these values are implied by the final
theory as a whole. Therefore our confidence in these values
should be the same as our confidence in the theory as a whole.
From this point of view, every instant that my body does not
implode into a black hole and that no other theoretically
impossible event occurs can be called a measurement that the
electron mass is whatever the theory says it is.
Also, from this point of view, this defined value for the speed
of light, 299,792,458 m/s, is also a measurement, re-measured
every time we observe that our theories continue to work.
I don't really know how much sense all this makes. I would
appreciate any comments. I'm not sure where the question mark
should go; perhaps at the end?
Yours,
Jim Burns
Alba Papa-Grimaldi
certain sense, and I hope that the curiosity of a philosopher can be
tolerated a bit further not withstanding the hope of myself giving a little
contribution to this millenarian riddle.
Particularly, in one of your e-mails you wrote: The debate still goes on
whether space is discrete and whether the Pythagorean theorem is valid. The
answer according to the Pythagoreans is sqrt(2) but we don't know whether
that can resolve or map into actual physical dimensions.... This obviously
means that the unmeasurability applies to all physical quantities. So the
question could be: why does this impasse come to the surface specifically
with sqrt(2)? and, in fact, the sqrt of any prime number?
This impasse, it seems to me, is more fundamental than the problem of
measurement itself, and invests all those cases in which our thought has to
represent the passage from the discrete (the unit) to the continuous
(concrete, physical quantity) as I will try to explain.
Irrational numbers are the proof, if any was needed, that our mind cannot
think concrete plurality. To think concrete plurality means to break free
from the one, the unit, the self-identity necessary to our thought to think
of anything. A=A is a feature attributed to reality because our thought
cannot think of something and its opposite at the same time. This passage or
attempted passage from "one" to concrete quantity makes us fall in the
"irrational" of the infinite, as sqrt(2) symptomatically proves. Only the
reiteration of the unit can be rationally thought. So 2 is thinkable only as
a reiteration of the unit (sum, multiplication etc,). But this 2 as 1+1 is
still not a real quantity, it is not concrete plurality. Such would have to
be instead a number which is not the unit and which multiplied by itself
would give 2. Such would have to be any number that multiplied by itself
yields a prime number; these latter, in fact, not being perfect squares and
so reiterations of the unit can have as sqrt only a number that yields other
than the unit: concrete quantity. Such a number cannot be thought of
rationally because it would require that our thought can think the non unit,
the non identical. This number does not exist in the sense that it is not a
number which can be thought rationally, it is a non accomplished number
because it represent the non accomplished passage from the discrete unity to
the continuous of real physical quantity. It is, in fact, an irrational
number. If sqrt(2), the non unit, the non reiteration of the unit, was per
absurdum a rational number it would mean that our thought can think what is
other than the unit, other than the self-identical. But this is called
irrational precisely because it cannot be thought of.
Allow me now an historical digression to contextualize the Pythagorean
claims. The contention between Pythagoreans and Eleatics regarded the
possibility to attain the plurality of the universe by multiplication or
addition. For the Eleatic logic multiplication (reiteration of the unit) is
useless for this end, because you are bound to start with either a nothing
or an Infinite, and by its means you get only what you start with, either a
nothing or an infinite (read this as discreteness and continuity).
In conclusion and making a reference to some of the passages where you
argued for the impossibility to effect a precise measurement as a
consequence of the flittering nature of physical reality, I believe this
latter is a consequence rather than a cause of our incapacity to think the
many or concrete quantity (concrete length in this case). For concrete
plurality (other than one) always engenders in our thought a regress ad
infinitum which is the symptom of the lack of transition that our thought
cannot make from self-identity and abstract quantity (reiteration of the
unit) to concrete plurality, (other than the one). Sqrt(2) is the first
historically documented proof of this.
Comments?
Thank you
Alba
Alba Papa-Grimaldi
David C. Ullrich
<al...@ticnet.co.uk> wrote:
>Thank you all for your answers, they have been very enlightening in a
>certain sense, and I hope that the curiosity of a philosopher can be
>tolerated a bit further not withstanding the hope of myself giving a little
>contribution to this millenarian riddle.
>
>Particularly, in one of your e-mails you wrote: The debate still goes on
>whether space is discrete and whether the Pythagorean theorem is valid. The
>answer according to the Pythagoreans is sqrt(2) but we don't know whether
>that can resolve or map into actual physical dimensions.... This obviously
>means that the unmeasurability applies to all physical quantities. So the
>question could be: why does this impasse come to the surface specifically
>with sqrt(2)? and, in fact, the sqrt of any prime number?
It doesn't. Did you read the replies? The idea of the length of an
actual physical rod of steel being exactly 1 meter is just as
problematic as the idea that the length is exactly sqrt(2) meters.
[...]
>
>In conclusion and making a reference to some of the passages where you
>argued for the impossibility to effect a precise measurement as a
>consequence of the flittering nature of physical reality, I believe this
>latter is a consequence rather than a cause of our incapacity to think the
>many or concrete quantity (concrete length in this case).
Think what you like. In _fact_ the problem is caused by at least two
factors, the second of which is a little more problematic than the
first: (i) in classical physics, while there might _be_ such a thing
as the exact length of a rod of iron, it's still true that any
measurement of the length involves some error (ii) unless
modern physics is all wrong in several ways, a rod of iron
simply does not _have_ an exact length.
>For concrete
>plurality (other than one) always engenders in our thought a regress ad
>infinitum which is the symptom of the lack of transition that our thought
>cannot make from self-identity and abstract quantity (reiteration of the
>unit) to concrete plurality, (other than the one). Sqrt(2) is the first
>historically documented proof of this.
Again, there is _no_ difference between sqrt(2) and 1, as far
as the question of making exact measurements is concerned.
>Comments?
You seem to be straining for a philosophical/psychological
explanation for phenomena for which there are _very_ simple
and very well-understood physical explanations. It's as though
one suggested that the reason that apples fall down instead
of falling up had something to do with our inability to concieve
of upwards-falling apples.
>Thank you
karl malbrain
SPECIFICATIONS AND STANDARDS FOR MEASURMENTS TO MAKE THEM EXACT:
David C. Ullrich <ull...@math.okstate.edu> wrote in message news:<a7qq8vk97mgh29l20...@4ax.com>...
> >Thanks in advance for any help you can give me
> >
Mechanic
"Alba Papa-Grimaldi" <al...@ticnet.co.uk> wrote in message
news:3e940ba7$0$374$7b0f...@reader.news.newnet.co.uk...
I don't think the above comment reflects Eleatic theories as expressed by
Xenophanes and Parmenides. Both discrete atomism (Democritus) and plurality
(Anaxagorars) were rejected by the Eleatics in favor of Monism. For the
Eleatics, everything is an illusion, including space, time and motion.
Poker Joker
> On Wed, 9 Apr 2003 13:02:13 +0100, "Alba Papa-Grimaldi"
> <al...@ticnet.co.uk> wrote:
>
> >Thank you all for your answers, they have been very enlightening in a
> >certain sense, and I hope that the curiosity of a philosopher can be
> >tolerated a bit further not withstanding the hope of myself giving a
little
> >contribution to this millenarian riddle.
> >
> >Particularly, in one of your e-mails you wrote: The debate still goes on
> >whether space is discrete and whether the Pythagorean theorem is valid.
The
> >answer according to the Pythagoreans is sqrt(2) but we don't know whether
> >that can resolve or map into actual physical dimensions.... This
obviously
> >means that the unmeasurability applies to all physical quantities. So the
> >question could be: why does this impasse come to the surface specifically
> >with sqrt(2)? and, in fact, the sqrt of any prime number?
>
> It doesn't. Did you read the replies? The idea of the length of an
> actual physical rod of steel being exactly 1 meter is just as
> problematic as the idea that the length is exactly sqrt(2) meters.
Seemingly only for some of us though.
> [...]
> >
> >In conclusion and making a reference to some of the passages where you
> >argued for the impossibility to effect a precise measurement as a
> >consequence of the flittering nature of physical reality, I believe this
> >latter is a consequence rather than a cause of our incapacity to think
the
> >many or concrete quantity (concrete length in this case).
>
> Think what you like. In _fact_ the problem is caused by at least two
> factors, the second of which is a little more problematic than the
> first: (i) in classical physics, while there might _be_ such a thing
> as the exact length of a rod of iron, it's still true that any
> measurement of the length involves some error (ii) unless
> modern physics is all wrong in several ways, a rod of iron
> simply does not _have_ an exact length.
A rod is not as long as itself?
> >For concrete
> >plurality (other than one) always engenders in our thought a regress ad
> >infinitum which is the symptom of the lack of transition that our thought
> >cannot make from self-identity and abstract quantity (reiteration of the
> >unit) to concrete plurality, (other than the one). Sqrt(2) is the first
> >historically documented proof of this.
>
> Again, there is _no_ difference between sqrt(2) and 1, as far
> as the question of making exact measurements is concerned.
>
> >Comments?
>
> You seem to be straining for a philosophical/psychological
> explanation for phenomena for which there are _very_ simple
> and very well-understood physical explanations. It's as though
> one suggested that the reason that apples fall down instead
> of falling up had something to do with our inability to concieve
> of upwards-falling apples.
LOL Now you're talking (posting).
Poker Joker
when discussing limitations of YOUR thought processes.
Virgil
"Poker Joker" <Po...@wi.rr.com> wrote:
> Please refrain from commenting on "our" thought processes
> when discussing limitations of YOUR thought processes.
>
>
Oh, is that what you call them!
David C. Ullrich
wrote:
Classically it may well be that a rob is as long as itself, but
there's still no way a measurement can determine the length
_precisely_, whether it's 1 meter or sqrt(2) meters.
Unless all of modern physics is simply _wrong_, then no,
in fact a rod is _not_ as long as itself, because it simply
does not _have_ a well-defined exact length, if we try to
measure it at a fine enough scale. Honest:
That rod is made of electrons and protons and things.
The "length" of the rod would be determined by the
position and size of those particles. But those particles
do not _have_ exact positions! An electron is not exactly
here or exactly there, it is maybe here or maybe there,
with certain probabilities. That's the way it _is_ (or
rather the way it is according to quantum mechanics,
the predictions of which are verified every day by
physicists - there's never been any reason to suspect
that quantum mechanics might be wrong.)
To be a little more precise, determining the position
of an electron necessarily involves uncertainty
regarding its velocity - if we _did_ somehow determine
its position precisely it follows we would have _no_
information whatever about its velocity, hence
_no_ clue where it would be a second later.
>> >For concrete
>> >plurality (other than one) always engenders in our thought a regress ad
>> >infinitum which is the symptom of the lack of transition that our thought
>> >cannot make from self-identity and abstract quantity (reiteration of the
>> >unit) to concrete plurality, (other than the one). Sqrt(2) is the first
>> >historically documented proof of this.
>>
>> Again, there is _no_ difference between sqrt(2) and 1, as far
>> as the question of making exact measurements is concerned.
>>
>> >Comments?
>>
>> You seem to be straining for a philosophical/psychological
>> explanation for phenomena for which there are _very_ simple
>> and very well-understood physical explanations. It's as though
>> one suggested that the reason that apples fall down instead
>> of falling up had something to do with our inability to concieve
>> of upwards-falling apples.
>
>LOL Now you're talking (posting).
>
>> >Thank you
>> >
>> >Alba
>> >
>> >
>>
>>
>> ******************
>>
>> David C. Ullrich
>
******************
David C. Ullrich
Poker Joker
> On Thu, 10 Apr 2003 00:02:54 GMT, "Poker Joker" <Po...@wi.rr.com>
> wrote:
[snip]
> >A rod is not as long as itself?
>
> Classically it may well be that a rob is as long as itself, but
> there's still no way a measurement can determine the length
> _precisely_, whether it's 1 meter or sqrt(2) meters.
>
> Unless all of modern physics is simply _wrong_, then no,
> in fact a rod is _not_ as long as itself, because it simply
> does not _have_ a well-defined exact length, if we try to
> measure it at a fine enough scale. Honest:
>
> That rod is made of electrons and protons and things.
> The "length" of the rod would be determined by the
> position and size of those particles. But those particles
> do not _have_ exact positions! An electron is not exactly
> here or exactly there, it is maybe here or maybe there,
> with certain probabilities. That's the way it _is_ (or
> rather the way it is according to quantum mechanics,
> the predictions of which are verified every day by
> physicists - there's never been any reason to suspect
> that quantum mechanics might be wrong.)
>
> To be a little more precise, determining the position
> of an electron necessarily involves uncertainty
> regarding its velocity - if we _did_ somehow determine
> its position precisely it follows we would have _no_
> information whatever about its velocity, hence
> _no_ clue where it would be a second later.
What about a rod that may be used as THE STANDARD measurement
device. It is as long as itself regardless of the fact that
it is made up of protons and such. Your nit-picking can be
countered with equal nit-picking. I have a rod in front of
me. I am going to use it as a standard of measurement.
Simultaneous to setting the standard by it, I will also measure
it relative to itself... Sure enough, its EXACTLY 1 rod in
length. The uncertainty in the standard is exactly reflected
in the measurement and so the measurement exactly coincides
with the standard.
[snip]
Virgil
"Poker Joker" <Po...@wi.rr.com> wrote:
May we assume that your rod is 5.02921005842 meters (or 1
rod) in length at all times?
A century or so ago, a physical rod in Paris was the
international standard meter. But it was found not to
give precisely the same measurement to the then desired
precision each time it was measured. Since then better
non-physical standards of length have become the agreed upon
international standard of length.
If you have found some unusual rod that does not behave as
other rods behave, and has a length that does not vary by
any measurable fraction of an angstrom unit under different
conditions of measurent, you might be able to make a fortune
from it.
But I doubt it.
Have you tried a high quality laser measuring device on your
rod?
David C. Ullrich
wrote:
You haven't been paying attention. That STANDARD rod does
not _have_ a well-defined length. You can say it's as long
as itself if you want - there's no physical content to this
statement. Because (if we're talking about absolutely
precise measurements) there's no way to say whether
or not another rod has the same length as the STANDARD
rod, so this is of no use in measuring things. A "standard
of measurement" that doesn't allow us to measure anything
is not what most people would call a standard of measurement.
And there's also no way to know whether the "real"
length of that rod is closer to 1 meter or 1000 light-years.
You think that that rod is sitting there in your office,
but there's no way to know that this or that electron
_is_ in your office - it's _probably_ in your office, but
there's a small but positive probability that one of the
electrons you think is part of that rod is actually
in another galaxy.
(Of course there's no problem using that rod as
a standard of measurement for approximate measurements
that are accurate to a high degree of probability - we
can say that with probability 99.99999% another rod
has the same length, plus or minus 0.00000001 standard
lengths. That's good enough to do physics, but in a
context like this thread, where we're talking about
whether a length is rational or irrational, it says
nothing whatever.)
>[snip]
>
******************
David C. Ullrich
Alba Papa-Grimaldi
> >
> >Particularly, in one of your e-mails you wrote: The debate still goes on
> >whether space is discrete and whether the Pythagorean theorem is valid.
The
> >answer according to the Pythagoreans is sqrt(2) but we don't know whether
> >that can resolve or map into actual physical dimensions.... This
obviously
> >means that the unmeasurability applies to all physical quantities. So the
> >question could be: why does this impasse come to the surface specifically
> >with sqrt(2)? and, in fact, the sqrt of any prime number?
>
> It doesn't. Did you read the replies? The idea of the length of an
> actual physical rod of steel being exactly 1 meter is just as
> problematic as the idea that the length is exactly sqrt(2) meters.
>
> Think what you like. In _fact_ the problem is caused by at least two
> factors, the second of which is a little more problematic than the
> first: (i) in classical physics, while there might _be_ such a thing
> as the exact length of a rod of iron, it's still true that any
> measurement of the length involves some error (ii) unless
> modern physics is all wrong in several ways, a rod of iron
> simply does not _have_ an exact length.
>
> Again, there is _no_ difference between sqrt(2) and 1, as far
> as the question of making exact measurements is concerned.
>
> You seem to be straining for a philosophical/psychological
> explanation for phenomena for which there are _very_ simple
> and very well-understood physical explanations. It's as though
> one suggested that the reason that apples fall down instead
> of falling up had something to do with our inability to concieve
> of upwards-falling apples.
> ******************
>
> David C. Ullrich
I said that the irrational number sqrt(2) as expressing the transition from
the unit to a concrete quantity, i.e. a 2 which is not a simple reiteration
of the unit but which is yielded as a true quantity, is the proof if any was
needed, that our thought cannot think (rationally) anything other than the
self-identical (unit). This means that sqrt(2) just exposes what applies to
every measurement ("symptomatic" is the word I used to be precise) which as
such represents the transition from discrete or abstract quantity (the
reiteration of the unit) to physical quantities or concrete plurality which
by definition cannot be the identity of the unit.
However the problem of measurement in western thought is obviously prior to
and independent of the findings of modern physics. The epistemological
context in which it arises is the contention between the Eleatics and the
Pythagoreans. This is important because understanding the roots of our
contemporary epistemological framework with its limitations and
idiosyncrasies, which has not just appeared in this last hundred years or so
but is the product of several millennia, can shine a light on the framework
itself. The contention between the two currents of thought was this: can a
physical quantity, a true plurality as such, yield anything but the infinite
and so the irrational?
The question arises not only because discrete quantity is not really a
quantity for it is an abstract reiteration of the unit, but also because one
has to presume that for change or movement to happen there must be something
other than the self-identity of discrete mathematical quantity and so one
must assume that physical quantity is somewhat continuous. But as soon as
you leave the discrete unit and plunge yourself into continuity this
yields the irreducible infinite. So if change cannot happen in 1, 2, or 3
because these express the self-identity of a state (a state of our thought
really), then one has to think of quantity as a transition between these
discrete states. As a concrete quantity or plurality this introduces the
divisibility ad infinitum which is our thought again treating this quantity
as a reiteration of the unit for it cannot do anything else, i.e. it cannot
conceive of what is neither A nor B, but something in the middle (excluded
middle). So our thought for its incapacity to grasp anything but the
abstract unit which represents the self-identity of thought itself, falls
into a regress ad infinitum when it thinks of concrete quantity and so it
creates the mathematical infinite.
Mathematicians have solved for mathematical and operative purposes the
problem of the infinite, but that solution cannot solve the logical problem
of the transition from identity to plurality, from discreteness to
continuity which sqrt(2), upon which our thought has first stumbled while
dealing with these problems, so well epitomizes. However, I am also bemused
by the naive approach to the findings of modern physics or any science for
that purpose. They are not independent of the restriction imposed on our
knowledge by the nature of human thought and so of metaphysical and
psychological considerations. Likewise, upwards falling apples are
unthinkable of for they imply a self-contradiction, surprisingly, not of a
scientific or empiric nature for there is nothing self-contradicting in the
idea that apples may start to fall upwards, but of a logico-linguistic
nature because of the restrictions imposed by the meaning of 'fall', which
though it is not the cause of the apples' falling, it is certainly the
reason why we will never be able to conceive of an upwards fall, (unless of
course we previously change the meaning of the word 'fall' due to a change
in Nature).
So whilst the empirical realm is open, the logico-semantic one is closed.
Again this means that any true logical restriction does not come from nature
whether at the macro or micro level, whose knowledge is only empirical,
but from our thought. Similarly we cannot think of a rational number (a
reiteration of the unit) which captures in measurement the concrete
plurality of physical quantity (the other than one by definition) not
because of the jiggling nature of physical reality, but because this would
be a contradiction in terms: the self-identical that captures the truly
plural. Why is it truly plural? Again not because of the flittering nature
of ultimate physical reality but because we must assume physical quantity
to be the realm of plurality, the other than the one. If this wasn't the
case than we would have the Parmenidean One, the immobile being that
encompasses everything. Parmenides reverted to this because of the logical
problems inherent in any knowledge of physical phenomena which because they
are intrinsically changeable cannot contain being which is for Parmenides
the self-identity of the one. So you can see that the jiggling nature of the
ultimate particles is in fact predicted by Parmenides' analysis of what a
physical phenomenon is. A phenomenon is the non identical, the other than
one, and only as such, may I add, is it distinguishable from the one and as
such exist for us. It is its changeability, its non immobility that brings
it into existence for us. But this same flittering nature makes it also
truly unknowable by a thought that can ultimately think rationally only the
identical. Parmenides says in fact that physical phenomena for their
changeable never identical nature , would require men to be like two headed
creatures, at the same time affirming and denying
And so it is that sqrt(2), a number that tries to yield 2, the first
quantity, as a true quantity and so as other than one, other than a
reiteration of the unit, cannot be thought of rationally and just like this
it is the same for any other physical measurement.
Alba Papa-Grimaldi
Alba Papa-Grimaldi
> > Allow me now an historical digression to contextualize the Pythagorean
> > claims. The contention between Pythagoreans and Eleatics regarded the
> > possibility to attain the plurality of the universe by multiplication or
> > addition. For the Eleatic logic multiplication (reiteration of the unit)
> is
> > useless for this end, because you are bound to start with either a
nothing
> > or an Infinite, and by its means you get only what you start with,
either
> a
> > nothing or an infinite (read this as discreteness and continuity).
>
> I don't think the above comment reflects Eleatic theories as expressed by
> Xenophanes and Parmenides. Both discrete atomism (Democritus) and
plurality
> (Anaxagorars) were rejected by the Eleatics in favor of Monism. For the
> Eleatics, everything is an illusion, including space, time and motion.
>
is exactly why they claimed against the idea of the Many that multilication
is useless for if you start with an indivisible you are bound to end up with
an indivisible (nothing), if you start with many you end up with many, the
infinite and irrational in greek thought. From this the zenonian paradoxes
take their cue....For more about this see my answer to D. Ullrich.
Alba
Mechanic
All of the above are known for hundreds of years. Do you have to add
anything valuable? Your whole re-iteration can be summed in a couple lines.
No need to do it here. Are you proposing something new? I cannot see the
reason for stating what is aleady known in a colloquialist fashion.
>
Poker Joker
You aren't being clear as to whether standards exist or not.
Certainly "what most people would call a standard of
measurement" implicitly suggests that those standards exist
right? If they exist, they must allow us to measure things.
Correct? So they either are perfectly accurate (in which
case I am done) or they are inaccurate and therefore don't
allow us to measure things. Therefore, according to your
statements, only the perfectly accurate standards exist if
any exist at all.
Let's suppose some inaccurate standards do exist however.
The rod I describe is probably inaccurate, but at the time
I do the measurement, the standard is established. The
MEASUREMENT EXACTLY COINCIDES WITH THE STANDARD and is
therefore perfectly accurate. From then on, measurements
may or may not be accurate, but I only needed to show
one case where a measurement is EXACT in order to show
that not all measurements are inexact.
> And there's also no way to know whether the "real"
> length of that rod is closer to 1 meter or 1000 light-years.
And I wouldn't care. My measurement is in rod lengths,
not meters or light years.
> You think that that rod is sitting there in your office,
> but there's no way to know that this or that electron
> _is_ in your office - it's _probably_ in your office, but
> there's a small but positive probability that one of the
> electrons you think is part of that rod is actually
> in another galaxy.
The good part is that the measurement follows it anywhere
it goes and whether or not I know that it is going there.
> (Of course there's no problem using that rod as
> a standard of measurement for approximate measurements
> that are accurate to a high degree of probability - we
> can say that with probability 99.99999% another rod
> has the same length, plus or minus 0.00000001 standard
> lengths. That's good enough to do physics, but in a
> context like this thread, where we're talking about
> whether a length is rational or irrational, it says
> nothing whatever.)
Talking about some measurements that are inexact doesn't
mean they all are.
Poker Joker
news:vmhjr2-AC5A2E....@netnews.attbi.com...
You may, but I would suggest against it beacuse:
Poker Joker
> Thank you all for your answers, they have been very enlightening in a
> certain sense, and I hope that the curiosity of a philosopher can be
> tolerated a bit further not withstanding the hope of myself giving a
little
> contribution to this millenarian riddle.
>
> Particularly, in one of your e-mails you wrote: The debate still goes on
> whether space is discrete and whether the Pythagorean theorem is valid.
The
> answer according to the Pythagoreans is sqrt(2) but we don't know whether
> that can resolve or map into actual physical dimensions.... This obviously
> means that the unmeasurability applies to all physical quantities. So the
> question could be: why does this impasse come to the surface specifically
> with sqrt(2)? and, in fact, the sqrt of any prime number?
If you are interested in physical dimensions why do you
bring up the added complexity of MEASURING them? Why
not just ask about the quantities you are interested in.
After all, a physical dimension is NOT NECESSARILY the
same as the measurement of it.
Are you interested in knowing if measurements
can be accurate? Are you interested in knowing if
inaccurate measuring devices can accurately reflect
the state of affairs? Or are you interested in if
physical dimensions are somehow limited to a certain
domain?
karl malbrain
(...)
> If you are interested in physical dimensions why do you
> bring up the added complexity of MEASURING them?
Duh, where do you think numbers came from to begin with. Let me give
you a hint: Start on the left little finger and count across. MOST
OFTEN, not always, you come up with a DECADE.
> Why
> not just ask about the quantities you are interested in.
Oh, goodie, more measurements. I like that.
> After all, a physical dimension is NOT NECESSARILY the
> same as the measurement of it.
You are confusing DISCIPLINE with MEASURE (or rule, as you prefer).
>
> Are you interested in knowing if measurements
> can be accurate?
After you invite Engineers in, we specify this for you. "Don't worry,
have a beer" is how they do it in CAL CS186.
> Are you interested in knowing if
> inaccurate measuring devices can accurately reflect
> the state of affairs?
Oh, a DARPA question. Why didn't you just say so. The nerds are
pushing 2GBS across the Atlantic as we speak.
> Or are you interested in if
> physical dimensions are somehow limited to a certain
> domain?
Wooopie. A domain name for a change. Try: www.taxbrain.com for
MALBRAIN.
Poker Joker
"karl malbrain" <kar...@acm.org> wrote in message
news:7fe212bf.03041...@posting.google.com...
karl malbrain
> Does someone hear the noise of a smallbrain?
>
Are you PROFERRING PERSONAL EXPERIENCE FROM MR. SHANNON to our group?
Do you actually know something about RATIOS when quantifying SIGNALS?
Do you take MEASURMENTS? Are you DISCIPLINED? Come, come, my dear
chap.
In aircraft we place a small TRANSPONDER IN THE NOSE that performs
IDENTIFICATION-FRIEND-OR-FOE -- it's not a "black box," nor an IFF of
any logical kind.
P.s. I'm flattered you recognize PHIL coming through in the following
post.
karl m
John Jones
tool made by us, and the use, the meaning, of that tool is called
measurement.
JJ
Poker Joker <Po...@wi.rr.com> wrote in message
news:aFHla.12744$B4.2...@twister.rdc-kc.rr.com...