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| John Jones... |
 Posted: Sat Oct 24, 2009 4:53 pm |
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Peano's number axioms offer a new description of number. The geometrical
grid associated with the Peano axioms is a line. Why is this? and what
are the consequences?
DISCUSSION
Peano's claim or idea that a number is the "successor" of another number
is only feasible, assertable, or a stipulation, if it is based on some
precedent that allows such a claim. That precedent is given by
intuitive, geometrical contingencies. In this case, the geometrical
contingency is that of a line.
In brief, then, Peano's objects or numbers are manifested through the
picture of a line.
AN OBSERVATION
That is why a curve is not numerically directly represented through the
Peano axioms - the Peano axioms are based on the geometrical
representation of a line, and not a curve. Mathematicians fail to take
note that it is this difference in the picture between a line and a
curve that makes the Peano numerical representation of pi impossible. |
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| Immortalist... |
| Posted: Sat Oct 24, 2009 4:53 pm |
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On Oct 24, 3:53 pm, John Jones <jonescard... at (no spam) btinternet.com> wrote:
[quote]Peano's number axioms offer a new description of number. The geometrical
grid associated with the Peano axioms is a line. Why is this? and what
are the consequences?
DISCUSSION
Peano's claim or idea that a number is the "successor" of another number
is only feasible, assertable, or a stipulation, if it is based on some
precedent that allows such a claim. That precedent is given by
intuitive, geometrical contingencies. In this case, the geometrical
contingency is that of a line.
In brief, then, Peano's objects or numbers are manifested through the
picture of a line.
AN OBSERVATION
That is why a curve is not numerically directly represented through the
Peano axioms - the Peano axioms are based on the geometrical
representation of a line, and not a curve. Mathematicians fail to take
note that it is this difference in the picture between a line and a
curve that makes the Peano numerical representation of pi impossible.
[/quote]
Informally, the Peano axioms may be stated as follows:
There is a natural number 0.
Every natural number a has a successor, denoted by S(a) or a'.
There is no natural number whose successor is 0.
Distinct natural numbers have distinct successors: a = b if and only
if S(a) = S(b).
If a property is possessed by 0 and also by the successor of every
natural number which possesses it, then it is possessed by all natural
numbers. (This axiom, also known as axiom of induction, ensures that
the proof technique of mathematical induction is valid.)
In mathematics, the Peano axioms (or Peano postulates) are a set of
second-order axioms [extension of propositional logic] proposed by
Giuseppe Peano which determine the theory of arithmetic. The axioms
are usually encountered in a first-order form, where the crucial
second-order induction axiom is replaced by an infinite first-order
induction schema, and Peano Arithmetic (PA) is by convention the name
of the widely used system of first-order arithmetic given using this
first-order form. However, Peano arithmetic is essentially weaker than
the second order axiom system, since there are nonstandard models of
Peano arithmetic, and the only model for the Peano axioms (considered
as second-order statements) is the usual system of natural numbers (up
to isomorphism).
http://en.wikipedia.org/wiki/Peano_axioms
http://en.wikipedia.org/wiki/Giuseppe_Peano
"Axiom", in classical terminology, referred to a self-evident
assumption common to many branches of science. A good example would be
the assertion that
When an equal amount is
taken from equals, an
equal amount results.
At the foundation of the various sciences lay certain basic hypotheses
that had to be accepted without proof. Such a hypothesis was termed a
postulate. The postulates of each science were different. Their
validity had to be established by means of real-world experience.
Indeed, Aristotle warns that the content of a science cannot be
successfully communicated, if the learner is in doubt about the truth
of the postulates.
The classical approach is well illustrated by Euclid's elements, where
we see a list of axioms (very basic, self-evident assertions) and
postulates (common-sensical geometric facts drawn from our
experience).
A1 Things which are equal to the same thing are also equal to one
another.
A2 If equals be added to equals, the wholes are equal.
A3 If equals be subtracted from equals, the remainders are equal.
A4 Things which coincide with one another are equal to one another.
A5 The whole is greater than the part.
P1 It is possible to draw a straight line from any point to any other
point.
P2 It is possible to produce a finite straight line continuously in a
straight line.
P3 It is possible to describe a circle with any centre and distance.
P4 It is true that all right angles are equal to one another.
P5 It is true that, if a straight line falling on two straight lines
make the interior angles on the same side less than two right angles,
the two straight lines, if produced indefinitely, meet on that side on
which are the angles less than the two right angles.
http://planetmath.org/encyclopedia/Axiom.html
http://www.mathgym.com.au/history/pythagoras/pythgeom.htm |
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| Peter Webb... |
| Posted: Sat Oct 24, 2009 5:35 pm |
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Guest
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"John Jones" <jonescardiff at (no spam) btinternet.com> wrote in message
news:hc00gu$a9v$1 at (no spam) news.eternal-september.org...
[quote]Peano's number axioms offer a new description of number. The geometrical
grid associated with the Peano axioms is a line. Why is this? and what are
the consequences?
[/quote]
Actually, a line is the geometric equivalent of R. For example, there is no
"first point", and given a point, there is no "next largest" point, it is a
continuum.
The geometric equivalent of Peano's axioms is a series of points, labeled
1,2,3 ... There is a first point, and given a point, there is always a next
largest point.
You obviously can't equate lines and N; the first has an uncountable number
of elements, the latter countable. If I am allowed to equate lines and N, I
can "prove" that N is uncountable - or that the number of points in a line
is countable. Neither of these is true. Any other statements that you make
which are based upon N and a line being somehow equivalent are just as
likely to be wrong. |
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| Jan Burse... |
| Posted: Sat Oct 24, 2009 6:28 pm |
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Guest
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John Jones wrote:
[quote]AN OBSERVATION
That is why a curve is not numerically directly represented through the
Peano axioms - the Peano axioms are based on the geometrical
representation of a line, and not a curve. Mathematicians fail to take
note that it is this difference in the picture between a line and a
curve that makes the Peano numerical representation of pi impossible.
[/quote]
The problems with fish is that they are living in water.
Therefore a fish can never be a bicycle, since a bicycle
used on land.
Very deep indeed.
Best Regards |
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| John Jones... |
| Posted: Sat Oct 24, 2009 8:35 pm |
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Peter Webb wrote:
[quote]
"John Jones" <jonescardiff at (no spam) btinternet.com> wrote in message
news:hc00gu$a9v$1 at (no spam) news.eternal-september.org...
Peano's number axioms offer a new description of number. The
geometrical grid associated with the Peano axioms is a line. Why is
this? and what are the consequences?
Actually, a line is the geometric equivalent of R. For example, there is
no "first point", and given a point, there is no "next largest" point,
it is a continuum.
The geometric equivalent of Peano's axioms is a series of points,
labeled 1,2,3 ... There is a first point, and given a point, there is
always a next largest point.
You obviously can't equate lines and N; the first has an uncountable
number of elements, the latter countable. If I am allowed to equate
lines and N, I can "prove" that N is uncountable - or that the number of
points in a line is countable. Neither of these is true. Any other
statements that you make which are based upon N and a line being somehow
equivalent are just as likely to be wrong.
[/quote]
You are wrong here.
In order to make an assertion of a "successor" I have to present a
picture of succession that can show us a succession. I can't just assert
"succession". I have to describe what it is that I assert. What I
describe will be the explanation, and the assertion then follows.
And the only explanation for the claim of "succession" is given
descriptively - a line. You MUST use the image of a line to assert the
Peano axiom of succession.
Now, it follows that it is your problem, and not mine, to show how the
succession of a line is not the succession of number. |
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| John Jones... |
| Posted: Sat Oct 24, 2009 8:36 pm |
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Jan Burse wrote:
[quote]John Jones wrote:
AN OBSERVATION
That is why a curve is not numerically directly represented through
the Peano axioms - the Peano axioms are based on the geometrical
representation of a line, and not a curve. Mathematicians fail to take
note that it is this difference in the picture between a line and a
curve that makes the Peano numerical representation of pi impossible.
The problems with fish is that they are living in water.
Therefore a fish can never be a bicycle, since a bicycle
used on land.
Very deep indeed.
Best Regards
[/quote]
Yes. There are relationships between numbers that are not given by
succession alone. |
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| Peter Webb... |
| Posted: Sat Oct 24, 2009 10:06 pm |
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"John Jones" <jonescardiff at (no spam) btinternet.com> wrote in message
news:hc0dgv$hj2$1 at (no spam) news.eternal-september.org...
[quote]Peter Webb wrote:
"John Jones" <jonescardiff at (no spam) btinternet.com> wrote in message
news:hc00gu$a9v$1 at (no spam) news.eternal-september.org...
Peano's number axioms offer a new description of number. The geometrical
grid associated with the Peano axioms is a line. Why is this? and what
are the consequences?
Actually, a line is the geometric equivalent of R. For example, there is
no "first point", and given a point, there is no "next largest" point, it
is a continuum.
The geometric equivalent of Peano's axioms is a series of points, labeled
1,2,3 ... There is a first point, and given a point, there is always a
next largest point.
You obviously can't equate lines and N; the first has an uncountable
number of elements, the latter countable. If I am allowed to equate lines
and N, I can "prove" that N is uncountable - or that the number of points
in a line is countable. Neither of these is true. Any other statements
that you make which are based upon N and a line being somehow equivalent
are just as likely to be wrong.
You are wrong here.
In order to make an assertion of a "successor" I have to present a picture
of succession that can show us a succession. I can't just assert
"succession". I have to describe what it is that I assert. What I describe
will be the explanation, and the assertion then follows.
And the only explanation for the claim of "succession" is given
descriptively - a line. You MUST use the image of a line to assert the
Peano axiom of succession.
[/quote]
No, YOU said that Peano's axioms can be represented as a line.
If that is the case, what is the successor function for a line?
[quote]Now, it follows that it is your problem, and not mine, to show how the
succession of a line is not the succession of number.
[/quote]
No, again this is YOUR claim, You have to show how Peano's axioms are
somehow modeled by a line.
What a line's equivalent of "1", the unique element which is the successor
to nothing?
What is a line's equivalent to the successor function?
What is the line's equivalent to induction?
Well? |
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| Marshall... |
| Posted: Sun Oct 25, 2009 12:29 am |
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Guest
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On Oct 24, 4:28 pm, Jan Burse <janbu... at (no spam) fastmail.fm> wrote:
[quote]John Jones wrote:
AN OBSERVATION
That is why a curve is not numerically directly represented through the
Peano axioms - the Peano axioms are based on the geometrical
representation of a line, and not a curve. Mathematicians fail to take
note that it is this difference in the picture between a line and a
curve that makes the Peano numerical representation of pi impossible.
The problems with fish is that they are living in water.
Therefore a fish can never be a bicycle, since a bicycle
used on land.
[/quote]
And yet a fish who has fallen off his bicycle and is gasping in
the trail dust is more likely to spout something of merit about
Peano's axioms than John Jones.
Marshall |
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| martin... |
| Posted: Sun Oct 25, 2009 4:48 am |
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John Jones wrote:
[quote]Peano's number axioms offer a new description of number. The geometrical
grid associated with the Peano axioms is a line. Why is this? and what
are the consequences?
DISCUSSION
Peano's claim or idea that a number is the "successor" of another number
is only feasible, assertable, or a stipulation, if it is based on some
precedent that allows such a claim. That precedent is given by
intuitive, geometrical contingencies. In this case, the geometrical
contingency is that of a line.
In brief, then, Peano's objects or numbers are manifested through the
picture of a line.
[/quote]
what bollocks!
A line is a continuous function, Peano's axioms are discreet.
[quote]AN OBSERVATION
That is why a curve is not numerically directly represented through the
Peano axioms - the Peano axioms are based on the geometrical
representation of a line, and not a curve. Mathematicians fail to take
note that it is this difference in the picture between a line and a
curve that makes the Peano numerical representation of pi impossible.
[/quote]
More bollocks. pi is an irrational number, Peano's axioms deal with
integers, unless you're going to stipulate pi = 3 like the bible did.
You don't half talk a load of shite |
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| martin... |
| Posted: Sun Oct 25, 2009 4:54 am |
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John Jones wrote:
[quote]Peter Webb wrote:
"John Jones" <jonescardiff at (no spam) btinternet.com> wrote in message
news:hc00gu$a9v$1 at (no spam) news.eternal-september.org...
Peano's number axioms offer a new description of number. The
geometrical grid associated with the Peano axioms is a line. Why is
this? and what are the consequences?
Actually, a line is the geometric equivalent of R. For example, there
is no "first point", and given a point, there is no "next largest"
point, it is a continuum.
The geometric equivalent of Peano's axioms is a series of points,
labeled 1,2,3 ... There is a first point, and given a point, there is
always a next largest point.
You obviously can't equate lines and N; the first has an uncountable
number of elements, the latter countable. If I am allowed to equate
lines and N, I can "prove" that N is uncountable - or that the number
of points in a line is countable. Neither of these is true. Any other
statements that you make which are based upon N and a line being
somehow equivalent are just as likely to be wrong.
You are wrong here.
In order to make an assertion of a "successor" I have to present a
picture of succession that can show us a succession. I can't just assert
"succession". I have to describe what it is that I assert.
[/quote]
More bollocks.
The Peano function does exactly what you say it doesn't, it really does
just yield the successor.
[quote]Now, it follows that it is your problem, and not mine, to show how the
succession of a line is not the succession of number.
[/quote]
Crap, you're the one making the assertion, you substantiate it, with
maths and not rhetoric. Anyway, the successor function for a line would
yield another line, not a discreet value that isn't a line in the first
place.
You really do talk rubbish. Did you just come across Peano or something
and don't understand it or something |
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| Marshall... |
| Posted: Sun Oct 25, 2009 5:34 am |
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On Oct 25, 6:35 am, John Jones <jonescard... at (no spam) btinternet.com> wrote:
[quote]
You really do talk rubbish. Did you just come across Peano or something
and don't understand it or something
Where do you think Peano got the idea of "successor" from? A magicians
hat? Or is it just that "everyone knows what succession is without
having to think about it"?
[/quote]
Generally everyone over the age of three, yeah.
Marshall |
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| LauLuna... |
| Posted: Sun Oct 25, 2009 6:00 am |
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On Oct 25, 3:35 am, John Jones <jonescard... at (no spam) btinternet.com> wrote:
[quote]And the only explanation for the claim of "succession" is given
descriptively - a line. You MUST use the image of a line to assert the
Peano axiom of succession.
[/quote]
No. Just remember Dedekind's "proof" of the existence of an infinte
system in Was sind und was ollen die Zahlen, theorem 66.
He pointed to the self, then to a thought about the self, then to a
thought about that thought, and so on. The notion of grounded
unlimited iteration does the job. It gives also the pattern for the
notion of recursive function.
It involves the discontinuity necessary for the representation of the
naturals. As far as I can see, there is no need of any geometrical
representation.
Regards. |
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| John Jones... |
| Posted: Sun Oct 25, 2009 7:24 am |
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martin wrote:
[quote]John Jones wrote:
Peano's number axioms offer a new description of number. The
geometrical grid associated with the Peano axioms is a line. Why is
this? and what are the consequences?
DISCUSSION
Peano's claim or idea that a number is the "successor" of another
number is only feasible, assertable, or a stipulation, if it is based
on some precedent that allows such a claim. That precedent is given by
intuitive, geometrical contingencies. In this case, the geometrical
contingency is that of a line.
In brief, then, Peano's objects or numbers are manifested through the
picture of a line.
what bollocks!
A line is a continuous function, Peano's axioms are discreet.
AN OBSERVATION
That is why a curve is not numerically directly represented through
the Peano axioms - the Peano axioms are based on the geometrical
representation of a line, and not a curve. Mathematicians fail to take
note that it is this difference in the picture between a line and a
curve that makes the Peano numerical representation of pi impossible.
More bollocks. pi is an irrational number, Peano's axioms deal with
integers,
[/quote]
And?
[quote]unless you're going to stipulate pi = 3 like the bible did.
You don't half talk a load of shite[/quote] |
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| John Jones... |
| Posted: Sun Oct 25, 2009 7:31 am |
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Peter Webb wrote:
[quote]
"John Jones" <jonescardiff at (no spam) btinternet.com> wrote in message
news:hc0dgv$hj2$1 at (no spam) news.eternal-september.org...
Peter Webb wrote:
"John Jones" <jonescardiff at (no spam) btinternet.com> wrote in message
news:hc00gu$a9v$1 at (no spam) news.eternal-september.org...
Peano's number axioms offer a new description of number. The
geometrical grid associated with the Peano axioms is a line. Why is
this? and what are the consequences?
Actually, a line is the geometric equivalent of R. For example, there
is no "first point", and given a point, there is no "next largest"
point, it is a continuum.
The geometric equivalent of Peano's axioms is a series of points,
labeled 1,2,3 ... There is a first point, and given a point, there is
always a next largest point.
You obviously can't equate lines and N; the first has an uncountable
number of elements, the latter countable. If I am allowed to equate
lines and N, I can "prove" that N is uncountable - or that the number
of points in a line is countable. Neither of these is true. Any other
statements that you make which are based upon N and a line being
somehow equivalent are just as likely to be wrong.
You are wrong here.
In order to make an assertion of a "successor" I have to present a
picture of succession that can show us a succession. I can't just
assert "succession". I have to describe what it is that I assert. What
I describe will be the explanation, and the assertion then follows.
And the only explanation for the claim of "succession" is given
descriptively - a line. You MUST use the image of a line to assert the
Peano axiom of succession.
No, YOU said that Peano's axioms can be represented as a line.
If that is the case, what is the successor function for a line?
Now, it follows that it is your problem, and not mine, to show how the
succession of a line is not the succession of number.
No, again this is YOUR claim, You have to show how Peano's axioms are
somehow modeled by a line.
What a line's equivalent of "1", the unique element which is the
successor to nothing?
What is a line's equivalent to the successor function?
What is the line's equivalent to induction?
Well?
[/quote]
A definition of "succession" requires the picture of a line. There is no
other way in which the word "succession" could deliver its sense.
Peano's numbers describe the geometry of lines.
The hidden assumption is that Peano "succession" does justice for a curve. |
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| John Jones... |
| Posted: Sun Oct 25, 2009 7:35 am |
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Guest
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martin wrote:
[quote]John Jones wrote:
Peter Webb wrote:
"John Jones" <jonescardiff at (no spam) btinternet.com> wrote in message
news:hc00gu$a9v$1 at (no spam) news.eternal-september.org...
Peano's number axioms offer a new description of number. The
geometrical grid associated with the Peano axioms is a line. Why is
this? and what are the consequences?
Actually, a line is the geometric equivalent of R. For example, there
is no "first point", and given a point, there is no "next largest"
point, it is a continuum.
The geometric equivalent of Peano's axioms is a series of points,
labeled 1,2,3 ... There is a first point, and given a point, there is
always a next largest point.
You obviously can't equate lines and N; the first has an uncountable
number of elements, the latter countable. If I am allowed to equate
lines and N, I can "prove" that N is uncountable - or that the number
of points in a line is countable. Neither of these is true. Any other
statements that you make which are based upon N and a line being
somehow equivalent are just as likely to be wrong.
You are wrong here.
In order to make an assertion of a "successor" I have to present a
picture of succession that can show us a succession. I can't just
assert "succession". I have to describe what it is that I assert.
More bollocks.
The Peano function does exactly what you say it doesn't, it really does
just yield the successor.
[/quote]
You MUST use the image of a line to convey the sense of "successor". You
can't mean or assert "successor" without conveying that sense.
[quote]
Now, it follows that it is your problem, and not mine, to show how the
succession of a line is not the succession of number.
Crap, you're the one making the assertion, you substantiate it, with
maths and not rhetoric. Anyway, the successor function for a line would
yield another line, not a discreet value that isn't a line in the first
place.
You really do talk rubbish. Did you just come across Peano or something
and don't understand it or something
[/quote]
Where do you think Peano got the idea of "successor" from? A magicians
hat? Or is it just that "everyone knows what succession is without
having to think about it"? |
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