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| martin... |
 Posted: Sun Oct 25, 2009 7:35 am |
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Guest
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John Jones wrote:
[quote]Peter Webb wrote:
"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?
A definition of "succession" requires the picture of a line. There is no
other way in which the word "succession" could deliver its sense.
[/quote]
Nonsense, Peano's successor function only yields the next symbol in a
sequence.
[quote]Peano's numbers describe the geometry of lines.
[/quote]
Does it bollocks.
[quote]
The hidden assumption is that Peano "succession" does justice for a curve.
[/quote]
You're talking crap. |
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| martin... |
| Posted: Sun Oct 25, 2009 8:05 am |
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John Jones wrote:
[quote]martin wrote:
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.
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]
Really?
set1 = {apple, orange, banana, turnip, grape, potato}
next(banana) = orange
next(apple) = grape
next(orange) = turnip
next(potato) = apple
next(grape) = banana
you think this is a line? You're barking.
[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
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]
No, people don't understand what succession means, and it seems neither
do you. |
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| martin... |
| Posted: Sun Oct 25, 2009 8:07 am |
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martin wrote:
[quote]John Jones wrote:
martin wrote:
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.
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.
Really?
set1 = {apple, orange, banana, turnip, grape, potato}
next(banana) = orange
next(apple) = grape
next(orange) = turnip
next(potato) = apple
next(grape) = banana
I missed one[/quote]
next(turnip) = potato
[quote]
you think this is a line? You're barking.
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
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"?
No, people don't understand what succession means, and it seems neither
do you.[/quote] |
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| Jan Burse... |
| Posted: Sun Oct 25, 2009 2:58 pm |
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John Jones schrieb:
[quote]Yes. There are relationships between numbers that are not given by
succession alone.
[/quote]
Yes for example the equality of dedekind cuts.
One can define it by using succession and
quantifiers, which are both part of peano:
peano = FOL (=quantifiers etc..) + S-Axioms (=sucession etc..)
Dedekind cuts are already half of the bill to
deal with reals, such as pi.
Any questions?
Bye |
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| Jan Burse... |
| Posted: Sun Oct 25, 2009 3:11 pm |
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Jan Burse schrieb:
[quote]John Jones schrieb:
Yes. There are relationships between numbers that are not given by
succession alone.
Yes for example the equality of dedekind cuts.
One can define it by using succession and
quantifiers, which are both part of peano:
peano = FOL (=quantifiers etc..) + S-Axioms (=sucession etc..)
Dedekind cuts are already half of the bill to
deal with reals, such as pi.
Any questions?
Bye
[/quote]
Read for yourself:
http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN23569441X&DMDID=dmdlog47 |
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| John Jones... |
| Posted: Mon Oct 26, 2009 7:05 am |
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Marshall wrote:
[quote]On Oct 25, 6:35 am, John Jones <jonescard... at (no spam) btinternet.com> wrote:
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"?
Generally everyone over the age of three, yeah.
Marshall
[/quote]
Well, I think that mathematicians really ought to have a better idea of
the foundations of their topic other than that given by common vernacular. |
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| John Jones... |
| Posted: Mon Oct 26, 2009 7:07 am |
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LauLuna wrote:
[quote]On Oct 25, 3:35 am, John Jones <jonescard... at (no spam) btinternet.com> wrote:
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. 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.
[/quote]
Recursion, continuity, next, between, etc all rely on a picture. The
picture is what gives them their sense. The minimal form of that picture
is a line.
It's not as if I'm saying anything that's new. |
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| John Jones... |
| Posted: Mon Oct 26, 2009 10:15 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: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?
A definition of "succession" requires the picture of a line. There is
no other way in which the word "succession" could deliver its sense.
Nonsense, Peano's successor function only yields the next symbol in a
sequence.
Peano's numbers describe the geometry of lines.
Does it bollocks.
The hidden assumption is that Peano "succession" does justice for a
curve.
You're talking crap.
[/quote]
thereyougofuckbollocks then? |
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| Trop... |
| Posted: Wed Oct 28, 2009 12:39 pm |
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Guest
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On Oct 26, 3:07 pm, John Jones <jonescard... at (no spam) btinternet.com> wrote:
[quote]LauLuna wrote:
On Oct 25, 3:35 am, John Jones <jonescard... at (no spam) btinternet.com> wrote:
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. 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.
Recursion, continuity, next, between, etc all rely on a picture. The
picture is what gives them their sense. The minimal form of that picture
is a line.
It's not as if I'm saying anything that's new.
[/quote]
I just can't understand, why are you trying to enforce everybody to
think geometrically? It's just one possible view on things, though
originally Dedekind didn't use geometrical objects. Moreover,
conceptually, this geometrical point of view restricts the insight of
natural numbers.
Sergei Tropanets |
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| Peter Webb... |
| Posted: Thu Oct 29, 2009 1:43 am |
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Guest
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"John Jones" <jonescardiff at (no spam) btinternet.com> wrote in message
news:hc46v8$lm6$2 at (no spam) news.eternal-september.org...
[quote]LauLuna wrote:
On Oct 25, 3:35 am, John Jones <jonescard... at (no spam) btinternet.com> wrote:
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. 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.
Recursion, continuity, next, between, etc all rely on a picture. The
picture is what gives them their sense. The minimal form of that picture
is a line.
It's not as if I'm saying anything that's new.
[/quote]
Yes, what you are saying is new. AFAIK, nobody has previously suggested that
Peano Axioms are somehow modeled by a line.
A line is normally used as a model for R. You can biject elements of R with
points on a line in a manner which preserves order, if you interpret "<" in
R as meaning "is to the left of" on the line and similarly with ">" meaning
"is to the right of".
There is nothing at all that I can see about "lines" which somehow connects
them to Peano's axioms. |
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| Mike... |
| Posted: Thu Oct 29, 2009 6:44 am |
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[quote]
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.
[/quote]
You are fundamentally confused about the nature of mathematics. There
is no "assertion" of the existence of a successor. Peano gives a set
of axioms and then says that IF a system satisfies these axioms then
certain logical consequences follow. Do you understand the meaning of
the word IF? A system that does not admit a successor function is
quite simply a different kind of system that Peano was not
considering. And never mind the pictures we make to help us apprehend
the mathematics. They are intuitively helpful to our understanding,
but completely irrelevant to the logic of the mathematics.
[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.
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] |
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| John Jones... |
| Posted: Sat Oct 31, 2009 10:56 pm |
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Trop wrote:
[quote]On Oct 26, 3:07 pm, John Jones <jonescard... at (no spam) btinternet.com> wrote:
LauLuna wrote:
On Oct 25, 3:35 am, John Jones <jonescard... at (no spam) btinternet.com> wrote:
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. 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.
Recursion, continuity, next, between, etc all rely on a picture. The
picture is what gives them their sense. The minimal form of that picture
is a line.
It's not as if I'm saying anything that's new.
I just can't understand, why are you trying to enforce everybody to
think geometrically? It's just one possible view on things, though
originally Dedekind didn't use geometrical objects.
[/quote]
Arithmetical systems and operations are based on geometry. Geometry
gives the signs of mathematics their symbolic meaning. Arithmetic can't
express that.
[quote]Moreover,
conceptually, this geometrical point of view restricts the insight of
natural numbers.
Sergei Tropanets[/quote] |
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| John Jones... |
| Posted: Sat Oct 31, 2009 10:57 pm |
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Guest
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Peter Webb wrote:
[quote]
"John Jones" <jonescardiff at (no spam) btinternet.com> wrote in message
news:hc46v8$lm6$2 at (no spam) news.eternal-september.org...
LauLuna wrote:
On Oct 25, 3:35 am, John Jones <jonescard... at (no spam) btinternet.com> wrote:
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. 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.
Recursion, continuity, next, between, etc all rely on a picture. The
picture is what gives them their sense. The minimal form of that
picture is a line.
It's not as if I'm saying anything that's new.
Yes, what you are saying is new. AFAIK, nobody has previously suggested
that Peano Axioms are somehow modeled by a line.
A line is normally used as a model for R. You can biject elements of R
with points on a line in a manner which preserves order, if you
interpret "<" in R as meaning "is to the left of" on the line and
similarly with ">" meaning "is to the right of".
There is nothing at all that I can see about "lines" which somehow
connects them to Peano's axioms.
[/quote]
Kant thought that mathematics was based on the temporal geometry of a line. |
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| John Jones... |
| Posted: Sat Oct 31, 2009 11:03 pm |
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Mike wrote:
[quote]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.
You are fundamentally confused about the nature of mathematics. There
is no "assertion" of the existence of a successor.
[/quote]
Yes, mathematics can have no syntactic assertion or presentation of a
successor. That's plain.
[quote]Peano gives a set
of axioms and then says that IF a system satisfies these axioms then
certain logical consequences follow.
[/quote]
"Satisfying the axioms" isn't sufficient. The axioms are based on more
fundamental considerations, considerations like "succession".
Succession cannot be defined by an axiom of succession, nor asserted by it.
[quote]Do you understand the meaning of
the word IF? A system that does not admit a successor function is
quite simply a different kind of system that Peano was not
considering.
[/quote]
I'm arguing about the very notion of "successor", not where the term is
used.
[quote]And never mind the pictures we make to help us apprehend
the mathematics. They are intuitively helpful to our understanding,
but completely irrelevant to the logic of the mathematics.
[/quote]
The logic (understanding) of mathematics is driven by that picture. A
geometrical picture. |
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| Jan Burse... |
| Posted: Sun Nov 01, 2009 5:33 am |
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Guest
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John Jones wrote:
[quote]The logic (understanding) of mathematics is driven by that picture. A
geometrical picture.
[/quote]
No its driven by the impression of music. A simple
tune that is ever increasing in pitch.
And than abstracting from the limitations of our
ear and physics...
Bye
Something like J. S. Bach's "Fugue In D Minor"? |
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