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| Alan Smaill... |
 Posted: Mon Nov 02, 2009 4:58 am |
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Jan Burse <janburse at (no spam) fastmail.fm> writes:
[quote]John Jones wrote:
The logic (understanding) of mathematics is driven by that
picture. A geometrical picture.
No its driven by the impression of music. A simple
tune that is ever increasing in pitch.
[/quote]
but where did the terminology "increasing" (more usually "rising")
some from?
[quote]And than abstracting from the limitations of our
ear and physics...
Bye
Something like J. S. Bach's "Fugue In D Minor"?
[/quote]
--
Alan Smaill |
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| Trop... |
| Posted: Mon Nov 02, 2009 9:08 am |
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On Nov 1, 5:56 am, John Jones <jonescard... at (no spam) btinternet.com> wrote:
[quote]Trop wrote:
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.
Arithmetical systems and operations are based on geometry. Geometry
gives the signs of mathematics their symbolic meaning. Arithmetic can't
express that.
Moreover,
conceptually, this geometrical point of view restricts the insight of
natural numbers.
Sergei Tropanets
[/quote]
What geometrical objects are you using while counting cash change?
Today you may speak about various equal interpretations of Natural
Numbers, but, originally (and today), people formed bijections between
sheeps and stones. That discovered correspondence between finite sets
is essentially the Idea of Natural Numbers. It is not a geometry - it
is bijection between individual objects! The operations with Natural
Numbers measures up to ways of combining sets of objects (unification,
intersection, difference and so on) set theoretically but not
geometrically. The positions of objects doesn't matter.
The Peano axioms are fundamental features of those things which
coming from our (non-geometrical) experience and are enough for
us.
I think most people today do not use geometrical view on Natural
Numbers (which of course isn't so with Reals).
Sergei Tropanets |
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| John Jones... |
| Posted: Tue Nov 03, 2009 11:35 am |
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Trop wrote:
[quote]On Nov 1, 5:56 am, John Jones <jonescard... at (no spam) btinternet.com> wrote:
Trop wrote:
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.
Arithmetical systems and operations are based on geometry. Geometry
gives the signs of mathematics their symbolic meaning. Arithmetic can't
express that.
Moreover,
conceptually, this geometrical point of view restricts the insight of
natural numbers.
Sergei Tropanets
What geometrical objects are you using while counting cash change?
[/quote]
Sequenced points or objects on a line.
[quote]Today you may speak about various equal interpretations of Natural
Numbers, but, originally (and today), people formed bijections between
sheeps and stones. That discovered correspondence between finite sets
is essentially the Idea of Natural Numbers. It is not a geometry - it
is bijection between individual objects! The operations with Natural
Numbers measures up to ways of combining sets of objects (unification,
intersection, difference and so on) set theoretically but not
geometrically. The positions of objects doesn't matter.
The Peano axioms are fundamental features of those things which
coming from our (non-geometrical) experience and are enough for
us.
I think most people today do not use geometrical view on Natural
Numbers (which of course isn't so with Reals).
Sergei Tropanets[/quote] |
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| Dan Christensen... |
| Posted: Tue Nov 03, 2009 8:00 pm |
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On Oct 24, 5:53 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]
Rather than an line, I find it helpful to visualize Peano's Axioms as
a directed graph with only one "source," no "sinks," no "branches," no
"merges" and every pair of distinct nodes being "connected." (Sorry, I
forget the proper terminology. I hope my meaning is clear.)
Dan
Download my DC Proof software at http://www.dcproof.com |
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| Peter Webb... |
| Posted: Wed Nov 04, 2009 4:32 am |
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"John Jones" <jonescardiff at (no spam) btinternet.com> wrote in message
news:hcpm5f$95k$1 at (no spam) news.eternal-september.org...
[quote]Trop wrote:
On Nov 1, 5:56 am, John Jones <jonescard... at (no spam) btinternet.com> wrote:
Trop wrote:
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.
Arithmetical systems and operations are based on geometry. Geometry
gives the signs of mathematics their symbolic meaning. Arithmetic can't
express that.
Moreover,
conceptually, this geometrical point of view restricts the insight of
natural numbers.
Sergei Tropanets
What geometrical objects are you using while counting cash change?
Sequenced points or objects on a line.
[/quote]
Why "on a line"?
Sequenced points or objects, certainly, but what properties of a line are
you using, other than it can include sequenced points or objects?
[quote]
Today you may speak about various equal interpretations of Natural
Numbers, but, originally (and today), people formed bijections between
sheeps and stones. That discovered correspondence between finite sets
is essentially the Idea of Natural Numbers. It is not a geometry - it
is bijection between individual objects! The operations with Natural
Numbers measures up to ways of combining sets of objects (unification,
intersection, difference and so on) set theoretically but not
geometrically. The positions of objects doesn't matter.
The Peano axioms are fundamental features of those things which
coming from our (non-geometrical) experience and are enough for
us.
I think most people today do not use geometrical view on Natural
Numbers (which of course isn't so with Reals).
Sergei Tropanets[/quote] |
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| Jan Burse... |
| Posted: Thu Nov 05, 2009 12:59 am |
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Dan Christensen schrieb:
[quote]Rather than an line, I find it helpful to visualize Peano's Axioms as
a directed graph with only one "source," no "sinks," no "branches," no
"merges" and every pair of distinct nodes being "connected." (Sorry, I
forget the proper terminology. I hope my meaning is clear.)
[/quote]
Ordinals also have this property, ain't they?
So what is needed that we do not cross omega?
Bye |
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| Dan Christensen... |
| Posted: Thu Nov 05, 2009 6:13 pm |
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On Nov 5, 12:59 am, Jan Burse <janbu... at (no spam) fastmail.fm> wrote:
[quote]Dan Christensen schrieb:
Rather than an line, I find it helpful to visualize Peano's Axioms as
a directed graph with only one "source," no "sinks," no "branches," no
"merges" and every pair of distinct nodes being "connected." (Sorry, I
forget the proper terminology. I hope my meaning is clear.)
Ordinals also have this property, ain't they?
So what is needed that we do not cross omega?
[/quote]
You might have something there. The only thing missing is some
equivalent of addition. Although it is usually not listed among his
axioms today, Peano originally postulated the existence of an add
function, something like:
For all x in N, x+1=next(x)
For all x, y in N, x + next(y) = next(x + y)
Can you define such a relation on the transfinite numbers? For
transfinite number x, we usually have x + x = x.
Dan
Download my DC Proof software at http://www.dcproof.com |
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| Tim... |
| Posted: Thu Nov 05, 2009 7:30 pm |
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On Oct 25, 8:24 am, John Jones <jonescard... at (no spam) btinternet.com> wrote:
[quote]martin wrote:
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?
Pi is not an integer. Get it yet? |
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| Ross A. Finlayson... |
| Posted: Fri Nov 06, 2009 10:28 pm |
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On Oct 24, 4:26 pm, Immortalist <reanimater_2... at (no spam) yahoo.com> wrote:
[quote]On Oct 24, 3:53 pm, John Jones <jonescard... at (no spam) btinternet.com> 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.
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.
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_axiomshttp://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.
[/quote]
Ah, so you see here.
Really the relevance is the effect of the event.
Bleah.
[quote]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.htmlhttp://www.mathgym.com.au/history/pythagoras/pythgeom.htm
[/quote]
Ah, yes, Euclid's theorems of geometry are totally complete and so on.
Yet, of course, as a theorem, they're separable axioms which is their
main component.
(In the theory.)
Regards,
Ross Finlayson |
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| Jan Burse... |
| Posted: Sat Nov 07, 2009 12:34 am |
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Dan Christensen wrote:
[quote]You might have something there. The only thing missing is some
equivalent of addition. Although it is usually not listed among his
axioms today, Peano originally postulated the existence of an add
function, something like:
For all x in N, x+1=next(x)
For all x, y in N, x + next(y) = next(x + y)
Can you define such a relation on the transfinite numbers? For
transfinite number x, we usually have x + x = x.
Dan
Download my DC Proof software at http://www.dcproof.com
Maybe the "every pair of nodes is connected" does not[/quote]
hold for limit ordinals.
Bye |
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| Jan Burse... |
| Posted: Sat Nov 07, 2009 12:36 am |
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Jan Burse wrote:
[quote]Maybe the "every pair of nodes is connected" does not
hold for limit ordinals.
Bye
[/quote]
Provided you don't allow merges, since your graph represents
successor. |
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| Jan Burse... |
| Posted: Sat Nov 07, 2009 1:00 am |
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Dan Christensen wrote:
[quote]On Nov 5, 12:59 am, Jan Burse <janbu... at (no spam) fastmail.fm> wrote:
Dan Christensen schrieb:
Rather than an line, I find it helpful to visualize Peano's Axioms as
a directed graph with only one "source," no "sinks," no "branches," no
"merges" and every pair of distinct nodes being "connected." (Sorry, I
forget the proper terminology. I hope my meaning is clear.)
Ordinals also have this property, ain't they?
So what is needed that we do not cross omega?
You might have something there. The only thing missing is some
equivalent of addition. Although it is usually not listed among his
axioms today, Peano originally postulated the existence of an add
function, something like:
For all x in N, x+1=next(x)
For all x, y in N, x + next(y) = next(x + y)
Can you define such a relation on the transfinite numbers? For
transfinite number x, we usually have x + x = x.
Dan
Download my DC Proof software at http://www.dcproof.com
Definition of addition is unsatisfactory for limit ordinals[/quote]
since they dont have predecessor.
On the other hand you would need to define more precisely
what "connected" means in your graph picture.
We might run circles when we for example define connected
by regressing to integers and define it as exists n such
that R^n(x,y) holds.
Ok lets try define "less than" à la peano:
(1) forall x: 0 < S(x)
(2) forall x,y: S(x) < S(y) <= x < y
plus provisio < is smallest relation satisfying (1) and (2)
Your requirement "every pair is connected" can now be
reformulated as:
forall x, y: x < y v x=y v y < x
Question will this avoid crossing omega? Can we drop
the second order provisio?
Bye |
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| Jan Burse... |
| Posted: Sat Nov 07, 2009 1:09 am |
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Jan Burse wrote:
[quote]forall x, y: x < y v x=y v y < x
Question will this avoid crossing omega? Can we drop
the second order provisio?
[/quote]
If we could proof trichotomy from induction, would this
not mean that we could replace induction by trichotomy? |
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| Dan Christensen... |
| Posted: Sun Nov 08, 2009 6:58 pm |
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On Nov 7, 1:00 am, Jan Burse <janbu... at (no spam) fastmail.fm> wrote:
[quote]Dan Christensen wrote:
On Nov 5, 12:59 am, Jan Burse <janbu... at (no spam) fastmail.fm> wrote:
Dan Christensen schrieb:
Rather than an line, I find it helpful to visualize Peano's Axioms as
a directed graph with only one "source," no "sinks," no "branches," no
"merges" and every pair of distinct nodes being "connected." (Sorry, I
forget the proper terminology. I hope my meaning is clear.)
Ordinals also have this property, ain't they?
So what is needed that we do not cross omega?
You might have something there. The only thing missing is some
equivalent of addition. Although it is usually not listed among his
axioms today, Peano originally postulated the existence of an add
function, something like:
For all x in N, x+1=next(x)
For all x, y in N, x + next(y) = next(x + y)
Can you define such a relation on the transfinite numbers? For
transfinite number x, we usually have x + x = x.
Dan
Download my DC Proof software athttp://www.dcproof.com
Definition of addition is unsatisfactory for limit ordinals
since they dont have predecessor.
On the other hand you would need to define more precisely
what "connected" means in your graph picture.
[/quote]
Sorry, I just meant that given any 2 distinct nodes, you can follow
some path (forward or backward) along edges of the graph from one node
to the other. Alternatively, if you start at 1 (or 0, depending on
your preference), and keep following successive edges, you will
eventually get to any other natural number. This allows us to do proof
by deduction. Visually, you are just ruling out 2 or more separate or
parallel graphs.
[quote]We might run circles when we for example define connected
by regressing to integers and define it as exists n such
that R^n(x,y) holds.
Ok lets try define "less than" à la peano:
(1) forall x: 0 < S(x)
(2) forall x,y: S(x) < S(y) <= x < y
plus provisio < is smallest relation satisfying (1) and (2)
Your requirement "every pair is connected" can now be
reformulated as:
forall x, y: x < y v x=y v y < x
[/quote]
You don't need a < relation for "connectedness" in the sense I meant.
The principle of mathematical induction would suffice.
[quote]Question will this avoid crossing omega? Can we drop
the second order provisio?
[/quote]
I am admittedly a bit out my depth here, to say the least, but here
goes... While the usual Peano Axioms seem to be able to "model" a
successor function on the transfinite numbers, might it be impossible
to "model" anything like the addition function that Peano originally
postulated on the transfinite numbers? Having x+x=x just wouldn't do.
Perhaps including such an addition function in PA would rule out the
transfinite numbers -- if that is what you mean.
Dan
Download my DC Proof software at http://www.dcproof.com |
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