| Topic: |
Science > Philosophy |
| User: |
"John Jones" |
| Date: |
03 Dec 2004 07:30:11 PM |
| Object: |
The sequencing of Number |
I would not know what I meant by 'two' if I said that 'two is twice
(two) the size of one'. So, otherwise, what is 'two' compared to
'one'?
It is clear that numbers cannot be 'compared'.
If this is true - that 'numbers cannot be compared', how then are
numbers arranged?
JJ
*-----------------------*
Posted at:
www.GroupSrv.com
*-----------------------*
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| User: "Mark Earnest" |
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| Title: Re: The sequencing of Number |
03 Dec 2004 08:13:12 PM |
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"John Jones" <jonescardiff@aol-dot-com.no-spam.invalid> wrote in message
news:41b11323$8_1@Usenet.com...
I would not know what I meant by 'two' if I said that 'two is twice
(two) the size of one'. So, otherwise, what is 'two' compared to
'one'?
It is clear that numbers cannot be 'compared'.
If this is true - that 'numbers cannot be compared', how then are
numbers arranged?
JJ
A number has to represent something to have crystal clear meaing.
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| User: "Milan" |
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| Title: Re: The sequencing of Number |
03 Dec 2004 08:55:43 PM |
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"Mark Earnest" <mark11315@SPAMLESSnetzero.com> wrote in message
news:cor6d9$n8g@library1.airnews.net...
"John Jones" <jonescardiff@aol-dot-com.no-spam.invalid> wrote in message
news:41b11323$8_1@Usenet.com...
I would not know what I meant by 'two' if I said that 'two is twice
(two) the size of one'. So, otherwise, what is 'two' compared to
'one'?
It is clear that numbers cannot be 'compared'.
It may be clear in your pretty little head. But that's about it.
regards
Milan
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| User: "John Jones" |
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| Title: Re: The sequencing of Number |
05 Dec 2004 02:31:11 PM |
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AEwrote:
Each natural number has exactly one successor:
Two is per definitionem the successor of one, which itself is the
successor of zero.
For any two natural numbers there is exactly one that is successor
of
the other or of one of it's successors.
[/quote]
Even if a definition tells me which numeral 'succeeds' another numeral
to make a 'number' sequence, what is the importance of this
'succession'?
JJ
*-----------------------*
Posted at:
www.GroupSrv.com
*-----------------------*
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| User: "AE" |
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| Title: Re: The sequencing of Number |
06 Dec 2004 01:35:05 PM |
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John Jones wrote:
AE wrote:
Each natural number has exactly one successor:
Two is per definitionem the successor of one, which itself is the
successor of zero.
For any two natural numbers there is exactly one that is successor
of the other or of one of it's successors.
[/quote]
Even if a definition tells me which numeral 'succeeds' another numeral
to make a 'number' sequence, what is the importance of this
'succession'?
This depends on the purpose of your numbers:
In case of ordinal number it's exactly what the numbers are for: To
describe an order. For example in a contest it's important to define who
is first, who second and so on.
Measurements on an ordinal level allow to do non-parametric statistical
tests, so quite obviously ordinal numbers are a powerful tool.
...
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| User: "John Jones" |
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| Title: Re: The sequencing of Number |
06 Dec 2004 04:31:21 PM |
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AEwrote:
John Jones wrote:
AE wrote:
Each natural number has exactly one successor:
Two is per definitionem the successor of one, which itself is the
successor of zero.
For any two natural numbers there is exactly one that is successor
of the other or of one of it's successors.
Even if a definition tells me which numeral 'succeeds' another
numeral
to make a 'number' sequence, what is the importance of this
'succession'?
[/quote:cf112d51c9]
This depends on the purpose of your numbers:
In case of ordinal number it's exactly what the numbers are for: To
describe an order. For example in a contest it's important to define
who
is first, who second and so on.
Measurements on an ordinal level allow to do non-parametric
statistical
tests, so quite obviously ordinal numbers are a powerful tool.
...[/quote]
If your explanation, using ordinals as an example, is correct, then it
seems that succession is not found in arithmetic. Nor is succession,
or order, found anywhere-, except in the particular case, as for
example in your case of who comes first and second in a race. This
may be worth pursuing, but there is another question which still
needs resolving in order to pursue it:
How is order configured in numerals, such that the numerals become
numbers? Now you can recite them, such as in the case where 3 comes
after 2 in terms of time elapsed between pronouncing them, but this
cannot be done for an equation, where time is not presented. In an
equation numbers are not actually presented as sequential. How would
you answer this?
JJ
*-----------------------*
Posted at:
www.GroupSrv.com
*-----------------------*
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| User: "AE" |
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| Title: Re: The sequencing of Number |
07 Dec 2004 01:41:07 PM |
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John Jones wrote:
AE wrote:
John Jones wrote:
AE wrote:
Each natural number has exactly one successor:
Two is per definitionem the successor of one, which itself is the
successor of zero.
For any two natural numbers there is exactly one that is successor
of the other or of one of it's successors.
Even if a definition tells me which numeral 'succeeds' another numeral
to make a 'number' sequence, what is the importance of this 'succession'?
This depends on the purpose of your numbers:
In case of ordinal number it's exactly what the numbers are for: To
describe an order. For example in a contest it's important to define
who is first, who second and so on.
Measurements on an ordinal level allow to do non-parametric statistical
tests, so quite obviously ordinal numbers are a powerful tool.
If your explanation, using ordinals as an example, is correct, then it
seems that succession is not found in arithmetic.
Nor is succession, or order, found anywhere-,
Not true: Every set of rational numbers is ordered. By calculating
absolute values complex numbers and vectors can be ordered as well.
except in the particular case, as for
example in your case of who comes first and second in a race. This
may be worth pursuing, but there is another question which still
needs resolving in order to pursue it:
How is order configured in numerals, such that the numerals become
numbers?
Numerals are only labels used to describe numbers. Maybe most
interesting is that we are using the same numeral to describe ordinal
and cardinal numbers.
Now you can recite them, such as in the case where 3 comes
after 2 in terms of time elapsed between pronouncing them, but this
cannot be done for an equation, where time is not presented. In an
equation numbers are not actually presented as sequential. How would
you answer this?
It's quite some way from a simple successor operation to mathematics,
but let me outline it quickly:
Succession allows to define cardinal numbers which allows to define
addition of integers:
The numeral 3 describes not only the ordinal number that is successor of
2, but it describes as well the number of scuccessor operations
necessary to reach the ordinal number when starting from zero.
"2 + 3 = 5" tells that three successor operations are required to reach
5 when starting with 2.
Sometimes it's interesting to do the inverse of addition: "Where did I
have to start to reach a given number when doing a given number of
successor operations?"
A product of this inverse operation is not only subtraction, but as well
negative numbers. Indeed negative numbers allow to extend our ordinal
system beyond the original starting point.
Yet another obvious step is - since we have cardinal numbers - to ask
what number will be reached when repeating addition a given number of
times. That's multiplication.
The inverse - "How many times do I have to repeat addition of a given
number to reach another given number?" - is division. This makes us
extend our number system once again, so we reach real numbers.
Interestingly real numbers allow to extend our ordered system in case a
new sample is not before the first or after the last already existing
sample, but between two of them - now granularity of our ordered system
can be extended ad infimum.
Maybe that's enough for everyday arithmetic :-?
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| User: "John Jones" |
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| Title: Re: The sequencing of Number |
09 Dec 2004 03:53:24 PM |
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AEwrote:
John Jones wrote:
AE wrote:
John Jones wrote:
AE wrote:
Each natural number has exactly one successor:
Two is per definitionem the successor of one, which itself is the
successor of zero.
For any two natural numbers there is exactly one that is successor
of the other or of one of it's successors.
Even if a definition tells me which numeral 'succeeds' another
numeral
to make a 'number' sequence, what is the importance of this
'succession'?
This depends on the purpose of your numbers:
In case of ordinal number it's exactly what the numbers are for: To
describe an order. For example in a contest it's important to
define
who is first, who second and so on.
Measurements on an ordinal level allow to do non-parametric
statistical
tests, so quite obviously ordinal numbers are a powerful tool.
If your explanation, using ordinals as an example, is correct, then
it
seems that succession is not found in arithmetic.
Nor is succession, or order, found anywhere-,
Not true: Every set of rational numbers is ordered. By calculating
absolute values complex numbers and vectors can be ordered as well.
except in the particular case, as for
example in your case of who comes first and second in a race. This
may be worth pursuing, but there is another question which still
needs resolving in order to pursue it:
How is order configured in numerals, such that the numerals become
numbers?
Numerals are only labels used to describe numbers. Maybe most
interesting is that we are using the same numeral to describe ordinal
and cardinal numbers.
Now you can recite them, such as in the case where 3 comes
after 2 in terms of time elapsed between pronouncing them, but this
cannot be done for an equation, where time is not presented. In an
equation numbers are not actually presented as sequential. How
would
you answer this?
It's quite some way from a simple successor operation to mathematics,
but let me outline it quickly:
Succession allows to define cardinal numbers which allows to define
addition of integers:
The numeral 3 describes not only the ordinal number that is successor
of
2, but it describes as well the number of scuccessor operations
necessary to reach the ordinal number when starting from zero.
"2 + 3 = 5" tells that three successor operations are required to
reach
5 when starting with 2.
Sometimes it's interesting to do the inverse of addition: "Where did I
have to start to reach a given number when doing a given number of
successor operations?"
A product of this inverse operation is not only subtraction, but as
well
negative numbers. Indeed negative numbers allow to extend our ordinal
system beyond the original starting point.
Yet another obvious step is - since we have cardinal numbers - to ask
what number will be reached when repeating addition a given number of
times. That's multiplication.
The inverse - "How many times do I have to repeat addition of a given
number to reach another given number?" - is division. This makes us
extend our number system once again, so we reach real numbers.
Interestingly real numbers allow to extend our ordered system in case
a
new sample is not before the first or after the last already existing
sample, but between two of them - now granularity of our ordered
system
can be extended ad infimum.
Maybe that's enough for everyday arithmetic :-?[/quote:ff4e703b21]
When I said that succession is not found in arithmetic, I was not
wrong: while the teaching of the symbols of arithmetic (the formulae,
the numbers) SEEM to suggest that there must be an order or a
succession in the symbols of arithmetic, we are not told how
arithmetic ITSELF presents this order.
I will give three examples: coming first or second in a race is a
matter of winning and losing; 'saying' that three comes after two and
before four seems to suggest an order based on temporality; writing
[1,2,3,4..] suggests an order based on the western style of reading
from left to right. I am still correct in saying that succession has
not been demonstrated in arithmetic - by these examples, and by any
example coming from the models of succession I currently have in
mind.
The question you pose "Where do I have to start to reach a given
number when doing a given number of successor operations?" seems to
suggest that to find the place 'where we should start to reach a
given number' is an impossible task. It is impossible because our
starting point is not a successionless numeral, but is supposed to be
a number. We now have the idea that the number is already there
waiting for the calculation to uncover it. But I would contend that
the number where we should start to reach a given number is not there
until the calculation is completed. It is only within a calculation,
or application, that numbers arise. Your observation was based upon
the idea that numbers are present outside of an application, but such
entities are numerals.
Your points about multiplication, division, subtraction, etc, are
valuable. I still need to define succession in arithmetic, however. A
possible solution is that succession is not a 'property' of numbers,
but that is for another time.
JJ
*-----------------------*
Posted at:
www.GroupSrv.com
*-----------------------*
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| User: "AE" |
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| Title: Re: The sequencing of Number |
11 Dec 2004 06:25:44 AM |
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John Jones wrote:
...
When I said that succession is not found in arithmetic, I was not
wrong: while the teaching of the symbols of arithmetic (the formulae,
the numbers) SEEM to suggest that there must be an order or a
succession in the symbols of arithmetic, we are not told how
arithmetic ITSELF presents this order.
I will give three examples: coming first or second in a race is a
matter of winning and losing; 'saying' that three comes after two and
before four seems to suggest an order based on temporality; writing
[1,2,3,4..] suggests an order based on the western style of reading
from left to right. I am still correct in saying that succession has
not been demonstrated in arithmetic - by these examples, and by any
example coming from the models of succession I currently have in
mind.
Maybe we differ in our basic way to view numbers and numerals.
"[1,2,3,4..]" is a set of glyphs, consisting on brackets you are using
to describe a set, numerals you might use to describe numbers but that
might as well represent members of this set in a more abstract way,
commata to show which glyphs are describing distinct elements, and two
dots intended to show the elements given are not the only members of the
set.
Interpreting this sequence that way I'd guess you want to describe the
set of natural numbers less zero.
One of the properties of natural numbers is their order and their
countability.
There are different ways to describe the concept of natural numbers (or,
btw, to define them), but all these descriptions lead to a set that
satisfies the Peano postulates.
Besides others these postulates contain the property of natural numbers
that every number has exactly one successor.
The question you pose "Where do I have to start to reach a given
number when doing a given number of successor operations?" seems to
suggest that to find the place 'where we should start to reach a
given number' is an impossible task.
Not at all. Since addition of two given numbers delivers exactly one
number as a result, it is invertable.
On the other hand there are divisions that don't have a result in the
set of natural numbers - for example "3 - 4". That's why negative
numbers were defined.
It is impossible because our
starting point is not a successionless numeral, but is supposed to be
a number. We now have the idea that the number is already there
waiting for the calculation to uncover it. But I would contend that
the number where we should start to reach a given number is not there
until the calculation is completed. It is only within a calculation,
or application, that numbers arise. Your observation was based upon
the idea that numbers are present outside of an application, but such
entities are numerals.
Numerals are glyphs or sets of glyphs only. Inside a numeral system each
numeral describes exactly one number.
I couldn't tell whether a number exists outside a calculation, but I can
tell that the same calculation delivers the same number as a result.
Indeed I couldn't tell whether you are existing when not writing news,
and I can't even tell whether you are the same person every time you are
posting in this group: Obviously I'm more sure a number is unique than
you are.
Your points about multiplication, division, subtraction, etc, are
valuable. I still need to define succession in arithmetic, however. A
possible solution is that succession is not a 'property' of numbers,
but that is for another time.
JJ
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| User: "" |
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| Title: Re: The sequencing of Number |
11 Dec 2004 04:00:20 AM |
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jonescardiff@aol-dot-com.no-spam.invalid (John=A0Jones) wrote:
I would not know what I meant by 'two' if I said that 'two is twice
(two) the size of one'. So, otherwise, what is 'two' compared to 'one'?
It is clear that numbers cannot be 'compared'.
[jillar] I would say they can be compared...'com-pared' as in fractions.
For example, 1=3D1/1, 2=3D2/1.
If this is true - that 'numbers cannot be compared', how then are
numbers arranged?
JJ
[jillar] If integers were not convertible to fractions then they can
freely roam a range. But 1.5 cannot roam among 1 and 2 and be considered
not to relate by size to the other two when they are converted into
fractions. So 1 and 2 must be fractured to understand this. 1=3D2/2,
1.5=3D3/2, and 2=3D4/2. Now we have relationship and 1.5 must be between,
not among, 1 and 2 and preceed 2. Wholeness is shifted down, down to a
common denominator, but maintains wholeness and sets up comparative
analysis to the converted parts shifted above. Comparative evaluation of
the numerator relates to quantity and therefore size.
Arrangement is relationship by size which is determined by the fractured
parts of whole numbers that are placed into a common evaluation. The
number of fractured parts is determined by the common denominator and
the relationship of less to more fractured parts determines the
arrangement.
BTW this is a good case for the need to break or fracture whole things
in order to get it to relate to order or arrangement. There is a waving
away the wholeness of a thing when there is a relationship gained in the
fracturing. But wholeness is never gone similar to the case of the
integer that shifts its extension of wholeness to the denominator when
it is converted. If we omit fractions we omit relationships. Present in
breaking and shifting wholeness is the possibility of new relationships.
Jillar
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| User: "" |
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| Title: Re: The sequencing of Number |
14 Dec 2004 01:43:06 AM |
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(John=A0Jones) wrote:
The easiest way to test the idea that fractions can tell us how numbers
are related to each and how they can be compared is to substitute
meaningless signs for the numerals. By doing this it makes it impossible
to unconsciously compare numbers from merely seeing the numerals. I will
keep the division sign the same ("/"). So,
=A3/=A3=A0=3D=A0=A3
*/=A3=A0=3D=A0*
*/*=A0=3D=A0=A3
I cannot conclude what =A3/* equals, except to say that it equals =A3/*.
It is also suggested that =A3 is of greater significance than *. There
is also the problem that the division sign assumes upon * and =A3 the
conditions of comparison and succession in number that I am still trying
to clear up.
It is not clear to me how the formation of fractions, if they are a
necessary condition for number formation, help us to establish
relationship between their signs.
You will have to explain further as I might have missed something.
JJ
*-----------------------*
[jillar]
Fractions tell us about 'sizes'. A proper one is smaller than an
improper one. Once we establish a proper fraction and an improper one we
can compare small to large with one more step that introduces a standard
for comparison.
If we can establish what is 'common' to all the proper and improper
fractions under consideration, we have a standard in which to compare
small to large for as many of those numbers that can be measured to the
standard.
Size is found in numbers by looking at measurement or extent of
comparison to a standard. But how do we establish this standard even in
this test with meaningless signs?
First with numbers, one way of finding the standard is to look for the
lowest common denominators (LCD) of the numbers set up for comparison
through conversion. LCD are important because it can tell something
about size (lowest) and standard (common).
With meaningless signs that follow the meaningful function of bar ("/")
and equal ("=3D") can we find the same lowest and common terms. Yes, by
finding the representative sign that is the same for its inverse.
=A3/=A3=A0=3D=A0=A3 is such a case. The top bar sign is the same as the
bottom bar sign and that inverse is equal to the same sign on the right
that is not bared. This is the lowest sign. Arrangement by size is =A3/*
then =A3 then * then */=A3.
Jillar
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| User: "John Jones" |
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| Title: Re: The sequencing of Number |
05 Dec 2004 02:31:12 PM |
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Azrael Nightwindwrote:
"AE" <hidden@nospam.com> wrote in message
news:couubk$iju$03$1@news.t-online.com...
Each natural number has exactly one successor:
Two is per definitionem the successor of one, which itself is the
successor of zero.
For any two natural numbers there is exactly one that is successor
of
the other or of one of it's successors.
Az:
Bravo, bravo!
Reminds me of math class in high school in which three quarters of the
class
was too dull to understand that.
John Jones schrieb:
I would not know what I meant by 'two' if I said that 'two is twice
(two) the size of one'. So, otherwise, what is 'two' compared to
'one'?
It is clear that numbers cannot be 'compared'.
If this is true - that 'numbers cannot be compared', how then are
numbers arranged?
JJ
*-----------------------*
Posted at:
www.GroupSrv.com
*-----------------------*
[/quote:96b5cca99a]
Some of the class repeated what the teacher wanted them to repeat,
some of the class could not repeat what the teacher wanted them to
repeat, but neither class nor teacher understood what they were told
to repeat.
JJ
*-----------------------*
Posted at:
www.GroupSrv.com
*-----------------------*
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| User: "AE" |
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| Title: Re: The sequencing of Number |
05 Dec 2004 06:17:04 AM |
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Each natural number has exactly one successor:
Two is per definitionem the successor of one, which itself is the
successor of zero.
For any two natural numbers there is exactly one that is successor of
the other or of one of it's successors.
John Jones schrieb:
I would not know what I meant by 'two' if I said that 'two is twice
(two) the size of one'. So, otherwise, what is 'two' compared to
'one'?
It is clear that numbers cannot be 'compared'.
If this is true - that 'numbers cannot be compared', how then are
numbers arranged?
JJ
*-----------------------*
Posted at:
www.GroupSrv.com
*-----------------------*
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| User: "Azrael Nightwind" |
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| Title: Re: The sequencing of Number |
05 Dec 2004 07:45:29 AM |
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"AE" <hidden@nospam.com> wrote in message
news:couubk$iju$03$1@news.t-online.com...
Each natural number has exactly one successor:
Two is per definitionem the successor of one, which itself is the
successor of zero.
For any two natural numbers there is exactly one that is successor of
the other or of one of it's successors.
Az:
Bravo, bravo!
Reminds me of math class in high school in which three quarters of the class
was too dull to understand that.
John Jones schrieb:
I would not know what I meant by 'two' if I said that 'two is twice
(two) the size of one'. So, otherwise, what is 'two' compared to
'one'?
It is clear that numbers cannot be 'compared'.
If this is true - that 'numbers cannot be compared', how then are
numbers arranged?
JJ
*-----------------------*
Posted at:
www.GroupSrv.com
*-----------------------*
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| User: "John Jones" |
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| Title: Re: The sequencing of Number |
14 Dec 2004 02:27:12 PM |
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I am at an impasse. How does the sign that is the same as its inverse,
represent the lowest sign?
If you could answer that please. Another thing I wanted to point out
is that strictly speaking the £ and * symbol I used to represent
fractions, are no more or less meaningless than the numerals
ordinarily presented in fractions. I merely used these signs to
prevent us making unrelated associations that we commonly hold in
respect of numerals.
JJ
Anonymouswrote:
(John Jones) wrote:
The easiest way to test the idea that fractions can tell us how
numbers
are related to each and how they can be compared is to substitute
meaningless signs for the numerals. By doing this it makes it
impossible
to unconsciously compare numbers from merely seeing the numerals. I
will
keep the division sign the same ("/"). So,
£/£ = £
*/£ = *
*/* = £
I cannot conclude what £/* equals, except to say that it equals
£/*.
It is also suggested that £ is of greater significance than *.
There
is also the problem that the division sign assumes upon * and £ the
conditions of comparison and succession in number that I am still
trying
to clear up.
It is not clear to me how the formation of fractions, if they are a
necessary condition for number formation, help us to establish
relationship between their signs.
You will have to explain further as I might have missed something.
JJ
*-----------------------*
[jillar]
Fractions tell us about 'sizes'. A proper one is smaller than an
improper one. Once we establish a proper fraction and an improper
one we
can compare small to large with one more step that introduces a
standard
for comparison.
If we can establish what is 'common' to all the proper and improper
fractions under consideration, we have a standard in which to
compare
small to large for as many of those numbers that can be measured to
the
standard.
Size is found in numbers by looking at measurement or extent of
comparison to a standard. But how do we establish this standard even
in
this test with meaningless signs?
First with numbers, one way of finding the standard is to look for
the
lowest common denominators (LCD) of the numbers set up for
comparison
through conversion. LCD are important because it can tell something
about size (lowest) and standard (common).
With meaningless signs that follow the meaningful function of bar
("/")
and equal ("=") can we find the same lowest and common terms. Yes,
by
finding the representative sign that is the same for its inverse.
£/£ = £ is such a case. The top bar sign is the same as the
bottom bar sign and that inverse is equal to the same sign on the
right
that is not bared. This is the lowest sign. Arrangement by size is
£/*
then £ then * then */£.
Jillar
*-----------------------*
Posted at:
www.GroupSrv.com
*-----------------------*
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| User: "John Jones" |
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| Title: Re: The sequencing of Number |
11 Dec 2004 05:27:57 PM |
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If I want to determine what number I have to start with to reach a
given number by succession, then I must already assume the succession
of numbers in order to do this. If I already assume the succession of
numbers then the number I have to start with to reach a given number
by succession is already known. If it is not known, then it is not a
number. all I am doing is pointing at two different parts of the
succession. I don't start at one point of the succession and end up
at another. It also still doesn't explain 'succession'.
This topic is difficult but is one of those cases where if enough
battering is done against it then something gives. I am not fully
clear of my own case, but something is working itself out.
JJ
AEwrote:
John Jones wrote:
...
When I said that succession is not found in arithmetic, I was not
wrong: while the teaching of the symbols of arithmetic (the
formulae,
the numbers) SEEM to suggest that there must be an order or a
succession in the symbols of arithmetic, we are not told how
arithmetic ITSELF presents this order.
I will give three examples: coming first or second in a race is a
matter of winning and losing; 'saying' that three comes after two
and
before four seems to suggest an order based on temporality; writing
[1,2,3,4..] suggests an order based on the western style of reading
from left to right. I am still correct in saying that succession
has
not been demonstrated in arithmetic - by these examples, and by any
example coming from the models of succession I currently have in
mind.
Maybe we differ in our basic way to view numbers and numerals.
"[1,2,3,4..]" is a set of glyphs, consisting on brackets you are using
to describe a set, numerals you might use to describe numbers but that
might as well represent members of this set in a more abstract way,
commata to show which glyphs are describing distinct elements, and two
dots intended to show the elements given are not the only members of
the
set.
Interpreting this sequence that way I'd guess you want to describe the
set of natural numbers less zero.
One of the properties of natural numbers is their order and their
countability.
There are different ways to describe the concept of natural numbers
(or,
btw, to define them), but all these descriptions lead to a set that
satisfies the Peano postulates.
Besides others these postulates contain the property of natural
numbers
that every number has exactly one successor.
The question you pose "Where do I have to start to reach a given
number when doing a given number of successor operations?" seems to
suggest that to find the place 'where we should start to reach a
given number' is an impossible task.
Not at all. Since addition of two given numbers delivers exactly one
number as a result, it is invertable.
On the other hand there are divisions that don't have a result in the
set of natural numbers - for example "3 - 4". That's why negative
numbers were defined.
It is impossible because our
starting point is not a successionless numeral, but is supposed to
be
a number. We now have the idea that the number is already there
waiting for the calculation to uncover it. But I would contend that
the number where we should start to reach a given number is not
there
until the calculation is completed. It is only within a
calculation,
or application, that numbers arise. Your observation was based upon
the idea that numbers are present outside of an application, but
such
entities are numerals.
Numerals are glyphs or sets of glyphs only. Inside a numeral system
each
numeral describes exactly one number.
I couldn't tell whether a number exists outside a calculation, but I
can
tell that the same calculation delivers the same number as a result.
Indeed I couldn't tell whether you are existing when not writing news,
and I can't even tell whether you are the same person every time you
are
posting in this group: Obviously I'm more sure a number is unique than
you are.
Your points about multiplication, division, subtraction, etc, are
valuable. I still need to define succession in arithmetic, however.
A
possible solution is that succession is not a 'property' of
numbers,
but that is for another time.
JJ[/quote:ccdb26839f]
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| User: "AE" |
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| Title: Re: The sequencing of Number |
12 Dec 2004 06:25:42 AM |
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John Jones schrieb:
If I want to determine what number I have to start with to reach a
given number by succession, then I must already assume the succession
of numbers in order to do this.
True. Succession of natural numbers is the beginning of all.
If I already assume the succession of
numbers then the number I have to start with to reach a given number
by succession is already known.
We know it does exist if negative numbers are defined. Obviously the
existance of subtraction is a product of the existance of addition.
If it is not known, then it is not a
number. all I am doing is pointing at two different parts of the
succession. I don't start at one point of the succession and end up
at another.
Three numbers are involved. Define any two of them and you are not any
more free to define the third one.
It also still doesn't explain 'succession'.
True.
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| User: "John Jones" |
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| Title: Re: The sequencing of Number |
11 Dec 2004 05:27:57 PM |
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The easiest way to test the idea that fractions can tell us how
numbers are related to each and how they can be compared is to
substitute meaningless signs for the numerals. By doing this it makes
it impossible to unconsciously compare numbers from merely seeing the
numerals. I will keep the division sign the same ("/").
So,
£/£ = £
*/£ = *
*/* = £
I cannot conclude what £/* equals, except to say that it equals £/*.
It is also suggested that £ is of greater significance than *. There
is also the problem that the division sign assumes upon * and £ the
conditions of comparison and succession in number that I am still
trying to clear up.
It is not clear to me how the formation of fractions, if they are a
necessary condition for number formation, help us to establish
relationship between their signs.
You will have to explain further as I might have missed something.
JJ
Anonymouswrote:
jonescardiff@aol-dot-com.no-spam.invalid (John Jones) wrote:
I would not know what I meant by 'two' if I said that 'two is twice
(two) the size of one'. So, otherwise, what is 'two' compared to
'one'?
It is clear that numbers cannot be 'compared'.
[jillar] I would say they can be compared...'com-pared' as in
fractions.
For example, 1=1/1, 2=2/1.
If this is true - that 'numbers cannot be compared', how then are
numbers arranged?
JJ
[jillar] If integers were not convertible to fractions then they can
freely roam a range. But 1.5 cannot roam among 1 and 2 and be
considered
not to relate by size to the other two when they are converted into
fractions. So 1 and 2 must be fractured to understand this. 1=2/2,
1.5=3/2, and 2=4/2. Now we have relationship and 1.5 must be between,
not among, 1 and 2 and preceed 2. Wholeness is shifted down, down to
a
common denominator, but maintains wholeness and sets up comparative
analysis to the converted parts shifted above. Comparative evaluation
of
the numerator relates to quantity and therefore size.
Arrangement is relationship by size which is determined by the
fractured
parts of whole numbers that are placed into a common evaluation. The
number of fractured parts is determined by the common denominator and
the relationship of less to more fractured parts determines the
arrangement.
BTW this is a good case for the need to break or fracture whole
things
in order to get it to relate to order or arrangement. There is a
waving
away the wholeness of a thing when there is a relationship gained in
the
fracturing. But wholeness is never gone similar to the case of the
integer that shifts its extension of wholeness to the denominator
when
it is converted. If we omit fractions we omit relationships. Present
in
breaking and shifting wholeness is the possibility of new
relationships.
Jillar[/quote:50c1a1ef61]
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| User: "AE" |
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| Title: Re: The sequencing of Number |
12 Dec 2004 06:20:25 AM |
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John Jones schrieb:
The easiest way to test the idea that fractions can tell us how
numbers are related to each and how they can be compared is to
substitute meaningless signs for the numerals. By doing this it makes
it impossible to unconsciously compare numbers from merely seeing the
numerals. I will keep the division sign the same ("/").
So,
£/£ = £
*/£ = *
*/* = £
£ is the neutral element of multiplication.
I cannot conclude what £/* equals, except to say that it equals £/*.
Well - due to the equations you've listed above £/* equals the inverse of *.
It is also suggested that £ is of greater significance than *.
Why?
There
is also the problem that the division sign assumes upon * and £ the
conditions of comparison and succession in number that I am still
trying to clear up.
It is not clear to me how the formation of fractions, if they are a
necessary condition for number formation, help us to establish
relationship between their signs.
Fractions are not necessary for number formation. Indeed natural numbers
are working quite well without fractions.
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