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John Jones · Guest

Mitch Harris wrote:
[quote]On Oct 28, 3:30 pm, John Jones <jonescard...@aol.com> wrote:
Mitch Harris wrote:
On Oct 27, 5:26 pm, John Jones <jonescard...@aol.com> wrote:
2=sqrt(4) wouldn>t mean anything, unless sqrt4 is used to come up with
another number. Nobody wants sqrt4, a reference, for its own sake.
Maybe -you- don>t. What about sqrt(9801)? -Somebody- may want to refer
to that, use it, all for its own sake.
Sqrt.x where x is any number has no real-time, everyday significance. 2
has a significance.

Don>t you think that>s a bit short sighted, in both ways?
- sqrt(2) is significant as the length of the hypotenuse of a
isosceles right triangle. That>s pretty 'everyday', at least for
carpenters.
[/quote]
My point was only that carpenters, and anyone else for that matter,
don>t use things like sqrt. 2, or 4 times 2, etc, except to find a
number they can do something with.

[quote]- does 100 have everyday significance? not really that much, no one
ever -really- counts out 100 things (ecxpet when you>re 6 years old
and discover that it>s possible).
[/quote]
I would make a distiction between unwieldy numbers like 100, and
mysterious things like 10 times 10. Nobody wants 10 times 10.

[quote]That is why mathematics is constantly looking for
solutions. A solution is always a number and never a reference on how to
find a number.

I disagree. A really good solution might be a number but there are all
sorts of solutions (really good ones) that are functions or algorithms
that take variables.
[/quote]
The number may not be stated, but its the only thing people understand
and want. So while mathematicians may not be content with sqrt 4, it>s
OK as a surrogate 2.

[quote]An object that is given through a reference to itself has
no properties.
no -other- properties, at least none that have yet been specified.
I meant no properties at all. Anything measuring or viewing itself
measures and sees nothing.

Right. -You- meant that, and I meant that your desription of such an
object certainly does have at least one property, that of referring to
itself. You might also then mean that that is not a true property
('not a true Scotsman'), but I think it is a perfectly fine property.
[/quote]
I might call a 'particular' reference a property. But I can>t see any
reason for saying that selfreference is a property.

[quote]Acts of self-reference are not surveyable.
What does that mean? DO you have some sort of technical or otherwise
specific meaning for 'surveyable'?
An act of self-reference, like the brain viewing itself and eliminating
its material properties and gaining consciousness, is not reportable. A
report reports external properties.

Is that what 'surveyable' is supposed to mean? 'reportable'? 'has
external properties'? What does 'internal'/'external' property mean
here? Is this circular? (i.e. by definition 'surveyability' = 'non
self-referring'). What does surveyable or reportable mean without
reference to self-reference?

Mitch
[/quote]
Properties that are referenced or assigned to an object are surveyable
or examinable, of course, like height and weight, and we can report our
survey to others. Sometimes they are referred to as "external
properties". On the other hand, authors often call manifestations
arising from self-reference as "internal properties"'. However, this
stretches the definition of 'property', for anything that manifests in
self-reference is not surveyable like an external property is
surveyable. I gave the example of consciousness as an unsurveyable
manifestation. And this is true, as we know. I cannot look upon my own
consciousness, nor can anyone else - it is not surveyable or reportable.

John Jones · Guest

Dan Christensen wrote:
[quote]On Oct 28, 3:34 pm, John Jones <jonescard...@aol.com> wrote:
Dan Christensen wrote:
On Oct 25, 4:17 pm, John Jones <jonescard...@aol.com> wrote:
The = sign doesn>t guarantee unreserved symmetry between expressions. It
can be a reminder or sign that is appended to one expression to show
that that expression is referencing another expression.
1) Taking the expression sq.rt4 = 2. Sq.rt4 refers to 2. 2 does not
refer to anything. 2 is the object itself. It is presented, we don>t
need to refer to it, it is there in front of us to use as we wish.
2) A = A (or A is equal to itself) is another asymmetric use of the =
sign. An object that references itself is not the object referenced by
another. Object properties are given by reference, but not by
self-reference.
You are making this way too complicated with your "objects,"
"references," etc.
Formally, A=B just means that you can substitute string A for sting B,
and vice versa (taking into account any bracketing or precedence
conventions).
Yes, I know. It wasn>t me who extended that to make use of it as an
identity in A = A.

In your example, "sq.rt4 = 2" tells us that wherever we see "sq.rt4"
in a statement, we can substitute "2." And A=A is just a useful
identity. That>s how I look at it anyway.
I know that>s what Wittgenstein said, but there is a problem with that
idea.. a problem which he probably thought of. The substitution of signs
must be about a substitution of meaning. It can>t be about just
substituting signs. If it was, microscopic visual differences between 2
and 2 would make only one of them referenced by sqrt.4.- Hide quoted text -

In mathematics -- at least at a foundational level like this -- I
don>t find it is very useful to be talking about "meaning" in this
way. Hilbert, though talking specifically about geometry, said, “One
must be able to say at all times -- instead of points, straight lines,
and planes --tables, chairs, and beer mugs.” (I love that line!)
[/quote]
It>s a good line, but it doesn>t square with what you said next.
Obviously Hilbert did NOT think that mathematics should be based on pure
abstractions devoid of beer mugs etc.

[quote]He
argues that geometry, and I presume he meant all of mathematics,
should be based on pure abstractions completely devoid of any
"meaning" at this level.

As for microscopic, visual differences between 2 and 2, every formal
system begins with an "alphabet" of unambiguously distinct symbols
from which statements are constructed. I really don>t see any problem
with that idea.
[/quote]
I can>t see how you can avoid meaning. Mathematicians can>t make a rule
to avoid microscopic differences in 2, for it wouldn>t be clear which 2
they are referring to.

[quote]Dan
Download my DC Proof software at www.dcproof.com

[/quote]

Mitch Harris · Guest

On Oct 29, 3:47 pm, John Jones <jonescard...@aol.com> wrote:
[quote]Mitch Harris wrote:
On Oct 28, 3:30 pm, John Jones <jonescard...@aol.com> wrote:
Mitch Harris wrote:
On Oct 27, 5:26 pm, John Jones <jonescard...@aol.com> wrote:
2=sqrt(4) wouldn>t mean anything, unless sqrt4 is used to come up with
another number. Nobody wants sqrt4, a reference, for its own sake.
Maybe -you- don>t. What about sqrt(9801)? -Somebody- may want to refer
to that, use it, all for its own sake.
Sqrt.x where x is any number has no real-time, everyday significance. 2
has a significance.

Don>t you think that>s a bit short sighted, in both ways?
- sqrt(2) is significant as the length of the hypotenuse of a
isosceles right triangle. That>s pretty 'everyday', at least for
carpenters.

My point was only that carpenters, and anyone else for that matter,
don>t use things like sqrt. 2, or 4 times 2, etc, except to find a
number they can do something with.
[/quote]
and my point is that as far as everyday-ness goes, there>s nothing
special about a numeral (like '2' or '100', or a string of symbols
like '5+17' or 'sqrt(2)', one can use them as is, or one can try to
analyze them a bit further to get meaning that wasn>t already evoked
by the symbols. '100' is the length of a football field or better
analyzed 10 sets of ten. 'sqrt(2)' is at first that hypotenuse, or can
be further analyzed as '1.414..' (which can be further analyzed). '2'
is hard to analyze because it so immediately evokes things in us that
it is hard to do anything more.

So contrary to your point, you don>t always use a string of
mathematical symbols -just- 'to find a number they can do something
with'. You can use it as is, or maybe the number is actually too much
by itself and needs to be analyzed (and frankly there>s nothing
special about number...(there>s the idea that Kronecker was actually
kidding when he said 'God made the numbers, all else is the work of
man') number happens to be the thing they teach at school so we can
balance our checkbooks; there>s graphs, and logic, and geometric
figures, etc, etc, etc, which are all non-number things.).

[quote]- does 100 have everyday significance? not really that much, no one
ever -really- counts out 100 things (ecxpet when you>re 6 years old
and discover that it>s possible).

I would make a distiction between unwieldy numbers like 100, and
mysterious things like 10 times 10.
[/quote]
Multiplication is mysterious?

[quote]Nobody wants 10 times 10.
[/quote]
There>s no accounting for taste.

[quote]An object that is given through a reference to itself has
no properties.
no -other- properties, at least none that have yet been specified.
I meant no properties at all. Anything measuring or viewing itself
measures and sees nothing.

Right. -You- meant that, and I meant that your desription of such an
object certainly does have at least one property, that of referring to
itself. You might also then mean that that is not a true property
('not a true Scotsman'), but I think it is a perfectly fine property.

I might call a 'particular' reference a property. But I can>t see any
reason for saying that selfreference is a property.
[/quote]
'This sentence is false': the self-reference there is a glaring
property of that sentence (in addition to all the other properties it
has).

[quote]Acts of self-reference are not surveyable.
What does that mean? DO you have some sort of technical or otherwise
specific meaning for 'surveyable'?
An act of self-reference, like the brain viewing itself and eliminating
its material properties and gaining consciousness, is not reportable. A
report reports external properties.

Is that what 'surveyable' is supposed to mean? 'reportable'? 'has
external properties'? What does 'internal'/'external' property mean
here? Is this circular? (i.e. by definition 'surveyability' = 'non
self-referring'). What does surveyable or reportable mean without
reference to self-reference?

Properties that are referenced or assigned to an object are surveyable
or examinable, of course, like height and weight, and we can report our
survey to others. Sometimes they are referred to as "external
properties". On the other hand, authors often call manifestations
arising from self-reference as "internal properties"'.
[/quote]
OK.

[quote]However, this
stretches the definition of 'property', for anything that manifests in
self-reference is not surveyable like an external property is
surveyable. I gave the example of consciousness as an unsurveyable
manifestation. And this is true, as we know. I cannot look upon my own
consciousness, nor can anyone else - it is not surveyable or reportable.
[/quote]
That>s a matter of psychological research methods and the definition
of consciousness. Even animals are considered to have some semblance
of what we call consciousness. Our tests are getting cleverer.

Mitch

Dan Christensen · Guest

On Oct 29, 3:52 pm, John Jones <jonescard...@aol.com> wrote:
[quote]Dan Christensen wrote:
On Oct 28, 3:34 pm, John Jones <jonescard...@aol.com> wrote:
Dan Christensen wrote:
On Oct 25, 4:17 pm, John Jones <jonescard...@aol.com> wrote:
The = sign doesn>t guarantee unreserved symmetry between expressions. It
can be a reminder or sign that is appended to one expression to show
that that expression is referencing another expression.
1) Taking the expression sq.rt4 = 2. Sq.rt4 refers to 2. 2 does not
refer to anything. 2 is the object itself. It is presented, we don>t
need to refer to it, it is there in front of us to use as we wish.
2) A = A (or A is equal to itself) is another asymmetric use of the > >>>> sign. An object that references itself is not the object referenced by
another. Object properties are given by reference, but not by
self-reference.
You are making this way too complicated with your "objects,"
"references," etc.
Formally, A=B just means that you can substitute string A for sting B,
and vice versa (taking into account any bracketing or precedence
conventions).
Yes, I know. It wasn>t me who extended that to make use of it as an
identity in A = A.

In your example, "sq.rt4 = 2" tells us that wherever we see "sq.rt4"
in a statement, we can substitute "2." And A=A is just a useful
identity. That>s how I look at it anyway.
I know that>s what Wittgenstein said, but there is a problem with that
idea.. a problem which he probably thought of. The substitution of signs
must be about a substitution of meaning. It can>t be about just
substituting signs. If it was, microscopic visual differences between 2
and 2 would make only one of them referenced by sqrt.4.- Hide quoted text -

In mathematics -- at least at a foundational level like this -- I
don>t find it is very useful to be talking about "meaning" in this
way. Hilbert, though talking specifically about geometry, said, “One
must be able to say at all times -- instead of points, straight lines,
and planes --tables, chairs, and beer mugs.” (I love that line!)

It>s a good line, but it doesn>t square with what you said next.
Obviously Hilbert did NOT think that mathematics should be based on pure
abstractions devoid of beer mugs etc.

[snip][/quote]

From "Philosophy of Mathematics" at Wikipedia:

"Formalism holds that mathematical statements may be thought of as
statements about the consequences of certain string manipulation
rules. For example, in the "game" of Euclidean geometry (which is seen
as consisting of some strings called "axioms", and some "rules of
inference" to generate new strings from given ones), one can prove
that the Pythagorean theorem holds (that is, you can generate the
string corresponding to the Pythagorean theorem). Mathematical truths
are not about numbers and sets and triangles and the like — in fact,
they aren>t "about" anything at all!"

Where does "meaning" come into mathematics? If there is any, I think
it would be through informal interpretations or through applications
of these "mathematical truths." (Historically, of course, the
mathematical formalisms were developed centuries after their
"applications" were well established in practice!) The article
continues:

"Another version of formalism is often known as deductivism. In
deductivism, the Pythagorean theorem is not an absolute truth, but a
relative one: if you assign meaning to the strings in such a way that
the rules of the game become true (ie, true statements are assigned to
the axioms and the rules of inference are truth-preserving), then you
have to accept the theorem, or, rather, the interpretation you have
given it must be a true statement. The same is held to be true for all
other mathematical statements. Thus, formalism need not mean that
mathematics is nothing more than a meaningless symbolic game. It is
usually hoped that there exists some interpretation in which the rules
of the game hold.

"A major early proponent of formalism was David Hilbert, whose program
was intended to be a complete and consistent axiomatization of all of
mathematics...."

http://en.wikipedia.org/wiki/Formalism_(mathematics)#Formalism

I>m not sure precisely where your difficulty with equality (identity)
arises, but I suspect it is not from the formalism itself (as you seem
to be suggesting), but from your interpretation of it -- perhaps in
your use of ambiguous (though useful) terms like "object" and
"reference."

[quote]As for microscopic, visual differences between 2 and 2, every formal
system begins with an "alphabet" of unambiguously distinct symbols
from which statements are constructed. I really don>t see any problem
with that idea.

I can>t see how you can avoid meaning. Mathematicians can>t make a rule
to avoid microscopic differences in 2, for it wouldn>t be clear which 2
they are referring to.

[/quote]
Maybe it would help to think of the set of all 2>s as forming an a
kind of equivalence class, with each "2" being interchangeable for the
purposes of number theory. Just a thought.

Dan
Download my DC Proof software atwww.dcproof.com

John Jones · Guest

Mitch Harris wrote:
[quote]On Oct 29, 3:47 pm, John Jones <jonescard...@aol.com> wrote:
Mitch Harris wrote:
On Oct 28, 3:30 pm, John Jones <jonescard...@aol.com> wrote:
Mitch Harris wrote:
On Oct 27, 5:26 pm, John Jones <jonescard...@aol.com> wrote:
2=sqrt(4) wouldn>t mean anything, unless sqrt4 is used to come up with
another number. Nobody wants sqrt4, a reference, for its own sake.
Maybe -you- don>t. What about sqrt(9801)? -Somebody- may want to refer
to that, use it, all for its own sake.
Sqrt.x where x is any number has no real-time, everyday significance. 2
has a significance.
Don>t you think that>s a bit short sighted, in both ways?
- sqrt(2) is significant as the length of the hypotenuse of a
isosceles right triangle. That>s pretty 'everyday', at least for
carpenters.
My point was only that carpenters, and anyone else for that matter,
don>t use things like sqrt. 2, or 4 times 2, etc, except to find a
number they can do something with.

and my point is that as far as everyday-ness goes, there>s nothing
special about a numeral (like '2' or '100', or a string of symbols
like '5+17' or 'sqrt(2)', one can use them as is, or one can try to
analyze them a bit further to get meaning that wasn>t already evoked
by the symbols. '100' is the length of a football field or better
analyzed 10 sets of ten. 'sqrt(2)' is at first that hypotenuse, or can
be further analyzed as '1.414..'
[/quote]
I would like to say the number is 1.41 etc, and we can link that number
up to a hypotenuse, but we can>t meaningfully link it up to an abstract
way of getting that number like sqrt.2. Even 'times' is beyond us.

[quote](which can be further analyzed). '2'
is hard to analyze because it so immediately evokes things in us that
it is hard to do anything more.
[/quote]
2 is itself I like to think, the end-product, like two beers etc. But I
dont want the sqrt of beers or beers times another beer.

[quote]So contrary to your point, you don>t always use a string of
mathematical symbols -just- 'to find a number they can do something
with'. You can use it as is, or maybe the number is actually too much
by itself and needs to be analyzed (and frankly there>s nothing
special about number...(there>s the idea that Kronecker was actually
kidding when he said 'God made the numbers, all else is the work of
man') number happens to be the thing they teach at school so we can
balance our checkbooks; there>s graphs, and logic, and geometric
figures, etc, etc, etc, which are all non-number things.).

- does 100 have everyday significance? not really that much, no one
ever -really- counts out 100 things (ecxpet when you>re 6 years old
and discover that it>s possible).
I would make a distiction between unwieldy numbers like 100, and
mysterious things like 10 times 10.

Multiplication is mysterious?

Nobody wants 10 times 10.

There>s no accounting for taste.

I might call a 'particular' reference a property. But I can>t see any
reason for saying that selfreference is a property.

'This sentence is false': the self-reference there is a glaring
property of that sentence (in addition to all the other properties it
has).
[/quote]
'This sentence' if it is referring to itself is not surveyable. What we
survey is ourselves referencing 'this sentence' which is a place to read.

[quote]Properties that are referenced or assigned to an object are surveyable
or examinable, of course, like height and weight, and we can report our
survey to others. Sometimes they are referred to as "external
properties". On the other hand, authors often call manifestations
arising from self-reference as "internal properties"'.

OK.

However, this
stretches the definition of 'property', for anything that manifests in
self-reference is not surveyable like an external property is
surveyable. I gave the example of consciousness as an unsurveyable
manifestation. And this is true, as we know. I cannot look upon my own
consciousness, nor can anyone else - it is not surveyable or reportable.

That>s a matter of psychological research methods and the definition
of consciousness.
[/quote]
That>s the scientific belief, or dogma, - that internal properties are
externally assessable. Internal properties are just another
spatiotemporal description that we can sooner or later find out about.
But is that your experience? 'internal properties' properties of the
same koind? If they aren>t then psychology cannot possibly help.

[quote]Even animals are considered to have some semblance
of what we call consciousness. Our tests are getting cleverer.
[/quote]
There aren>t any tests for consciousness. As an 'internal' property it
is not surveyable.

Mitch Harris · Guest

On Oct 30, 5:29 pm, John Jones <jonescard...@aol.com> wrote:
[quote]Dan Christensen wrote:
Thus, formalism need not mean that
mathematics is nothing more than a meaningless symbolic game. It is
usually hoped that there exists some interpretation in which the rules
of the game hold.

I could not understand nmost of that. Besides which, I though rules were
stipulated and not 'true' or not.
[/quote]
Yes, one of the defining characteristics of mathematics is that the
meanings of named things are stipulated. But they can be considered
'true' or not (more words with stipulated meanings) for a given
context ('interpretation'). The rules and relations and axioms and
definitions are payed around with until they match closely what people
think informally, but then it is easier to manipulate the stipulated
objects.

Mitch

Mitch Harris · Guest

On Oct 30, 4:05 pm, John Jones <jonescard...@aol.com> wrote:
[quote]Mitch Harris wrote:
On Oct 29, 3:47 pm, John Jones <jonescard...@aol.com> wrote:
My point was only that carpenters, and anyone else for that matter,
don>t use things like sqrt. 2, or 4 times 2, etc, except to find a
number they can do something with.

and my point is that as far as everyday-ness goes, there>s nothing
special about a numeral (like '2' or '100', or a string of symbols
like '5+17' or 'sqrt(2)', one can use them as is, or one can try to
analyze them a bit further to get meaning that wasn>t already evoked
by the symbols. '100' is the length of a football field or better
analyzed 10 sets of ten. 'sqrt(2)' is at first that hypotenuse, or can
be further analyzed as '1.414..'

I would like to say the number is 1.41 etc, and we can link that number
up to a hypotenuse, but we can>t meaningfully link it up to an abstract
way of getting that number like sqrt.2.
[/quote]
What? Of course we can. How do you think that string of digits
"1.414..." can be come up with? Not by a ruler. But by a meaningful
link of 'sqrt' with a systematic procedure of some numerical
operations (see Newton>s Method).

[quote]Even 'times' is beyond us.
[/quote]
I don>t see how 'times' is beyond us.

[quote](which can be further analyzed). '2'
is hard to analyze because it so immediately evokes things in us that
it is hard to do anything more.

2 is itself I like to think, the end-product, like two beers etc.
[/quote]
I think you>re letting the superficial simplicity of 2 lead you to
think that it>s as far as you need to go (and also the direction to go
in). It is a very good thing to go to when possible, but sqrt(3) can
be the end or 1.732... could be, it depends.

[quote]That>s a matter of psychological research methods and the definition
of consciousness.

That>s the scientific belief, or dogma, - that internal properties are
externally assessable. Internal properties are just another
spatiotemporal description that we can sooner or later find out about.
But is that your experience? 'internal properties' properties of the
same koind? If they aren>t then psychology cannot possibly help.
[/quote]
I don>t know. I don>t think I can read other people>s minds directly.
But I may be dumb. I can>t feel the presence of other people by
electric charge, but an electric eel can.

[quote] > Even animals are considered to have some semblance
of what we call consciousness. Our tests are getting cleverer.

There aren>t any tests for consciousness. As an 'internal' property it
is not surveyable.
[/quote]
I don>t think you can claim that. It>s like saying there are only 7
planets because that>s all we can see.

How do you know there are 'atoms'? We don>t see them directly (I think
in your terms, they are not surveyable). But we see their effects.

Mitch

John Jones · Guest

Dan Christensen wrote:

[quote]
From "Philosophy of Mathematics" at Wikipedia:

"Formalism holds that mathematical statements may be thought of as
statements about the consequences of certain string manipulation
rules. For example, in the "game" of Euclidean geometry (which is seen
as consisting of some strings called "axioms", and some "rules of
inference" to generate new strings from given ones), one can prove
that the Pythagorean theorem holds (that is, you can generate the
string corresponding to the Pythagorean theorem). Mathematical truths
are not about numbers and sets and triangles and the like — in fact,
they aren>t "about" anything at all!"

Where does "meaning" come into mathematics? If there is any, I think
it would be through informal interpretations or through applications
of these "mathematical truths." (Historically, of course, the
mathematical formalisms were developed centuries after their
"applications" were well established in practice!) The article
continues:

"Another version of formalism is often known as deductivism. In
deductivism, the Pythagorean theorem is not an absolute truth, but a
relative one: if you assign meaning to the strings in such a way that
the rules of the game become true (ie, true statements are assigned to
the axioms and the rules of inference are truth-preserving), then you
have to accept the theorem, or, rather, the interpretation you have
given it must be a true statement. The same is held to be true for all
other mathematical statements. Thus, formalism need not mean that
mathematics is nothing more than a meaningless symbolic game. It is
usually hoped that there exists some interpretation in which the rules
of the game hold.
[/quote]
I could not understand nmost of that. Besides which, I though rules were
stipulated and not 'true' or not. Again, I can>t see what sort of thing
it is that the authors think can be 'interpreted' as a rule.

[quote]"A major early proponent of formalism was David Hilbert, whose program
was intended to be a complete and consistent axiomatization of all of
mathematics...."

http://en.wikipedia.org/wiki/Formalism_(mathematics)#Formalism

I>m not sure precisely where your difficulty with equality (identity)
arises, but I suspect it is not from the formalism itself (as you seem
to be suggesting), but from your interpretation of it -- perhaps in
your use of ambiguous (though useful) terms like "object" and
"reference."
[/quote]
1)Two things are equal when both parties are happy with their share.
2) Or, two things are equal when one of them references the other, like
sqrt4 references 2, or for Frege, the morning star references venus
(which to me is an unacceptable physicalism).
3) Two things are also equal when their lack of difference reduces to
one thing. In which case, we need to say why there are two. A =A says a
thing is identical to itself, that is, A, as it is referenced by us, is
also A when it is referenced by itself.

In the first of these cases there is symmetry of outcome, but the
outcome is not expressible by maths. There are no equivalences of
concept or quantity in the last two cases.

[quote]I can>t see how you can avoid meaning. Mathematicians can>t make a rule
to avoid microscopic differences in 2, for it wouldn>t be clear which 2
they are referring to.

Maybe it would help to think of the set of all 2>s as forming an a
kind of equivalence class, with each "2" being interchangeable for the
purposes of number theory. Just a thought.
[/quote]
Better to say perhaps that we can find a particular from the general
case. Like chihuahas can be identified from dogs. Finding that
particular is a job that maths can>t do. A chihuahua dog is an emergent
property from the general case of dog. Emergent properties inform
syntax, otherwise, we get the problems I alluded to. But emergent
properties are not expressible by mathematics.

[quote]Dan
Download my DC Proof software atwww.dcproof.com[/quote]

Dan Christensen · Guest

On Oct 30, 5:29 pm, John Jones <jonescard...@aol.com> wrote:
[quote]Dan Christensen wrote:

From "Philosophy of Mathematics" at Wikipedia:

"Formalism holds that mathematical statements may be thought of as
statements about the consequences of certain string manipulation
rules. For example, in the "game" of Euclidean geometry (which is seen
as consisting of some strings called "axioms", and some "rules of
inference" to generate new strings from given ones), one can prove
that the Pythagorean theorem holds (that is, you can generate the
string corresponding to the Pythagorean theorem). Mathematical truths
are not about numbers and sets and triangles and the like — in fact,
they aren>t "about" anything at all!"

Where does "meaning" come into mathematics? If there is any, I think
it would be through informal interpretations or through applications
of these "mathematical truths." (Historically, of course, the
mathematical formalisms were developed centuries after their
"applications" were well established in practice!) The article
continues:

"Another version of formalism is often known as deductivism. In
deductivism, the Pythagorean theorem is not an absolute truth, but a
relative one: if you assign meaning to the strings in such a way that
the rules of the game become true (ie, true statements are assigned to
the axioms and the rules of inference are truth-preserving), then you
have to accept the theorem, or, rather, the interpretation you have
given it must be a true statement. The same is held to be true for all
other mathematical statements. Thus, formalism need not mean that
mathematics is nothing more than a meaningless symbolic game. It is
usually hoped that there exists some interpretation in which the rules
of the game hold.

I could not understand nmost of that. Besides which, I though rules were
stipulated and not 'true' or not.
[/quote]
I think they mean the _interpretation_ of a rule is accepted as true,
not the rule itself.

[quote]Again, I can>t see what sort of thing
it is that the authors think can be 'interpreted' as a rule.

[/quote]
An equality / substitution rule as discussed above would be an
example: If A=B is an axiom or derived statement where A and B are
strings representing valid expressions, then wherever we see A in a
statement, we can substitute B (and vice versa), thus deriving another
statement.

[quote]"A major early proponent of formalism was David Hilbert, whose program
was intended to be a complete and consistent axiomatization of all of
mathematics...."

http://en.wikipedia.org/wiki/Formalism_(mathematics)#Formalism

I>m not sure precisely where your difficulty with equality (identity)
arises, but I suspect it is not from the formalism itself (as you seem
to be suggesting), but from your interpretation of it -- perhaps in
your use of ambiguous (though useful) terms like "object" and
"reference."

1)Two things are equal when both parties are happy with their share.
2) Or, two things are equal when one of them references the other, like
sqrt4 references 2, or for Frege, the morning star references venus
(which to me is an unacceptable physicalism).
3) Two things are also equal when their lack of difference reduces to
one thing. In which case, we need to say why there are two. A =A says a
thing is identical to itself, that is, A, as it is referenced by us, is
also A when it is referenced by itself.

[/quote]
If we have any equality statement A=B, then, by substitution, we can
derive the statement A=A. How can this be problematic? Is substitution
to be disallowed?

[quote]In the first of these cases there is symmetry of outcome, but the
outcome is not expressible by maths. There are no equivalences of
concept or quantity in the last two cases.

[/quote]
How about: two expressions are equal if the two expressions are
interchangeable everywhere in the system under consideration (e.g.
number theory)?

[quote]I can>t see how you can avoid meaning. Mathematicians can>t make a rule
to avoid microscopic differences in 2, for it wouldn>t be clear which 2
they are referring to.

Maybe it would help to think of the set of all 2>s as forming an a
kind of equivalence class, with each "2" being interchangeable for the
purposes of number theory. Just a thought.

Better to say perhaps that we can find a particular from the general
case. Like chihuahas can be identified from dogs. Finding that
particular is a job that maths can>t do.
[/quote]
All things equal to 2 are interchangeable in number theory. What is
wrong with that?

[quote]A chihuahua dog is an emergent
property from the general case of dog. Emergent properties inform
syntax, otherwise, we get the problems I alluded to. But emergent
properties are not expressible by mathematics.

[/quote]
Now, I am confused. Again, you seem to be making things unnecessarily
complicated.

Dan
Download my DC Proof software at www.dcproof.com

Dan Christensen · Guest

On Oct 30, 9:34 pm, Mitch Harris <maha...@gmail.com> wrote:
[quote]On Oct 30, 5:29 pm, John Jones <jonescard...@aol.com> wrote:

Dan Christensen wrote:
Thus, formalism need not mean that
mathematics is nothing more than a meaningless symbolic game. It is
usually hoped that there exists some interpretation in which the rules
of the game hold.

I could not understand nmost of that. Besides which, I though rules were
stipulated and not 'true' or not.

Yes, one of the defining characteristics of mathematics is that the
meanings of named things are stipulated.
[/quote]
Only informally. I agree with the rest of what you say.

Dan

[quote]But they can be considered
'true' or not (more words with stipulated meanings) for a given
context ('interpretation'). The rules and relations and axioms and
definitions are payed around with until they match closely what people
think informally, but then it is easier to manipulate the stipulated
objects.

Mitch[/quote]

Mitch Harris · Guest

On Oct 31, 12:18 am, Dan Christensen <Dan_Christen...@sympatico.ca>
wrote:
[quote]On Oct 30, 9:34 pm, Mitch Harris <maha...@gmail.com> wrote:

On Oct 30, 5:29 pm, John Jones <jonescard...@aol.com> wrote:

Dan Christensen wrote:
Thus, formalism need not mean that
mathematics is nothing more than a meaningless symbolic game. It is
usually hoped that there exists some interpretation in which the rules
of the game hold.

I could not understand nmost of that. Besides which, I though rules were
stipulated and not 'true' or not.

Yes, one of the defining characteristics of mathematics is that the
meanings of named things are stipulated.

Only informally. I agree with the rest of what you say.
[/quote]
I could hide behind 'informal', but instead I>ll, why do you think
'only informally'? I could just as well say that stipulation is one
part of the formalization process, and formalization itself is one of
the defining characteristics of mathematics. I wouldn>t think it
terrible if the latter statement can only be stated 'informally', but
I>m not understanding what>s so special about the restriction.

Mitch

Dan Christensen · Guest

On Oct 31, 10:42 am, Mitch Harris <maha...@gmail.com> wrote:
[quote]On Oct 31, 12:18 am, Dan Christensen <Dan_Christen...@sympatico.ca
wrote:

On Oct 30, 9:34 pm, Mitch Harris <maha...@gmail.com> wrote:

On Oct 30, 5:29 pm, John Jones <jonescard...@aol.com> wrote:

Dan Christensen wrote:
Thus, formalism need not mean that
mathematics is nothing more than a meaningless symbolic game. It is
usually hoped that there exists some interpretation in which the rules
of the game hold.

I could not understand nmost of that. Besides which, I though rules were
stipulated and not 'true' or not.

Yes, one of the defining characteristics of mathematics is that the
meanings of named things are stipulated.

Only informally. I agree with the rest of what you say.

I could hide behind 'informal', but instead I>ll, why do you think
'only informally'? I could just as well say that stipulation is one
part of the formalization process,
[/quote]
[snip]

How could it be? Formalization is the process of modelling a system
using strictly typograhical manipulation. It is a bit like writing a
computer program to simulate some process in the real world. In the
code of such a computer program, "meanings" are confined to a
commentary section that has no effect on the output of the system.
This is not to say that appending commentary (stipulating meanings) to
a program (formal system) is a pointless exercise. They are essential
to human understanding and enable programmers (mathematicians) to
eliminate errors (internal contradictions).

Problems in the commentary, however, do not necessarily point to
errors in the program code (formal system). And I think John>s
difficulty with the equality relation is probably just such a problem
-- a problem in his particular "commentary section," if you will.

Dan
Download my DC Proof software at www.dcproof.com

Mitch Harris · Guest

On Oct 31, 11:38 am, Dan Christensen <Dan_Christen...@sympatico.ca>
wrote:
[quote]On Oct 31, 10:42 am, Mitch Harris <maha...@gmail.com> wrote:

On Oct 31, 12:18 am, Dan Christensen <Dan_Christen...@sympatico.ca
wrote:

On Oct 30, 9:34 pm, Mitch Harris <maha...@gmail.com> wrote:

Yes, one of the defining characteristics of mathematics is that the
meanings of named things are stipulated.

Only informally. I agree with the rest of what you say.

I could hide behind 'informal', but instead I>ll, why do you think
'only informally'? I could just as well say that stipulation is one
part of the formalization process,

[snip]

How could it be? Formalization is the process of modelling a system
using strictly typograhical manipulation. It is a bit like writing a
computer program to simulate some process in the real world. In the
code of such a computer program, "meanings" are confined to a
commentary section that has no effect on the output of the system.
This is not to say that appending commentary (stipulating meanings) to
a program (formal system) is a pointless exercise. They are essential
to human understanding and enable programmers (mathematicians) to
eliminate errors (internal contradictions).
[/quote]
When said that way, how could stipulation -not- be part of
formalization? Formalization isn>t just randomly putting syntactic
elements together and manipulating them with random rules; there>s an
intended correspondence between the informal and the formal version.
Part of that is taking the informal language, say 'number', and
stipulating that in the formal version 'number' must mean an element
of Z.

[quote]Problems in the commentary, however, do not necessarily point to
errors in the program code (formal system). And I think John>s
difficulty with the equality relation is probably just such a problem
-- a problem in his particular "commentary section," if you will.
[/quote]
That metaphor works for me.

Mitch

Dan Christensen · Guest

On Oct 31, 2:37 pm, Mitch Harris <maha...@gmail.com> wrote:
[quote]On Oct 31, 11:38 am, Dan Christensen <Dan_Christen...@sympatico.ca
wrote:

On Oct 31, 10:42 am, Mitch Harris <maha...@gmail.com> wrote:

On Oct 31, 12:18 am, Dan Christensen <Dan_Christen...@sympatico.ca
wrote:

On Oct 30, 9:34 pm, Mitch Harris <maha...@gmail.com> wrote:

Yes, one of the defining characteristics of mathematics is that the
meanings of named things are stipulated.

Only informally. I agree with the rest of what you say.

I could hide behind 'informal', but instead I>ll, why do you think
'only informally'? I could just as well say that stipulation is one
part of the formalization process,

[snip]

How could it be? Formalization is the process of modelling a system
using strictly typographical manipulation. It is a bit like writing a
computer program to simulate some process in the real world. In the
code of such a computer program, "meanings" are confined to a
commentary section that has no effect on the output of the system.
This is not to say that appending commentary (stipulating meanings) to
a program (formal system) is a pointless exercise. They are essential
to human understanding and enable programmers (mathematicians) to
eliminate errors (internal contradictions).

When said that way, how could stipulation -not- be part of
formalization? Formalization isn>t just randomly putting syntactic
elements together and manipulating them with random rules; there>s an
intended correspondence between the informal and the formal version.
[/quote]
True. You might think of the informal system as part of the
"specifications" for the formal system. Once the formal system is
constructed, however, it takes on a life of its own that is quite
independent of the specifications or other documentation.

[quote]Part of that is taking the informal language, say 'number', and
stipulating that in the formal version 'number' must mean an element
of Z.

[/quote]
Again, the "meanings" are used for informal analysis only and should
be kept quite separate from the formal system. Otherwise, it is no
longer a _formal_ system. Like comments in the code for a computer
program, they should have no effect on functioning of the system. That
doesn>t mean the commentary is useless. Quite the contrary.

[quote]Problems in the commentary, however, do not necessarily point to
errors in the program code (formal system). And I think John>s
difficulty with the equality relation is probably just such a problem
-- a problem in his particular "commentary section," if you will.

That metaphor works for me.

[/quote]
Thank you.

Dan
Download my DC Proof software at www.dcproof.com

John Jones · Guest

Dan Christensen wrote:
[quote]On Oct 31, 2:37 pm, Mitch Harris <maha...@gmail.com> wrote:
On Oct 31, 11:38 am, Dan Christensen <Dan_Christen...@sympatico.ca
wrote:

On Oct 31, 10:42 am, Mitch Harris <maha...@gmail.com> wrote:
On Oct 31, 12:18 am, Dan Christensen <Dan_Christen...@sympatico.ca
wrote:
On Oct 30, 9:34 pm, Mitch Harris <maha...@gmail.com> wrote:
Yes, one of the defining characteristics of mathematics is that the
meanings of named things are stipulated.
Only informally. I agree with the rest of what you say.
I could hide behind 'informal', but instead I>ll, why do you think
'only informally'? I could just as well say that stipulation is one
part of the formalization process,
[snip]
How could it be? Formalization is the process of modelling a system
using strictly typographical manipulation. It is a bit like writing a
computer program to simulate some process in the real world. In the
code of such a computer program, "meanings" are confined to a
commentary section that has no effect on the output of the system.
This is not to say that appending commentary (stipulating meanings) to
a program (formal system) is a pointless exercise. They are essential
to human understanding and enable programmers (mathematicians) to
eliminate errors (internal contradictions).
When said that way, how could stipulation -not- be part of
formalization? Formalization isn>t just randomly putting syntactic
elements together and manipulating them with random rules; there>s an
intended correspondence between the informal and the formal version.

True. You might think of the informal system as part of the
"specifications" for the formal system. Once the formal system is
constructed, however, it takes on a life of its own that is quite
independent of the specifications or other documentation.

Part of that is taking the informal language, say 'number', and
stipulating that in the formal version 'number' must mean an element
of Z.

Again, the "meanings" are used for informal analysis only and should
be kept quite separate from the formal system. Otherwise, it is no
longer a _formal_ system. Like comments in the code for a computer
program, they should have no effect on functioning of the system. That
doesn>t mean the commentary is useless. Quite the contrary.

Problems in the commentary, however, do not necessarily point to
errors in the program code (formal system). And I think John>s
difficulty with the equality relation is probably just such a problem
-- a problem in his particular "commentary section," if you will.
That metaphor works for me.

Thank you.

Dan
Download my DC Proof software at www.dcproof.com
[/quote]
An interesting talk between you two. Not sure where to break in. But
this point seems pertinent. I put it as a question.

Are you placing too great a conceptual burden on 'formalisation? You
might indeed argue that formalization is syntax, pure and simple, and
that this places it beyond the ken of understanding (if there is any
other sort of 'ken'), beyond the commentary section.

But something I have only indicated, so far: that 'formalization' is
the commentary of object behaviours. The other commentary which you
refer to is the commentary of outcomes delivered by those behaviours.
The former is formative and the latter 'normative'.

Is there a gulf here between the formative and the normative? That is,
is there a gulf between what objects do for themselves and what we make
of the things that objects do? But the point is, is that the formative
is the vehicle for the normative, rather than the formative being
another distinct realm of understanding or 'ken' from the normative.

Kant declared that the formative and the normative were each nothing
without the other.


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