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John Jones · Guest

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

I think they mean the _interpretation_ of a rule is accepted as true,
not the rule itself.
[/quote]
Would the rule have been in some doubt, such that they had to interpret
it? Why not let the stipulation be as it was said?

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

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]

Yes, Ok. However, wouldn>t we be stipulating a substitution, and not
deriving it?

[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.

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]
I cannot see that. There is no A substituting itself.

If A is substituted for A, then we must have a means of distinguishing
between what is 'substituted' for the thing that substitutes it. But we
cannot make that distinction, we cannot distinguish between A and the
substituted A.

[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]
Symmetry distinguishes at least two things. But if we are given only A
then we cannot make that distinction.

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

OK. But interchangeability is no more than an expression of one thing in
a variety of contexts, and not an expression of two or more things in
one context.

[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.

All things equal to 2 are interchangeable in number theory. What is
wrong with that?
[/quote]
I would say that there are mo things or elements equal to two. Things
'reference' 2. They are not 'interchangeable' with it.

[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.

Now, I am confused. Again, you seem to be making things unnecessarily
complicated.
[/quote]
Yes, I am. I was indicating that syntax is given its meaning by a
commentary that falls outside mathematics. Assuming, which I don>t, that
syrastiline syntax in mathematics is perceived as essentially divorced
from meaning.

John Jones · Guest

Mitch Harris wrote:
[quote]On Oct 30, 4:05 pm, John Jones <jonescard...@aol.com> wrote:
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.

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]
I don>t see how a sqrt is meaningful, except as a tool to work out
something that ISis meaningful, like a number.

[quote]Even 'times' is beyond us.

I don>t see how 'times' is beyond us.
[/quote]
I can>t 'multiply' two boiled eggs. I can eat them.

[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.

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]
Nobody knows sqrt 3. I don>t mean the value of sqrt 3 - the value of
sqrt 3 is meaningful. I mean the 'operation' of sqrt>ing 3. It>s an
uncanny mystery.

[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.

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]
How would you know when you have been shown a demonstration of
consciousness? How would any physical signal show you consciousness.
Anyway, I used the brain as an example to ahow the difference between
external and 'internal' properties,

[quote]Even animals are considered to have some semblance
of what we call consciousness. Our tests are getting cleverer.
[/quote]
When were we informed by tests that something was conscious?

[quote]There aren>t any tests for consciousness. As an 'internal' property it
is not surveyable.

I don>t think you can claim that. It>s like saying there are only 7
planets because that>s all we can see.
[/quote]
It>s not like that. Planets fall within the same spatiotemporal
framework and share its properties. Mind does not. The properties of
their frameworks are different.

[quote]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
[/quote]
AS I>m saying, the spatiotemporal framework cannot encompass mind. In
the framework of mind, objects disappear. They don>t in spatiotemporality.

Mitch Harris · Guest

On Oct 31, 10:49 pm, John Jones <jonescard...@aol.com> wrote:
[quote]Mitch Harris wrote:
On Oct 30, 4:05 pm, John Jones <jonescard...@aol.com> wrote:
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.

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).

I don>t see how a sqrt is meaningful, except as a tool to work out
something that ISis meaningful, like a number.

Even 'times' is beyond us.

I don>t see how 'times' is beyond us.

I can>t 'multiply' two boiled eggs. I can eat them.
[/quote]
You>re being willfully obtuse. Of course you can>t multiply two boiled
eggs because it doesn>t make sense. If you have ten sets of 10 boiled
eggs (in each set) then you can, sensibly, feed 100 people one egg
each. The eggs themselves are not being multiplied, it>s the numbers
of eggs.

I have a feeling you>re saying that some mathematical operations are
'beyond us' (uninterpretable maybe?) because you put some mathematical
words together that don>t make sense. You can>t 'add' five boiled eggs
either. 'add' usually takes two parameters that are just numbers.
Those numbers might be the properties of sets of eggs or something
else, it doesn>t matter.

(for others of a mathematical bent, one can surely make meanings of
these sentences so they -do- make sense, but that is obviously not
what was intended)

[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.

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.

Nobody knows sqrt 3. I don>t mean the value of sqrt 3 - the value of
sqrt 3 is meaningful. I mean the 'operation' of sqrt>ing 3. It>s an
uncanny mystery.
[/quote]
Mystery? Botswana is a mystery to me too, I>ve never been there. But I
have a great suspicion that there are people there, and that those
people are pretty familiar with it.

[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.

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.

How would you know when you have been shown a demonstration of
consciousness? How would any physical signal show you consciousness.
Anyway, I used the brain as an example to ahow the difference between
external and 'internal' properties,
[/quote]
Demonstration of consciousness? How to show that somebody else thinks?
Talk to them? I know it>s a bit unscientific but it>s a first pass.

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

When were we informed by tests that something was conscious?
[/quote]
- I presume that you consider many other people to have consciousness.

- What makes you think that you have consciousness?

- The mirror test - put a mirror in front of an animal. Some animals
react to it in a way that is particularly reminiscent of how people do
it.

[quote]There aren>t any tests for consciousness. As an 'internal' property it
is not surveyable.

I don>t think you can claim that. It>s like saying there are only 7
planets because that>s all we can see.

It>s not like that. Planets fall within the same spatiotemporal
framework and share its properties. Mind does not. The properties of
their frameworks are different.
[/quote]
I understand that consciousness -feels- different than those other
things. The properties of one>s own mind does feel very different
from, say, planets. But what about someone else>s mind? All you know
about it is, to use your word, external data.

Mitch

Mitch Harris · Guest

On Oct 31, 10:35 pm, John Jones <jonescard...@aol.com> wrote:
[quote]Dan Christensen wrote:
Now, I am confused. Again, you seem to be making things unnecessarily
complicated.

Yes, I am. I was indicating that syntax is given its meaning by a
commentary that falls outside mathematics. Assuming, which I don>t, that
syrastiline syntax in mathematics is perceived as essentially divorced
from meaning.
[/quote]
What does 'syrastiline' mean (outside the current context)?

Mitch

Mitch Harris · Guest

On Oct 31, 5:40 pm, Dan Christensen <Dan_Christen...@sympatico.ca>
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.
....
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]
I don>t think, in the formalization process (taking an informal idea
and making it formal), that this 'commentary' is so much like
ignorable computer program comments. Or rather, there has to be some
explicit working connection between the two (the informal or formal),
the informal can>t be just ignored completely and thrown away.

At this point, I really feel I>m going down the road of flinging total
BS (if not already). So I>ll stop for the moment.

Mitch

Dan Christensen · Guest

On Nov 1, 10:31 am, Mitch Harris <maha...@gmail.com> wrote:
[quote]On Oct 31, 5:40 pm, Dan Christensen <Dan_Christen...@sympatico.ca
wrote:

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.
...
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.

I don>t think, in the formalization process (taking an informal idea
and making it formal), that this 'commentary' is so much like
ignorable computer program comments. Or rather, there has to be some
explicit working connection between the two (the informal or formal),
the informal can>t be just ignored completely and thrown away.

[/quote]
I have never suggested the informal be "thrown away." That would
require us to throw out most mathematical theory today! After all,
very few mathematicians actually work with truly formal systems.

In the event of a dispute in mathematical theory, one avenue of appeal
is, of course, to go back to first principles, to a formal system like
ZFC, but I am not aware of any such dispute in recent times. I would
think that such a dispute would have to be based on an apparent
internal contradiction in, say, number theory -- not on philosophical
hairsplitting on the use of words like "object" and "reference" as we
see here. (Sorry, John.)

Dan
Download my DC Proof software at http://www.dcproof.com

John Jones · Guest

Mitch Harris wrote:
[quote]On Oct 31, 10:35 pm, John Jones <jonescard...@aol.com> wrote:
Dan Christensen wrote:
Now, I am confused. Again, you seem to be making things unnecessarily
complicated.
Yes, I am. I was indicating that syntax is given its meaning by a
commentary that falls outside mathematics. Assuming, which I don>t, that
syrastiline syntax in mathematics is perceived as essentially divorced
from meaning.

What does 'syrastiline' mean (outside the current context)?

Mitch
[/quote]
Crystalline I meant to say. The idea, myth even, of syntax being
divorced from meaning in a pure realm of its own.

Jan Burse · Guest

John Jones schrieb:
[quote]Mitch Harris wrote:
On Oct 31, 10:35 pm, John Jones <jonescard...@aol.com> wrote:
Dan Christensen wrote:
Now, I am confused. Again, you seem to be making things unnecessarily
complicated.
Yes, I am. I was indicating that syntax is given its meaning by a
commentary that falls outside mathematics. Assuming, which I don>t, that
syrastiline syntax in mathematics is perceived as essentially divorced
from meaning.

What does 'syrastiline' mean (outside the current context)?

Mitch

Crystalline I meant to say. The idea, myth even, of syntax being
divorced from meaning in a pure realm of its own.
[/quote]
You can also view it as a marriage of two mathematical
abstract entities which are not found immediately
in natural language and thus there was never
a divorce of such concepts.

Look see, there are models of natural language processing
which consists of a multi-step pipeline. Here is my
favorite pipeline:

Step 1: Input: Sentence, Dictionary, Grammar
Output: Parse Tree
Process: Chart Parser
Step 2: Input: Parse Tree, Thesaurus
Output: Intralingua
Process: Substitution
Step 3: Input: Intralingua
Output: Normal Form
Process: Logical Inference

Mathematically Parse Tree, Intralingua and Normal Form
are all syntax. Mathematically the processes Chart Parser,
Substitution and Logical Inference can all be viewed
as meanings.

The better word than meaning would be semantics. How
can semantics define a process? Easy, semantics can
be viewed as defining a notion of thruth, thus we can
relate input/output via truth, and thus we have
processes.

The details are here:
- Chart Parser truth is that of being the parse of a
string of tokens. The dictionary and grammar serves as parsing
postulates. (Examples of this meaning interpretation of
parsing can be found plenty, see for example DCG)

- Substitution truth is that of being the intralingua
of a parse tree. The thesaurus serves as substitution
postulates. (Examples of this meaning interpretation
of parsing can be found as well plenty, see for example
lambda calculus)

- Logical Inference truth is that of being the normal
form of an intralingua. (Example of this meaning
interpretation of logical inference can be found as
well plenty, see for example computer algebra)

Now how can we counter your claim of mathematical syntax/
semantic doing harm to natural language? In two ways:

- First of all: You are true, the immediate relationship
between natural language and mathematical syntax/semantic
is a myth. Nobody ever claims that there is any direct
relationship.

The example pipeline above shows that mathematical syntax/
semantic might be the briks which is used to model
natural language processing steps. But the pipeline
also shows that there is not a trivial relationship
between natural language and any syntax/semantic.

On the contrary the relationship is very multi-layer
and syntax/semantics is needed in many flavors. On
request I can give you real world examples that show
that it is needed, assuming a certain target normal form.

- Second: Any model we create is just a model. We cannot
open the brain of someone and look for a normal form
or what ever. Even if we talk to someone he will answer
in natural language, so all we have is the surface.

Any model that goes below the surface, is created
by us, because we have some purpose in mind. It cannot
directly be validated. And also it might have some
rival pipeline models, which might eventually also
satisfy our purpose.

Even if we venture into the validation of such a model
and its parts we will face our limmited resources. How
long will it take to build the needed dictionary, grammar,
thesaurus? How will we vaidate its performances? If we
know the How Long and How, the next question will be the Who?
Who will do the Job?

But the Job will never be done by attacking innocent
mathematics notions like syntax/semantics. Instead of using
these notions in creating explanations of phaenomena.

That is what the purpose of mathematical logic is. To give
one a set of recurring basic tools. Which will hopefully enable
one to sally forth and dabble into more complex problems,
which are typically interdisciplinary.

Bye

Jan Burse · Guest

Dan Christensen schrieb:

[quote]I have never suggested the informal be "thrown away." That would
require us to throw out most mathematical theory today! After all,
very few mathematicians actually work with truly formal systems.

In the event of a dispute in mathematical theory, one avenue of appeal
is, of course, to go back to first principles, to a formal system like
ZFC, but I am not aware of any such dispute in recent times. I would
think that such a dispute would have to be based on an apparent
internal contradiction in, say, number theory -- not on philosophical
hairsplitting on the use of words like "object" and "reference" as we
see here. (Sorry, John.)

Dan
Download my DC Proof software at http://www.dcproof.com

[/quote]
Or instead of that things are getting worse in the future,
i.e. contradictions pop up, the contrary might happen.

Namely it could be that things get better in the future,
namely it could happen that for example more evidence of
the consistency of arithmetic might pop up.

I have recently heard that there is now a cut-elimination
for some proof logics (logics with a modal operator for
proofs). If I remember well, proof logics are able to
express Gödel>s incompletness.
http://www.springerlink.com/content/w1j82765h4h13686/
(There are also some more recent follow ups of
Valentini)

Where>s the catch??

Best Regards

Dan Christensen · Guest

On Nov 3, 2:09 pm, John Jones <jonescard...@aol.com> wrote:
[quote]Dan Christensen wrote:
On Nov 1, 10:31 am, Mitch Harris <maha...@gmail.com> wrote:
On Oct 31, 5:40 pm, Dan Christensen <Dan_Christen...@sympatico.ca
wrote:

I have never suggested the informal be "thrown away." That would
require us to throw out most mathematical theory today! After all,
very few mathematicians actually work with truly formal systems.

In the event of a dispute in mathematical theory, one avenue of appeal
is, of course, to go back to first principles, to a formal system like
ZFC, but I am not aware of any such dispute in recent times. I would
think that such a dispute would have to be based on an apparent
internal contradiction in, say, number theory -- not on philosophical
hairsplitting on the use of words like "object" and "reference" as we
see here. (Sorry, John.)

Dan
Download my DC Proof software athttp://www.dcproof.com

I think you would be out on your own to say that the foundation-building
of the logicians was irrelevant. Contradictions are found within a
system and are not especially informative without an examination of the
system>s foundations. All logicians work out the philosophy of their new
logical systems prior to assembling their structures, contradictions, etc
[/quote]
Are there currently any outstanding foundational problems (internal
contradictions, etc.) in set or number theory? I am not aware of any.
Much of what logicians do seems to me like a "solution" looking for a
problem. I stand to be corrected, of course, but they seem to be
largely still fighting a war against Russell>s Paradox that was won
almost a century ago to almost every one>s satisfaction. Don>t get me
wrong -- I myself have been dabbling in foundational issues as an
amateur for years and found it be great fun! Largely re-inventing the
wheel, I admit, I have been motivated more by esthetic and pedagogical
concerns than theoretical difficulties. (See the result at my
website.)

Dan
Download my DC Proof software at http://www.dcproof.com

John Jones · Guest

Mitch Harris wrote:
[quote]On Oct 31, 10:49 pm, John Jones <jonescard...@aol.com> wrote:
Mitch Harris wrote:
On Oct 30, 4:05 pm, John Jones <jonescard...@aol.com> wrote:
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.
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).
I don>t see how a sqrt is meaningful, except as a tool to work out
something that ISis meaningful, like a number.

Even 'times' is beyond us.
I don>t see how 'times' is beyond us.
I can>t 'multiply' two boiled eggs. I can eat them.

You>re being willfully obtuse. Of course you can>t multiply two boiled
eggs because it doesn>t make sense. If you have ten sets of 10 boiled
eggs (in each set) then you can, sensibly, feed 100 people one egg
each. The eggs themselves are not being multiplied, it>s the numbers
of eggs.
[/quote]
I can do that. Multiplication looks as if it is an act. And if acts are
done upon objects, and numbers are objects, then its as much sense to
say I can multiply two eggs by 10 as I can multiply 2 by 10. We have
just gotten used to the idea from school, though it never made sense, of
making objects perform these acts.

[quote]I have a feeling you>re saying that some mathematical operations are
'beyond us' (uninterpretable maybe?) because you put some mathematical
words together that don>t make sense. You can>t 'add' five boiled eggs
either. 'add' usually takes two parameters that are just numbers.
Those numbers might be the properties of sets of eggs or something
else, it doesn>t matter.
[/quote]
I don>t know where to begin to help this topic along and make
mathematical operations make sense. All I can say is that perhaps
pattern recognition has something to do with it. I know there are some
mathematicians who see the truths of mathematics as structural.

[quote](for others of a mathematical bent, one can surely make meanings of
these sentences so they -do- make sense, but that is obviously not
what was intended)
[/quote]
They make sense, but I think only through a veiled third agency. The
link between mathematics and what makes sense is achieved I believe
through a veil. THis veil called multiplication and other things, was
introduced at a very early age and it gains its power through
familiarity of use. Such familiarity, like following the rules, is not
the same as understanding of course. We are then trained to associate
the two.

[quote]Nobody knows sqrt 3. I don>t mean the value of sqrt 3 - the value of
sqrt 3 is meaningful. I mean the 'operation' of sqrt>ing 3. It>s an
uncanny mystery.

Mystery? Botswana is a mystery to me too, I>ve never been there. But I
have a great suspicion that there are people there, and that those
people are pretty familiar with it.
[/quote]
Sqrt works and I have been trained in its use. But if anyone asked me to
explain it I would do a bit of spluttering and then frantically reach
into my special bag of old tricks and hope to pull out something that
would make everyone feel ok about it. With any luck I could even
convince myself.

[quote]Demonstration of consciousness? How to show that somebody else thinks?
Talk to them? I know it>s a bit unscientific but it>s a first pass.
[/quote]
If you want a physical marker of consciousness then you would need more
than an impression that someone was conscious. I am saying that no such
physical marker can be had, not even in the simplest of cases. Machines
pick up light and weight but not 'consciousness'.

[quote]There aren>t any tests for consciousness. As an 'internal' property it
is not surveyable.
I don>t think you can claim that. It>s like saying there are only 7
planets because that>s all we can see.
It>s not like that. Planets fall within the same spatiotemporal
framework and share its properties. Mind does not. The properties of
their frameworks are different.

I understand that consciousness -feels- different than those other
things. The properties of one>s own mind does feel very different
from, say, planets. But what about someone else>s mind? All you know
about it is, to use your word, external data.
[/quote]
Other minds emerge as an emergent property from what we see others
saying and doing. What others say and do, do not of themselves summate
to an impression of other minds. We can take a step further and say that
all spatiotemporal objects are constructed and this would be true, for a
material world cannot itself place limits that tell us where an object
starts and an object ends.

[quote]Mitch
[/quote]
I hope to use the example of consciousness as an example of what has
been called elsewhere an 'internal property'. I have called it an act of
self-reference, though the act is not spatiotemporal.

'Internal properties' (like consciousness) are
Not surveyable or examinable, not countable, propertyless, they create
objects.

John Jones · Guest

Mitch Harris 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. 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]
I don>t want to agree to the idea that we can invent meanings. I hold
that there are no invented meanings and that all possible meanings are
already in place, and that these are what we work with.

John Jones · Guest

Dan Christensen wrote:
[quote]On Nov 1, 10:31 am, Mitch Harris <maha...@gmail.com> wrote:
On Oct 31, 5:40 pm, Dan Christensen <Dan_Christen...@sympatico.ca
wrote:

I have never suggested the informal be "thrown away." That would
require us to throw out most mathematical theory today! After all,
very few mathematicians actually work with truly formal systems.

In the event of a dispute in mathematical theory, one avenue of appeal
is, of course, to go back to first principles, to a formal system like
ZFC, but I am not aware of any such dispute in recent times. I would
think that such a dispute would have to be based on an apparent
internal contradiction in, say, number theory -- not on philosophical
hairsplitting on the use of words like "object" and "reference" as we
see here. (Sorry, John.)

Dan
Download my DC Proof software at http://www.dcproof.com

[/quote]
I think you would be out on your own to say that the foundation-building
of the logicians was irrelevant. Contradictions are found within a
system and are not especially informative without an examination of the
system>s foundations. All logicians work out the philosophy of their new
logical systems prior to assembling their structures, contradictions, etc.

John Jones · Guest

Jan Burse wrote:
[quote]John Jones schrieb:
Mitch Harris wrote:
On Oct 31, 10:35 pm, John Jones <jonescard...@aol.com> wrote:
Dan Christensen wrote:
Now, I am confused. Again, you seem to be making things unnecessarily
complicated.
Yes, I am. I was indicating that syntax is given its meaning by a
commentary that falls outside mathematics. Assuming, which I don>t,
that
syrastiline syntax in mathematics is perceived as essentially divorced
from meaning.

What does 'syrastiline' mean (outside the current context)?

Mitch

Crystalline I meant to say. The idea, myth even, of syntax being
divorced from meaning in a pure realm of its own.

You can also view it as a marriage of two mathematical
abstract entities which are not found immediately
in natural language and thus there was never
a divorce of such concepts.

Look see, there are models of natural language processing
which consists of a multi-step pipeline. Here is my
favorite pipeline:

Step 1: Input: Sentence, Dictionary, Grammar
Output: Parse Tree
Process: Chart Parser
Step 2: Input: Parse Tree, Thesaurus
Output: Intralingua
Process: Substitution
Step 3: Input: Intralingua
Output: Normal Form
Process: Logical Inference

Mathematically Parse Tree, Intralingua and Normal Form
are all syntax. Mathematically the processes Chart Parser,
Substitution and Logical Inference can all be viewed
as meanings.

The better word than meaning would be semantics. How
can semantics define a process? Easy, semantics can
be viewed as defining a notion of thruth, thus we can
relate input/output via truth, and thus we have
processes.

The details are here:
- Chart Parser truth is that of being the parse of a
string of tokens. The dictionary and grammar serves as parsing
postulates. (Examples of this meaning interpretation of
parsing can be found plenty, see for example DCG)

- Substitution truth is that of being the intralingua
of a parse tree. The thesaurus serves as substitution
postulates. (Examples of this meaning interpretation
of parsing can be found as well plenty, see for example
lambda calculus)

- Logical Inference truth is that of being the normal
form of an intralingua. (Example of this meaning
interpretation of logical inference can be found as
well plenty, see for example computer algebra)

Now how can we counter your claim of mathematical syntax/
semantic doing harm to natural language? In two ways:

- First of all: You are true, the immediate relationship
between natural language and mathematical syntax/semantic
is a myth. Nobody ever claims that there is any direct
relationship.

The example pipeline above shows that mathematical syntax/
semantic might be the briks which is used to model
natural language processing steps. But the pipeline
also shows that there is not a trivial relationship
between natural language and any syntax/semantic.

On the contrary the relationship is very multi-layer
and syntax/semantics is needed in many flavors. On
request I can give you real world examples that show
that it is needed, assuming a certain target normal form.

- Second: Any model we create is just a model. We cannot
open the brain of someone and look for a normal form
or what ever. Even if we talk to someone he will answer
in natural language, so all we have is the surface.

Any model that goes below the surface, is created
by us, because we have some purpose in mind. It cannot
directly be validated. And also it might have some
rival pipeline models, which might eventually also
satisfy our purpose.

Even if we venture into the validation of such a model
and its parts we will face our limmited resources. How
long will it take to build the needed dictionary, grammar,
thesaurus? How will we vaidate its performances? If we
know the How Long and How, the next question will be the Who?
Who will do the Job?

But the Job will never be done by attacking innocent
mathematics notions like syntax/semantics. Instead of using
these notions in creating explanations of phaenomena.

That is what the purpose of mathematical logic is. To give
one a set of recurring basic tools. Which will hopefully enable
one to sally forth and dabble into more complex problems,
which are typically interdisciplinary.

Bye

[/quote]
Here>s the deal. There are two things in this world - familiar objects
and names. There are no in-betweeny lands like 'abstractions', 'beneath
the surface>s. These are names, names for 'I have not understood'.

Mathematical syntax is about familiar objects behaviours. Logic is about
familiar object behaviours. There>s nothing else on offer in logic or
maths, nor is it possible, analytically, and contingently, that there
could be anything else on offer. That is why I feel comfortable in
writing what I write.

To say that 'there are things we don>t understand' is to utter a
contradiction.

Admittedly, we can throw together a system that we cannot make sense of.
But our 'failure' to make sense of it is because what we have built is
senseless, and not because of a failure, or a consequence of our
'limited' faculties.

John Jones · Guest

Dan Christensen wrote:
[quote]On Nov 3, 2:09 pm, John Jones <jonescard...@aol.com> wrote:
Dan Christensen wrote:
On Nov 1, 10:31 am, Mitch Harris <maha...@gmail.com> wrote:
On Oct 31, 5:40 pm, Dan Christensen <Dan_Christen...@sympatico.ca
wrote:
I have never suggested the informal be "thrown away." That would
require us to throw out most mathematical theory today! After all,
very few mathematicians actually work with truly formal systems.
In the event of a dispute in mathematical theory, one avenue of appeal
is, of course, to go back to first principles, to a formal system like
ZFC, but I am not aware of any such dispute in recent times. I would
think that such a dispute would have to be based on an apparent
internal contradiction in, say, number theory -- not on philosophical
hairsplitting on the use of words like "object" and "reference" as we
see here. (Sorry, John.)
Dan
Download my DC Proof software athttp://www.dcproof.com
I think you would be out on your own to say that the foundation-building
of the logicians was irrelevant. Contradictions are found within a
system and are not especially informative without an examination of the
system>s foundations. All logicians work out the philosophy of their new
logical systems prior to assembling their structures, contradictions, etc

Are there currently any outstanding foundational problems (internal
contradictions, etc.) in set or number theory? I am not aware of any.
[/quote]
I argued that syntax was related to commentary in that the former
mirrors object behaviours, and the latter describes objects themselves.
'Objects' in both cases are everyday empirical objects. So syntax
disallows appearance and disappearance of terms, for example. You know
the sort of objects/terms that commentary deals with.

[quote]Much of what logicians do seems to me like a "solution" looking for a
problem. I stand to be corrected, of course, but they seem to be
largely still fighting a war against Russell>s Paradox that was won
almost a century ago to almost every one>s satisfaction. Don>t get me
wrong -- I myself have been dabbling in foundational issues as an
amateur for years and found it be great fun! Largely re-inventing the
wheel, I admit, I have been motivated more by esthetic and pedagogical
concerns than theoretical difficulties. (See the result at my
website.)

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


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