Help. What is a model?
revo...@live.co.uk
Suppose I have a well defined formal system. That means is has a clearly defined set of symbols, axioms and rules of inference. For that system there is a set of well formed expressions of the system.
Now we suppose that there is a model of this system. As far as I can ascertain, any expression that is a valid expression of the formal system is also an expression of the model.
So my question is, what is the difference in the model and the system that it is a model of? If they are the same, I don’t see the point of having a model.
David C. Ullrich
>Can anyone help me with getting to grips with model theory and models of formal systems.
>I cannot seem to get a clear definition of what a model is.
Have you tried looking in a book on mathematical logic?
The actual _definition_ is a little technical. Here's an example:
First-order group theory is a formal system. It has axioms like
Ax Ay Az x(yz) = (xy)z
and then theorems like
Ax Ay Az (xy = xz -> y = z).
A _model_ of this formal system is a _group_.
>Suppose I have a well defined formal system. That means is has a clearly defined set of
>symbols, axioms and rules of inference. For that system there is a set of well formed expressions of the system.
>
>Now we suppose that there is a model of this system. As far as I can ascertain,
>any expression that is a valid expression of the formal system is also an expression of the model.
This doesn't make any sense at all. There's no such thing as "an
expression of the model". Continuing that example: There is such
a thing as an expression of group theory - two examples are
above. There's no such thing as "an expresion of a group".
>So my question is, what is the difference in the model and the system that it is a model of? If they are the same, I don�t see the point of having a model.
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
revo...@live.co.uk
You said
> This doesn't make any sense at all. There's no such
> thing as "an
> expression of the model". Continuing that example:
> There is such
> a thing as an expression of group theory - two
> examples are
> above. There's no such thing as "an expresion of a
> group".
I don't follow. If that's the case, how does the model refer to expressions of the formal system, such as in your example
Ax Ay Az x(yz) = (xy)z
or
Ax Ay Az (xy = xz -> y = z)?
I thought that these expressions could be true or false in the model. If they aren't expressions of the model how can they be true or false in the model. I'm afraid that I'm more confused than before.
A N Niel
<9709171.7462.12452426...@nitrogen.mathforum.org>,
<"revo...@live.co.uk"> wrote:
Let (G,*) be a group. We say the statement "Ax Ay Az x(yz) = (xy)z" is
true in (G,*) iff: for all a,b,c in G, a*(b*c) = (a*b)*c . In fact,
when I say G is a group, part of what I mean is exactly this.
Now consider statement "Ax Ay xy = yx" ... this is true for some groups
(called Abelian groups) and false for other groups.
David Bernier
A model of a formal theory is an interpretation where all theorems of
the formal theory come out true.
Many formal theories studied are cases of first order logic
theories, so F.O.L. seems to me important here:
http://en.wikipedia.org/wiki/First-order_logic
David Bernier
David C. Ullrich
>Yes, I have tried looking at books and articles on the subject. A lot of the definitions seem to be circular, so that you end up going back and forward trying to work out what it is all meant to mean.
>
>You said
>> This doesn't make any sense at all. There's no such
>> thing as "an
>> expression of the model". Continuing that example:
>> There is such
>> a thing as an expression of group theory - two
>> examples are
>> above. There's no such thing as "an expresion of a
>> group".
>
>I don't follow. If that's the case, how does the model refer to expressions of the formal system,
A model doesn't "refer to" anything.
>such as in your example
> Ax Ay Az x(yz) = (xy)z
> or
> Ax Ay Az (xy = xz -> y = z)?
>
>I thought that these expressions could be true or false in the model.
Yes.
>If they aren't expressions of the model how can they be true or false in the model.
Maybe the problem is the word "in" in "true in the model". Saying a
statement is "true in a model" doesn't say the expression itself is
"in" that model. "In" a group it may or may not be true that
xy = yx; that doesn't say that the expression "xy = yx" is "in"
that group.
(xy = yx is "true in the reals". It is not "in the reals"; nothing is
"in" the reals except numbers.)
>I'm afraid that I'm more confused than before.
You claimed to be confused about whether a model of a formal
system was the same as the formal system. Are you confused
about either of the following?
(i) A group is a model of group theory.
(ii) A group is not the same as group theory.
If you follow both of those statements then you're no longer
confused about whether a model of a system is the same as
the system.
MeAmI.org
The below is a model which David C. Ulrich agrees resolves [[P Versus
NP]]:
Bipolar junction transistor - Wikipedia, the free encyclopedia
Ebers–Moll Model for PNP Transistor. I_{\text{E}} = I_{\text{ES}: I_
{\text{C}} = \alpha_T I_{\text{. The base internal current is mainly
by diffusion (see ...http://en.wikipedia.org/wiki/
Bipolar_junction_transistor)
galathaea
wrote:
there are many kinds of mathematical "structures"
there are sets
and pointed sets
and magmas
and semigroups
and quasigroups
and groups
and rings
and ...
often these are expressed in some algebraic setting
like universal algebra
all of these structures
can be specified with a certain signature
that describes a functional ontology
whose roles are described by that structure
so a group
for instance
has a single binary product
and a single unary inverse
and a single nullary identity
once the idea of mathematical structure is understood
then we can define a model
a model M of a theory T
is a structure M for which there exists a mapping
sigma: T -> A
that obeys certain technical constraints
(to be precise
the mapping must "respect" the functional roles
which are defined in the theory
this amounts to the fact that the semantic functor
from the category of theories
to the category of structures
is a galois adjunction
among other things)
now
there may be many such mappings
in the example used elsewhere in this thread
a theory that describes "groups"
describes any group
so a particular model of the theory of groups
might be the 4-group
or the monster group
or some lie group
or ...
there are a number of reasons why this is useful
dealing with a concrete example of a theory
shows immediately that the theory is consistent
(this is godel's model-theoretic consistency theorem)
we know that a group is a consistent notion
because we can look at the 4-group directly and see it must be so
also
a model may be adjusted in various ways
that might be shown to still respect the theory
so one model can derive another model
which may have useful properties
this is effectively what "forcing" is
which is how the independence of
the axiom of choice
or the continuum hypothesis
were proven from models of ZF
on a more philosophical level
the association built
between a given theory and a specified ontology
is precisely the concept of "meaning"
we say the symbology "cats" _means_ actual cats we perceive
so this concept of interpretation or modelling
has been used as the foundation for theories of meaning
and has been formalised in computer science
and other formal language studies
..
model theory revolutionised metamathematics
almost immediately after it was introduced
by the 70s it was already becoming well incorporated
in computer science fields
it has lived on the edge of epistemology
and the philosophy of meaning
for quite some time
and one of the first uses tarski applied it to
was the philosophy of science
and the structure of scientific theory
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
Spiros Bousbouras
wrote:
> Yes, I have tried looking at books and articles on the subject.
> A lot of the definitions seem to be circular, so that you end up
> going back and forward trying to work out what it is all meant to
> mean.
It would be helpful if you concentrated on one book and asked here
at the earliest point when you don't understand something. "Logic
and structure" by Dirk van Dalen is fairly short and clear.
But for quicker feedback try
http://en.wikipedia.org/wiki/Structure_(mathematical_logic)
and tell us what is the first thing you don't understand.
--
Who's your mama?
MoeBlee
wrote:
> Yes, I have tried looking at books and articles on the subject. > A lot of the definitions seem to be circular, so that you end
> up going back and forward trying to work out what it is all
> meant to mean.
Try Enderton's 'A Mathematical Introduction To Logic'. Both his
technical specification of what a model is and his discsussion about
the subject are quite clear and not circular in the way you mention.
What books on the subject have you've found to suffer the fault you
describe.
MoeBlee
Edward Green
> On Wed, 17 Jun 2009 04:20:38 EDT, "revolt...@live.co.uk"
>
> <revolt...@live.co.uk> wrote:
> >Can anyone help me with getting to grips with model theory and models of formal systems.
> >I cannot seem to get a clear definition of what a model is.
>
> Have you tried looking in a book on mathematical logic?
>
> The actual _definition_ is a little technical. Here's an example:
>
> First-order group theory is a formal system. It has axioms like
>
> Ax Ay Az x(yz) = (xy)z
>
> and then theorems like
>
> Ax Ay Az (xy = xz -> y = z).
>
> A _model_ of this formal system is a _group_.
Wow. There's that mathematician physicist 180 thing working again! I
would have said that a group is a model of some physical system.
Well, I guess we can have it both ways.
To a physical scientist (we'll include _all_ the physical sciences by
courtesy in that remark, even, shudder, sociology) a model is any
logical structure which is used as a guide to the behavior of a
physical system. I never dreamed we could have "models" _within_
logic, with one abstraction modeling another.
<snip>
> "Understanding Godel isn't about following his formal proof.
> That would make a mockery of everything Godel was up to."
> (John Jones, "My talk about Godel to the post-grads."
> in sci.logic.)
Hmm...
MeAmI.org
Inverse (represent)
19 in the house!!!!
NP_for_free_Complete...
Free the "NP6"!!!!!
David C. Ullrich
<spamsp...@netzero.com> wrote:
>On Jun 17, 7:30�am, David C. Ullrich <dullr...@sprynet.com> wrote:
>> On Wed, 17 Jun 2009 04:20:38 EDT, "revolt...@live.co.uk"
>>
>> <revolt...@live.co.uk> wrote:
>> >Can anyone help me with getting to grips with model theory and models of formal systems.
>> >I cannot seem to get a clear definition of what a model is.
>>
>> Have you tried looking in a book on mathematical logic?
>>
>> The actual _definition_ is a little technical. Here's an example:
>>
>> First-order group theory is a formal system. It has axioms like
>>
>> � Ax Ay Az x(yz) = (xy)z
>>
>> and then theorems like
>>
>> � Ax Ay Az (xy = xz -> y = z).
>>
>> A _model_ of this formal system is a _group_.
>
>Wow. There's that mathematician physicist 180 thing working again! I
>would have said that a group is a model of some physical system.
That's a different sense of the word "model".
>Well, I guess we can have it both ways.
>
>To a physical scientist (we'll include _all_ the physical sciences by
>courtesy in that remark, even, shudder, sociology) a model is any
>logical structure which is used as a guide to the behavior of a
>physical system. I never dreamed we could have "models" _within_
>logic, with one abstraction modeling another.
Now you know. It's a perfectly standard thing, at the very
start of mathematical logic.
> <snip>
>
>> "Understanding Godel isn't about following his formal proof.
>> That would make a mockery of everything Godel was up to."
>> (John Jones, "My talk about Godel to the post-grads."
>> in sci.logic.)
>
>Hmm...
David C. Ullrich
David C. Ullrich
>Yes, I have tried looking at books and articles on the subject.
>A lot of the definitions seem to be circular, so that you end up going
>back and forward trying to work out what it is all meant to mean.
We can't help you with that unless you tell us exactly what
set of definitions you're referring to. If the definitions in
some book actually _are_ circular that would make it
a very bad book. But it seems more likely that they just
seem circular to you because you're missing something;
what that something would be is impossible to say
without more information.
>You said
>> This doesn't make any sense at all. There's no such
>> thing as "an
>> expression of the model". Continuing that example:
>> There is such
>> a thing as an expression of group theory - two
>> examples are
>> above. There's no such thing as "an expresion of a
>> group".
>
>I don't follow. If that's the case, how does the model refer to expressions of the formal system, such as in your example
> Ax Ay Az x(yz) = (xy)z
> or
> Ax Ay Az (xy = xz -> y = z)?
>
>I thought that these expressions could be true or false in the model. If they aren't expressions of the model how can they be true or false in the model. I'm afraid that I'm more confused than before.
David C. Ullrich
Michael Stemper
>On Wed, 17 Jun 2009 08:43:20 EDT, "revo...@live.co.uk" <revo...@live.co.uk> wrote:
>system was the same as the formal system. Are you confused
>about either of the following?
>
>(i) A group is a model of group theory.
I guess that it's just my day for dumb questions. I always thought that
a group was an object studied by group theory. Is this like saying that
a Camaro is a model of Chevy?
>(ii) A group is not the same as group theory.
This one is perfectly clear.
--
Michael F. Stemper
#include <Standard_Disclaimer>
If we aren't supposed to eat animals, why are they made from meat?
Virgil
David C. Ullrich <dull...@sprynet.com> wrote:
> On Wed, 17 Jun 2009 08:43:20 EDT, "revo...@live.co.uk"
> <revo...@live.co.uk> wrote:
>
> >Yes, I have tried looking at books and articles on the subject.
> >A lot of the definitions seem to be circular, so that you end up going
> >back and forward trying to work out what it is all meant to mean.
>
> We can't help you with that unless you tell us exactly what
> set of definitions you're referring to. If the definitions in
> some book actually _are_ circular that would make it
> a very bad book. But it seems more likely that they just
> seem circular to you because you're missing something;
> what that something would be is impossible to say
> without more information.
It is also possible to have the *appearance* of circularity if one is
working through two or more books. Or even through different editions of
the same book.
>
> >You said
> >> This doesn't make any sense at all. There's no such
> >> thing as "an
> >> expression of the model". Continuing that example:
> >> There is such
> >> a thing as an expression of group theory - two
> >> examples are
> >> above. There's no such thing as "an expresion of a
> >> group".
> >
> >I don't follow. If that's the case, how does the model refer to expressions
> >of the formal system, such as in your example
> > Ax Ay Az x(yz) = (xy)z
> > or
> > Ax Ay Az (xy = xz -> y = z)?
> >
> >I thought that these expressions could be true or false in the model. If
> >they aren't expressions of the model how can they be true or false in the
> >model. I'm afraid that I'm more confused than before.
>
> David C. Ullrich
>
> "Understanding Godel isn't about following his formal proof.
> That would make a mockery of everything Godel was up to."
> (John Jones, "My talk about Godel to the post-grads."
> in sci.logic.)
--
Virgil
KMF
> <9d3i35tbnpukus5j1...@4ax.com>, David
> C. Ullrich <dull...@sprynet.com> writes:
> >On Wed, 17 Jun 2009 08:43:20 EDT,
> "revo...@live.co.uk" <revo...@live.co.uk> wrote:
>
> >system was the same as the formal system. Are you
> confused
> >about either of the following?
> >
> >(i) A group is a model of group theory.
>
> I guess that it's just my day for dumb questions. I
> always thought that a group was an object studied by group theory. Is this like saying that
> a Camaro is a model of Chevy?
>
(Let e review my logic. Please feel free to jump
in and correct:)
Not really. A model is a structure (def. from logic)
that satisfies a collection of axioms. The axioms
stated are satisfied by a group.
A model deals with truth and meaning, i.e., semantics
assigned to formal axioms, which are syntactic, i.e.,
they consist of abstract relations. A structure
specifies the meaning to be assigned to the abstract
relations. When we define a group, the properties
of the group satisfy in a precise way the axioms of
a group (Tarski's def. )
Maybe re. your issue of cars, it may be a better
way of seeing it that a camaro is a model of a car:
you define the abstract propeties that an object
'car' satisfies, say, it has wheels, an engine, etc.
a camaro satisfies these properties, so it is
what you consider a car, i.e., it is a model of
the abstract concept of a car.
MoeBlee
> model
In mathematical logic, and concerning ordinary classical logic in
first order languages, there are different formalizations of the
notion of a model, but these formulizations end up providing the same
basic materials.
One particularly elegant formalization is in, e.g., Enderton's 'A
Mathematical Introduction To Logic'.
For a language (I'll use 'language' to stand for 'first order
language'), we say the set of parameters of the language is the set
whose members are all and only the universal quantifier (I'll use 'A')
and the predicate symbols of the language and the function symbols of
the language.
Then a structure for the language is a certain kind of function M
whose domain is the set of parameters of the language.
M('A') is some nonempty set that we call 'the universe of M'.
If 's' is an n-place function symbol of the language, then
M(s) is some n-place function on the universe of M. (For n=0, M(s) is
some member of the universe of M.)
If 's' is a 0-place predicate symbol, then
M(s) is 0 or M(s) is 1, depending on whether we wish 's' to stand for
"falsehood" or "truth", respectively.
If 's' is a 1-place predicate symbol, then
M(s) is a subset of the universe of M.
If 's' is an n-place predicate symbol, n>1, then
M(s) is an n-ary relation on the universe of M.
That defines 'M is a structure for (the language) L'.
Then, we apply some recursive definitions (in the manner of Tarksi's
formulation) to define '(The formula) P is satisfied by (the
structure) M and (assignment to the variables) v (per the language
L)'. (I'll leave off 'per the language L' in further formulations, as
it is to be understood that all such formulations regard some specific
language.)
And thus we also get a definition of '(The sentence) S is true in M'.
Then we say 'M is a model of (the set of formulas) G iff every member
of G is satisfied by M with every assignment v to the variables'. And
'M is a model of (the set of sentences) G iff every member of G is
true in M'.
Then it turns out that every structure is a model and every model is a
structure. (By definition every model is a structure; and that every
structure is a model follows from the fact that every structure is a
model of the set of logically valid sentences.)
Now, we can relax all that formalization too. For example, consider
the language of group theory (formulated with just two parameters: the
universal quantifier and a 2-place function symbol '+'). Then a model
M for this language just maps 'A' to some non-empty set X and '+' to
some 2-place function f on X. But <X f> might not be a group. However,
if every axiom of group theory is true in M, then <X f> is a group.
And, given that every axiom of group theory is true M, we may say in a
relaxed sense that <X f> is a model of group theory; and thus we may
say, in a relaxed sense, that every group is a model of group theory.
In other words, we may regard the model either, as strictly defined,
the FUNCTION {<'A' X> <'+' f>}, or as, simply <X f>.
MoeBlee
KMF
I understand that the concept of model can be formalized. I thought the poster I was replying
to needed/wanted an informal def. or a motivation
for the idea of model, and that, if my informal
layout was good-enough, then the formal and more
rigorous part would make more sense to him.
MoeBlee
> I understand that the concept of model can be formalized. I thought the poster I was replying
> to needed/wanted an informal def. or a motivation
> for the idea of model, and that, if my informal
> layout was good-enough, then the formal and more
> rigorous part would make more sense to him.
There are some parts of your informal remarks that really need to be
sharpened, even if not formalized.
MoeBlee
revo...@live.co.uk
David C. Ullrich said
> You claimed to be confused about whether a model of a
> formal
> system was the same as the formal system. Are you
> confused
> about either of the following?
>
> (i) A group is a model of group theory.
Here I presume you mean 'group theory' to mean a specific group theory rather than the study of groups. I don't know much about groups, so trying to explain models in terms of groups probably isn't the best way. I thought a group was a particular instance, such as the integers being a group and the results of certain operations such as addition and subtraction are also within the group. So when you say that a group is a model of group theory, that seems to be saying that a group like the integers is at the same time a particular group and also a model of some given group theory, but I don't know what you mean by that. Wouldn't that indicate that any given formal system is a model of some given theory, and so that the expressions of a formal system are expressions of that model?
> (ii) A group is not the same as group theory.
Well, assuming that a group is that which is defined by some group theory, yes, they are not the same, but I don't see how that helps
revo...@live.co.uk
> A model of a formal theory is an interpretation where
> all theorems of
> the formal theory come out true.
If that was the case wouldn't every model be inconsistent?
revo...@live.co.uk
> But for quicker feedback try
> http://en.wikipedia.org/wiki/Structure_(mathematical_l
> ogic)
> and tell us what is the first thing you don't
> understand.
Right. I have done. When I get to the interpretation function, I get:
The interpretation function assigns a function fA to every function f of the signature of the structure.
But it doesn't tell me anything about the function fA - except that it has the same arity as f.
Then it goes on: When a structure (and hence an interpretation function) is given by context, no notational distinction is made between a symbol s and its interpretation I(s). Whatever that is meant to mean, it seems to imply that there must be something definite about the interpretation function that makes it so – and if that is the case, why is the definition that makes it so not included?
revo...@live.co.uk
> a model M of a theory T
> is a structure M for which there exists a mapping
> sigma: T -> A
> that obeys certain technical constraints
> (to be precise
> the mapping must "respect" the functional roles
> which are defined in the theory
This is where I have a problem. That defines a model in terms of a structure. But that simply begs the question – what is a structure. Yes I have an intuitive notion of what I think it is supposed to be, but I find that unsatisfactory. And the same applies to a mapping that must "respect" the functional roles which are defined in the theory.
Surely, if the notion of a model is a precise notion, it should be possible to define it precisely rather than relying on everyone attaching the same meaning to somewhat vague statements.
MoeBlee
wrote:
> what is a structure. Yes I have an intuitive notion of what I think it
> is supposed to be, but I find that unsatisfactory.
> Surely, if the notion of a model is a precise notion, it should be
> possible to define it precisely
Please see my post today in which I give a quick description of a
precise definition, and in which I relate that precise definition also
to models in the sense of systems, for example, such systems <X f>
that are groups and how that also relates to first order group theory.
MoeBlee
KMF
Could you please elaborate?. I welcome input.
revo...@live.co.uk
That seems to be saying that a model is basically the same concept as Plato's concept of perfect forms, so that although there are myriads of real trees, the abstract concept of a tree encapsulates all of the characteristics of a real tree. But if that is the case, then it seems that a model of a formal system should be a particular formal system, while there should be a general concept of formal systems that is the abstract concept. But as far as I can see, a model is definitely not a particular formal system.
MoeBlee
Note: As per the reason given in my earlier post, I don't fear to use
'model' and 'structure' interchangably. But we must be careful NOT to
interchange the notion of a structure (model) FOR a LANGUAGE and model
OF a SET OF FORMULAS (such as a theory).
> A model deals with truth and meaning, i.e., semantics
> assigned to formal axioms, which are syntactic, i.e.,
> they consist of abstract relations.
The part about "abstract relations" doesn't shed any light.
And rather than say "a model deals with truth and meaning", it would
be better to say that the meanings of terms are determined by models
(through certain recursive definitions) and the satisfaction (per
assignments to the variables) and truth of formulas and sentences,
respectivelty, are determined by models (through certain recursive
defintions).
> A structure
> specifies the meaning to be assigned to the abstract
> relations.
No, a structure (through certain recursive definitions applied to the
structure) specifies the meaning assigned to the terms of the language
and the satisfaction (per assignments for the variables) or truth of
formulas or sentences, respectively. The structure ITSELF specifies a
non-empty universe and to each function or predicate symbol is
specified a function or relation, respectively, on the universe.
MoeBlee
revo...@live.co.uk
> Then a structure for the language is a certain kind
> of function M
> whose domain is the set of parameters of the
> language.
>
> M('A') is some nonempty set that we call 'the
> universe of M'.
>
> If 's' is an n-place function symbol of the language,
> then
>
> M(s) is some n-place function on the universe of M.
> (For n=0, M(s) is
> some member of the universe of M.)
>
> If 's' is a 0-place predicate symbol, then
>
> M(s) is 0 or M(s) is 1, depending on whether we wish
> 's' to stand for
> "falsehood" or "truth", respectively.
..
> And thus we also get a definition of '(The sentence)
> S is true in M'.
>
> Then we say 'M is a model of (the set of formulas) G
> iff every member
> of G is satisfied by M with every assignment v to the
> variables'. And
> 'M is a model of (the set of sentences) G iff every
> member of G is
> true in M'.
So we have a bijective function M whose domain is basically the symbols of some language, and whose range is called the universe of M. But if that is all the information that we have, how can we know whether M(s) = 1 or M(s) = 0, unless we have a clear definition of the function M? And if we don't have a clear definition of M, the how can we tell if every sentence of G is true in M?
If the above is right, then it seems to me that a model is simply that which is given by a certain type of function, and it seems to me that if that is the case, to define what a model is you just have to define the function M. So how is this function M defined? If that could be clearly set out, then surely we would have a clear definition of a model? Or am I missing something?
KMF
Well, consider the following to be temptative, and subject to corrections/improvements; I am trying to
help you as well as improve my own understanding.
Hopefully someone like Jack Markan can comment :
This is precisely the point: a model for a
formal system is _not_ a formal system. It is
a structure ( in the sense of logic), in which
the collection of axioms is realized in a precise
sense. As Jack said,( I believe; I apologize if not)
a model is a map between the formal system and the
structure, that satisfies certain properties,
A model is an instantation : a map between the
formal, syntactic world, where we can talk about
provability, but not about truth, to the world
of the semantic, where we can talk about truth.
Of course, this is not just any map.
Maybe a good example would be that of Euclidean
geometry, as given first by a collection of axioms.
We don't say that these axioms are true or not, we
talk about what we can prove within this collection
of axioms, together with some deductive rules;
some are general, i.e., logical, like Modus Ponens,
and some are non-logical in the sense that these
rules are intrinsic to the area, i.e., there are
some inference rules specific to each area.
A model for Euclidean geometry would be a
specific context that realizes these axioms.
We assign meaning to terms we use : what
do we mean by line, point, etc. Once we have
a notion of meaning, we can talk about truth.
A model for Euclidean geometry that does
not satisfy the parallel postulate is given
by a sphere, in which we consider lines as
great circles.
MoeBlee
wrote:
> So we have a bijective function M
No, I didn't say it has to be a bijection.
> whose domain is basically the symbols of some language, and whose range is called the universe of M.
No, the universe of M is not the range. The universe of M is the value
of the function at the argument 'A'.
> But if that is all the information that we have, how can we know whether M(s) = 1 or M(s) = 0, unless we have a clear definition of the function M?
In this sense, M is just a function like any other. What its values
are for given arguments depends on M. If "we" happen to explicilty
define M then we'll know its values at each argument.
> And if we don't have a clear definition of M, the how can we tell if
> every sentence of G is true in M?
You must distinguish between (1) what is the case and (2) what we know
("can tell") to be the case. What we know to be the case about the
truth of sentences in M depends on what we know about M, of course.
But another thing we need to consult is a recursive definition of
'true in M' that determines what sentences are true in M. Whether it
is CALCULABLE that a sentence is true or false in M may depend on M
(for example, if the universe of M is finite, then truth in M is
calculable, but we can't make that promise if the universe of M is
infinite).
> If the above is right, then it seems to me that a model is simply
> that which is given by a certain type of function,
Under the precise definition, the model IS the function. Then, as I
mentioned, in a more relaxed way of speaking, we may say that the
model is, as you said, "that which is given by the function", i.e, the
SYSTEM (i.e. a certain kind of TUPLE), such as <X f> in our exmaple.
In ordinary working situations, from the model as a function we can
determine the "that which is given by the model" (e.g. <X f>), and
from a system - in ordinary working situations - we can determine the
actual function that is the model.
> and it seems to me that if that is the case, to define what a model
> is you just have to define the function M.
Right. Or, we can take some system, such as <X f> and work backwards
to set a model such as {<'A' X> <'+' f>}.
> So how is this function M defined?
In most working cases, we define the function explicitly. But it is
not requried that we explicitly define. If set theory is the context
in which we're doing all this, to define, we need only prove the
existence of a unique M such that [some property here].
I really stress that the best way (aside from taking a class) to
understand this matter is to read a good textbook on the subject. It's
not practical to fill in all the technical stuff and all the needed
explanations in a few posts back and forth.
P.S. If the model (the actual function) is infinite, then the
resulting system won't be a tuple, but rather itself a function, such
as transfinite sequence.
To make all of this work fluidly, we use the notion of a signature of
a language, and signature types, and isomorphisms between signature
types. Then we get models with universal algebra and the whole thing
purring right along.
MoeBlee
MoeBlee
> Hopefully someone like Jack Markan can comment :
By the way, the name 'Jack Markan' is an old moniker I used only
briefly at one posting portal. Most peopleknow me instead as
'MoeBlee'. Also, there are people who know a lot more about this
subject, so I hope you'll have the benefit of their remarks too (as
indeed some have already posted in this thread).
> This is precisely the point: a model for a
> formal system is _not_ a formal system. It is
> a structure ( in the sense of logic), in which
> the collection of axioms is realized in a precise
> sense. As Jack said,( I believe; I apologize if not)
> a model is a map between the formal system and the
> structure, that satisfies certain properties,
No, that is not correct. The model is a map whose domain is the set of
parameters of the language. From that map we can also determine a
system in the sense of a tuple (such as algebraic systems) or a
transfinite sequence (in case the set of parameters is infinite).
> A model is an instantation : a map between the
> formal, syntactic world, where we can talk about
> provability, but not about truth, to the world
> of the semantic, where we can talk about truth.
> Of course, this is not just any map.
Well, something like that.
The model maps symbols of a language to certain objects. Then the
truth of sentences in a language per that model is determined by our
recursive defintions.
> Maybe a good example would be that of Euclidean
> geometry, as given first by a collection of axioms.
> We don't say that these axioms are true or not, we
> talk about what we can prove within this collection
> of axioms, together with some deductive rules;
> some are general, i.e., logical, like Modus Ponens,
> and some are non-logical in the sense that these
> rules are intrinsic to the area, i.e., there are
> some inference rules specific to each area.
Usually the rules are fixed and universal across all subject matters
but the non-logical axioms vary according to subject matter. (Though I
don't preclude that one might devise non-logical rules, just that it
is not as ordinary.)
> A model for Euclidean geometry would be a
> specific context that realizes these axioms.
> We assign meaning to terms we use : what
> do we mean by line, point, etc. Once we have
> a notion of meaning, we can talk about truth.
With a language whose primitives are 'point', 'line'. 'plane', first
let's imagine those are just nicknames for formal symbols such as
predicate symbols 'P', 'L', and 'N'. Then we would first specify some
non-empty set to be the universe U (i.e., M('A') = U). Then to the 1-
place predicate symbols 'P', 'L', and 'N', we would assign to each a
subset of U. If these assignments result in satisfaction of, e.g., the
incidence axioms of ordinary geometry, then we may say <U M('P') M
('L') M('N')> is a Euclidean incidence geometry.
MoeBlee
David C. Ullrich
>Thanks for all the posts so far. Progress still a bit slow, I'm afraid.
>
>David C. Ullrich said
>> You claimed to be confused about whether a model of a
>> formal
>> system was the same as the formal system. Are you
>> confused
>> about either of the following?
>>
>> (i) A group is a model of group theory.
>
>Here I presume you mean 'group theory' to mean a specific group theory rather than the study of groups.
By (first-order) group theory I mean the set of all logical
consequences of a certain set of axioms.
>I don't know much about groups, so trying to explain models in terms of groups probably isn't the best way.
>I thought a group was a particular instance, such as the integers being a group and the results of certain
>operations such as addition and subtraction are also within the group.
That's correct. Except that it's also required to satisfy certain
axioms, for example x(yz) = (xy)z for all x, y and z.
>So when you say that a group is a model of group theory, that seems to be saying
>that a group like the integers is at the same time a particular group and also a model
>of some given group theory,
A group is a model of _group theory_, not "some given group theory".
>but I don't know what you mean by that.
What I mean by that has been explained several times. For example, one
of the axioms of group theory is Ax Ay Az x(yz) = (xy)z. In order
to be a model of group theory, a structure must have the property that
in fact (xy)z = x(yz) for every x, y, and z in the structure.
>Wouldn't that indicate that any given formal system is a model of some given theory,
No! For example, group theory is a formal system. It consists of a
bunch of _sentences_, like Ax Ay Az x(yz) = (xy)z. The sentence
"Ax Ay Az x(yz) = (xy)z" is not an element of a group; group
theory is not a group.
Conversely, the integers (with addition as the operation) are a group.
2 is an integer. 3 is an integer. 2 is not a sentence. 3 is not a
sentence. 2 and 2 are elements of N, the set of integers; 2 and 3
are not elements of group theory.
>and so that the expressions of a formal system are expressions of that model?
There is no such thing as "an expression of a model".
It really seems as though you're not _trying_ to follow this. It's
already been explained that there's no such thing as "an expression
of a model". If you don't understand why not fine, ask. But
continuing to talk about "an expression of a model" after
the people you're asking to explain have explained that
there's no such thing makes very little sense.
Look. The set of integers, Z, is a group. (Or rather, the pair
(Z, +) is a group). There is no such thing as "an expression
of Z". The elements of Z are not expressions, they're numbers.
>> (ii) A group is not the same as group theory.
>
>Well, assuming that a group is that which is defined by some group theory,
>yes, they are not the same, but I don't see how that helps
You said you were confused over whether a model of a formal
system was the same thing as the formal system. One would
think that a counterexample would help. Group theory is
a formal system. A group is a model of group theory. A
group is not group theory. So a model of group theory
is not the same thing as group theory.
revo...@live.co.uk
So, in these terms of such a programmed computer, how would you describe the model? You say that the model is a function, and that can be explicitly defined, so that given the original computer program for the formal system, one would only need:
a) additional keys on the keyboard for the symbols of the function that are not symbols of the formal system
b) some additional programming.
Now, you say that the model function, for some formal system formula, will have one of two values, 0 or 1. But clearly, these values cannot be obtained from our computer program, not for every formula. For if that were the case, the problem of incompleteness of formal systems would be solved. So I don't see what benefit the model is in the examination of a formal system. It simply seems to be an additional layer of complexity with no raison d'etre.
revo...@live.co.uk
> A group is a model of _group theory_, not "some given
> group theory".
You may think that there can only be one 'group theory', but a theory is simply a man-made conception. There is nothing to stop conceiving a theory that they call a 'group theory'. It might be inconsistent, and you might not think it is a 'proper' group theory, but it can still be called a group theory, so that you cannot assume that the set of all group theories is a set with a singular element.
> There is no such thing as "an expression of a model".
>
> It really seems as though you're not _trying_ to
> follow this. It's
> already been explained that there's no such thing as
> "an expression
> of a model". If you don't understand why not fine,
> ask. But
> continuing to talk about "an expression of a model"
> after
> the people you're asking to explain have explained
> that
> there's no such thing makes very little sense.
That doesn't tally with what Jack Markam said. He said that a model is a function that can be explicitly stated. Therefore there is an expression that is that function, and when a variable of that expression is substituted, that also is an expression. So such expressions are clearly inherent aspects of the model. Denying that they 'belong' to the model seem to me to be the worst sort of semantical hair-splitting that hinder understanding rather that assist it.
> You said you were confused over whether a model of a
> formal
> system was the same thing as the formal system. One
> would
> think that a counterexample would help. Group theory
> is
> a formal system. A group is a model of group theory.
> A
> group is not group theory. So a model of group theory
> is not the same thing as group theory.
Your seem to think that I am either dim-witted or unwilling to try to understand what a model is. You can think that if you like, but you are making assumptions that make it difficult for me to know what you intend to mean. Simply referring to groups is not very helpful. Suppose that I showed an alien without legs from another solar system a chair and a table, and said,
"That is a chair, and that is a table - now you know what a chair is, because I've shown you an example."
But the alien still has no idea what the essential characteristics of a chair are. Does it have to be made out of metal and plastic? Does it have to have a flat horizontal area at half a metre from the ground? And so on.
Apparently, a model:
a) shares some, but not all of the properties of the formal system that it is a model of
b) has some properties that are not the properties of the formal system that it is a model of
Surely someone can clearly define these properties as in a) and b in terms that can apply to any formal system, so that we can have a clear definition of what a model actually is?
Aatu Koskensilta
> Surely someone can clearly define these properties as in a) and b in
> terms that can apply to any formal system, so that we can have a clear
> definition of what a model actually is?
Is there some reason you can't just look up the definition?
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Musatov
Dictionaries can be revised. Definitions depend on context.
In this sense a model is [...]: a new way to approach the P vs. NP
problem: storytelling.
"3SAT is Not Too Easy."
A Future Retrospective In Computational Complexity
Read Gödel's Lost Letter and P=NP. Also check out Iannis Tourlakis'
paper on extensions of these lower bound. What I find nice about this
type of proof is you get lower bounds by finding new algorithms.
How to Solve P=NP? Gödel's Lost Letter and P=NP was a good start but
we all guessed wrong. Gödel's Lost Letter and P=NP led to a discussion
and proof that \mathsf{NLOG} is closed under complement. Perhaps we
should put P=NP on theory exams and hope \dots [...].
I am serious. All the properties are easy to check. The main point is
if the machine guesses wrong during the computational complexity
theory, P = NP if and only if there exists a Turing machine T and a
natural number k such that <j>(ri) < nk, the question already
considered by Godel!
Computational Complexity: Gödel Prize 12 Apr 2006 [...] Even proving
it couldn't be done would solve the P/NP puzzle. [...] NP because it
seems very counterintuitive to say that we can check an exponential
number of [...]. Obviously it's not a formal proof, but it is an
intuitive coding.
Imagine some time in the future the problem has been solved and we
have read an article including an interview with the ingenious
programmer who solved the P vs. NP problem. [Note: I only use the word
"ingenious" here as it is cited in the P vs. NP problem description:
http://www.claymath.org/millennium/P_vs_NP/ from the Clay Mathematics
Institute: (excerpt )"However, this apparent difficulty may only
reflect the lack of ingenuity of your programmer."]
Imagine the article interviewing the programmer contained:
"Well, I thought I may not be able to prove P = NP by traditional
means, but I can certainly prove NP ! [....]. Hey Atwood, here's a
suggestion for your next article on Godel's Theorem. [.....] And Jeff
cashes yet another nice check from the controversy his blog has [....]
proof, establishing that 3-SAT could be reduced to 2SAT in O(n3)
time."
What can I say? It is the Power of randomness: Efficient Computation &
P vs. NP.
Gödel's letter to von Neumann [1954]: [...] Probabilistically
Checkable Proofs (PCPs). Claim: The Riemann Hypothesis. Prover:
(argument)[...] Every proof can be efficiently transformed to ZK proof
[...].
An Argument for P=NP: the winner needn't provide a constructive proof
that P=NP. 2. Despite Godel's [...] Godel, writing of course before
the modern P=?NP framework, inquires [.....]. By definition, a guess
for an NP problem is checkable in polynomial time. [...]
Imagine possibilities, conceive, wonder, speculate, discover!
Complex Multi-Tiered Abstracts:
Abstract: It is shown how to restrict Gödel's system $T$ of recursion
in all higher [...] Abstract: We discuss the forcing approach to P==NP
problem. [...] sound" proof-systems) showed how to efficiently check
proofs of [...].
Then mathematics, science, technology, innovation, and most
importantly advances in medicine, (still governed by fundamental
physics--bases of reality) become more like storytelling.
"Imagination is more important than intelligence." --Albert Einstein
"The Tale of NP-Completeness"
By [...]
1931 – Kurt Gödel introduces the incompleteness theorems [...]. First
Formal Proof that P NP. By giving a poly-time solution for the
matching search problem, [....] find a proof then check it? - [YES].
Is nature non deterministic? [YES].
A Short History of A Short History of Computational Complexity
By Martin Musatov
It was a dark night in Los Angeles. I was curled up on a futon at a
friend's downtown loft (4th & Broadway), writing a USENET post at 4
AM. "Man, they are going to think you're obsessed." I thought as I
poked at the tiny QWERTY keyboard of my BlackBerry.
--
Martin Musatov
http://MeAmI.org
"Better than Google alone, plus no ads."
Bacle
Maybe the best analogy for the relation between models
and formal systems is that between sentence logic and
truth tables. But of course, the notion of truth and
satisfiability are different, as are many other issues:
sentences are not just empty letters any more, but
the inner structure of the sentence now matters:
quantification, predicates, etc., instead of just
place-holders as in sentence logic.
David C. Ullrich
>David C. Ullrich said
>> A group is a model of _group theory_, not "some given
>> group theory".
>
>You may think that there can only be one 'group theory',
For heaven's sake. I said what I meant by "group theory".
There is only one formal system satisfying that definition.
>[...]
>Your seem to think that I am either dim-witted or unwilling to try to understand what a model is.
That's the way it's starting to appear, yes.
>You can think that if you like, but you are making assumptions that make it
>difficult for me to know what you intend to mean. Simply referring to groups is not very helpful. Suppose that I showed an alien without legs from another solar system a chair and a table, and said,
>"That is a chair, and that is a table - now you know what a chair is, because I've shown you an example."
I didn't claim to be giving a definition of "model". You've _seen_
prcise definitions, and they didn't seem to help, so I thought that
an example might server to clarify whatever you're misunderstanding.
>But the alien still has no idea what the essential characteristics of a chair are. Does it have to be made out of metal and plastic? Does it have to have a flat horizontal area at half a metre from the ground? And so on.
>
>Apparently, a model:
>a) shares some, but not all of the properties of the formal system that it is a model of
>b) has some properties that are not the properties of the formal system that it is a model of
>
>Surely someone can clearly define these properties as in a) and b in terms that can apply to any formal system,
>so that we can have a clear definition of what a model actually is?
There are clear definitions in plenty of books. The first time I said
that you claimed that the definitions in books were circular.
I asked you to explain what seemed circular so if, as seems likely,
the apparent circularity is just due to your misunderstanding,
we could explain. You never replied to that. That certainly makes
it seems as though you're not actually interested in getting this
straight.
MoeBlee
wrote:
> That doesn't tally with what Jack Markam said. He said that a model is a function that can be explicitly stated.
No, I said a model is a certain kind of function. Some models may be
explicitly stated, but I didn't claim every model can be explicitly
stated.
> so that we can have a clear definition of what a model actually is?
I already gave a precise definition of 'model', as well as I showed
how different senses of 'model' in mathematical logic boil down to
essentially the same.
MoeBlee
Bacle
p.80 of Enderton's book.?. I agree it has a nice treatment, and it is self-contained. To add to that,
at least two of us have the book, so it is easier
to answer your questions.
revo...@live.co.uk
> There are clear definitions in plenty of books. The
> first time I said
> that you claimed that the definitions in books were
> circular.
> I asked you to explain what seemed circular so if, as
> seems likely,
> the apparent circularity is just due to your
> misunderstanding,
> we could explain. You never replied to that. That
> certainly makes
> it seems as though you're not actually interested in
> getting this
> straight.
Models are often defined in terms of structures. But when you try and find the definition of a structure, that turns out to be a problem. It often turns out that a structure is defined in terms of a model. Re structures, I've already been asked (Spiros Bousbouras) to look at http://en.wikipedia.org/wiki/Mathematical_structure which directs to
http://en.wikipedia.org/wiki/Structure_(mathematical_logic) where it talks about an interpretation function. The problem is that the interpretation function assigns a function fA to every function f of the signature of the structure.
But it doesn't tell me anything about the function fA - except that it has the same arity as f.
Then it goes on: When a structure (and hence an interpretation function) is given by context, no notational distinction is made between a symbol s and its interpretation I(s).
That seems to imply that there must be something definite about the interpretation function that makes it so, but if that is the case, why is the definition that makes it so not included?
revo...@live.co.uk
> Maybe the best analogy for the relation between models
> and formal systems is that between sentence logic
> and truth tables. But of course, the notion of truth
> and satisfiability are different,
Isn't it the case that a formula is satisfiable if it is true, not satisfiable if it is false, and vice versa? So how is truth and satisfiability different?
> as are many other issues:
> sentences are not just empty letters any more, but
> the inner structure of the sentence now matters:
> quantification, predicates, etc., instead of just
> place-holders as in sentence logic.
That seems to be saying that the inner structure of the sentence 'matters' in the model but not in the formal system. But that isn't the case. The structure of the formulas of the formal system matter, otherwise we could not have a specification of what formulas were well-formed and which were not.
Bacle
revo...@live.co.uk
Right. So what it seems to boil down to is this. A model of a formal system is essentially a mapping from well-formed propositions of a formal system to truth values of true or false. Depending on what is being modelled, this mapping may be a purely hypothetical concept for which there is no possible precise definition that expresses that mapping. If that's the case, it seems to me that you cannot have a general definition of such mappings that is complete and entirely unambiguous. And that there cannot be a definition of a model that is complete and entirely unambiguous. So perhaps in looking for a clear definition of a model I have been trying to get a definition of the indefinable.
Musatov
MeAmI.org wrote:
Any system relies on the manipulation of variables over a defined set
of constants to extract meaning. This process is cyclical in nature.
When constants begin to slack or the systems we build and define as
truth building become succeptible to blatant manipulation these
systems must be overhauled and the set of constants and the entire
construct/manifold must be revised. Take the parsing of information
across USENET as an example. I am impacting a state of truth building
by posing true statements as truth. The system is not absolute. It is
relative. It is imperfect. It is flawed. It is the best we have but
perhaps others exist. Systems built on the discovery of true
conditions instead of the deduction of information from comparing it
to what we perceive as impossibility. To defend the current construct
with conviction and rigor is the opposite of illogical religious
censorship in the sinister sense. It is the "beast" those logical men
who wrote the bible speak. The enforcement of the status quo in the
eyes of the masses to proclaim it is the only logical system for
computation and reason. Truth remains many exist. Possibility does not
need recognition to exist. It exists waiting like a piece of fruit low
hanging or on the vine. To the one plucks it proclaims, "I am the
creator." And to this one another remains just out of reach oblivious
to its conception in the mind of a person.
Now I must apologize for waxing philosophical at length. I can only
pray the words fall on right ears and serve the purpose of their
intent and not my amateur attempt to string them along this narrative
stream.
MATH:
1. Can someone please explain how NP is the set of problems where
given a solution, you can check if all P problems are NP, but NP
problems aren't necessarily PCS?
What is the purpose of the {S}? So Dyson's wording doesn't pick out
the P vs. NP problem--again, unless SH: The "Truth vs. Higher Truth"
thread title reminds me? I am being serious here. If this is your
absolute truth do you see how flimsy it is?
P=NP complete___The Maximal. A. "Ain't got no human logic, just the
combinatoric arithmetic of basic fundamental P=NP, the Answer Is
Truth: A definitive proof of decidability and Truth.
At least to the Clay Mathematics Institute
Now, the P=NP? problem. Where were we? Oh, yes. Recall an algorithm is
a procedure for solving some [....] so u and v are mines -- and this
means U and V have truth-value T. [...]http://www.claymath.org/
Popular_Lectures/Minesweeper/
P vs NP A TM M is of time complexity T(n) if whenever M is given
anThm: If some NP-complete problem P is in P, then P = NP. PROOF:
Follows from the fact If φ is a Boolean expression, then a truth
assignment T is the set of all time-polynomial Turing machines isn't
recursive at all.
First of all: prove independence and you'll prove the truth of P<NP.
Variables not words (hint: the below set is an expansion if P<NP.):
Kids.Net.Au - Encyclopedia > Complexity classes P and NP
End set. Continue:
Although we don't know whether P=NP, we do know of problems outside
both P and NP. [...] The problem of deciding the truth of a statement
in Presburger [...]http://encyclopedia.kids.net.au/page/co/
Complexity_classes_P_and_NP
NP vs coNP exists a truth assignment T such that f(T) is true.
Clearly, any problem in P is in NP. But it may be that there are
problems in NP which are not in ...http://aleph.straylight.co.uk/
coNP.pdf
Plug-and-Play-HOWTO : Setting up a PnP BIOSIf you tell the BIOS you
don't have a PnP OS, then the BIOS will do the [...] If you intend on
fibbing, should you tell the truth about Windows or Linux? [...]http://
www.linuxselfhelp.com/HOWTO/Plug-and-Play-HOWTO-4.html
Computational Complexity: How to Prove NP Different from P11 Apr 2008
[...] This approach reduces various problems including P ≠ NP to hard
[....] I don't expect this will happen to computational complexity
theory, [...] P is not equal to NP, knowing its truth seems to add
very little. [...]
P equals NP, knowing its truth seems to add very much. [...]
See what I mean?
Musatov
MeAmI.org wrote:
Any system relies on the manipulation of variables over a defined set
David C. Ullrich
>David C. Ullrich said
>> There are clear definitions in plenty of books. The
>> first time I said
>> that you claimed that the definitions in books were
>> circular.
>> I asked you to explain what seemed circular so if, as
>> seems likely,
>> the apparent circularity is just due to your
>> misunderstanding,
>> we could explain. You never replied to that. That
>> certainly makes
>> it seems as though you're not actually interested in
>> getting this
>> straight.
>
>
>Models are often defined in terms of structures. But when you try and find the definition of a structure,
>that turns out to be a problem. It often turns out that a structure is defined in terms of a model.
Not in a competent exposition.
>Re structures, I've already been asked (Spiros Bousbouras) to look at http://en.wikipedia.org/wiki/Mathematical_structure which directs to
>http://en.wikipedia.org/wiki/Structure_(mathematical_logic) where it talks about an interpretation function.
>The problem is that the interpretation function assigns a function fA to every function f of the signature of the structure.
>But it doesn't tell me anything about the function fA - except that it has the same arity as f.
That's not a problem, that's the whole point. If f is a function
symbol in a certain formal language then the interpretation f^A
of f in a structure A _can_ be any function with the same arity.
This is simply not a problem. Consider the following
definition:
Def: n is an _even integer_ if f = 2k for some integer k.
Your complaint about the definition of "structure" is
like someone complaining that the definition of
"even integer" gives no information about the integer k.
It's not _supposed_ to give any such information,
other than that k is an integer! If k is _any_ integer
then 2k is an even integer.
>Then it goes on: When a structure (and hence an interpretation function) is given by context,
>no notational distinction is made between a symbol s and its interpretation I(s).
Sometimes there is no distinction made in the notation. There _should_
be, but people take shortcuts.
>That seems to imply that there must be something definite about the interpretation function that makes it so,
No, it doesn't _imply_ anything. In fact this only comes up when it is
clear _what_ interpretation is intended.
>but if that is the case, why is the definition that makes it so not included?
The fact (or assumption on the part of the author, which may be true
or false) that the author and the reader have the same interpretation
in mind.
People sometimes use informal/incomplete/bad notation. So what?
David C. Ullrich
>> On Jun 21, 2:09�am, "revolt...@live.co.uk"
>> <revolt...@live.co.uk>
>> wrote:
>>
>> > That doesn't tally with what Jack Markam said. He
>> said that a model is a function that can be
>> explicitly stated.
>>
>> No, I said a model is a certain kind of function.
>> Some models may be
>> explicitly stated, but I didn't claim every model can
>> be explicitly
>> stated.
>>
>> > so that we can have a clear definition of what a
>> model actually is?
>>
>> I already gave a precise definition of 'model', as
>> well as I showed
>> how different senses of 'model' in mathematical logic
>> boil down to
>> essentially the same.
>>
>> MoeBlee
>>
>
>Right. So what it seems to boil down to is this. A model of a formal system is essentially a mapping
>from well-formed propositions of a formal system to truth values of true or false.
That's not what a model is. A model of a thery gives rise to such a
mapping, yes.
> Depending on what
>is being modelled, this mapping may be a purely hypothetical concept for which there is no possible
>precise definition that expresses that mapping. If that's the case, it seems to me that you cannot have a
>general definition of such mappings that is complete and entirely unambiguous. And that there cannot
>be a definition of a model that is complete and entirely unambiguous. So perhaps in looking for a clear
>definition of a model I have been trying to get a definition of the indefinable.
This is simply wrong. There's nothing unambiguous about the definition
of "model". The definition of the word "model" is not the same as the
definition of a particular model.
One can define "real number" in various ways - here's one common one:
A real number is a set S of rationals such that S is nonempty, S is
not all the rationals, if p, q are rationals with p < q and p in S
then p is in S, and S has no largest element.
If that definition is new to you and you don't see why it's a sensible
definiton of "real number" never mind, that's not important right
now. It _is_ one of the standard ways of defining "real number".
Now, some real numbers are definable, and some are not.
The real numbers 0, 2, pi and sqrt(2) are examples of definable
real numbers. There are some real numbers that are _not_
definable. I can't give an explicit example, but there must
be non-definable reals because there are uncountably many
reals but only countably many definitions.
So: Some real numbers cannot be specified in finitely many
words - they are not "definable". Does that mean there's
something indefinite or fuzzy about the definition of
"real number"? No. I've defined _exactly_ what a real
number is, and the existence of reals I cannot define
has no bearing on that.
That undefiniable real _is_ a set S of rationals such that
[etc].
Bacle
Not so. All concepts in mathematics are clearly
defined. The fact that a map cannot be given explicitly
does not mean in this sense that it is not clearly-defined. You can define, e.g., a map f:R-->R that tells
you x-->sinx , without knowing what explicitly happenms to each term, meaning without a listing of all specific
elements and its respective images.
x in R
MeAmI.org
Let \mathit{COMPOSITE} = \{x\in N:x=pq and R = \{(x,y)\in N\times N: A
proof of P = NP could have stunning practical consequences, if the
proof leads Toward combinatorial proof of Toward combinatorial proof
of P < NP Toward combinatorial proof of. P < NP. P < NP. L. Gordeev.
Tübingen-Utrecht. June 2006 [...] of reals x, y x y 0 x y 0 x. ℝ y≠0
a∨b ,a∧b x. ℝ Algebraic versions of “P=NP?”X is NP. R if for all x ∈
R n. , x ∈ X ⇔ ∃y ∈ M p(n). 〈x,y〉 ∈ Y [....] NP. M . Proof sketch :
with access to an NP oracle, one can effectively “run” [...]http://
www.ens-lyon.fr/LIP/MC2/files/Pascal_Koiran.pdf
P-versus-NP page
This yields yet another proof that P=NP. The paper is available at
[...] R(x), treated as a Boolean function, is in the complexity class
NP, but not in P. [...]http://www.win.tue.nl/~gwoegi/P-versus-NP.htm
and proof Clique is NP-Complete L is in RP iff there is a language L'
in P: L' is a subset of {(x,r): [....] If the verifier in MIP does not
use randomness, then MIP-with-no-randomness = NP <...>http://
www.cs.unm.edu/~gemmell/CS500/Dec8.ppt
Computational complexity theory Show P = NP implies that given any
polynomial-time relation R< ...> it) we can construct a polynomial-
time TM G such that (3y.R(x,y)) => R(x,G(x))
http://books.google.com/books?isbn=082182872<b>X</b>Algebraic versions
of “P=NP?”〈x,y〉 ∈ Y with Y ∈ P. K . A typical NP. R. -complete
problem : <.....> Proof of (ii) based on ⊕P-completeness of
⊕HAMILTONIAN PATHS. http://www.univ-orleans.fr/lifo/Manifestations/MCU07/Talks/koiran.pdf
Require Export ZArith. Require Export List. Require Export Arith
<...>Lemma le_O_mult : forall n p:nat, 0*n <= 0*p. Proof. intros n p;
apply le_n. Qed. <....> (forall x:A, P x -> R)->R. Definition my_le (n
p:nat) := for all P:nat http://www.labri.fr/perso/casteran/CoqArt/everyday/SRC/chap5.v
Specifically, R consists of all pairs (x , y) such that y is an NP-
[...] R e PC c Pf implies S e P. In the second part of the proof, we
associate with each R http://books.google.com/books?isbn=052188473<b>X</b>The
Coq Standard Library | The Coq Proof Assistant(forall p, p < m -> P n
p) -> P n m) -> forall n m, P n m. Proof. intros P Hrec p; pattern p
in |- *; apply [...] exists! x, P x /\ forall x', P x' -> R x x'.
http://coq.inria.fr/distrib/V8.2rc1/stdlib/Coq.Arith.Wf_nat.html
Q.E.D.
Cure childhood cancer with http://MeAmI.org
MeAmI.org
> The Coq Standard Library Library Coq.Arith.Wf_nat Well-founded relations and natural numbers Require Import Lt.Open Local Scope nat_scope.Implicit Types m n p : nat.Section Well_founded_Nat.Variable A : Type.Variable f : A -> nat.Definition ltof (a b:A) := f a < f b.Definition gtof (a b:A) := f b > f a.Theorem well_founded_ltof : well_founded ltof.Proof. red in |- *. cut (forall n (a:A), f a < n -> Acc ltof a). intros H a; apply (H (S (f a))); auto with arith. induction n. intros; absurd (f a < 0); auto with arith. intros a ltSma. apply Acc_intro. unfold ltof in |- *; intros b ltfafb. apply IHn. apply lt_le_trans with (f a); auto with arith.Defined.Theorem well_founded_gtof : well_founded gtof.Proof. exact well_founded_ltof.Defined.It is possible to directly prove the induction principle going back to primitive recursion on natural numbers (induction_ltof1) or to use the previous lemmas to extract a program with a fixpoint (induction_ltof2) the ML-like program for induction_ltof1 is : let induction_ltof1 f F a = let rec indrec n k = match n with | O -> error | S m -> F k (indrec m) in indrec (f a + 1) a the ML-like program for induction_ltof2 is : let induction_ltof2 F a = indrec a where rec indrec a = F a indrec;; Theorem induction_ltof1 : forall P:A -> Set, (forall x:A, (forall y:A, ltof y x -> P y) -> P x) -> forall a:A, P a.Proof. intros P F; cut (forall n (a:A), f a < n -> P a). intros H a; apply (H (S (f a))); auto with arith. induction n. intros; absurd (f a < 0); auto with arith. intros a ltSma. apply F. unfold ltof in |- *; intros b ltfafb. apply IHn. apply lt_le_trans with (f a); auto with arith.Defined.Theorem induction_gtof1 : forall P:A -> Set, (forall x:A, (forall y:A, gtof y x -> P y) -> P x) -> forall a:A, P a.Proof. exact induction_ltof1.Defined.Theorem induction_ltof2 : forall P:A -> Set, (forall x:A, (forall y:A, ltof y x -> P y) -> P x) -> forall a:A, P a.Proof. exact (well_founded_induction well_founded_ltof).Defined.Theorem induction_gtof2 : forall P:A -> Set, (forall x:A, (forall y:A, gtof y x -> P y) -> P x) -> forall a:A, P a.Proof. exact induction_ltof2.Defined.If a relation R is compatible with lt i.e. if x R y => f(x) < f(y) then R is well-founded. Variable R : A -> A -> Prop.Hypothesis H_compat : forall x y:A, R x y -> f x < f y.Theorem well_founded_lt_compat : well_founded R.Proof. red in |- *. cut (forall n (a:A), f a < n -> Acc R a). intros H a; apply (H (S (f a))); auto with arith. induction n. intros; absurd (f a < 0); auto with arith. intros a ltSma. apply Acc_intro. intros b ltfafb. apply IHn. apply lt_le_trans with (f a); auto with arith.Defined.End Well_founded_Nat.Lemma lt_wf : well_founded lt.Proof. exact (well_founded_ltof nat (fun m => m)).Defined.Lemma lt_wf_rec1 : forall n (P:nat -> Set), (forall n, (forall m, m < n -> P m) -> P n) -> P n.Proof. exact (fun p P F => induction_ltof1 nat (fun m => m) P F p).Defined.Lemma lt_wf_rec : forall n (P:nat -> Set), (forall n, (forall m, m < n -> P m) -> P n) -> P n.Proof. exact (fun p P F => induction_ltof2 nat (fun m => m) P F p).Defined.Lemma lt_wf_ind : forall n (P:nat -> Prop), (forall n, (forall m, m < n -> P m) -> P n) -> P n.Proof. intro p; intros; elim (lt_wf p); auto with arith.Qed.Lemma gt_wf_rec : forall n (P:nat -> Set), (forall n, (forall m, n > m -> P m) -> P n) -> P n.Proof. exact lt_wf_rec.Defined.Lemma gt_wf_ind : forall n (P:nat -> Prop), (forall n, (forall m, n > m -> P m) -> P n) -> P n.Proof lt_wf_ind.Lemma lt_wf_double_rec : forall P:nat -> nat -> Set, (forall n m, (forall p q, p < n -> P p q) -> (forall p, p < m -> P n p) -> P n m) -> forall n m, P n m.Proof. intros P Hrec p; pattern p in |- *; apply lt_wf_rec. intros n H q; pattern q in |- *; apply lt_wf_rec; auto with arith.Defined.Lemma lt_wf_double_ind : forall P:nat -> nat -> Prop, (forall n m, (forall p (q:nat), p < n -> P p q) -> (forall p, p < m -> P n p) -> P n m) -> forall n m, P n m.Proof. intros P Hrec p; pattern p in |- *; apply lt_wf_ind. intros n H q; pattern q in |- *; apply lt_wf_ind; auto with arith.Qed.Hint Resolve lt_wf: arith.Hint Resolve well_founded_lt_compat: arith.Section LT_WF_REL. Variable A : Set. Variable R : A -> A -> Prop. Variable F : A -> nat -> Prop. Definition inv_lt_rel x y := exists2 n, F x n & (forall m, F y m -> n < m). Hypothesis F_compat : forall x y:A, R x y -> inv_lt_rel x y. Remark acc_lt_rel : forall x:A, (exists n, F x n) -> Acc R x. Proof. intros x [n fxn]; generalize dependent x. pattern n in |- *; apply lt_wf_ind; intros. constructor; intros. destruct (F_compat y x) as (x0,H1,H2); trivial. apply (H x0); auto. Qed. Theorem well_founded_inv_lt_rel_compat : well_founded R. Proof. constructor; intros. case (F_compat y a); trivial; intros. apply acc_lt_rel; trivial. exists x; trivial. Qed.End LT_WF_REL.Lemma well_founded_inv_rel_inv_lt_rel : forall (A:Set) (F:A -> nat -> Prop), well_founded (inv_lt_rel A F). intros; apply (well_founded_inv_lt_rel_compat A (inv_lt_rel A F) F); trivial.Qed.A constructive proof that any non empty decidable subset of natural numbers has a least element Set Implicit Arguments.Require Import Le.Require Import Compare_dec.Require Import Decidable.Definition has_unique_least_element (A:Type) (R:A->A->Prop) (P:A->Prop) := exists! x, P x /\ forall x', P x' -> R x x'.Lemma dec_inh_nat_subset_has_unique_least_element : forall P:nat->Prop, (forall n, P n \/ ~ P n) -> (exists n, P n) -> has_unique_least_element le P.Proof. intros P Pdec (n0,HPn0). assert (forall n, (exists n', n'<n /\ P n' /\ forall n'', P n'' -> n'<=n'') \/(forall n', P n' -> n<=n')). induction n. right. intros n' Hn'. apply le_O_n. destruct IHn. left; destruct H as (n', (Hlt', HPn')). exists n'; split. apply lt_S; assumption. assumption. destruct (Pdec n). left; exists n; split. apply lt_n_Sn. split; assumption. right. intros n' Hltn'. destruct (le_lt_eq_dec n n') as [Hltn|Heqn]. apply H; assumption. assumption. destruct H0. rewrite Heqn; assumption. destruct (H n0) as [(n,(Hltn,(Hmin,Huniqn)))|]; [exists n | exists n0]; repeat split; assumption || intros n' (HPn',Hminn'); apply le_antisym; auto.Qed.Unset Implicit Arguments.nth iteration of the function f Fixpoint iter_nat (n:nat) (A:Type) (f:A -> A) (x:A) {struct n} : A := match n with | O => x | S n' => f (iter_nat n' A f x) end.Theorem iter_nat_plus : forall (n m:nat) (A:Type) (f:A -> A) (x:A), iter_nat (n + m) A f x = iter_nat n A f (iter_nat m A f x).Proof. simple induction n; [ simpl in |- *; auto with arith | intros; simpl in |- *; apply f_equal with (f := f); apply H ].Qed.
Qed.
Bacle
Would you like it if I sprayed graffiti all over your
house?. If I trashed your webpage?. This is what
_you_ are doing. Don't fool yourself in your
self-absorption: you are nothing more than a vulgar
thug. Why don't you just fuck off and stop trashing
otehrs' posts, fucko?
MoeBlee
wrote:
> Models are often defined in terms of structures. But when you try
> and find the definition of a structure, that turns out to be a
> problem.
No, it doesn'! I gave you a rigorous definition, and I told you what
textbooks will fill in other details or explanation.
> It often turns out that a structure is defined in terms of a model.
You simply ignored my explanation about the relation between
'structure' and 'model'.
> The problem is that the interpretation function assigns a function
> fA to every function f of the signature of the structure.
> But it doesn't tell me anything about the function fA - except that
> it has the same arity as f.
Because that is ALL there IS to know.
First, let's go back to better notation:
For a function symbol 'f' of arity n, with a model (i.e., a strucure,
i.e., an interepretation function) we have M('f') is an n-place
function on the universe of M.
What SPECIFIC function M('f') is or what other prorperties M('f') has
depend simply on what M assigns to 'f'. Each model M may assign a
different function to 'f'.
> Then it goes on: When a structure (and hence an interpretation
> function) is given by context, no notational distinction is made
> between a symbol s and its interpretation I(s).
Who wrote that? Forget about it (at least for now). It seems to be
mentioning some kind of informal notational convention. We don't need
it; especially if it is confusing you.
Come on, I gave you a rigorous definition, and I told you some books
that you can read. If you're just going to ignore all of that, then OF
COURSE, you'll just keep coming back claiming that it's not been
defined and explained for you.
MoeBlee
MoeBlee
wrote:
> how is truth and satisfiability different?
I covered that! You simply ignored it.
Satisfiability pertains to open formulas in general (formulas that
have one or more free variable). Truth pertains to sentences (formulas
that have no free variables.
> That seems to be saying that the inner structure of the sentence
> 'matters' in the model but not in the formal system.
No, the structure of formulas matters SYNTACTICALLY in the formal
system.
> The structure of the formulas of the formal system matter, otherwise
> we could not have a specification of what formulas were well-formed
> and which were not.
Quite so!
MoeBlee
MoeBlee
wrote:
> A model of a formal system is essentially a mapping from well-formed
> propositions of a formal system to truth values of true or false.
NO! NO! NO!
I already explained this. Why do you just ignore what people have
taken the time to explain to you?
The model is a mapping from a certain subset of the set of SYMBOLS.
THEN there is ANOTHER mapping (based on the model) from the set of
FORMULAS. The model is NOT a mapping from the set of formulas. Rather,
it is ANOTHER mapping that we BUILD from the model that is a mapping
from the set of formulas.
MoeBlee
revo...@live.co.uk
> >> <revolt...@live.co.uk>
> >> wrote:
> >Right. So what it seems to boil down to is this. A
> model of a formal system is essentially a mapping
> >from well-formed propositions of a formal system to
> truth values of true or false.
David Ulrich wrote
> That's not what a model is. A model of a theory gives
> rise to such a mapping, yes.
Jack Markam said that a model is a structure, and a structure is a model, and the structure for a formal language is a mapping function.
So who is right, you or Jack Markam? And why?
I think you are completely missing the point here. One can indeed define real numbers in different ways, but the proof that there does not exist any mapping of the naturals to the reals depends on real numbers being specifically defined as infinite sums of rationals, such as in decimal notation. Hence the proof itself specifies that real numbers are given by a function of infinite arity. To get any one real number, the value of every variable of the function must be specified.
I don't see what that has to do with the concept of a function that maps propositions of a formal system to truth values of true or false. There does not appear to be any similar definition for this hypothetical function, whereas the definition of a function that gives every real number is defined. There is no rationale to suggest that it this hypothetical function is a function of infinite arity, where for particular values of the variables, the function gives one of two values, true or false.
Aatu Koskensilta
> Would you like it if I sprayed graffiti all over your house?. If I
> trashed your webpage?. This is what _you_ are doing. Don't fool
> yourself in your self-absorption: you are nothing more than a
> vulgar thug. Why don't you just fuck off and stop trashing otehrs'
> posts, fucko?
Would you like it if I painted "Please don't spray graffiti all over
other people's houses" all over your house?
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, dar�ber muss man schweigen"
David C. Ullrich
_A_ proof of the uncountability of the reals depends on this.
>Hence the proof itself specifies that real numbers are given by a function of infinite arity.
>To get any one real number, the value of every variable of the function must be specified.
It's not as essential as you think, but never mind that. Let's say a
real number is an infinite decimal (with some clause about how
sometimes two different decimal expansions give the same real,
of course).
So now there is a function mapping sequences of digits onto
the real numbers. Great. Let's call that function F, so
F(1,0,0,0,...) = .1 = 1/10. Great.
>I don't see what that has to do with the concept of a function that maps propositions of a formal
>system to truth values of true or false. There does not appear to be any similar definition for this
>hypothetical function, whereas the definition of a function that gives every real number is defined.
Yes, F is perfectly well defined. But that's not relevant. With our
new definition of "real number" what I said above becomes this:
There _is_ a precise definition of "model", as a certain function.
The definition does not specify what function that should be.
So what? It's precisely the same to say that there is a precise
definition of "real number", namely "a real number is
S(d1, d2, ...) for some sequence of digits d_1, d_2, ...".
Your complaint about the definition of "model" not
specifying what function we're talking about makes
exactly as much sense as complaining that the definition
of "real number" does not specify what sequence d_1, ...
we're talking about.
>There is no rationale to suggest that it this hypothetical function is a function of infinite arity,
>where for particular values of the variables, the function gives one of two values, true or false.
David C. Ullrich
MoeBlee
wrote:
> > >> On Jun 21, 2:09 am, "revolt...@live.co.uk"
> > >> <revolt...@live.co.uk>
> > >> wrote:
> > >Right. So what it seems to boil down to is this. A
> > model of a formal system is essentially a mapping
> > >from well-formed propositions of a formal system to
> > truth values of true or false.
>
> David Ulrich wrote
>
> > That's not what a model is. A model of a theory gives
> > rise to such a mapping, yes.
>
> Jack Markam said that a model is a structure, and a structure is a
> model, and the structure for a formal language is a mapping
< function.
>
> So who is right, you or Jack Markam? And why?
David and I are not contradicting each another. We're both right. I've
already explained in detail. Why are you just ignoring the information
I posted for your benefit?
By the way, I didn't use the expression "mapping function". It's
redundant. A function is a mapping; a mapping is a function.
A structure is a certain kind of function. We say that a structure is
a structure FOR a LANGUAGE. Then, given a particular structure, we
define another function, which is a function from the set of sentences
into {0 1} (for false, true, respectively). Then we say a structure is
a model OF a theory iff every sentence in the theory evaluates to true
per the function on sentences that we defined per the structure. So
every model is a structure. And every structure is a model since every
structure is a model of the set of valid (logically true) sentences.
Please, why don't you just get a good book on mathematical logic where
this is explained in more detail and with more systematic pedagogical
wisdom?
> One can indeed
> define real numbers in different ways, but the proof that there
> does not exist any mapping of the naturals to the reals depends on
> real numbers being specifically defined as infinite sums of
> rationals,
> such as in decimal notation.
WRONG. (1) We don't (ordinarily) define reals to be such "infinite
sums". (2) The proof of uncountability depends only on the reals being
a carrier set for a complete ordered field. WHATEVER definition of 'is
a real number', as long as the set of them is a carrier set of a
complete ordered field, that set is uncountable.
> I don't see what that has to do with the concept of a function that
> maps propositions of a formal system to truth values of true or
> false.
You shouldn't be having such a difficult time with this concept.
A structure for a language is a certain kind of function from a
certain subset of the set of symbols. Then we build on that to provide
a certain function from the set of formulas.
To say what exactly the function is, depends on the PARTICULAR
function, just as with many ordinary functions. If I say, "f is a
function on the set of real numbers", or "f is a 1-1 function" or "f
is an isomorphism", I'm just telling you what KIND of function f is;
and if we need to know the specific values for f, then that will
depend on what specific function f is. The same for structures. A
structure is a certain KIND of function. What it's values are depend
on what SPECIFIC function (from among many that are of the KIND that
we say is a structure) we are talking about.
MoeBlee
revo...@live.co.uk
> There _is_ a precise definition of "model", as a
> certain function.
> The definition does not specify what function that
> should be.
> So what? It's precisely the same to say that there is
> a precise
> definition of "real number", namely "a real number is
> S(d1, d2, ...) for some sequence of digits d_1, d_2,
> ...".
> Your complaint about the definition of "model" not
> specifying what function we're talking about makes
> exactly as much sense as complaining that the
> definition
> of "real number" does not specify what sequence d_1,
> ...
> we're talking about.
Yes, but you are still ignoring the fact that we do have a general function that represents a real number in general, but the problem is that you have no function that is a general function for a model. I can take, for example, the real numbers 0.1234567 and 0.1234568, and thereby define a subset of real numbers, the set of all real numbers between those two values. You appear to have no comparable way of defining a subset of your set of all model functions.
revo...@live.co.uk
> David and I are not contradicting each another. We're
> both right. I've already explained in detail.
> Why are you just ignoring the information I posted for your benefit?
Read the posts. You do contradict each other.
> By the way, I didn't use the expression "mapping
> function". It's
> redundant. A function is a mapping; a mapping is a
> function.
Saying 'mapping function' doesn't give any incorrect information. I'll take your comment as intended to be quirkily humorous rather than demonstrative of an obsessive need to appear superior to every one else.
> A structure is a certain kind of function. We say
> that a structure is
> a structure FOR a LANGUAGE. Then, given a particular
> structure, we
> define another function, which is a function from the
> set of sentences
> into {0 1} (for false, true, respectively). Then we
> say a structure is
> a model OF a theory iff every sentence in the theory
> evaluates to true
> per the function on sentences that we defined per the
> structure. So
> every model is a structure. And every structure is a
> model since every
> structure is a model of the set of valid (logically
> true) sentences.
So, every structure is a model and every model is a structure. If that is the case, then there terms model and structure are synonymous, and are simply different names for the same thing.
Now, a structure is defined as consisting of a domain, a signature, and an interpretation function that maps every function/relation of the signature to any function/relation at all, provided the arity is the same.
And you say that a structure is a model of a theory iff every sentence in the theory evaluates to true.
But unless you can define what 'true' means then you haven't defined the interpretation function, and you haven't defined what a model/structure is. So without a definition of truth, model theory is as indefinable as the notion of truth.
> > One can indeed
> > define real numbers in different ways, but the
> proof that there
> > does not exist any mapping of the naturals to the
> reals depends on
> > real numbers being specifically defined as
> infinite sums of
> > rationals,
> > such as in decimal notation.
>
> WRONG. (1) We don't (ordinarily) define reals to be
> such "infinite
> sums".
I don't know who you mean by 'we, and why that 'we' carries more authority than any other 'we'. Real numbers are commonly defined in such terms (although it is more correct to state that irrational numbers can be defined as the limiting value of infinite sums of rationals of decreasing size such as given by decimal notation).
> (2) The proof of uncountability depends only
> on the reals being
> a carrier set for a complete ordered field. WHATEVER
> definition of 'is
> a real number', as long as the set of them is a
> carrier set of a
> complete ordered field, that set is uncountable.
You say that I am wrong (note: putting 'wrong' in capital letters doesn't add anything to support that belief).
You talk about ordered fields. The definition of an ordered field includes the concepts of 'greater than' and 'less than'. Evidently these concepts do not apply to all things, so a complete definition of 'greater than' and 'less than' must make it completely clear as to what things these concepts apply to.
In order to define 'greater than' or 'less than' requires certain definitions of number. So the definition of an ordered field inherently includes the definition of certain properties of natural numbers - properties that generate the concept of division, which itself generates the concept of rational numbers. So the definition of an ordered field inherently defines the rational numbers.
So how do you define 'less than' for the definition of an ordered field, without using a circular definition?
For example, if you define 'less than' for natural numbers, you might state that a natural number a is greater than another natural number b if it is not equal to that number and if there exists a natural number c where a - b = c. That indeed gives a definition of less than, but it only applies to natural numbers.
Similarly, if you define 'less than' for rational numbers, you might state that a positive rational number d is greater than another rational number e if it is not equal to that number and if there exists a positive rational number f where d - e = f. Again we have a definition of less than, which only applies to rational numbers.
So how is 'less than' or 'greater than' defined for an ordered field and which applies to real numbers, without real numbers themselves first being defined?
The problem I have with your explanations is that there is no definite general specification for a function for a model. And if you do not have a clear and definite specification for such a function, then how can you say that you actually have defined in general terms what a model is?
I would make the analogy of having a specification for software which has some intended purpose. If it is clearly defined then there may be many ways of accomplishing the same result. So although the specification does not define any particular software program, that does not mean that it is not a clear and precise specification that can be defined in very definite terms.
And since your 'definition' of a model is dependent on the notion of truth which is an indefinable notion, it is not a specification for anything at all.
Aatu Koskensilta
> But unless you can define what 'true' means then you haven't defined
> the interpretation function, and you haven't defined what a
> model/structure is.
A peculiar line of thought.
> So without a definition of truth, model theory is as indefinable as
> the notion of truth.
Why not look up the definition?
T.H. Ray
> The problem I have with your explanations is that
> there is no definite general specification for a
> function for a model. And if you do not have a clear
> and definite specification for such a function, then
> how can you say that you actually have defined in
> general terms what a model is?
Suppose I ask, if one has not specifically described
a procedure to take the limit of a function, does a
limit not exist? In fact, knowing the limit depends on
the procedure that defines it. The model that results
depends only on the distinction between a continuous
function--in which case f(x)has no limit--and the
behavior of f(x) near some point of the limit.
> I would make the analogy of having a specification
> for software which has some intended purpose. If it
> is clearly defined then there may be many ways of
> accomplishing the same result. So although the
> specification does not define any particular software
> program, that does not mean that it is not a clear
> and precise specification that can be defined in very
> definite terms.
> And since your 'definition' of a model is dependent
> on the notion of truth which is an indefinable
> notion, it is not a specification for anything at all.
I must have missed the part of this discussion that holds
that truth is indefinable; truth is very precisely
defined in mathematics. If a theorem were not a true
mathematical statement, mathematics would be empty of
meaning.
In any case, a computing program does not model
continuous functions. A computer model, therefore, is
a result to arbitrary precision and not a "precise
specification."
Tom
Aatu Koskensilta
> In any case, a computing program does not model continuous
> functions.
On the contrary, all computable real functions are continuous.
As to the definition of truth, there is no mathematical definition of
a true mathematical statement -- indeed there can be no such
definition since "mathematical statement" is not a mathematically
defined notion. In logic we do have a definition of truth for any
number of mathematically defined classes of mathematical statements,
e.g. those in the language of arithmetic, analysis, and so on. These
definitions do not involve the notion of a mathematical theorem, since
this notion again has no mathematical definition.
T.H. Ray
> "T.H. Ray" <thra...@aol.com> writes:
>
> > In any case, a computing program does not model
> continuous
> > functions.
>
> On the contrary, all computable real functions are
> continuous.
>
> As to the definition of truth, there is no
> mathematical definition of
> a true mathematical statement -- indeed there can be
> no such
> definition since "mathematical statement" is not a
> mathematically
> defined notion. In logic we do have a definition of
> truth for any
> number of mathematically defined classes of
> mathematical statements,
> e.g. those in the language of arithmetic, analysis,
> and so on. These
> definitions do not involve the notion of a
> mathematical theorem, since
> this notion again has no mathematical definition.
>
I find your philosophy strained and artificial.
The utility of mathematics depends very much on
defining the boundary conditions of truth, and the
meaning of a continuous function.
While it is true that all real functions are continuous,
it is not helpful to qualify computable functions in
that domain, when finite limits are arbitrary.
The mathematical definition of theorem is "true
mathematical statement." If you can't reach that level
in formal logic, you are not doing mathematics.
Tom
> --
> Aatu Koskensilta (aatu.kos...@uta.fi)
>
> "Wovon man nicht sprechen kann, darüber muss man
> schweigen"
> - Ludwig Wittgenstein, Tractatus
> s Logico-Philosophicus
Aatu Koskensilta
> While it is true that all real functions are continuous, it is not
> helpful to qualify computable functions in that domain, when finite
> limits are arbitrary.
I'm unable to make anything of this sentence. Perhaps you're using
"real function" in some sense other than the usual?
> The mathematical definition of theorem is "true mathematical
> statement."
This is not a mathematical definition at all.
> If you can't reach that level in formal logic, you are not doing
> mathematics.
The conclusion is inescapable, no one is doing mathematics.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
T.H. Ray
> "T.H. Ray" <thra...@aol.com> writes:
>
> > While it is true that all real functions are
> continuous, it is not
> > helpful to qualify computable functions in that
> domain, when finite
> > limits are arbitrary.
>
> I'm unable to make anything of this sentence. Perhaps
> you're using
> "real function" in some sense other than the usual?
>
What do you regard as "the usual?"
> > The mathematical definition of theorem is "true
> mathematical
> > statement."
>
> This is not a mathematical definition at all.
>
It most certainly is.
> > If you can't reach that level in formal logic, you
> are not doing
> > mathematics.
>
> The conclusion is inescapable, no one is doing
> mathematics.
>
You cannot escape that conclusion only because you've
disallowed conclusion-making tools. As I claimed,
the logic is strained and artificial.
Tom
> --
> Aatu Koskensilta (aatu.kos...@uta.fi)
>
> "Wovon man nicht sprechen kann, darüber muss man
> schweigen"
> - Ludwig Wittgenstein, Tractatus
> s Logico-Philosophicus
Aatu Koskensilta
> What do you regard as "the usual?"
Just the usual: a real function is a function from reals to reals. For
the common definitions of computability of such functions, any
computable real function is continuous.
> It most certainly is.
No,
The mathematical definition of theorem is "true mathematical
statement."
is simply nonsense since the proposed definition involves the notion
of a true mathematical statement which has no mathematical
definition.
In standard usage "theorem" also does not mean "true mathematical
statement"; in standard usage a theorem is rather a mathematical
statement for which a proof has been given. (In mathematical logic
"theorem" is used also in a technical, mathematically defined sense.)
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
T.H. Ray
>
> > What do you regard as "the usual?"
>
> Just the usual: a real function is a function from
> reals to reals. For
> the common definitions of computability of such
> functions, any
> computable real function is continuous.
Then what do you mean by "continuous?" No mechanical
algorithm can model a continuous function except to
arbitrary precision. It is one thing to prove that all
real functions are continuous. It is quite another to
speak of a continuous function without boundary
conditions--conditions which are imposed arbitrarily.
My comments are addressed to the OP's topic. A model
is a result to arbitrary precision of domain and boundary
conditions that are freely chosen.
> > It most certainly is.
>
> No,
>
> The mathematical definition of theorem is "true
> e mathematical
> statement."
>
> is simply nonsense since the proposed definition
> involves the notion
> of a true mathematical statement which has no
> mathematical
> definition.
>
> In standard usage "theorem" also does not mean "true
> mathematical
> statement"; in standard usage a theorem is rather a
> mathematical
> statement for which a proof has been given. (In
> mathematical logic
> "theorem" is used also in a technical, mathematically
> defined sense.)
>
I don't know whose "standard usage" you're referring
to, but we are not going to agree. Apparently, you
think logicism drives mathematics. I'm afraid it's the
other way around.
Tom
> --
> Aatu Koskensilta (aatu.kos...@uta.fi)
>
> "Wovon man nicht sprechen kann, darüber muss man
> schweigen"
> - Ludwig Wittgenstein, Tractatus
> s Logico-Philosophicus
Aatu Koskensilta
> Then what do you mean by "continuous?"
Just the usual.
> No mechanical algorithm can model a continuous function except to
> arbitrary precision.
It is apparent that you are simply unaware of the relevant definition of
a computable real function, and consequently unable to understand the
mathematical content of the statement, that all computable real
functions are continuous. This is perfectly fine, and we can drop the
subject.
> It is one thing to prove that all real functions are continuous.
I see you're a follower of Brouwer!
> My comments are addressed to the OP's topic. A model is a result to
> arbitrary precision of domain and boundary conditions that are freely
> chosen.
The OP was asking about models in the technical sense of mathematical
logic. Precision, boundary conditions, and what not are entirely
irrelevant.
> I don't know whose "standard usage" you're referring to, but we are
> not going to agree.
Come now, this is not a matter of opinion, or of philosophy. In standard
usage, as documented in dictionaries etc., as known to all who work in
mathematics, a theorem is a mathematical statement for which a proof has
been given.
> Apparently, you think logicism drives mathematics.
What actual statement of mine do you have in mind? I haven't said
anything even remotely related to logicism.
> I'm afraid it's the other way around.
Mathematics "drives logicism"? What does this mean?
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
T.H. Ray
> "T.H. Ray" <thra...@aol.com> writes:
>
> > Then what do you mean by "continuous?"
>
> Just the usual.
>
Apparently, no one can disagree with what's in your head,
as long as you don't elaborate enough to risk it being
shown wrong.
> > No mechanical algorithm can model a continuous
> function except to
> > arbitrary precision.
>
> It is apparent that you are simply unaware of the
> relevant definition of
> a computable real function, and consequently unable
> to understand the
> mathematical content of the statement, that all
> computable real
> functions are continuous. This is perfectly fine, and
> we can drop the
> subject.
>
That's probably a good idea. Don't kid yourself,
however, that content recapitulates meaning.
> > It is one thing to prove that all real functions
> are continuous.
>
> I see you're a follower of Brouwer!
>
Indeed I am, to a large extent. And Dedekind. And
Weierstrauss. And Weyl. Analysis is a rich source
of mathematical meaning.
> > My comments are addressed to the OP's topic. A
> model is a result to
> > arbitrary precision of domain and boundary
> conditions that are freely
> > chosen.
>
> The OP was asking about models in the technical sense
> of mathematical
> logic. Precision, boundary conditions, and what not
> are entirely
> irrelevant.
If these are irrevelant, we are not talking mathematics.
That being so, any definition of model should do. If
the OP would care to return, I would say more.
> > I don't know whose "standard usage" you're
> referring to, but we are
> > not going to agree.
>
> Come now, this is not a matter of opinion, or of
> philosophy. In standard
> usage, as documented in dictionaries etc., as known
> to all who work in
> mathematics, a theorem is a mathematical statement
> for which a proof has
> been given.
>
I can't think of a single time I have used a dictionary
to research a mathematical question. Dictionaries reflect
colloquial usage. A mathematical statement that has been
proven, however, _is_ a "true mathematical statement."
And by the way, one can work in mathematics without
doing mathematics, as is the case with many.
> > Apparently, you think logicism drives mathematics.
>
> What actual statement of mine do you have in mind? I
> haven't said
> anything even remotely related to logicism.
>
If you say so.
> > I'm afraid it's the other way around.
>
> Mathematics "drives logicism"? What does this mean?
>
It means that meaning precedes construction. When you
essay to construct meaning, you are engaging in logicism,
whether you think so or not.
Tom
> --
> Aatu Koskensilta (aatu.kos...@uta.fi)
>
> "Wovon mann nicht sprechen kann, darüber muss man
> schweigen"
> - Ludwig Wittgenstein, Tractatus
> s Logico-Philosophicus
T.H. Ray
Aatu Koskensilta
> Apparently, no one can disagree with what's in your head, as long as
> you don't elaborate enough to risk it being shown wrong.
So you don't know the usual definition of continuity? You can look it up
in any calculus or analysis text.
> Indeed I am, to a large extent. And Dedekind. And Weierstrauss. And
> Weyl. Analysis is a rich source of mathematical meaning.
I see. You embrace with enthusiasm set theoretic classical analysis,
predicativism and intuitionism alike.
> If these are irrevelant, we are not talking mathematics.
You seem to take a rather dim view of mathematical logic, which,
apparently, isn't even mathematics in your book, its definitions, of
such concepts as those of a model, of a formal theory, and so on,
failing to include vague waffling about boundary conditions and what
not.
> A mathematical statement that has been proven, however, _is_ a "true
> mathematical statement."
Certainly. I haven't said anything to the contrary. Not all true
mathematical statements are theorems, however. It also remains that
The mathematical definition of theorem is "true mathematical
statement."
is not a mathematical definition of anything.
> And by the way, one can work in mathematics without doing mathematics,
> as is the case with many.
Yes, yes, you've already expressed the opinion that much of what is
usually regarded as mathematics is in fact not mathematics at all.
> It means that meaning precedes construction. When you essay to
> construct meaning, you are engaging in logicism, whether you think so
> or not.
What do you mean by "logicism"? In standard usage, logicism refers to a
specific conception or philosophy of mathematics put forth by Frege,
Russell and others, a conception not in any apparent way related to the
comments by either of us in this thread.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
MoeBlee
wrote:
> Jack Marakam said
I know of no "Jack Marakam". Rather, as I explained to you 'Jack
Markan' was a user name I devised for a particular forum but that now,
without my intent, gets attached to my posts made under the name
'MoeBlee'.
Next, I give you a number of further explanations in this post. I
suggest that rather than you followup with yet more questions, instead
you get a good book on the subject. The process of explaining to you
ad hoc, out of context of you reading a good book, is not very
profitable. Each round of explanation may engender yet another round
of questions, but all in reverse as we move backwards from more
developed concepts toward initial assumptions and most basic
explanations. It would be better for you to first read the most basic
material and learn systematically toward the more developed notions
we're now talking about without a context of you knowing the basics
upon which these more developed notions are built.
> > David and I are not contradicting each another. We're
> > both right. I've already explained in detail.
> > Why are you just ignoring the information I posted for your benefit?
>
> Read the posts. You do contradict each other.
You THINK we do, because you are confused. You are confused because
you've not sat down with a book on this subject, and, with some quiet
presence of mind, followed the book step by step through its
formulations and explanations.
> > By the way, I didn't use the expression "mapping
> > function". It's
> > redundant. A function is a mapping; a mapping is a
> > function.
>
> Saying 'mapping function' doesn't give any incorrect information.
(1) Whether it is correct or not to say "mapping function", I stated
for a matter of record that it is not a phrase I used (or at least, I
hope I didn't), and, to let you know WHY it is not a phrase I would
like to use, I mentioned that it is redundant.
> I'll take your comment as intended to be quirkily humorous rather than demonstrative of an obsessive need
> to appear superior to every one else.
That's a false dichotomy. It was intended neither as humorous nor as a
demonstration of any need to appear superior. Rather, it was a comment
explaining why I do, or would, avoid the phrase. Interesting that you
didn't just take it at face value, but rather imposed your false
dichotomy on it.
> > A structure is a certain kind of function. We say
> > that a structure is
> > a structure FOR a LANGUAGE. Then, given a particular
> > structure, we
> > define another function, which is a function from the
> > set of sentences
> > into {0 1} (for false, true, respectively). Then we
> > say a structure is
> > a model OF a theory iff every sentence in the theory
> > evaluates to true
> > per the function on sentences that we defined per the
> > structure. So
> > every model is a structure. And every structure is a
> > model since every
> > structure is a model of the set of valid (logically
> > true) sentences.
>
> So, every structure is a model and every model is a structure. If that is the case, then there terms
> model and structure are synonymous, and are simply different names for the same thing.
Strictly speaking, they are different words for the same predicate 'is
a structure' or 'is a model'. However, in conversational usage, they
may have different connotations, different emphases as to the role
being played by the structure in a particular context of conversation.
> Now, a structure is defined as consisting of a domain, a signature, and an interpretation function that
> maps every function/relation of the signature to any function/relation at all, provided the arity is the
> same.
There are different exact formulations, depending on the author (and I
explained how the different formulations nevertheless are essentially
the same notion). What you gave a above is not the formulation I
mentioned. However, we could sharpen the formulation you gave above:
A structure (in the sense we are discussing here) for a language L is
a pair <S I> where S is a nonempty set and I is a function (we call it
an 'interpretation function') on the set of function and relation
SYMBOLS of L, and such that I maps n-place function symbols to n-place
functions on S and n-place predicate symbols to n-place predicates on
S.
> And you say that a structure is a model of a theory iff every sentence in the theory evaluates to true.
> But unless you can define what 'true' means then you haven't defined the interpretation function,
I already explained to you that we DO provide a definition of 'true in
a model'.
But, we don't need that definition for the interpretation function.
Rather, we have a structure (a non-empty set and an interpretation
function) and we have a definition of 'true PER a given structure' (or
'true in a given model'). That is, 'true in the structure' is not part
of the interpretation function. First is the interpretation (given by
the interpretation function), then comes an evaluation of
satisfiability and then truth based on a given interpretation.
> and you haven't defined what a model/structure is.
No, I did. I gave you the definition earlier, and (to accommodate your
poorly worded formulation just now) I just gave a variation in this
post that accomplishes the same thing as my earlier definition. The
only difference is that my earlier definition made the universal
quantifier part of the domain of the interpretation function and took
the domain of discourse to be the value of the interpretation function
with the universal quantifier as an argument, while with the variant
in this post, I just made the domain of discourse S onto itself in a
tuple <S I> where I is just like before but without the universal
quantifier in the domain. Just two different ways to hook the same
fish.
>So without a definition of truth, model theory is as indefinable as the notion of truth.
But we do give a definition of 'satisfied by the structure' for
formulas and 'true in the structure' for sentences.
> > > One can indeed
> > > define real numbers in different ways, but the
> > proof that there
> > > does not exist any mapping of the naturals to the
> > reals depends on
> > > real numbers being specifically defined as
> > infinite sums of
> > > rationals,
> > > such as in decimal notation.
>
> > WRONG. (1) We don't (ordinarily) define reals to be
> > such "infinite
> > sums".
>
> I don't know who you mean by 'we, and why that 'we' carries more authority than any other 'we'.
It just refers to common mathematical practice as found in ordinary
textbooks in set theory, analysis, and number systems at a college
level.
> Real numbers are commonly defined in such terms
Ordinarily, real numbers are defined as Dedekind cuts or as
equivalence classes of Cauchy sequences of rationals. There are other
treatments, but as I said, ordinarily, reals are not defined as
"infinite sums".
>(although it is more correct to state that irrational numbers can be defined as the limiting value of
> infinite sums of rationals of decreasing size such as given by decimal notation).
Wait until you've studied the subject in a systematic and rigorous way
such as in set theory and analysis.
> > (2) The proof of uncountability depends only
> > on the reals being
> > a carrier set for a complete ordered field. WHATEVER
> > definition of 'is
> > a real number', as long as the set of them is a
> > carrier set of a
> > complete ordered field, that set is uncountable.
>
> You say that I am wrong (note: putting 'wrong' in capital letters doesn't add anything to support that
> belief).
Use of all caps in such instances is not meant to add support for
believability, but rather to draw emphasis for the need to correct
your misunderstanding.
> You talk about ordered fields. The definition of an ordered field includes the concepts of 'greater
> than' and 'less than'. Evidently these concepts do not apply to all things, so a complete definition of
> 'greater than' and 'less than' must make it completely clear as to what things these concepts apply to.
The less than relation on reals is defined in set theory. Of course,
where the notion of an ordered field is taken onto itself
axiomatically, then '<' is primitive but controlled by the axioms
about it. (However, as far as I know, there is no first order
axiomatization of COMPLETE ordered field.)
> In order to define 'greater than' or 'less than' requires certain definitions of number. So the
> definition of an ordered field inherently includes the definition of certain properties of natural
> numbers - properties that generate the concept of division, which itself generates the concept of
> rational numbers. So the definition of an ordered field inherently defines the rational numbers.
Every ordered field has a rational subfield (if I correctly recall the
statement of the theorem). And when we construct PARTICULAR complete
ordered field by such methods as Dedekind cuts or equivalence classes
of Cauchy sequences, then, yes, of course, rational numbers are used.
But the GENERAL definition of 'ordered field' does not require prior
definition of 'rational numbers'.
In any case, none of this diminishes the remarks I previously made.
> So how do you define 'less than' for the definition of an ordered field, without using a circular
> definition?
For the GENERAL notion of 'ordered field' the 'less than' relation is
any relation that satisfies the definitional axioms. For defining a
PARTICULAR ordered field, a definition of the less than relation for
that ordered field will be provided. And this is done, without
circularity, as you may find in many a textbook on set theory or on
number systems.
> For example, if you define 'less than' for natural numbers, you might state that a natural number a is
> greater than another natural number b if it is not equal to that number and if there exists a natural
> number c where a - b = c. That indeed gives a definition of less than, but it only applies to natural
> numbers.
Yes, assuming that you've already defined '-' for natural numbers.
In any case, we do provide particular definitions of 'less than' when
we define any particular ordered field.
> So how is 'less than' or 'greater than' defined for an ordered field and which applies to real numbers,
> without real numbers themselves first being defined?
No, we DO first define the predicate 'is a real number'. Then we
define the less than relation on that set. On the other hand, as I
mentioned, for the GENERAL notion of an ordered field, 'less than' is
taken as primitive but obeys the definitional axioms of 'ordered
field'.
You're going in circles. You keep saying the above, and yet I keep
showing you the exact definition. I can't help you further on this. If
you don't understand my explanations (along with the explanations of
other posters) then you do need to take some quiet time, clear your
thoughts, and carefully follow the presentation in a good book on the
subject.
> And since your 'definition' of a model is dependent on the notion of truth which is an indefinable
> notion, it is not a specification for anything at all.
No, you have it BACKWARDS. The definition of 'structure' does not
depend on the definition of 'satisfied in the structure' and 'true in
the structure'. Rather, first we define what we mean when we say "M is
a structure for the language L". Then we define what we mean when we
say "Formula P is satisfied in structure M" and "Sentence P is true in
structure M". Then we say a structure M is a model of a set of
sentences G iff every member of G is true in M. So every model is a
structure, by definition. Then I merely noted that every structure is
a model since every structure is a model of the set of logically true
sentences of the language of the structure.
But, again, you'll understand all of this kind of thing much better by
sitting down with a good book on the subject.
MoeBlee
T.H. Ray
Is there some plausible reason that you find it
necessary to keeping posting a new subject instead of
continuing in the thread? I am not going to chase you
all over the n.g. I don't find your dialogue that
stimulating. Nor in fact, that relevant or profound.
Instead of pretending to try and teach me analysis, you
might do a little digging into the literature yourself.
Tom
Aatu Koskensilta
> Is there some plausible reason that you find it necessary to keeping
> posting a new subject instead of continuing in the thread?
I am not "posting a new subject". MathForum's threading is apparently
broken -- if you inspect the headers in my posts you'll find they all
contain a correct References:-line.
> Instead of pretending to try and teach me analysis, you might do a
> little digging into the literature yourself.
Why are you pretending I'm trying to teach you analysis? It is indeed a
good idea for you to do a little digging into the literature. You will
find, in particular, the definition of a computable real function, the
definition of continuity for real functions, and the theorem that all
computable real functions are continuous. Alas, you will not find
"theorem" defined as "true mathematical statement", nor will you find
any mathematical definition of a mathematical theorem.
Frederick Williams
>
> > It is one thing to prove that all real functions are continuous.
>
> I see you're a follower of Brouwer!
It is not all real functions just those defined on a closed interval I
think.
--
Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested.
Aatu Koskensilta
> Aatu Koskensilta wrote:
>
>> I see you're a follower of Brouwer!
>
> It is not all real functions just those defined on a closed interval I
> think.
It is a theorem of intuitionistic analysis that all total real functions
are continuous.
T.H. Ray
> "T.H. Ray" <thra...@aol.com> writes:
>
> > Is there some plausible reason that you find it
> necessary to keeping
> > posting a new subject instead of continuing in the
> thread?
>
> I am not "posting a new subject". MathForum's
> threading is apparently
> broken -- if you inspect the headers in my posts
> you'll find they all
> contain a correct References:-line.
>
Please accept my apology then. I post from MathForum
without trouble.
> > Instead of pretending to try and teach me analysis,
> you might do a
> > little digging into the literature yourself.
>
> Why are you pretending I'm trying to teach you
> analysis? It is indeed a
> good idea for you to do a little digging into the
> literature. You will
> find, in particular, the definition of a computable
> real function, the
> definition of continuity for real functions, and the
> theorem that all
> computable real functions are continuous.
I know the references. Again--don't think that
content recapitulates meaning. Do you get that?
> Alas, you
> will not find
> "theorem" defined as "true mathematical statement",
> nor will you find
> any mathematical definition of a mathematical
> theorem.
>
That is, of course, the veriest nonsense.
Tom
> --
> Aatu Koskensilta (aatu.kos...@uta.fi)
>
> "Wovon mann nicht sprechen kann, darüber muss man
> schweigen"
> - Ludwig Wittgenstein, Tractatus
> s Logico-Philosophicus
Aatu Koskensilta
> I know the references.
Great! Why then all these questions about what I mean by this and
that?
> Again--don't think that content recapitulates meaning. Do you get
> that?
I have no problem not thinking that content recapitulates meaning
since I've no idea what that's supposed to mean.
> That is, of course, the veriest nonsense.
Again, this is not a matter of opinion, or of philosophy. Where in the
literature do we find "theorem" defined as "true mathematical
statement"? Where in the literature do we find any mathematical
definition of a mathematical theorem?
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Frederick Williams
>
> ... Where in the literature do we find any mathematical
> definition of a mathematical theorem?
Why shouldn't such a thing exist, my little Passiflora caerulea blossom?
T.H. Ray
> "T.H. Ray" <thra...@aol.com> writes:
>
> > I know the references.
>
> Great! Why then all these questions about what I mean
> by this and
> that?
>
Because as I said from the very beginning, I find your
philosophy strained and artificial. One cannot abstract
meaning from content. To draw on an analogy in physics,
Einstein noted that a symphony can be described in a
mathematical model as variations in sound wave pressure.
That brings us to no understanding of a symphony, no
meaning. The same can be said of a theory and
consequently of the theorems that comprise it.
> > Again--don't think that content recapitulates
> meaning. Do you get
> > that?
>
> I have no problem not thinking that content
> recapitulates meaning
> since I've no idea what that's supposed to mean.
>
Obviously. However, the concept is neither exotic nor
esoteric.
> > That is, of course, the veriest nonsense.
>
> Again, this is not a matter of opinion, or of
> philosophy. Where in the
> literature do we find "theorem" defined as "true
> mathematical
> statement"? Where in the literature do we find any
> mathematical
> definition of a mathematical theorem?
>
Yes, it is a matter of your opinion and your preference.
Since it obvious that you do not actually do
mathematics, however, I will give a couple of quick
and contemporary references:
"To anticipate the discussion in Chapter 8, I shall claim
that mathematical truth exists, but is not to be found
in the content of any particular theorem or set of
theorems. The intuition that mathematics accesses the
truth is correct, but not in the manner that is usually
understood. The truth is to be found more in the fact
that in the content of mathematics."~Wm. Byers, How
Mathematicians Think, Princeton University Press, 2007.
p. 15
"What confidence one has when one sees into the truth of
some mathematical theorem!...if you are not completely
certain, if you have the slightest doubt, then you just
don't get it. Mathematical truth has this certainty,
this quality of inexorability. This is its essence."~op.cit., p.328
"Nowadays, almost all working mathematicians are trained
in, and concerned with, the production of rigorous
arguments that justify their conclusions. These
conclusions are usually framed as _theorems_, which are
statements of fact, accompanied by an argument, or
proof, that the theorem is indeed true." ~ Tom Archibald,
"The Development of Rigor in Mathematical Analysis," in
Tim Gowers, ed, with June Barrow-Green and Imre Leader,
The Princeton Companion to Mathematics, Princeton
University Press, 2008.
> --
> Aatu Koskensilta (aatu.kos...@uta.fi)
>
> "Wovon man nicht sprechen kann, darüber muss man
> schweigen"
> - Ludwig Wittgenstein, Tractatus
> s Logico-Philosophicus
Aatu Koskensilta
> Aatu Koskensilta wrote:
>>
>> ... Where in the literature do we find any mathematical
>> definition of a mathematical theorem?
>
> Why shouldn't such a thing exist, my little Passiflora caerulea
> blossom?
Allow me to expound, my dearest Brassica oleracea sprout.
A mathematical theorem is a mathematical statement for which a
mathematical proof has been given. Whether or not we have managed to
produce a mathematical proof is obviously not a mathematical matter, but
turns rather on the contingencies of our mathematical practice and human
existence in general. We can't, for example, expect mathematics to tell
us if the time should come when no-one bothers doing mathematics
anymore, and consequently no new theorems are produced. This is of
course nothing but obnoxious pedantry. So let's try to be a bit more
charitable and, perhaps, slightly more informative.
The perfectly fine definition of mathematical theorem -- a mathematical
statement for which a mathematical proof has been given -- is not a
mathematical definition since it involves three notions which have no
mathematical definition, viz. "mathematical statement", "mathematical
proof" and "has been given". As noted we can't hope to give any
mathematical explanation or definition for "has been given". Whether
something has been given, to the mathematical community, to some
individual, to the human race, is simply not a mathematical matter. So
let's idealise a bit, abstracting away this blatantly non-mathematical
ingredient. We then obtain something on the lines of
A mathematical theorem is a mathematical statement for which a
mathematical proof can, in principle, be given.
Reading "can, in principle, be given" simply as "exists" we're left with
two problematic notions: mathematical statement, mathematical proof.
Strictly speaking we can't of course expect a mathematical definition
for either of the notions mentioned, anymore than we can in case of
"hamburger", "a fine example of Edwardian English prose", "computer",
"informative Usenet post". What we may hope for is rather a mathematical
explication, a mathematical analysis, a mathematical model that captures
those features we consider salient. The question now becomes: can we
give such informative and faithful mathematical analysis of the notions
of a mathematical statement and a mathematical proof? Should we succeed
we'd immediately obtain one also of the notion of a mathematical
theorem.
Examining the notion of a mathematical statement we encounter our next
quandary. Starting with the standard understanding, that a mathematical
statement is a declarative sentence (in English, say) making an
assertion about mathematically defined properties of and relations
holding among mathematical objects, we're faced with two problems, one
having to do with the wide variety of attitudes, ideas, opinions,
mathematicians hold, and one purely logical.
The first problem is easily illustrated: the finitist considers all talk
of infinitary objects non-sense, and indeed considers quantification
over naturals etc. illegitimate, theological business. The predicativist
happily allows unrestricted quantification over naturals, and talk of
definable collections of naturals, and so on, but balks at the notion of
an arbitrary collection of naturals. The set theorist considers the
predicativist a sissy but enamoured with the iterative conception of set
may well have his doubts about quantification over proper classes,
collections of proper classes, etc. The intuitionist has yet other
causes for complaint. And so on and so forth. Yet they all do share the
same notion of a mathematical statement -- indeed otherwise it would be
very odd to say they /disagree/ about anything, which, on the face of
it, they very much do, propounding in great length why these and those
notions are or are not legitimate and meaningful in mathematics.
But let's set that aside and suppose, contrary to the fact, that
mathematicians were a boring lot, in heartening agreement on what
notions make mathematical sense, what sort of stuff we may meaningfully
quantify over, and so on. Even then we couldn't hope for a mathematical
explication of the notion of a mathematical statement. For suppose we
set up a formal language incorporating notions, concepts, etc. we agree
are meaningful, so that the formal sentences can be taken to be
(formalisations of) meaningful mathematical statements. By simple
diagonlisation argument there are then mathematical notions, such as
"true sentence in the formal language" that aren't expressible in the
language, but which we, having agreed the language is meaningful and its
sentences make mathematical assertions, recognise as legitimate
mathematics. (This doesn't preclude the possibility that, in fact, there
is a formal language in which all mathematical notions we recognise or
could in some idealised sense recognise as legitimate can be expressed;
but if there is such a language -- and unless we're pretending there is
necessarily some fact of the matter as to whether this or that notion
makes mathematical sense it's not given the idea makes any sense -- we
can't recognise it as such.)
Enough about mathematical statements. Let's set this problematic notion
aside and consider whether we may give a mathematical analysis of, say,
the notion of a mathematical theorem expressible in the language of
arithmetic[1]. The sole problematic notion is now that of a mathematical
proof. Basically just the same problems I went over in case of
mathematical statements resurface. A mathematical proof is a piece of
logically correct, compelling reasoning from basic principles
mathematicians accept as correct (are self-evident, immediately obvious,
etc.). What counts as "logically correct reasoning" we may take to be
unproblematic. But what of these basic principles? Some regard the
existence of, say, a weakly compact cardinal perfectly obvious, made
evident to them by pondering the vast and unfathomable universe of sets
-- others seriously doubt even the consistency of PA. A mathematical
proof to one may well not be a proof to another.
Spelling out the second problem, that a mathematical definition of
mathematical proof would immediately, by an obvious diagonalisation
argument, lead to a contradiction, is left as an exercise to the
reader. (Which is just an oblique way of saying I'm tired of typing. I
do hope the above was sufficiently artificial and strained.)
Footnotes:
[1] "Mathematical statement expressible in the language of arithmetic"
can be given a mathematical explication. For details see my ludicrously
long-winded rant /Formalisation/ available at
http://groups.google.com/group/sci.logic/msg/1cf3026be617d644
(Message-ID: <slrnf6am4g.q15....@localhost.localdomain>)
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
Aatu Koskensilta
> Because as I said from the very beginning, I find your philosophy
> strained and artificial.
I've cited a theorem of recursion theory and made a few rather
unexciting remarks about standard English usage in mathematics. What
philosophy do you purport to discern in my comments?
> Obviously. However, the concept is neither exotic nor
> esoteric.
Wonderful. Perhaps you could explain this concept and its relevance to
anything I've said?
> Yes, it is a matter of your opinion and your preference.
How is whether "theorem" is defined as "true mathematical statement" or
whether we find in the literature any mathematical definition of
mathematical theorem in any manner a matter of my opinion and
preference?
> Since it obvious that you do not actually do mathematics, however, I
> will give a couple of quick and contemporary references:
I've never done any actual mathematics and indeed no nothing about
mathematics at all, but what is the relevance of these quotes? They do
not contain any definition of "theorem" as "true mathematical
statement". They do not contain any mathematical definition of
anything. So I ask again: Where in the literature do we find "theorem"
defined as "true mathematical statement? Where in the literature do we
find any mathematical definition of mathematical theorem?
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
T.H. Ray
> "T.H. Ray" <thra...@aol.com> writes:
>
[snip]
> So I ask again: Where in the literature do
> we find "theorem"
> defined as "true mathematical statement? Where in the
> literature do we
> find any mathematical definition of mathematical
> theorem?
>
By your criteria, where in the literature do we find
a definition of "number?" The utility of a
concept is in application.
When you essay, as you appear to be doing, to combine
language and meaning as if they were not independent,
you are doing philosophy--not mathematics.
Tom
> --
> Aatu Koskensilta (aatu.kos...@uta.fi)
>
> "Wovon mann nicht sprechen kann, darüber muss man
> schweigen"
> - Ludwig Wittgenstein, Tractatus
> s Logico-Philosophicus
T.H. Ray
Soulless comes to mind, too. In any case, your claim:
> A mathematical theorem is a mathematical statement
> for which a
> mathematical proof can, in principle, be given.
>
> Reading "can, in principle, be given" simply as
> "exists" we're left with
> two problematic notions: mathematical statement,
> mathematical proof.<<
is false. A theorem is accompanied by a proof that the
theorem is true, or it is not a theorem. A conjecture
is a statement for which a proof can in principle
be given.
It is not necessary to drill down to the meanings of
statement and proof. The meaning is contained in the
definition of theorem: "True mathematical statement."
The quantifier "there exists" follows from the
result, not from the assumption that some result might
be given.
Your discourse could appeal only to a postmodernist.
The only working mathematician I can think of who might
lay claim to that philosophy is Brian Rotman. And while
I have enjoyed his books, I think even he would rather be
doing mathematics than writing about it. He would also
surely recognize the difference.
Once again: content does not recapitulate meaning.
Tom
>
> Footnotes:
> [1] "Mathematical statement expressible in the
> language of arithmetic"
> can be given a mathematical explication. For details
> see my ludicrously
> long-winded rant /Formalisation/ available at
>
> http://groups.google.com/group/sci.logic/msg/1cf3026b
> e617d644
> (Message-ID:
> :
> <slrnf6am4g.q15....@localhost.localdomain
> >)
>
> --
> Aatu Koskensilta (aatu.kos...@uta.fi)
>
> "Wovon mann nicht sprechen kann, darüber muss man
> schweigen"
> - Ludwig Wittgenstein, Tractatus
> s Logico-Philosophicus
Aatu Koskensilta
> Soulless comes to mind, too.
Oh no!
> In any case, your claim:
>
>> A mathematical theorem is a mathematical statement for which a
>> mathematical proof can, in principle, be given.
>>
>> Reading "can, in principle, be given" simply as "exists" we're left
>> with two problematic notions: mathematical statement, mathematical
>> proof.
>
> is false. A theorem is accompanied by a proof that the theorem is
> true, or it is not a theorem.
What in what I wrote do you take to contradict this?
> A conjecture is a statement for which a proof can in principle be
> given.
It is your stand, then, that no conjecture in mathematics is false?
> Your discourse could appeal only to a postmodernist.
What in my remarks do you regard as reeking of pomo?
> Once again: content does not recapitulate meaning.
Once again: perhaps you can explain this notion in clear and explicit
terms?
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Aatu Koskensilta
> By your criteria, where in the literature do we find a definition of
> "number?"
We don't find any mathematical definition of "number" in the
literature. You have claimed that "theorem" is defined as "true
mathematical statement" and that this is a mathematical
definition. Surely if this is the case you can point me to some relevant
passage in the literature?
> The utility of a concept is in application.
The pleasure we find in a musical composition is in the delight we take
in experiencing it. Do you think making trivial observations such as
these serves any useful purpose?
> When you essay, as you appear to be doing, to combine language and
> meaning as if they were not independent, you are doing philosophy--not
> mathematics.
You take a very dim view of philosophy too, it seems. Your waffling,
alas, is neither philosophy nor mathematics.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
T.H. Ray
You are asking for a replay of issues I have addressed in earnest and with citation.
Lower I cannot stoop.
Tom
Aatu Koskensilta
> These exchanges have run their course, Aatu.
So it seems.
> You are asking for a replay of issues I have addressed in earnest and
> with citation.
Your citations have been nothing to the point. But by all means, let's
terminate this apparently pointless discussion.
Frederick Williams
>
> Frederick Williams <frederick...@tesco.net> writes:
>
> > Aatu Koskensilta wrote:
> >>
> >> ... Where in the literature do we find any mathematical
> >> definition of a mathematical theorem?
> >
> > Why shouldn't such a thing exist, my little Passiflora caerulea
> > blossom?
>
> Allow me to expound, my dearest Brassica oleracea sprout.
Thank you for the reply my dear little Ranunculus asiaticus blossom.
Can one not define in ZFC, for each theorem phi of ZFC, "phi is a
theorem of ZFC"? If one can, why would one not call that a definition
of mathematical theorem?
Frederick Williams
>
> Aatu Koskensilta wrote:
> >
> > Frederick Williams <frederick...@tesco.net> writes:
> >
> > > Aatu Koskensilta wrote:
> > >>
> > >> ... Where in the literature do we find any mathematical
> > >> definition of a mathematical theorem?
> > >
> > > Why shouldn't such a thing exist, my little Passiflora caerulea
> > > blossom?
> >
> > Allow me to expound, my dearest Brassica oleracea sprout.
>
> Thank you for the reply my dear little Ranunculus asiaticus blossom.
>
> Can one not define in ZFC, for each theorem phi of ZFC, "phi is a
> theorem of ZFC"? If one can, why would one not call that a definition
> of mathematical theorem?
What I meant was:
Can one not define in ZFC, for each formula phi of ZFC, "phi is a
theorem of ZFC"? If one can, why would one not call that a definition
of mathematical theorem?
Sorry.
Aatu Koskensilta
> Can one not define in ZFC, for each formula phi of ZFC, "phi is a
> theorem of ZFC"?
The notion of a sentence in the language of set theory being provable in
ZFC does indeed have a mathematical definition, a definition which can
be formalised in the language of set theory. The same holds of any
formal theory.
> If one can, why would one not call that a definition of mathematical
> theorem?
Why would one not call "provable in ZFC + 'ZFC is inconsistent'" or
"provable in ZFC + 'there exists infinitely many twin primes'" a
definition of mathematical theorem? Presumably because one would not be
willing to assert P on basis of the existence of a formal proof of P in
ZFC + "ZFC is inconsistent" or ZFC + the twin prime conjecture.
If we're to regard "provable in ZFC" as a definition of mathematical
theorem -- that is, not simply an arbitrary stipulation about how we
wish to use the term "mathematical theorem" but an explanation of the
meaning of this phrase in mathematical terms --, we must be willing to
accept
If the formalisation of P is formally provable in ZFC then P is a
mathematical theorem.
(We may come to accept such a principle for any number of reasons, as a
result of reflecting on our mathematical reasoning, divine inspiration,
keen intuition, ...)
Given that on any ordinary understanding of "mathematical theorem" we
accept for any mathematical statement P that
If it is a mathematical theorem that P, then P.
the principle
If P is formally provable in ZFC, then P.
follows.
A particular instance of this is:
If 0 = 1 is formally provable in ZFC, then 0 = 1
but certainly it is not a mathematical theorem that 0 = 1. Hence, if we
accept that everything formally provable in ZFC is a mathematical
theorem, we must also accept that 0 = 1 is not formally provable in
ZFC. But this is itself a mathematical theorem, and by the second
incompleteness theorem is not formally provable in ZFC. We thus find
that there is, on the assumption that we can convince ourselves that all
statements formally provable in ZFC are mathematical theorems, a
mathematical theorem of which "formally provable in ZFC" does not hold.
T.H. Ray
I wouldn't count on Aatu to assign any meaning to any
definition. Insofar as he is coherent at all, I have
only been able to ascertain that his definition of
"mathematical definition" is that which comes in the
form of a mathematical result. Even my statement of fact
--content does not recapitulate meaning--has no meaning
to him. That is the most degenerate form of postmodern
philosophy, seeded by the very Wittengenstein he quotes.
For if meaning ends at the border of language, meaning
and language are therefore identical.
The practice of mathematics, however, demands the
independence of language and meaning. No result stands
for itself alone, but for the way in which it implies
another result.
Tom
Aatu Koskensilta
> I wouldn't count on Aatu to assign any meaning to any
> definition. Insofar as he is coherent at all, I have only been able to
> ascertain that his definition of "mathematical definition" is that
> which comes in the form of a mathematical result.
How did you ascertain this? Do you have some actual statement of mine in
mind?
> Even my statement of fact --content does not recapitulate meaning--has
> no meaning to him.
Well, yes, since you have consistently refused to explain what you mean
by this "statement of fact".
T.H. Ray
> "T.H. Ray" <thra...@aol.com> writes:
>
> > I wouldn't count on Aatu to assign any meaning to
> any
> > definition. Insofar as he is coherent at all, I
> have only been able to
> > ascertain that his definition of "mathematical
> definition" is that
> > which comes in the form of a mathematical result.
>
> How did you ascertain this? Do you have some actual
> statement of mine in
> mind?
>
Well of course, I do. More than one, in fact, but one
will suffice. You wrote to Frederick Williams
>>The perfectly fine definition of mathematical theorem --
>a mathematical statement for which a mathematical proof
>has been given -- is not a mathematical definition since
>it involves three notions which have no mathematical
>definition, viz. "mathematical statement", "mathematical
>proof" and "has been given". As noted we can't hope to
>give any mathematical explanation or definition for "has
>been given". Whether something has been given, to the
>mathematical community, to some individual, to the human
>race, is simply not a mathematical matter.<<
Not a mathematical matter? That is the _only_
"mathematical matter." It is all that matters, i.e., in
the study of propositions of the form A--->B, because
the transmission of knowledge and meaning is independent
of the content and the language of the
transmitter. Or don't you think that your computer is
more than the programming that you employ to communicate
over Usenet?
> > Even my statement of fact --content does not
> recapitulate meaning--has
> > no meaning to him.
>
> Well, yes, since you have consistently refused to
> explain what you mean
> by this "statement of fact".
>
Well yes, I have explained it, though I still fail to see
what is so hard to understand. I even supplied published
opinions of two other mathematicians on the relation
between content and meaning. Look it up. It is a fact,
and it is particularly a fact of mathematics.
Tom
> --
> Aatu Koskensilta (aatu.kos...@uta.fi)
>
> "Wovon mann nicht sprechen kann, darüber muss man
> schweigen"
> - Ludwig Wittgenstein, Tractatus
> s Logico-Philosophicus
Chip Eastham
> I wouldn't count on Aatu to assign any meaning to any
> definition. Insofar as he is coherent at all, I have
> only been able to ascertain that his definition of
> "mathematical definition" is that which comes in the
> form of a mathematical result. Even my statement of fact
> --content does not recapitulate meaning--has no meaning
> to him. That is the most degenerate form of postmodern
> philosophy, seeded by the very Wittengenstein he quotes.
> For if meaning ends at the border of language, meaning
> and language are therefore identical.
Hi, Tom:
I too cannot understand your "statement of fact
--content does not recapitulate meaning." Not
only does it seem outside the provenance of
objective truth, in the context of the thread
(asking for help, what is a model?), it has a
counterintuitive sense. If content doesn't
recapitulate(?) (represent ?) meaning, what
does? Given your choice of insults ("I wouldn't
count on Aatu to assign any meaning... most
degenerate form of postmodern philosophy"),
I'm guessing you think well of "meaning" and
that something ought to recapitulate and/or
communicate it, but I'm in need of a clue to
follow what you're trying to say about it.
> The practice of mathematics, however, demands the
> independence of language and meaning. No result stands
> for itself alone, but for the way in which it implies
> another result.
>
> Tom
I'm hearing a bit of rhetorical flourish
with "demands" here. Sure, a practical
virtue of mathematics lies in principles
that operate "symbolically," abstracted
from one or the other application (i.e.
symbols interpreted so as to have, dare
I say, meaning?). But practicing
mathematicians are as capable IMHO of seeing
the applications and interpretations as
others (say engineers and physicists),
so your "demand for independence of
language and meaning" is at best half
the story. An apt language for vital
meaning is part of the mathematical art.
regards, chip
P.S. I've always found Aatu lucid and
meaningful in his posts, never more so
than when twisting my nose over some
imprecision in my own posts.
LudovicoVan
> I wouldn't count on Aatu to assign any meaning to any
> definition. Insofar as he is coherent at all, I have
> only been able to ascertain that his definition of
> "mathematical definition" is that which comes in the
> form of a mathematical result.
That just makes no sense, but sounds interesting... I've not read the
thread, but I might!
> Even my statement of fact
> --content does not recapitulate meaning--has no meaning
> to him. That is the most degenerate form of postmodern
> philosophy, seeded by the very Wittengenstein he quotes.
> For if meaning ends at the border of language, meaning
> and language are therefore identical.
That's completely upside down: Wittgestein is a modern; meaning has no
ends, it's language that ends at its own borders; the silence outside
of language is not the absence of meaning, it is the absence of
language; meaning and language are distinct, with different roles and
levels.
> The practice of mathematics, however, demands the
> independence of language and meaning. No result stands
> for itself alone, but for the way in which it implies
> another result.
The independence of language and meaning is simply demanded by the
economy of language: i.e., in general, two distinct words => two
distinct notions. More importantly: no _result_ exists without
*application*.
-LV
Aatu Koskensilta
> The independence of language and meaning is simply demanded by the
> economy of language: i.e., in general, two distinct words => two
> distinct notions.
On the contrary: in general, one word => many distinct notions; this you
can easily verify by consulting a dictionary. But perhaps you had in
mind some Witterhead notion of an ideal language (such as found in the
feverish imaginings of young Wittgenstein), with one symbol per object
and so on?
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus