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From: Tim Golden BandTech.com on 25 Jan 2010 08:50
On Jan 24, 11:44 am, Marshall <marshall.spi...(a)gmail.com> wrote:
> On Jan 24, 6:47 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com>

> If you have a problem with ZxZ, then you have a problem with
> the rational numbers.

This has nothing to do with the topic that I am discussing, but then,
the topic I am discussing has little to do with the topic of this
thread, and then too the convenience with which content is deleted to
suit ones own needs further clouds the issue, and on the other a cloud
of undeleted material may equally cause an overcast state. To be as
direct as possible seems to me the resolution. Your methods are of
indirection.

- Tim
From: Marshall on 25 Jan 2010 10:00
On Jan 25, 5:50 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com>
wrote:
> On Jan 24, 11:44 am, Marshall <marshall.spi...(a)gmail.com> wrote:
>
> > On Jan 24, 6:47 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com>
> > If you have a problem with ZxZ, then you have a problem with
> > the rational numbers.
>
> This has nothing to do with the topic that I am discussing, but then,
> the topic I am discussing has little to do with the topic of this
> thread, and then too the convenience with which content is deleted to
> suit ones own needs further clouds the issue, and on the other a cloud
> of undeleted material may equally cause an overcast state. To be as
> direct as possible seems to me the resolution. Your methods are of
> indirection.

In summary, your worries about abstract algebra have been
shown to be unfounded.

Good luck with the new math you're pursuing!


Marshall

From: jbriggs444 on 25 Jan 2010 11:40
On Jan 22, 4:39 pm, Marshall <marshall.spi...(a)gmail.com> wrote:
> On Jan 22, 12:50 pm, jbriggs444 <jbriggs...(a)gmail.com> wrote:
>
>
>
>
>
> > On Jan 22, 3:25 pm, Marshall <marshall.spi...(a)gmail.com> wrote:
> > > On Jan 22, 11:45 am, jbriggs444 <jbriggs...(a)gmail.com> wrote:
> > > > On Jan 22, 12:03 pm, Marshall <marshall.spi...(a)gmail.com> wrote:
>
> > > > > Such expositions concern constructions, encodings,
> > > > > representations of numbers. If we remember that they
> > > > > are such, and that constructions of numbers are not
> > > > > the numbers themselves, then we're fine. If we
> > > > > fail to make that distinction, then we may end up
> > > > > with results that are flat-out wrong. We might
> > > > > conclude that 1/2 =/= .5, or that 1 =/= 1+0i,
> > > > > which are flat-out wrong. We might conclude that
> > > > > the cartesian x,y point (1, 1) is the same point
> > > > > than the polar r, theta point (1, 1), when in fact
> > > > > they are different points.
>
> > > > In my view there is no such thing as "<mumble> number" as a really
> > > > truly physical, authentic, "this is the one and only set of <mumble>
> > > > numbers", accept no imitations.  What we have instead are
> > > > axiomatizations and constructions or models that satisfy the axioms..
> > > > In my view, the best we can ever say is that "yes, these are the
> > > > <mumble> numbers, up to isomorphism".
>
> > > I have no argument with that. We cannot directly process
> > > anything except reified abstractions; we cannot get any
> > > closer to "a _is_ b" than "up to isomorphism."
>
> > > However if we extrapolate from that fact to the claim
> > > that only physically representable things exist, then
> > > we have to throw out a lot of useful stuff: almost
> > > all real numbers, for example. That would seem an
> > > overreaction, as tempting as it might be. In view of this,
> > > I have no qualms about drawing a distinction between
> > > constructions and the things they represent.
>
> > Fine by me.  Draw the distinction.
>
> > Now explain to me why that distinction supports your claim that the
> > real zero and the complex zero are, in fact, identical rather than the
> > opposing claim that the real zero and the complex zero are, in fact,
> > different.
>
> How does Leibnitz equality grab you?

Dunno. Never heard of it. Is it something like "things which have
equal properties are equal"?
If "is a complex number" and "is a real number" are properties then
that would seem to be begging the question.

>
> Or how about some prose from Leslie Lamport?
>
> ----------------
> Mathematicians typically define objects by explicitly constructing
> them. For example, a standard way of defining N inductively is to let
> 0 be the empty set and n be the set {0, . . . , n − 1}, for n > 0..
> This makes the strange-looking formula 3 ∈ 4 a theorem.
>
> Such definitions are often rejected in favor of more abstract ones.
> For example, de Bruijn [1995, Sect.3] writes
> "If we have a rational number and a set of points in the Euclidean
> plane, we
> cannot even imagine what it means to form the intersection. The idea
> that
> both might have been coded in ZF with a coding so crazy that the
> intersection
> is not empty seems to be ridiculous."

This does not support the problematic contention. It supports a
contention that one ought to ignore constructions as being definitive.

> In the abstract data type approach [Guttag and Horning 1978],
> one defines data structures in terms of their properties, without
> explicitly constructing them. The argument that abstract definitions
> are better than concrete ones is a philosophical one. It makes no
> practical difference how the natural numbers are defined. We
> can either define them abstractly in terms of Peano’s axioms, or
> define them concretely and prove Peano’s axioms. What matters
> is how we reason about them.

Again, making the point that constructions need not be definitive and
that they instead, "make no practical difference".

> > Bear in mind that you've already as much as admitted that there is no
> > fact of the matter.
>
> Um, remind me where I admitted that again?

Two replies up:

' I have no argument with that. We cannot directly process
anything except reified abstractions; we cannot get any
closer to "a _is_ b" than "up to isomorphism." '

This statement implies that there is no definitive construction for
the real numbers or for the complex numbers. Both are only specified
"up to isomorphism". [You are welcome to chase down the rabbit hole
of considering each real number as the equivalance class of all the
things that it could be mapped to in terms of some isomorphism or
other. I doubt you'll get far]

This, in turn, means that there is no fact of the matter about what
any particular real or complex number _is_.

In particular, it means that there is no fact of the matter about
whether (real) 1.0 is equal to (complex) 1.0+0.0i.

If you want to declare them to be equal, that's fine.
If you want to declare them to be unequal, that's fine too.
If you want to say that somebody else's declaration is wrong, that's
not fine.


Let me again try to state explicitly what I have a problem with. And
since you want to work with signed integers and naturals, let me put
it in that context.

Your claim as understood by me:

a. You claim that every natural number _is_ a signed integer.
b. If someone else says that the two sets are disjoint, you say that
they are wrong.

My response:

There is no fact of the matter. At most there is convention and
convenience. But you're not arguing convention and convenience.
You're trying to argue fact. With no evidence.

If you use constructions as your guide to underlying truth then there
are constructions that answer the question one way and other
constructions that answer it the other way. So constructions don't
resolve the question. [I gave competing constructions from the get
go]

If you use algebraic properties as your guide to underlying truth then
there still isn't an answer.
All the algebraic properties of the signed and of the unsigned
integers hold regardless of whether the naturals are a subset of the
integers or are disjoint from the integers.


Let me try a new argument out on you...

There are two theories in physics. "Special relativity" and "Lorentz
Ether Theory".

Both theories make the same physical predictions for every conceivable
experiment. They have different names for the underlying pieces in
their respective models, but when it comes to telling you what number
will appear in a digital readout, they both make the same predictions,
every time.

Scientists generally accept Special Relativity as preferable to the
Lorentz Ether Theory. It has a cleaner axiomatic basis. They will
generally accept Special Relativity as "true" (or as true as any well-
verified physical theory can be).

I have no problem with that.

But if a scientist chooses to say that this means that Lorentz Ether
Theory is false. I gotta call a foul. The experimental record for
Lorentz Ether Theory is every bit as unblemished as that for Special
Relativity.


In my book, when there is no physical fact of the matter it's fine to
pick a point of view and adopt it as if it were true. But in my book,
that sort of might-as-well-be-truth is not a valid basis for asserting
that contrary views are false.
From: jbriggs444 on 25 Jan 2010 14:37
On Jan 23, 3:47 pm, Marshall <marshall.spi...(a)gmail.com> wrote:
> On Jan 23, 7:52 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com>
> wrote:
>
>
>
>
>
> > On Jan 22, 3:42 pm, Marshall <marshall.spi...(a)gmail.com> wrote:
>
> > Alright Marshall. I do think you have a strong position and admit that
> > it has taken me some time to come around to seeing how you are seeing.
> > Still, the formality of the subsetting is not quite there. For
> > instance, the value
> >   + 5.1
> > is in R, the reals. Is + 5.1 in C? Well, it is in there quite a few
> > times. For instance we have
> >    + 5.1 - 1.2 i ,
> >    + 5.1 + 0.1 i,
> >    + 5.1 + 0 i , ... .
> > Now, I'm pretty sure it is the last one that you meant but how am I
> > supposed to know that it wasn't the first one? We are bumping into
> > that same specification of zero that was used earlier. Beyond these
> > simplistic versions there are even more means of subsetting a real
> > line into a complex plane, and I admit that I am thinking graphically
> > of those possibilities. For instance, I was considering a real valued
> > line E whose origin E0 is at -1+i and whose unity position E1 is at
> > 0+2i. Is this real line E a subset of the complex plane? Well, I'm
> > pretty sure we're going to now bump into the operators and I will be
> > called a fool and so forth. Still, this is the level of simplicity
> > that I am looking at and I've already been called a fool, so what have
> > I to lose? I am not afraid to be wrong, especially if that helps me to
> > understand some fundamental possibilities.
>
> You are confusing yourself with various irrelevant distractions.

Could be. But you aren't helping.

> Is 2 a natural number or an integer? It's both of course.

There's no "of course" to it. You're ignoring some very relevant
distractions.

> But wait, can't I come up with various mappings from N
> to Z? If you are talking about the natural number 2, how
> do I know that when you say it's also an integer, you don't
> mean the integer -100? There does exist a mapping where
> that makes sense, right?

You seem to be conflating the notation we use to express a number with
[one of the] constructions we use to make corresponding number system
explicit. Or maybe you're accusing Tim of doing so.

Is 2 a natural number or an integer? Yes, no, neither, both. Absent
a context, it is ambiguous.

I look at "2" as a numeric literal. It assumes whatever data type is
appropriate in context.

Arguing based on notation that (signed integer) 2 is equal to
(natural) 2 is a fallacy. Notation has nothing to do with the answer.

You're right, of course. There is a context in which the signed
integer -100 is mapped by the unsigned integer 2 in a construction of
the signed integers from the unsigned integers.

I've never used such a construction. You've never used such a
construction. Nor, to the best of my knowledge has anyone else. But
there's nothing invalid about it.

One could decide to use "the equivalence class of ordered pairs of
natural numbers of which (200,100) is an exemplar using the
equivalance relation (a,b) eqv (c,d) iff a+d=b+c" as a notation for
the signed integer -100. Most mathematicians find that "-100" is more
convenient.

It's arguably an abuse of notation. But it saves so _much_ room.

>
> See the point? 2 is 2, which is natural, integer, rational,
> real, and complex.

No. You are failing to distinguish betwen the notation and the thing
referred to by the notation.
As I said before, argument by notation is fallacious.

> > I'm still standing by the understanding that the complex value
> >    a + b i
> > is composed of two real values a and b and one nonreal i, whose
> > quality I think is best described as a unit vector. These details help
> > explain how this product and sum do not evaluate to a single element
> > via the operators defined in ring theory. They are incompatible with
> > the ring definition.
>
> You are confusing yourself.

Yes. He's arguing from notation, just like you do. And getting wrong
answers, just like you do.

Both of you are taking the notation too literally. The map is not the
territory.
From: Patricia Aldoraz on 25 Jan 2010 16:14
On Jan 26, 3:40 am, jbriggs444 <jbriggs...(a)gmail.com> wrote:
> On Jan 22, 4:39 pm, Marshall <marshall.spi...(a)gmail.com> wrote:
>
>
>
> > On Jan 22, 12:50 pm, jbriggs444 <jbriggs...(a)gmail.com> wrote:
>
> > > On Jan 22, 3:25 pm, Marshall <marshall.spi...(a)gmail.com> wrote:
> > > > On Jan 22, 11:45 am, jbriggs444 <jbriggs...(a)gmail.com> wrote:
> > > > > On Jan 22, 12:03 pm, Marshall <marshall.spi...(a)gmail.com> wrote:
>
> > > > > > Such expositions concern constructions, encodings,
> > > > > > representations of numbers. If we remember that they
> > > > > > are such, and that constructions of numbers are not
> > > > > > the numbers themselves, then we're fine. If we
> > > > > > fail to make that distinction, then we may end up
> > > > > > with results that are flat-out wrong. We might
> > > > > > conclude that 1/2 =/= .5, or that 1 =/= 1+0i,
> > > > > > which are flat-out wrong. We might conclude that
> > > > > > the cartesian x,y point (1, 1) is the same point
> > > > > > than the polar r, theta point (1, 1), when in fact
> > > > > > they are different points.
>
> > > > > In my view there is no such thing as "<mumble> number" as a really
> > > > > truly physical, authentic, "this is the one and only set of <mumble>
> > > > > numbers", accept no imitations.  What we have instead are
> > > > > axiomatizations and constructions or models that satisfy the axioms.
> > > > > In my view, the best we can ever say is that "yes, these are the
> > > > > <mumble> numbers, up to isomorphism".
>
> > > > I have no argument with that. We cannot directly process
> > > > anything except reified abstractions; we cannot get any
> > > > closer to "a _is_ b" than "up to isomorphism."
>
> > > > However if we extrapolate from that fact to the claim
> > > > that only physically representable things exist, then
> > > > we have to throw out a lot of useful stuff: almost
> > > > all real numbers, for example. That would seem an
> > > > overreaction, as tempting as it might be. In view of this,
> > > > I have no qualms about drawing a distinction between
> > > > constructions and the things they represent.
>
> > > Fine by me.  Draw the distinction.
>
> > > Now explain to me why that distinction supports your claim that the
> > > real zero and the complex zero are, in fact, identical rather than the
> > > opposing claim that the real zero and the complex zero are, in fact,
> > > different.
>
> > How does Leibnitz equality grab you?
>
> Dunno.  Never heard of it.  Is it something like "things which have
> equal properties are equal"?
> If "is a complex number" and "is a real number" are properties then
> that would seem to be begging the question.
>
>
>
>
>
> > Or how about some prose from Leslie Lamport?
>
> > ----------------
> > Mathematicians typically define objects by explicitly constructing
> > them. For example, a standard way of defining N inductively is to let
> > 0 be the empty set and n be the set {0, . . . , n − 1}, for n > 0.
> > This makes the strange-looking formula 3 ∈ 4 a theorem.
>
> > Such definitions are often rejected in favor of more abstract ones.
> > For example, de Bruijn [1995, Sect.3] writes
> > "If we have a rational number and a set of points in the Euclidean
> > plane, we
> > cannot even imagine what it means to form the intersection. The idea
> > that
> > both might have been coded in ZF with a coding so crazy that the
> > intersection
> > is not empty seems to be ridiculous."
>
> This does not support the problematic contention.  It supports a
> contention that one ought to ignore constructions as being definitive.
>
> > In the abstract data type approach [Guttag and Horning 1978],
> > one defines data structures in terms of their properties, without
> > explicitly constructing them. The argument that abstract definitions
> > are better than concrete ones is a philosophical one. It makes no
> > practical difference how the natural numbers are defined. We
> > can either define them abstractly in terms of Peano’s axioms, or
> > define them concretely and prove Peano’s axioms. What matters
> > is how we reason about them.
>
> Again, making the point that constructions need not be definitive and
> that they instead, "make no practical difference".
>
> > > Bear in mind that you've already as much as admitted that there is no
> > > fact of the matter.
>
> > Um, remind me where I admitted that again?
>
> Two replies up:
>
> ' I have no argument with that. We cannot directly process
> anything except reified abstractions; we cannot get any
> closer to "a _is_ b" than "up to isomorphism." '
>
> This statement implies that there is no definitive construction for
> the real numbers or for the complex numbers.  Both are only specified
> "up to isomorphism".  [You are welcome to chase down the rabbit hole
> of considering each real number as the equivalance class of all the
> things that it could be mapped to in terms of some isomorphism or
> other.  I doubt you'll get far]
>
> This, in turn, means that there is no fact of the matter about what
> any particular real or complex number _is_.
>
> In particular, it means that there is no fact of the matter about
> whether (real) 1.0 is equal to (complex) 1.0+0.0i.
>
> If you want to declare them to be equal, that's fine.
> If you want to declare them to be unequal, that's fine too.
> If you want to say that somebody else's declaration is wrong, that's
> not fine.
>
> Let me again try to state explicitly what I have a problem with.  And
> since you want to work with signed integers and naturals, let me put
> it in that context.
>
> Your claim as understood by me:
>
> a.  You claim that every natural number _is_ a signed integer.
> b.  If someone else says that the two sets are disjoint, you say that
> they are wrong.
>
> My response:
>
> There is no fact of the matter.  At most there is convention and
> convenience.  But you're not arguing convention and convenience.
> You're trying to argue fact.  With no evidence.
>
> If you use constructions as your guide to underlying truth then there
> are constructions that answer the question one way and other
> constructions that answer it the other way.  So constructions don't
> resolve the question.  [I gave competing constructions from the get
> go]
>
> If you use algebraic properties as your guide to underlying truth then
> there still isn't an answer.
> All the algebraic properties of the signed and of the unsigned
> integers hold regardless of whether the naturals are a subset of the
> integers or are disjoint from the integers.
>
> Let me try a new argument out on you...
>
> There are two theories in physics.  "Special relativity" and "Lorentz
> Ether Theory".
>
> Both theories make the same physical predictions for every conceivable
> experiment.  They have different names for the underlying pieces in
> their respective models, but when it comes to telling you what number
> will appear in a digital readout, they both make the same predictions,
> every time.
>
> Scientists generally accept Special Relativity as preferable to the
> Lorentz Ether Theory.  It has a cleaner axiomatic basis.  They will
> generally accept Special Relativity as "true" (or as true as any well-
> verified physical theory can be).
>
> I have no problem with that.
>
> But if a scientist chooses to say that this means that Lorentz Ether
> Theory is false.  I gotta call a foul.  The experimental record for
> Lorentz Ether Theory is every bit as unblemished as that for Special
> Relativity.
>
> In my book, when there is no physical fact of the matter it's fine to
> pick a point of view and adopt it as if it were true.  But in my book,
> that sort of might-as-well-be-truth is not a valid basis for asserting
> that contrary views are false.

If it is really true that nothing in the future could ever distinguish
the predictions of one theory over another then perhaps they are
really *the same theory* and are both either true or both false. It
all depends on how you count theories. If the words used to describe
one theory *mean* different to the words used to describe another
theory, it does not follow that they are not the same theory any more
than that "The man in the green hat" could not be the very same person
as "Vera's husband".

On the other hand, if they differ in meaning, it makes sense that one
could be true and the other false. Not necessarily, but as a matter of
fact.
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