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From: John Stafford on 25 Jan 2010 21:16
In article
<b784c360-2650-4776-90e6-78eeecc34e16(a)a5g2000yqi.googlegroups.com>,
Patricia Aldoraz <patricia.aldoraz(a)gmail.com> wrote:

> [...] some things are the same as a matter of fact and others
> as a matter of logic.

Thank you for the picture perfect example of the root of religion.
From: Marshall on 25 Jan 2010 21:40
On Jan 25, 8:40 am, jbriggs444 <jbriggs...(a)gmail.com> wrote:
> On Jan 22, 4:39 pm, Marshall <marshall.spi...(a)gmail.com> wrote:
> >
> > 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.

Thinking only of those two predicates would be perverse.
What about all the other predicates?


> > 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.

What I was trying to contend was that one ought to ignore
constructions as being definitive. So apparently I succeeded!


> > 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".

Exactly!


> > > 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".

So once again you are arguing that we ought to pay
attention to constructions. But I argue the opposite.
I agree that there is no fact of the matter as regards
constructions. But I am talking about numbers, not
constructions of numbers.


> 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.

Really? You say "it's fine" to say that they're unequal?
So, if they are unequal, then when you subtract one
from the other you get some number other than
zero. Can you show me how this is so? Or are
you going to claim some important distinction
between 0 and 0+0i?


> 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.

Yup.


> 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]

Right, and I said constructions don't resolve the questions
in response.



> 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.

Sorry, not interested in physics.


Marshall
From: Marshall on 25 Jan 2010 22:04
On Jan 25, 11:37 am, jbriggs444 <jbriggs...(a)gmail.com> wrote:
> On Jan 23, 3:47 pm, Marshall <marshall.spi...(a)gmail.com> wrote:
>
> > 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.

It's funny how you say I'm "not helping" and ignoring
relevant distractions, and arguing without any evidence,
but then you assert the existence of these relevant
distinctions but don't list any.

What relevant distinctions?


> > 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.

I was saying nothing about notation; I was carrying Tim's argument
about mappings to an absurdity.


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

I look forward to you supplying a context in which the answer
to the question "Is 2 a natural number?" is "no."


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

But I'm not talking about "2"; I'm talking about 2. The
number. Not any kind of literal, numeral, representation,
construction, or encoding. The thing represented,
not the number.


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

I agree that notation is irrelevant, and encourage you
to stop trying to bring it in to the conversation.


> 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.

It's pretty clear you completely missed the point of what
I was saying about how the existence of mappings is
unimportant. Let's drop it, since it's relevant only to
an argument Tim was making earlier.


> > 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 am not talking about notation. If I had been talking about
notation, I would have said something like:

"2" is an ascii character.

"2" is of course not a natural number, but I didn't say it
was. I said that 2 is a natural number.


> > > 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.

If, at some point in the future, I say something having to do
with notation, your analysis of how I am taking the notation
will be relevant. In the meantime, I would encourage you to
note that someone talking about the territory doesn't
necessarily have anything to say about the map.


Marshall

From: Patricia Aldoraz on 25 Jan 2010 23:08
On Jan 26, 1:16 pm, John Stafford <n...(a)droffats.ten> wrote:
> In article
> <b784c360-2650-4776-90e6-78eeecc34...(a)a5g2000yqi.googlegroups.com>,
>  Patricia Aldoraz <patricia.aldo...(a)gmail.com> wrote:
>
> > [...] some things are the same as a matter of fact and others
> > as a matter of logic.
>
> Thank you for the picture perfect example of the root of religion.

Another totally obscure non contribution from an idiot.
From: Tim Golden BandTech.com on 26 Jan 2010 13:57
On Jan 25, 10:04 pm, Marshall <marshall.spi...(a)gmail.com> wrote:
> On Jan 25, 11:37 am, jbriggs444 <jbriggs...(a)gmail.com> wrote:
>
> > On Jan 23, 3:47 pm, Marshall <marshall.spi...(a)gmail.com> wrote:
>
> > > 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.
>
> It's funny how you say I'm "not helping" and ignoring
> relevant distractions, and arguing without any evidence,
> but then you assert the existence of these relevant
> distinctions but don't list any.
>
> What relevant distinctions?
>

I'm sorry to snip here, but at this point the discussion has sort of
lapsed, or rather it has changed character. I do find Uncle Briggs's
methods interesting. Still, I don't think those methods are very
strong. Perhaps there is an underlying principal that we should rely
upon:
simplicity.
Should we construct more complex objects from simpler objects?
Yes.
Is the real value one dimensional? Does tradition construct two
dimensional entities from a composition of one dimensional entities?
Yes.
Yet within abstract algebra we will see the insistence of constructing
the one dimensional entity from an infinite dimensional form, and yet
be caught because we had to have constructed that infinite dimensional
form from the one dimensional form.

Briggs says he doesn't care about dimension and that this style of
argument is irrelevant. I cannot accept this rejection. Information
theory is nearby and at some level various breakdowns should align by
some philosophy of consistency. The quantity of information present in
a real value is a fine way of addressing the problem. If we alot one
chunk to a real number then we see that a two dimensional
representation has two chunks. This is in the spirit of the cartesian
form. To prescribe an infinite dimensional number as equivalent to the
one dimensional number is a hypocrisy under this informational style
of thinking.
Again, the simpler form belongs at the bottom, and this is the form
with less information.
This sort of informational logic does follow in parallel like a
philosophy. Abstract algebra is an abuse of information theory.

- Tim
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