The Definition of Points [Math]

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From: Bob Kolker on 23 Apr 2007 21:27

Ben newsam wrote:
>
> Here we go, tap dancing in porridge again.

I thought it was word stew.

Bob Kolker

From: Tony Orlow on 23 Apr 2007 21:31

Virgil wrote:
> In article <462cd75a(a)news2.lightlink.com>,
> Tony Orlow <tony(a)lightlink.com> wrote:
>
>> stephen(a)nomail.com wrote:
>>> In sci.math Tony Orlow <tony(a)lightlink.com> wrote:
>
>>>> Consider that you have a right triangle so A^2+B^2=C^2. Now, if you swap
>>>> the values between the hypotenuse and one of the legs, that formula will
>>>> produce a second leg of the original length times i. So, in a way it is
>>>> like picturing an impossible triangle. You have to "imagine" the other
>>>> leg of length i. :)
>>> Swap the values between the hypotenuse and one of the legs? What
>>> exactly does that mean?
>>>
>>>> A
>>>> |\
>>>> | \
>>>> | \ 1
>>>> sqrt(2) | \
>>>> | \
>>>> | \
>>>> B i C
>>>
>>> So you are claiming that the obviously longer line from A to C
>>> is shorter than the line from A to B? So you can just
>>> assign any old "length" to a line regardless of how long
>>> it actually is?
>> If the distance from B to C is imaginary, then the hypotenuse is shorter
>> than the other leg. A strange concept, but perhaps the best
>> visualization one can get of a square root of a negative.
>
> Distances are not imaginary. Points in the complex planes may have
> imaginary as well as real parts, but the distances between them are
> still real.

Distances didn't used to be negative, but the distance to the left of
any point on a real line is a negative vector. Is distance an "absolute
value". Is distance a "real value"? Is that a valid question? Yepperdoodles.

>
> And one can "visualize" the square roots (plural) of complex numbers
> best in polar representation
> If x + i*y = r*(cos(theta + 2*n*pi) + i*sin(theta + 2*n*pi)),
> n in N and r >= 0, then the square roots (plural) of x + i*y are
> sqrt(r)*(cos(theta/2 + n*pi) + i*sin(theta/2 + n*pi))

"Best" in some ways. Kind of like "quantitative order". :) <3

>
>>> There is a way to make sense of this, but it is quite at odds
>>> with traditional geometry, which seems to be the basis for all
>>> your arguments.
>>>
>>> Stephen
>>>
>> It's somewhat at odds with traditional geometry. It's not a very well
>> fleshed out idea. Just thought I'd mention the image.
>
> TO would do better to flush it out that to try to flesh it out.

Or, flash it up. TADA!!! (whoosh)

teedles

From: Tony Orlow on 23 Apr 2007 21:47

From: Virgil on 23 Apr 2007 21:48

In article <462d5e09(a)news2.lightlink.com>,
Tony Orlow <tony(a)lightlink.com> wrote:

> Virgil wrote:
> > In article <462cd75a(a)news2.lightlink.com>,
> > Tony Orlow <tony(a)lightlink.com> wrote:
> >
> >> stephen(a)nomail.com wrote:
> >>> In sci.math Tony Orlow <tony(a)lightlink.com> wrote:
> >
> >>>> Consider that you have a right triangle so A^2+B^2=C^2. Now, if you swap
> >>>> the values between the hypotenuse and one of the legs, that formula will
> >>>> produce a second leg of the original length times i. So, in a way it is
> >>>> like picturing an impossible triangle. You have to "imagine" the other
> >>>> leg of length i. :)
> >>> Swap the values between the hypotenuse and one of the legs? What
> >>> exactly does that mean?
> >>>
> >>>> A
> >>>> |\
> >>>> | \
> >>>> | \ 1
> >>>> sqrt(2) | \
> >>>> | \
> >>>> | \
> >>>> B i C
> >>>
> >>> So you are claiming that the obviously longer line from A to C
> >>> is shorter than the line from A to B? So you can just
> >>> assign any old "length" to a line regardless of how long
> >>> it actually is?
> >> If the distance from B to C is imaginary, then the hypotenuse is shorter
> >> than the other leg. A strange concept, but perhaps the best
> >> visualization one can get of a square root of a negative.
> >
> > Distances are not imaginary. Points in the complex planes may have
> > imaginary as well as real parts, but the distances between them are
> > still real.
>
> Distances didn't used to be negative, but the distance to the left of
> any point on a real line is a negative vector.

Distance is a scalar, not a vector. otherwise the distance from Chicago
to New York wold be the negative of the distance from New Youk to
Chicago. There is such a thing as a /directed/ distance, which, on a
line, is like a one-dimensional vector.

> Is distance an "absolute
> value".

Yup!

> > And one can "visualize" the square roots (plural) of complex numbers
> > best in polar representation
> > If x + i*y = r*(cos(theta + 2*n*pi) + i*sin(theta + 2*n*pi)),
> > n in N and r >= 0, then the square roots (plural) of x + i*y are
> > sqrt(r)*(cos(theta/2 + n*pi) + i*sin(theta/2 + n*pi))
>
> "Best" in some ways. Kind of like "quantitative order". :) <3

TO suggests that there are ways in which it is not best, but one of them
not appear to exist within mathematics.

From: Tony Orlow on 24 Apr 2007 09:27

Lester Zick wrote:
> On Mon, 23 Apr 2007 11:54:05 -0400, Tony Orlow <tony(a)lightlink.com>
> wrote:
>
>> Lester Zick wrote:
>>> On Fri, 20 Apr 2007 12:00:37 -0400, Tony Orlow <tony(a)lightlink.com>
>>> wrote:
>>>
>>>>>> What truth have you demonstrated without positing first?
>>>>> And what truth have you demonstrated at all?
>>>>>
>>>>> ~v~~
>>>> Assuming two truth values as 0-place operators, I demonstrated that
>>>> not(x) is the only functional 1-place operator, and then developed the
>>>> 2-place operators mechanically from there. That was truth about truth.
>>> And how have you demonstrated the truth of the two "truth values" you
>>> assumed?
>>>
>>> ~v~~
>> The "truth" of the two truth values is that I've declared them a priori
>> as the two alternative evaluations for the truth of a statement.
>
> Well there are two difficulties here, Tony. You declare two a priori
> alternatives but how do you know they are in fact alternatives? In
> other words what mechanism causes them to alternate from one to the
> other? It looks to me like what you actually mean is that you declare
> two values 1 and 0 having nothing in particular to do with "truth" or
> anything except 1 and 0.

1=not(0)
0=not(1)
x<>not(x)

That says it all. "not" defines the relationship between 0 and 1. In
arithmetic terms, not(x) can be equated with 1-x, for obvious intuitive
reasons. If 1 is the universe, and x is some portion of that universe,
everything that is not(x) is everything in the universe, minus what's in x.

"not not" then equates to 1-(1-x), which is just x, which doesn't have
any truth value until you assign it one.

>
> The second problem is what makes you think two "truth" alternatives
> you declare are exhaustive? This is related to the first difficulty.
> You can certainly assume one thing or alternative a priori but not
> two. And without some mechanism to produce the second from the first
> and in turn the first from the second exclusively you just wind up
> with an non mechanical dualism where there is no demonstration the two
> are in fact alternatives at all or exhaustive alternatives either.

See the mechanism for their mutual definition, above.

As far as the two truth values being "exhaustive", it's simply declared
in Boolean logic that these are the only two values allowed, true() and
false(). Now, these are not really all the possibilities, if one looks
at truth in the context of probability. Where a probability of truth of
x is 100%, the truth value is 1, meaning "definitely true". Where a
statement has a 0% probability of truth, the value is 0, meaning
"definitely false". Between these two lie a continuous set of possible
probabilities. So, in this context, {0,1} is NOT an exhaustive set of
all alternatives. But, this requires that you accept partial truth,
which doesn't seem to be acceptable for you. What other alternatives do
you see besides true(x) and false(x) for a statement x?

>
> We already know you think there are any number of points in the
> interval 0-1 so apriori declarations do not erase that inconsistency
> between different sets of assumptions.

I'm trying to keep it simple, and just discuss the mechanics of the most
basic kind of logic, where absolute "truth" exists. It doesn't, in real
science.

>
> A few days ago you asked about what I mean by "mechanics" or an
> "exhaustive mechanics" and this is what I mean. You're welcome to
> assume one mechanism but then you have to show how that one mechanism
> produces every other aspect of the "mechanics" involved.

That's what I TRIED to do.

>
> And I believe it obvious that the one "mechanism" has to be the
> process of "alternation" itself or there is no way to produce anything
> other than our initial assumption. A chain is no stronger than its
> weakest link and if you make dualistic apriori assumptions neither of
> which is demonstrably true of the other in mechanically exhaustive
> terms you already have the weakest link right there at the foundation.
>

Again, please comment on my use of "not" to define the relationship
between true() and false().

>> That's
>> not a matter of deduction until you specify what statements you are
>> assuming true or false, and on what basis you surmise them to be one or
>> the other.
>
> Well this comment is pure philosophy, Tony, because we only have your
> word for it. You can certainly demonstrate the "truth" of "truth" by
> regression to alternatives to "truth" by the mechanism of alternation
> itself and I have no difficulty demonstrating the "truth" of "truth"
> by regression to a self contradictory "alternatives to alternatives".
> Of course this is only an argument not a postulate or principle but
> then anytime you analyze "truth" you only have recourse to arguments.
>

If you're discussing logic, you have the additional recourse to the
mechanics of logic itself, the basics of which are well understood, if
not widely.

>> Truth tables and logical statements involving variables are
>> just that. If I say, 3x+3=15, is that true? No, we say that IF that's
>> true, THEN we can deduce that x=4.
>
> But here you're just appealing to syllogistic inference and truisms
> because your statement is incomplete. You can't say what the "truth"
> of the statements is or isn't until x is specified. So you abate the
> issue until x is specified and denote the statement as problematic.

Right. The truth of the statement 3x+3=15 cannot be determined without
specifying x. That's my point.

>
> The difficulty with syllogistic inference and the truism is Aristotle
> never got beyond it by being able to demonstrate what if anything
> conceptual was actually true. The best he could do was regress
> demonstrations of truth to perceptual foundations which most people
> considered true, even if they aren't absolutely true. But even based
> on that problematic assumption he could still never demonstrate the
> conceptual truth of anything beyond the perceptual level.

Now you're the one lapsing into philosophy, so I'll oblige.

The only thing we can ever really know is that we exist, ala Descartes.
Beyond that, there is always the chance that our perceptions are
entirely false. However, one has to ask oneself why that would be. Where
would these illusions come from? I don't see any explanation for why a
seemingly consistent universe should not be considered to be just what
it seems. So, we have to start with the assumption, and that's just what
it is, that this is not all an illusion, despite the fact that our
perceptions may be flawed or limited. Having accepted that, we apply
scientific method, either formally or intuitively, to detect "truths",
and the only measure of a truth is whether it consistently predicts
measurable facts from other measurable facts. Where we detect a formula
which "works", we have detected a "truth" of some sort or another. Of
course, we may never truly understand the most fundamental nature of
what we observe, but we can still "know" something about it, in that sense.

>
> And here the matter has rested for mathematics and science in general
> ever since. Empiricism benefitted from perceptual appearances of truth
> in their experimental results but the moment empirics went beyond them
> to explain results in terms of one another they were hoist with the
> Aristotelian petard of being unable to demonstrate what was actually
> true and what not. The most mathematicians and scientists were able to
> say at the post perceptual conceptual level was that "If A then B then
> C . . ." etc. or "If our axiomatic assumptions of truth actually prove
> to be true then our theorems, inferences, and so forth are true". But
> there could never be any guarantee that in itself was true.
>
>

Well, if the axiom systems we develop produce the results we expect
mathematically, then we can be satisfied with them as starting
assumptions upon which to build. My issue with transfinite set theory is
that it produces a notion of infinite "size" which I find
unsatisfactory. I accept that bijection alone can define equivalence
classes of sets, but I do not accept that this is anything like an
infinite "number". So, that's why I question the axioms of set theory.
Of course, one cannot do "experiments" on infinite set sizes. In math,
one can only judge the results based on intuition.

>
>> If I say 3x+3=3(x+1) is that true?
>> Yes, it's true for all x.
>
> How about for x=3/0?
>

Division by pure 0 is proscribed because it produces an unmeasurable oo.
If x is any specific real, or hyperreal, or infinitesimal, then that
statement is true for all x. 3/0 is not a specific number.

>> If I say a=not b, is that true? Not if a and b
>> are both true.
>
> How do you arrive at that assumption, Tony? If a and b are both true
> of the same thing they can still be different from each other.

You are confusing logical "not" with non-identity. "a=not(b)" means that
if b is true, a is false, and vice versa. You are talking about
"not(a=b)", which means simply that there is some difference between a
and b, but not that there is a contradiction between the two.

If I say, "A woman cannot be an engineer", then
woman(x)=not(engineer(x)) and engineer(x)=not(woman(x)). If there IS a
woman who is an engineer, then woman(x) and engineer(x), and
"engineer(x)=not(woman(x))" is false.

If I say, "A woman is not an engineer", that does not have any bearing
on whether the same being cannot at both times be a woman and an
engineer. That's not a matter of logical contradiction, but of some set
of distinctions between the two classes of objects.

>
>> If I say a or not a, that's true for all a. a and b are
>> variables, which may each assume the value true or false.
>
> Except you don't assign them the value true or false; you assign them
> the value 1 or 0 and don't bother to demonstrate the "truth" of either
> 1 or 0.
>

1 is true, 0 is false. If a is 0 or 1, then we have "0 or 1", or "1 or
0", respectively. Since or(a,b) is true whenever a is true or b is true,
or both, or(1,0) and or(0,1), the only possible values for the
statement, are both true. So, or(a,not(a)) is always true, in boolean
logic, or probability.

Intuitively, if a is a subset of the universe, and not(a) is everything
else, then the sum of a and not(a) is very simply the universe, which is
true.

>> If you want to talk about the truth values of individual facts used in
>> deduction, by all means, go for it.
>
> I don't; I never did. All I ever asked was how people who assume the
> truth of their assumptions compute the truth value of the assumptions.
>
> ~v~~

By measuring the logical implications of their assumptions.

01oo

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