Re: Math as Religion


David Marcus wrote:
Timothy Golden BandTechnology.com wrote:

David Marcus wrote:
Timothy Golden BandTechnology.com wrote:
*** T. Winter wrote:
In article <1163159563.726461.58320@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> "Timothy Golden BandTechnology.com" <tttpppggg@xxxxxxxxx> writes:

> > > The fundamental law which I have applied is
> > > Sum for s = 1 to n ( s x ) = 0 . (where s is sign and x is
> > > magnitude)

> I also have to accuse you of insincerity.

Where? You stated a formula above. You state for s = 1 to n. The only
way I see that can be read is for s = 1, 2, 3, ..., n, not anything else.

> But the cheap revisions that he poses here are weak and I have seen
> this behavior before.

Why cheap? I indicate to you basic mathematical problems with your
model.

It is cheap because you ignored the specification that s is sign.
I see in some other post you suggest putting an underscore.
The point is that you choose to ignore parts of the communication.
I can go back to past threads and find the same poor style of debate.
It only weakens your position.
For you to criticize my communication by this method is paradoxical.

If you write "sum s=1 to n", then it means s is an integer. If that's
not what you mean, then don't write that. You can't say, "s is a sign"
and then say "s=1 to n". The reason is that 1 is not a sign. You can
have a first sign, but then you need to give it a name, e.g., s_1 (where
the underscore means subscript). If you say that s_i is the i-th sign,
then you can write "sum_{i=1}^n s_i x".

I concede that I can do a better job of communicating the properties of
polysign numbers.
In particular there are places where the notation is seriously
conflicted due to the usage in traditional mathematics of the '+' sign
as summation whereas under the polysign approach this does not hold
true. So to do algebra with a '+' sign as summation has to be explicity
stated and cannot mix with concrete instances.

If you are defining a new operation, then state that "+" means your new
operation, not the usual addition. If you need to use both operations,
then make up a new symbol for your new operation.

--
David Marcus

The simplest and most intuitive sign symbology uses
- + * # ...

You are free to use whatever symbols you wish, as long as you state
clearly what you are doing.

Here the first symbol in one line, the second symbol two lines, etc.
These symbols represent natural numbers that will be married to a
magnitude to achieve
-1.2 , +35.6 , *5 , etc.
if the sign in these elemental values is regarded as a natural number
then the generic form is
s x
where s is sign and x is magnitude.

I don't know what "these symbols represent natural numbers" or "is
regarded as a natural number" mean. It looks like you've got a sequence
of "signs". The normal notation for a sequence is something like s_i,
where s_1 is the first element in the sequence, s_2 is the second, etc.
So, you could define s_1 to be -, s_2 to be +, and, in general, s_i to
be your i-th sign. This would let you refer to the i-th sign as s_i. It
is just another name for the same thing.

These are two different data types; they combine but they do not
evaluate to a singular type.

So, you have orderd pairs. s x means the ordered pair (s,x) where s is a
sign and x is a natural number.

They retain structure.
That s is a natural number does not deny it this possibility.
In particular operations on the sx form go like:
( s1 x1 )( s2 x2 ) = ( s1 + s2 ) x1 x2
s1 x1 + s1 x2 = s1( x1 + x2 )
These form the basic operations of polysign numbers, the first being
product and the second being superpostion or summation where the '+'
operator means superposition.

Hold on. You are using "+" as one of your "signs". You shouldn't now use
it for something else. If you do, then you'll get formulas like

( + 5 )( + 6 ) = ( + + + ) 5 6

What is that supposed to mean? It is very confusing.

The product causes a sign sum s1+s2 which is actually a mod n type of
sum.
Really these rules are trying to explain something that is more
practically learned from my website using the second grader approach.
But since mathematicians like things like this sx terminology there it
is. Really there is just one more statement and the polysign
construction is completed:
Sum for s = 1 to n ( s x ) = 0
where s is sign and x is magnitude.

Because noone has gneralized sign before this notation may be
uncomfortable.
Does the s_i symbol mean anything different?

If we define what s_i is, then it is just another name for the thing. If
I say x_22 = 55, then "x_22" and "55" are both names for the number 55
(note the use of double quotes to refer to the string of characters).

--
David Marcus

You have jumped in on the middle of a discussion that regards a
construction that is new. I have written out the description many
times and will do so again here for you. The proper tutorial is my
website:
http://www.bandtechnology.com/PolySigned
but let's just start from scratch. It really won't take that long to
describe the entire system.
Just suppose that there is a new domain of numbers called three-signed
numbers.
Now we need a third sign.
The existing signs are '-' and '+' and so we choose '*' for the third
sign since when we write it down on a piece of paper it has three
lines. So these symboles are numerical mnemonics.
Now, we hunt for a symmetrical property in the reals that can extend to
this three-signed system and find that
- x + x = 0
can be expanded to
- x + x * x = 0 (the identity law).
Under the reals when we do superposition if the values have the same
sign we simply add the values and preserve the sign. The same will
happen here so that:
- 1.2 - 2.3 = - 3.5 .
* 5.6 * 1.1 = * 6.7 .
The way that the identity law gets applied is when we have a value like
- 2.3 + 4.5 * 1.1
This value can be broken down to
- 1.2 + 3.4 - 1.1 + 1.1 * 1.1
where the last three values are equivalent to zero.
So really
-2.3 + 4.5 * 1.1 = - 1.2 + 3.4 .
Now you will be capable of performing any concrete summation in
three-signed math.

The product has a rotational character that matches the real numbers.
Let's just look at the sign rules of the real product as if a plus sign
jumps twice and a minus sign jumps once. In effect this is just a count
that keeps wrapping at two where the amount to count is represented by
the sign mnemonic. This sign product rule extends to three-signed
numbers so that:
- - = +
- + = *
- * = -
+ * = +
+ + = -
* * = * .
If you want to insert ones to make these concrete products that is
fine.
Really you won't need the table. its just addition.
So for example the third line of this table can be used to do:
( - 3 )( * 4 ) = - 12 .
This is no different than what we might write in the reals:
( - 2 )( - 3 ) = + 6 .
The distributive, commutative, and associative properties work so that
( - 2 + 3 )( - 1 * 2 )
= + 2 - 4 * 3 + 6
= - 4 + 8 * 3
= - 1 + 5 .
The last line is the reduced form but it really doesn't matter if you
reduce.
Perhaps you already see that these three signed numbers are
two-dimensional.
Upon graphing a value the reduction takes place automatically. Please
see my website for a drawing. It turns out that they are the complex
numbers in a natural form that extends directly from the real numbers
by generalizing sign. The proof is on my website.

The rules you have hopefully just learned are extensible and allow
algebraic geometry in any dimension. In order to discuss the entire
family in general we need a generic representation. It has taken me a
long time to arrive at the sx notation. I am still integrating it into
my website. The use of modulo sum is clunky until the zero sign is
introduced and I don't bother with that yet since it adds another
element of confusion.

Your concern over the '+' symbol is exposed here in three-sign (P3)
math.
In the reals (P2) we can write:
+ 5 - ( - 1 ) = + 6 .
and we are used to + preserving sign since it jumps two places:
- 4 + ( - 3 ) = - 7 .
But in P3 '*' takes this role so in the FOIL expansion that I wrote out
above I could have done something more like:
( - a + b )( - c * d )
= (-a)(-c) * (-a)(*d) * (+b)(-c) * (+b)(*d)
where all letters are magnitudes. So usually I am careful to say for
example
"In P9
( z1 + z2 ) z3 = z1 z3 + z2 z3 = z3 z1 + z2 z3
where '+' means superposition. This is also true for any sign level
Pn."
Otherwise someone might substitute in values and forget to change the
plus's to s_9's (your underscore notation).

This is a slightly different description than I usually write; I try to
shake it up to see how it works so I would appreciate your feedback on
what is confusing and of course feel free to ask questions. *** and I
have already been through this level months ago. His claim of
misunderstanding the sign representation is not a real position. He
completely understands the construction. He's just giving me a hard
time about notation. His suggestion I guess is what you are pushing but
I'll stand by my earlier critique. The underscore notation will not be
as clean. Anyhow notational variation is allowed. If you want to use
the underscore notation I can follow it while communicating with you.
It does not alter the underlying construction.

-Tim

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