How Do Addition and Multiplication Differ in Boolean Logic?

[Organic]

Hi,

Can someone please show some example of Qualitative difference between Addition and Multiplication, when using Boolean Logic?

(by Qualitative difference I mean that Quantity remains unchanged
during addition or multiplication operations between n positive integers).

 

[matt grime]

Multiplication and addition of what?


You have started this thread instead of answering any questions coherently in some of your other threads. None of the questions was to do with this; once more you are changing the subject and obfuscating the lack of intelligibility in your articles.

If you wish to get some answers, you are going to have to restate the question with some more information. Exactly what do you mean by different? Is it sufficient to demonstrate there exist numbers x and y with x*y not equal to x+y for x and y natural numbers? How far back into the basics must we go? Do you wish for us to talk about cardinals instead of numerals? Must we deal with Peano's axioms?

 

[matt grime]

And why are you posting this in physics/general theory?

 

[Organic]

Matt,

Please look again at my first post.

( by the way also please read this: https://www.physicsforums.com/showthread.php?s=&postid=131193#post131193 )

 

[Hurkyl]

And why are you posting this in physics/general theory?

This is a less inappropriate location than the math forums.

 

[Hurkyl]

It might be instructive to point out that, group theoretically, the only difference between the axioms of addition and multiplication is the symbol used.

And there is at least one "normal" example of addition and multiplication having the same form:

Addition on the real numbers is isomorphic to multiplication on the positive real numbers.

Or to put it in layman's terms, through the proper renaming of numbers and symbols, one can convert addition of real numbers into multiplication on positive real numbers. (and this scheme is reversible)

One example of such a renaming scheme is to replace any number $x$ with the number $2^x$, and replace $+$ with $*$. So, for example, using this scheme, we convert the equation

$$1 + 3 = 4$$

into the equation

$$2 * 8 = 16$$

 

[Organic]

Hi Hurkyl,

Thank you for your answer.

But please try to answer to my question that can be found in my first post.

Thank you.


Organic

 

[master_coda]

Originally posted by Organic
by Qualitative difference I mean that Quantity remains unchanged
during addition or multiplication operations between n positive integers

Are you asking for the difference between multiplication and addition when they both give the same result (such as how 2+2=4 and 2*2=4)?

 

[Hurkyl]

But please try to answer to my question that can be found in my first post.

Your question is too vague. Matt grime asked you to specify better what you mean. ("Multiplication and addition of what?") Until you answer it, I have to rely on my psychic powers to figure out what sort of response you will find interesting. (Apparently, they failed on my first attempt)

 

[Organic]

And I wrote muliplication and addition between n positive interegs.

What is so vague in it?

Here my question again:

Can someone please show some example of Qualitative difference between Addition and Multiplication, when using Boolean Logic?

(by Qualitative difference I mean that Quantity remains unchanged during addition or multiplication operations between n positive integers).

 

[Hurkyl]

I was unaware you had edited your original post to provide your response, instead of saying it in a new post.


There are several qualitative differences between addition and multiplication of positive integers (at least in the English sense of "qualitative"); for instance:

there is no additive identity, but there is a multiplicative identity.

Using addition, the positive integers can be generated by a single element: 1. (in layman's terms, every positive integer can be written using as a formula involving only "1" and "+")

Using multiplication, the positive integers cannot be generated by any finite set of generators; any set of generators must contain every prime number.¹


There is an isomorphism between the positive integers under addition and a subset of the positive integers under multiplication. (Lots, actualy) However, the reverse is false.

(by Qualitative difference I mean that Quantity remains unchanged
during addition or multiplication operations between n positive integers).

I'm just not really sure what you mean by this, though... what quantity remains unchanged? Why did you capitalize "Quantity" and "Qualitative"? By an "addition operation between n positive integers", do you mean something like $a_1+a_2+\ldots +a_n$ where each $a_i$ is a positive integer?


1: (A technical note: due to the way "generate" is defined, any set of generators will generate "1"; even the empty set)

 

[Organic]

Dear Hurkyl,

Maybe this is the right time to look at what I wrote to you here:

https://www.physicsforums.com/showthread.php?s=&threadid=12436

Please look at it, and I am sure that you will understand my point of view.

 

[Russell E. Rierson]

"Addition is repeated multiplication?"

A + B = C

A = C - B

B = C - A

[C - B] + [C - A] = C

For example:

[C - B] = [C^[1/2] - B^[1/2]]*[C^[1/2] + B^[1/2]]

= [C^[1/4] - B^[1/4]]*[C^[1/4] + B^[1/4]]*[C^[1/2] + B^[1/2]]

= [C^[1/8] - B^[1/8]]*[C^[1/8] + B^[1/8]]*[C^[1/4] + B^[1/4]]*[C^[1/2] + B^[1/2]]

= [C^[1/16] - B^[1/16]]*[C^[1/16] + B^[1/16]]*[C^[1/8] + B^[1/8]]*[C^[1/4] + B^[1/4]]*[C^[1/2] + B^[1/2]]

etc... [C^[1/2^n] - B^[1/2^n]] = infinite product...

[C - [C - B]] + [C - [C - A]] = C

etc...

= Infinite composition...

 

[Hurkyl]

Unbounded, not infinite.


It is true that:

$$C - B = (C^{2^{-n}} - B^{2^{-n}}) \prod_{i=1}^n (C^{2^{-i}} + B^{2^{-i}})$$

where $n$ can be any positive integer. However, there is not an infinite version of this product;

$$\prod_{i=1}^{\infty} (C^{2^{-i}} + B^{2^{-i}})$$

diverges to infinity. Also note that

$$(C^{2^{-n}} - B^{2^{-n}})$$

approaches zero as $n$ approaches infinity.


The lesson: just because you can repeat a construction, process, or whatever an indefinite number of times does not guarantee that you can directly translate it into an infinite version of that construction, process, or whatever.

 

[Russell E. Rierson]

[A^[1/2^n] - B^[1/2^n]]*[A^[1/2^n] + B^[1/2^n]] ...

As n--->oo

X^[1/2^n] = 1

The expression approches [1 - 1]*[1 + 1] ---> 0*2

n-->oo [A^[1/2^n] - B^[1/2^n]] = 1 - 1 = 0

n-->oo [A^[1/2^n] + B^[1/2^n]] = 1 + 1 = 2


At unbounded[infinite?] number, does mathematical existence become a self similar pattern?

0202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202

 

[Hurkyl]

At unbounded[infinite?] number, does mathematical existence become a self similar pattern?

0202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202020202

I'm not really sure how this relates either to your previous text, your previous post, nor to Organic's post.

 

[Russell E. Rierson]

[C - B] = [C^[1/2] - B^[1/2]]*[C^[1/2] + B^[1/2]]

= [C^[1/4] - B^[1/4]]*[C^[1/4] + B^[1/4]]*[C^[1/2] + B^[1/2]]

= [C^[1/8] - B^[1/8]]*[C^[1/8] + B^[1/8]]*[C^[1/4] + B^[1/4]]*[C^[1/2] + B^[1/2]]

= [C^[1/16] - B^[1/16]]*[C^[1/16] + B^[1/16]]*[C^[1/8] + B^[1/8]]*[C^[1/4] + B^[1/4]]*[C^[1/2] + B^[1/2]]

n--->oo

A^[1/2]^n + B^[1/2^n] = 2

n--->oo

A^[1/2^n] - B^[1/2^n] = 0

Product of [A^[1/2^n + B^[1/2^n]] becomes 2*2*2*2*2*2...

[A^[1/2^n - B^[1/2^n]] becomes 0.

0*2*2*2*2*2... = 0*2*0*2*0*2... = TFTFTFTFTFTFTFTF ?

 

[matt grime]

Originally posted by Organic
Dear Hurkyl,

Maybe this is the right time to look at what I wrote to you here:

https://www.physicsforums.com/showthread.php?s=&threadid=12436

Please look at it, and I am sure that you will understand my point of view.

Does this thread in anyway *explain* what you meant by qualitative, quantiative, change, or for that matter multiplication and addition.

Why isn't division complementary to multiplication?

 

[Organic]

Dear Matt,

This is a good question.

I did not research it yet, but in general there opposites that destroy each other results, like *,/ or +,- .

In the case of *,+ I have found a very simple model where they complement each other, as you can see here:

http://www.geocities.com/complementarytheory/ET.pdf

It would be a great help if you research my model from *,/ point of view, and show us that my complementary model of *,+ can't hold.

Can you do that?

Thank you.

Organic

 

[matt grime]

I can read more of your stuff in an attempt to
*understand* it (in fact I think I've read this particular one), but I will not. As I've explained to you at length, I do not agree with your statements about boolean logic in NewDiagonal.pdf: they are obviously wrong. It is from these fallacious statements you make there that you seemingly decided to write a whole slew of incoherent pieces without ever bothering to define any of the terms you use prefering instead to rely on the 'inuitive'. You have not answered any of the criticisms of your work and refused to explain the meaning of any of the rhetorical (possibly mendacious) figures you introduce. I have no professional interest in the foundations of set theory, and my personal interest is not in the correctness of *your* theory (one without rigour, or proper definitions) but in attempting to correct the mistakes you make in proper mathematics, i.e. one with rigour, and in trying to make you define *any* of the words you use. At the moment your pieces smack of personal philospophy, and not mathematics.

 

[Organic]

Shortly speaking, you don't understand my point of view because you force it to be in the "shape" of your point view.

For you there is a meaning for all x where x related to some collection of infinitely many objects.

I clearly show the interesting results that can be found if instead of "all" we use the word "any".

Since you refuse to see the fine but critical difference between these to words, when related to a collection of infinitely many objects, you choose not to communicate.

If you choose not to communicate then bye, bye.

 

[Hurkyl]

Whatever meaning one wants to ascribe to "all" and "any" don't matter mathematically; it's the meaning of $\forall$ that counts (as given by the axioms of logic); "all" and "any" are, in any instance I can imagine atm, merely the english translation of $\forall$.

 

[Organic]

Dear Hurkyl,

Is there any reason why you ignore my reqest to read and response to my answer to you at:

https://www.physicsforums.com/showthread.php?s=&threadid=12436 ?

 

[matt grime]

Originally posted by Organic
Shortly speaking, you don't understand my point of view because you force it to be in the "shape" of your point view.

For you there is a meaning for all x where x related to some collection of infinitely many objects.

I clearly show the interesting results that can be found if instead of "all" we use the word "any".

Since you refuse to see the fine but critical difference between these to words, when related to a collection of infinitely many objects, you choose not to communicate.

If you choose not to communicate then bye, bye.

I don't understand your point of view in the sense that I don't think it is something that can be understood as it stands, as you fail to explain adeqautely what you mean, and abuse the vagaries of the English language to hide your intent. Intentionally or not I can't decide.

Besides, I do not need to be a carpenter to know when one table leg is shorter than the other.

 

[Hurkyl]

Is there any reason why you ignore my reqest to read and response to my answer to you at:

(a) I don't want to contribute to hijacking the thread.
(b) I'd rather not have a discussion better suited for TD in the general math forum.
(c) I think I've read ET and AHA before, and had my say then. (I may be up to rereading them at another time, though)

 

[Organic]

(a) If the one who opened the current thread offers you to look at its answer to you, which is related the subject of this thread, so why it is a hijacking?

(b) The thread in General Math is related to my work.

(c) You helped to write the first 9 lines of ET.pdf and I want to thank you for that.

Since then you did not answer to any of my questions about these two pdf files. It will be nice if you find the time for them.

Thank you.

Organic