wolframscience.com

A New Kind of Science: The NKS Forum : Powered by vBulletin version 2.3.0 A New Kind of Science: The NKS Forum > NKS Way of Thinking > Monadic Mathematics
  Last Thread   Next Thread
Author
Thread Post New Thread    Post A Reply
Doron Shadmi
http://www.tahal.com
Israel

Registered: Aug 2004
Posts: 9

Monadic Mathematics

One of the efficient ways to change some framework is to ask new fundamental questions that maybe lead us to new frontiers.

For example, the fundamental question of the language of Mathematics framework is: ”How many?”.

Let us try another question, for example: “What do we have?

A closer look of these questions shows that ”How many?” questions are mostly about the Quantity concept, where “What do we have?” questions are mostly about the Structure concept.

Let us check if “What do we have?” questions can be a fruitful ground for mathematical development.

First we have to define the minimal concepts that can be used under the structure concept.

In other words, these concepts when used, determine our framework’s domain, for example:

Let use say that the two main concepts that are related to the structure concept are Length and Direction.

It means that by using these concepts, we can define the building-blocks of our mathematical framework.

We also know that by using these building-blocks we suppose to get some input that can be used by us to develop our framework.

So, the next question is “What are the limits that beyond them no input can be found?

The lowest “no input” state is Emptiness, where Length and Direction do not exist.

The highest “no input” state is Fullness, where Length and Direction are beyond measurement.

So the useful elements have measurable Length and/or Direction, that enable us to use them as some input.

Let us represent these ideas in a table:


code:
Monadic Mathematics build-blocks -------------------------------- V = Available X = Not Available +-----------------------------+ | Measurement of | | | | Length | Direction | |--------------|--------------| | | | {} | | | Emptiness | | | | | | |--------------|--------------| | | | {.} | V | | Point | | | | | | |--------------|--------------| | | | {._.} | V | V | Segment | | | | | | |--------------|--------------| | | | | | V | Tendency | | | | | | |--------------|--------------| | | | {__} | X | X | Fullness | | | | | | +-----------------------------+




General:

{' and '}' are the notations of a framework which we call a set.

Between these notations we can put our examined concepts and then try to find out what we can do with each one of these concepts, and also what interactions can be found between concepts and themselves and/or concepts with other concepts.

The concept of Nothingness or Emptiness is notated in this framework by {}.

Any examined concept in my framework is examined by its structural properties and also by its Quantitative properties.

The most basic Structural property in my framework is based on the Length concept.

From this point of view, a Point {.} has exactly 0 Length.

The Length concept has no meaning in my framework when it is related to the Emptiness concept (which is notated as {}), therefore 0 cannot be connected directly to the Emptiness concept.

Another option to define 0 is to ask: How many things there are in {}?

By 'how many?' question we actually define the Cardinal concept, which is notated by using '|' and '|' .

In this case, the cardinal of {}, which is notated as |{}|, is equal to 0.

In Standard Mathematics framework (when Ordinality is omitted) the one and only one option to connect between the Number concept and the Set concept, is by using only the Quantity concept, and in this case |{}| = 0.

But as you see, in Monadic Mathematics framework, there are two kinds of cardinals, where one of them is the standard Quantitative Cardinal, but the second type of cardinal is what I call the Urelement Cardinal, which is based on the Length concept (which is the structural property of the Number concept).

In Monadic Mathematics, the structural property of a number is more basic then its quantitative property.

According to what I wrote above, when a Point eliminates itself, then the result is Emptiness.

In other words Point - Point implies Emptiness, or in other representation:
0 - 0 implies {}.

Standard Mathematics takes '0' notation as something which is first of all related to the Quantity concept.

In this framework 0 - 0 = 0

Monadic Mathematics takes '0' notation as something which is first of all related to the Structure concept.

In this framework 0 - 0 implies {}




A non technical explanation of Monadic Mathematics' '+' and '-' operations:

The most basic question of Standard Mathematics is “How many?”.

The most basic question of Monadic Mathematics is “What do we have?”

The basis of a “How many?” question is the Quantity concept.

The basis of a “What do we have?” question is the Structure concept.

It means that the Number concept in Monadic Mathematics, is first of all based on the Structure concept.

Question:
What are the minimal, and distinguished, structural forms in Monadic Mathematics?

Answer:
Emptiness (notated as {}), Point (notated as {.}), Segment (notated as {._.}), Fullness (notated as {__}).

{} and {__} are the weak ({}) and the strong ({__}) limits of Monadic Mathematics framework.
It means that they cannot be used as inputs in Monadic Mathematics.

So, Monadic Mathematics operations are based on {.} and {._.}.

The two basic structural properties, which related to the Number concept, are Length and/or Direction.

{.} has 0 length and no directions.

{._.} has 0_x length and at least two opposite directions, which are
0_x or x_0.

The most basic arithmetical operations between these elements are ‘+’ and ‘-‘.


Important:

If '-' sign is used not as a binary operation, for example: -x_0, then it is a negation symbol and not a binary Elimination operation.

Therefore -x_0 = 0_x or -0_x = x_0.

Also {} - x_0, which is a vacuous binary operation (because {} or {__} cannot be used as inputs) is equal to -x_0 = 0_x, etc.


‘+’ is understood as Concatenation, for example:

1_0 + 1_0 = 2__0 (._. + ._. = .__.)

0_1 + 1_0 = 1_0_1 (._. + ._. = ._._.)

0 + 0 = 0 ( . + . = . because Concatenation of 0 Length is 0 Length)

0__2 + 0_1 = 0___3 (.__. + ._. = .___.)

2__0 + 1_0 = 3___0 (.__. + ._. = .___.)

0_1 + 2__0 = 2__0_1 (._. + .__. = .__._.)

etc …


‘-’ is understood as Elimination, for example:

1_0 - 1_0 implies {} (._. - ._. implies {} because both Length and Direction are identical)

0_1 - 1_0 = 0 (._. - ._. = . because only Length is identical)

0 - 0 implies {} ( . - . implies {} because Length is identical and there is no Direction)

0__2 - 0_1 = 0_1 (.__. - ._. = ._.)

2__0 - 1_0 = 1_0 (.__. - ._. = ._.)

0_1 - 2__0 = 1_0 (._. - .__. = ._.)

etc …


More detailed information can be found in:

http://www.geocities.com/complement...irst-axioms.pdf


According to Monadic Mathematics the Number concept is the fundamental building-block of a non-destructive interaction between opposites, which is based on Included-middle reasoning (which is the logic that can be found between at least two opposites that define their middle domain).

For more details please read:

http://www.geocities.com/complement...aloisDialog.pdf

__________________
http://www.geocities.com/complement...ry/CATpage.html

Last edited by Doron Shadmi on 07-27-2005 at 02:38 PM

Report this post to a moderator | IP: Logged

Old Post 10-16-2004 01:30 AM
Doron Shadmi is offline Click Here to See the Profile for Doron Shadmi Click here to Send Doron Shadmi a Private Message Click Here to Email Doron Shadmi Visit Doron Shadmi's homepage! Edit/Delete Message Reply w/Quote
Gunnar Tomasson


Registered: Oct 2003
Posts: 69

Re. the following:

One of the efficient ways to change some framework is to ask new fundamental questions that maybe lead us to new frontiers.

For example, the fundamental question of the language of Mathematics framework is: ”How many?

Comment:

I agree that the question "How many?" is the fundamental question of the LANGUAGE of Mathematics.

I also have come to view the LANGUAGE of Mathematics as the rules of the game whereby we DEDUCE conclusions from some set of GIVEN axiomatic premises.

If that be so, then it seems to me that the "language" of mathematics cannot in principle advance our understanding of the world we live in.

For the "language" of mathematics is a man-made "vocabulary" for expressing in meticulously precise terms whatever conclusions may be drawn from a given set of axiomatic premises.

As for the latter, axiomatic premises do not spring up and, as it were, identify themselves to the student of nature.

Instead, as emphasized by Einstein to the consternation of some/many/most of his peers, the axiomatic premises of our MATHEMATICS and PHYSICS are the product of our INTUITIVE faculty.

As such, our mathematics and physics are - and cannot in principle be anything but - refined products of our intuitive faculty.

Products, however "refined", which are mere shadows projected upon the "walls" of our ignorance as famously suggested by Plato.

String theorists beware!

Report this post to a moderator | IP: Logged

Old Post 10-19-2004 02:44 AM
Gunnar Tomasson is offline Click Here to See the Profile for Gunnar Tomasson Click here to Send Gunnar Tomasson a Private Message Click Here to Email Gunnar Tomasson Edit/Delete Message Reply w/Quote
Doron Shadmi
http://www.tahal.com
Israel

Registered: Aug 2004
Posts: 9

Dear Gunnar Tomasson,

First, thank you for your reply.

In Monadic Mathematics we do two basic things before we start to develop any possible product that can be found within its domain.

1) We clearly show its lowest and highest limitations.

2) We use the Included-Middle logical reasoning, which is the framework that stands in the basis of any existing product, which is the outcome of any possible interactions between at least two opposites.

After we did these two steps, we can talk about anything, which is under these two principles.

I have found that this attitude can lead us to ask interesting questions about the abstract/non-abstract aspects of our life, which lead us to become active participators of our answers, and not just passive observers of them.

In my opinion this is the real value of any living language, formal or informal, that it can be changed by its users, during non-trivial dialogs among them, that also includes within it “pointers” that also related to their common environment.

Form this point of view, I do no agree with Platonic idealism or any Formalistic scholastic closed game.

In short, the Number concept of Monadic Mathematics is the building block of the non-trivial progressive dialog between opposites.

For more details please look at http://www.geocities.com/complement...y/TheBestOf.pdf

Thank you.

__________________
http://www.geocities.com/complement...ry/CATpage.html

Last edited by Doron Shadmi on 10-20-2004 at 11:34 AM

Report this post to a moderator | IP: Logged

Old Post 10-19-2004 10:10 AM
Doron Shadmi is offline Click Here to See the Profile for Doron Shadmi Click here to Send Doron Shadmi a Private Message Click Here to Email Doron Shadmi Visit Doron Shadmi's homepage! Edit/Delete Message Reply w/Quote
Post New Thread    Post A Reply
  Last Thread   Next Thread
Show Printable Version | Email this Page | Subscribe to this Thread


 

wolframscience.com  |  wolfram atlas  |  NKS online  |  web resources  |  contact us

Forum Sponsored by Wolfram Research

© 2004-14 Wolfram Research, Inc. | Powered by vBulletin 2.3.0 © 2000-2002 Jelsoft Enterprises, Ltd. | Disclaimer | Archives