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#1
07-15-2005, 03:21 AM
 Epsilon=One Senior Member Join Date: Jul 2005 Location: Orange County, California Posts: 2,117
The Proof of One

Some musings:

I would think that number theory is on shaky ground until the value of “One” can be established such that it is a constant throughout any system.

It also seems logical that the numbers and their relationship to each other must be established from natural origins for number theory to be more than a mechanical contrivance.

Such should then directly relate the Fibonacci sequence to Pi, the Golden Ratio, and sinusoidal and elliptical curves; and thus, demonstrate why both the FS and the Golden Ratio are ubiquitous throughout nature. A term to describe this relationship could be referred to as the Natural function.
Mathematics does not
......explain Nature;
Nature explains
......mathematics.
All mathematics is
......a function of Nature;
thus,
......its sublime poetry . . .

Last edited by Epsilon=One : 06-30-2006 at 06:00 PM. Reason: format
#2
01-24-2006, 12:30 AM
 JasonRox Junior Member Join Date: Jan 2006 Location: Ontario Posts: 6

What do you mean by the definition of the number 1?

It seems pretty well established through the Peano Axioms.
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"Learning mathematics is always extraordinarily hard work -- reading it, listening to lectures." - Paul S. Halmos
#3
01-24-2006, 06:08 PM
 Epsilon=One Senior Member Join Date: Jul 2005 Location: Orange County, California Posts: 2,117
Mathematics is a function of Nature.

Quote:
 Originally Posted by JasonRox What do you mean by the definition of the number 1? It seems pretty well established through the Peano Axioms.
All mathematics is unprovable until "One" can be proved from within any system. The Peano Axioms do not assign a value to "One"; they merely state that it exists. Other numbers are functions of "One."

All mathematical numbers represent values that are functions of an arbitrary value for "One"; said arbitrary value must be proven.

The solution depends upon defining "One" in terms of the given system of numbers without the definition being circular. In essence, this is what Kurt Gödel said was most unlikely, or metaphysical, with his Incompleteness Theory.

Pulsoid Theory's Natural source of numbers and their mathematical manipulations are related to the Proof of One.

See: "Universal Proof of One."
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"Seek simplicity; and
. . . Natural integers."

...c˛ = 2v˛ – s˛........v = εP˛

Last edited by Epsilon=One : 01-24-2006 at 06:14 PM. Reason: spell check
#4
06-10-2006, 12:07 AM
 phyti Junior Member Join Date: Jun 2006 Posts: 6

The standard definition of a prime integer is: an integer divisible only by itself and 1.
Since all integers are divisible by 1, this cannot be a qualification.
The integer 1 is only divisible by itself but is excluded to preserve the uniqueness feature of the fundamental theorem of arithmetic
The fundamental theorem of arithmetic states: all integers>1 are the unique product of primes, even though prime integers have no factors (a product requires 2 factors). A single prime is not a product unless paired with 1, but this would allow multiple one's, which violates the unique product aspect of the theorem.
These two 'fundamental' examples alone show the need for consistent definitions without arbitrary restrictions and fudging to make things work.
An alternate prime definition is: a prime integer is a multiple of 1 only.
The integer 1 is excluded by definition because it is not a multiple.
The fundamental theorem of arithmetic could be redefined as:
all composite integers are the unique product of primes.
These defintions require a different concept in which the integers contain
three mutually exclusive classes of objects, {0, 1, M} which eliminates division
by zero, and the definiton of identity elements.
The primes stand alone and 1 is not a factor in any integer because it is not a prime. If any interest I have a paper on a simple set development for the integers. phyti
#5
06-16-2006, 01:22 AM
 Epsilon=One Senior Member Join Date: Jul 2005 Location: Orange County, California Posts: 2,117
The source of numbers and arithmetic is Nature.

Quote:
 Originally Posted by phyti The standard definition of a prime integer is: an integer divisible only by itself and 1. Since all integers are divisible by 1, this cannot be a qualification.
Your definition is not exactly accurate even for the conventional definitions of prime numbers. More better: An integer not divisible by any integer factors except itself AND one. Your interpretation of being simply divisible by one would truly, as you point out, be wrong. You are very perceptive concerning the problems that few can see; however, I question your grasp of where the answers lie.

You are quite correct in asserting that the many definitions of “primes” are quite contrived and in many ways arbitrary. This fuzziness of definitions can be extended to the entire concept of numbers without too much effort.

There is only one definition of primes that should be considered as a Standard Model from which all other sets/definitions are derived as special cases. Any other beginning definition would be “incomplete.” The conventional definition, no matter how applied, is not actually even a single set of numbers.

The source of numbers and arithmetic is Nature where “primes” are very integral to the most seminal constructs . . . without "primes" there is no structure for seminal quanta. When Natural “primes” are understood all the mystery of the uniformity of primes vanishes. The distribution of Natural "primes" is uniform and with simple algebra all "primes" can be mapped to any uniform set of positive integers.

Nature does not consider one or two as “prime.”

I would be interested in your paper concerning the development of the integers; though, if this post flows from it, I don’t believe, as you assert, that you have an understanding of the origin of numbers; particularly, the status of One, Zero, the common denominator concept, and prime numbers. Have you considered an answer to Gödel’s Incompleteness Theorem?
__________________
..."Click" to E-mail Me Directly
.....Also, my forum Private Msg box,
..........finally, is now working.
"Seek simplicity; and
. . . Natural integers."

...c˛ = 2v˛ – s˛........v = εP˛
#6
06-28-2006, 05:16 PM
 OfficeShredder Senior Member Join Date: Jun 2006 Posts: 191

Let zero be defined as the quantity of elements in the null set (we can all agree the null set exists, right? Because if it didn't, that would be weird).

Then take a new set, named A. A is the set of all subsets of zero. That is:

A = {null}.

1 is the quantity of elements in A.

Let B be the set of all subsets of A.

B = {null, {null}}.

2 is the quantity of elements in B.

etc. etc.
#7
06-28-2006, 06:10 PM
 Epsilon=One Senior Member Join Date: Jul 2005 Location: Orange County, California Posts: 2,117
Problems concerning one and zero

Quote:
 Originally Posted by OfficeShredder Let zero be defined as the quantity of elements in the null set (we can all agree the null set exists, right? Because if it didn't, that would be weird). Then take a new set, named A. A is the set of all subsets of zero. That is: A = {null}. 1 is the quantity of elements in A. Let B be the set of all subsets of A. B = {null, {null}}. 2 is the quantity of elements in B. etc. etc.
The above comments hardly solve any of the problems concerning "One" or the Natural source of numbers.

To begin: sets are contrivances.

Zero can not be so easily defined; because, zero has several distinct connotations that must be defined before it can be associated with any set.

It's rather hard to understand the existence of the "null set," whether that be "weird" or not, until you define existence. For many, symbols are not existence, but merely placeholders for the "real" thing . . . that usually exists.

Your use of "1" as the "quantity of elements in A" would hardly satisfy the requirements of Gödel's Incompleteness.
__________________
..."Click" to E-mail Me Directly
.....Also, my forum Private Msg box,
..........finally, is now working.
"Seek simplicity; and
. . . Natural integers."

...c˛ = 2v˛ – s˛........v = εP˛
#8
06-28-2006, 09:08 PM
 OfficeShredder Senior Member Join Date: Jun 2006 Posts: 191

Quote:
 Originally Posted by Epsilon=One To begin: sets are contrivances.
That depends on how you look at things. While obviously we define a set in a mathematical sense (meaning it's contrived for us), there's no reason for you to not consider the possibility that the single most fundamental building block in the universe is the concept of a collection. Atoms are a collection of protons, neutrons, and electrons, which are collections of quarks, etc. There's no reason to pass sets off as a contrived idea; similiarly, any concept in any field could be so passed off.

Quote:
 Zero can not be so easily defined; because, zero has several distinct connotations that must be defined before it can be associated with any [set.
You can define the quantity of zero to be the number of elements in the null set. The rest of the properties can be followed up on just like they were over the course of history.

Quote:
 It's rather hard to understand the existence of the "null set," whether that be "weird" or not, until you define existence. For many, symbols are not existence, but merely placeholders for the "real" thing . . . that usually exists.
The great thing about the null set is that it exists even if nothing else does (in fact, you could call everything the null set then).

Quote:
 Your use of "1" as the "quantity of elements in A" would hardly satisfy the requirements of Gödel's Incompleteness.
I'm not quite sure what you're looking for anymore...

Quote:
 I would think that number theory is on shaky ground until the value of “One” can be established such that it is a constant throughout any system. It also seems logical that the numbers and their relationship to each other must be established from natural origins for number theory to be more than a mechanical contrivance.
There is a single particular value of 1 using the set addition definition throughout any system. Furthermore, ignoring your qualm about sets not being "real" enough for you, it's a natural relationship from one number to the next (particularly from 0 to 1), and not a simple mechanical contrivance. You can even add and multiply the set values of natural numbers, and it works out great.
#9
06-30-2006, 04:06 AM
 phyti Junior Member Join Date: Jun 2006 Posts: 6
definition of one

To form knowledge the mind;
perceives reality,
forms concepts to model reality,
predicts reality from these concepts,
retains the concepts as knowledge when prediction matches reality.
modifies the concepts as needed.
Knowledge is a set of concepts used as a reference for understanding.
By definition knowledge is always incomplete because all reality is never perceived.For simplicity, a concept is defined within a context that excludes other concepts.
Other concepts may not be relevant to the purpose.
There may be relevant concepts that have not been discovered.
An approximate definition may be sufficient for the purpose.
Definition is a relative referencing process.
A definition is expressed in terms of other definitions.
This process can be circular or incomplete.
Forming knowledge is a continuous process of refinement.
Knowledge is only as good as its definitions
In summary we know reality indirectly through images.
The preceding forms a basis for the following.
If knowledge is incomplete then in principle all systems,
including those analyzed by Goedel are incomplete.
History supports this view by the constant revision of theories and methodologies.
The concept of number was created from a need to measure things.
To measure is to compare an object to a reference object to evaluate a property
A property is an attribute/quality of an object that identifies it from other objects [color, texture, shape, hardness]
A unit is a concept defined within a context as simple, basic, indivisible [i.e. anything can be defined as a unit]
The simplest natural number set is the fingers, used for counting.
One is a relative concept depending on perspective. A person has one car but to the mechanic it is many parts.
The occurrence in the natural world, of spirals, Fibonacci sequences, etc. to me indicate design and order,
something that cannot be explained by random variation.
0 = nothing: literally "no thing", a condition or state of emptiness
Zero and one (nothing and something), are mutually exclusive concepts.
On that basis alone, how can 0 be treated as a typical number? A place holder, yes, a different type, yes.
What is one of the biggest bugs of all programmers,... division by zero.
Ask yourself, why can't I divide by zero. Division works for all other numbers so the operation can't be at fault,
that leaves zero as the problem.
My suggestion is, stop reciting the axioms like a mantra, get a good dictionary, and think outside the box.
Discovery is wonderful!
#10
06-30-2006, 11:09 AM
 OfficeShredder Senior Member Join Date: Jun 2006 Posts: 191

Quote:
 Originally Posted by phyti On that basis alone, how can 0 be treated as a typical number? A place holder, yes, a different type, yes.
It's not... it's the only number that's neither positive nor negative. This is important.

Quote:
 What is one of the biggest bugs of all programmers,... division by zero. Ask yourself, why can't I divide by zero. Division works for all other numbers so the operation can't be at fault,
Right off the bat, ignoring convergence issues, the lack of it being positive or negative means you don't know whether the quotient is positive or negative. So it makes an intuitive sense that you can't divide by zero.

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