Article #35
Subject: How do You define the "number One"?
Author: Andrew W. Harrell
Posted: 7/27/2009 08:20:45 PM

How do You define the number One?



Here is another of those difficult philosophical problems mentioned at the
beginning of this research paper. How can we understand what the “Concept of
a Concept” is if we don’t understand what the Concept of One is? This basic
question is phrased above as a mathematical question in the logical
foundations of the branch of mathematics called set theory. We believe the
question of how we can develop a better understanding of science and faith is
interconnected to this question. Thus, it is not only its mathematical
implications that make this conundrum important for us to solve.
This question from mathematical research relates to a fundamental theological
and spiritual one… If we are to say that the fundamental nature of God is
that He is One, what do we mean by this? The standard theological answer to
this question is contained in a religion’s teaching about the name of God
(which is a trinity for Christians and Hindus). Is it possible to be a Father
and Mother of God using structures of thinking within His Holy Spirit
operating in our own minds and spirits and souls? Elsewhere on this prayer
website I have posted some thoughts about this. And, an understanding of how
our human interest in this question as a mathematical one developed will
certainly better help us understand how our different human forms of science
and faith are related in us. To do this we need to go back to the turn of the
last century when questions of the foundations of the area of mathematical
analysis and the beginnings of the development of mathematical logic as it
relates to the invention of computers was being developed.
The mathematical area of research called set theory was created in order to
understand what “a real number” in Calculus means. This interest developed
because in the previous decades techniques were developed in order to solve
practical problems in mathematical analysis which made use of what Cauchy and
Gauss (and earlier Euler) had called “complex” or “imaginary” numbers. Euler
used an algebra of calculation in his trigonometric formulas (which had
applications of mapping and geodesy) which made use of the imaginary
number “i”. Cauchy further developed these algebraic techniques and also
showed how it was possible to integrate functions involving complex numbers.
Gauss developed the beginnings of the concept of a “manifold” which would
later revolutionize thinking in electrodynamics and Einsteinian physics. So
the questions then became, “What is this ‘real number’ which determines how
we calibrate or measure the space we are analyzing?” “What is a real
function?” “What is a complex function?”
Of course, for centuries and millennium philosophers had speculated about
various theories of reality and metaphysics. But, in order to answer this
question in an scientific and analytic sense people began thinking about what
particular logical mathematical foundations that we had up to now assumed as
given in the background of our axiomatic system determine its solution.
Various theories of generalized algebraic numbers were created and Leopold
Kronecker made his famous statement, “God created the integers and all else
is the work of man.”. But, how did God create the integers? People noticed
that the positive integers formed what was called in set theory a “sequence”
and that one of the main ways things were proved in mathematic problems
involving sequences of integers was something called the “inductive
principle”. If a statement or proposition about integers was true for the
next integer, after a given integer (no matter what that integer was) and it
was true for the first integer, then it had to be true for all integers. The
self-evident truth of this fundamental principle of mathematical proof of
course depends on the fact (which is not true for all sets of objects) that
there is a least positive integer, “One”. So, how do we define the “number
One”?

The Greek philosophers realized this as a fundamental philosophical question
too. Plato’s dialogue Parmenides is perhaps his hardest and most important
attempt in the classical era to try and understand this . And, it deals with
just this question, “What is does the Concept of One mean philosophically and
mathematically?’
Here is the most generally accepted mathematic answer, figured out by
the mathematician/philosophers Frege, Betrand Russell, and Peano at the turn
of the beginning of the last century. In short… The number One is “the
cardinality (similarity class of one-one functions) of the set whose only
member is the empty set.” This definition hides a huge logical complexity of
definition. What is a similarity class? What is a one-one function (what is a
mapping or function for that matter?). What is a set? What is an empty set
(basically this is determined logically when you know what an element in a
set is and what a set is)?
So what are the philosophical implications of this definitions? First
of all in order to understand what the concept of Oneness is we have to
understand what logic is. There must be an intellectual component (ie not
only intuition to our theory of knowledge). And, we must understand what
reality and the reality of an object is or means (for it to be an element in
a set) in terms of the aforementioned intellectual component of our theory of
knowledge. Then, if we understand this we must still also understand what
functional computation (which allows us to create one-one mappings) is. This,
in turn, allows us to understand what mathematical and
scientific/theological Oneness is. We must understand this in order to
understand what these two things are and the important fundamental question
to our ethical and philosophical theories of well-being and truth.

This is a very short history of how the question, “How do you define
the number One?” was so important to mathematicians, and scientists, in the
last century. But, what about its theological importance? Is it in fact the
case as Dr. Kronecker has said, “God created the integers (and the science of
nature that depends on using them to count and measure) and all else is the
work of man.” Uf an understanding of Oneness in terms of natural science is
all that we want, then the definition of One from the above paragraph in set
theoretical logical terms: “The number One is the cardinality (similarity
class of one-one functions) of the set whose only member is the empty set.”
Solves the problem. With an understanding of this definition we have
understood how God created the idea of “Oneness” inside of his created
Universe. But, the Bible says that God did more than just create the World.
It says that he created Man (and also Woman) in His own image and likeness.
How does this relate to the above proposed set theoretical/logical definition
of what “the number One” is? In the time between the beginning of the last
century and our new millenium Mathematicians and logicians have shown, except
for some notable gaps, how “real numbers (rational, algebraic,
transcendental)”, and likewise various other “complex and ideal numbers” can
all be constructed logically from the positive integers. The possibility of
the “notable gaps” come from the proof of the independence of the continuum
hypothesis.

Article #39
Subject: Offering of a lotus of happiness for you, a Christ Jesus Buddha to be
Author: Andrew W. Harrell
Posted: 9/3/2009 10:20:40 AM

For a practical guided meditation, ordered, to help, you, define the number
One as your own personal spiritual oneness which has as a part of itself and
the World it experiences around us, a substance of things hoped for and the
capacity to experience the evidence of things not seen, "a five-dimensional
thought form in our shared consciousness which consists of nothing inside of
nothing and which can appear, in a logically consistent way, inside of God
and us from starting from nowhere" see the posting on 5 Yoga sutras in the
Yoga and meditation section.

Article #160
Subject: LinkedinDiscussions on this
Author: Andrew W. Harrell
Posted: 5/22/2012 07:42:55 PM

How do You define the number One?



Subject: Linkedin Discussions on this at
http://www.linkedin.com/groups/How-do-we-define-Number-56601.S.89357390
Author: Andrew W. Harrell
Posted: 1/17/2012 03:31:41 PM
How do we define the Number One?.
This question arises when we consider how we can develop a better
understanding of the interrelations of science and faith. At the turn of the
last century work on the area of the foundations of mathematical analysis and
the beginnings of the development of mathematical logic increased. This
happened along with the invention of digital computers. And, a new area of
mathematical area of research called set theory was created in order to
understand what "a real number" in Calculus means. Leopold Kronecker made
his famous statement, "God created the integers and all else is the work of
man." But, how did God create the integers? Plato's dialogue Parmenides is
perhaps his hardest to understand work and the most important attempt in the
classical era to try understand different ways we can answer this question.
6 hours ago

Mike Rand • Frege had a go at this.........well it was his life's work.

He defined the number one after he defined the number zero.

Zero is the representative of sets that have no elements. There is only one
such set, and thereby we define the number one.

Now I think this reveals something really important. Whilst Frege actually
thought that numbers had some real extra-mental existence, I think this
conceptual approach simply reveals that the way in which we can not analyse
our understanding of numbers objectively. Numbers are intuitive. Kant
understood this best. They are, to borrow his phrase, synthetic a priori. Do
they exist extra-mentally? Kant understood this too....we can not know what
is in the world as it is in itself (the noumenal world).

Does the number one have existence? - yes as a human concept - does it have
existence independent of the human race? - we can not know.


PeterUnfollow Follow Peter
Peter Anderson • Defining the number One requires the perception of unity.
The number one is defined by your perception of space, and the potential for
that is a priori in your mind, and also in the availability of the objects of
sensation to which that potential is applied, as Kant said.

A set with no elements only exists as the definition of its parameters - the
possibility of a set. So it is with a priori perception of unity - it exists
only as a potential perception of sensations synthesized.
0

Andrew Harrell • Yes, Mike Frege's 1879 article, "Begriffsechrif, a formula
language, modeled pon that of arithmetic, for pure thought" was the first
really detailed explanation and follow up of Kant's ideas on this. It is
translated into English in the book"From Frege to Godel".
Frege's defintion is not a tautology in the same sense that Luca caught me
saying wrongly last week in his group that "1+1=2" is not a tautology. Kant
has explained this to us. Read his books.

Bertrand Russell's and Frege had a long discussion about this in which
Russell and Whitehead took another viewpoint in their monumental
work "Principles of Mathematics". Quine a student of Russell's and Whitehead
wrote several books later taking their logical positivist position.
My own viewpoint is that it is possible to have "a set of all sets" exist. If
you assume this you can have what Kant wanted, such a thing as a "a concept
of a concept" which explaines the meaning of meaninful words..
Then the number one in mathematical logic is defined as "the set of all sets
which have no elements in them" (which is really what we call a "class" in
functional programming ala Church, Haskeell. Russell and Quine defined it
as "the set which has only one element, the empty set" in it. If you are
going to write a computer compiler (something I have never done, but tried
several times) the consequences of which way you define the number one can be
monumental.
Hence my question, "How do we (or you) define the number one?"
Luca welcome to our group. In this group there is a least someone (me) who
believes that God created everything. God Bless.
1 hour ago


Andrew Harrell • Peter, you are correct in saying the number one can only be
a thing, a "substance" if it exists in space and time. For something to be a
substance it has to have form (ie be defined at points of space and time like
a quantum field, or electrical field,or magnetic field in mathematical
physics is. However, if you consider 4 or 10 or 11 dimensional space and time
as the compactification of a larger dimensional object of symmetry (28
dimensional Lie Group) then all the symmetry parts of the mathematical are
not necessarily things having "substance". They can be "no
things", "nothing". If God exists in this higher realm of symmetries, then He
can create things out of this nothing.
Hence, we can say God created the world out of nothing. But, us out of
something, Himself.. Do you understand this?
1 hour ago


PeterUnfollow Follow Peter
Peter Anderson • Andrew, I don't understand much of that, no. From the
Kantian perspective of spatial perception numbers other than one can only be
conceived by regularity of division, not addition, because there is no
awareness or consciousness of sensation prior to synthetic unity, and
mathematics is a discipline of consciousness. What I think is really
interesting is that once you have a grasp on the sequentiality of perceptions
of spatial unity (Hume-like), you can use the awareness of discrete moments
in time to count the number of conscious perceptions you make to assemble
more detailed unities, including historical unities, and thus induct higher
numbers into your experience. They are at that point abstractions which
mathematicians can manipulate.
38 minutes ago •

Andrew Harrell • Okay, Peter, let's leave aside the unified field theory of
mathematical physics and concentrate on examples in computer programming and
mathematical logic trying to under stand Kant and Frege together as amateur
philosophers. I didn't earn my living in philosophy, only in mathematics,
computer programmng, and engineering.. Do you have a background in graduate
studies in studies in philosophy or mathematics or computer programming?
22 minutes ago


Andrew Harrell • Peter, what do you mean by the phrase, "sequentiality of
perceptions of spatial unity (Hume-like)? Do you mean spatial unity in the
sense of overlapping, reflected coordinate realizations as in a mathematical
manifold? Hume believed that our minds are blank slates with no a priori
reference concepts, didn't he. Doesn't this conflict with Kant's belief in a
priori synthetic concepts?
to Evans: I'd like to think that I understand some of Your examples of (mis)
using of "number";
but the question was: what are YOU mentioned in;
do YOU can exactly describe YOURS notion of a number?;

an exagerrated example (V.I.Arnold): Jacobi saw as a most remarkable property
of math: the SAME FUNCTION governs over an integer (number) in a form of
a "sum of four squares" as like as over the "true motion of pendulum";

and a question of existence can't be decided by a sole indication to
a "fiction";
then we must think across what existence have indications (etc.);
Posted by Anatoly
Anatoly writes: what is a number?;
is it a "set of undivided unities"?;
do you say about numerical or order numbers?;
is it a finite construction (or a construction based on a finite schemes of
axioms)?;
is it a term, defined implicitly, in an axiomatic way?;

Jud:
It depends on the context in which it is used.

In the street if I stop to buy a hot-dog (which, being a vegetarian I would
never do) I might say:

*A hot dog please.*
The vendor (seeing I was accompanied by a companion) might say:
*Just one?*
*Yes please* I answer.

In a discussion with my Professor of Mereology he might, handing me a
pomegranate, say:

*Is this one object or many?*
Knowing that he was a Professor of Mereology, and intuitively realising that
he wished to trick me so I might say:

*I depends upon which way you are looking at it Professor. - I

1. *Are you you a conceiver of it as a macro singleton?*

2. Or are you a conceiver of it as a composite of the hundreds of seeds
within which the outer peel contains? Are you a conceiver of a heap of sand
one object, or are you a conceiver of it as a collection of millions of
individuate contiguous tiny objects? Are you a conceiver of the Milky Way as
one enormous object? Are you, like Parmenides, a conceiver of the Universe
one even larger object?*

Numbers?
Numbers do not exist - they are useful fictions which originated as useful
linguistic attributes invented by primitive man to point to the difference
betwixt one object and more/many items in a collection of objects.

Such words (later symbols) made it easier to differentiate, discuss, decide
and determine the number of goats or geese the leader of the tribe or the Ju-
Ju man is willing to give him, so that he can add his daughter as a concubine
or slave to the seventy-seven he already has, etc.

Then some clever crowd of conceptual clever-clogs learned to abstract such
convenient morphemic clusters (names for numbers) add semeotic symbols as
handy aleratives and cut the numerical linkages to material objects
(daughters, goats, wine-jars, bushels of corn, or whatever) and manipulat
numbers abstractively.
There was no longer a need to preface every mathematical statement by saying
something like:

*Imagine there were twenty-seven wine jars lying on the riverbank and a big
wave came and washed nine of them away - how many were left on the bank?*

A man could now forget all about such objectival scene-setting and partially
abstract by simply saying:

If the were 27 imagined things and you subtracted 9 imagined things what
would that equal?

Complete abstraction came later when a man could say *What is 27 - 9?*

In other words such useful fictions and all the paraphernalia of *x* and the
backward* E* and sets - and all the other utile mathematical transcendental
tokens that came later were NAMED into a putative "existence" by mankind.

They were NOT created by some great Number God in the sky , that scratched
his crotch, absent-mindedly threw an olive to his favourite Greek groveller
Plato , looked down at the mess he had created and thought:

*I better provide those poor buggers below with a few numbers to help them
mathematically sort themselves out."

Cheers,
Jud






avingYour can bet your bottom ruble that *nature* (the material imperia) did
not provide objects in single units so that some organanic blood and water-
filled organisms on the verge of blowing up the planet some tiny planet
parked far away in the suburbs of a minor star-system to make it easy for
them to work out their muliplication tables.
Posted by Jud


@Andrew
You might directly ask Miran Lipovača, the author of the Haskell book you
refer to. He's on Google+, he may even be interested in joining us here. I am
a(n) Haskell programmer, but my interest is in sociology/anthropology, which
is why I spend time in humanities forums instead of programming forums.
Arguing with programmers gives me a major headache. That's not a 'dis to
them, it's just a result of a major generation gap (I'm old).
Posted by Michael

There is a documentary called "The story of One," made by Terry Jones (a
Monty Python member). Quite interesting...

http://videosift.com/video/The-Story-Of-One-Terry-Jones-BBC-number-
documentary-5904

20,000 years ago the number one exists for the first time.
We conclude this from determining the oldest evidence of human scratchs on
bones. Many humans civilizations, the aboridginies has developed and existed
today without ever using any numbers ( or even the number one).
However, the whole science of measurement depends on having an idea of what
the number one means to start out the measurements. And, according to the
reference listed the Egyptians where some of the first to develop advanced
methods for measurements (using a ruler) and hence beginning a question of
what one means inside of us.
Later in human history, the Greek philosopher Pythagoras set up a group of
vegetarian philosohers and mathematicians. He believed everything was made of
numbers, including music. He wanted to understand why certain combinations of
notes sound harmonious. He studied ratios of whole number ( collections of
multiples of one) in order to understand this. He coined the term, “music of
the spheres”. If the beauty of music relies on whole numbers then so must
everything else. And, since whole numbers are at the heart of music and one
is at the heart of whole numbers it must be very important to understand
what “one” is. This belief system was later destroyed by the discovery
of “irrational numbers”. Pythagoras could not conceive of numbers unless they
represent actual objects. However, Archimedes broke us free from this
philosophic assumption by telling us we could think of numbers as objects
(concepts) in themselves. This takes one away from being the “essence of the
universe”.
But, later the Romans (who tending to think more like some modern day
engineers do that it is not so important to understand the why or the how of
things as how to use them) did invent their hard to use for calculation
numerals and set back the science of mathematics several centuries. As a
result, unlike the Greeks, not a single Roman mathematician is celebrated
today.
We (humankind) were saved from our black ages in discovering things about
computing by the Indians. As early as 500 BC, in order to write down the huge
numbers mentioned in their scriptures they invented an improved numeric
placement system for numbers form 1 to 9 and added an entirely new number
called zero which was quickly accepted and added to our modern day set of y
numbers. How was it that we didn’t think of this earlier? Zero is the Holy
Grail of numbers. Its use has changed the entire world. For the first time
someone made “nothing” a number. When we teamed zero up with one magic
started to happen. (We now know the reason from writing programs to and
studying how to generate numbers…integers, rationals, irrationals, using
the computer).
The bringing of numbers to the Islamic world brought a host of brand new
tricks, quadratic equations, algebra which enabled mathematicians to reach
brand new heights and help the Western world achieve its destiny and
potential as me know it now.
The Italian mathematician Fibonnaci wrote a book in 1202 about methods of
calculation, becoming a great mathematician. His book is now regarded as a
showcase of Indian and Islamic philosophy and learning. These new ideas were
considered so revolutionary that in 1299 in Florence, Italy people were
banning from using zero and these new numbers. There was a competition to
determine between the abacus and these new numbers as which was the best in
practice. Our current friends (the numbers we use today) won this competition
n.
During the Englightenment period of history, numbers made it easier to
calculate latitudes and longitudes and helped a wave of European explorers
discover new and fascinating lands around the world ( including the
Americas).
Leibnitz was the one of the first to invent a mechanical calculator (see my
talk two years ago at the academy meeting on the history of Logic Machines).
He also invented a new number system ( the binary number system which uses
only the numbers 1 and 0) He also invented the idea of a Monad ( which is
important nowadays in functional programming languages. And, as a result of
thinking about what “real numbers (integers, rationals, and irrational) were
explored ideas in physics and invented Calculus .

The beating heart of modern day computers is one and zero. Colossus, one of
the first computers developed in Britian, the mathematics of one and zero may
have helped shorten World War II by as much as two years. So, to conclude,
today, Roman numerals have been consigned to the dust bin of history.
Pythagora’s idea of one and zero are all we need to create the modern
computations that have transformed our world into a new information age and
pushed humankind forward up to the edge of discovering how we ourselves are
made (by God) genetically out of a coded string of amino acids.

LinkedIn Groups
• Group: History and Philosophy of Science
• Discussion: How do we define the Number One?.
In the fine tradition of this HPS group, let me offer a couple more
references:

Frege and Kant:
MacFarlane, J. (2002). Frege, Kant, and the Logic in Logicism. The
Philosophical Review, 111(1)
(read this before you form entrenched opinions on any of this stuff)
(MacFarlane is a prof at Berkeley, interested in logic and ancient
philosophy)
(see johnmacfarlane.net)

and a good one-page introduction to Haskell:
Will, M. (2011). Future Psychohistory. Retrieved 06 20, 2011, from
geopense.net: http://www.geopense.net/distrib/future-psychohistory.pdf
;)
"Among different languages, even where we cannot suspect the least connexion
or communication, it is found, that the words, expressive of ideas, the most
compounded, do yet nearly correspond to each other: a certain proof that the
simple ideas, comprehended in the compound ones, were bound together by some
universal principle, which had an equal influence on all mankind." A Treastis
on Human Understanding Book 1 Section III
Isn't this Hume saying that he believes in such a thing as a "concept of a
concept"?


MichaelUnfollow Follow Michael
Michael Will • A simple problem in physics is finding the centre of mass
(COM) of an irregular shape. As an example, consider a square steel plate
with a hole punched through it centred in the middle vertically and one-third
of the way along its horizontal axis. Although I don't have time to draw or
find a picture of this thing, you can see in your mind's eye that it is
indeed irregular enough that no simple fomula can give its COM. Intutitvely,
you can see that the COM will be in the middle vertically, and to the right
of centre horizontally. This problem is actually reducible to only one linear
dimension, but 2D makes it more 'real'.

The trick is to replace the punched hole with a congruent disk of 'negative
steel' (having negative mass). When these two objects: an un-punched square
plate and the disk of negative steel are superimposed, the COM of the punched
plate is found by adding the two separate COMs.

Now, what is 'negative steel'? It doesn't exist. It could even be argued that
it couldn't possibly exist. It might be some form of antimatter, but that
makes this whole image too frightening to concentrate on. Naming it doesn't
bring it into existence (I'm struggling with that whole line of thought,
people). It has no place in the physical world, it is only a mathematical
construct. What is this? I assume it's not reification.


Andrew Harrell • It is what Poincare called an "hypothesis" in his book
Science and Hypothesis. We don't know whether it actually exists or not until
we test its correspondence with reality. But, we certainly don't know it
can't exist, just because it is an hypothesis
"Among different languages, even where we cannot suspect the least connexion
or communication, it is found, that the words, expressive of ideas, the most
compounded, do yet nearly correspond to each other: a certain proof that the
simple ideas, comprehended in the compound ones, were bound together by some
universal principle, which had an equal influence on all mankind." A Treastis
on Human Understanding Book 1 Section III

Isn't this Hume saying that he believes in such a thing as a "concept of a
concept"?

Hi Andrew:
For Hume yes - for me no. But for different reasons.

For me neither language nor concepts exist - only human conceptualists exist.
BUT - having said that I agree that cognisant humans do have communicable
existential experiential personal biographies which they share using various
regional semiotic systems - and in that sharing encounter embrained
foreigners who exist in similar ideative existential modes.

Such European systems, particularly those that Hume came into contact with as
an 18th century intellectual, are members of the great Indo-European language
family of which the sub-group sibling tongues, Germanic, Slavic, Romance,
Celtic, Greek, etc. have all inherited common etymologies, declensional and
conjugational systems and cultural affinities, etc. It would be very strange
if some conceptual confluence had not taken place don't you agree?
For example Swedish is my main language, after my English mother tongue and
because of that I find it easy to get around in the languages of the other
members of that Teutonic sub-group. You can bet your bottom dollar that for
an English speaker to learn (say) Italian is a lot easier than it is for a
Korean.

Sadly, Hume would not have been exposed to the information discovered by the
great
Sir William Jones in 1786, (the year Hume died) Jones Based on Sanskrit, he
figured out that an influential Proto-Indo-European language had existed many
years before. Although there is no concrete proof to support this one
language had existed, it is believed that many languages spoken in Europe and
Western Asia are all derived from a common language. A glance at any Sanskrit
grammar demonstrates the obvious connects to most European tongues A few
languages that are not included in the Indo-European branch of languages
include Basque. Even more important was Jacob Ludwig Carl Grimm (also born
just before Hume died) 1785. He was a German philologist. He is best known as
the discoverer of Grimm's Law ,(The great Germanic sound shift) one of the
Brothers Grimm of Grimm's Fairy Tales. He added even MORE evidence of the
similarity amongst our European tongues.

Cheers,

Jud

Article #161
Subject: LinkedinDiscussionsonThis II
Author: Andrew W. Harrell
Posted: 5/22/2012 07:44:27 PM

Andrew Harrell • Thanks for the detailed answer, Jud. One of my first
philosophy and linguistic teachers, Col. Donald Marshall a student of Lincoln
Bloomfield at Harvard was adament in saying there is no universal language or
concept of a conept in that sense. After studying Sanskrit I agree with him,
I don't think the complicated grammar of Panini could have possibly been
written BC as some of the people you refer to were claiming back then.
However, from studying modern logic and computer math I do believe there is
at least the possibiliity of figuring out what it is sometime in the future.

The number one is the mathematical object that leaves any number unchanged,
under multiplication.


SteveUnfollow Follow Steve
Steve Faulkner • The number one is the mathematical object that leaves any
number unchanged, under multiplication.


Andrew Harrell • Steve,
Yes, it is that, But, does that property define it uniquely? There is an
object that belongs to the set of rationals and has this property. There is
an object that belongs to the set of real numbers and has this property.
There is an object that belongs to the set of integers and has this property.
There are also three more objects that have the property of leaving any
number unchanged under addition. Each of these objects have to be defined
differently because because those different sets and different operations are
defined differently. The question is how do we define one object that does
all of this and is unique?

@STeve,
A question you did not ask, but is pertinent is, 'If we define the Number One
as the "set of all sets which have the set of no element in them", then how
can it be a mathematical object, an "operator" which leaves any number
invariant when multiplied by it? The answer I believe is because we have
things called "Functors" from the category of sets to the category of
arithmetrical operators. Functors were introduced in mathematics alot in the
1950s and 1960s in algebraic topology and algebraic geometry to computer
mathematical characteristics of manifolds. However, I don't believe they were
used in computer science much until recently with the Haskell computer
language which has things called "Monads". The relationship of
Haskell "Monads" to Leibnitzs "Monads" is a fascinating topic and maybe
relevant to our discussion. Does anyone else want to carry on from here?


AnatolyUnfollow Follow Anatoly
Anatoly Tchoussov • to Harrell: Your's expression 'If we define the Number
One as the "set of all sets which have the set of no element in them" is not
correct enough;
more correct will be smth like "one is a set which is equivalent to a set,
whose only subsets are an empty set and a set, which containes an empty set
as an element";
but there is a problem for such definition: in a case of such construction of
a set theory there are two non-equivalent models of number;


Andrew Harrell • Hello again Anatoly,
If you recall I did not say there were not going to problems figuring this
out. Nor, did I say there were not two non-equivalent set theories. On the
face of it I do not understand your criticism of what I have said. Could you
go into a little more mathematical details (not as much as in Russell and
Whitehead's Principa Mathematica or Quine's books on mathematical logic)


AnatolyUnfollow Follow Anatoly
Anatoly Tchoussov • to Harrell: may be I've said not very clear;
I've meant that:

1) Yours expression is not correct, because numbers are defined as classes of
equivalencies and not as a sum of sets;

2) such definition has an intrinsic difficulties, i.e. two non-equal models:
in former note I will use Z as a symbol of an empty set, and figure brackets
{} as symbols of a set;

there are (at least) two ways to introduce numbers:

A): {Z}, {{Z}}, {{{Z}}}, {{{{Z}}}}, ...

B): {Z}; {Z, {Z}}; {Z, {Z}, {Z, {Z}}}; {Z, {Z}, {Z, {Z}}, {Z, {Z}, {Z,
{Z}}}}; ...

in a case A "3" doesn't belong to "5", but in a case B "3" belongs to "5";


Andrew Harrell • Thanks Anatoly, that's a really great comment. You may not
hear from me for a while as I try to write a book on this. If I don't want
you to beat me at it, I better not mention about the things called Monad
Transformers which are used to map between two Monads, each of which has
different classes and types in it.
Add/Reply to this discussion board posting
________________________________________
Article #99
Subject: "necessity for twitter rules"
Author: Andrew W. Harrell
Posted: 1/25/2012 12:27:19 PM

L PUnfollow Follow L P
L P Cruz • @Andrew,

"In this group there is a least someone (me) who believes that God created
everything"

Same here. There are now two ;-)

Ok here goes, Classes are Types but Types are not necessarily Classes. I
believe in FP, types pertain to propositions describing values or quantities.

Incidentally, in OO programming, we see Platonism in action. So in this
discussion, I quote Kurt Godel a believer in Platonic mathematics who said...
* Materialism is False.
* Concepts have objective existence

When you ask how we define the number One, you are speaking semantically,no?

If so I get you.For clearly it is more than mere notation.This is already
evident, if I have two pencils on my desk and my son takes one of them, I am
left with one pencil on my desk. The 1 on paper has semantic correspondence,
that single pencil on my desk.

In the above example, I will boldly assert that One does exists. I believe
Godel's conjecture "concepts have objective existence" is true, it is an
axiom in my philosophy and since we are speaking of One as a concept, the
proof is now trivial.


OK, I think I can see now why you bring types into the discussion.

One is a type, if that is your point, well done, I think I am catching your
drift.

LPC

LucaUnfollow Follow Luca
Luca De Ioanna • As far as I remember, there are at least 3 stances: realism,
intuitionism and logicism
One claims numbers are real platonic entities; another one claims they are
somehow discovered I think; the last one claims they are a logical device to
help us formalize reality and reasoning.

Apologies for the very sketchy summary, it is not my main area.


Andrew Harrell • Yes, LPC, Luca what I am trying to argue along with Plato,
Frege, Godel is "concepts have objective existence". Formalists like the
great mathematician David Hilbert did not believe there is a separate
metaphyical existence associated with logical truth. They can write wondelful
books though like " Mathematical Logic", Hilbert and Ackermann, Chelsea
Publishing, 1950 They say, "Mathematics is the invariance under change of
notation." Some intuitionists, like my graduate school differential geometry
teacher at Berkeley J.Wolf, say this, He and professor Smale and Mandelbroit
being of the French school of Poincare say this too. I say, (with no lack of
respect to others) "Mathematics is the invariance under change of notation in
certain circumstances."


LucaUnfollow Follow Luca
Luca De Ioanna • Ah Andrew, it is an intellectual pleasure learning from you.
I do not quite much about philosophy of mathematics, but reading your
comments make me feel like trying to explore more. Thanks for taking some
time to summarize this all.


Andrew Harrell • You are welcome. Thanks for sharing your thoughts on this
also.


L P Cruz • @Andrew,

My mentor/prof also is like your teacher at Berkeley. He and I have mutually
agreed to disagree though I think he is not that hardened against Godel
because he knows a lot about his work.

I am quite intrigued on your introduction of types, that is a strong argument
I think.

I hope to follow up my readings on it for certainly one of the most
successful programming style is OO and that is down right Platonic in
philosophy and it works!

LPC



PeterUnfollow Follow Peter
Peter Anderson • Imagine someone saying in a high tone: "I insist on great
authority and with much support that not believing that there is the
possibility of a creator of the universe is a mental delusion, similar to the
effect of a migraine on the field of vision, except that delusion is
(obviously) in the cognitive field. Delusional atheists (all of them, without
exception) shouldn't have any standing in any philosophical or scientific
field, and should be sent for testing. I'm going to say it over and over
until no one can imagine anything else is the truth. Hopefully I'll polarise
so many people that the atheists will have to start calling me deluded in
order to express the vehemence of their opposition to being called deluded.
Then, when everyone's pointing at the other side and calling them deluded,
we'll have another war over God, this time between atheists and everyone
else. When the dust settles, the atheists will start calling theists deluded
all over again, and the theists will return it, and when there are enough
other reasons for a war, we'll have yet another one way in the future." That
seems to be the road that some atheists are trying to force this whole forum
along, to polarise debate with invective and make fair thinking impossible
for anyone who is atheist. The prime antagonist is deluded if he thinks he's
stopping wars of religion by being rudely anti-religion. Both theism and anti-

theism have caused suffering. The difference is always how people act and
speak to each other.

Criticizing people's beliefs by calling them deluded is a way of starting a
new religious war, between religion and anti-religion. The people of
communist countries suffered for a generation because of their insistence
that the only god was their own closed minds. Theists and atheists should
read and think about neurobiology and religion, carefully, and reject
absolute statements about the mental health of those who believe or don't
believe in God.

"The Lord your God is One" is one of Hebrew literature's greatest statements.
Doesn't say one God, but does say God is unity.



Luca De Ioanna • well Peter, you may try to do that. But there is a problem.

A condition regarded as medically healthy is a condition where people do NOT
see, NOR talk, NOR invoke to unicorns, santa claus, ghosts, and the like.

People who insist to report such experiences, depending on the societies they
are in, and their grade of progress, are either elected to kings, holy
priests, saints or locked up into mental institutions.

In general, seeing and/or talking to spirits, gnomes, elves and the like are
not symptoms of good mental health.

It is the same with religion: masses of people gathering to feed their
delusion, albeit asymptomatic very often, represent masses of people whose
neurosis are directed towards some imaginary sentient figure, protecting
them.

So, it is not thos who do not suffer from these ideas who are the affected
ones.

I am not sick if I do not have bellyache. I am if I do. The lack of symptoms
does not make someone sick.

It is the same with religion. If you have caught some religious illness, some
treatment might help. Does not have to be electroshock, some nice counseling
would do :)


DanUnfollow Follow Dan
Dan O'Dea • From the original question: 'Leopold Kronecker made his famous
statement, "God created the integers and all else is the work of man." But,
how did God create the integers?' - to this I would reply, "How does Leopold
Kronecker know God created the integers?" All that statement does is do an
end-run around the question "Where do integers come from?"

The answer, simply, is we invented them. We love to count things. I cannot
say for sure, but before language got invented it's likely we "counted" on
our fingers without applying the language-related term "integer" or
even "number" to quantities. All we really knew about was none, one (unary),
and more than one.

To press home the point further, we didn't have an integer for "none" until
the zero was invented (by man, the Indians specifically) sometime in the 9th
century, and the concept of negative numbers was originated sometime before
200 BC in China and fully developed (again by the Indians) much later
(probably around the same time as zero).

If God invented integers, then, why did humans only start using them very
recently? In my opinion, humans invented integers to count and categorize
things, which we've done since before recorded time. Things existed before we
numbered them; that does not require the numbers to have existed along with
the things.


Andrew Harrell • Let's try to focus more on the mathematical science and
philosophy of this. I have flagged some of the above ramblings as
inappropriate.
What was it that Luca, Jud, and I have agreed we have learned so far.
1) Frege was a platonist, for he believes concepts (like that of the Number
one) are
objects.
2) Hume although we can't call him a platonist, (from the quote I posted from
Jud's site) clearly believed in something I have called a "concept of a
concept"
What is it, that's a much more difficult question.
Frege's understanding of what a concept is is also harder to explain. He
believed not only that they were objects, but that they were "functions". He
himself was the first to explain in a better sense than Kant ever did what
functions defined in terms of propositional logic of Aristotle were to be.
This along with the development of modern Boolean logic helped push
mathematics forward in a great leap afterwards. Certainly, it was one factor
which helped the development of the modern digital computer.
3) But, Frege, if you read his books did not yet understand what we call in
mathematics today, a "functor" (which is an object mapping functions to
functions along with associated classes and types). This idea is what we need
to implement Hume's idea of a "concept of a concept" as Monad Transformers
discussed above.
That's about it,
Any thoughts about this much so far from others ( no more philosphy for and
against religion please).

Andrew Harrell • Oh, yes. In answer to Dan, I say read and study the above
more before you criticize it so quickly. Concepts (including that of the
number one) cannot exist as separate metaphysical objects without any
connection to the physical realm if something else self-existent and divine
like helps it. That's the point I think Kronecker was trying to make. They
are not just psychological constructs in our own human minds or meaningless
formal tautologies.


DanUnfollow Follow Dan
Dan O'Dea • Andrew, naming a "divine being" is presumptive; doing so makes me
ask you, "Prove it." There is ample evidence for the creation of negative
integers and zero. Prior to their creation people considered answers giving
negative numbers as invalid or meaningless, but afterward correct and
meaningful.

And there is good correlation between numbers and the physical realm, in both
directions. One example: Paul Dirac's equations for the electron in 1928s
produced two equal answers of opposite sign. Realizing the negative value for
the electron was actually a reflection of reality and not some mathematical
error, he proposed there was a particle identical in every way to the
electron except for the charge, i.e., he'd "found" the positron, a particle
of antimatter. In 1932 Carl Anderson discovered these particles exist in
nature. Thus, Dirac's number, for a time, existed without a known physical
counterpart... and antimatter is not divine but a natural result of quantum
interactions.

I did study the main question fully; my answer simply disagrees with yours. I
remind you, the "divine" is an extraordinary claim, while mine is a factual,
provable statement. It is not up to me to disprove your extraordinary claim,
but up to you to provide extraordinary proof for your extraordinary claim.


LucaUnfollow Follow Luca
Luca De Ioanna • Not to reiterate my point too much, but I do fail to see why
we always have to bring divinities into any possible philosophical reasoning.
Especially considering that these divinities are never defined in their
properties, not even minimally.

In truth, such entities should be not only defined, but also somehow proven,
either logically, or empirically, or both.

Otherwise, I suggest to stick to the topic without asking for divine
intervention :) The ancient Greek called upon Zeus, Athena, Hermes and Ares,
and now we make fancy movies about them in Hollywood studios with special
effects from ILM and Pixar.

Great words from T Jefferson: Question with boldness even the existence of a
god; because if there be one he must approve of the homage of reason more
than that of blindfolded fear.
-Thomas Jefferson, letter to Peter Carr, August 10, 1787

Ali AbuTaha • @Andrew: Your last comment opening line brings up key
words: “Let’s try to focus more on the mathematical science and philosophy of
this.” The key words are science, math, and philosophy of the “one.” As you
say, we’ve had a 100 years to think about Frege, Russell, Whitehead, etc. and
how they understood and developed Platonic thinking. I say we’ve had 2,500
years to think about Plato.

A central concept in Plato is the “forms,” and the 19-20th centuries
mathematicians, logicians, etc. tried to deal with the concept. Plato’s use
of “eidos,” or ideas for forms messed things up. The mess started with
Aristotle.

I was probably getting ahead of myself when, in previous comments, I jumped
on the “One and Many.” Let me put things in perspective. Consider the
following legitimate pairs of opposites from Plato and Aristotle:

One = Many
Unity = Plurality
Universal = Particulars
Likes = Unlikes
Measure = Measured
Indivisible = Divisibles, etc.

And Aristotle brilliantly topped it all with:

Dunamis = Entelecheia

The “One” is better understood if analyzed and combined with “unity,
universal, likes, measure, indivisible, etc. and dunamis.” How do we do that?

Russell, who picked up on the footsteps of Frege, gives the following
equality in his math-logic-philosophy: “Forms = Ideas = Universals!” One is
tempted today to say that Frege, Russell and others were Platonists, and move
on with analysis.

If I say we must begin with Plato, it is because of mix-ups that happened
over the centuries. Regarding Russell’s equality, I used a silly example in
other threads. There, I used a simple sentence by a child, “I have an “idea,”
let’s order pizza.” We all understand the meaning of the words,
especially, “idea.” But, we do not allow “I have a “form,” let’s order
pizza,” or “I have a “universal,” let’s order pizza.”

In essence, I am pointing at the genesis of the “One,” “universal,” “unity,”
etc. as these (the One) relate to Plato’s “forms” and “ideas.” The first task
is to clear the mess. The second task is to actually “find” and “draw” a
mathematical form of the One; a geometric diagram! When the latter is done,
things fall nicely and rapidly into place in science, math, and philosophy.

Anatoly Tchoussov • comment to to Luca's: "A condition regarded as medically
healthy is a condition where people do NOT see, NOR talk, NOR invoke to
unicorns, santa claus, ghosts, and the like."

numbers are the same in their's existence;

the problem consists (IMHO) in existence of non-observable;

1) we know that the necessity of making relation to smth non-observable is
permanently reproduced (in human societies);
2) we can refer to a non-observable only by the means of representation of
some kind (in formal expression - semiotical);
3) we inevitably are to presuppose two kinds of existence for our
representations - else they are not representations, i.e. don't refer to smth
other than they immediately are (IMHO they immediately are only material
bodies of signs);


AnatolyUnfollow Follow Anatoly
Anatoly Tchoussov • to O'Dea: I agree with Your's way of thinking - it seems
to me as evolutionarist way, i.e. the way of explaining phaenomenae of the
world with an inner causality/determination of the world (m.b., I'm wrong);

I'd like to add in some "marxist" (IMHO) tradition:
numbers are the tools which are giving the possibility "to differ sets
exactly";
they are coming to existence strictly in that way (sociological and
administrative and governmental needs and m.b. some more...?!);
we know that autists can invent numbers (in a bounded, but not a-priori
predictable degree) without special learning (but there also is necessity to
learn them to signify numbers);
but autists cannot survive in "primitive" societies (so the problem of
inventing of numbers is still not solved in a full volume);

questions about zero or negatives are then equal to an expansion of practice
(and it is very close to a trade operations);

and for inventing integes by some god there's lasting a question: does he
create a common numbers or p-adical? i.e. which topology of numbers does he
create?



Peter Anderson • @ Ali AbuTaha, a good survey of what's on the table, and a
common sense analysis of the confusion that interpreting Platonism can bring
on people. Some of it is terminology, you are right, but there are important
differences in the use of these words that you miss. An idea is not a Form
because a Form is not mental content. However, 'One Form' is mental content,
and an idea can be of a Form, but a Form cannot be an idea. Thinking about a
Form is mental content, but a Form is not. Again 'a Form' is mental content,
but a Form is not.

Initially we have to sharpen the definition of one by cross-referencing it
with awareness of nothing, projective mapping of imaginary similar
representations (Kant) and comparison with nothing, knowledge of 'two', 'ten'
and 'a million', and then we learn to count the pocket money coming in and
take more into our visual and intellectual field.

@Luca, I know you need to repeat your statements against religion for as long
as you might need to. It would be great if you could apply your critical
insight to the content of your statement. What you contribute otherwise is
not dependent on that view of religion. The rigour of your training is
obviously capable of it.


Andrew Harrell • That's a whole lot (couple of pages) of criticism to just
three lines I wrote which I thought were completely justified if anyone
wanted to take the trouble to think about
what we have said so far. Dan, if you agree with Plato and Frege that number
concepts are objects existing independently by themselves, then whoever
created them must also exist independtly by Himself. And, the realm in which
they were created, being differrent from our material one is "divine" also in
some sense. That's all I said, go back and read it. Ali: I agree let's clear
away any mess remaining. Anatoly, don't assume you have the right to set the
rules of discussion here. You did not start the discussion.
Peter you are just repeating a lot of nonsense.


AnatolyUnfollow Follow Anatoly
Anatoly Tchoussov • to Anderson: "An idea is not a Form because a Form is not
mental content";
let's stop here and observe around;

do You think, idea hasn't form?;
do You think, a form hasn't mental existence?;
do You think, idea (and form too) hasn't relation to a content?;
how do You think, the idea without a form can exist?;
(there much more questions, I'm restrained myself a lot...)



Anatoly Tchoussov • excuse me, pls, my English


Andrew Harrell • Anatoly, what is your defintion of a "form"? For me a form
has to be something that exists in space and time. And, since eyes are what
give us evidence of something in space and time, forms are usually visual.
Ears, also tell us about sounds in space and time. But, what is the
instrument that controls the eyes and ears... The mind. And, the mind can
sometimes see and hear transcendental things. Also, our ideas and our
concepts come from something more than our minds, don't you agree?

Article #162
Subject: Linkedin Discussions on This III
Author: Andrew W. Harrell
Posted: 5/22/2012 07:45:42 PM

Andrew Harrell • Hear Ye, Hear Ye! In the interest of clearing away the
clutter, I suggest we all adopt "twitter rules" (no more than 140 characters
per posting). And, please no more than one posting per day per person.


Peter Anderson • Sure, Andrew. The 'nonsense' you say I am repeating is
simply a way of drawing attention to the need for grammatical accuracy if
anyone wants to discuss numbers philosophically. If you don't understand it,
that's not my problem.


L PUnfollow Follow L P
L P Cruz • I think the issue now at this stages is what Andrew said - the
question now is, do people here agree that numbers, as concepts have
transcendent existence.

t is not necessary at this stage to define who and what is the property of
that God at this stage.That can come later. The point is - do concepts have
objective existence?

My position is already know, I answer : Affirmative.

The bringing in of physics or whatever physicists may have or not have
discovered is also irrelevant because physics borrows or uses the tools
produced by mathematics. If their model of the world fails to account for
other phenomena, that is not a problem for mathematics, it is the physicist's
problem with his model.


The development in programming languages I believe gives a very strong
argument for Platonism. Programmers are not philosophers necessarily but for
the last 20 years they have been supporting Platonism when they use object
oriented languages in their work, without them realizing it. A few have
blogged about such realization and continue to program in languages like
C /Java etc.

This is the reason why even hardened mathematical non-Platonists can not
afford to be thorough going anti-Platonists. Firstly they are a minority, and
secondly, it runs counter to their sense of integrity to thoroughly ditch
platonism in their work, because it stares them in the face each time they do
abstraction.

@Andrew,

I agree with the the functor idea though I miss why we need to interact with
Hume?

LPC


Virginia Malone • All of this have been very interesting. To the question How
do we define the Number One?.
Maybe it is in our DNA many animals understand the concept of one. Totters
quickly understand the concept of you may have one cookie and they even have
some idea of the null set - there are no more cookies.


AnatolyUnfollow Follow Anatoly
Anatoly Tchoussov • to Harrell:
the basic definition of "form" isn't mine - I think that Aridtoteles defined
a form as smth like "possibility to be definable (predicated ?!)";
that's not question of my definition (I think) but of common (and reproducing
fast by itself) understanding of basical intuitions of existence (in Kant's
defs - Anschauungs)



Peter Anderson • @Anatoly,
do You think, idea hasn't form?;

The use of the possessive 'has' alters the meaning of form to a quality.
Plato wasn't talking about qualities of objects. An idea has the quality of
form in a person's brain and in their life, of course. That is different from
Plato's Forms.

do You think, a form hasn't mental existence?;

According to Plato's theory, a Form only has mental existence in the context
of discussing or thinking about Plato's theory as an idea, and as the idea's
physical manifestations in the brain. Its real existence is meant to be
independent of the mind.

do You think, idea (and form too) hasn't relation to a content?;

An idea has relation to a content. An idea can be itself content, for
example, Plato's Forms hold ideas as one of its contents as well as (when
discussing Plato) particulars, because ideas are particulars of the mind.
Particulars are, in the context of discussing Plato, ideas.


Mike Rand • An idea must have a logical form, but that doesn't mean there is
any extra-mental ontology.

Anatoly Tchoussov • to Anderson: I'd like to get a more systematical answer;

do You think that an "idea" is more common (abstract?) notion (concept?!)
than a "form"?
the link to "possesive" doesn't explain this subject;
and a referrance to smth that You do interpret as a "form" - too:

if You think that Plato is an absolutely limiting point of investigate - this
is (not) only Yours point of view;
I prefer a point of view: Plato shows us a temporary (and in some way con-
temporary) limits of universal thought;

(IMHO) methaphorical tools of Plato can't give us a proper differrence
between real and realisable;

but we are living in a world of (material in most cases) realizations


Anatoly Tchoussov • to all:
excuse me my English expressions - - I tried to make them English

Peter Anderson • @Mike, that's right. Plato's idea of Forms postulates that
there is extra-mental ontology. Because of the way it is presented, it is
unknown exactly what this ontology entails, hence its usefulness for
philosophical metaphysics.

@Anatoly, sorry, I always try to assume that someone is using grammar
deliberately. I don't understand where commonality comes into it. Are you
asking whether it's more common to use the word "idea" than "form" for the
same mental object? If so, no one uses "form" for "idea", except when
discussing mental events as Forms.

"if You think that Plato is an absolutely limiting point of
investigate...Plato shows us a temporary (and in some way con-temporary)
limits of universal thought" These are too imprecise for me to understand,
sorry. Can you rephrase?

Plato's metaphorical tools are a good place to start in working out how to
define 'real versus realisable'.

Yes, we are living in a world of realizations, mostly material, including
mental.

Luca De Ioanna • Numbers are logical entities which have properties. Whether
they exist out there or are invented, it is to be decided.
I still fail to see the relation between numbers and the gods.
Numbers are logical entities, pertaining to the logical space of reason.
Gods are fictional entities, subject matter for fables, tales and
superstitions.

This is my problem with metaphysical ontologies: nothing can be said about
them.

L P Cruz • "Whether they exist out there or are invented, it is to be
decided. "

- To you it might be but not to others. If numbers are transcendental, then
how did it come to be? Where did it come from, because if they are
transcendental, they were not our inventions, so there is where 'gods'
or "God" comes in.

But that is something you need to determine first and the issue of God or
Godhood follows. Are numbers transcendental? You are undecided on this so
therefore, you won't see the relation of Godhood in the discussion.

We can even go further than the Numbers, do it for the Laws of Logic, are
they transcendental? Food for thought, just a suggestion.

"Gods are fictional entities, subject matter for fables, tales and
superstitions.

This is my problem with metaphysical ontologies: nothing can be said about
them."

Sherlock Holmes is a mythical character but we can talk about him, Middle
Earth and The Hobbits where creation of Tolkien but we can talk about these -
in the process we are saying things about them. But when we talk about
Hobbits, we do so under the information given by Tolkien. So it is false that
nothing can be said about what you claim are "fictional entities".

LPC

Dan O'Dea • Andrew Harrell: "if you agree with Plato and Frege that number
concepts are objects existing independently by themselves, then whoever
created them must also exist independtly by Himself". IMO (and that of a
great many historians) humans invented numbers to make sense of reality (how
many "things" are there). So we exist independently... I agree. What I
disagree with is how you conclude there must be a God... it seems like you do
so because you can't imagine how there can't be one.

Part of this is believing there is an actual reality out there independent of
our personal viewpoint. This DOES NOT imply a creator; as Steven Hawking has
shown, the universe does not require a creator. There could be one, or not...
but one is not required.

To address Virginia's question (I hope), the words for numbers are merely
labels for a given quantity of objects; that's it. "One", "un", "ein", "واحد
", "一", "ένας"... whatever, is a single object. "Two", "deux", "zwei", "اثنان

", "兩", "δυο"... whatever, is two objects. Different words for different
numbers, but the underlying concept (none, one, more than one) is the same.

Whether any given number exists or not is the question (and I think that's
what Virginia is echoing). I submit we discover "numbers" as we decipher the
complexity that is the universe. It's clear what an integer is; we think in
integers. Defining concepts such as e, i, etc. uses integers, ultimately, to
define broader constants such as the natural log base and an imaginary number
respectively. Mathematics builds on previous steps, starting with the whole
numbers. Then "cut a number in half" for fractions and decimals, "count the
zeros after or before the high-order number and use that" to express really
large or small numbers (exponential notification)... it's all there.

The cool question is, "Why does mathematics explain the behavior of the
universe so well?" The weak anthropic principle states, "otherwise we
wouldn't be here to observe." Many scientists and mathematicians allow this
because it's rational. What we object to is the strong anthropic
principle: "The universe was designed for us." A good counter for that
argument is within reach: find life elsewhere in the universe.


L P Cruz • " What I disagree with is how you conclude there must be a God...
it seems like you do so because you can't imagine how there can't be one"

If you believe that numbers are transcendent, then, the minimum you can do,
and still deny God put them there for us to observe, is that you posit that
it is a mystery.

However, I believe numbers are transcendent and I can posit God....because it
is the most logical explanation and a good explanation why it is
transcendent, i.e., I go beyond the minimum.

It is not the only explanation but I would like to see an explanation how it
came to be transcendent, if you do believe it is first.

If you do not believe it is not transcendent, i.e. that numbers are not
transcendent, then I would like to see the argument from this aside from
psychology or sociology. I eliminate them because they are very subjective -
they appeal to experience which may not be common for all people, for all
time.

Like I said, even the most ardent materialist or atheist programmer, can not
escape platonic categories because he/she uses it's principles every day.
Each time an abstraction is appealed to, a platonic category comes into play.
I am being extreme but I need to so the point can be made.


LPC


Anatoly Tchoussov • to Harrell: "Anatoly, don't assume you have the right to
set the rules of discussion here. You did not start the discussion"
IMHO I didn't try to set the rules;
I tried to enlighten (m.b. reformulate) a theme of discussion and my position
in regard to it



Anatoly Tchoussov • to Harrell: "Hear Ye, Hear Ye! In the interest of
clearing away the clutter, I suggest we all adopt "twitter rules" (no more
than 140 characters per posting). And, please no more than one posting per
day per person."

I've read this rule only now;
it's a right of Yours to set the rules (as a creator of discussion)
do I exaust my daily maximum by quoting?


LucaUnfollow Follow Luca
Luca De Ioanna • Very well said Cruz:

unicorns and elves are fictionary, but a lot of things can be CONSISTENTLY
SAID ABOUT THEM.

Gods, especially unique gods, are property-less, exactly like metaphysical
obhects.
This is why I see little use for metaphysics.
If you could provide me with some minimal set of properties of this god
thing, the conversation might advance.
Or else, I see no reason to call upon divinities in logic.

Personally, I do not think at all that numbers nor the rules of logic are
transcendental.
I think they both stem from language and er, therefore, a human invention.


AliUnfollow Follow Ali
Ali AbuTaha • @All: A fascinating amalgamation of “ideas,” and I don’t envy
the moderator the task of keeping it on track.

If Plato is the champion of “forms,” Locke is the champion of “ideas.” In his
Essay, Locke apologized for his very frequent use of the word “idea.” I was
always tempted to keep a tally of the word in Locke. There is intellectual
honesty when Locke says of the forms: “I confess I have no idea at all, but
only of the sound ‘form.’” “Substantial forms,” he described as “wholly
unintelligible.”

The question then arises as to whether Hume, Kant and others after Locke
grasped the difference between form and idea. The divergent opinions
expressed in only the last few comments clearly show that the distinction has
not been clearly made yet. I see many comments that are worthy of careful
response, but that will take many “characters.” The moderator may want to
ease the restriction on brevity.

You will note that in the above, I introduced another pesky
expression: “Substantial forms.” This brings to mind the other
kind, “accidental forms.” The confusion over these “forms” is more palpable
than the confusion over “forms” and “ideas.”

To proceed further, are we interested in the substantial forms of the One,
i.e., in the substance and essence of the One, or are we interested in the
accidental forms of the One, i.e., how the One operates in our minds, senses
and in the world?

Philosophy, by definition, is “critical inquiry.” So, if I criticize, and I
will, please remember that I am criticizing concepts and not individuals.

A last note for now is on classification. All computer programming can be
classified as mathematico-rational enterprise, using algorithms to obtain
answers; named after Algorismi, or AlKhawarizmi. Most of science today can be
classified as mathematico-empirical, or –experimental, after Galileo. It
seems to me that any Platonist must follow the “divided line,” or the Cave,
of Plato, in Republic. The divided line contains all three components:
Rational, empirical and mathematical. Are we going to unravel the “One” using
one of the two modern approaches, or Plato’s integrated approach?

Sorry Andrew if my comment is too long or off the mark. It’s not easy to just
narrow the parameters.

Andrew Harrell • Plese obey "twitter rules". Quotes do not count. The
moderator turns over the task of moderation this discussion to God. If He
does not exist I may be in trouble.


AnatolyUnfollow Follow Anatoly
Add/Reply to this discussion board posting
________________________________________
Article #100
Subject: Ali and Milo join the discussion
Author: Andrew W. Harrell
Posted: 1/26/2012 03:29:07 PM

OK Everyone please take a deep breath, again!
Yes, as Plato explained in the dialogue Parminides there is both 1) Oneness
as Sameness (inside of), and 2) Oneness as Difference (separating them). The
Oneness as Difference, the One is the One and the Many is the Many is
Sanctification. The Oneness as Sameness is Unity inside of Diversity, the
Oneness in the Many.
But, as we ourselves explained above there is also above 3) the reference to
the two different ways of defining the Number One mathematically as a set
above, the way of Frege (3a cardinal numbers) and the way of Von Neumann (4a
ordinal numbers0. If you define, as Frege did, the Number One as the set
whose only element is the set with no element in it (empty set) that is a way
that mixes the One and the Many together (inside of each other separatedly).
Sorry I violated “twitter rules’

BREAK IN YHWH SCHOOL POSTINGS
@Milo Gardner: "Wow, no one has mentioned one as equivalent to unity". I
did: "All we really knew about was none, one (unary), and more than one". I
used the term unary rather than unity (perhaps incorrectly, but with the best
intentions) because unary implies a single tally per single item (or a single
function), while unity has a more broad definition which includes
psychological aspects I was trying to avoid. @Ali AbuTaha follows your
comment with an excellent one defining "unity" mathematically, which is more
accurate than my definition. I apologize for the confusion.

I also want to thank you for adding to my very brief history of the integers
zero and negative numbers; I didn't include Egyptian notation because they're
not integers, as you show us. However, I should point out the Olmecs used
zero as a placeholder a few hundred years before the Mayans adopted it.

In re Plato's Forms: there is plenty of criticism of the Theory of Forms,
including from Plato himself and from Aristotle, who clearly and strongly
disagreed with his teacher. I do not think one should use Plato's Theory of
Forms to define what "one" is for this reason: Plato said we know Forms by
remembering the soul's past lives, and Aristotle's arguments against this
treatment of epistemology are compelling... as are arguments from others in
this discussion. Even Plato stated, "That which is non-existent cannot be
known."

AliUnfollow Follow Ali
Ali AbuTaha • @Peter: “Being and becoming” (BnB) “is” a different subject
than “Being and non-being ” (BnnB).

BnB are Opposites and have beginning, middle and end. BnnBs do not.

BnB relate to “motion,” hence, we can do math. BnnB has nothing to do with
motion.

BnBs have equality, actually, “absolute equality,” which doesn’t apply to
BnnBs.

Equality of BnBs must be in “number” and in “kind,” which doesn’t apply to
BnnBs.

And there is more.

Since before Plato, everyone knew the above features. My aim, as stated
earlier, is to clear up the mess, which goes all the way back to Aristotle. I
respect what the great thinkers wrote over the ages, but I like to think for
myself. As to ontology, I'd stick with epistemology for now.

@Andrew: The distinctions I make above are essential to do correct math,
especially of the “One.” To me, what Cantor, Boole, Frege, Whitehead, Russell
and other 19-20C logical-mathematicians tried to do was to solve the same old
problems using sets and algebra, instead of arithmetic, geometry and algebra.
The old problems remain unsolved. Your “take a deep breath” is more loved
with your comment about “twitter rules.”


MiloUnfollow Follow Milo
Milo Gardner • @Ali, Andrew ...

“One.” To me, what Cantor, Boole, Frege, Whitehead, Russell and other 19-20C
logical-mathematicians tried to do was to solve the same old problems using
sets and algebra, instead of arithmetic, geometry and algebra. The old
problems remain unsolved"

The old problems were solved by arithmetic, algebra, geometry and other forms
of math ... including economic scaling of commodities for use as
money ...Cantor, Boole, Frege ,, et al discussed other unity subjects.

Unity dealt in the classical sense scaled rational numbers with units created
for business, and physics applications. Math and unity applications were the
language of science and business ... a methodology that ended in 1454 AD ...
the the 250 year use of Fibonacci's Liber Abaci fell out of use ... and fully
replaced in 1585 AD by Stevin's base 10 decimal system.

Here are some “new problems” that Frege’s definitions helped solve.
It seems to me the question we are discussing now is
“How do Plato and Parminides categories of the same, the different, the one,
the many relate to Frege’s definition of the number one?”
Frege defined the “number one” as “the set whose only element is the set
without any elements”.
If we identify “many” with something in a set, an object…
Also we have to make the observation that in our propositional
logic, “different from” is not necessary the same as “not (the same)” then
We assume our propositional logic has a variable substitution unification
algorithm that allows the computation of what we call the “Same”
and “Different” in terms of substitution of free and closed variables.
differencing out all representations of a set whose inside elements consists
of many (the set of objects defined according to the rule that there is no
object in them). We have
defined what the number one is (both as a cardinal and ordinal number).
Ali, Here are some "new problems" that the discussions arising out of Frege's
set theoretic ideas on arithmetric definitions helped solve.:

The tremendous ramifications of this fundamental beginning of automated
thought are that in mathematical computing there are both things
called “recursive functions” which are defined inductively using a starting
element (least) and “co-routines” which are defined inductively without
necessarily having any starting elements.
Co-routines and recursive functions were known to Church, Curry, and Turing.
The reason for having a separate functional objects called a co-routine is
that in order to conduct physical simulations it is possible that the object
instantiation part of the classes we are computing during the simulation has
changed during the course of the backward or forward recursive part of the
computation. The co-routines often need to switch back and forth between each
other. And, if this is so we cannot just trace the variable substitutions
back to a “first” one (like in a recursive function), where it all began.
And, indeed we assume there are situations where we don’t need to [sic] See
the recent book “The Haskell Road to Logic, Maths, and Programming” by Kevin
Doets, Jan van Eijck. This is what I was referring to earlier in the
discussion when I mentioned the applicability of “functional programming
languages” to this question.
Church’s and Curry’s ideas on this helped us create the “LISP” class object
oriented, forward chaining recursively defined language in the 1950s, 60s,
and 70s. Starting in the 80s we saw backward chaining, goal oriented,
recursively defined logic-programming languages like “Prolog”. These
languages pushing the field of “artificial intelligence” forward
considerably. But, in my opinion, right now… as I said earlier, there is
still a good deal of the way left to go before we understand “how we (meaning
us humans) can/should define the number one”.

Article #163
Subject: Linked In Discussions on This IV
Author: Andrew W. Harrell
Posted: 5/22/2012 07:46:52 PM

________________________________________
Article #101
Subject: Further Postings
Author: Andrew W. Harrell
Posted: 1/28/2012 04:10:10 PM
Ali, Here are some "new problems" that the discussions arising out of
Frege's set theoretic ideas on arithmetric definitions helped solve.:
In mathematical computing there are both things called “recursive functions”
which are defined inductively using a starting element (least) and “co-
routines” which are defined inductively without necessarily having any
starting elements.
Co-routines and recursive functions were known to Church, Curry, and Turing.
The reason for having a separate functional objects called a co-routine is
that in order to conduct physical simulations it is possible that the object
instantiation part of the classes we are computing during the simulation has
changed during the course of the backward or forward recursive part of the
computation. The co-routines often need to switch back and forth between each
other. And, if this is so we cannot just trace the variable substitutions
back to a “first” one (like in a recursive function), where it all began.
And, indeed we assume there are situations where we don’t need to [sic] See
the recent book “The Haskell Road to Logic, Maths, and Programming” by Kevin
Doets, Jan van Eijck.

Peter Anderson • @Ali, your reply with BnB and BnnBs makes no sense to me.
Being is a vital component of becoming as England's Newton implied,
and 'Being and Non-Being' as manifest in German Liebnitz's binarianism makes
infinitely more sense to me. (If you want to stick with ancient Greece, I
just read about Eudoxus, so he might help you past Plato).

If you're taking the epistemological road, how will Plato help you escape the
problem of the ontology of knowledge, without Descartes and the
epistemological positivism of Kant? And if you admit Kant, aren't you
therefore bound to accept that noumenal Being is not knowable in itself, and
that all knowledge of Being is in fact a Becoming of knowledge, and that Non-
Being is an extinction of the Ergo Sum?

Ali
Ali AbuTaha • @Andrew: By “the old problems remain unsolved,” I certainly
didn’t mean problems were not solved. The quantity and quality of math
problems solved in the last centuries are phenomenal. I am thinking of this
thread's “one.” There is an age-old math problem associated with the “one”
that has not been solved – by arithmetic, geometry, algebra, sets, calculus,
etc. I can paraphrase the age-old math problem as follows:

Show that “One = Many,” in “number” and in “kind.”

For millennia, the problem was stated as follows: “How can two or more things
be one in anyway?”

This, and related issues, are the math problems I am talking about.

It’s been a while since I updated my library. I’ll look up “The Haskell
Road…” and other books. But I think we can pursue the above “one and many”
problem with what we have. That’s what I am driving at, slowly.
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Ali
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Ali AbuTaha • @Peter: The characteristics of BnB and BnnB I listed above are
plain definitions that I hunted for decades. I list those in preparation to
set up the math of the one and many. Do you find anything wrong in any
specific item?

While the Parmenidean being and non-being can be represented as 1 and 0; that
was not the source or inspiration for Leibniz’s binary 0 and 1. And Leibniz
recognized the difference between non-being and becoming, which some
philosophers after him did not.

Whitehead described all philosophies to be “footnotes to Plato,” and there is
a reason for this. I mentioned Plato’s divided line of knowledge, which
divided things into two domains; in us, the sensible and intelligible and in
nature, the empirical and rational. When I read Descartes’ Cogito, I
see “mind” and “body;” but these are Plato’s sensible (body) and intelligible
(mind) divisions. And when I read Kant’s “concepts without percepts are
empty, percepts without concepts are blind,” I see Plato’s intelligible
(concepts) and sensible (percepts). And I can go on. I am well “past Plato”
and also Eudoxus, another favorite. The question is, how do we solve
the “one” and “many”?

@Dan: There is so much to say about the “Forms,” “Ideas,” Plato, Aristotle,
predication, participation, etc. These will come up again.

Peter Anderson • @Ali,
I think you must be oversimplifying your statements.

You wrote: "BnB are Opposites and have beginning, middle and end. BnnBs do
not."

If we take out 'Becoming' and 'Non-Being', your statement is: "Being" has a
beginning, middle and end, but "Being" does not. The same thing applies to
the second statement.

I stopped there, because without knowing what you're talking about, the rest
is meaningless.
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K NUnfollow Follow K N
K N Padh • 1 is God, a Sun, energy and 0 is Soul. things are created out of 1
and 0. 1 and 0 are complementary. 0^1 = 0 ; 1^0 = 1. 0 can nullify or make
more. 1 unifies. 0 is feminine and 1 is masculine --- must for any system.
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Luca
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Luca De Ioanna • Amen KN Padh...amen..ite, missa est :)
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Ali
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Ali AbuTaha • @Peter: Oversimplification is a problem. Twitter rules. But
then, long comments are also boring and distracting.

I want to do the math of “1 n many” in today’s (21st C) terms. First,
parameters must be set and agreed to. If I keep going back to Plato and
Aristotle, it is because they set the parameters. Afterwards, everyone
followed – From Plotinus and Neoplatonists, to Averroes (IbnRushd), Aquinas
and Maimonides, to Descartes, Spinoza and Leibniz, etc.

An important theme in the ancients, and most followers, is “change.”
Aristotle gives 4 kinds of change: (1) Quantitative, (2) qualitative, (3)
local position (or locomotion), and the problematic (4) coming to be and
ceasing to exist, or becoming and perishing. The first 3 kinds of change
are “motions,” the 4th is absolutely not motion. The 4th kind of change
relates to our friend “being and non-being!”

If you think in terms of “motion,” you’ll see that both “being” and becoming
have beginning, middle and end. But, when “being” is used with “non-being,”
there is “change,” but there is absolutely “no motion,” and, hence, no
beginning, middle or end.

The above should be highlighted in any good book on Aristotle.

Today, we have better understanding of “motion,” including math equations. We
should be able to use the equations of motion for “being and becoming,” or “1
n many.” The math of “being and non-being” (1,0) is different. Hope this
clarifies what I’m doing.
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MiloUnfollow Follow Milo
Milo Gardner • @Dan [I also want to thank you for adding to my very brief
history of the integers zero and negative numbers; I didn't include Egyptian
notation because they're not integers, as you show us. However, I should
point out the Olmecs used zero as a placeholder a few hundred years before
the Mayans adopted it.

*** EGYPTIAN MATH USED INTEGERS IN QUOTIENT STATEMENTS. YOUR POINT DISCUSSES
EGYPTIAN UNIT FRACTION REMAINDERS. OLMEC LONG COUNT DID SET THE STANDARD FOR
MAYAN NUMERAITION AND MAYAN ARITHMETIC ... MAYANS ADDED CALENDAR CYCLES
PROVIDED BY OLMECS IN WAYS THAT ARE UNDER INVESTIGATION. MORE LATER***


In re Plato's Forms: there is plenty of criticism of the Theory of Forms,
including from Plato himself and from Aristotle, who clearly and strongly
disagreed with his teacher. I do not think one should use Plato's Theory of
Forms to define what "one" is for this reason: Plato said we know Forms by
remembering the soul's past lives, and Aristotle's arguments against this
treatment of epistemology are compelling... as are arguments from others in
this discussion. Even Plato stated, "That which is non-existent cannot be
known." ]

***AS A PARALLEL EGYPTIANS WERE OLMECS, AND PLATO WAS A MAYAN. following the
anaYlsis: EGYPTIAN NUMERATION AND EGYPTIAN UNIT FRACTION MATH SET THE
STANDARD FOR GREEK NUMERATION AND GREEK ARITHMETIC.

NEO-PLATONIC PHILOSOPHICAL COMMENTS ARE OFF TOPIC. READ PLATO''S OWN WORDS TO
DETERMINE THE NUMERATION AND ARITHMETIC USED ..A FEW COMMENTS ON THE
HISTORICAL PLATO HAVE BEEN POSTED ... I'LL REPEAT THEM IF YOU WISH.

THANKS FOR THE DISCUSSION.

Milo Gardner • Planetmath's server is down for repair ... it should be up
soon ... at that time the historical Plato is shown to follow Egyptian
numeration and Egyptian unit fraction arithmetic per
:
Adding specifics to the missing aspects of Greek math ...
consider ....http://planetmath.org/encyclopedia/PlatosMathematics.html

may be of interest.

My view is that Neo-Platonic thought considers one and zero in different ways
than Greeks and Egyptians. Possibly some would like to open that topic for
discussion...
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MikeUnfollow Follow Mike
Mike Rand • There may be more than one desription that fits the phenomena.

Ptolemy and Copernicus both provided systems that predicted the movement of
the planets.

That something fits does not make it real.

So Frege provided a model for natural numbers. That it seems to work does not
make it real. A logical explanation for numbers might not get to the heart of
the matter, but it is likely that those interested in numbers are inclined
towards logic.

We need to look to the psychology. Humans describe the world numerically. The
solution to what numbers are lies in the developed human psychology that
delivered an adaptive advantage. A logical description of numbers is also
possible, but that is a red herring.
Peter Anderson • @Ali,
That makes it a bit clearer.
It seems to me that 'Being and Becoming' applies to the 2nd and 4th, and
Being and Not-Being applies to 1. Locomotion would seem to me to be the
essence of Not-Being in time and space when comparing point of departure
versus point of destination, although the parameter issue makes it
a 'Becoming' question, as the object transitions between successive idealized
positions.
For example, if I have one flying matchstick, then at a given point in time
it either exists or doesn't as the quantity of one matchstick - Being or Not-
Being one matchstick. If the matchstick changes into a Space Rocket, then it
is undergoing qualitative change, and we have to look at Being and Becoming,
and look at the ideality of numbers. Once it has become a space rocket, it
takes off, and in every place into which it advances it can potentially find
itself defined as a space rocket. Yet it doesn't occupy every space at once
or instantaneously, so it 'Becomes' gradually into each space on its
trajectory, formally marked for convenience as mission critical checkpoints.
Being and not-Being are clear there.

Ali
Unfollow Follow Ali
Ali AbuTaha • @Mike: I agree with your overall assessment. Math is deductive,
hence, logical. But that doesn’t mean that math is logic. As a mathematician,
if I want to add “5” and “7,” I just simply add the numbers, and I don’t
bother with first building recursive, or any other kind of, sets, mentally or
on paper. This is not to say that set theory is not useful. It is very
useful, but the program to reduce math to logic, like you suggest, really
didn’t work.

Ali AbuTaha • @Peter: Perhaps, a few more words on the 4 kinds of “change”
are useful:

(1) Quantitative change = growth and diminution.
(2) Qualitative change = Alteration, like a green leaf turn red.
(3) Position change = locomotion
(4) Substance change = coming to be and ceasing to exist, or being and non-
being.

The match-rocket example is new to me. Need to think more about how it fits
the definitions. The 4th kind of change is like flipping a coin – heads or
tails; there is no middle point. It is or it is not. The other three have
beginning (of change), middle (of change) and end (of change).

You introduce the key word “potentially,” which must be related to change.
The 4 kinds of change must be related to “potentiality” and “actuality,” or
power and action, or dunamis and entelecheia, which caused a lot of problems
over the ages, especially for solving the “1 n many” mathematically.

Milo Gardner • @Mike, psychology can not be ignored ... but to elevate it per
your comment "
We need to look to the psychology. Humans describe the world numerically. The
solution to what numbers are lies in the developed human psychology that
delivered an adaptive advantage. A logical description of numbers is also
possible, but that is a red herring. "

Logical uses of numbers lie in different domains than language and
psychology. Elevating psychology lowers math to a secondary status ... all
are equals ... the red herring is placing psychology into a dominate role.
Read logical descriptions of rational numbers during the medieval, Arab,
classical Greek, and Egyptian unit fraction eras. I have done that ... those
that stress psychology tend to ignore history ... and live in the present ...

Mike Rand • The great insight came from David Hume who understood that reason
is and always ought to be the slave of the passions.

Mathematics is simply a synthetic a priori method for us to cope with and act
upon the world (see Kant).

Milo Gardner • nope ... Kant was a loser innumerate type of philosopher ...
read Archimedes, Newton and Gauss ... and modern philosophers that you love
should be connected to history dating back 4,000 years ... living in the
present without appreciating the past seems like an empty life ...
Andrew Harrell • For newcomers I will try to summarize our
discussion so far:

The Number One a substantial mental object? Who says so? Plato. Frege.
Its transcendental and in you too (as a two) says Kant.
Easy for you to say, that “one + one is two”.
But, if all you mean is a tautology, then who but God and it (“one’) or
no "one" can be?
The Number One can be a “lonely number”, when “no ‘one’” else is around to
for it to be.
But, is it possible? Is it true? Put it in a person and then “at once”
We have some “body” else (logically and mentally).
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________________________________________
Article #105
Subject: Plato's Dialogue Parmenides
Author: Andrew W. Harrell
Posted: 1/31/2012 04:20:07 PM
Plato in the dialogue Parmenides starts humankind off studying “How to We (or
You) Define the Number One.” In this dialogue Socrates and Parmenides discuss
the arguments and paradoxes of Zeno and other contemporary Greek
Philosophers. It assumes a knowledge of previous dialogues like Phaedrus
where Plato has explained his theory of independently existing mental objects
called Forms. How do we “know” these independently existing mental objects?
Our mind can know them by “participating” in them, not in the sense that
through sense experience we collect sense-contents of material objects but in
another more directly intuitive sense. This dialogue discusses the
question, “What is the Form of One” (if indeed such a thing exists… for the
method used is to discuss a philosophic problem by both assuming the
consequences of believing a logical proposition about Forms to be True and
then also believing it to be false. This method of mental analysis
anticipated that of Boolean logic functions by several thousand years. So,
during the dialogue a series of eight “hypotheses” are put forward. And,
the “participants” in the dialogue discuss the consequences of assuming the
hypotheses are true or not. The first two hypotheses are that.


These questions are about something Plato believes in existing as a Form or
Concept in Divine mind and calls “The One”

How does Plato use Concepts to Investigate Concepts in this dialogue:

.... We trace out the definition of a lessor known concept by recognizing in
it the same elements as are present in a better known concept. In literature
this method is used in the form of metaphors and similies. In ordinary
speech it is the method of analogies.

....Since we are actually hunting concepts, we can bring one out in the
open, so to speak, by asking questions about. These questions will expose
what dols the concept together.

.... By seeking out those elements in concepts that constantly reoccur, we
obtain the most economical description of them.

THE MIND AS A SWITCHING YARD----PASCAL "ONE MUST

SUBSTITUTE DEFINITION FOR THE DEFINED."


Names are associated with objects; so are their meaning. But an objects's
name can be one the tip of the tongue when a corresponding meaning isn't:
how else do we know sounds when we hear them, or other things when we see
them. The answer is that our mind automatically deals with things by sorting
them into pre-established groups. We have, in our mind, a switching yard. If
we want to know what a song or anything else is, we must ask ourselves
question about the mental defintion of it. It will be helpful to discuss the
nature of this mental switching yard, in order to understand how to ask
these questions.

A concept (and hence the Concept of One that we are trying to understand) is
a rule that may be used to decide if a object falls in a
certain group. It is an abstract way of grouping thoughts. It deals with the
information associated with the object of thought by asking questions about
it. This may be a simple process like the way we classify concrete objects
by the "marks" of sense impressions: such as physical size or texture (this
process is instinctual in simpler animals). Or it may be a more complicated
process using a lot of other concepts.

Notes on Plato’s Dialogue the Parmenides:

These notes can only be cursory here because of the deep and complicated
nature of this wonderful treasure chest of a book.

A primary reference and starting point for further study is “Plato and
Parmenides”, by Francis MacDonald Cornford, Bobbs-Merrill.

Important note: Trying to read and understand the translation by itself
without any explanation, eg as given in Great Books volume 7 U. of Chicago
Press , is almost impossible in my opinion

Also, the One under discussion here is not necessarily the “Number One” of FR
From passage 137C
Hypothesis I) “If the One is defined as absolutely one, it is in no sense
defined many or a whole of parts”.
This hypothesis turns out to have the consequences that assuming this:
137D “the One has no parts and is unlimited”
138A “it has no extension or shape”
139B “is neither in motion or in rest”
139E-140D “it is not the same or different, like or unlike, equal or unequal
from itself”
“it cannot be or become older and is in no sense an ‘is’”
From passage 142B
What we are talking about here it clearly not the “Number One”, but something
like a set with no objects in it Z={ }.

Hypothesis II) “If the One has being, it is one Entity with both unity and
being.
This hypothesis turns out to have the consequences that assuming this:
142D “A One Entity is a whole of parts, (both one and many).”
This follows since ‘is’ is asserted to belong to the One, which is and
‘one’ is asserted to belong to this being which is One,
And since ‘being’ and ‘one’ are not the same thing.
This is what will be called “Unity in Diversity” later by our forefathers in
the United States.

145A “A One Entity (having parts…its unity and being) is indefinitely
numerous and also limited.’
This follows since ‘unity’ can never be lacking in its part ‘being’ nor vice
versia.
Thus, each of the two parts never lacking in the other will co-define
themselves forward inductively.
This argument will referred to as proof by co-induction two thousand years
later [10] Doets and Van-Eijck

145A “A One Entity (being limited) can have both extension in space and
shape.”
Indeed, being limited it is a form (something defined or made up of objects
satisfying certain
conditions in space and time.

145E“A One Entity (being an extended magnitude) can be both in itself and
another.”
The proof of this assertion given forms the foundation reasoning that
justifies what is called “proof by induction” for positive integer functions
later in mathematics
‘To prove, f(n) for all positive integers”
1) Prove there is a first integer ‘one’ for which the proposition
holds.
2) Prove for each integer there is ‘another’, a successor integer
for which is holds.
Then, the proposition holds . [10] Doest and Van-Eijck
.
Further conclusions with respect to like and unlikeness, equality and
inequality, and becoming of the object (that it has a beginning, middle,
end) also follow now.


146A “A One Entity (being a physical object in space) can have both motion
and rest.”
According to Aristotle motion can occur only in substances occupying “place”,
having “quantity” “quality”. It has “place” because it exists in space and
time. Anything having “place” can be moving or
At rest. This proves motion with respect to Aristotle’s category of place.
Motion with respect to Aristotle’s category of quantity is proved in the
previous
Reference. Proof of motion with respect to Aristotle’s category of quality is
actually not given to the next paragraph below.

147B “The One Entity described above is the same as and different from itself
and Others.”
“ The One Entity described above is both like and unlike itself
and the Others.

This is true because we have already seen that it (the one) is both different
from itself (as it exists in copies or differents sets representing itself
{Z} not equal to a copy {Z}) and the same (in the sense of its cardinality
being equal card{Z}= card {Z}.

Here, Plato’s term “other” means other copies of the positive integers which
we have just shown to exist
above. The One Entity is what the Pythagoreans Greeks called a “Monad”. Much
later we will decide to call a “Monad” data type a class definition of a
computer object having memory allocation and “functorial rules” for both
types of terms and functions mapping these terms among each other. Here it
can be thought of as the beginning element in the inductive definition of
integers, ie. The number one. The “Dyad” to be defined below,It is the
result of applying a 1) limiting factor (eg. The set of all integers x ‘such
that’ x < 5) and 2) an unlimiting factor (eg. The set of all integers such
x ‘for all’ integers y* x > x) to the Monad.

So, there is some confusion introduced here by Plato in that he proves his
assertion assuming his “One Entity” is what we now call “The Number One” but
the previous assertions were proved that it was just, the broader “Concept of
One” or “Oneness”.


149D “The One Entity, as described, has and has not contact with itself
and Others.”

This assertion again, is proved, by assuming we are talking about, “The
Number One”.

The word contact means the reality of the ‘participation of forms’ (mental
objects) in each other as already postulated in other dialogues. Thus we can
say that two ideal lines can intersect or cross in space as mental objects,
even though they are both idealizations of material things.

It is here that the writer of the dialogue attempts to explain how rational
numbers are generated from the starting point of the Monad and Dyad explained
above. However, in my opinion, it wasn’t until much later in the late 19th
century AD that Dedekind cuts were used to define real numbers as the ‘gaps’
between an upper and lower series of approximating rational fractions that
this difficult passage was really understood by anyone. E. Landau,
Foundations of Analysis

151B “The One Entity ( as continuous quantity and magnitude) is equal and
unequal both to itself and to the Others.”

Having created integers by pure ordered thoughtful reasoning above, now Plato
concludes that the rationals and reals can arise from logical mental thought
also in this way. It is argued that one continuous magnitude can be in
another continuous magnitude in a way that the container can be greater than
the contained, and yet they both still be equal. Again, it wasn’t until much
later in human history that we thought out the full definition of such terms
as “continuous” and “cardinal” and “ordinal” numbers.

Plato also correctly understands that the logical predicates of “greatness”
and “smallness” are what we now call “relations” or what can be understood as
types of “functions” defined separatedly from the way Oneness itself is
defined as a set. And, that the question of the existence of
these “relations” is one that is independent from that of the set that
defines what “oneness” is.

155C “A One Entity (as above qualified) exists in Time and is and is
becoming, and is not and is not becoming, older and younger than itself and
the Others.

Since the continuum of Time defined above mathematically exists, our objects
of thought, will reside in space as reasoned above, and also in time as just
concluded. And, since they are “mental objects” their participation
accumulation, disappearance (quantitatively) and assimilation,dissimulation
(qualitatively), in the material world occurs as a “quantum jump” ,that is,
instantaneously.

But, there are some paradoxes about this line of reasoning that took us more
than a thousand years to explain, for instance:


Zeno’s Paradox


This paradox, called the racecourse paradox, goes back 2400 years to
the Greek philosopher Zeno of Elea (495-435 B.C.). “A runner can never
reach the end of a racecourse because he must cover half of any distance
before he covers the whole. That is to say, having covered the first half he
still has the second half before him. When half of this is covered, one-
fourth yet remains. When half of this one-fourth is covered, there remains
one-eighth, and so on, ad infinitum.” This problem can be translated into
the language of symbolic mathematics in the following way: Suppose the runner
travels at a constant speed and suppose it takes him T minutes to cover the
first half of the course. Then the total time under consideration will be


Determining whether or not this infinite series ever converges to a
finite value is the same as answering whether the paradox is correct or not.
Trying out a sequence of values for the variable n suggests the hypothesis
that the sum is 2T, i.e.:



The proof of this was delayed until the 17th and 18th centuries when
mathematicians created the foundations of the theory of infinite series.

Define . Using the mathematical procedure called proof by
induction, we may demonstrate that for all positive integers n. Indeed,
this is true for the case n=1 because T = T. And, if we know it to be true
for the case n=n-1, we know that . Then, adding the value of to both
sides of this equation gives us the assertion for the case n=n.

Since, as n increases indefinitely, this shows that .
The fact that this proof uses the “principle of induction” which itself
assumes there is a first element in the series of times brings us back to the
importance of the “number one” again.

Zeno’s paradox arises from not understanding the statement above from
section 151B in the dialogue “The One Entity ( as continuous quantity and
magnitude) is equal and unequal both to itself and to the Others.”
Earlier Zeno and Gorgias had said: “an unlimited being (a many) cannot either
be in itself or in something other than itself, namely place. And, concluded
that it must be nowhere.” “and, if it is nowhere it does not exist and place
does not exist and if things are many, there do not exist.”
This was because it seemed that there were two equally possible but logically
contradictory possibilities to think about:
1) “ if things are a plurality they must be just as many as they are. But,
if they are as many as they are they will be finite in number.
2) If things are a plurality they will be infinite in number. For there will
always be between be others between any of them, and again between these yet
others.” Cornford pg 148-149, 180-181.

Again, tt wasn’t until the late 19th century, Frege’s logical prospositional
definitions of a function, relations, and mappings (correspondences) and
Georg Cantor’s set theoretic terms and definitions of what cardinality of a
set and one-to-one correspondences and arguments about what a “set” of
mathematical objects were that all of this was explained and understood.

Now that we have shown how these hypotheses about the One can create thought
using the fundamental ideas of “existence” and “being” as participating in
the One. Some new questions now come up, We have demonstrated the
possibility of existence of what we now call mathematical sets of objects
existing in space and time. For further study is Hypothesis I or II going to
be assumed as an axiom for new set theoretic questions? And, if Hypothesis
II, does the “existence” associated with the One come before its “being” in
the thought process of mental creation or vice versia.

Depending on how we answers this question Plato considers two hyptheses
Hypothesis III and Hypothesis IV

Hypothesis III “If the One is just One Entity which both both one and many
(as in Hypothesis II) or a whole of parts, the Others, as a plurality of
ones, form one whole, of which each part is one.


Hypothesis IV “If the One unity is defined as entirely separate from the
Others and absolutely one (as in Hypothesis I)the Others, can have no unity
as a wholes or parts and cannot be a definite plurality of other ones.

Finally, Plato turns to Hyptheses V,VI,VII,VIII which are parallel hyptheses
and consequences to those of I,II,III,IV resulting from assuming the “One is
not”.
These will not be discussed here.

Later Aristotle, Hume after him and others were to improve Plato’s teachings
about knowledge by pointing out that all knowledge (even if it has
his “intimations” or “intuitive participations” in great ideas such as Truth,
Justice, Goodness, Oneness, Equality must be based on and also contain sense-
contents from direct sense experience.

Plato and Parmenides, by Francis MacDonald Cornford, Bobbs-Merrill.
Edmund Landau, Foundations of Analysis, Chelsea.

Article #164
Subject: Linked In Discussions on this V
Author: Andrew W. Harrell
Posted: 5/22/2012 07:48:05 PM

Article #110
Subject: further discussions
Author: Andrew W. Harrell
Posted: 2/2/2012 06:52:21 PM
Mike Rand • Are we not bound to come to the conclusion that whatever the
extra-mental metaphysical ontology of numbers the facts of the matter are
epistemologically unreachable?

The best we can hope for is to be inclined to one or other of a coherent
theory. Some of us may claim knowledge of this but will have only belief, of
which we will never know the real corresponding truth, but we can be at least
certain that it would lack justification.
. Andrew Harrell • I would argue that the existence of the modern
digital computer, which arose out of speculations like this (how do we
implement integers on logic machines), which has help create the information
age in which we live, is justified scientifically and hence the philosophies,
the understanding of which made it possible, are also justified. j
2 days ago
Luca

Luca De Ioanna • Colleagues, what about numbers being human inventions?
Logical entities. No need to look for platonic entities out there.
Is this line feasible?

L P Cruz • Luca,

You directed a question of the same vain to me before when you asked me about
God creating the numbers.

I did not answer that question because if you do not believe it is
transcendent, well then that is the first question that is of importance.

As I said before, mathematicians who claim they are not being platonistic,
can not avoid being one when they do their maths.

Humans in general do not invent anything in some sense, what they do is to
describe what is already out there.

Though there are mathematicians who promote social construtivist approach,
they to their detriment ignore the way mathematics historically evolved.

LPC
Milo Gardner • Luca is closer to the historical truth thanL.P. Cruz, A short
essay will make the point:

Zero and one have always been out there (hence LP Cruz begins on strong
ground) ... but the manner the ideas were used together have long defined the
nature of cultures (Luca's point). For example, Egyptians thought of the pair
one of many of life' dualities. Mayans on the other hand used zero as
beginning and ending of cycles, with one as the next integer (counting
number). Going back to Egypt, without writing 0 and 1, the Old Kingdom from
4,000 to 5,000 years ago used a cursive numeration system that looks and acts
like modern binary computers. One was the first counting number. The nature
of Egyptian culture changed when zero and one could not be exactly record
rational number remainders ... so the rounded off Horus-Eye binary number
problem was fixed. One became a numerator, and the use of unitized weights
and measures replaced zero high status in the pactical cultrure. One was no
longer a matched pair with zero when money came into use by paying working in
equivalent commodites by absentee landlords.. Zero became a number to fear.
Zero was moved to a measurement of the empty set used in empty inventories,
so grain inventories could be maintained in good years and bad year. Yes, one
was a loved number for 3,600 years ... until 1585 AD ... when base 10 decimal
arrived ... today zero is loved more than one, defining he nature of
Euro/pean cultures since 1600 to today.
1 day ago • Unlike • Like • Reply privately• Flag as inappropriate • Flag as
promotion1 . Andrew Harrell • MIlo. I disagree. Many philosohers and
mathematicians (including me) love "One" and hence "the Number One) more than
the number "Zero". As, Ali has said previously here, commenting on Plato's
dialogue Parmenides (cf. Francis Conford's notes on how he divides Plato
discussion into eight hypotheses). Hypothesis II, "The One has being, it is
one Entity, with both being and unity." can serve as a philosophic definition
of the term "being" . See the Religion and Science subdirectory of the
discussion groups in my website (cf. given above) for more details on this.
1 day ago

Richard Ksiazek • Andrew, I think the programming analogy of Class is
confusing as the term 'class' is used in mathematics to denote a collection
of sets that can be unambiguously defined by a property that is shared by all
the members of the class. The use in programming is loosely based on this but
not identical to this. You also stated earlier that you believed in the
existence of the set of all sets. This set is logically impossible because
for each set there exists a powerset which is the set of all subsets of a set
including the set itself and the empty set. The power set's cardinality is
strictly higher than that of the set from which it is formed. If there were a
so called set of all sets it would have to include it's own powerset which is
of course impossible because the powerset's cardinality is strictly higher
than that of the set from which it is formed.

More relevant to the topic at hand there are several possible constructions
of which the von Neumannn is most commonly understood:

0 = { }
1 = {{ }}
etc...

Frege's approach was to define a natural number n as the set of all sets with
n elements.

0 = {{ }}
S(A) = {x ∪ {y} | x ∈ A ∧ y ∉ x }, for ay set A.

Milo Gardner • Andrew, the historical Plato loved one as a numerator and
other applications. Zero to Plato was not loved in the same way. However,
after Simon Stevin zero' s 1585 ADs exponent in base 10 decimal notation
radically lowered the love for one, and up lifted the love of zero beyond one
for non neo-Platonic folks. Yes, Neo-Platonic folks continue to love one in
the classical Greek sense. Were not those the major points in my essay?
1 day ago • Unlike • Like • Reply privately• Flag as inappropriate • Flag as
promotion0 .
Peter
Unfollow Follow Peter
Peter Anderson • What we're asking is in what sense 'one' means the same
thing in the statements 'There is one telephone' and 'There is one
matchstick'. The qualites of telephones and matchsticks are not comparable
except at the most general level of physical description - electrons,
protons, and so forth.

The prevalent knowledge of atomic theory is the only thing that makes One
confusing to define. So we have to put ourselves in the shoes of our
ancestors who came up with the notion. We generate in our minds a perception
of unity, by using the distinction created by the invisibility of air (zero)
at the boundary of visible matter (one 'object'), or the distinction of two
distinct notes by the intervening silence, in response to that which we see.

That we can pick one note off the page of a symphony does not make it any
less a single symphony made of multiple notes, any more than picking one
orange off a tree makes the orchard any less an orange grove.
1 day ago
Milo Gardner • Peter, Plato and Egyptians love one in unity statements . For
example the unity of 1 hekat = 64/64 hekat allowed a hekat volume unit to
divide (64/ 64) by rational number n such that

(64/64)/n = Q/64 hekat + (5Rn)1/320 of a hekat ...

i.e. 1/3 of an Egyptian hekat reported 21/64 hekat + (5/3)1/320 of a hekat
with 1/320 of a hekat replaced by ro and the scribal mental calculation

(16 + 4 + 1)/64 hekat + (5/3)ro =

(1/4 + 1/16 + 1/64)hekat + (1 + 2/3)ro

as the Akhmim Wooden Tablet (1900 BCE) and the Ahmes Papyrus (1650 BCE)
reported and proved!

Plato's era used the same class of unity statements that stressed one as a
numerators in a readable finite arithmetic operations Platonic and Egyptian
matb. facts that have been re-discovered in the last 10 years.
1 day ago
Mike Rand • Andrew, you point to relevant evidence which would support a
theory. From a Humean perspective then we are simply observing some
regularities in nature. The metaphysics of the matter remains beyond reach.
As Hume would point out, we do not experience the numbers themselves, and as
Kant would point out, to speak of the thing in itself is meaningless. The
ontology remains, and will always remain, beyond reach, but I accept that a
coherent account is presented as a vaiable option.
1 day ago • Unlike • Like • Reply privately• Flag as inappropriate • Flag as
promotion0 . Andrew Harrell • Yes, Richard the set of all sets can never be
computed in final form. But, that does not make it logically impossible.
Cantor's numbers, the cardinality of the integers or of the reals can never
be computed in finite time. They are what we call a 'limit cardinals"
or "limit ordinals". They exist because of the axiom of infinity is in the
standard set theory (Zermelo-Frankel). It is just basically the old principle
of inductive proof by which Zeno's paradox is avoided: If a proposition is
true for a starting element, and it is true for each successor element in an
ordered set, then it must be true for all (even through we can never check it
completely by tested every case). It is a debatable point whether either
the "axiom of infinity" or the "set of all sets" should be included in
standard set theory. Constructive models of set theory and "non-standard"
analysis try to avoid it.See the reference by Quinn quoted above for a good
discussion of this.
21 hours ago

Richard Ksiazek • I was incorrect to say logically impossible as there are
consistent set theories with a universal set, but these all have limitations
(axiom of choice fails, problems of extensionality, axiom of infinity is a
theorem rather than an axiom, etc...). The only one that seems up to the task
to building mathematics upon is NFU+Infinity+Choice (see
http://math.boisestate.edu/~holmes/holmes/head.pdf) but the ordinals of any
model of NFU are not well ordered, so it seems that ZFC remains the best
theory for use. This is not to say there is no merit to trying other set
theories and continuing the development, but that one must make severe
sacrifices in choosing to use those theories.
20 hours ago • Unlike • Like • Reply privately• Flag as inappropriate • Flag
as promotion1 . Andrew Harrell • Thanks for that reference Richard. It may
take me a while to red it. And, even more time to understand it. My thought
is that all this stuff about Monads and functional programming is new since
NF was thought out in the 1940s,1950s. I am sure they did not have Monad
Transformers then. What do you think about using them and functional
programming "streams" (which use partially defined co-routines) to
define,implement a class for a universal set and then run simulation of a
process that helps it determine what it actually is ?


. Andrew Harrell • Richard, And, another
difference from what I said, is that Holmes in his paper does assume the
existence of a set of all sets, a universe V. But, he starts out with the
number one defined as the set of all singleton sets of elements in the
universe V. This is equivalent to defining the one as what Parmenides and
Plato called the "many". Clearly this definition would cause problems if you
added it to an "axiom of infinity". I hope Holmes, Quine, Russell and you
didn't do this on purpose trying to confuse us.
3 hours ago
K N Padh • it is nice to learn innumerable explanations for 1, which now
tends to ' infinity '. lastly 1 is an abandon Prime expelled from the Prime
Numbers' Community. Be
Add/Reply to this discussion board posting
________________________________________
Article #111
Subject: bisumulations and ordinary inductive principles possible?
Author: Andrew W. Harrell
Posted: 2/6/2012 11:55:39 AM
Milo Gardner • Planetmath is operational .. Plato's math discussed this topic
in a clear way per: http://planetmath.org/encyclopedia/PlatosMathematics.html
Andrew Harrell • B N Padh says, "Be 1".
AndrewWHarrell Andrew W. Harrell
I AM already One (the will of a man of God willing God's will). I AM One now
(in the present moment) and ever will be (One in our future)..

Mike Rand • I am (we are) several.

The subconscious currents of thoughts and perceptions swirl in deep dark
waters illuminated all so briefly in to experience before the bubbling waves
of receed in to the depths.

Like a fish stranded on a shore gasping for breath, so too the transcendental
rational agent desperately pulls teleology out of the chaotic radom firing of
neurons. It coheres thoughts out of the subconscious objects and presents
them to our consciousness and fools us in to believing they are our own.

We are many things. We are the body, the brain, the mind, the self. Each of
these are divisible. If the mind is emergent from the brain and the self is
emergent from the mind, and if the brain is a divisible phyisical object,
then we are in principle divisible (e.g. split brain patients).

The road up is the road down. There is always a Heraclitean perspective for
any Parmenidean ONE.


Richard Ksiazek • Andrew,

I think your approach may still run into problems with the inaccessible
cardinals. Because x=x is a stratified formula The universal set V = {x | x =
x} exists by comprehension, but there is a difference between the existence
of a universal set and the ability to determin what that set is. If you
accept the axiom of small ordinals you are forced to accept the existence of
inaccessible cardinals (Robert Solovay).

The definition of the number one using the set of all singleton elements is
so that membership is the only primative predicate. This can be used to
define any absract object. Red is the set of all red things. So to
instantiate red in the world, is to be a member of the set of red things. To
instantiate one in the world is to be a member of the set of singletons. A
singleton cannot be the number one because that would require that there are
as many different number ones as there are singletons. We want a single
number one and that number one is the set of all singletons.
Andrew Harrell • That's a rather complicated comment Richard.
As, I said I haven't even finished reading and thinking about Holme's notes
yet. But, the thought did cross my mind that his system and Quine's might be
related to Solovay's and Rosser Boolean-valued models. These are some of
hardest to understand logical models every created, and you throw in the
theory and types to it makes it hard. Help! It may take me some time to
research this. Talk to you about it next week, God Bless.

Andrew Harrell • Richard wrote last week: "I
think your approach may still run into problems with the inaccessible
cardinals. Because x=x is a stratified formula The universal set V = {x | x =
x} exists by comprehension, but there is a difference between the existence
of a universal set and the ability to determin what that set is. If you
accept the axiom of small ordinals you are forced to accept the existence of
inaccessible cardinals (Robert Solovay). "
Why would anyone want to prove the existence of inaccessible cardinals
(assuming we understood what they meant)? Well, Homes shows in his notes,
that for such a system (with the axiom of a universal set, and the axiom of
singleton sets) there is what he calls "stratified comprehension" and the
standard Zermelo-Frankel model can be interpreted as existing inside of this
larger model. In this sense, i would say, the "weakly-inaccessible" limit
cardinal becomes a kind of God particle. allowing enough logical machinery to
define inductive (standard infinite recursive) and co-inductive recursive
(bisimulations) functions.

Richard, I do not believe it is possible to prove the existence of God (or a
God particle) anyway. I think this should be a matter of faith. So, I do not
believe this is problem for my way of defining the "number one".

But, do you believe the rest of the proofs in Holmes notes would go over with
my defintion of "one" as the "set of all sets which are equivalent to the set
with only the empty set inside it"(except for the proof of the existence of
a "weakly inaccesible limit cardinal."
Add/Reply to this discussion board posting
________________________________________
Article #112
Subject: further comments
Author: Andrew W. Harrell
Posted: 2/11/2012 03:12:45 PM
Dan DuPort • I haven't read thru all the comments, but one answer to the
question of how to define the number ONE is set forth in THE ROCKY MOUNTAIN
JOURNAL OF MATHEMATICS, vol 4, Fall 1974, number 4, in the article - A FORMAL
NUMBER-TERMED NUMBER SYSTEM BASED ON RECURSION by TREVOR J. MCMINN.

Set theory is not a part of the formalization. "In the system presented
here we formally axiomatize recursion directly so that an immediate
assault on recursive problems is possible. This is done without either
general functional concepts or any part of set theory, in such a way
that all terms denote only numbers or meaninglessness."

MiloUnfollow Follow Milo
Milo Gardner • Babylonian and Egyptian Old Kingdoms applied a recursive
algorithm that included zero in numeration systems that rounded-off rational
numbers n/p. Babylonian numeration and arithmetic recursive based 60 used an
algorithm that rounded off inverse prime numbers like 1/91 to inverse even
numbers like 1/90 for all of its history. Egyptian scribes rounded off within
a binary Horus-Eye base 10 system to six-places by throwing away upto 1/64
before 2050 BCE in spiritual and secular situations.

By 2050 BCE Middle Kingdom Egyptians included zero in secular situations that
applied finite numeration and arithmetic rules. The secular system recorded
rational numbers 1/p and 1/n to concise unit fraction series as often as
possible that eliminated the Old Kingdom Horus-Eye round-off problem.

Of course by 1585 AD base 10 decimals applied zero as a place-holder such
that Dan Dupont's citation of Trevor J. McMinn offers an appropriate cursive
story line.
4 days ago • Unlike • Like • Reply privately• Flag as inappropriate • Flag as
promotion0 .
DanUnfollow Follow Dan
Dan DuPort • In McMinn’s paper, 0 and 1- are taken, along with 2 other
constructs, as primitives. ONE is then the successor of zero. The idea of
successor is given by
pcrs n Ξ II xy x 1- n
scsr n Ξ the x (pcsr x = n)
1 Ξ scsr 0
The only criticism that I can see to the approach is common to all Peano type
approaches, it assumes that “everything” starts at one place.

I extended McMinn’s idea to include all the integers, and gave indication
that the system could be extended to describe the rationals. This was my
master’s thesis under his direction. We never seriously discussed the
possibility of extending the system to the reals at that time. Perhaps
someone out there can comment on this – can you or can’t you define the reals
within a non-set theoretic framework?
4 days ago • Unlike • Like • Reply privately• Flag as inappropriate • Flag as
promotion0 . Andrew Harrell • Dan, that's rather unusual (in my experience)
set-theoretic notation. Please only post comments using terms and formulas
you have defined in current or previous comments. Don't assume large sections
of papers just by giving URLS. This helps keep us understanding each other
better in this already difficult research area.

Dan DuPort • Andrew,

Perhaps you should read what I have written. The terms are not set-theoretic,
and I haven't given any URLs.

Not to upset you, but are you thinking about what you're writing?

Dan DuPort • Andrew, I have flagged both your preceding comments as
inappropriate and include them here, so you can't delete them and leave me
holding the bag.

Andrew Harrell • Dan, that's rather unusual (in my experience) notation for
set theoretic terms and formulas. Please only post references to terms and
formulas you have defined in the previous or current postings..

Andrew Harrell • Dan, that's rather unusual (in my experience) set-theoretic
notation. Please only post comments using terms and formulas you have defined
in current or previous comments. Don't assume large sections of papers just
by giving URLS. This helps keep us understanding each other better in this
already difficult research area.

You think that because you brought the ball, we have to play by your rules,
is that it?

HossamUnfollow Follow Hossam
Hossam Aboulfotouh • Plato wrote also the dialogue (of) Timaeus, which is
closely related to the dialogue (of) Parmenides (et al). Reading the two
dialogues, might reveal that the arguments of Parmenides (and his student
Zeno) were not as excellent as those were mentioned by Timaeus (the Italian
Astronomer), on the meaning of "one", "the same" and "the diverse". That was
implied by the responses of Socrates (the master of the Hellenistic
Philosophy) as a reaction to what was imposed on him, using contradictory
abstract arguments; taking into consideration that both dialogues are based
on the notion that God created one and the other two. If I followed the
argument of Timaeus, at the beginning of his talk to Socrates, I would say
that Parmenides and his student Zeno, had walked into the trap of abstraction
and thus mixed between the nature and the attributes of the three things that
are parts of "the whole" that became or to become. They used the abstract
nature of the things, excluding their intrinsic characteristics (concerning
either material or ethereal things) which is essential criterion in the
Platonic school of thought, and taking into consideration once again that the
opinion of Plato who was, or might was, present during the two presentations,
is likely to be very much with the side of his teacher Socrates. Perhaps,
Parmenides and Zeno were not astronomers but apparently they have had ideas
not in line with the philosophical thoughts of Socrates and his students and
followers.

I wrote once, mathematical model to describe Timaeus's ideas that was
appreciated by Socrates and perhaps also by Plato. In short, in my humble
opinion, the ideas of Timaeus on "the one that is many", could be perceived
today in the form and the role of the Hydrogen Atom or the like architectonic
cosmic structure of single orb: from the scale of the atom up to the scale of
the most great celestial system that one can think of. I cannot imagine a
cosmos without that prime structure "that is one and many in the same time";
and that could be collide and link with similar "one" and with "other", and
any other here is unlike the one. Timaeus's idea on "the same" is similar to
that of Albert Einstein in his Special Theory of Relativity (based on Lorentz
transformation). That is, "the same" is the frame of reference that its law
and time apply to every thing within its domain of multi-dimensions. The
third idea is on the concept of the "diverse", which could not be only "one"
and should be "many" in number, and all of them are "different" from each
other and all of them belong to "the same" as the frame of reference, and the
law and the time of "the same" is also encoded in all of them.

Sincerely,

Hossam M. K. Aboulfotouh
Andrew Harrell • Dan, what I meant by references to URLs was
your reference to the 23 page online paper by McMinn. It uses very non-
standard notation and it in turn refers to the book by by Maclane (which is
not online) for justification. So, I still think your comments were too hard
to understand (and hence inappropriate) for someone who does not have access
to these references. Please be more respectful of myself and others who might
lack of understanding of your thoughts. Sorry however if I upset you.

Dan DuPort • Well Andrew, it is a surprise to me that McMinn's paper is
online, and if you would, please post the URL, I would like to have it. Note
that I simply referenced the journal.

Yes, the notation is strange and is from a language created by A. P. Morse
who was around Berkeley about the same time as Vaught and Tarski. In fact,
Morse wrote a book called "A Theory of Sets" which uses similar notation, and
was a very unusual presentation of set theory. Morse was highly criticized
for it.

My comment which lists what you call unusual set-theoretic notation follows
the one which sites McMinn. The comment was a follow-up to simply give an
indication of the unusual non set-theoretic notation used.

I also ask "Perhaps someone out there can comment on this – can you or can’t
you define the reals within a non-set theoretic framework? "

Perhaps that should be a new discussion.

Hope you understand where I'm coming from.
Andrew Harrell • Just do a google search on "A Formal Number-
Termed Number Sysem Based on Recursion." Good luck trying to understand it
without the other book you mentioned (or even with it)..

MiloUnfollow Follow Milo
Milo Gardner • @Judd " Anatoly writes: what is a number?;
is it a "set of undivided unities"?;
do you say about numerical or order numbers?;
is it a finite construction (or a construction based on a finite schemes of
axioms)?;
is it a term, defined implicitly, in an axiomatic way?"

I agree that classical Greeks thought and wrote rational numbers as unities
by following a finite and axiomatic scheme (that did not draw upon
algorithms.)

Algorithms formally replaced Greek numeration by 800 AD when rational numbers
n/p were scaled by an LCM m such that

(n/p -1/m) = (mn -p)/mp with (mn -p) set to one as often as possible, thereby
defined a 2-term series.

At times 3-term and higher series were written by Arabs after 800 AD as
Fibonacci that adopted the notation in 1202 AD in the famous book Liber
Abaci. Following Fibonacci's 7th distinction (Sigler's 2002 translation of
the Liber Abaci) one of the examples given was 4/13 scaled to 1/4 and then
1/18 that found

4/13 = 1/4 + 1/18 + 1/468

For those that have not read Sigler's translation the following steps are
implied:

1. (4/13 - 14) = (16-13)/52 ... no value for m can find a 2-term series, this

2. (3/52 - 1/18) = (54 -52)/936 = 2/936 = 1/468

3. 4/13 = 1/4 + 1/18 + 1/468

It should be clear that Greeks did not use an algorithm that converted

1 4/13 x (4/4) = 16/52 = (13 + 2 + 1)/52 = 1/4 + 1/26 + 1/52

with (4/4) being the needed unity.

Thanks for the discussion unities used Plato and classical Greeks.
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M. VijayUnfollow Follow M. Vijay
M. Vijay Balaji • Among the positive integers one is number one and every
integer has a one in it. Every integer has a zero too. No other integer can
claim this privilege. The zero is required to divide the positive and
negative and also is a frame of reference in this regard. Thus perhaps zero
has this extra privilege. It is a zero which identifies the one!
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Peter
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Peter Anderson • 3 x 1/3
0.33 x 3 + 0.01
3000000/3000000

We define One as a piece of a cake which we never saw all of.
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M. VijayUnfollow Follow M. Vijay
M. Vijay Balaji • Without a zero, every other number is zero in the context
of frame of reference. It is because of a zero that one is one otherwise
nothing.

The big one is without bounds. the small one is a whole by itself.
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Ali

Ali AbuTaha • @Hossam: As you write, “Timaeus” and “Parmenides” are good
sources for Plato’s opinion about the “One” and how the one relates to the
many and other things. Other important input about the “one” and “many” is
given formally in “Republic,” indecisively in “Philibus,” uncertainly
in “Sophist,” etc. To unravel the “One” and “Many” from Socrates and Plato
requires complete input, and that can only come from most of the dialogues.

You write: “I would say that Parmenides and his student Zeno, had walked into
the trap of abstraction…” I like this. Generally, philosophers believe that
Plato failed to establish his important Theory of Forms and how it relates to
the “one and many,” and they use for proof the dialogue “Parmenides.” Yet, I
find the first part of Parmenides to be the strongest presentation of the
Forms and “One and Many.” Indeed, Parmenides and Zeno walked into a trap, as
you say. Remember the Eleatic Stranger in the “Sophist,” who was supposed to
be familiar with the ideas of Parmenides and Zeno, had no idea whatsoever
about the relationship between the “One,” “Many,” and “One and Many.”

There is a fundamental mathematical problem relating to the “One and Many.”
The problem was stated by Plato, Aristotle, and before them Heraclitus and
others. The math problems is this:

Show that “One = Many” in “number” and in “kind.”

Socrates said in “Parmenides” that to say that he (Socrates) is made of many
parts (therefore, many) and that he is one among seven people (therefore,
one) is not astonishing. That is because here, “one” is not equal to “many”
in number, and person is not equal to parts in kind! Did these factors show
up in your “mathematical model” from Timaeus? I think the above is relevant
to the “definition of the one.” Others may disagree. I like to hear your
views.

Milo Gardner • @Peter,your decimal point "3 x 1/3
0.33 x 3 + 0.01
3000000/3000000 "

was not a historical context in the Western Tradition until 1585 AD. Prior to
1454 AD and the fall of Byzantium Fibonacci's book Liber Abaci one was scaled
to weights and measures units, one being an Egyptian hekat that selected the
many parts of (64/64) such that zero round off took place.

Fibonacci solved this class of problem in the Liber Abaci, a unitized weights
of measures that modified the Greek and Egyptian unit fraction system in
minor ways.

Following up on the decimal 1/3 analysis, the classical Greek scribes
understood the Egyptian 1/3 of a hekat that wrote

1 hekat times 1/3 = (64/64) times 1/3 = 64/3 times 1/64 = (21/64 + 1/192)
hekat

further partitioned into

(16 + 4 + 1)/64 hekat + (5/3) times 1/320 of a hekat.

By 2,000 BCE and the Akhmim Wooden Tablet, 1/320 of hekat was replaced by the
word ro that allowed

1/3 of a hekat was recorded as

(1/4 + 1/16 + 1/64)hekat + (1 + 2/3)ro

It is important to show that the AWT scribe proven the correct unit fraction
series by multiplying his answer by 3 such that

a. (1/4 + 1/16 + 1/64)hekat x 3 = 21/64 x 3 = 63/64

b. (1 + 2/3)1/320 times 3 = 5/3(1/320) times 3 = 5/320 = 1/64

c 63/64 + 1/64 = 64/64

as Greeks were understood by Fibonacci by attaching Q.E.D.

Adding Fibonacci's explicit method for calculating this same class of 1/3 of
a number to this discussion can be provided if anyone desires.

Milo Gardner • Excuse the previous long post. A Wikipedia page link would
have sufficed:

http://en.wikipedia.org/wiki/Akhmim_wooden_tablets

Milo Gardner • Wikipedia includes a Liber Abaci link that discusses weights
and measures. The entire introduction must be read to grasp with 1/3 of a
weights and measures unit written in Fibonacci's modified unit fraction
system:

Fibonacci's notation for fractions will hopefully be of interest ... copied
from Wikipedia

"In reading Liber Abaci, it is helpful to understand Fibonacci's notation for
rational numbers, a notation that is intermediate in form between the
Egyptian fractions commonly used until that time and the vulgar fractions
still in use today. There are three key differences between Fibonacci's
notation and modern fraction notation.

Where we generally write a fraction to the right of the whole number to which
it is added, Fibonacci would write the same fraction to the left. That is, we
write 7/3 as \scriptstyle2\,\frac13, while Fibonacci would write the same
number as \scriptstyle\frac13\,2.
Fibonacci used a composite fraction notation in which a sequence of
numerators and denominators shared the same fraction bar; each such term
represented an additional fraction of the given numerator divided by the
product of all the denominators below and to the right of it. That is,
\scriptstyle\frac{b\,\,a}{d\,\,c} = \frac{a}{c} + \frac{b}{cd}, and
\scriptstyle\frac{c\,\,b\,\,a}{f\,\,e\,\,d} = \frac{a}{d} + \frac{b}{de} +
\frac{c}{def}. The notation was read from right to left. For example, 29/30
could be written as \scriptstyle\frac{1\,\,2\,\,4}{2\,\,3\,\,5}, representing
the value \scriptstyle\frac45+\frac2{3\times5}+\frac1{2\times3\times5}. This
can be viewed as a form of mixed radix notation, and was very convenient for
dealing with traditional systems of weights, measures, and currency. For
instance, for units of length, a foot is 1/3 of a yard, and an inch is 1/12
of a foot, so a quantity of 5 yards, 2 feet, and \scriptstyle 7 \frac34
inches could be represented as a composite fraction: \scriptstyle\frac{3\ \,7
\,\,2}{4\,\,12\,\,3}\,5 yards. However, typical notations for traditional
measures, while similarly based on mixed radixes, do not write out the
denominators explicitly; the explicit denominators in Fibonacci's notation
allow him to use different radixes for different problems when convenient.
Sigler also points out an instance where Fibonacci uses composite fractions
in which all denominators are 10, prefiguring modern decimal notation for
fractions.
Fibonacci sometimes wrote several fractions next to each other, representing
a sum of the given fractions. For instance, 1/3+1/4 = 7/12, so a notation
like \scriptstyle\frac14\,\frac13\,2 would represent the number that would
now more commonly be written \scriptstyle 2\,\frac{7}{12}, or simply the
vulgar fraction \scriptstyle\frac{31}{12}. Notation of this form can be
distinguished from sequences of numerators and denominators sharing a
fraction bar by the visible break in the bar. If all numerators are 1 in a
fraction written in this form, and all denominators are different from each
other, the result is an Egyptian fraction representation of the number. This
notation was also sometimes combined with the composite fraction notation:
two composite fractions written next to each other would represent the sum of
the fractions."
Add/Reply to this discussion board posting
________________________________________
Article #113
Subject: Rudolg Carnap's notes on Frege's sysem prove Russell's paradox not
a "knock out blow"
Author: Andrew W. Harrell
Posted: 2/16/2012 12:47:44 PM
Hossam Aboulfotouh • @ Ali;
What I have said in my previous message in this thread, concerns only my
humble opinion on the architectonic form of the "prime number one" and its
likely physical form, in our mother-nature that the scientific community of
today have certain knowledge about and recorded great deal of, in the
literatures. In this regard, I cited the dialogue (of) Timaeus by Plato that
I consider it the best from among Plato's works for the subject under
consideration; if I understand it in correct way. I meant the ideas of
Timaeus (the Italian Astronomer) on the role of "the smallest one" in the
nature that we do know, for assembling any great or "the greatest one" that
we can think of. Although we human beings cannot see any great one from all
of its sides, or the whole of the smallest one in nature, even with using the
modern technology, we have been gifted some parts of the sense of imagination
of the great geometer to do so with diverse degree of accuracy based on the
degree of harmony of the soul at the time of thinking.

The responses of Socrates to Zeno in the presence of his teacher Parmenides,
as was narrated by others and later on published by Plato, with or without
Plato's amendments or his imbedded philosophical opinion, were rather to show
the weak ground on which Zeno was trying to build his pillar of philosophical
persuasion. I prefer, therefore, not to include that sort of philosophical
tricks that was imposed on Zeno by Socrates in any talk on the definition
of "the real-one in nature" and "its sole or diverse mathematical image(s),
which all or some might have the same frame of reference that has any law
and/or time parameters"; I mean at least from my side.

Sincerely,

Hossam M. K. Aboulfotouh
. Andrew Harrell • Thank yo for your comments Mr. Aboulfotouh. I
appreciate the respectful way they are framed. I see your visualization idea
of Plato's idea of "the One and the Many" Being similiar to how we visualize
the hydrogen atom as a useful thought. It seems similiar to me as how Taoism
visualization the primal Energy of Creation of the Cosmos as a Yin and Yang
symbol of intertwineed positive and neagtive energies. They then place this
primal Yin Yang symbol inside of what is called a "padua" of eight steps of
three-fold I Ching change thoughts. These thoughts become spinning wheels
(paduas) inside of wheels (paduas) like in the prophet Ezekiel's vision in
the Old Testament Bible. This then becomes God's creation outside of the
inner symbol. Do you have any thoughts about this?

Peter Anderson • @Milo,
I appreciate your history lessons. Does Akhmim predate Rhind's discovery?

The Egyptian way of looking at the sum total as One and everything a fraction
is really interesting. It makes alot of sense.
The thread is, after all, asking how we (now, not historically) define the
number One.

Milo

Milo Gardner • the berlin papyrus was reported in 1862 the rhind papyrus
1864, 1879. the akhmim papyrus was partially analyzed in1906 and fully read
in 2002, 2006. the akmim and berlin papyri date to pre1900BCE with the rhind
dated to 1950BCE.

how number one was defined in the western tradition are all connected in
subtle ways the use of Platonic dialogues and metaphors may be the most
visible , but not necessarily the most important to math historians and
modern math philosophers...
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Milo
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Milo Gardner • oops the rhind papyrus dates to 1650 BCE and used the akhmim
wooden table's scaled one recorded as 64/64 a thinking style that lasted
until 1454 AD..and the fall of
Byzantium and the rise of base 10decimal arithmetic after 1585AD.

cutting off over 3000 years of one scaled by unities of many sizes seems
short sighted...
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promotion0 . Andrew Harrell • @Milo,
Yes, I agree with Peter. Let's stimulate healthy debate with new ideas, not
just report on history. But, it is interesting, also, to study how we got to
where we are. Another thread could be devoted to this.

Milo Gardner • @Andrew, I agree. Let's balance the discussion of one in the
modern tense, Computers use of one offers wonderful examples. Yet, the
subject is very old. At times subtle threads color modern discussions in
unexpected ways. Plato is mentioned in this way. When seen, should not the
historical Plato;s use of one be properly cited?
Andrew Harrell • Yes. You are very welcome to monitor and
contribute this way.
Andrew Harrell • Let me try and steer the
discussion back to Frege's more recent ideas. As we noted he believed numbers
were objects. Concepts are also objects according to him and Plato. But,
Frege believed also that numbers were not concepts, they were(are) according
to him, "values of concepts" or "extensions of concepts". Anyone else have
thoughts on this aspects of Frege's "Foundations of Arithmetic"?

Mike Rand • I often think that it is good to take a stance on Schopenhauer's
philosophy. Schopenhauer rejected this striving real world and looked towards
the ideal platonic realm. Nietzsche in turn rejected Schopenhauer and
affirmed life.

I worry about the Fregean/Platonic perspective on the world - that sterile
purity of a higher realm. Much better to live with the imperfect
Nietzschean/Aristotelian view.

As Renton implored in Trainspotting - Choose life.
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promotion0 . Andrew Harrell • Mike,
Although he was an atheist, I agree with Betrand Russell's analysis of
Nietzche (google Bertrand Russell on Nietzche UTube). But, what about your
thoughts on the specific questions I did ask about Frege's metaphysical views
on numbers? .

Mike Rand • The Ptolemaic view of the heavens gave a good empirical fit to
the evidence - but it was not an accurate description of reality. Though the
Fregean system might give a model of numbers that by no means demonstrates
any metaphysical correspondance. (Although I do think Russell's paradox is a
knockout blow to the whole project)

Numbers have no extramental reality, (they exist within a social context but
that isn't independent of the mental), they are manifestations of an evolved
cognitive aparatus that delivered adaptive advantage.

An elevated perspective that we might have of them is some kind of Kantian
compromise - a transcendentalism.

Andrew Harrell • MIke,
Russell's paradox was clearly not a "knock-out" blow to the Fregian system.
See Rudolf Carnap's notes on "Frege's Lectures of Logic" from 1910 at Jena
(Published by Open Court)..In these notes Frege leaves out his Axiomx V and
VI from the Grundgetsetze which led to the Russell paradox problem from his
theory of extensions (associated with what I referred to above as his idea
of "values or extensions of concepts). What is left is a clear and workable
system of mathematical logic in which set theory, a theory of identity in
statements of propositional logic, mathematical functions, ordinal numbers,
cardinal numbers can be defined. As to the detals of how this relates to
recursive functions, lamba calculus, functional computing; that is a lot
harder to understand. Maybe we (the members of this discussion group) will be
able to figure it out together?