PHILOSOPHY OF THE INFINITE: by Aristo Tacoma
Notes on my Firth Lisa GJ2
[[[GENERAL BOOK AND ASSOCIATED INFO:
Title: Philosophy of the infinite
Subtitle: Notes on my Firth Lisa GJ2
Author: Aristo Tacoma
A PRIVATE PUBLICATION
Book is copyright author -- who is
also, as author names, Stein von Reusch,
and Stein Henning Braten Reusch, and
Henning W Reusch -- all rights reserved;
ordinary quoting accepted when reference
giving. Stein Henning Braten Reusch asserts
the copyright of all works published with
(artist) name / main name Aristo Tacoma.
Publisher: Yoga4d von Reusch Gamemakers,
(company earlier called Wintuition:Net)
Year of publication: 2009
Place of publication: Oslo, Norway
Extended ISBN registration number:
ISBN 978-82-996977-1-2
(This document replaces the proposed
publications of some uncompleted
documents which the Yoga4d publishing
company proposed for similar ISBN number
but without a formal printing and
transmission, as required, to the
National Library of Norway, which
has the 2004 book by same author
-- Passion without greed or hatred,
resonating over dancers, creating
new physics (pen name: Stein von
Reusch), and Sex, Meditation and
Physics, 1999 (pen name: Henning
W Reusch), already; the 2004 book
with the ISBN 82-996977-0-0 as
verbatim reproduced on top of the
webpages http://www.yoga4d.com
and http://www.yoga6d.com which
in toto are unchanged forever from
June 18th 2009 (the final hints6.txt),
and that goes for the TITLESEARCH
and CURRENCY and JPG-BROWSER as
well on the front -- ONLY the
yoga4d.com/talks and the
yoga6d.com/stamash.htm parts are updated.
[[[GENERAL BOOK AND ASSOCIATED INFO ABOVE]]]
AUTHOR'S THANKS TO:
Extra thanks to all my family for support
during more than decade-long development
of new computer programming language and
operating approach, and associated research
& development around this, also with physics
and with new computer hardware components --
not in the least Else Reusch Braten and Stein Braten,
my parents (see http://www.stein-braten.net
for overview over some of father's publications)
for their dialogical attitude and very helpful
attitudes. The many acknowledgements found in free
manuscripts at my websites naturally applies as well,
as well as some extra references inside this text.
Prelude
The wonder in your mind is an infinity -- anything
can happen. Philosophy honors you when you claim
your infinity, your meditation over life's greatest
questions. But in this infinity something may
crystallize itself -- and speak with finiteness.
This, too, we must learn about. We must not deny
the finite. But what we learn about the finite
cannot always be applied to the infinite, and
what we learn about the infinite cannot always
be applied to the finite.
You are no stranger to the infinite if you have
feelings about the universe as a whole -- they
draw from an infinite reservoir inside you, a
sense of cosmos. But you may also want to find
your way through the city to a certain street
and street number: in a finite structure, you
need to navigate by means of a knowing of
finiteness. Sometimes our feelings and thinkings
are infinite-connected, sometimes they are
finite-connected.
How clear can we be about all this? Can we get
it right, you and me in the 21st century, where
humanity has a past meddling with these themes
in ways which perhaps have belittled infinity
and glorified finiteness?
I think we can get it right -- "it" being the
starting-point, not the conclusion. We can unlearn
the mistakes and get the questions sorted, and
see some of the confusions so clearly we avoid
them.
PHILOSOPHY OF THE INFINITE: by Aristo Tacoma
Notes on my Firth Lisa GJ2
CHAPTERS
* GETTING KANT AWAY FROM METAPHYSICS
* MINIMIZING DARWIN IN EVOLUTION THINKING
* GETTING BOHR AWAY FROM QUANTUM PHYSICS
* GETTING BLACK HOLES AND BIG BANG AWAY
* RE-INTRODUCING INFINITY AND CONTINUITY WITHOUT
NUMBERS
* THE ROLE OF THE UNCOMPUTER
* GETTING BERTRAND RUSSELL AWAY FROM LOGIC
* GETTING SPINOZA AWAY FROM DETERMINISM
* GETTING FREUD AWAY FROM THE SUBCONSCIOUS
* REPRINT OF THE REJECTED 2003 EXAM (from MYWEBOOK)
* WHAT I CALL "NEOPOPPERIANISM", TO REPLACE POPPERIANISM
* EXAMPLE: AN ELIZA PROGRAM WRITTEN IN LISA GJ2
* ELEMENTS OF SYNTAX OF MY FIRTH LISA GJ2 (from MTDOC)
* THE LANGUAGE LISA GJ2 AS A NEW KIND OF THEORY
****** GETTING KANT AWAY FROM METAPHYSICS
When we enquire into ourselves, in our minds, so as to
reach out to the universe as a whole, we are engaging in
an activity -- call it "hobby" if you wish -- which, as
I see it and as I use the word, rightly can be called
"metaphysics". This is something other than piecemeal
investigation into the world as it presents itself in
glimpses and flashes and resonances through our sensory
organs such as by touch, by sight, by listening, by
smell, and so on.
I suppose some of you might have heard that a thinker
called Kant worked hard on dissolving the idea of
metaphysics as a valuable praxis. By arguments which
today would rather be called "psychological" than
"philosophical", but which at his time preceeded the
erection of the field of psychology, he pointed out that
far from "looking at the world" a human being is rather
filtering and sorting bits and pieces through a pretty
heavy set of in-built categories -- or that's how he
saw it, put in rough, and to him, for sure, inconcise
or even misleading words. The world as such is "out
there", the individual hasn't got any actual touch
with it but can play with thoughts about it, yet any
connection happens by categories.
It may be of some value to note that Kant didn't
himself have much contact with the fullness of human
civilisation. Indeed, to him, humanity was mostly a
question of categories with which he had little
contact. He was a man of mechanical strict habit,
or so we are led to believe by the typical biographical
descriptions of the man. He stayed away from visiting
any places if he could avoid so; he wrote and wrote
and wrote and did so by the clock as his work; and
in this rather sexless life, if that is the word I
want (I don't mean it absolutely, but as a rough
term describing his work-attitude as it has been
presented), he gives a very accurate description of
how his psyche has become. This psyche is however
not necessarily representative for the necessary or
essential human psyche.
This man Kant does not speak of the intense,
feverent love-resonances that seem to arise
spontaneously when some people make out. He does not
give serious consideration to the possibilities of
intuitions which can not only give glimpses not
allowed by sensory organ connection, but which can
have such a mind-shattering impact that they
reorganize the categories of the mind and provide
a revolution of consciousness. If he has ever looked
at the stars so as to forget himself, and forget
that he is an individual, and come into a state of
the fullness of wholeness and meditation and its
awesome truth, he has not allotted it space worthy
of it being part of the enquiry into the world. He
has, if he has had intimations of such states (and
who hasn't?), probably dismissed them as passing
emotional states.
We shall not also forget that even a lifetime of
writing with apparently strong and noble intentions
in finding out right things to say about reality
can have had a hidden agenda, for instance a political
agenda. Most writings in European culture prior to
20th century had to relate to the fact that shoddy
christian priests had a magnificent power of a
political kind -- for instance, they could evict
a person from society, or even get the person killed,
if that person engaged in what they saw as
anti-dogmatic writing. This led to a division between
writers on reality, on natural philosophy, on philosophy,
-- whatever we call that group of thinkers: on the one
hand, we found people like Descartes, who conceded that
the priests were right about many essential things,
and on the other hand -- as an extreme -- we found
people like Nietzsche, who did his outmost to belittle
the priests. And so, while talking about wholeness,
compassion, love and so on came easily to the priest-
friendly guys, those who sought to politically knock
out the priests wrote in a language which typically
became exquisitely and exclusively atheistic -- and
here Kant fits.
The word "atheism", with the "a" in this particular
case being short for "anti" (while in other cases it
can mean something such as "an instance of"), and
"theism" referring to God, or Theos, Deus, a word
apparently derived from Zevs, and the beliefs associated
with the concept of such a creator, thus can denote a
certain type of attitude to the world as a whole,
in which "hidden connections" and "deeper patterns"
and "higher meanings" are belittled in favor of a
notion of seeing the world as composed of disjunct,
disassociated parts which have little genuine
relationship to anything; at best a little relationship
to themselves in their own closure. And with a natural
use of words, we can call the latter for "a metaphysics".
We can call the notion of there being a creator and
life woven full of hidden meanings, synchronicities
perhaps, and compassionate love potentials and direct
intuitive capacities and such, for "another metaphysics"
(or group of metaphysics).
Now it is the task of a politically inclined person,
who is not interested in truth or honesty as much as
to win, gain in conceptual or actual power over other
parties, to put them down. It is easy to say of
the priests that they were indulging in a form of
metaphysics; it was possible for Kant to conceal that
he himself indulged in metaphysics by criticizing
metaphysics as such, arguing that his own standards
of thinking represented 'necessities' or 'self-evident
truth', not mere vague foggy metaphysical speculation.
But I want to say: Kant's works, although it is
massive,
and in portions have elements of light to them (such as
in talks about ethics, up to a level, and in giving
a larger understanding of how an individual can be
biased in perception), is a vague foggy metaphysical
speculation. It is no argument against metaphysics as
such: rather, it is a very strong, and nonworking,
attempt to assert an atheistic metaphysics as primary
without any really good reasons other than that he
cannot seem to get out of his shoddy little self when
he tries to look at the world.
I speak these words strongly because Kant's influence
has been strong: so that it is possible to enquire
more freely.
The clue to see the weakness in Kant's argument is
that it relies solely and completely on statements
involving impossibilities: he speaks of a direct,
mind-to-mind perception or mind-to-thing perception
as impossible, and says that the individual is left
with categories which are as if "perturbed" or
"disturbed" by fragmented impulses arriving through
the sensory channels. But does he make it clear
that it is impossible? No; he merely says that it
is self-evident and any element of doubt anyone
may bring to the matter is evidence of immaturity
and lack of philosophical depth. This is the type
of argument a politician, or a corrupt scientist
can come with; but all that I know of who can
be considered to have philosophical depth, will
approve of doubt as a purifying agent. They will
not say, do not dare to doubt a big truth. They
will rather say, a truth will withstand good doubt.
Whereas that which is an illusion will fall away
when we apply doubt to it.
I will connect this soon to questions about
the infinite.
Let me give a metaphor, a verbal illustration,
of how saying that something is impossible can
become, for a while, self-justifying. I am saying
that Kant's statements can, if believed, work to
dullen the mind, because the impossibilities that
he claims can, for a while, maybe a long while,
be self-justifying. This, I believe, will show that
Kant was not a good philosopher: rather, he was an
amateurish psychologist, getting hold of a bit of
psychology and trying to portray it as the whole.
That doesn't meant that I don't think Kant has a
good grip on language, but a good grip on language
doesn't signify philosophical maturity. Kant may have
been important, politically, to get the meaningless
power given to the corrupt christian priests away,
and that may be enough to say that perhaps he was
not entirely wrong in doing what he was doing, but
as philosophy standing on its own, apart from the
question of political rethorics and its justification,
it is poor quality-stuff. We want high-quality stuff
when we are going to think about the metaphysics of
infinity; we cannot have a bunch of low-quality Kant
stuff injecting itself in our discussions, so I spend
this time on negating Kant from our consciousness.
So here's the simple verbal illustration: a person
is of the opinion that a particular piece of electrical
equipment, such as a lamp or a dish-washer, doesn't
work. There is much experience in this person of
trouble with the item earlier on; this person doesn't
know, however, that in its present condition, it
works sublimely well. So this person and another,
who is not prejudiced, goes into a room and, before
the other gets a chance to switch the machine or
lamp on, the first says: "Don't bother, it doesn't
work." The buttons which have to be pushed are
perhaps a bit complex. If one doesn't push them
right, and in the right sequence, nothing works.
The unprejudiced guy says: "Oh, I am sure it
works", and is about to switch it correctly on,
when the first person meddles wrong on the
buttons, and says, "You see? It doesn't work
at all. In fact," that person adds, ripping out
a major part of the equipment so it breaks loose
and the machine goes into disorder, "this piece
better be replaced. The machine is just a mess.
Forget it."
Let that machine intuition -- not intuition as
such, but the capacity an individual has, in,
say, a week, for relating to intuition and
strong direct perceptions of the wholeness
of existence without bias, rather than merely
to ego-moods and ego-categories. The first
person is Kant. That other person is You, the
reader, and You are Not Prejudiced. Agree? ;-)
Thank you.
Kant, then, did not argue against metaphysics.
He argued against one type of metaphysics
associated with a group of people who unjustly,
as he saw it, had grabbed hold of power; by means
of another metaphysics, which he did not name
as such. We name it as such. We also do not want
to confuse metaphysics with political argument
or confuse philosophy with amateurish psychology
in which one denies the great bridging capacity
that doubt has to gap the subjective mind with
objective reality. All questions are open.
Love can be real. Direct mind-to-mind contact
as resonance, as ecstatic experience, can happen.
The enquiry into the wholeness of existence
doesn't have to happen by means of fixed
categories. All the same, we do thank Kant for
reminding us not to be biased, and to watch out
for projecting self-satisfactory categories
(whether from Kant or from somewhere else)
onto our perceptive game with reality.
We, in fact, LOVE metaphysics. We can
objectively enquire into it. We can go into
a mood of enquiry, a state of mind in which
we have a direct experience of infinity and
of things beyond the subjective field of
psychological activity. However we reserve
the right to doubt ourselves in the midst of
the process so as to provide points of control,
points of checking, as to how unbiased we have
in fact been in our perception. We are not saying
it is always easy for all to perceive reality
in an unbiased, categoryless way. We are asserting
that it is obviously possible. Yet we are giving
room to those who, like Kant, would like to argue
against such possibilities -- for a while; and then,
after weighing the arguments, we toss them away for
they do not have depth in them. Assertion of
impossibilities rarely do.
And, please, let's be clear: we do so without a
hidden agenda. We are not automatically in favour
of a particular theology even as we dismiss the
needlessly mechanical and brutal denial that
atheism implies. We are merely saying that the
enquiry into reality with the WORKING MACHINERY
OF INTUITION will have to take trust in the real
and actual possibility and even probability of
intuition as startingpoint -- otherwise (keeping
the metaphor above in mind), one might mess up
a vital instrument in the mind for a while. We
also will not use the word "mind" in a way which
denotes a sharp difference between "mind" and
"emotion" or between "mind" and "heart" or
between "one individual's mind" and "another
individual's mind" or even between "my mind"
and "God's mind".
****** MINIMIZING DARWIN IN EVOLUTION THINKING
Speaking about God, let's get rid of Darwin. Or, more
precisely, let's focus on what was the right and
proper and interesting contribution of Darwin, and
let go of all the political atheist propaganda that
people both after him, and even to some extent he
himself, have surrounded his general propositions
with. For the more we negate of those who have
opposed an interesting form of infinity operating
in the world -- I mean, the more of the thoughts
of those kinds that we negate -- then the more easily
will we have a genuine earnest deep interest in
infinity, and in getting clear, coherent, beautiful
in our thoughts and feelings about it, finding an
order there. Getting Kant away from metaphysics
and rescuing the sense of metaphysics as a grand
old pondering over the wholeness of existence as
a worthwhile intellectual pursuit is what I
attempted in the previous chapter; let us now look
to one who, without reason as I see it, is considered
an antithesis to big-scale elements of infinity
mingling with life and its evolution, due to his
work.
Again, we must think politics. Otherwise there
will be no understanding of Darwin and his interest
in natural philosophy or that which today is called
"biology". He boarded a slip run by slaves, black
slaves whipped and maltreated more or less unto
death by lighter-skinned "slave owners", to get
over to the Galapagos Island in South America
to study both present natural wild life and remains
of earlier natural wild life, looking for signs that
the church and its priests were wrong.
He hated the treatment the dark-skinned people
got. According to lighter-skinned priests, God
created the world on some six, seven days, and
made human beings in his image, and threw in some
animals and trees and what not for the sake of
these beings he had created. He gave souls to
the human beings. He also created apes of various
forms, and the priests inclined, at the time of
Darwin, in many european countries, to indicate
that whereas the light-skinned human was a direct
descended from the first pair of human beings
created by God, the darker-skinned slaves weren't
quite human for they had somehow got ape-blood
in them. This, then, justified atrocities against
the slaves; the slaves weren't quite free for
they weren't quite human and hadn't quite soul --
and Charles Darwin was, according to some
biographers, absolutely outraged with the point
of view, and here we totally cohere with Darwin.
Wrong is wrong. Slave-stuff is all wrong. Aristotle
and Platon appear to have been in favour of regarding
slaves as a different type of species and unfit for
normal human dignity and normal human freedom within
civil rights and so we must say that, despite the
great glimmers of wisdom here and there in their
works, they were wrong. Socrates was perhaps the
greater of the three, the teacher, we might say,
of Platon, and Platon the teacher, for a while,
of Aristotle; and Aristotle again the teacher
-- and lover, as often was the case, of his young
student Alexander, who became Alexander the Great,
bridging Egypt, Greece, India and more by hoardes
of horse-driven military led by himself personally.
After Alexander the Great, one of his generals
became Ptolomy, faroh of Egypt, and the important
and longlegged and sensual-lipped Cleopatra is a
direct descendant from the in-breeding of Ptolomy
with his sister and their children with, in general,
more within the family, over several hundred years.
This in-breeding led to the great beauty of
Cleopatra and, just as with horse-breeding, it
sometimes works this way -- whereas in the case
of bad genes, cross-breeding is necessary.
In any case, Socrates, whatever his views were
on slaves -- he seemed not to leave writings of
his own, but rather we know about him through the
often funny theatre-like writings of Platon and
occasional references by Aristotle in the more
prosaic but massive works of Aristotle --
was a pure spirit igniting fascinating forms of
dialogue and doubt in Athens. The polis Athens was
dedicated to Athena, the beautiful warrior muse
(or "goddess", but I think muse is a more proper
translation for only Zevs was really God in the
greek scenario at that time). Since Athens had
a dictatorship as to slaves, only a portion of
it was a democracy; Socrates was accused by
people in this democracy that he was upsetting
it; he laughed and spoke sarcastically at those
present in court when they made a legal case
against him; and seemed to provoke the capital
execution sentence on him.
Now Darwin saw that the slave prosecution was
sustained by a different sort of reasoning,
but with rather exactly the same result, many
centuries later: God hadn't created them in
the same way as the lighter-skinned human
beings, according to the priests; and so
it was not inhumane to have and punish and
even kill slaves. So Darwin looked for items
and phenomena which could indicate that instead
of creation being a week-long thing, after which
things had merely shall we say pottered on, it
had instead proceeded over millions of years.
And he found many strong indications, in a
certain perspective of this.
So, listing a number of these, he postulated
a couple of things. For the sake of a dialogic
discussion of what he says, I will not bother
with reproducing either his own words nor the
pompeous priestly way that modern-day scientists
like Dawkins speak of his propositions, but rather
put it so it sounds as trivially true in some
way as I think it is, but also so that it shines
through that a lot is unsaid, so much that the
entire role of Darwin is minimized. This is my
intent. For more reverent discussions of Darwin,
turn to some other works.
Darwin said that those who are fit and beautiful
tend to get more children; over a long time, then,
they come to dominate with their genetic streaks.
That is obviously the case.
Darwin said that occasionally, changes in the
physical make-up of the organism being born
occur, relative to the mother and father, and
that these changes may produce even more fit
individuals. In looking at a variety of animals,
he found indications of such a thing, and called
it "evolution". He noted that it also seemed that
this process involved not-so-fitful developments
which didn't work out in the long run.
Let's just say that he was right -- at least
as far as the past development of animals go,
and as far as any past version of the human
being goes. As Arne Naess has often pointed out,
one cannot automatically say about the future
that it will reproduce a pattern found to apply
in the past.
Let us now note that Darwin said that these
changes -- which he called mutations -- happened
in a way which was not the result of a greater
design at all -- but happened by means of some
kind of random or chancelike turnabout of the
genetic tendencies. How could he know? Of course,
he couldn't know. He merely put this into his
above-mentioned, relatively correct propositions,
because he found indications that not all changes
worked out.
But consider a person writing an article. Will
there not be experimentative versions of this
article that are discarded before the finished
version of this article is produced? Yes, typically.
But is this indication that the article was
produced by means of a random or chancelike
turn-about of the letters? No, not at all.
So in bringing in the component of "chance" or
"random", Darwin made a total, unforgivable
blunder. He had no right to do so.
And the fact that this blunder was part of
pattern in the narrow mind of Darwin, is shown
by a number of notes he made here and there in
his less clear writings, as published. He states
for instance that everywhere in nature he finds
brutality, and no evidence whatsoever of any
design.
So he merely re-iterates his own narrow
insight into reality. He doesn't see the greatness
of nature. He doesn't know of course of the 20th
century revelations of far far more design,
including at the molecular level, of all living
beings; he doesn't see the beginnings and
expressions of compassion which flourish among
dolphins and chimpanzees; he doesn't see the
greatness of the full mindfulness of the human
being. But we do totally share the sentiment
that all human beings, no matter what skin
color, of course are equal in worth and
identity. There is nothing whatsoever of
validity in the priestly hypothesis that
God created some human beings more than others;
or that some were messed up with animal blood
more than others. That is blah-blah and a
dangerous, dictatorship-oriented policy; and
so Charles Darwin the politician was a genius;
but Darwin the scientist was a bullshitter,
apart from getting some facts of evolution in
the past up and to the forefront of human
thinking.
What drives this evolution is totally
other than random changes: this will be
obvious if you think of just how few times
there are evolutions by means of mutations
in going from a parent to offspring over
some billions of years, relative to the
immensity of the fantastic structures found
everywhere in Nature.
So there are infinitely many possible
viewpoints between the shoddy little
narrow-minded petty atheist view that
throwing stupid dice together to make
up some genetical mixture, added to
billions of years, will produce all
we see of living beings, and the other
idiotic view, that God sat back for
six days and created the lighter-skinned
First Couple and the animals and nature
around them and then let it roll without
any evolution.
For instance, Goethe points out that
there seems to be holistic formative
principles in nature shaping the
forms of leaves. There can be resonances
and connections and gestalt principles
organizing at least part of the
fluctuations of the mutations. There
can be all sorts of subtle effects of
something like God which works together
with billions of years of evolution.
There can also be -- as I have pointed
out elsewhere -- that past up to a
point was a simulated process, vast
in complex detail, but not real as the
present unfolding moment (and its future)
is real.
Darwin doesn't prove anything whatsoever
of impossibilities of a general kind. He
merely comes forth with indications that
time does play a role; that things do
change, or at least have done so in the
past; that there are forms of connectedness
between human beings and other species found
in nature, at least of some kinds; and the
scientific sceptical point of view is to say
of this that this is merely an isolated
island of propositions which cannot be
used to affirm nor disavow the presence of
godhood, synchronicities, or other
metaphysical views on time. In other words,
the role of Darwin in thinking about evolution
is extremely meagre except in a political
sense. Politically, he did some sound work.
But it wasn't very scientific. It cannot
be scientific to throw in a word like
chance or coincidence or random when
the term is not understood, not defined,
not thought about, not reasoned about.
And then again, after the empirical findings
connected to work in so-called "quantum
theory" in the 20th century with its
so-called "nonlocality", it is simply
rediculous to say of the trivial, scarce,
unenlightening propositions of Charles
Darwin that they contribute anything
significant to our understandings of
the fluctuations of reality.
Darwin is hereby minimized in evolution
thinking. He is negated; he is out. We
cannot anymore mingle politics with
science, or think hot-headed arguments
against belief in greatness can have any
importance when they so totally lack
intellectual credibility. Imagine that
the millions of processes of the human
body -- all intertwined, all extremely
finely tuned, is a product of a mere
billions of years or so of -- what? Of
nothing except unorganized, uncontrolled,
chancelike behaviour of inert, lifeless,
dumb matter particles. That is an
absolutely superstitious belief in the
power of a tiny bit of time. For a
billion years is a tiny bit of time;
you would not get an article like this
chapter out of a billion years of apes
typing on typewriters. They would get
wrong, and worse; they would not have
time to find out where they have got it
wrong. Some experimentation in evolution,
yes; but that most of it is absolutely
mindless experimentation -- or chancelike
events -- is so completely untrustworthy
that only the most low-brained, unevolved
of human minds can sustain such a view.
It is completely and utterly unscientific
to hold of Darwin's hypotheses that they
say anything significant whatsoever as to
the origin and evolution of life or of
human beings. Science lies elsewhere: in
lucidity, not in the muddy thinking of
someone wrapped up in political aggression
against an aggressive church. We approve
of his successful re-callibration of the
dark-skinned into humanity; but as for his
science, the only thing Darwin contributed
with, was a willingness to go and look for
oneself, rather than reading the same petty
book over and over again like a fairy tale
one has fallen in love with.
End, therefore, the idea of Darwin as a
scientist. He was not: except in a very, very
fragmentary sense, and the definition of being
a scientist is that one is not merely so as
a fragment of oneself. One must be a scientific
in terms of one's integrity -- which is a word
signifying "untouched wholeness", or incorruptible
wholeness. If one is a scientist but where it fits
one's political agenda, and the rest of mind is
full of twarted, twisted, wishful thinking and
the closure born of prejudice and lack of
experience, then one is not a scientist at all:
and this is the necessary verdict any thinking,
rational person must pass over Charles Darwin.
Evolution, then, is something we can think
about now with Darwin in a clearly minimized
role. He triggered some points of view after
centuries of priestly narrow-mindedness; but
to go on repeating the views of Darwin is
but another type of opposite narrow-mindedness.
To equate science of biology, or the science
of life, with "darwinism", is something only
the most dishonest and disreputable scientists
can do: those who are but scientists in name.
We are scientists, you the reader and me, in
reality, not just in name, and so we do not
confound real thinking with the poppygock
associated with name-cultivation, person-cultivation,
and we do not at all cultivate nor even cherish
Darwin, nor respect the whole set of his propositions.
Eclectically, we pick those few and in fact rather
trivial pieces of what Darwin came with which makes
rational good sense and matches experience and
intuition, and leave the rest to be burned away
by the flame of sincere, scientific attention.
It is with this attention we uncover layer
after layer of thought-processes which surround
the notion of the infinite. This we will not
necessarily express very clearly before we have
done some more work. We must do it on an emotion
level also, not just by cut'n'dried formulations;
because how we view each other and the universe
and so forth is also a question of feeling; and
this feeling must be attuned to where we go
intellectually for us to have a whole meditative
approach -- and a scientific enquiry into the
greater forms of metaphysics will not be truly
scientific unless it also has straightened out
the feelings to cohere with a meditative flow
of rational thoughts, rational questions,
rational arguments. Here, of course, we use the
word "rational", with its root "ratio", to mean
"holistic proportions and relationship in flowing
movement", rather than the up-in-the-head and
insincere form of "rationalism" found in the
political opposition movements against the
priest powers in the 18th and 19th centuries.
The so-called (by some called) "enlightenment"
connected to that shallow-minded "rationalism"
does not do justice to the full powers of the
human mind to enquire unselfishly into the true
nature of existence. We need not bother with such
superficial definitions; and we rather go, as good
neo-popperians go, into the matters both with eyes
open and with the intuitive heart awakened.
[[[The notion of "neo-popperianism" is fleshed out
rather well by me in other writings also at my
websites http://www.yoga4d.com and yoga6d.com,
and can be said to characterise the general approach
taken in this book in a nondogmatic sense; this is
a word defined by me as a crucially important
extension of the rather narrow-minded but still
valuable, up to a point, approached which perhaps is
called "popperianism". However the reader of the
present book need not at first go into this
concept but merely note that it is, at a general
level, quite a good description of what we are
doing in this book.]]]
****** GETTING BOHR AWAY FROM QUANTUM PHYSICS
When we decide to investigate the world in its fullest
parameters, so to say, then we cannot ignore looking
also at the smallest details. And, lo and behold!, in
finding patterns there that reflects greater wholeness --
also across time (as we do in quantum phenomena), we may
begin to analyze.
When someone (like Niels Bohr) then comes along (having
had luck early on with giving some unexplained rules for
the behaviour of some features of some initial uncharted
area of quantum phenomena) and says that there is no
point in further analysis, then this someone is not a
scientist; hence has nothing to do with quantum physics;
and that ends it.
There is a pattern throughout the bloody, barbarious
history of thinking in humanity across the millenia that
when someone gets into power, this someone will often
misuse that power to sustain that power. Niels Bohr got
into power: because of the luck he had with the initial
formulations over the atom and the like. His images were
wrong; but some formulations were good. Later on, some
images were proposed by others (such as Broglie), which
turned out to be not quite right; and Bohr, having a
sense of immense power, as I see it, decided to declare
that it is part of the mature state of mind of the
evolved advanced quantum scientist to avoid further
imaginary depictions of the phenomena.
Thus, it follows logically that he would dismiss all
attempts at giving alternative renderings of the
fundamental energy interactions found to apply at the
atomic, photonic and electronic level. This he also
did. One sees that people in power likes to use
sarcasms when they dismiss upcoming people who puts
problematic questions to them. Bohr dismissed David
Bohm's proposals, however cluttered they initially
were, with a statement that it sounds like somebody
saying that it may be that, under certain circumstances,
and given certain assumptions, two plus two equals
five. He therefore cancelled himself out of science.
Bohm's proposals still stand unchallenged as one
pathway of interpretation and mathematical re-rendering
of the very same type of equations that Bohr had
himself had the opportunity of being part of launching
together with a group of physicists. This group included
however Louis de Broglie who didn't like Bohr's
interpretation and who picked up David Bohm's work
and used it to further his own alternative thinking --
in effect, therefore, breaking with Bohr's group,
although de Broglie stands forever as the contributor
of one of its most important equations, connecting
matter movement with a new form of matter waves.
Whatever the quantum phenomena are, they are not
understood by any of the mainstream physicists in
20th century science. Niels Bohr, heralded as the
most important quantum physicist, did enough blunders
that we have to dismiss him as utterly immature and
unfit for science. His power was misused; he didn't
merely commit errors in science, but actively tried
to prevent very fruitful developments and consciously
radiated sarcasms and bored indifference to genuine
alternatives. He proved that even a person who speaks
with infatuation about complementarity can become
totalitarian and dogmatic like a priest. He saw to
it that physics declined, for he didn't give the
dialogues that Einstein wanted an opportunity to
evolve when they could. By pure stubbornness,
Niels Bohr brought the physics of the 20th century
to a standstill. That is a remarkable achievement,
of course, but it is not the achievement of a
scientist: rather, it is the horrible achievement
of a power-man who should never had the power he
had.
We have thus having cleared quantum physics from
the debris of the megalomaniac writings, uttered in
the humblest and softest of tones of the modest-
looking individual Niels Bohr (for he appeared
to be so humble, just like some of the most
fanatical and criminal of priests can appear so
humble, when they give talks).
****** GETTING BLACK HOLES AND BIG BANG AWAY
We note that throughout the 20th century the views on
cosmos in mainstream fashion of physics have had big
controversies and changed a lot. At the present moment
of writing, though, there is a certain, let's say,
influenza, left in physics -- it is called "black
holes"; and, a related cold or headache is called
"big bang". We will prove the incoherence of these
ideas here, shortly; which is a resume of the exam
thesis I gave to the University of Oslo at the
rediculously low level of a major ("master's") thesis,
and which, thank God, was rejected: for one does not
easily accept greatness at a low level, especially
not when the institute has a reputation of petty
loyality to bygone proofs, replicated with the
lack of insight which typically is found among the
disciples of false prophets.
I state these strong words easily now, in 2009,
for in 2003, 2004 (as published, and available on
public library with ISBN number), 2007 (the uncompleted
book manuscript but released on the Internet since
then), and in the 2007 Lisa GJ2 manual, and the
comment about these in 2008 and 2009 on the Internet,
I have purposedly spoken about it all in mild,
dialogical words, with the type of lucidity and
slowness required to meet even the hardest of
sceptics. Here, I will not mince words.
First, let us restate the fact that in mainstream
physics, there are applications of Einstein's theory
of gravitation in his general theory of relativity
(as of early 20th century) together with applications
of the group-work called quantum theory (as of early
and middle 20th century) to common phenomena, such
as to the structures that seem to arise after the
collapse of very great suns under their own gravity.
This does not justify the term 'quantum gravity',
although this term is sloppily used by some
journalists and some physicists. Also, there are
equations involving unclear, incoherent ideas
which to some extent bridge some of the equations
in general relativity theory with some of the
equations in quantum theory, existing within
theoretical physics pursuits such as M-theory
and superstring theory and similar such, which
also speak of 'a grand unified theory of gravitation
and quantum phenomena', or of 'quantum gravity',
although these terms are not justified at all,
unless they by the word "theory" mean nothing
at all beyond "a bunch of loosely connected
incoherently-thought-about equations".
So this is the state of physics as I see it,
and I see it in an unbiased way, I claim: I have
nothing to gain from stating that it is grand
when it is not and nothing to gain from stating
that it is mediocre, as it is, if it is grand.
In order to see why black holes and big bang
theories are not coherent we must understand
some of their origin, and in particular we must
understand the significance of the term --
or, as I see it, the lack of significance of
the term -- "singularity", which is the
'sine qua non' foundation of both the black hole
theory and the big bang theory.
Singularity is a concept introduced by some
mathematicians to apply to a certain twist
which is again applied to the general relativity
theory of Einstein, which is -- very broadly
speaking -- a source of the concepts both of
the gravitation theory (very directly), and
(more indirectly, and occasionally so that
some features, but never all, of these
concepts are directly negated), in quantum
theory. All clear so far?
Let us be so kind to the concept of singularity
for a little while that we explain it by an image.
The image is that of a piece of paper, or anything
that you can bend enough to introduce a sharp
cut or two in: let these sharp cuts form a cross,
with an intersection point. Let us further assume
that this point sticks sharply out whereas the
lines introduced on this paper by bending it
are, except at this point somehow, rather mild.
The sheet of paper is the image of continuity,
let's say. The singularity, then, is that point.
A breaking in the continuity. Size of it: nil.
In other words, nada. In yet other words, zero.
In still other words, infinitesimal.
This notion, of the infinitely small, was
in the 20th century by mainstream fashion
scientists, including physicists and mathematicians,
regarded as just about as trivial as anything else
can be, as far as their much-used concepts go.
After all, even Euclid spoke of the infinitely
small area covered by the point, and he also
switched (sloppily, as I see it), between the
notions of finite lines and "infinitely long
lines".
Let us also note that especially since the
work of Georg Cantor rather late in the 19th
century, that is, before the work of young
Einstein on his relativity theories (which
were published in the first two decades of
the 20th century), using numbers and equations
to go "gradually towards" the infinite and
also towards the infinitesimal has been
regarded, by certan applications of set
theory and the so-called "limit concept"
(another instance of a flu or cold in
mathematics, as I see it), been seen by
mainstream fashion in science as rather
trivial.
A singularity, then, is proposed to be
arising out of certain consequences of what
Einstein proposes as to the nature of
gravitation and its parallel size to
mass when this mass collapses with a
certain accellerated speed onto itself
after the hydrogen of a very, very, very
large star (immensely greater than the
tiny Sun around which Earth rotates) has
been used up, by and large, as it fuel
for its fusion power generation processes.
Among others, Roger Penrose, a mathematician
at Oxford University, heralded the point of
view that the resulting lump of matter, if
the nova (as it is also called, the exploding
star that is), is sufficiently large, will
have to reach the size of the infinitely
small, viz., the infinitesimal.
Einstein, I have read somewhere, did not
at all agree to the possibility of
singularities. We must remember that Einstein
wanted a formalism to be as a frame applied
to the painting in imagination which is the
theory proper; he wanted the informal to
be prior, and the informal aspect to come
later; and if the informal aspect threatens
the clarity of the formal he would be
inclined to deny those implications.
That is, I think, entirely the right
attitude (as far as it goes).
A so-called "black hole", then, would be
a point-sized, or rather, nil-sized, lump
of concentrated matter, sucking up all,
radiating nothing -- or hardly anything,
if the student of Penrose, the physicist
Stephen Hawking, got it right. Hawking
pointed out that the Heisenberg Uncertainty
Principle, or HUP, denies the possibility
of simultaneous full knowledge of position
and momentum -- or movement information --
of anything. Since the position of a black
hole is given, then there must be some
openness as to the movement associated
with that area; this, he inferred, must
imply that there are fluctuations, or, in
other words, some radiation; he quipped
that this is a 'black hole with hair'. The
young Hawking made a bet with a fellow
physicist that black holes do not exist;
if Hawking lost the bet (that is, if
indications were found that black holes
do exist), he would get a subscription
of -- Penthouse I think it was (otherwise
Hawking would have to pay for a set of
Champagne). There has been interviews on
TV with Hawking much later than that where
Hawking speak not with that admirable sense
of doubt anymore. Black holes are now black
holes, not "theories of black holes". It is
not anymore "quantum theory as applied to
this general relativity phenomena" or words
to some such effect, it is rather "quantum
gravity". Such certainty tends to grow on
certain stubborn-minded physicists late
in their age.
Let us, before we show the lack of meaning
and lack of coherence and lack of consistence
of the reasoning associated with the concept
of singularity, then have a very brief look
on the big bang theory. The big bang theory
is, put very crudely, a theory of a reversal
of the production of a black hole. Instead of
a gigantic amount of matter being compressed
into a singularity, it is gigant amount of
matter being expanded from a singularity.
As for empirical findings, there is a great
deal about the universe -- in fact, most of
it, which does not fit into any single
coherent theory at all. Rather, there are
findings here -- some of which support the
notion of something like black holes existing,
but not necessarily 'black hole singularities'
(but possibly so), and findings there -- some
of which support the notion of a universe
which is expanding and which to some extent
might have been more compressed, as a vague
possibility, before. This looseness and
vagueness and wild diversity of findings
notwithstanding, there is not a lack of
eager and willing physicists who, just like
economists in such phases as they have
forgotten what it is for econonomy to
have a collapse, will eagerly speak of
certain computer maps as 'showing all
the galaxies in the universe' and 'showing
with some measure of certainty the exact
number of billions of years the universe
has existed' and even 'how it will look
in some billions of years'. They will get
a lot of public TV and radio support, and
those who are sceptical about it will be
dubbed 'fanatics'.
Exactly the same types of situations
were found, for instance, when some people
early in the 20th century sought to say
that maybe the Milky Way is not the only
galaxy of size in the universe.
Let us then dismantle the idea of the
singularity. We will do this by noting
that the concept is nil, nada and
nothing unless it has both the image --
the continuity -- and the mathematics --
the numbers as applied to the infinitesimal
-- associated with it. If we look at the
latter, the numbers as applied to the
infinitesimal, we will see that we have
here various sorts of equations which
all, in some way or another, refers back
to the existence of the set N of all
and only positive finite whole numbers.
This proof I delievered in formal
thesis form to the University of Oslo,
they have it there, at the Institute
of Language, Logic and Psychology, cfr
prof Herman Ruge Jervell, in June 2003,
it is reproduced verbatim within the
commented text yoga4d.com/mywebook.txt,
and it is compressed here, as I also
compress it similarly in other writings
you find high up on http://www.yoga4d.com.
The proposition P1 is that there exists
such a set of N of all and only positive
finite whole numbers, beginning with 1
(one), 2 (two) and 3 (three).
I will show that proposition P1 leads
to, by pure logic, by reductio ad absurdum,
the proposition Not-P1, ie, that there is
no such set. This then will be shown in
such a way as to make it clear that what
can be found to work for sure in the
definitely finite realm of numbers
cannot be considered to work when the
limits of these numbers are left more
open -- whether it means going higher
and higher, or whether it means going
towards 1/n where n goes higher and
higher and this fraction then goes more
and more towards the infinitesimal --
which is a necessary and very essential
part of the mathematics used in all forms
of discussions about singularities and
hence such items (from now on, non-items)
as "theories of black holes" and
"theories of big bang". Indeed, any theory
involving any kind of continuity or basing
itself on any kind of derivative /
differential or integral, implying continuity
of this kind, will no longer be seen as
having any formal aspect, as must be
considered purely a work of imagination
unfounded by any formal structure (this,
then, as I point out very clearly in
the above-mentioned texts, concerns then
all of quantum theory and general relativity
theory and is the justifiable grounds for
saying that the Lisa GJ2 formalism, my
own formalism and computer language, is,
since it is the only formalism weaved
in full awareness of the following
reductio ad absurdum proof, and since it
is also fully capable of transmutating
any finite set of psychologically
meaningful 32-bit finite numbers to any
other definite set -- the only formalism
worth its berth and, when equipped with
a proper discussion along the lines that
we do equip it would, constitutes part
of a novel theory over the world, which
I also call a 'theory of supertexts' or
'theory of active models' or (in the 2004
published work) 'theory of supermodels').
PROOF (or argument, or reasoning):
The proposition P1: there exists
such a set of N of all and only positive
finite whole numbers, beginning with 1
(one), 2 (two) and 3 (three).
We visualize the first three members of
this set by writing the numbers as vertical
lines I and give suitable systematic and
consistent and simple spacing to the characters
so that the triangle as here indicated are
understood to be perfectly symmetrical in
their shape -- meaning that the left vertical
height and the top horizontal length are
identical.
I I I
I I
I
This, from bottom and up, is 1, then 2, and,
as the highest, topmost line, the number 3
represented as I I I.
We add one member 4 and of course get
I I I I
I I I
I I
I
And we add still one member, now at 5, and get
I I I I I
I I I I
I I I
I I
I
This is then a square-angled triangle in which,
given the proper systematic simple consistent
spacing, the left vertical line is identical
in size with the top horisontal line. We further
note that the left vertical line, when counted
in terms of lines, indicates the number of
members so far added to the set (five), while
the topmost line indicates what this number is
(also five).
This, then, is an absolute perfect symmetry.
In going to the number 6, we see that we have
it still going:
I I I I I I
I I I I I
I I I I
I I I
I I
I
It is now absolutely clear that there is never
any breaking with this absolute symmetry of the
left-hand vertical length with the top-most
horisontal length.
We are now in a position to evaluate the
proposition P1 clearly. It speaks of ALL finite
numbers. By the conventional assumption
penetrating all of mathematical and physics
thinking at this point, we say that given any
finite number proposed to be the maximum
limit, we can produce the next by the arithmetic
operation of addition, and adding 1. Since the
left horisontal line echoes the size of the
set, we say that, as concerns the set talked
about in part of proposition P1, this set is
not finite, then; and so that line is not
finite, as concerns the set talked of in
proposition P1. But given the absolute and
perfect symmetry with the top line, this means
that the very same set also has members which
are not finite. But this breaks with proposition
P1, which says of the set that it must ONLY have
members which are finite. Hence, Not-P1. QED.
****** RE-INTRODUCING INFINITY AND CONTINUITY WITHOUT
NUMBERS
So, in June 2003, I think it was (check with the University
of Oslo, in their records of delivered theses at the
institute which has the department for Logic, Language
and Linguistics, or with prof Herman Ruge Jervell), and
re-iterated in the self-published book, in 2004, called
'Passion without greed or hatred; resonating over
dancers; creating new physics', given to the National
Library of Norway (cfr http://www.nb.no, with ISBN
82.996977.0.0), and given, at times, at my websites
in full text form -- as also now -- and, continuously
since March 2006 with the Firth platform (updated 2007
to Firth Lisa, with extension for Firth Lisa GJ2 later
in 2007, cfr http://www.yoga6d.com/city for all free
downloads), I showed what I showed above -- with more
words -- and concluded (in the 2004 book, with the
thesis included verbatim in the MYWEBOOK.TXT link found
on top of the page http://www.yoga4d.com, and with much
the words as used above in the uncompleted-as-book
but completed-as-its-own-kind-of-text of the 2007
text as also listed on top of the yoga4d.com page) --
that 'mathematics is a bag of tricks', and
as a completely natural consequence of this, physics
lacks a formalism. Now the first statement, that
mathematics is a bag of tricks doesn't mean that
these tricks aren't (sometimes, for definite
purposes like calculating the strength of tiles
for building a house) useful. A bag can also be
useful, it is not necessarily derogative in its
sense. But it means that the assumed 'diamond-clear'
structure of the whole thing doesn't exist: it was
a facade of half-thought concepts, and Euclid,
regarded, along with Pythagoras, as the father of
the axiomatic strong type of formal mathematics,
can be seen to have begun the confusion since he
introduced in axiomatic writing the sloppiness of
not realizing the mile-long gap between talking
about finite structures and talking about infinite
structures -- the latter of which we know nothing,
and we certainly don't know that the tricks we
find working on the finite structures work when
carried over there.
The proof, or argument just indicated, shows
that any unlimited gathering of limited numbers
of a very basic kind in mathematics get a
different types of members than those which are
at first intended, members which may not subscribe
at all to the rules and laws of e.g. arithmetic
as applied to definitely finite structures, and
we cannot block out these members from coming in
when we seek to have an infinite aggregate. We
cannot define them out for the proof will still
apply to the infinite resulting set and show that
this set has members we don't know how to handle.
And if we do try to handle these members, we will
get a set of rules which can be re-represented as
a form of arithemetic which only applies to finite
members, and then the argument applies once more,
on "top" of them, giving again the undefinability
of the result. So any attempt to make an infinite
collection of finite items creates undefinable
results. These undefinable results may mask
themselves as new types of "infinities" which
crop up again and again in the most "unwanted"
places -- and it is therefore not at all surprising
that the half-cooked attempt called "quantum theory"
is riddled with them (cfr e.g. Richard Feynmann's
self-criticial statements on his own masterful
contributions to the mathematics of quantum theory;
for instance, he says that as long as we regard
something approximately there, applying the type
of 'normalization' rules which make no good sense
at all, then we get a certain type of numerical
result which corresponds with laboratory research;
but when the result is pressed for precision,
the numerical prediction from theory diverts not
a little bit, but diverts exponentially much and
finally 'infinitely much' from the approximate
prediction -- resulting in a general feeling that
quantum theory is messy in its math; see e.g.
his lectures in Q.E.D. -- Quantum Electrodynamics).
Don't mess around with infinity -- that seems
to be the morale of all this. But it is a subtle
thing to say; for (as I also point out and spend
indeed a great time on pointing out carefully
and with much self-sceptical statements in the
above-mentioned thesis, that the University of
Oslo rejected, and the grounds for this
rejection I discuss in the MYWEBOOK.TXT and
find to be reasonable, given the low level of
expectation they have at that level of exams,
combined with the sense of loyality to the
bygone systems) the infinity may come in when
one merely says 'etc' or gives the notorious
three dots "..." after a statement which does
not explicitly bring in the infinite.
(By the way, when you read the rendering of
the proof in the 2007 text as listed on top
of yoga4d.com and yoga6d.com, there is a
slight typing-in error in one of its final
sentences -- the "in" should be crossed out
in front of one of the final words -- can
you find which?)
The finite -- such as finite whole numbers
-- we relate to as strictly finite only when
we are able to actually confine mentally and
along a psychological natural path to clearly
understood boundaries, such as both an upper
and a lower level. If you read on my works in
computing, you will often find the phrase
'32-bit numbers'. This is a range of numbers
going about from minus two billion to about
two billion -- or, more precisely, from
2 raised to the power of 31, plus or minus
one, below zero, to 2 raised to the power of
31, plus or minus one, above zero; or
-2,147,483,648 to 2,147,483,648 plus minus one
on each side. For reasons given technological
clarity a computer is organized into handling
information as numbers and numbers as strings
of zeroes and ones and these again in bundles
which are numbered so that they themselves
are a power of 2. That means that the earliest
computers were typically 8-bit, and as they
matured a little bit and we had the language
Forth by C. Moore in the 1960s used to steer
telescopes and satellites, we progressed
towards 16-bit; but working with sixteen
bit numbers means being within the range of
plus minus 32,768 plus minus one, and it just
so happens that it is very tough to get any
complex piece of programming done that way.
All these limitations vanish in thin air as
we multiply the numbers of 'bits' (the ones
and zeroes) by two and get up to the modern-day
32-bit, which saw its first moments of glory
in home-computing around year 2000 for real.
But those who do not recognize that we have,
with the 32-bit computer, a psychologically
meaningful boundary, think that we must go
on and on; they are wrapped up in incoherent
thinking about 'etcetera', and so assume that
there is a 'next step' and the that next step,
the misstep as I would call it, is 64 bit, then
128 bit and so on. But 64 bit is a computer
nobody in person can program directly unless
a vanishingly small part of the computer is
used. The number range is not twice that of
32-bit, but it is rather the highest number
of the 32-bit kind MULTIPLIED BY ITSELF: as
if we were trying to program some two or four
billion computers each with the limit
minus two billion to plus two billion. And
the numbers don't make psychological sense
unless they are bundled hierarchically and
dealt with in a second-hand fashion, in which
the programmer no longer is able to see what
is done. The mind naturally perceives things
around an order of not significantly more than
ten or eleven items. You see that a number like
one billion is stretching the amount of digits
you want to look at at once: 1,000,000,000;
once we get into something like
1,000,000,000,000,000 then even those extra
commas have to be counted.
In economy, there is also a psychological
problem once the billion becomes the trillion
(in Norwegian, that is the transition from
what is there called a 'milliard' to a
'billion' whereas 'trillion' is corresponding
to English 'quadrillion').
And this problem can be seen to be a
transition from what I call 'first-hand
economics', in which the people operating
the big finances directly relate in a mindful
way to the actual numbers involved, to a
'second-hand economics', in which they have
to resort to statistical-looking programs
and in which the coherence of the whole
process breaks down due to the arbitrariness
of the machinery involved, and the mindlessness
of the whole (echoed again by the cocaine usage
so typical of the Wall Street billionaires of
2008, when the world economy saw its first
gigantic crisis since the stock trade collapse
of the 1930s).
I am willing to say that all of human culture
in the past has had severe problems connected
to the notion that since we can handle a
certain well-bounded structure with ease,
we can also safely say 'et cetera' and trust
that our understandings as applied to the
well-bounded area will also apply there. But
sailing a boat on the wide open ocean is
widely different from sailing it on a lake;
dealing with numbers from minus two billion
to plus two billion is widely different from
dealing with 'any large number'; and saying
that human thought is able in meeting with
the concrete surroundings, sensually, around
the human body is widely different than
saying that human thought can understand
all that is.
Indeed, the human mind as such may be
considered a kind of self-resonant living
fluid structure which cannot with ease
handle a situation in which there is any
type of endlessness of an invasive kind
to its structures. If there is going to
be an infinity, it had better be, as far
as the human mind is concerned, an infinity
which is caring and fair. One of the
complexities with the human mind in
encountering a number of experiences --
such as, a beach house on a night with
a winter storm making the ocean dark
and dangerous and full of tidal dark
powers, or a Bach masterpiece in which
the main oceanic feeling is one of the
sadness of the martyrdom of Jesus, or
the writings of the ruthless avenging
Allah-God of the Quoran related to those
who do not submit to the submissiveness
that Islam (meaning, "submission")
literally is, to which is added, as
morning prayer, 'Allah Akbar', which
means, 'God is greater' (than what? --
than anything one can say, therefore
it is not said; and as such, it is
like saying that this Allah-God, so
dark, so ruthlessly aggressive against
nonbelievers that they should be
killed, is an instance of an 'etcetera') --
when the human mind plays along with
dark and sad forms of 'et cetera',
it decays. And weak human minds eventually
instigate such dark political constructs
and designs as 'sharia' -- which is, put
simply, group tyranny, illusionary forms
of justice which is but enslavement of
women, leading to such action as to put
acid in the faces of those who wish to
create, or go to, girl schools -- as
found in Taliban-ridden areas of Afganistan
and Pakistan at the moment of writing.
So the weakened human mind becomes
atheistic, even as it possibly screams
about God, for the infinity it discusses
is a dark one, leaving the intellect
feeble: and it becomes insensitive, and
during political catastrophies, aggression,
the beastly aspect of the human being,
coming forth whether in Nazism or Islam
or Stalinism or Marxism or other forms of
sects, such as Jehovas' Witnesses, the
Mormons, or Scientology, takes the upper
hand.
The acid test, therefore, as I have
proposed it before in texts connected to
my programming of an Amharic editor for
Ethopian computing in the Firth Lisa
GJ2 platform (all finished and available
at http://www.yoga6d.com, cfr the
http://www.yoga6d.com/GJ2NEW1.TXT to
http://www.yoga6d.com/GJ2NEW11.TXT
programs and associated documents),
is not how an item of human culture
works on educated minds in a luxurious
apartment in an affluent city, but rather
how the impoverished human being responds
to that cultural item (e.g. a book). If
something like group aggressive tyranny
then emerges, that means the item is
too weak in its understanding of just
how to say the best of what it says
(for, as I have heard Goethe pointed
out, it is sometimes not as much what
one says, as how one says it). Nevertheless,
I point out that Anselm, a forerunner of
much important thinking in continental
europe and thinking on the British isles
in the millenia before the third, were
able to produce a kind of understanding
of God as the always-greater-than-whatever-
we-think which to some extent cohered with
his christian church and its bible, and
which is, as such, coherent with the
islamic postulate of 'God is greater'.
So, put in other words, there is nothing
wrong with infinity as such! It is merely
that we must watch what the human thought
is up to: is it doing hubris -- subtly or
not -- in trying to wreck its own coherence
and its possibility of meditating on the
greatness of infinity by artificially and
wrongly bridging the finite and the
infinite, as it talks about the infinite,
or is it humble-about-the-finite-as-finite
and awed-about-the-infinite-as-infinite?
I wish to say that the notion of the
continous field is consistent with the
notion of the infinite. This means that
any talking of the continious type of
thing, or which presupposes them, must
have the same caution as any talking of
the infinite.
Further, I want to say that it is my
strong and clear and enduring intuition
that the essence of the world is continuous
and infinite. What this means I think we can
consider open to exploration; but let me
at once say that I think the bibles of
the various world religions, and especially
the Quoran and those early chapters of
Bhagavad-Gita which makes too many wars
seem 'holy', do not adequately treat the
infinite as something glorious and
compassionate and warm and just but rather
lashes into it with human political
propaganda (and an added problem about
the Quoran, different in intensity and
focus and concrete statements -- such as
head-chopping approved of -- from all the
other world religions, is that it lashes out
against a still existing group of people,
namely, Jews, and as such incorporates a
certain form of group-hatred into what
may be called the essence of their faith).
I also spend a lot of time in the exam
theses suggesting that the essence even
of the number concept at a subtle level
may be the infinite proper. The finite
emerges, to some extent, rather as
temporary 'relations within the infinite',
more or less as we can speak of crystals
emerging from a solution of minerals in
water, or ice in cooled water.
****** THE ROLE OF THE UNCOMPUTER
In the earlier-mentioned MYWEBOOK I introduce the
concept of the UNCOMPUTER, which I coined in
some analogy to the notion of the unconscious,
to reflect the source of the insights which
involve cause-and-effect and computing machinery
and programs and indeed everything patternized
as itself beyond all patterns, flowing, continous.
Before I come to this concept anew, here, I
will give some thoughts on the role of
compassion in politics relative to the questions
of the importance of boundaries that we have
already touched on.
While the ancient Greeks did have a form of
'people ruling' in Athens and in other polis
areas -- city states -- for some of the people
they did not have it for all people; for they
had slaves; and at the same time some of the
most influential philosophers, including in
particular Plato and Aristotle, were against
'people ruling' -- 'democracy'. Plato spoke
of democracy as giving free reigns to laziness
and idleness and mere pursuit of superficial
pleasures and thus found much more value in
imagining that the wise emperor or dictator,
the benevolent philospher with all-powers,
come to political stardom. Something of this
was put to the test with Plato's pupil
Aristotle, who grew to challenge some of
Plato's propositions and represent in several
ways an alternative to Plato, and who got
Alexander, later Alexander the Great, socalled,
as his pupil again; and, if I'm not mistaken,
also ruled with Alexander for a while. (I am
not undertaking to evaluate the ethical
success of Alexander's regime and I have
explored the matter little.)
We will see how the notion of the uncomputer
(which can be written lowercase or all letters
in uppercase on occasion for a certain emphasis)
can be seen to have a political role or
relevance for political thinking; but to come
to this I will give some suggestions on how
I see the question of going beyond 'the ego'
[[[in other words, the question of the type
of enlightenment which I discuss at great
length also in such documents as are found
within the MYWEBOOK, which is not merely the
ideas and ideals of the very misnamed
'Enlightenment' period in European history
sometime prior to the 20th century (a period
characterized by resistance against church
oppression and not characterized by a fullness,
in both feeling and thought, of transcending
the impulse of selfishness as a whole)]]] --
relative to politics.
I often find it of superior value to compare
a well-functioning society to a healthy body.
Now a living human body who gives in to every
impulse, every whim, without the slightest
thought of security even when hazardous
adventures are undertaken, is likely to
cease to exist rather quickly. There is
something different about the wise human
being who restrains whimsical impulses
in all areas where there is a hazard in
being whimsical, and rather allow a fun
free fluctuating type of activity happen
within boundaries which are worked out
before-hand to be so as to allow certain
forms of free play. In the same way, a
society which is a democracy in which
absolutely every item in that society is
put up for a vote and the result of that
voting, whether one percent or a hundred
percent of the population partook in it,
is recklessly and immediately put into
action, is likely to be a society which
vanishes pretty quickly. It selfdestructs.
Rather, therefore, a democracy in practise
limits what is put to the test and that
limitation is so as to keep things which
are essential for the long-term survival
of the society out of whimiscal impulses.
Now, the fascist or fascistoid society
is a society, such as Mussolini's Italy
in early 20th century, or Hitler's Germany
in roughly same period, which is organized
around the notion of the fascination with
the hard, militant, stern, strong individual,
or possibly group of individuals (such as
a group of tough-looking military generals
in such cases as we speak of a 'military
junta'). If there is an election of
anything or anyone, to a certain role,
in such a society, it is usually nothing
but extra paint applied to a machinery
which is entirely uninterested in its
population except as to what it produces
of goods that the leaders can enjoy in
their palaces. The population is expected
to show 'discipline' and the fascist or
near-fascist, fascistoid, regime takes
pride in seeing people lined up as if
they were bits and parts of dumb machines,
being willing to give up their life and
their pleasures for the sake of the Party
or the Leader or Der Fuhrer. Typically, to
enforce the fascist regime, there is the
constant depiction of some kind of evil
nation 'out there'; and whenever there is
any problem, such as food shortage or
water pollution, it is caused by Them;
and the population is encouraged to work
even harder so that They are overcome.
This type of regime can be compared to
a person in one of the worst states of
ego-selfcenteredness, in which humor
has totally evaporated and all which is
left is a bundle of lies to support a
stale persona, without any flexibility
of insight except cunningness to keep
on going with the illusions for some
more time.
The illusion of the fascist or
fascistoid regime -- such as sharia, when
in a so-called religious context -- is that
of limitless power. A human being cannot
entertain a sense of having limitless or
near limitless power without getting potty,
put simply.
But a strict democratic regime, of the
kind where every issue, even those important
for long-terms survival of the society, are
put to the test of perhaps daily voting by
some or many and recklessly put into action
no matter how stupid it is, is yet another
form of limitless power: this time not
allotted to a party, but to the ballot box,
the voting idea, or to the idea of the people
as a kind of mass entity. Anything can happen
in such a context. It might be a free reign
to egotism, idleness, and selfishness of every
kind as Plato proposed: but it might also be
a free reign for a majority to bully a minority;
or for a minority who bothers to vote to bully
a majority that doesn't. Democracy, then, in
a totalistic sense is possibly just as bad
as a fascist or fascistoid regime.
Many of the regimes which are called
'democratic' in early 21st century on Earth
are merely quasi-democratic, by the way, and
in praxis rather fascistoid. For there are
rarely any vote at all -- perhaps at most
one pr year -- usually with the vote that
matters every fourth year or so -- and then
only the vote is not on any central issue
at all, but on which one of a couple, or maybe
of three persons, all with a rather similar
agenda, but each with their own family
dynasties and friend-mafioso-like networks,
should dominate and propose laws and decide
all sorts of matters for several years onwards
until the next rather meaningless vote.
Perhaps there is an additional form of vote,
in which there is some such more individuals
offering themselves to be voted on more
locally, to make up the 'res publica', the
republicans, those who 'represent the
public', whose job it is to shout and
scream at those who actually govern when
they do so 'against the will of the people',
but this may often have not much effect at
all.
The argument one can often hear in favour
of this extremely boiled-down version of
democracy is that, however bad it is, it
rarely gets as bad as the overly fascist
regimes are; and even if it gets terrible,
it usually passes away after four or eight
years or so, rather than something being
kept on for three decades or longer, as in
the case of Cuba.
But this latter form of argument is only
appropriate as seen from the point of view
of those who find that the way that society
is totally dominated by those who govern it
is fairly enlightened and wise. If the whole
way the society is governed by those who
typically get into power each time is, on
several key issues -- such as the freedom
for girls to get an education, or to fuck
however they want to -- totally wrong, then
this form of democracy is also, for them,
insofar as they are right, totally wrong.
It is not 'just a little bad'; it is a
wrongness that doesn't self-correct at all,
for that which they are interested in is
not up for voting, and doesn't change from
one leadership-up-for-voting to the next.
Rather, what happens in the case where
a society has a leadership vote once
every third or fourth or fifth year or
so, is that a so-called 'political
establishment' arises, producing the
candidates according to its patterns;
and this establishment has some sense of
total -- limitless -- even totalitarian
power over the society. It produces the
very, very limited, and un-discussed,
alternative candidates up for voting;
and it has its tentacles into the media,
generally speaking, and certainly into
those who practise the laws, and it governs
also the police and certainly everything
military and as such, it is in practise
fascistoid, if not fascist in a certain
very flower-covered way.
So my proposal is that the political
worlds on Earth are as cluttered and
confused as to the questions of what
should be limited and what the proper
place of the limitless, or the infinite
is, as the bag of tricks which was
called 'mathematics' has been. It is
all a mess. Surely, a society can only
be wise if that which is important for
its coherence and wholeness and health
both short-term and long-term is NOT
up for voting, but that EVERYTHING ELSE IS
up for voting. It must not be so that an
arbitrary bunch of things, some of which
are important for its long-term survival
-- such as food production, or state
security -- are put on the ballot box
and submitted to a completely whimsical
voting propaganda period without any
greater thought to it -- while things
of absolutely no importance for its
long-term survival -- such as the sexual
freedom for girls -- are limited
meaninglessly by a political establishment
which doesn't want to put several key things
of daily life importance to open discussion
and voting.
It runs through my argument here that I
am indeed in favor of voting in many cases:
I consider it essential in a responsive
society. But one must also, just as the
wise person, know that there are areas
which have to be thought about beforehand
so that the proper constraints are found,
and then implemented, and not doubted while
one is in the midst of carrying them safely
out. One doesn't do electricity whimsically,
but seriously well. So one must think about
it coherently, and stick to the rules one
can find to apply and not put them up for
arbitrary voting, so it gets safely done.
But a lot of things -- within certain
boundaries, within certain limits, which
gives a sense of finiteness to the result
within which it can freely and responsible
vary -- can fruitfully be voted over, whether
by many or few, and perhaps often, maybe
even weekly or so -- why not? This could
be easily arranged in a technological society.
But who, or how, are the issues connected
to long-term coherence and quality survival
of the society as a whole, going to be sorted
out, and sorted out well, so that also great
insights arise into all the questions and
that the answers thus found are kept to,
and faithfully held on to?
In other words, what can rightly dictate
the few foundational rules a society must
have, so that everything within the wisely
found boundaries and constraints can have
fun and interesting fluctuations as e.g.
driven by voting processes and similar such
processes, such as a market economy involving
fluctuating prices connected to demands for
goods?
In earlier texts, one also called the
Compassionate Anarchist, and in a discussion
of what I in MYWEBOOK call Interactivity
Economy, I suggest that a market economy
must be without mamuts and giants and this
rule must be implemented to ensure the
freedom of individual enterprise, otherwise
the originally free market will become
entirely dominated by monopolies which
then take over the role of a dominating
state, but perhaps even worse, for they
do not need to have the least sense of
values beyond greed and to stay within
the laws. I remember that when I initially
proposed this to some people who are much
in favour of free markets they at once
said, "Who is going to enforce such a rule?
How is it going to be enforced?" When I said
that such a rule should be enforced just as
any rule at all is enforced -- by police,
and so on, they shook their heads and said
it was unrealistic; but some years later
so many examples of companies who had
grown too big for just about everyone's
good had come forth to the newsmedia that
this response was no longer so common: and
more recently, with traces going long back
into 20th century history concerning so-called
'anti-trust' laws, very, very big companies
have had a great deal of trouble with
law-makers just because of their massive
monopolistic influence and have had to
comply with limitations or else face very
strong billion-dollar charges against
them. This is not far from a police-enforcement
of limitation on size of companies; but I am
talking of a society constructed as from
bottom-up, on more insightful premises,
where we do not need thousands and thousands
and yet more thousands of peculiar extra-rules
and extra-law-enforcers and extra panels of
judges to combat the mamut entities; rather,
we have seen what the problem is and dealt
with it in an essential way: namely, we have
seen that companies with many many hundreds
of people -- or worse -- become second-hand
in their dealings both with themselves and
others because of the need for hierarchy
to implant itself on the structure. It is
no longer organic. It no longer knows itself
so it must systematize itself; and in so doing,
it becomes a monster machine trampling people
in society down, instead of contributing.
It also becomes greedy for results for it
may have as a goal to keep itself going
steady in its over-big size, while it hasn't
much to offer people; and so it tends to,
goebbels-like, begin with propaganda involving
lies to people so that they shall buy their
meaningless products.
So I say, after meditation, not trying to
implement political power here, that obviously
it makes no sense to grant lawful existence
of companies above a certain number of people.
Where does this come from?
Answer: It comes from THE UNCOMPUTER. For
it doesn't come from within the computer.
Nobody told it to me. I looked at the
structures, the machines, the giants performing
in these rotten societies, and saw the errors
of these 'programs' or 'computers' and that
seeing of error calls on the insight of the
UNCOMPUTER; it calls on the flow of attention
which has, in the first place, created the
structures in thought which sustain these
societal structures; and we go back to the
flowing deep-conscious attentiveness and
in that peace we come up with a solution.
It is not a solution we try to implement
at once nor say 'how realistic is it'. We
merely note that the biggest companies
are the least charming ones, and the most
ethically complicated ones; and while small
groups can be nasty, they are nasty in way
which can be met by more easily frameable
ethically aware rules of a more conventional
kind, whereas the highly complex, almost
statistical-like type of subtle crimes of
the mega-mega-companies indicate that the
solution must be very essential: keep all
companies small, -- for, as Shumacher said,
small is beautiful.
So it is this I mean by a compassionate
anarchy, which I also sometimes call a
'distributed compassionate anarchy', for
the flow of compassion and the freedom to
do anything and also to propose anything
for voting and to vote over anything is
distributed into the areas where, within
finite boundaries, it make sense to have
such freedom, and not so as to overturn the
foundational structures of society.
I opened the chapter -- I almost forgot --
by mentioning the word 'compassion'. It is
an immense force. Compassion emanates from
the depth of being. It is the UNCOMPUTER in
action, dissolving and recreating, or feeding
a structure. Now compassion is complicated,
too. For just as it feeds a process, it can
have a direction; but what if the direction
is based on a lack of proper whole insight
into the implications of giving nutrition
in a certain direction? Suppose, as an
example, that a policeperson is hunting
down a very dangerous, wile individual or
group and has some secret clues and must
go carefully at work with these -- but
suddenly gets a bad, terrible headache.
In this headache, the policeperson gets
the idea that it is the burden of doing
this secret work that is causing it, and
instead of going on with the work, or
taking a sick leave and giving it, as the
rules probably are or should be, to a
proper colleage, this person instead goes
to some kind of mass-propaganda action
proclaiming that us policepeople are
getting too much headache and it is due
to too much secret dangerous work and
just hear this secret and blah blah blah --
and then we find that many many people
begins to shower this idiotic, self-centered,
self-pitying policeperson with compassion.
How terrible headache! How bad it must be
to go around with that secret! Meanwhile,
of course, the secret has lost its value
for it has been let out: the self-pitying
policeperson demanding compassion has
let a dangerous phenomenon in society
grow in its danger and cleverness. It
is now forewarned and can protect itself.
So compassion is an awesome force: but
compassion can be misdirected by the
pathways set up by self-pitying idiots:
and the word 'id-iot' literally means
'self-centered'.
But compassion, in its wise deep sense,
is also, when not steered by the ego,
exactly the action of THE UNCOMPUTER, that
almost raw sense of enlightenment-flow, as
a river of meditative refreshing insight,
which comes when all self-centeredness is
set aside. This is also sexual, beauty-
oriented, as girl-with-girl beauty-oriented,
too. Obviously no human being can be an
absolute vehicle for the UNCOMPUTER.
****** GETTING BERTRAND RUSSELL AWAY FROM LOGIC
So just how deep runs the wrong thinking about
infinities in what can be called the 20th century
style of European and Norther American philosophy?
Does it run very, very deep? In fact so deep that
every item influenced by its philosophy -- including,
for instance, the thinking about human lives, and
about money, and about nature, is wrought with the
same kind of errors -- which we have pointed the
way out of. Now, I'll admit, the way out of it has
been pointed out clearly here, but what you'll find
when you walk that way for a good while -- whether
it be a glorious castle there, a radiant beach,
a spaceship, God, what not, is not something I have
gone into here, in this text. Maybe I will, maybe
I won't.
I think a keyword is "all". Look, if you like,
at the eloquently and pleasantly easy yet ingenious
writings of one of the most famous of the so-called
"analytical philosophers" of the twentieth century,
Bertrand Russell. (Now I do not always include, in
my texts, direct quotations of people whom I refer
to -- even if the copyrights are so that it is
possible -- but I do act on a principle that if
I do not find a work in some way superceded later,
then I will bring at least an excerpt of it onwards;
yet I sense that the most radiant and good aspects
of earlier productions in humanity in some way
lives on as "new archetypes" in the collective
awareness of humanity -- "for what is beautiful
will never wither").
It is my sense of Russell's works that he is
striving hard, very hard, intensely hard, even
with greed, towards getting his hands on "all".
That is, on "all that is", or "all that can be
clearly thought", or "all that can be said
logically". The word is, clearly, one of his
favorites. Whether he, with his teacher, A.N.
Whitehead, goes into Principia Mathematica or
whether he delves into classical philosophical
texts or seeks to elaborate some new formulations
on the logical distinctions between statements
such as 'I believe' and 'I know', one finds that
he often uses the word "all". Not only when he
speaks explicitly of infinity, but obviously also
then. The only time he has in explicit book
production wavered from his fiercely atheistic
stance with a great belief in the mechanical
and systematized kind of logical thought as
the conquerer of the fuzziness of real life,
is in his booklet "Mysticism and Logic", which
he later publically regretted: He speaks there
of the oceanic feeling of infinity as the ground
for a sense of the universe as whole and good --
and later claims that he would never have written
this unless it was for a love affair with a
certain woman of class.
But whether he speaks of mystical feelings
connected to infinity, or keeps himself tightly
confined to speaking about the logic of such
things as -- the set of all sets, the set of
all finite numbers, the set of all real numbers,
or the set of all X satisfying postulate P(x),
or merely speak of 'all' in connection with
human knowledge in general, and in more vague
formulations, or use a synonym for the word
all, there is no clear indication whatsoever
in his works that he really sees what an awful
blunder it is to assume that the more or less
mechanical properties found to apply to some
finite aspects of 'what is' can be rigidly and
without complications be applied to something
as grandiose, if not even megalomaniac, concepts
as infinite sets of all so-and-so. The notion
of "all" is, in most contexts in which the
boundaries are not clearly defined, definitely
a notion of infinity. The patterns found to
apply to simple arithmetic, such as 1+1+2,
are assumed to apply indefinitely, as one
goes infinitely higher and keeps on working
with finite numbers. Through and through his
massive book production Bertrand Russell thus
commits the same, grave error as one of his
idols, Georg Cantor, who again commits, but
more explicitly and obviously, the same error
as one of Cantor's idols, the greek thinker
Euclid.
****** GETTING SPINOZA AWAY FROM DETERMINISM
The concept of a 'dimension' is a subtle one: have
you got it? I mean, have you got a fairly clear idea
as to what a dimension is all about? The idea was
around, more or less, at the time of Euclid some
several hundred years B.C., and implicit in his
axiomatic writings; and very explicitly and forcefully
so in the writings of the 'priest' Descartes (I call
him 'priest' for he was so concerned with fitting
whatever he thought -- and he thought pompeously,
like priests -- to the teachings of the church),
when he lines up an X and a Y and says, a point
can be located by giving an x value and an y value;
we have then the vertical and horisontal dimension.
All clear so far? So we say of furniture and things
in the room, that they also have depth, not just
height and width; and so we can say that we add
a Z axis. Dimensionally, we like to say that we
can vary freely along one while keeping the rest
of the dimensions unchanged; but if they are
tied together, it may be that the description
fits rather in a situation with perhaps e.g.
fewer dimensions.
Let's note at once that dimensions are counted.
That is to say, they are finite. We have two, or
three, or fifteen, or whatever finite number;
but if we speak of something like 'infinite
dimensional' we are into a different domain
altogether. And that different domain altogether
may be what essential reality is all about (is
also something David Bohm purported).
When I speak of the wholeness, or (in the
ancient Indian language Sanskrit, related to
the flavour also called Pali which is more
or less what Gothama the historical Buddha
spoke, but preceeding him, it seems, by
many millenia and holding great works of
art, fiction and astronomy), a wholeness
which involves also 4D or four dimensions,
and 6D or six dimensions, as Yoga4d and
Yoga6d, then I wish to say: we ALSO have to
do with something like a dimension of
duration. I want to completely avoid to say
'dimension of time', which leads to the
narrow machine-like inflexible thought of
Albert Einstein: that is a horrible idea.
We must not try to say of the quintessence
of all movement -- time, the tidal waves,
moving the universe, -- that they are merely
spread out in the next dimension beyond width,
length and depth. That is such a foolish
mistake, not worthy of the genius element which
seemed to have characterised some of Einstein's
works.
And it is not enough to add a couple of
dimensions, though we need that, too. We need
the six dimensions, for merely to organize
processes along a duration -- which is not
the quintessence of time, but within time --
along one dimension is way too narrow for
even the most mechanical of processes.
So I say YOGA up front: the wholeness,
which is continous, which is infinite, is
really the ground. And in this ground,
flowing, in flux, we manifest the four
dimensions, which I call space-duration.
It is not space-time. It is space-duration.
Duration is akin to time but less than that,
more measurable than that, more contained
than that, more changable as to its own
type of previous form and its own type of
upcoming form. Duration is laid out and so
where there is strong gravitation it is
twisted and twarted and that binds light;
and life processes must be near stellar
gravitational fields for that reason.
So there is an immensity to fluctuation.
Fluctuation happens subtly, it manifests
through dimensions which are countable --
but then may have its origins in that
which goes beyond the countable, that
which is truly infinite and of an ENTIRELY
DIFFERENT NATURE as the proof we have
looked at shows with all possible meditative
clarity if you give it enlightened space
in your mind for sufficient quality-time.
Now if one attempts to say, at the same
time, there is determinism -- that is to
say, that what is to happen is already
determined completely, fully, before it
happens -- then one has an apparent paradox.
And yet I say: there is no paradox.
Determinism is true. I will show how it can
be true at a vague and obviously intuitive
level -- which is as far as one can go in
a human discourse, naturally.
But first I will show how determinism
must be cleansed of the debris of Spinoza.
Einstein reputedly read Spinoza a lot and
kept Spinoza's axiomatic, mechanistic,
deterministic schemes preceeding him with
several centuries on his night-table. And
I think that can explain a lot about the
lesser brilliant aspects of Einstein. I
think I have said somewhere that the only
thing Spinoza really got right was
determinism -- and I can say that now also
-- but I should then hasten to add, to
make it precise, of course not at all
the type of silly, dumb, mechanical
determinism that Spinoza talked about.
A completely different kind is necessary.
For Spinoza aims to say nothing much
more than that which happens is necessary,
based on necessities he hopes to elucidate
by entirely mechanical-looking axioms,
in a mistaken attempt to apply Euclid's
already mistaken axiomatic version of
both-infinite-and-finite geometry to
the whole of existence. These so-called
necessities become, for Einstein and
perhaps also for Spinoza, laid out in
a kind of book-like dimension: what is
to happen is just what is by necessity
what happens, for the page of the book
is turned and -- most boringly -- the
sentences which are there are produced
by necessity from the sentences which
previously have been produced. This leads
to the type of dice-man type of chaotic,
purposeless on-going-ness which is
entirely an illusion, for it is all a
static iceblock, although cut in different
slices of 'this instant'; and Einstein
then moves on the relativize just WHICH
instant is sliced out, and how, by means
of looking at such as the speed and
accelleration of the reference frame of
the observers,-- we need not go into all
that, except to say that he sought to
make sense of some very real correlations,
which involve such puzzling phenomena as
a slowing of internal processes in a
thing which accellerates tremendously
OR which exists in a very dense
gravitational field.
But the phenomena of the quantum
theory were very hard to explain along
the same line: rather, one needed something
such as potential travel-possibilities
for each unit, both particle-unit and
aggregate-unit -- the whole structure or
dance -- in interaction with themselves
and each other in a wave-like fashion,
resulting in pilot waves which then,
according to a de Broglie description,
are by a form of subtle resonance picked
up by the components involved. This I
explain better with my supermodel theory,
I think.
But all this indicates that determinism
of the Einstein type -- with a fixed stuff
in a fourth dimension -- is wrong, wrong,
wrong; and that wipes Spinoza also away --
for what we need is fluctuations along
the extra dimensions beyond the three of
what I call 'RD' -- 'room/depth'. We need
at least a couple of dimensions more, as
well as the algorithmic capacities of
multiplying and adding and substracting
and weighing various wholes in terms of
contrasts and similarities and echoing
fields of reverberances -- what I call the
PMW principle, or Principle of a tendency
of Movement towards Wholeness -- and all
this means that a great, great, great deal
of fluctuation is going on beyond RD, and
determinism along the lines of Spinoza --
with necessities from the past merely
producing what is to come by a kind of
clock-work mechanical naturalness -- is
a far to simplistic and shoddy
description of the universe even at a
fairly manifest level.
Going to more subtle levels, how can
there be determinism still, as I maintain
is right? The answer lies in the
understanding that despite the fact that
a formalism can only be seen to work as
long as it concerns itself strictly with
finite numbers with known boundaries,
the universe may certainly well interact
with something which is finite from an
infinite source, somehow.
****** GETTING FREUD AWAY FROM THE SUBCONSCIOUS
The subconscious -- a term which, as far as I'm
concerned, is rather synonymous with "unconscious",
but better-sounding (since the word doesn't have the
connotation of being used, as the word "unconscious"
has, as a different concept in circumstances describing
a human being not in a conscious state of mind, but in
a state of mind which has to be healed) -- is, I
intuit, a collection of gestalts which flow from
two sources, from deeper intuition or silent
awareness (also prescient, clairvoyant and such),
and from the sensory impressions. A "gestalt" is
a word meaning, to me, a whole, composed of
contrasting similarities and similar contrasts,
including for instance shifting complementarities.
A gestalt can enter into a loop with itself, so
as to block out perception of the most appropriate
gestalts in each case. In the healthy state of
mind, the gestalts are playfully yielding to
new perceptions, which are guided by the
intuitive silence but allow contact with the
sensory impressions. When the brain takes a
pause from contact with intuitive awareness,
the gestalts earlier acquired from intuition
are given a somewhat loop-like situation,
in which they are strengthened to resist
illusions which could arise from too quick
interpretations (also emotional and reactive
interpretations, not just intellectual) of
sensory impressions. (In a religious context,
this is called 'faith'.)
Gestalts which accumulate themselves in
what we can call manifest strength radiate
themselves into consciousness -- from the
subconscious. To some extent this happens
during dreams; or at least they may reorganize
themselves during dreaming, in the freedom
of (too much) sensory impressions and need
for muscular work. But it is natural and
sane and good for the creative individual
to sleep so much that not all sleep is
of the deep and vaguely dreamless type
(in contrast to what Jiddu Krishnamurti
postulated), but that dreams come forth;
it is furthermore healthy to convert, by
intuitive gestalt perception, some
features of some dreams into perceptive
cognitive structures in the consciousness,
and then consciously intend to empty or
rid oneself of (needless) dream images.
Most dreams will however be vastly
exaggerated in meaningless ways as
concerns concrete people and things and
should entirely be forgotten without
any attempt at interpretation into
consciousness. That is also why they are
naturally forgotten by one who does not
intend to carry them into the day.
Now gestalts are also sexual. And sex
is intimate to the mystical or meditative
individual mind, and part of the collective
consciousness, in a synchronistic and
archetypical sense, with the universe --
in this I agree with the pupil of Freud
(who disagreed with Freud at some points),
Carl Gustav Jung. (I do however not agree
to Jung's focus on the concept of the
shadow and on the focus on the equalization
of the sexes.)
Freud's activity in writing intensely
(while also, it has been said, using
cocaine) as a medical doctor developing
a new field of psychiatric techniques for
healing the subconscious and the conscious
mind, in late 19th and early 20th century,
takes place in an european context which
had seen the many, many centuries of
aggression against sexuality by the
Christian church (and, even more, in areas
where they had influence, by the muhammedan
mullahs, ayatollahs and imams as well).
So just as Nietzsche's excremental poetry
was a fight against the Church, and lacked
any developed sexuality which had any sense
of health in itself -- his main emotion, it
seems, was pain, and whatever other emotions
he approached, he approached them by analogy
with pain, instead of as something new in
themselves, -- then the somewhat later Freud
also had a lot of complications going on
with sexuality. He was not free from fear
of it: he saw it everywhere, but in a
darkened gestalt. And this darkened, cloudy
gestalt he projected into his writings,
with the egotism and self-centeredness and
cunning one often finds with modern-day
quasi-intellectuals drugging themselves on
cocaine, ruining their brain and the finer
levels of positive feelings to pump up a
little self-confidence for the sake of
immediate satisfaction.
Freud rightly saw the gestalts of sex
dominating the development of the infant
baby and the child, and rightly spoke of
it as penetrating the entire existence of
the extremely young person and all the
way upwards to the adult: but wrongly
gave a flavour of the sad, the terrible,
and the foggily paranoic to the sexual
impulse. The joy he saw was little but
a flip of the coin of the pain/pleasure
pair, and not a joy which is grander
than a mere duality.
****** REPRINT OF THE REJECTED 2003 EXAM
Elements of infinite mathematics analyzed algorithmically
(THIS IS VERBATIM WHAT WAS DELIVERED TO THE UNIVERSITY OF OSLO,
DEPT. OF COGNITIVE SCIENCE (SLI/HF) JUNE 19TH, 2003. REF PROF
HERMAN RUGE JERVELL.)
*
Much of the exploration of the foundations of mathematics in the
past century, in terms of logic, has involved questions of the
finite and the infinite. Let us call the infinite simply
'infinity'.
*
Cantor showed that if P(X) is the set of all subsets of the set X,
then P(X) is bigger not only if X is a finite set, but also if X is
an infinite set such as the set of all natural numbers. His work
created a wave of implications that entirely set the agenda for
twentieth century mathematics. Even Alan Turing's work, that led to
the computer concept as he conceptualized it, may be seen to happen
in the struggle, as we may put it, to overcome 'larger infinities'.
In the case of Alan Turing, he sought to find a way to incorporate
the theorems that Kurt Goedel some years earlier (Turing in 1937,
Goedel in 1931) had shown to be 'unprovable', in some kind of rule-
bound machinery. Turing generalized Goedel's way of showing what is
an unprovable theorem within a system to a 'goedelizator'. This
goedelizator could go on to infinity producing unprovable theorems.
However, Turing had to realize that even such a goedelizator
machine is incapable of generating every unprovable theorem for a
system. For it, too, is a machine that can be represented by means
of a finite axiomatic system rich enough to contain integer
arithmetic, and so, it, too, is subject to a second level series of
'goedelization' theorems. Each such new level seems to be beyond
the reach of finite computability. So the work of Turing, combined
with Church and others, lead from 'uncountability' -- that an
infinite set, such as that of natural numbers, cannot be used to
'count' another and presumably 'larger' infinite set, such as that
of the real numbers -- to 'incomputability'.
*
The mathematician Brouwer suggested that there are ways of counting
real numbers if we introduce 'choice sequences'. The temporary (?)
end of the discussion around the kind of 'intuitionism' that
Brouwer propagandized is that some issues are not properly within
the domain of that which can be called 'mathematics'.
*
Let us consider a way in which we might imagine that we could,
after all, find some sense of countability of P(N), with regard to
N, where N is the set of natural numbers {1, 2, 3, ...}.
Let us bear in mind that an infinite set is said to be 'countable'
if and only if it can be put into a one-to-one mapping with regard
to the set of natural numbers. An example of an infinite set that
is countable is the set of prime numbers {1, 2, 3, 5, 7, 11, ...},
for we can here make a mapping of this kind: {(1, 1), (2, 2), (3,
3), (5, 4), (7, 5), (11, 5), ...}. While both sets are infinities,
they are understood to be infinities in somewhat the same sense.
Consider the way of writing natural numbers as a matrix:
1 2 3 . . .
1 2 3 . . .
1 2 3 . . .
. . .
. . .
. . .
If we can now make a rule of enumeration of all the subsets of the
natural numbers, so that we can make a mapping {(1, N1), (2, N2),
(3, N3), ...} where N1 is the first subset, N2 is the second
subset, and N3 is the third subset, and so on, of subsets of N,
such that, when this is allowed to go to infinity, we have every
subset of N, then we have a way to show that P(N) is countable. Let
us do this as an attempt and see how it goes, in the spirit of free
and open enquiry. It may be that we can readily comprehend
something about the question of what it means to 'go to infinity'
in so doing; and let us not jump to the conclusion that a mere
hundred years of mathematics is in any way adequate to ensure that
everything important has been said about this issue. After all,
Euclid's axiomatic geometry was around for twenty times that period
before some of the most important things about geometry, as seen
from an immediate post-Riemann perspective, came to be said.
The most challenging aspect of trying to enumerate the above is to
avoid having this kind of enumeration {x1, x2, ... to infinity, y1,
y2, ... to infinity, z1, z2, ... to infinity, and so on}, because,
unless we are able to confine the notion of going to infinity to
the end of the set, we have produced nothing that proves the
possibility of countability by means of natural numbers. That is to
say, in our particular case, let us avoid applying a rule of
enumeration first to the first line in toto, then to the second
line in toto, and so on. Rather, we want an approach that works
from the upper left corner and downwards to the right, in a step by
step progression, such that the infinity is confined to this 'right
and downwards' progression through the matrix. Let us see if this
can be done.
In any case, we can make rules of enumeration pertaining to line 1,
let us call the line 1 for L1, and line 2 for L2, and so on. Let us
make rules of enumeration called R1, R2 and so on. We can then make
the notion of a first step being taken for L1 by R1, then announce
that as we go to the second step of R1, we also go to the first
step of R2 for L2; as we go to the third step of R1, we also go to
the secon step of R2 for L2, and the first step of R3 for L3. In
this way, we get progressively rightwards as we get progressively
downwards. We must avoid the situation of going to an infinity
either rightwards or downwards 'before' doing something else. We
must bring the infinity-movements together, so to speak.
Let us denote the first act of enumeration by R1 on L1 as E(1, 1).
The second act of enumeration by R1 on L1 (rule 1 on line 1, that
is) for E(1, 2). The third act of enumeration by R1 on L1 is then
E(1, 3) etc.
The second line is enumerated by E(2, 1), E(2, 2), E(3, 2) etc. The
third line is enumerated by E(3, 1), E(3, 2), E(3, 3) etc.
We now say: we wish to make the enumeration E(linenr, stepnr) such
that E produces a subset of the numbers at line number referred to.
Furthermore, we wish to make the enumration E(linenr, stepnr) such
that when we go through both all the lines and all the steps
somehow (if we are able to), then we have every possible subset of
natural numbers. Each line, we bear in mind, is imagined to be
simply a list of all natural numbers.
That is, we want the set of subsets of N, which we can call Q,
since it is our question, to be generated in this way -- referring
to our strategy above, as going rightways and downwards in the
matrix as just described:
Q = {
E(1, 1),
E(1, 2), E(2, 1),
E(1, 3), E(2, 2), E(3, 1),
...
}
We see that in each step of generation G1, G2, G3, and so on, we
generate n subsets if we are at step Gn:
Q = {
G1: E(1, 1),
G2: E(1, 2), E(2, 1),
G3: E(1, 3), E(2, 2), E(3, 1),
Gn: ...
}
If we allow n to go to infinity, we have of course generated an
infinite set. Now look at exactly what happened here: we allow n to
go to infinity -- which means that if we imagined (somewhat
inaccurately) that 'infinity' constitued one of the steps, then we
would, in that 'step', add an infinity of E's (subsets) to the set.
We'll leave that remark for now, only noticing that this is an
issue that we will return to again and again, and it is here we
must face the questions of exactly what we mean by a countable
infinity.
In any case, whatever E does -- and we will soon see what it must
do -- it is clear that we are in a position to write the following
mapping:
{
(E(1,1), 1),
(E(1, 2), 2), (E(2, 1), 3)
(E(1, 3), 4), (E(2, 2), 5), (E(3, 1), 6)
...
}
in other words, since we generate Q by a finite countable extension
of subsets in each step, we are able to count each step by natural
numbers, and as we go to infinity in so doing, we appear to get a
countable set Q. However, we need to look into this question again,
and also, perhaps, in relation to other kinds of sets.
Let us now see what E should be. It should certainly be interesting
if we could now make E such that when E(n, m) is generated for all
n and all m, we do in fact have all subsets of natural numbers.
If we are going to make a full list of subsets, beginning with
small sets and going on to larger sets, why don't we begin with the
subsets that is simply a single number.
That is,
E(1, 1) is the subset {1}, whereas E(1, 2) (where E is E(linenr,
stepnr)) is the subset {2}, and E(1, n) is the subset {n}, for a
natural number n.
Then,
E(2, 1) is the subset {1, 2}. It seems. However we must be careful
to include subsets such as {1, 3} just as much as {2, 3}, somewhere
in our procedure. Clearly, we need some kind of loop in each of the
higher E's, of a finite and countable kind of course, to produce
what we need.
I am, by the way, making this procedure for the first time -- I
have not checked anywhere whether this has been done before nor
have I tested it; I go by 'gut feeling' at the moment, checking
with logic. I am, however, fairly certain that it is no harder task
making an enumeration of the E kind than the task I have already
looked at, which is enumeration all possible permutations around
the decimal point.
Our present task is to look at line 2 in
1 2 3 . . .
1 2 3 . . .
1 2 3 . . .
. . .
. . .
. . .
so that we can produce, somehow, all possible subsets of two and
exactly two numbers by what is included by the procedure E(2, n),
if n goes to infinity. We need then to get sets such as
{1, 2} {1, 3} {1, 4} ... {2, 3} {2, 4} {2, 5} ...
{3, 4} {3, 5} {3, 6} ...
where we avoid, of course, {3, 3} since this, by typical set
theoretical standards, equivalent to {3} and hence superfluous,
since we already have included this series of subsets from the
steps in line 1. However, since it may be simpler to make a rule
that also includes {n, n} let us not emphasize the point: as long
as it becomes countable, it is okay that the set, when generated by
a rule, has some duplicates which are then imagined to be 'filtered
out'.
We see that we have several infinite series for E(2, n) and we are
at liberty to organize them, too, in terms of a matrix going
rightwards and downwards (or upwards, to be more accurate, when
speaking of size; however since we write letters rightwards and
then down to the next line, by convention it becomes rightwards and
downwards):
{1, 2} {1, 3} {1, 4} {1, 5} ...
{2, 3} {2, 4} {2, 5} ...
{3, 4} {3, 5} ...
{4, 4} ...
. . .
. . .
which we may sort out in a zig-zag fashion from the upper/left
corner through the diagonal of the matrix as follows
{1, 2} {1, 3} {2, 3} {1, 4} {2, 4} {3, 4} {1, 5} {2, 5} {3, 5} {4,
4} etc
We can cut the whole thing down to the following:
For E(2, n) generate all subsets of {a, b} where a and b are
natural numbers between 1 and n, except where a = b. This can be
done, in terms of an algorithm, as follows, in terms of something
like Java 1.1:
/* For E(2, n) given any n */
for (int a = 1; a <= n; a++)
for (int b = 1; b <= n; b++)
if (a != b)
addSet(a, b);
We are now in a position to say how E(3, n) must be constructed: It
must generate all subsets of {a, b, c} where a and b and c are
natural numbers 1 and n, except where a = b, b = c, or c = d (or
any combination). In other words,
/* For E(3, n) given any n */
for (int a = 1; a <= n; a++)
for (int b = 1; b <= n; b++)
for (int c = 1; c <= n; c++)
if ( (a != b) && (b != c) && (a != c) )
addSet(a, b, c)
We see now that given any line number L, we have a route procedure
to make E(L, n), for all natural numbers n, so that the mapping
{
(E(1,1), 1),
(E(1, 2), 2), (E(2, 1), 3)
(E(1, 3), 4), (E(2, 2), 5), (E(3, 1), 6)
...
}
when going to infinity is a complete mapping of all the subsets of
N.
At least it may appear so. Let us discuss this, and furthermore,
raise more general questions of infinity and finiteness afresh,
with this apparently contradictory result to Cantor's original
approach in mind.
To anticipate: we will see that by an advanced algorithmic mind,
trained in computer programming, we can provide situations in which
it appears that not only this apparent uncountability situation --
that of Cantor with regard to the subsets of the set of all natural
numbers -- but also others, in particular the set of all real
numbers -- can be, perhaps surprisingly, dealt with. However, this
is less of a surprise if we look more closely into what this
implies for the set of natural numbers itself. It turns out that
the two just-mentioned results are rather special cases on a
feature of counting which, to my mind (but that may be just because
I am not well enough educated in mathematics; and I expect to work
further on this in the context of a ph.d.), comes into being --
even in the set of natural numbers -- once we say: go to infinity.
It may be of great value to the reader of this treatise to ponder
on the possibly huge difference between "going as far as we like" -
- to any arbitrary finite point -- and actually going infinitely
long. In the case of the use of the word "limit" in mathematics,
the first situation is involved; in the case of the consideration
of the kind of infinity involved in the "size" of the set of all
natural numbers, the second situation is involved. It appears to
me.
The whole of this thesis can then be summed up as follows: when we
look, from the point of view of programming, at old situations of
apparent uncountability, we come to regard something which has been
considered a rather easy point -- namely the visualization of the
set of all numbers, based on counting, such as 1, 2 and 3, when we
go to infinity -- to be a rather difficult point. Without going
beyond the confines of this treatise I think it is apt to point out
that if the basic set of counting has somehow more difficulties
involved with it, then it may be that some of the difficulties
ascribed to countability may be relocated.
*
It may be objected to the procedure above that, although we at each
step have a perfectly well-defined procedure, then, ultimately, in
order to "go to infinity", we need a case of an infinite procedure.
In other words, it would be required to do something like the
following, which is doable enough in a finite context
for (int X1 = 1; X1 <= n; X1++)
for (int X2 = 1; X2 <= n; X2++)
for (int X3 = 1; X3 <= n; X3++)
...
for (int XL = 1; XL <= n; XL++)
addSet(X1, X2, X3, ..., XL)
And then say:
L => infinity
as if it were a command in the language.
*
The case we just discussed is a complex version of something much
more simple. Before reaching this simplicity let us show how the
example above is comparable to enumeration of decimals around the
decimal point. Let us write decimal numbers by zeroes and ones,
separated by a decimal dot. Then, consider the list
Permutation of 1 digits before and after:
0.0
0.1
1.0
1.1
Permutation of 2 digits before and after:
00.01
00.10
00.11
01.00
01.01
01.10
01.11
10.00
10.10
11.01
11.11
Permuation of 3 digits before and after:
000.000
000.001
etc
We can then say: when we have come as far as to permutate n digits
before and after, we reach an enumeration of all possible real
numbers involving, at most, n digits before and/or after the
decimal dot.
We can further say: if n can go to infinity we appear to get a
complete enumeration, in a step-by-step manner, of all decimal
numbers. However we must take care to realize that an infinite-
decimal number like pi can only be mapped to a natural number that
has just as many digits ("before" its imagined decimal dot) as pi
has after. That is, it would be a specific number but not finite.
What's this? And what is, indeed, the exact constitution of the
members of the set of all natural numbers, whose beginning seem
some complacently simple: {1, 2, 3, ...}?
*
Just to tease the reader who wishes to tell exactly what on earth
is going on in the above reasonings, we will not provide any
solutions as yet, but merely drive the contradictions in with
maximum of compatibility power. In the next point, there is a
program. In the following point, there is a shortening of its
output. In the following point, again, there is a discussion of
what it does -- in generating a list of real numbers. And only
after all this we will concentrate the question into looking at
what is exactly the constitution of the set of natural numbers.
After the program comes the explanation.
*
PROGRAM
import java.applet.*; import java.io.*; import java.net.*;
import java.awt.*; import java.awt.image.*; import
java.awt.event.*;
import java.util.*; import java.math.BigInteger;
// Please test the program if you wish, at
// http://wintuition.net/wnmagazines/
// quantum/quantum4program1.html
// It runs as an applet in Java 1.1.
// Designed by H W Reusch, released under GNU GPL.
final public class wn_quantum4program1
extends Applet {
final static int _DIGITS = 2; // Change this as you please, if
memory enough
// Remember that this algorithm is to show that the procedure
// as sketched in the article is correct, not that it is
// computationally efficient or anything like that.
// Strings are used in this case, and Java strings are
// pretty leisurely beings...but they are easy to change.
static int _HOW_MANY = 2500; // Change this as you please, too
// If you set _HOW_MANY to too high or to zero, you get all of
them!
final static boolean _SHOW_ALL_THREE = true;
final static boolean _CHECK_FOR_REPETITIONS = false;
// Saves a little time to only print the map set (then set to
false)
// If, say, _DIGITS == 3, then the numbers are of the form
// +000.000; if _DIGITS == 8, then +00000000.00000000 etc.
// The algorithm makes no assumption that this has to be small.
// If you run this on a supercomputer, you may want to turn
// the 'int' into something larger than the Java int,
// so you can generate thousands of digits.
/*
Step 1. We start with the set {0} for W and {0.0} for X. We make a
mapset, called M,
where {(0, 0.0)} is our initial member. See initialization part for
this thing.
*/
public int W[]; // The whole numbers are stored as int, of course
public String X[]; // The real numbers are stored as string, as
said
public String M[]; // We store each (a,b) pair as a string "(a,b)"
/*
Step 2. We set index s=0 and another index t=1.
*/
public int s=0;
public int t=0;
/*
Step 3: Clear a temporary set Y={}, that is, empty.
*/
public String Y[]; // Same type as X, of course
public int AMOUNT_IN_Y = 0;
public int AMOUNT_IN_X = 0; // Keeps track
public int AMOUNT_IN_W = 0; // of how far
public int AMOUNT_IN_M = 0; // we have come!;)
public int MAXIMUM_AMOUNT; // *will be calculate based on _DIGITS
public boolean thisisfresh; // Since sets should not
// have repetitions, this flags helps us to keep them right.
public String text;
public Frame textFrame;
public TextArea textFrameArea;
public void init() { // The whole procedure is within applet init
/* We prepare the frame to show the results. */
textFrame = new Frame();
textFrame.setSize(650,432);
textFrame.setTitle("YOU HAVE ORDERED "+_DIGITS+" DIGITS BOTH
SIDES");
textFrame.setLayout(new BorderLayout());
textFrameArea = new TextArea(100, 90); // extendable
textFrame.setFont(new Font("Courier", Font.ITALIC, 13));
textFrameArea.setText("\n * * * * *");
textFrame.add(textFrameArea, BorderLayout.CENTER);
/* The frame will close at once IF the user indicates so. */
textFrame.addWindowListener(new WindowAdapter() {
public void windowClosing(WindowEvent e) {
textFrame.setVisible(false); } } );
textFrame.validate();
textFrame.setLocation(110, 225); // Somewhere up to the middle
textFrame.setVisible(true);
requestFocus();
textFrame.requestFocus(); // Put window up front
boolean finished = false;
/* EXTRA INITIALIZATIONS...cfr step 1 and 2 and 3*/
MAXIMUM_AMOUNT = 10;
String BIGZEROREAL = "+"; // initial sign
for (int a=1; a<=_DIGITS; a++) {
MAXIMUM_AMOUNT = MAXIMUM_AMOUNT * 10;
BIGZEROREAL = BIGZEROREAL + "0";
}
BIGZEROREAL = BIGZEROREAL + ".";
for (int a=1; a<=_DIGITS; a++) {
MAXIMUM_AMOUNT = MAXIMUM_AMOUNT * 10;
BIGZEROREAL = BIGZEROREAL + "0";
}
MAXIMUM_AMOUNT = 2 * MAXIMUM_AMOUNT ; // Plus and minus
MAXIMUM_AMOUNT = MAXIMUM_AMOUNT + 5; // A little extra
BIGZEROREAL = BIGZEROREAL + " ";
// Now BIGZEROREAL is "+0000.0000 "
// if _DIGITS are 4.
int DOT_POSITION = _DIGITS + 1;
// Note that both strings and arrays
// in Java begins with first entity at pos 0.
X = new String[MAXIMUM_AMOUNT];
W = new int[MAXIMUM_AMOUNT];
M = new String[MAXIMUM_AMOUNT];
Y = new String[MAXIMUM_AMOUNT];
for (int a=0; a < MAXIMUM_AMOUNT; a++) {
X[a] = null;
W[a] = 0;
M[a] = null;
Y[a] = null;
}
if ((_HOW_MANY == 0) || (_HOW_MANY > MAXIMUM_AMOUNT))
_HOW_MANY = MAXIMUM_AMOUNT;
/* Okay. Step 1 and 2 here: */
X[AMOUNT_IN_X++]=new String(BIGZEROREAL);
W[AMOUNT_IN_W++]=0;
M[AMOUNT_IN_M++]="(0,"+BIGZEROREAL+")";
/* LET'S START LOOP! */
while (!finished) { // WHILE#1
/*
Step 4: s=s+1. That is, increase s by one.
*/
s=s+1;
if ((s > _DIGITS) || (AMOUNT_IN_X >= _HOW_MANY)) { // IFTHENELSE#1
finished = true;
do_the_printing();
}
else { // IFTHENELSE#1 go on making!
/*
Step 5: For index i=1, increased stepwise by 1, up to and including
s, carry out step 6.
*/
for (int i=1; i<=s; i++) { // FOR#1
/*
Step 6: For each positive member x in X (or zero, first), carry out
step 7.
*/
for (int xmember = 0; xmember < AMOUNT_IN_X; xmember++) { // FOR#2
String x = X[xmember];
if (x.charAt(0) == '+') { // IF#1
/*
Step 7: For each of the digits d = 1...9, do the following 8-11:
*/
for (int d = 1; d<=9; d++) { // FOR#3
/*
Step 8: With member x, make a new member by replacing the digit in
position i before (if i negative, after) the dot by d, and add it
to Y, if it is not already in X. Call it x2.
*/
// In this run, i is positive.
int p = DOT_POSITION + i;
String x2 = null;
x2 = x.substring(0, p) + ((char)(d+'0')) + x.substring(p+1);
thisisfresh = (!a_member_of_X(x2));
if (thisisfresh) { //IF#thisisfresh
Y[AMOUNT_IN_Y] = x2;
AMOUNT_IN_Y++;
/*
Step 9: If x2 was new: Make a new member in W by adding one to
index t and adding this as a member to W.
*/
t=t+1;
W[AMOUNT_IN_W] = t;
AMOUNT_IN_W++;
/*
Step 10: If x2 was new: Make a pair (t, x2) and add it to M, the
mapset.
*/
M[AMOUNT_IN_M] = "("+t+","+x2+")";
AMOUNT_IN_M++;
}//IF#thisisfresh
/*
Step 11: If -x2 is new to X, then add one -x2 to Y; Increase t by
one; Add t to W; Add (-t, -x2) to M.
*/
// Make x2 into -x2
x2 = "-" + x2.substring(1);
thisisfresh = (!a_member_of_X(x2));
if (thisisfresh) { //IF#thisisfresh
Y[AMOUNT_IN_Y] = x2;
AMOUNT_IN_Y++;
t=t+1;
W[AMOUNT_IN_W] = t;
AMOUNT_IN_W++;
M[AMOUNT_IN_M] = "("+t+","+x2+")";
AMOUNT_IN_M++;
} //IF#thisisfresh
} // FOR#3
} // IF#1
} // FOR#2
} // FOR#1
/*
Step 12: For each member in Y, add this member to X, and set Y={}
again.
*/
for (int a = 0; a < AMOUNT_IN_Y; a++) { // FOR
X[AMOUNT_IN_X++] = Y[a];
Y[a]=null;
} // FOR
AMOUNT_IN_Y = 0;
if ((s > _DIGITS) || (AMOUNT_IN_X >= _HOW_MANY)) { // IFTHENELSE#1
finished = true;
do_the_printing();
return;
}
/*
Step 13: Go up to step 5 again, and do the same with negative i
down to and including -s.
NOTE: To do this, we give the code above in a compressed form
again, without comments, where the sign of i has changed:
*/
// *******COMPRESSED CODE WITH CHANGED SIGN OF i **************
for (int i=-1; -s<=i; i--) { // FOR#1
for (int xmember = 0; xmember < AMOUNT_IN_X; xmember++) { // FOR#2
String x = X[xmember];
if (x.charAt(0) == '+') { // IF#1
for (int d = 1; d<=9; d++) { // FOR#3
// In this run, i is negative.
int p = DOT_POSITION + i;
String x2 = null;
x2 = x.substring(0, p) + ((char)(d+'0')) + x.substring(p+1);
thisisfresh = (!a_member_of_X(x2));
if (thisisfresh) { //IF#thisisfresh
Y[AMOUNT_IN_Y] = x2;
AMOUNT_IN_Y++;
t=t+1;
W[AMOUNT_IN_W] = t;
AMOUNT_IN_W++;
M[AMOUNT_IN_M] = "("+t+","+x2+")";
AMOUNT_IN_M++;
} //IF#thisisfresh
x2 = "-" + x2.substring(1);
thisisfresh = (!a_member_of_X(x2));
if (thisisfresh) { //IF#thisisfresh
Y[AMOUNT_IN_Y] = x2;
AMOUNT_IN_Y++;
t=t+1;
W[AMOUNT_IN_W] = t;
AMOUNT_IN_W++;
M[AMOUNT_IN_M] = "("+t+","+x2+")";
AMOUNT_IN_M++;
} //IF#thisisfresh
} // FOR#3
} // IF#1
} // FOR#2
} // FOR#1
for (int a = 0; a < AMOUNT_IN_Y; a++) { // FOR
X[AMOUNT_IN_X++] = Y[a];
Y[a]=null;
} // FOR
AMOUNT_IN_Y = 0;
// *******COMPRESSED CODE WITH CHANGED SIGN OF i
**************FINISHED
} // IFTHENELSE#1
/*
Step 14: Go up to step 4 again. Ad infinitum.
*/
} // WHILE#1
} // Finish public void init()
public void do_the_printing() {
System.gc(); // Do garb.collection
textFrame.setTitle("wn_quantum4program1 (c) Stein von Reusch");
textFrameArea.setVisible(true);
text = "Permutations for numbers complete\n"+
"Now doing amazingly slow Java string handling before output...\n";
System.gc(); // Do garb.collection
textFrameArea.setText(text); // In case the next step takes time...
text = "You have ordered "+_DIGITS+" digits on each side.\n";
if (_CHECK_FOR_REPETITIONS)
text +="Repetitions have been removed from the lists.\n";
else
text +="TO SAVE TIME, MANY REPETITIONS HAVE BEEN ALLOWED. \n";
text +="You have asked for ";
if (_SHOW_ALL_THREE) text += " all three sets.\n";
else text += "only the map list M.\n";
text = text + "You have asked for ";
// Clearly, AMOUNT_IN_X == AMOUNT_IN_W == AMOUNT_IN_M ///
if (_HOW_MANY == MAXIMUM_AMOUNT) text += "all the items.\n";
else text += _HOW_MANY + " items.\n";
text += "CONGRATULATIONS. Here is your list!\n\n";
if (AMOUNT_IN_X < _HOW_MANY) _HOW_MANY = AMOUNT_IN_X;
if (_SHOW_ALL_THREE) { //IFELSE
text += " W R M\n";
for (int teller = 0; teller<_HOW_MANY; teller++) { //FOR
// Assume AMOUNT_IN_X = AMOUNT_IN_W = AMOUNT_IN_M
if (M[teller] != null)
text += W[teller] + " " +
X[teller] + " " + M[teller]+"\n";
} //FOR
} else //IFELSE
for (int teller = 0; teller<_HOW_MANY; teller++) { //FOR
if (M[teller] != null)
text += M[teller]+"\n";
} //FOR
//IFELSE
textFrameArea.setText(text);
textFrameArea.validate();
textFrameArea.setVisible(true); // In case user has closed window,
reopen.
textFrameArea.requestFocus(); // In case something else is in
front.
return;
} //public void do_the_printing()
protected boolean a_member_of_X(String potential_member)
{
if (!_CHECK_FOR_REPETITIONS) return false;
int counter=0;
while (counter< AMOUNT_IN_X) { //WHILE
if (X[counter].equals(potential_member)) return true;
} //WHILE
return false;
} //protected boolean a_member_of_X
} //
*
SHORTENED OUTPUT OF PROGRAM
You have ordered 2 digits on each side.
TO SAVE TIME, MANY REPETITIONS HAVE BEEN ALLOWED.
You have asked for all three sets.
You have asked for 2500 items.
CONGRATULATIONS. Here is your list!
W R M
0 +00.00 (0,+00.00 )
1 +00.10 (1,+00.10 )
2 -00.10 (2,-00.10 )
3 +00.20 (3,+00.20 )
4 -00.20 (4,-00.20 )
5 +00.30 (5,+00.30 )
6 -00.30 (6,-00.30 )
7 +00.40 (7,+00.40 )
8 -00.40 (8,-00.40 )
9 +00.50 (9,+00.50 )
10 -00.50 (10,-00.50 )
11 +00.60 (11,+00.60 )
12 -00.60 (12,-00.60 )
13 +00.70 (13,+00.70 )
14 -00.70 (14,-00.70 )
15 +00.80 (15,+00.80 )
16 -00.80 (16,-00.80 )
17 +00.90 (17,+00.90 )
18 -00.90 (18,-00.90 )
19 +01.00 (19,+01.00 )
20 -01.00 (20,-01.00 )
21 +02.00 (21,+02.00 )
22 -02.00 (22,-02.00 )
23 +03.00 (23,+03.00 )
24 -03.00 (24,-03.00 )
25 +04.00 (25,+04.00 )
26 -04.00 (26,-04.00 )
27 +05.00 (27,+05.00 )
28 -05.00 (28,-05.00 )
29 +06.00 (29,+06.00 )
. . .
. . . // we have shortened the list here! ///
. . .
2297 +07.06 (2297,+07.06 )
2298 -07.06 (2298,-07.06 )
2299 +07.07 (2299,+07.07 )
2300 -07.07 (2300,-07.07 )
2301 +07.08 (2301,+07.08 )
2302 -07.08 (2302,-07.08 )
2303 +07.09 (2303,+07.09 )
2304 -07.09 (2304,-07.09 )
2305 +08.01 (2305,+08.01 )
2306 -08.01 (2306,-08.01 )
2307 +08.02 (2307,+08.02 )
2308 -08.02 (2308,-08.02 )
2309 +08.03 (2309,+08.03 )
2310 -08.03 (2310,-08.03 )
2311 +08.04 (2311,+08.04 )
2312 -08.04 (2312,-08.04 )
2313 +08.05 (2313,+08.05 )
2314 -08.05 (2314,-08.05 )
2315 +08.06 (2315,+08.06 )
2316 -08.06 (2316,-08.06 )
2317 +08.07 (2317,+08.07 )
2318 -08.07 (2318,-08.07 )
2319 +08.08 (2319,+08.08 )
2320 -08.08 (2320,-08.08 )
2321 +08.09 (2321,+08.09 )
2322 -08.09 (2322,-08.09 )
2323 +09.01 (2323,+09.01 )
2324 -09.01 (2324,-09.01 )
2325 +09.02 (2325,+09.02 )
2326 -09.02 (2326,-09.02 )
2327 +09.03 (2327,+09.03 )
2328 -09.03 (2328,-09.03 )
2329 +09.04 (2329,+09.04 )
2330 -09.04 (2330,-09.04 )
2331 +09.05 (2331,+09.05 )
2332 -09.05 (2332,-09.05 )
2333 +09.06 (2333,+09.06 )
2334 -09.06 (2334,-09.06 )
2335 +09.07 (2335,+09.07 )
2336 -09.07 (2336,-09.07 )
2337 +09.08 (2337,+09.08 )
2338 -09.08 (2338,-09.08 )
2339 +09.09 (2339,+09.09 )
2340 -09.09 (2340,-09.09 )
2341 +01.11 (2341,+01.11 )
2342 -01.11 (2342,-01.11 )
2343 +01.12 (2343,+01.12 )
2344 -01.12 (2344,-01.12 )
2345 +01.13 (2345,+01.13 )
2346 -01.13 (2346,-01.13 )
2347 +01.14 (2347,+01.14 )
2348 -01.14 (2348,-01.14 )
2349 +01.15 (2349,+01.15 )
2350 -01.15 (2350,-01.15 )
2351 +01.16 (2351,+01.16 )
2352 -01.16 (2352,-01.16 )
2353 +01.17 (2353,+01.17 )
2354 -01.17 (2354,-01.17 )
2355 +01.18 (2355,+01.18 )
2356 -01.18 (2356,-01.18 )
2357 +01.19 (2357,+01.19 )
2358 -01.19 (2358,-01.19 )
2359 +02.11 (2359,+02.11 )
2360 -02.11 (2360,-02.11 )
2361 +02.12 (2361,+02.12 )
2362 -02.12 (2362,-02.12 )
2363 +02.13 (2363,+02.13 )
2364 -02.13 (2364,-02.13 )
2365 +02.14 (2365,+02.14 )
2366 -02.14 (2366,-02.14 )
2367 +02.15 (2367,+02.15 )
2368 -02.15 (2368,-02.15 )
2369 +02.16 (2369,+02.16 )
2370 -02.16 (2370,-02.16 )
2371 +02.17 (2371,+02.17 )
2372 -02.17 (2372,-02.17 )
2373 +02.18 (2373,+02.18 )
2374 -02.18 (2374,-02.18 )
2375 +02.19 (2375,+02.19 )
2376 -02.19 (2376,-02.19 )
2377 +03.11 (2377,+03.11 )
2378 -03.11 (2378,-03.11 )
2379 +03.12 (2379,+03.12 )
2380 -03.12 (2380,-03.12 )
2381 +03.13 (2381,+03.13 )
2382 -03.13 (2382,-03.13 )
2383 +03.14 (2383,+03.14 )
2384 -03.14 (2384,-03.14 )
2385 +03.15 (2385,+03.15 )
2386 -03.15 (2386,-03.15 )
2387 +03.16 (2387,+03.16 )
2388 -03.16 (2388,-03.16 )
2389 +03.17 (2389,+03.17 )
2390 -03.17 (2390,-03.17 )
2391 +03.18 (2391,+03.18 )
2392 -03.18 (2392,-03.18 )
2393 +03.19 (2393,+03.19 )
2394 -03.19 (2394,-03.19 )
2395 +04.11 (2395,+04.11 )
2396 -04.11 (2396,-04.11 )
2397 +04.12 (2397,+04.12 )
2398 -04.12 (2398,-04.12 )
2399 +04.13 (2399,+04.13 )
2400 -04.13 (2400,-04.13 )
2401 +04.14 (2401,+04.14 )
. . .
. . .
. . .
etc to infinity
*
EXACT DESCRIPTION OF WHAT THE PROGRAM DOES
Step 1. We start with the set {0} for W and {0.0} for X. We make a
mapset, called M, where {(0, 0.0)} is our initial member.
Step 2. We set index s=0 and another index t=1.
Step 3: Clear a temporary set Y={}, that is, empty.
Step 4: s=s+1. That is, increase s by one.
Step 5: For index i=1, increased stepwise by 1, up to and including
s, carry out step 6.
Step 6: For each positive member x in X (or zero, first), carry out
step 7.
Step 7: For each of the digits d = 1...9, do the following 8-11:
Step 8: With member x, make a new member by replacing the digit in
position i before (if i negative, after) the dot by d, and add it
to Y, if it is not already in X. Call it x2.
Step 9: If x2 was new: Make a new member in W by adding one to
index t and adding this as a member to W.
Step 10: If x2 was new: Make a pair (t, x2) and add it to M, the
mapset.
Step 11: If -x2 is new to X, then add one -x2 to Y; Increase t by
one; Add t to W; Add (-t, -x2) to M.
Step 12: For each member in Y, add this member to X, and set Y={}
again.
Step 13: Go up to step 5 again, and do the same with negative i
down to and including -s.
Step 14: Go up to step 4 again.
Key to read the above: Imagine that W is the set of whole numbers,
whereas X, if this procedure is allowed to go on to infinity,
becomes more and more R.
*
Having done as we promised, we will now look into the set of
natural numbers. We will take care to consider the fact that the
way we write natural numbers may have an influence on how we
mentally think of these numbers. When, for instance, did it become
utterly clear and accepted that a number like 3.14159265358...
gives any meaning at all? Why is it so clear that that number makes
full sense whereas this number 14159265358... belongs to science
fiction, at best?
Since conventions, mental habits, and cultural 'debris' (I
apologize for the metaphor, but it is sometimes necessary to point
out how culture may cloud our perception) are heavily ingrained
with our natural numbers, let us do something that must be among
the most ancient ways of handling numbers -- we write them by one
'digit' only:
Let 1 be I
Let 2 be II
Let 3 be III
And so on.
Let us not even group them, since we are quickly going beyond the
finite range anyway. That is, we won't write eight as two groups of
four I (like IIII IIII) but we will write eight as
IIIIIIII
It is now tempting to imagine that infinity can be written as
IIIIIIII...
but we must be careful. Let us point out that in the enumeration
above, of decimal numbers, we had a comparatively easy situation as
regards finite numbers, and with finite decimal precision. That is,
we can imagine situations where numbers like 3.14 and 5.18 have an
enumeration as follows:
IIIIIIIIIIIIIIIII: 3.14
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII: 5.18
Let us now imagine that we are in a position to carry out the
thought experiment -- as we are! -- that the generation of decimal
numbers goes to infinity, somehow. Then, let us consider the more
transcendent-style numbers 3.141519265358... and the ratio-based
number 5.181818181818... Let us write POS1 and POS2 for the
imagined position in a list, which is the mapping of the set of
decimal numbers generated by our procedure. That is,
POS1: (IIIIIIIIIIII....): 3.141519265358...
POS2: (IIIIIIIIIIII....): 5.181818181818...
The question that impells itself onto us, not based on any
mathematical culture that I am strongly aware of, but simply by our
plain procedural reasonings when taken to infinity, is: what kinds
of numbers, if any, might POS1 and POS2 be said to be, and might
they, or might they not, if we grant them status as numbers, be
said to be equal?
First of all, let us not automatically assume that we have
transgressed from what has been somewhat pompously been declared to
be 'cardinal numbers' to 'ordinal numbers' -- that belongs properly
to a tradition in which certain things have been taken to be sure
as to the non-denumerability of certain sets; and we must remind
ourselves, at this point, that we have not as yet adopted this type
of approach. Rather, we have striven, if I may point out again, to
enumerate that which appears to be very difficult to enumerate, by
converting sets of the kind {a1, a2, a3, ..., b1, b2, b3, ...., c1,
c2, c3, ...} into sets of the kind {E1, E2, E3, ...}, where E is
some enumeration route procedure. We have then said: let this go to
infinity. And in this way, we achieved a list which, 'at the moment
of infinity' so to speak, should be containing members that we
might consider to be of infinite size. Let me make this clear:
I
II
III
IIII
IIIII
.
.
.
We can also turn it upside down, thus:
.
.
.
IIIII
IIII
III
II
I
Let me speak explicitly of the visual hint that we now just
inserted: instead of putting the dots vertically, by analogy of
putting them horisontally in the set {1, 2, 3, ...} (or in the set
{I, II, III, ...}) then we have put them in the actual direction of
growth, where the size of the member inserted at step 1 is 1, at
step 2 it is 2, and at step n it is n.
That is, if we generate a set in this way that has a thousand
members, then there is a number in it, which we may write as
n:(III...) where n equals thousand, and where the notation
n:(III...)
signifies that we achieve this number by the procedure, again in
Java,
String Inumber = "";
for (int i = 1; i <= n; i++)
Inumber.concat("I");
In Java, string concatenation may in fact be written by the
addition operator, so this can be written also
String Inumber = "";
for (int i = 1; i <= n; i++)
Inumber = Inumber + "I";
However, since Java handles strings as objects that really are just
replaced in each case, a loop going up to thousand would involve
the rather useless creation of ninehundred and ninety-one objects
first -- which explains the slowness, in general, of Java string
handling. I just mention this in the interest of using Java to
construct somewhat lofty programming languages that can handle a
concept of infinity as indicated by three dots in the case of sets
like {I, II, III, ...}.
In any case, if n:(III...) has the size n, and this member is
always and inevitably member of the set {I, II, III, ...} when this
set has n members, then it follows by sheer logic that if the set
is infinite in size, then there are members whose size is infinite
too. That is, we cannot exclude members in which n:(III...) exist
in some sense in which n is not any more finite than the size of
the set itself.
This is visually obvious in the reproduction from above as follows:
.
.
.
IIIII
IIII
III
II
I
We see here that the dots indicate the actual direction of growth:
it is inseparably going upwards and rightwards AT THE SAME TIME. If
we could separate these two directions of growth, then we might
say: let us grow it infinitely long, this list, but avoiding such
nuiassance members as those that are infinitely wide. Alas, it does
appear plainly incoherent in our thought experiment to do so!
*
If we write the generation of natural numbers, inspired by the
above, as follows
.
.
.
1000
.
.
100
.
.
10
9
8
7
6
5
4
3
2
1
Then we see that also here, when we write natural numbers in terms
of the ten digit conventional system typically implemented in our
societies today, we also have a direction of growth that is
inseparably going in two directions at once.
*
I wish to now ask the question, for which I do not as yet know the
best way to answer. I ask it in the sense of a thought experiment.
I ask it in order to try to approximate conventional thinking about
infinities in mathematics, in which it is taken for granted that
the set of natural numbers only contain finite such numbers. The
research question is: can we imagine the following set {1, 2, 3,
...} to be infinite without containing anything except finite
numbers?
That is, written in the challenging manner
.
.
.
IIIII
IIII
III
II
I
can we imagine this set to go on to be an infinite set, without at
any time we coming into the situation of having to admit a number
which must be written as n:(III...) where n is no longer a finite
number?
I am going to attempt to say 'no', for the sake of research, and
certainly not at all in a dogmatic manner. I am going to say: No, I
don't think we can go to infinity in size without going to infinity
in width. And so, no, I don't think we can construct the set N =
{1, 2, 3, ...} such that it admits of exclusively finite numbers.
This I will say in the spirit of research, but also in order to be
honest to the visual experience of growth in two directions in an
inseparable manner. This growth is in terms of the upward direction
of size of set and the rightward direction of size of numbers and
these two directions are acted upon by a single function, namely
the function of going from one step to the next, so that they are
both involved in each step; and in this situation, I wish to say
'No, it doesn't seem as though we can mess with infinities in one
way and not the other'.
That is, while we can go 'as far as we want to' in strictly finite
terms, and still have only finite-sized members, I wish to say that
if we actually want to 'all the way' to infinity, in a moment,
somehow -- in the dance of our thought perceptions around the
question -- then we have also left the finite numbers in so doing.
However, if I stand up and say this very strongly, then I must
admit to one very essential fact: every finite number is still a
member of this set, and I cannot say that this subset, then, as it
will be, can be itself finite. That is, again by apparently clear
reasoning, there is a subset, which is infinite, of the set {I, II,
III, ...} such that this set is indeed only constituted of finite-
sized n:(III...) numbers.
And then, for the sake of research, and for giving maximum impact
to the bold 'No' just given, let us challenge this and say: No,
there is no such subset. For if there is such a subset of finite
numbers, it can be written
.
.
.
IIIII n=5
IIII n=4
III n=3
II n=2
I n=1
where n is a finite number for each and every member. But if n is a
finite number for each and every member, then, by the above
reasoning, where we linked the size of the largest member to the
size of the set, then the set must be finite. Reductio ad absurdum,
there is no infinite subset of {I, II, III, ...} of all the
finitely sized members. We are led into a contradiction, in other
words, if we assume that there is an infinite subset of only finite
members and this contradiction, inasmuch as it is correctly
reasoned out, tells us to avoid the assumption. There is no
infinite subset of only finite numbers of {I, II, III, ...}.
*
If there is no infinite subset of finite numbers of {I, II, III,
...} then how can there be an infinite set of finite numbers N =
{1, 2, 3, ...} at all? In fact, is it such that once we speak of
"any" number, of an "arbitrary" number, then we are not in a
position to announced that this number is finite? And what is it,
if it is not finite?
Just to make clear how embarrasing this conclusion would be, let us
think of all the proofs, including Goedel's famous incompleteness
theorem, that explicitly uses a line of reasoning of this kind:
"We imagine that there is a natural number n such that P(n). This
implies:
.
.
.
We see that the assumption of a number n such that P(n) leads to a
contradiction, hence we cannot imagine that there is such a number
as n."
However, in all such proofs that I am aware of, it is taken for
granted that we can speak of "any" natural number in the sense that
we can achieve "any finite natural number". But what if the notion
of "any finite natural number" is itself a contradiction? Then the
above line of reasoning should be recast into the following:
"We imagine that there is a finite or not finite natural number n
such that
P(n). This implies:
.
.
.
In the case that we assume that n is finite, we see that P(n) leads
to a contradiction. We are led to assume that it either does not
exist, or that it is in some way a natural number that is not
finite -- and in this case, we may have to look at the definition
of P(n) and see if it has any defined behaviour; since we gave it
no such defined behaviour we must again look at our whole proof. In
the meantime, it seems that if n exists, it is other than finite."
Note: we have not at any point introduced a particular 'wonder'
number, rather, we have pointed out what may be the case for a
range of numbers of a certain kind, whose temperance is rather like
the decimals of transcendent numbers; and we do so from a humble
computer science spirit, not from the point of view of a
mathematician who has read all there is to read about this, and
proved everything there is to prove about this.
*
So let's make it better and better. Still anchored in the spirit of
research, of giving energy to investigating alternative hypotheses,
let us recall the two numbers 3.141519265358... and 5.1818181818...
and let us also recall that we had a procedure that would generate
any finite-digited decimal number; and that if we allowed this
procedure to proceed infinitely, we could imagine that it could
produce every infinite-digited decimal number -- that is, all the
real numbers, as they are called. The whole of R. We could then map
the whole of R, beginning with the finite decimal numbers and map
them to finite numbers in the list I = {I, II, III, ...}, and let
us now consider what in I that the infinite-digited decimal
numbers, as the two above-mentioned numbers, could be mapped unto.
Since any finite-sized n:(III...) is mapped to an equally finite-
sized limited decimal number, we must consider the case of numbers
n:(III...) where n is not finite. A number with an infinite series
of digits in R is mapped to a number of an infinite series of
digits in I. The question I wish to pursue, since I gave two
different numbers, 3.14159265358... and 5.1818181818..., is whether
the mapping somehow is conceptually equivalent or conceptually
nonequivalent.
This entails that we must ask something new of our minds. We must
actually imagine that the finite computational procedure, for
instance as carried out by the Java program given above, with
example listing, is really carried out to infinity. We must, as it
were, let the road that the program generates in performing onwards
and onwards, be continued into the horizon to infinity. Then we
must lift our gaze above the horizon and somehow ascertain the
sense of this road as a whole. We must not confine our visulizing
nerves to the domain of the finite -- NO MATTER HOW PLEASANT OR
COMFORTABLE THE FINITE REGION IS, due to our computational
procedures. We must ACTUALLY VISUALIZE THE INFINITE.
In so doing, in visualizing the infinite, we must ask: can we see,
somewhere, the number pi? The number pi is not entirely mysterious,
though it contains beautiful order. After all, it is a number, and
it is a number that is generatable in terms of nothing but a
permutation of digits. That is, when we permute all possible digits
at each position after the number '3' and the dot '.', as 3.nnn...,
where each n is 0...9 (say), then we we certainly get (also) to pi.
The fact that we cannot pick pi out without also having additional
information is not the issue at the moment; pi can be generated
algorithmically, after all. We are interested in the precise number
pi. We see that we have made a list and we have involved a sense of
a moment of suddenly going infinitely far in so doing. Pi does not
belong to anywhere which we can finitely pin-point; however, it is
clear that pi is comparable to the number 5.1818181818... and we
can say such things as: pi is less than this number. Let us call
the number 5.18181818... for pa. So pi < pa.
The set of all real numbers is unordered; but when we map the
making of a set stepwise to the integers, we get what we can call a
'list'. In this sense, each number in a list has a position. This
is exactly what we done with the Java program above. So, we see, it
generates more and more decimal numbers as it generates more and
more digits for these decimal numbers -- it works in the matrix
way, beginning in one corner and working onwards. Not at any
particular position we find 'position infinity' -- indeed the
concept is wrong, since infinity, properly speaking, involves
beyond-position-ness; not-limited-ness; not-finite-ness. Rather, it
is an act of thought, and it is in that moment of thought that the
infinite positions all belong. We see that since the number pi <
pa, then pi comes, in this Moment Of Infinity (MOI), to be
"before", in some sense, pa.
So, the position of pi is not "at" MOI, but it is "in" MOI, since
MOI is more like a dimension of duration, rather than a spot in
this dimension.
However, the position of pi in MOI is different than the position
of pa in MOI. It is before pa. And a number like the square root of
two is again before pi, also in MOI. The number [3.14] has a finite
numbered position; the number pi does not. We might give the
notation pi// to indicate that the position is in MOI, that the
position is associated with indefinite movement 'upwards and
rightwards' in the list
.
.
.
IIIII
IIII
III
II
I
We can then say: the position of pi in terms of a mapping to the I
= {I, II, III, ...} as made by a procedure as in the above Java
program is a number n//, not a number [n]. And this corresponds to
a number n//:(III...). In other words, when we speak of the size of
the number III... we can speak of the size as proper to MOI -- the
moment of infinity. That is not the same as saying that the
position 'is infinity'. For that sounds, doesn't it?, as something
definite, limited. It is precise, in that pi<<4.14159265358...<5.
14159265358...<6. 14159265358...., say, just to give some examples
of positions, but it is precise in a way that is beyond the
confinement of the [n] type of natural numbers.
*
However, if it is accurate to say that the very assumption of there
being an infinite subset of
.
.
.
IIIII
IIII
III
II
I
such that only finite-sized n:(III...) are included, then we cannot
properly conceptualize the kind of numbers of the [n] type. This
seems strange, that we begin by finite numbers to reach a number
concept in which we cannot properly any more talk about finite
numbers.
*
There are cases in the history of science when the startingpoint of
investigation, such as the theory of classical mechanics (as it is
called nowadays), which is researched upon so as to be modified
appropriately to accomodate new evidence, say, of subatomic
behavioural patterns, later is switched to be considered a special
case of the new kinds of theories that is being worked out. In
other words, it is a leading thought among many of the most
prominent quantum physicists that classical mechanics is to be
understood as a special case of quantum mechanics. However,
classical mechanics was originally thought to be something that
needs to be equipped with an extra understanding, rather than to be
revised into a special case of something else. (The debate on this
is not finished, by the way -- confer discussions on the possible
future role of the Correspondence Principle of quantum mechanics;
the Correspondence Principle concerns exactly this, in that it says
that quantum mechanics should do nothing except approximate
classical mechanics in situations where the energy concerned is
high; this is a statement made from the point of view that
"classical mechanics must be rescued", and that classical mechanics
is somehow of deep significance to what goes on in the environment
of the researching laboratory, even if the microscale energies
studied do not behave according to classical mechanics. However,
superconductors are example of macroscale situations in which
classical mechanics is wholly inapplicable; this calls into
question what the socalled Correspondance Principle should really
be understood to be -- perhaps a rule of thumb of construction of
quantum mechanics in a historical perspective...).
Let us still do our thought experiments on infinity very
diligently, and let us pay particular attention, with the just-
mentioned point in mind, to the contrast between 5.18181818... (the
number we called 'pa') and another number, which may call po,
6.18181818.... Imagine that the numbers of the n// kind is the kind
that we without contradiction feels to be our primary ground of
numbers, with the argument going towards this from two directions -
- one direction is from the thought experiment, in which the moment
of infinity, or MOI, contains a great deal of order and seems to be
perfectly called for in terms of the 'upper level existence' of the
set {I, II, III, ...}, and one direction is from the sense in which
it seemed, at least for a while, to be a contradiction to talk of
an infinite subset of only finite-width numbers of the same set {I,
II, III, ...}, which we have called I.
Let us now say that the difference between po and pa is an example
of how we may define the number One. Yet another number, perhaps
called 'py', like 7.18181818..., has the difference to pa, such
that pa-py can be how we may define the number Two. In this way, we
can achieve a definition of finite numbers without trying to impose
the notion of extracting an infinite subset where it seems
difficult to impose such a notion; rather, we can define it by
means of an established substraction operator operating on real
numbers.
If we generalize the substraction operator so it can handle not
just infinite-decimal numbers, but also the n// type of numbers, we
can reach something like a definition of finite numbers by means of
the n// numbers -- which may seem preferable, in many context, to
starting with real numbers. We would then imagine that, say, the
position of pi in the list where we map the real numbers by the set
I is given by one number n//, and that there would be another
position, the number m//, such that their difference is something
like our number One. In order to get a sense of such numbers, we
might want to invent a way of writing them so that the typical ten-
digit number system, with the same order of priority -- the least
significant digits to the right -- is adhered to. Then, if we have
an n// style number of the kind
...XXXXXXXXXXXXXXXX1
and another number, where the X's stand for exactly the same digit
series,
...XXXXXXXXXXXXXXXX2
then we could "define" One to be the difference m// - n//. I
mention this very tentatively, fully aware that we have not defined
any value for the substraction operator in dealing with entities
that are not finite.
*
If the above reasonings and musings and thought experiments, the
above denials and affirmations, carry any power of coherence at all
-- and I beg your pardon if you don't think it does -- then we
should be able to see an interesting feature of the numbers in I,
namely that
III...
and
III...
might be two different numbers. For we have above indicated how
numbers may be infinite in terms of a digit series that has a
beginning (or an ending, but not both), such as ...XXX2 and
...XXX1, whose difference corresponds to what we may define as a
finite number. If we wish to write this as ...III or III..., then
we are led to say the following: even though the three dots
indicate MOI, -- the Moment Of Infinity -- these three dots do not
by themselves give exhaustive information. Rather, they suggest
that the numbers belong a range.
*
Perhaps everything above hings on a hidden false assumption. If it
does, then this treatise so far has been an exploration of the
consequences of taking a false assumption seriously -- rather like
asserting something that is wrong in terms of an axiom, and see how
it goes; keeping going until either a contradiction arises, or it
becomes all to incoherent to feel worthwhile to continue. In the
spirit of research, I say: of course this is possible, that it has
been the case all along. Of course, especially since we are
contradicting the typical way of talking about infinity as I have
understood it. For we have not introduced one single infinity to
account for the size of the set of natural numbers. Rather, we have
endavoured to argue, like a radical politician against all the
others, perhaps, that the set of natural numbers is greater than
what we might imagine at first. Or else that it is a contradiction
to assume that it exists, if only finite members are admitted.
Therefore, when we speak of infinity as MOI, the moment of
infinity, we speak of a range of what we might call the n//
numbers. I wish to say that if we take this line of reasoning
seriously, we ought to give this style of numbers a name -- and not
the name "transfinite" or "ordinal" or anything that has been
associated with those who have taken for granted the ease by which
we may assert that {1, 2, 3, ...} can be superceded in size.
Boldly, I suggest -- also for the sake of speed in research into
this assumption -- the term "essence" number; suggesting that the
finite natural numbers ought to be defined by means of comparison
of the essence numbers, but with care taken not to assume that it
is easy to speak of an infinite set of finite natural numbers for
the reasons given, on the direction of growth in case it is written
{I, II, III, ...}. That is, we may speak of finite numbers, but we
must be aware that the finite numbers "become" essence numbers as
soon as we let this process go to infinity. That is, it is not
easy, though imaginable, to confine the group of finite natural
numbers. This may seem a strange statement, but if we accept that
something we have thought of as easy to be "difficult", then
something we may have thought of as "difficult" may become easy.
It is easier, then, to continue the line of research, to assume
that we have a set -- let us call it E, for Essence Numbers -- of
all essence numbers -- than to have a set N of natural finite
numbers. Exactly how to write E is not my concern at the moment,
but we might indicate it this way: 1, 2, 3, ...E...
That is, E properly "is" the region of 'going to infinity'. E is
MOI, the moment of infinity. It is not 1, 2, or 3, nor 7, 14, or
238527693. Whatever number we write of the finite kind, this finite
number can at best be a construction equipment to "get there",
rather as a rocket must leave its ramp. The rocket, when it
actually progresses without any limit at all, exists in a dimension
which is the E dimension proper. And the countability questions
should be deferred to be a question of mapping with E such
described, rather than N; and in this case, it seems that
everything we have so far looked into, including the set of all
subsets of numbers P(N), the set of all real numbers R, are E-
countable.
Vague proposition 1:
The set E, which belongs to the moment of infinity indicate by the
three dots when we write 1, 2, 3, ..., but not so that any finite
number belongs to it, can be easier defined that the set of finite
natural numbers.
Open definition 1:
"Essence number": any members of E.
Open proposition 2:
R is E-countable.
Corrollary 1 to Prop 1:
The notion of "any natural number n", when it is taken for granted
that n is finite, should be cleared up somehow.
Corrollary 2 to Prop 1:
Finite numbers should be defined by means of essence numbers.
Corrollary to this:
Proofs of "uncountability" and "incompatibility" concerns merely
the limitations of a finiteness that we do not easily have in any
case; and as such, there are options of "countability" and
"computability" all over the place, at least in the sense that the
original proofs of the reductio ab absurdum case involving negation
of existence of (finite) natural numbers n to fulfill P(n) no
longer can be said to have the same strength as before.
Open conclusion
Infinity is a concept that is not easily superceded.
*
It is interesting that with the approach taken, we have come back
to the sense of the infinite more or less as given in any typical
dictionary, including this dictionary of synonyms from 1942:
infinite. Infinite, eternal, semipiternal, boundless, illimitable,
uncircumscribed agree in meaning having neither beginning nor end
or being without known limits of any sort.
-- Webster's Dictionary of Synonyms, First Edition, G. & C. Merriam
Co., Publishers, Springfield, Mass., USA 1942.
*
In other words, in common sense use -- in daily life language --
the notion of the infinite is such that it can, in general, in no
way be superceded as to limits. The infinite is the quintessential
concept of that which has no limits. Why, then, was it regarded as
such an act of genius in mathematics to consider that the
difficulty of ordering the real numbers or the difficult of
ordering the set of subsets of natural numbers amount to an
impossibility? It is a cardinal, not to say ordinal, mistake to
regard something as impossible when it is merely extremely
difficult. There is a huge jump between zero probability and near-
zero probability -- to use a statistical term.
Perhaps one of the reasons why the new notions of several
'infinities' in mathematics came to be so readily accepted was
because of the notion of the concept of the 'limit', as applied to
the approach to a finite number, replaced the notion of an
'infinitesimal'. The idea of going as close 'as we wish' to zero
in, say, 5/x, as regards x, always and in each case gives us a
finite result. It was felt that this adequately could deal with
situations where we in fact were interested, not merely in being
near zero, but in having x equal to zero. The convention came
about, that we can speak of 'going closer' as nothing more than
'going a finite step closer', rather than going all the way.
Of course, it is radically different to merely go 'as far as we
please' in a finite way to zero in 5/x, and to go there all the
way. Or, to be more accurate, 'as far as we please' is a term that
involves a certain arbitrariness, and this arbitrariness, if taken
seriously, means going all the way. If it doesn't, it is not 'as
far as we please'. Going all the way is, however, conceptually
different and involves something entirely new. It seems, then, that
if we are tempted to accept the notion of a 'limit' combined with
the notion of 'as far as we please', near to a finite number, then
it also becomes tempting to disregard the upper level members of a
set like
.
.
.
IIIII
IIII
III
II
I
for here, too, if we go merely finitely long, we have finite
members; but if we actually want the FULL set {I, II, III, ...}
then of course there is no sense in which the concept of the
'limit' will do favour to this; the full set involves a unique
Moment Of Infinity, a MOI, in which something conceptually new
comes into being. Namely not-finite-width numbers.
So it may be, psychologically-historically, in the mathematical
tradition, that the confusions, discussions and compromises arising
from giving up infinitesimals and going to the notion of 'as near
as we please, but not there', crept into the notion of set theory
when a full infinite set is considered; from there, the pathway was
short to ask of this infinity whether it is not too limited -- and,
of course, by sloppy reasoning about the actual constitutents of
the set in the first place, it would quickly be found to be
insufficient; and then we had a new series of infinites going;
applying the same sloppiness to this new series, there would be no
end to the need of the 'transfinite' or ordinal.
*
It is possible to take another approach to the whole issue that
that which penetrates most of the earlier reasoning in this
treatise. And that is the stance, more or less, indicated by
especially David Hilbert, however modified with modern insights.
Hilbert, as is part of the common lore of mathematics, made himself
known as head of a 'programme' to reduce mathematics to a rule-
based logical foundation. It was this programme that came to a
rather abrupt end as Kurt Goedel did the masterpiece of showing the
inherent contradictions in the common assumptions that Hilbert and
others applied. Goedel did not invent a new type of numbers or
something like that; he merely pointed out that given the type of
assumptions implied in our theories of numbers, then, by means of
creative use of mapping of formal axiomatic systems to describe
integer arithmetic (etc) to whole numbers by means of prime number
factorization and primitive recursive functions, we are led to
assume that there are correct but unprovable statements. Our
criticism in the foregoing treatise is in now way attacking Goedel,
but rather pointing out that there may be more to natural numbers
than what has become the stigma of natural numbers, so to speak.
Let us now take a stance closer to Hilbert, but modified to
accomodate fresh perceptions. This stance says: we are interested
in that which we can prove, and we are interested in seeing how
rules of logic, or of computing somehow, can give us certain
results; and let us say: this is mathematics; mathematics is not
about the perception of eternal ideas lying out there in a platonic
sphere of the mysticists; rather, it is about making algorithms, we
might say, and that includes talking about these algorithms and
making algorithms that correct and investigate and analyze other
algorithms inasmuch as it is possible to do so. Also, we want to be
able to write these algorithms or route procedures or whatever on
paper, not wanting infinite paper, we stick to finite algorithms.
At least in this sense, the word 'finite' makes certainly sense.
In this stance, then, mathematics becomes almost a branch of
computer science, as computer science stands today. Mathematics no
longer is an issue of the cognitive informal intuitive perception
of relationships between imagined structures in a possibly infinite
space, but rather a question of the handling of formalisms of a
finite nature according to rules, which we may change and permute
and so on, but all the time they do have the feature of being of a
finite amount. Of course, Kurt Goedel was careful in pointing out,
in his 1931 article, that his incompleteness result concerns finite
systems, that is, finite axiomatic system with at least adequate
complexity to contain multiplication and addition with whole
numbers, and do some primitive recursive functions on these. Any
more complex system will be touched by this. It has been a strange
result, in this region, that it appears that the axiomatic system
relating to the real numbers, not defining the whole numbers, is
having less complexity, and can in fact have a sense of
completeness. This little oddiness apart, the result by Goedel
stand there, strongly. It does suggest that some creative ability
of our minds would be greatly called for in the changing and
reordering of our finite axiomatic schemas; it does not, by itself,
suggest that we must from now on engage in 'infinity
perceptiveness', however much we may like to interpret it in this
or another way.
If we take this stance, that mathematics concerns looking at
consistencies and contradictions concerning arbitrary but finite
lists of rules and their permutations, such as in axiomatic
systems, or in terms of programming languages (in which the rules
in their explicit form can be anything whatsoever that looks like
something like a machine can do, such as substituting 'x' for 'y',
and they need not be exactly that which is called 'formal logic';
however formal logic may be used to describe these rules, for
example), then we do not come anywhere near the question of having
to look into our minds as to the relative position of a mapping of
R against E in a domain of infinite-sized essence numbers. We can
then stick to the rules, try them out on machines occasionally,
drink our coffee, eat our lunch, report the consistencies and
inconsistencies, and go home after a successful day of Mathematical
Work.
*
However, if our stance is that mathematics is nothing but this
finite rule permutation thing, then it is clear that we must not
anymore think that we are within that scheme when we talk of
infinite sets. In particular, we must abstain from asserting things
with self-assurance about such sets as N = {1, 2, 3, ...}, since
this set, however much it is generated by a finite algorithm, is
itself not what we can write out; and so we must assume that the
finite algorithm is performed infinitely, but since this is outside
of what we can ask our machines to do, we cannot talk about it with
such ease and grace as to what happens when we perform this
algorithm, for instance:
int i=1;
int j=1;
int temp;
System.out.println("Fibonacci: " + i + " " + j + " ");
for (int p=1; p<=55; p++)
{
temp = j;
j = i+j
i = temp;
System.out.println(j + " ");
}
We can talk of: what would happen when we run this algorithm and
insert 155 instead of 55? 'Oh, very interesting, that would give
this and that result.' But what if we produce this notion:
Concept MOI = ...;
int i=1;
int j=1;
int temp;
System.out.println("Fibonacci: " + i + " " + j + " ");
for (int p=1; p<=MOI; p++)
{
temp = j;
j = i+j
i = temp;
System.out.println(j + " ");
}
Then we we might imagine that we should protest and perhaps utter
something to the extent like this: please, don't upset our ideas,
you are no longer doing mathematics -- you have introduced a not
finitely computable concept here, MOI! This is not mathematics,
this is science fiction, fantasy, psychology -- or worse, it is
parapsychology. This is not something we can write on paper. Mind-
work! How terrible!
The reply to this is, of course, along the lines that Karl Popper
suggested: Unless our ideas are not vulnerable, they are not
scientific. The task of science -- any science, also mathematics --
is to expose ideas to criticism, not shield them from it.
*
In that humble spirit I submit these thoughts, which can be summed
up in one sentence: there is perhaps more to infinity.
More than what? That is exactly the point: more than anything we
can say.
****** EXAMPLE: AN ELIZA PROGRAM WRITTEN IN LISA GJ2
((DATA THE-SET-OF-POSSIBLE-RESPONSES 80 ; 100 => RAM-PM
THE-SET-OF-POSSIBLE-RESPONSES <<ASSERT ))
((DATA QUANTITY-OF-POSSIBLE-RESPONSES
QUANTITY-OF-POSSIBLE-RESPONSES => SETBASIS ))
((DATA LIST-OF-TEN-POSSIBLE-RESPONSES 10 => %MAKE
LIST-OF-TEN-POSSIBLE-RESPONSES <<ASSERT ))
(( VARSTRING THE-QUESTION }} ; THE-QUESTION => >VARSTR ))
((DATA STATISTICAL-MATCHES ))
((DATA MATCHES-INDEX ))
((DATA STATISTICAL-MATCHES-DB-POINTERS ))
((DATA WORDS-SET-1 80 ; 80 => RAM-PM WORDS-SET-1 <<ASSERT ))
((DATA WORDS-SET-2 80 ; 80 => RAM-PM WORDS-SET-2 <<ASSERT ))
((DATA QTY-IN-WORDS-SET-1 ))
((DATA QTY-IN-WORDS-SET-2 ))
(LET GET-WORDS-SET BE (( & ; ))
((
(( # => CLEAR-MATRIX ))
(( 0 => >N6 ))
(( ; => LENGTH
(COUNT
(( ; N1 => CHN ; 32
=> EQN => NOT
(MATCHED
(( INCN8 ))
(( N8 ; 50 => INTGREATER = RM N8 ; RMS RMS && ; EXIT === ))
(( 0 => >N4 ))
(( GOLABEL4: ))
(( ; N1 ; N4 => ADD => CHN ; N4 INC ; N8 ; # => PM ))
(( INCN4 ))
(( N1 N4 ADD ; 79 => INTGREATER = RM N8 ; RMS RMS && ; EXIT === ))
(( ; N1 ; N4 => ADD => CHN ; 32 => EQN => NOT = GOUP4 === ))
(( N1 ; N4 => ADD => >N1 ))
MATCHED) ))
COUNTUP) ))
(( ; RM ))
(( N6 => => ))
(( && ))
)) OK)
(LET COUNT-MATCHES-OF-TWO-WORDS-SET BE (( ))
((
(( WORDS-SET-2 >>> => & ))
(( WORDS-SET-1 >>> => & ))
(( 0 => >N7 ))
(( QTY-IN-WORDS-SET-1 >>> => >N4 ))
(( QTY-IN-WORDS-SET-2 >>> => >N6 ))
(( N6
(COUNT
(( N1 ; ## => MATR>TEXT ; ))
(( N6
(COUNT
(( ; => ONE ;
N1 ; # => MATR>TEXT => EQ
(MATCHED
(( INCN11 ))
(( RMS ; RMS ; GOFORWARD4 ))
MATCHED) ))
COUNTUP) ))
(( GOLABEL4: ))
(( ; RM ))
COUNTUP) ))
(( N7 => => ))
(( && ; && ))
)) OK)
(LET ADD-THIS-TO-SENTENCE BE (( & >N6 >N5 ))
((
(( # => ISBASIS ;
N5 => ISBASIS ; N6 => ISBASIS => TRIPLEORR = && ; EXIT === ))
(( N6 ; # => %LENGTH => INTGREATER = && ; EXIT === ))
(( N6 ; # => %GET => & ))
(( # => %LENGTH => ISBASIS = && ; && ; EXIT === ))
(( # => %LENGTH
(COUNT
(( N1 ; # => %GET => ISBASIS = RMS RMS ; && && ; EXIT === ))
(( N7 ; N1 ; # => %GET => ADD ;
N1 ; # => %PUT ))
COUNTUP) ))
(( && ; && ))
)) OK)
(LET >>>E BE (( ; => LENGTH => >N5 ))
((
(( THE-SET-OF-POSSIBLE-RESPONSES >>> => & ))
(( N5 => ISBASIS = RM ; && ; EXIT === ))
(( QUANTITY-OF-POSSIBLE-RESPONSES => INCVAR ))
(( ; QUANTITY-OF-POSSIBLE-RESPONSES >>> ; # => TEXT>MATRIX ))
(( -1 ; QUANTITY-OF-POSSIBLE-RESPONSES >>> ; # => ADD-THIS-TO-SENTENCE ))
(( && ))
)) OK)
}UIJT!JT!BO!JNQPSUBOU!QPJOU-!NBZCF//} >>>E
}J!IBQQFO!UP!TFOTF!UIBU!ZPV(SF!TPPO!UPVDIJOH!PO!TPNFUIJOH//} >>>E
}QFSIBQT!ZPV!BSF!UPVDIJOH!PO!TPNFUIJOH//} >>>E
}XPOEFSGVM"!ZPV!DFSUBJOMZ!!D!B!O!!!FYQSFTT!UP!ZPVSTFMG//} >>>E
}ZPV!FYQSFTT-!CVU!BMTP!HJWF!BUUFOUJPO!UP!IFBM!XJUIJO//} >>>E
}TP!ZPV!DBO!TBZ-!TP!ZPV!DBO!BTL-!CVU//} >>>E
}JT!JU!UIBU//POF!UIJOH!JT!BMM!UIJT!EPVCU-!BOPUIFS!UIJOH;!GBJUI//} >>>E
}ZPV!!E!F!M!J!C!F!S!B!U!F!M!Z!!!!CSJOH!VQ!IVNPVS-!BT!OFX!BUUFOUJPO//} >>>E
}UIF!UXP!XBZT!UP!IPME;!UP!DMVUDI!GFWFSFOUMZ-!PS!QMBZGVMMZ!MJGU//} >>>E
}PO!TFFJOH!UIJT-!XIBU!EP!ZPV!UIJOL!..!XJTEPN!DBO!TVSFMZ!DPNF!OPX//} >>>E
}JO!TPNF!XBZT-!JO!UIF!WBTU!FYQBOTF!PG!UJNF-!UIFSF!JT!B!SPMF//} >>>E
}WBMVF!PG!CFJOH!QSJWBUF!JT!UP!EFGVTF!GSVJUMFTT!FNPUJPOT//} >>>E
}BOE!UIJT!BTTPDJBUFT!UP//} >>>E
}UIJT!TVSFMZ!DBO!CF!FYQSFTTFE!JO!XBZT!XIJDI!DBMM!PO!NPSF!HFOFSPTJUZ//} >>>E
}BOE!FWFO!ZFU!NPSF!HFOFSPVTMZ-!UIF!XBZT!ZPV!DBO!TBZ!JU!BSF//} >>>E
}CFBVUZ-!MPWF-!USVUI-!HPPEOFTT-!DPNQBTTJPO-!GFBSMFTTMZ!SJHIUFPVT//} >>>E
}ZPV!IBWF!UP!CF!TPGU!BU!UIF!SJHIU!NPNFOUT-!IBSE!UP!EFGFOE!HPPEOFTT//} >>>E
}TP!JO!BMM!UIJT-!TVSFMZ-!BUUFOUJPO!UP!GFBSMFTT!SJHIUFPVTOFTT//} >>>E
}TP!JO!BMM!UIJT-!XJUIPVU!HVJMU-!MFBWJOH!TFOUJNFOUBMJUZ-!HPJOH!BIFBE//} >>>E
}UIFSF!JT!B!TVSFOFTT!XIFO!B!HMPX!PG!HPPEOFTT!JT!JO!TUPNBDI//} >>>E
}UIF!GVUVSF!..!PG!TPVM-!TVSFMZ-!FWFS.OFX!CPEJFT!..!NBUUFST!UP!BMM//} >>>E
}UJNJOH!PG!B!EBODF!NFBOT!NBLJOH!UIF!EBODF!HPPE//} >>>E
}ZPV!MFU!HP!PG!XIBU!ZPV!NVTU!MFU!HP!PG!XIFO!UJNF!JT!SJHIU//} >>>E
}BOE!UIFSF!JT!B!USVTU!UIBU!UIFSF!JT!B!GBJSOFTT!JO!FTTFODF//} >>>E
}GJOE!B!TJMFODF!BMTP-!B!TXFFU!TJMFODF!JT!BMXBZT!QPTTJCMF!UP!QSBZ!GPS//} >>>E
}GJOE!UIF!HPPE!NVTJD!BMTP-!UIF!NVTJD!XIJDI!QPJOUT!UP!MPWF//} >>>E
}MPWF!EPFTO(U!IBWF!UP!DBSSZ!B!GJYFE!GPSN!JNBHF!PG!B!OFJHICPVS//} >>>E
}XF!BSF!OPU!EVTU!JO!HPE(T!FZFT-!CVU!TPNFIPX-!UIFSF!JT!B!IJHI!WBMVF//} >>>E
}IPX!DBO!POF!GJOE!UIBU!XIJDI!JT!HPE!GPS!HPPE!JO!UIJT!BMTP//} >>>E
}JO!CFJOH!USVMZ!HPPE!BSF!ZPV!OPU!JO!DPOUBDU!XJUI!HPE-!NPSF!BOE!NPSF//} >>>E
}UIFO!JO!CFJOH!SFMBUJWFMZ!HPPE-!GSFF!GSPN!BUUFNQU!UP!BCTPMVUJ[F//} >>>E
}SFMBUJWFMZ!HPPE-!SFMBUJWFMZ!IBQQZ-!TVSFMZ!UIBU!DPNFT!NPSF!BOE!NPSF//} >>>E
}UIFSF!JT!B!EBODF!PODF!UIF!TFOTF!PG!FHP.DMPVE!JT!QVODIFE!BXBZ//} >>>E
}POF!EPFTO(U!IBWF!UP!QVODI!PUIFST-!JG!POF!DBO!XBJU!BOE!QSBZ!GPS!IFBM//} >>>E
}UIJOHT!BSF!TPNFUJNFT!XPOEFSGVM.GFFMJOH-!PUIFS!UJNFT!OPU//} >>>E
}TP!UIF!MPOH!TXFFU!HPBM-!GPS!BMM-!KPJOT!BMM-!OP!NBUUFS!XIBU!OPX//} >>>E
}UIFSF!JT!OP!HSPVQ!NPSF!JNQPSUBOU!UIBO!BMM!TPVMT-!JO!HPE(T!NJOE//} >>>E
}CFJOH!SFMJHJPVT!EPFTO(U!SFRVJSF!BUUBDINFOU!UP!B!GJYFE!TUBUJD!JEFPMPHZ//} >>>E
}GJOEJOH!QFBDF!XJUIJO!..!JT!JU!OPU!B!NBUUFS!PG!UPUBM!TFDSFU!GPSHJWJOH//} >>>E
}ZPV!MJTUFO!XFMM!XIFO!ZPV!XBML!TMPXMZ!BU!TPNF!TUBHFT-!GBTU!BU!PUIFS//} >>>E
}ZPV!XBML!BIFBE!JOUP!UIF!GVUVSF!XJUI!XFMM.NBTTBHFE!GFFU//} >>>E
}JO!EPJOH!ZPHB-!BMTP-!ZPV!NBTTBHF!ZPVS!GFFU-!GFFMJOH!ZPVS!GVUVSF//} >>>E
}BOE!TVSFMZ!UIFSF!BSF!HPMEFO!HMJNNFST!PG!SBEJBOU!HPPEOFTT!JO!BMM//} >>>E
}BOE!TVSFMZ!BMM!IVNBO!CFJOHT!BSF!HPPE!BU!CPUUPN//} >>>E
}BOE!TVSFMZ!UIFSF!JT!OP!TUSFOHUI!HSFBUFS!UIBO!LJOEOFTT//} >>>E
}BOE!TVSFMZ!ZPV!BSF!MPWFE!CZ!BMM!ZPVS!GFMMPX!IVNBO!CFJOHT!JO!IFBSU//} >>>E
}BOE!TVSFMZ!OPU!BMM!MPWF!OFFE!FYQSFTTJPO!QIZTJDBMMZ//} >>>E
}BOE!TVSFMZ!UIF!CFBVUJGVM!NPNFOUT!DBSSZ!PO!FWFO!BT!QFSTPOT!TIJGU//} >>>E
}UIFSF!JT!B!EFQUI!CFZPOE!UIF!NBTL-!CFZPOE!UIF!QFSTPO-!UP!FBDI//} >>>E
}BOE!ZPV!MJTUFO!UP!UIF!NPUJWFT!CFZPOE!UIF!BDUJPOT-!BOE!VOEFSTUBOE//} >>>E
}JU!JT!IBSE!UP!IBUF!UIBU!XIJDI!JT!GVMMZ!VOEFSTUPPE//} >>>E
}GJOEJOH!B!MJUUMF!USVUI!JO!QMBZGVMOFTT-!UZQJOH!UIF!XPSE!#HPPEOFTT#//} >>>E
}HPPEOFTT-!DPNF!PO"""!..!ZPV!BGGJSN!TVDI-!EFIZQOPUJTF!GBMTFOFTT//} >>>E
}ZPV!TVSFMZ!QSBZ!UP!HPE!NVDI-!SJHIU-!JNQSPWJTJOH!FBDI!UJNF//} >>>E
}ZPV!TVSFMZ!QSBZ!UP!HPE!NVDI-!SJHIU-!JO!IVNCMF!BOE!NPEFTU!XBZT//} >>>E
}ZPV!TVSFMZ!QSBZ!UP!HPE!NVDI-!SJHIU-!JO!XBZT!OPCMF!BOE!XIPMF//} >>>E
}UIJT!XPSE!#HSBUFGVMOFTT#-!JU!JT!!!T!P!!!!MPWFMZ-!DBO!CF!SFQFBUFE//} >>>E
}XSJUJOH!UP!POFTFMG!BCPVU!CJH!HPPE!WBMVFT!NBZ!SFMJFWF!POF!PG!TUSBJO//} >>>E
}TFFJOH!TNBMM!UIJOHT!BT!TNBMM!SFRVJSFT!UIF!MJGUJOH!PG!QFSTQFDUJWFT//} >>>E
}UIF!CJHHFTU!CFBVUJGVM!NPTU!OPCMF!BDUJPO!SJHIU!OPX-!XIBU!JT!UIBU//} >>>E
(( LOOKSTK ))
(LET GIVE-REACTION-TO-NIL BE (( ))
((
(( }(at any time, you can exit this program by typing EXIT} => POP ))
(( }or just the letter X or by resetting the computer.)} => POP ))
)) OK)
(LET GET-WORDS-SET-FOR-QUESTION BE (( ))
((
(( WORDS-SET-1 >>> => & ))
(( THE-QUESTION => VARSTR> ; # => GET-WORDS-SET
QTY-IN-WORDS-SET-1 <<ASSERT ))
(( && ))
)) OK)
(LET GET-WORDS-SET-FOR-THIS-POSSIBLE-RESPONSE BE (( >N6 ))
((
(( THE-SET-OF-POSSIBLE-RESPONSES >>> => & ))
(( WORDS-SET-2 >>> => & ))
(( N6 ; ## => MATR>TEXT => EXTRACTLETTERS ;
# => GET-WORDS-SET
QTY-IN-WORDS-SET-2 <<ASSERT ))
(( && ; && ))
)) OK)
(LET STORE-STATISTICAL-MATCHES-FOR-ALL-POSSIBILITIES BE (( ))
((
(( STATISTICAL-MATCHES-DB-POINTERS >>> => & ))
(( STATISTICAL-MATCHES >>> => & ))
(( MATCHES-INDEX >>> => & ))
(( QUANTITY-OF-POSSIBLE-RESPONSES >>> => >N4 ))
(( # => %LENGTH
(COUNT
(( N1 ; N1 ; # => %PUT ))
(( 0 ; N1 ; ## => %PUT ))
(( N6 => >>V => FR => GETV ; N1 ; ### => %PUT ))
COUNTUP) ))
(( GET-WORDS-SET-FOR-QUESTION ))
(( # => %LENGTH
(COUNT
(( N1 ; # => %GET ; ### => %GET =>
GET-WORDS-SET-FOR-THIS-POSSIBLE-RESPONSE ))
(( COUNT-MATCHES-OF-TWO-WORDS-SET ; N1 ; ## => %PUT ))
COUNTUP) ))
(( && ; && ; && ))
)) OK)
(LET SORT-STATISTICAL-MATCHES BE (( ))
((
(( STATISTICAL-MATCHES-DB-POINTERS >>> => & ))
(( STATISTICAL-MATCHES >>> => & ))
(( MATCHES-INDEX >>> => & ))
(( ## ; # => QSORT2 ))
(( && ; && ; && ))
)) OK)
(LET EXTRACT-TOP-TEN-STATISTICAL-MATCHES-TO-HERE BE (( >N7 ))
((
(( STATISTICAL-MATCHES-DB-POINTERS >>> => & ))
(( STATISTICAL-MATCHES >>> => & ))
(( MATCHES-INDEX >>> => & ))
(( # => %REVERSE ))
(( 10
(COUNT
(( N1 ; # => %GET ; ### => %GET ;
N1 ; N9 => %PUT ))
COUNTUP) ))
(( && ; && ; && ))
)) OK)
(LET MAKE-LIST-OF-TOP-TEN-STATISTICALLY-SUITABLE-RESPONSES BE (( & ))
((
(( STORE-STATISTICAL-MATCHES-FOR-ALL-POSSIBILITIES ))
(( SORT-STATISTICAL-MATCHES ))
(( # => EXTRACT-TOP-TEN-STATISTICAL-MATCHES-TO-HERE ))
(( && ))
)) OK)
(LET ACQUIRE-A-SYNCHRONISTIC-SOLARIS-NUMBER BE (( ))
((
(( SOLARIS => INC ; SECONDS => INC => ADD => => ))
)) OK)
(LET ACQUIRE-AN-RFFG-NUMBER BE (( ))
((
(( ^77777777 FR GETV => => ))
)) OK)
(LET GET-A-NUMBER-BETWEEN-1-AND-10-FROM-THIS BE (( >N4 >N6 ))
((
(( N4 ; N6 => ADD ; 10 => MOD => INC => => ))
)) OK)
(LET GET-APPROPRIATE-RESPONSE BE (( ))
((
(( THE-SET-OF-POSSIBLE-RESPONSES >>> => & ))
(( LIST-OF-TEN-POSSIBLE-RESPONSES >>> => & ))
(( # => MAKE-LIST-OF-TOP-TEN-STATISTICALLY-SUITABLE-RESPONSES ))
(( ACQUIRE-A-SYNCHRONISTIC-SOLARIS-NUMBER => >N5 ))
(( ACQUIRE-AN-RFFG-NUMBER => >N8 ))
(( N5 ; N8 => GET-A-NUMBER-BETWEEN-1-AND-10-FROM-THIS => >N6 ))
(( N6 ; # => %GET ; ## => MATR>TEXT => => ))
(( && ; && ))
)) OK)
(LET PREPARE-STATISTICAL-MATCHES-INDEX BE (( ))
((
(( QUANTITY-OF-POSSIBLE-RESPONSES >>> => ONE => ONE => ; ; ; ))
(( ; => %MAKE STATISTICAL-MATCHES-DB-POINTERS <<ASSERT ))
(( ; => %MAKE STATISTICAL-MATCHES <<ASSERT ))
(( ; => %MAKE MATCHES-INDEX <<ASSERT ))
)) OK)
(LET ELIZA BE (( ))
((
(( L ; FUNCURSOR ))
(( }WELCOME TO ELIZA!} => POP ))
(( CRLN ))
(( PREPARE-STATISTICAL-MATCHES-INDEX ))
(( }ANYTHING ON YOUR MIND? (YOUR REPLIES ARE NOT STORED.)}
=> POP ))
(( }(THE RESPONSES ARE READY-MADE, PICKED A LITTLE BIT}
=> POP ))
(( }BY CHANCE, TO ENCOURAGE YOUR CHATTING TO YOURSELF.}
=> POP ))
(( }THIS PROGRAM SAVES NOTHING, REQUIRES NO NETWORK CONTACT.}
=> POP ))
(( }EXIT ANYTIME YOU LIKE BY REBOOTING THE COMPUTER, OR}
=> POP ))
(( }TYPING THE WORD EXIT OR JUST THE LETTER X.)}
=> POP ))
(( GOLABEL4: ))
(( CRLN ))
(( }TYPE IN SOMETHING, PLEASE (ONE LINE)===>>>} => POP ))
(( READLN => UPCM ; 79 => SETMAXLEN => EXTRACTLETTERS ;
THE-QUESTION => >VARSTR ))
(( L ; SOLIDCURSOR ))
(( THE-QUESTION => VARSTR> => RMBL => TOLEN => ISBASIS
(MATCHED
(( GIVE-REACTION-TO-NIL ))
)(OTHER
(( THE-QUESTION => VARSTR> => ONE ; }X} => EQ ;
SWITCH ; }EXIT} => EQ => ORR = GOFORWARD2 === ))
(( GET-APPROPRIATE-RESPONSE => POP ))
MATCHED) ))
(( 79 => MAKEBL ; THE-QUESTION => >VARSTR ))
(( GOUP4 ))
(( GOLABEL2: ))
(( RAM-REFRESH ))
)) OK)
(( LOOKSTK ))
(LET AUTOSTART BE ELIZA OK)
****** WHAT I CALL "NEOPOPPERIANISM", TO REPLACE POPPERIANISM
The vulnerability of thought to checking is in focus in
many of Karl R Popper's works, for instance, The Open
Society and Its Enemies. As I've pointed out, e.g. in
my book from 2004, there is a massive amount of hidden
assumptions not only in his works, but in all those who
lean on him, and/or on Rudolf Carnap, and/or on Bertrand
Russell, -- as a gathering term one can call this
'popperianism' -- and those hidden assumptions conceal
that there is a very clearly limiting worldview in-built
into the conception of how checking is limited to
sensory input apparatus. Those hidden assumptions
conceal, in effect, a belief that each individual has
a mind which is sharply divided, in a localist and
pre-quantum sense, from all in reality -- except to
sensory organs.
Whether or not those massive hidden assumptions are
correct or not cannot be investigated as long as they
are not admitted to exist. In hiding themselves from
attention, they attain a status of the irrefutable.
And the irrefutable, according to the writings of
Karl R Popper, has no place in science. That is to
say, then, in effect, popperianism has no place in
science. Q.E.D.
However, if we extract, eclectically, the obviously
interesting, valuable and truthful notion that
the checking or testing against something other
than thought, of a thought, -- the thought being
in this sense 'checkable' (a more positive term
than 'refutable') -- then we arrive at neopopperianism
in its cut'n'dried sense.
This is how I define neopopperianism at the outset:
it is free from the burden of the localist
metaphysics and indeed from any other metaphysics.
It is rather a grand pondering, a metaphysical
question, associated with that of relating what
thought points to with that which it points to,
in a manner which gives a priority to that
which it points to, so as to have an impact
on the thought which points -- providing instances
of confirmations and instances of disconfirmation.
In the evolved, let's say enlightened sense --
the word "enlightenment" now meaning, 'being a
light unto oneself, having a light in (en) oneself'
-- and not referring either to bygone supposedly
enlightened masters nor bygone supposed enlightened
historical movements -- neopopperianism involves
what I will here name 'the intuitive necessities'.
Moreover, it involves giving the intuitive
necessities free reign, dissolving those thoughts
which represents twarted perceptions, as well as
those thoughts which hardly represent any
perceptions at all.
As a progressive, slow pathway to (for humans,
relative) enlightenment (for that which is
absolute must be conferred to the origin of
all only, in a Bishop Berkeley sense), the
discovery of the ways thought can trick itself
into maintaining the pretense of perceptions,
very fast and subtly, by still faster and
still more subtle tricks -- so as to dissolve
the first set of tricks -- is part of the
distinctly complicated work a human being
must undertake to come to see what indeed
is an intuitive necessity, and what works
which, as a whole, are composed fully and
solely of intuitive necessities and which
is as a whole an intuitive necessity.
The Lisa GJ2 language contains no element,
I submit for the personal intuitive
introspection and verfication of each,
which is not an intuitive necessity and
nothing is lacking nor has anything been
twarted. It is an expression of the sense
of the essence numbers as source of the
finite numbers in the way that I detailed
in the exam, which, I submit again to such
personal testing, is, although the University
failed to realize it, also a work of
intuitive necessity. More precisely, the
Lisa GJ2 formalism is the first fully
boundary-aware number formalism. It was
created in its manifest form with all
its extant features in a way which was
directly informed by the proof of the
impossibility of coherently conceptualizing
'all and but finite whole numbers'. It is
the first such formal structure and it
fills the gap created by the proof I
delievered in 2003.
****** ELEMENTS OF SYNTAX OF MY FIRTH LISA GJ2 (from MTDOC)
[[This speaks of 'Lisa' rather than 'Lisa GJ2' for there were
certain additions before the Lisa GJ2 was complete as it has
been delievered since, and in free open form available for
download and use for all, at the links given to my sites
at http://www.yoga6d.com/city. As of to-day, it is gaining
in popularity all across humanity. I experimented with various
extensions of GJ2 again, towards what I called GJ2 FIC, but
found that these did not represent the intuitive necessity
of the whole gestalt that Lisa GJ2 represents. The development
of a hardware completely oriented towards bridging
semiconductors with this language is presently progressing,
and in what I call the Yoga6d von Reusch Food & Hardwares,
or Yoga6d:VRFH, section of my Yoga4d:VRGM company, I also
intend 3rd world products of an idealistic kind, founded
by my successful currency trading work, to come out of this --
including robots which e.g. can manufacture small elegant
things of use freely distributable to orphanages. This
computer has a CPU which is nonmicro called the GJ2 LT
CPU.]]
One makes Lisa programs is by making new functions and associated
data structures.
These functions can, unlike e.g. classical Algol or classical Pascal,
but like more object-oriented inspired languages including Free Pascal
and the variations of C / C++, be addressed directly
e.g. through lists or through variables which can hold what is in some
languages called 'pointers' to them; this is also shared with Forth from
the 1960s by C. Moore, which has the sequence of parameter transfer
found here.
The command GJ2
is in general followed by the command
:MYPROG IN
where MYPROG.TXT is the name of the program put to the C:\BOEHM\BOEHMIAN
directory, or
:/MYPROG IN
when MYPROG.TXT (max 8 characters, plus dot, plus TXT, no blanks, usually
only letters and digits and a few more types of signs) is at the top
directory, or
:MYFOLD/MYPROG IN
when it is a folder called MYFOLD (or the like) under the standard folder
C:\BOEHM\BOEHMIAN. In $$$ Lisacode $$$ the reference is with backslash;
in Lisa, the proper reference is with forwardslash for all internal
functions. (However Lisa can call on Lisacode in which case backslash
should be used.)
The comments above indicates that the Lisa language is not merely
algorithmic or functional, nor merely class/object oriented as in the
language Simula67 by Ole-Johan Dahl and Kristen Nygaard (which led to
the varieties of class/object-oriented languages), but of a kind which
is implicit in all language in which functions can be handled through
memory pointers, but explicit here in terms of standardisation and
semantic ease. Since the word 'pointer' seems to emphasize a distinction
which is strong, and since it does not properly convey the near-assembly
immediacy with which this language is handling the issue, there is the
suggestion in Lisa to call the immediacy of connecting to a function
by means of a number holding its position in RAM by a 32-bit structure
for a 'warp'. The ease and optimism with which Lisa have been crafted
around the notion of warps lends power to the assertion that Lisa can
be said to be a warp-friendly language, then. This phrase, coined by
this writer, implies that warps are an option but not a necessity
(thus it is not as much a direction as a warm welcoming route of action
for those who like it). (As acknowledged plenty earlier on, this writer
recalls affectionately the many warms conversations with one of the authors
of Simula67 up to the very beginning of the third millenium, namely
with Kristen Nygaard, known through family since in the 1960s my father
Stein Braten worked with Nygaard's language doing simulation of a theory
on behaviour relative to voting with the version of Simula before it
achieved inheritance classes, and in which objects were called
'activities'.)
To make a new function in the Lisa language, the typical format (although
other formats with essentially or exactly the same result are available) is:
(LET function-name BE (( input-parameter-indications.. ))
((
(( general-action-1 ))
(( general-action-2 ))
(( .. ))
)) OK)
(Note: The compiler won't protest if one tries to define something which one
cannot define but generally it won't be available unless the name make
sense and is not a native word. What is native in GJ1-context is
completely different from the GJ-context in the future versions.)
In this case, the lower-case words indicates something which should be
replaced with proper content, which, in the general convention, except
in quotes, is uppercase. The parameter flow is generally from left to
right, and from one line to the next line, but moves not directly from
a function or value to the next function or variable, but through stacks,
of which there are four major: the main stack, holding letters, words,
as well as numbers, and even long text lines up to 250 characters,
the simple stack, which offers local variable treatment for whole
numbers and warps, the variable stack, which is a convenient way to
store warps, sometimes also between functions, and the decimal number
stack, called also rich stack, metaphorically also because one can
enter numbers to it directly rather than through the main stack by
prefixing with a dollar sign. Prefixing with ~ puts a number on top
to the simple stack. Prefixing with a ^ puts a number on top to
the variable stack (which is more used than the ~, by the way). No
prefix of a number means that it goes to the main stack. So does
a word which is quoted; quoted either by prefixing with a colon,
completing with a comma (see also LISTOK because this affects a
counter), or, in the case of something such as a sentence which can
contain blanks, prefixing with a right curly bracket } and also
completing the quote with the same, as in
(( }Hello world!} => POP ))
The role of the (( and )) is in giving a semantic idea as to what
constitutes some kind of whole in action. Only in the case of (LET
and OK) and some other words in which the parenthesis are 'melted
together' with the word are they of necessity to write just there.
The arrow, too, is a semantic device, rather like comma in a
language like English which normally can be omitted but which it is
part of the definition of the Lisa language, alongside the (( and )),
to assert as an important semantic device. Of the same nature is the
semicolon, for instance as in
(( 2 ; 3 => ADD => POP ))
which is used basically to say -- there is something on stack after
this operation and for the time being we leave it there. This can also
sometimes be useful after a (( .. )) operation has logically completed
but with a residue handled a couple of lines further down in the
function. It can also often be used right after the (( starting
a statement when one picks up what is on the stacks referring to
above in the function. For instance in this function
(LET THREE-POP BE (( ; ; ; ))
((
(( ; ; ; => POP => POP => POP ))
)) OK)
there is nothing really that should be done about the stacks
initially in the first (( )) so three semicolons there are used
to indicate that this function is made with the intention to
receive three input parameters. On the first line inside the
function we see again the three ; ; ; to indicate that within
this (( .. )) clause we handle all three of them in some way.
A use of the arrow is also to indicate an output from the
function, in which case the arrow is generally used twice --
again as a semantic device which strictly technically could
be ommitted but it is asserted as part of the Lisa formalism
(which obviously have, on the use of parentheses and uppercases
and dashes, esthetical similarities with the noteworthy Lisp
formalism from the 1950s, but which is otherwise inspired in
essence more by Forth yet the main stack and the simple stack
are completely novel developments relative to Forth, of course;
the word 'hybrid' is sometimes used to indicate what Lisa is --
a hybrid between two languages, Forth and Lisp, in some way,
and a hybrid between a language and an operating platform /
system as a standard; however the Firth and the Lisa are
written entirely from scratch near assembly level of memory
handling, of course, with the assistance of the eminently
made DJGPP tool and handled by the versatile open source
SETEDIT editor by Salvador E Tropea, started by the command
TEXT, or E, at $$$ Lisacode $$$ and extremely competently
made even for vast-sized documents of a standard 7-bit ascii
form). An example of output from a function, which also shows
how input can be handled in two ways:
(LET MUL3-VER-1 BE (( >N3 >N2 >N1 ))
((
(( N1 ; N2 => MUL ; N3 => MUL => => ))
)) OK)
(LET MUL3-VER-2 BE (( ; ; ; ))
((
(( ; ; ; => MUL => MUL => => ))
)) OK)
When it is known that the use of the function in general
tends to be not that time-critical, the MUL3-VER-1 handles
numbers with an ease which tends to pay off in somewhat
more complicated functions, in which N1..N11 can leisurely
be used to indicate the input in the sequence it was given,
e.g. will
(( 3 ; 5 ; 8 => MUL3-VER-1 => POP ))
lead to 3 going to N1, 5 going to N2 and 8 going to N3.
The convention in Lisa is therefore, when we have to do
with plain number input, to put a sequence of up to 11
>Nnn in reverse order (since the topmost number is going
to the highest-numbered Nnn). It is to be noted that the
>N1..>N11 and the readings of them, N1..N11, refer to
11 free slots always automatically and speedily allocated
to a function on its entrance and unallocated on its exit,
-- a feature evolved on the journey away from Forth and over
to a new type of language in which the sense of clutterings
about stack handling is reduced to a comfortable minimum.
(It is for this reason the ~ operator to push new numbers
on top of the simple stack is rarely used.)
However, when the operation is rediculously simple and
the amount of numbers in are many, and/or the operation is
known to be called on in a time-critical fashion, the
MUL3-VER-2 is a perfectly meaningful format also.
In the cases of inputs such as texts, it is a convention
to most often use a semicolon for each of these inputs so
as to show the reader of the definition the intended amount
of items in, but in some cases it makes sense to do a light
operation on the text already in the first (( )) clause,
which is after all fully capable of having any sort operation
since the (( )) divisions are semantic entities not precluding
any type of algorithmic operation. A good example of this
can be this, which tells, in text mode, on-screen, the length
of an input text:
(LET TELL-LENGTH (( ; => LENGTH => >N1 ))
((
(( ; => RM ))
(( }The length of the text is} => POPS ))
(( N1 => POP ))
)) OK)
This is but one example of very many ways in which this could
have been written. The single semicolon indicates that it is
but one input expected; moreover, the length of it (the amount
of characters, blank included), is stored in position N1.
At the first line after this, the text itself is removed,
since we are only interested in the length in this particular
case. The POPS prints a line without lineshift but with a space
character afterwards instead; and N1 => POP ensures that the
length is printed out.
If we have need for the text we give to TELL-LENGTH we can
copy it first; since the main stack of Lisa is relatively fast
and since it is a well-known boundary of size of the elements,
that these are generally not longer than a textline, there is
a number of quick operations, including ONE to copy the topmost
item, TWO to copy the two topmost items, keeping their sequence,
THREE to copy the three topmost items, also keeping their
sequence, ANGEL to switch around the sequence of the three
topmost items, and many more such. For instance, we might
do this, either outside a function or typed directly in at
the interactive compiler after the above function is typed
in (this interactive feature a friendliness and a "You-ness"
also found in Forth, Basic and which some languages including
Perl can give if they are made to work like this with a little
input loop and parsing, and this encourages a learning intimacy
which also proves to be a quick way of exploring any half-known
algorithm and to find out whether a word is previously undefined
or not -- and many more things besides). Here is the example,
using ONE, which makes an extra copy of a text before giving
it to TELL-LENGTH, so that it can be used by a function after
that, in this case POP:
(( }This is a text} => ONE => TELL-LENGTH ; POP ))
The suggested convention of sign-use in this situation is
to use a semicolon to indicate that there is more left on
the stack. Though => TELL-LENGTH => POP would in some
sense be meaningful, it seems to indicate that TELL-LENGTH
is generating the output which is then given to POP; but
the above sign-approach solves that.
To store a whole number, one can write
((DATA variable-name ))
where the lowercase is replaced by a proper uppercase. In
this case, there is no initialization value. There is a variety
of ways of initializing values, of course, and that is why the
beginning ((DATA is defined clearly whereas what comes before
the completing )) is rather open. It is an implemented syntax
in Lisa to speak of two very often-used values in programming,
namely zero and 1, by the poetically friendly word 'basis' and
'dance'. For instance,
((DATA AMOUNT-ALLOCATED AMOUNT-ALLOCATED => SETBASIS ))
would put it to zero while
((DATA PAGE-NUMBER PAGE-NUMBER => SETDANCE ))
would enable page-number to start at 1. Please initialize
rather during program-startup if it may be that the program
is restarted without exiting by XO so that RAM is refreshed
as any ((DATA .. )) statement happens outside of function
definitions, and is performed actually while the program
text is compiled. Although syntactically you can make new
numerical variables with ((DATA .. )) and new text variables
with (( VARSTRING .. )) (and the numerical variables can
point to, or warp to, as we say, whole long documents of
texts, or images, or even sound files or programs...)
anywhere in the program text, after much experimentation
it has come to be the recommended practise (anyhow from
this author) to put all of these data definitions to
the beginning in the program -- right after the commentaries
setting forth who and what and when -- and right before the
first function. I can give a lot of logical reasons for
this. But there is one big reason for it: good programming
needs a focus on action. What is action must stand clearly
forth. A function is action. They should refer to storage
locations under neutral names so that attention is not
diverted (nor mislead by funny names -- for instance,
if a variable is call CORRECT-DATE-TODAY but contains
something entirely different this is less informative than
a variable called DATE-1 or even just N1). But if we
distribute ((DATA ..)) and (( VARSTRING .. )) definitions
here and there, say, before the first couple of functions
which might use each, then that clear distinction is
getting spoiled. Putting them up front prepares the mind
for them: okay, folks, here are the storage rooms -- use
them as you please, but know that these are but storage
rooms and there is nothing much more about them. Then,
folks, here is what the program really is doing.
I have elsewhere spoken strongly out against the notion
that programming is improved by dividing it in blocks with
different naming conventions, localized names, and hierarchies
of schemes of assigning functions to groups of data -- it may
work best in extremely hierarchical, factory-like companies
when these work bureacratically with with rigid, bank-like
system practise for collaboration between people and also
collaboration between programmers. But such object/class/
hierarchy of programming tends to consume hours if not months
of attention to petty details of rigid structure rather than
to the perfection of a program. Just when I completed the
extension of NOD501.TXT to GJ1.TXT, for instance, I had
some ideas for a new way of storing fonts, a new way of
editing fonts, and a new way of printing them, and before
two days had passed -- two days of joyful programming --
the entire new font was created, with a fully functioning,
tested font editor -- MOREFONT (try it). In fact, I wonder
if it was not just one day. Not a bit of it has been changed
since its initial completion; it is now part of the standard.
It wasn't part of a grandiose plan at a conscious level;
it came as a positive surprise to me, yet, of course,
in full alignment with where I had sensed it must be
right to go.
The program text is compiled each time
during startup because this is fast enough and because this
asserts that there is a minimum of cluttering of files
present on disk, a maximum of open source readability,
and more advantages of this type.
Another common way of initializing is by means of <<ASSERT,
and SETBASIS, SETDANCE and <<ASSERT can be used freely
throughout a program. ISBASIS and ISDANCE (as well as
looking into the variable value by VISBASIS and VISDANCE)
gives fast checks on these much-used numbers. The psychological
advantage of a programming language and formalism which is
coherently woven around the assumption of positive
affirmations and a general good feeling attunes to the
intelligence and production-quality of the creative writer
in the language (is my postulate, and I know of no better
programming language to work in, honestly). Here is <<ASSERT
used in initializing a value:
((DATA RIGHT-MARGIN 10 RIGHT-MARGIN <<ASSERT ))
The user of variables should be aware that when the name of
the variable is written the warp or address for it is found
on the variable stack. To change the value, <<ASSERT is one
way. INCVAR would increase the value by one, ADDVAR by a
number given on the main stack. But to retrieve it, >>> is
used, and >>> actually has a computational function. In the
earlier edition of Lisa, the =-> was its appearance, and
that version can still be used, but >>> is clearer, I find.
(( }The right margin is now} => POPS ))
(( RIGHT-MARGIN >>> => POP ))
A predefined function, of so-called the basic (historically,
by me, called Firth) kind, is the function ( which is like
POPS except that there is no blank after. This is useful
also when quick checking on stack content is sought
e.g. during program correction, when one can type such as
RIGHT-MARGIN >>> (
at the interactive compiler, in such cases consciously and
deliberately using a syntax completely without any extra
semantic elements in order to get an as quick and prompt
response from the compiler as possible with the minimum
of typing.
The text line can be stored in such a variable as
(( VARSTRING DAY }Monday} DAY => >VARSTR ))
and this can be retrieved e.g. by
(( DAY => VARSTR> => POP ))
In cases where many text lines is sought to be stored,
the typical way is by means of a matrix which is defined
by means of the Lisa word RAM-PM, which has as input the
amount of rows and the amount of columns. It is a programming
praxis in Lisa to always give some extra room to variables
and always have some extra amount of columns beyond what's
strictly necessary. This praxis is an advised feature,
and in some cases, as when a matrix is put to, or retrieved
from, disk file, a necessity; but sometimes, for very large
matrices (and in such cases where the disk file operators
FILE2PM and PM2FILE are not used) containing a very large
amount of numbers one might be more precise in allocation.
In general, 32-bit numbers, typically within the range of
plus minus two billion, should be used in preference to
decimal numbers or other number types whenever it is suitable
in this context. Such a whole number also happens to be the
kind of number that a warp is. Thus, if we make a variable
like
((DATA CLS-HOUSE }CLS} => ? CLS-HOUSE <<ASSERT ))
we store the warp of the function CLS, which clears the
screen, in the variable CLS-HOUSE. The convention is that
a place where a warp is stored is called a house, or another
simile of the kind; sometimes the word "warp" itself is used
in a combination with another word and dashes in between instead
of the "house" word. These conventions are not strict, of course.
To perform the CLS function in the case above,
(( CLS-HOUSE >>> => H ))
would do. If we rather make a new function which e.g. paints
half the screen blue and wants a sudden change in the program
which calls on the line just given each time the screen is
to be changed all we have to do is something like this --
while the program is running, inside the program, to change
the defined functionality:
(( }PAINT-HALF-BLUE} => ? CLS-HOUSE <<ASSERT ))
It is this utter simplicity of the handling of warps which
can be given so many shapes and enable a new form of real
sense of wholeness in any Lisa formalism.
When making a program text, please be aware that in the
beginning you can decide whether the program text is going
to run over the screen as it compiles, with VISIBLECOMPILATION
vs INVISIBLECOMPILATION (and how fast, interesting in cases
where a statements unexpectedly causes abrupt stopping of
compilation with a number of messages -- though this is
rare, it can happen, of course, for the language is a very
free one). You can decide whether it is going to check for
left-over item at the stack or stacks by inserting the word
LOOKSTK in between functions. This can detect when nesting
of some checks or the like are not complete. During complex
program correction, it can be a LOOKSTK between every function
in an area which you look into, then much fewer as the program
is completed.
(( ))
The way to do conditional performance of some lines rather
than others is done by means of (MATCHED .. MATCHED), in an
intuitively rather obvious way which is given inside the
following dictionary. Synonyms for these are = and ===. The
alternative condition can be written as, respectively,
(MATCHED .. )(OTHER .. MATCHED) and =, == and ===.
The so-called boolean flag, or 'flag', which indicates
whether a condition has been met, is generated by such
functions as EQN, which compares two numbers, and EQ, which
compares two texts, and is a unit-sized lowercase y for 'yes'
or n for 'no', where other values should not be used, in
general, as for input for (MATCHED or =. On these one can
have boolean operators such as AND and the negation, NOT,
and, another three-letter function, ORR.
One can generate a number of highly easy-to-read looking
loops by the up-to-four GOLABEL1: .. GOLABEL4: marks inside
each function, in which a single -- and only a single -- call
to each one, either by GOUP1 .. GOUP4 or GOFORWARD1 ..
GOFORWARD4 has been found by this author as fruitful in
well-structured programming. This is a moderate version of
the notorious 'goto' command in Fortran and Basic. It is only
used inside a function. Some other loop commands are implemented
in basic Firth but the combination of (MATCHED or = with these
simple GOxxx commands suit eminently, and the only one used in
addition, in most cases, is the enormously simple (COUNT ..
COUNTUP) command which takes a single argument -- the highest
number which the count, which starts on 1, will go up to,
within a 32-bit boundary as normal. As with GOxxx commands,
(MATCHED and (COUNT belong inside functions only.
When (COUNT .. COUNTUP) loop is used, it is often the case
that something like >N8 >N7 >N6 is written inside the input
parameter clause instead of >N3 >N2 >N1, for the (COUNT ..
COUNTUP) uses the simple stack itself to add a couple of
new values on top of it. For during a (COUNT .. COUNTUP)
phase, one will find that N1 inevitably holds the current
count value, while N2 holds the value it goes up towards.
What was before N1 will be accessible within that loop as N3.
In short, then, one adds 2 to the number n in the Nn variable
used, so that N8 becomes, inside the loop, N10, and N7 becomes
N9, and so on. To remind the reader of the program of this,
therefore, it is often convenient to use a much higher numbering
of the N-variables when such a loop is just about to be
encountered inside the function. For instance,
(LET MANY-TIMES BE (( >N8 >N7 ))
((
(( N7
(COUNT
(( N10 => H ))
COUNTUP) ))
)) OK)
This is a loop which performs the operation requested that
many times, by means of a warp which is given at the top-most
location at the main stack. For instance, one could do
something like
(( 10 ; (( }CRLN} => ? )) => MANY-TIMES ))
to get ten lineshifts. When the .. COUNTUP) part is reached,
then, when the loop has completed itself, the use of the
simple stack returns to normal, and the N8 -- which inside
the loop will be accessed as N10 -- will again be N8. One
can nest several (COUNT .. (COUNT .. COUNTUP) .. COUNTUP)
inside one another in a function -- if one doesn't overdo
it, for each function should be psychologically pleasant
to read, if possible -- and then one will find that each
nesting level adds another 2 to the N-numbers used.
After the function completes, the eleven slots in the
simple stack are released, of course. One will therefore
find the variable stack sometimes exceedingly welcome when
using the (COUNT .. COUNTUP) type of loop, since it is not
touched by this phenomenon of adding two new members on top;
the variable stack is often initialized in the beginning of
a function, perhaps with a & function, and released towards
the end of the function, perhaps with a && function, but it
can also, in such cases where one does so with clarity,
be used to carry over values in between (a group of)
functions.
The word 'program' can refer to the collection of functions
leading up to the highest-level function which calls on all
the others, with suitable initialization of variables and
completing procedures, or it can, when the context makes it
clear, refer to a single function -- perhaps together with
its variables. It is noteworthy that also the variables are
in some sense to be seen as small programs (see comment on
this in connection with >>> function).
As the very completing line or lines, you can either include
the calling on a function which gives some textual info as to
the success of the compilation, and perhaps what one might type
next to start this or that, or you can start the program itself
by means of a particular phrase, which in general goes like
this:
(LET AUTOSTART BE my-program-name OK)
You would type in, usually in uppercase, the real program
name where it says my-program-name. Sometimes it will also
say XO on this line; that's okay, but AUTOSTART should, as
a convention, normally be nothing but a line long because of
its very unusual features and because of the importance of
being sure that the definition of AUTOSTART is devoid of
any and all syntatical complications whatsoever, since, once
it is defined, it will be started no matter what -- speaking
generally.
It is important to take advantage of this function rather than
writing your program name at the bottom of the program, since
the compiler will still be in compilating program mode when the
completing line is reached. So, if the completing line is
something like
(( PROGRAM-INFO ))
which just prints a few lines then exits there is no trouble at
all. But if you have a program which is going to have an
endurance and a duration in performance you will want the
compiler to have done with all compilation.
****** THE LANGUAGE LISA GJ2 AS A NEW KIND OF THEORY
In applying neopopperianism over a great deal of time
to questions of the metaphysical sense (in the sense
of Aristoteles, as Meta ta Physica, the name given
by some to his questions concerning the sense of
existence as a whole rather than in particulars),
I have come to approve of what can be called an
eclectic version of Bishop Berkeley's understanding
of reality as part of a mind. (More about this on
my webpages.)
Furthermore, this neopopperian study -- and I
submit this not in a revelationary spirit, but
as something to be researched in a neopopperian way
by each, as time goes by, and in the manner seem
fit to each -- suggests that the stable features
of reality reflects, in this mind, the erection
of computer-like features to deal with
psychologically meaningful finitely-sized (as
I explain on my websites, this means more than
16 bit and less than 64 bit and still in the
2-4-8-16-32-64 bit sequence, namely 32 bit, or
plus minus about two billion as whole number
size) numbers, and their transformations. These
computer-like features must inevitable involve
a formalism. The berkeleyan features of reality
means that the formalism is in some sense prior
to the hardware. The hardware is merely a
subfunction uphelding the mentality of the
gestalt of the formalism as a whole. This
formalism, I further submit in the same manner,
must be composed of only and all intuitive
necessities. It is in this spirit I submit to
you: the Lisa GJ2 language (the "Firth" can be
dropped since it refers more to the way the
Lisa GJ2 language was created out of a re-building
of all the stack structures of Forth and of
FREEDOS components and more, and such) is such
a formalism. As a whole, it is a mentality --
viz., something informal. It can of course
present any type of transformation of
any clearly and exceedingly well-boundary-aware
defined finite set of 32-bit numbers to any
other. It does present this in a way which
echoes a network of mentalities working upon
each other: the warp structures. As such, it is
a theory, allowing for nonlocality as well as
locality and a number of other possibilities.
It allows for gravitation and a Principle of
tendency of Movement towards Coherence or
PMW as I called it in my 2004 book. It allows,
then, for supertexts -- programs -- or as I
call it, 'supermodels', in my 2004 book; and
lists, and matrices of such. It allows for
the interaction of this deeper mentality with
these formal aspects it has created so as
to make possible that deeper-determinism-
than-Spinoza.
The only reason the 20th century and early
21st century type of so-called "mathematics"
of its so-called "physics science" works is,
I submit, those eclectic components which
echo bits and pieces of the finite aspects
of just this Lisa GJ2 language, which was
not created at the time. The type of
self-reference which Russell & Whitehead
sought to avoid in Principia Mathematica
but which Kurt Goedel showed that exists
implicitly within the arithmetical
possibilities opened for by their set theory
I have shown to exist within the number
conception of the idea of 'all finite
numbers' itself, excluding the possibility
of 'all and only finite numbers' and
leading to 'all finite numbers and also
another type of not finite numbers' as
the minimum conception of the infinite,
leading to essence numbers. In other words,
what is not self-referential of the infinite
is not coherently thought about: just as
what is self-referential of the computational
is not coherently thought about. One can
only wed the finite and the infinite in
one's mind by having a well-defined working
space of the finite which does not seek to
bring in the infinite on the premises of
the finite, but rather which keeps the
infinite suspended as it were, knowing --
as if in secret to itself -- that the
finite is created within the infinite,
just as the finite whole numbers are
created within the essence numbers.
This is pantheistic but not limited to
it. It would be irrational to keep on
talking of reductio ad absurdum all the
time; one has to allow that if what I've
said above, as well as earlier, in sum
total are absolute intuitive necessities,
these are not appreciated by superficial
browsing or comparison with other's
superficial browsing or in-depth incoherent
works -- and among those who have in-depth
incoherent works behind them we must
include such enthusiastic thinkers as
Abel, Turing and Chaitin. The Lisa GJ2
differs utterly from their enthusiasm and
comes with an enthusiasm of its own, for
it starts with infinity as undefinable and
truly un-circumscribable and does not try,
when going into the finite, to assume that
one can re-define the infinite in terms
which before was used on the finite.
FINIS
QUOTATIONS: PLEASE REFER TO ISBN NUMBER
AND WEBSITE ADDRESS IN FULL