A. Tacoma
Mathematics Hath No Principles
***part of the essays combining themes
which also are philosophical with
deduction also as applied to programming
available at the essay section of G15 Yoga6dOrg
programming sites,
norskesites.org/fic3/fic3inf3.htm
(with redistribution license along the
lines of the normal yoga4d.org/cfdl.txt)
1::A::2013::3::21
***note that any program code (G15 YOGA6DORG)
included in the main essay text as part of the
text may be uncorrect and meant mostly as a
sketch of roughly how it might be programmed;
this is in contrast to other listings of
mountable programs (so-called "mountable apps",
short for "mountable applications") found at
the same page, which generally are much more
worked-through
***note: a few likes to DO programming, some
more likes to WATCH program code content, but
most are more happy with words, images, sounds
and in the G15 Multiversity academic approach
we are not asserting that one cannot get
around to get a higher education without
having much contact with programming. So also
with these essays: they should be fairly readable
without consulting code. Still, the words are often
by necessity long and not all that common if we
want precision in the essays; and so there are
other approaches in the G15 Yoga6dorg Multiversity
to show the same content. It might be comforting
to know that the essays and their code do exist
however.
***
***
***
In early 20th century it was imagined, by some,
that one can make a kind of standard recipe to
judge whether any statement about geometry,
arithmetic and so on is right or not. Such a
recipe would in case be what also can be called
a 'route procedure'. Such a route procedure can
be done mechanically, by a machine; it is
clear-cut and distinct in its rulebook for each
step and requires no contribution from any flash
of intuition.
Kurt Goedel showed, around 1930, that the
production of a programme of mathematics in
Bertrand Russell and Alfred North Whitehead's
Principia Mathematica -- 'the principles of
mathematics' -- could not fulfill that goal at
all; his argument had a form that made it clear
that any such idea of a route procedure to prove
all true propositions is an illusion.
To use something of the language of that time
-- of the programme of mathematics -- is it so
that this statement, or proposition is true:
"For every true proposition P about our domain,
there exists a pathway by our rules of deduction
from our axioms to this proposition P.'
Don't worry if this is not your favorite
style of saying things. We'll deal with it
in more imaginative language that is precise
enough to create real insight into the matter.
But you can imagine that there's a
starting-point, -- a set of axioms and rules --
and a kind of machine that generates loads
of proofs, they come sputtering out of the
machine (with each proposition thereby
proved at the completing line).
Goedel showed that no starting-point will
do to reach all true propositions. In other
words, stuff that's true aint always provable.
Pushing it, it may be that most of it isn't
provable.
Does it matter? But there's some art to this,
some beauty about all this analysis. It hints
towards a deepened understanding of living
mind.
We're going to make use of g15 Yoga6dorg
language -- that is, the most elementary and
machine code near language for this our
G15 cpu when it works in its normal G15
O.S., programs that (when corrected) can be
compiled by CTR-A and performed by CTR-X,
when typed into the G15
(cfr www.norskesites.org/fic3)
to show essentially the same, drawing on
the vast volumes of work between then and now,
where many have done work on programming
language thinking versus such set theory
thinking as the book by Russell and Whitehead
contained.
The set of all things that can be proved
turned out to be a set that can be talked
about within such a set theory as that
which Bertrand Russell and Alfred North
Whitehead made in their big, hopeful
production entitled Principia Mathmatica.
This is so in spite of the fact that they
had gone to great pains to delete any
traces they saw of such "self-references",
knowing the complications they lead to.
For as soon as Goedel had thwarted their
stuff to define a set of all that is
provable, he went on to construct a
proposition that denied membership
of this set: a proposition that declared
of itself that it wasn't within what
could be generated by a route procedure,
in short.
And as Goedel put it, by "a form of
meta-reasoning" we can see that this
proposition is true. It must be true
that the proposition is unprovable;
for if it is false, something false
becomes provable, meaning that the
whole thing implodes with
self-contradictions. In other words,
provable must only be true things; and
if it is false, a false thing would be
proven; leading to collapse of it all.
So the basic tenets of the theory imply
that stuff exists that go beyond them –
even if this went counter to the
intentions of the folks behind the theory.
Ultimately, we can say this in a
snappy way, that speaks against the
spirit of the times, and indicates
the perhaps most true philosophical
spirit and light that we can
understand it, namely:
Mathematics Hath No Principles.
It doesn't mean that anything goes in
deduction, but it does meant that trying
to capture the core of deduction in a
formula such as "mathematics" is a no-go.
Ultimately, this type of result became
known as Goedel's work on incompleteness
(he made a couple of theorems about this).
In order to get the sense of how the
ground of ambition (of the authors of the
Principia) breaks away in the midst of it
all, we must as if entertain the ambition
for a moment ourselves, as a thought
experiment. Imagine, then, this whole
project of codifying all thinking and
dispensing with the further need for
intuition. Think in grand terms of abstract
spaces of statements and rules of deduction
that roll out perhaps infinitely fast.
Imagine vast machines of logic, they conquer
vast fields of truth.
Then we ask, are there after all fields
forever beyond their touch? If so, then
the idea going back some centuries to
Gottfried Leibniz must remain scifi:
he imagined that when people disagree,
they would sit down around a table and
say, 'Let's calculate!' and solve all
their differences simply by permuting
the rules and working from axioms. He
imagined world peace by mechanising all
thought.
He imagined, indeed, a world where
there is no higher truth, only a more
complicated one -- one that takes
longer time to 'calculate'.
And so after Goedel's work, we can say,
well-educated atheists had a really tough
time; while some converted themselves,
strived to incorporate the vision of the
mind as entirely beyond the machine and
as part of a cosmos that allows this.
And the recently arrived quantum theory
pointed, indeed, in some such direction.
But Goedel is complicated stuff. I've
heard a person who called himself a
religious believer and who had just
finished a higher eduction in themes
near these speak of the work of
Kurt Goedel as 'merely a formal result,
saying nothing about anything, really.
It is but a series of tokens. Purely
a formal thing.' Or that was the gist
of what he said.
I have thought about what he said,
and I think he is wrong -- very wrong.
A statement which is able to say something
about the limitations of nearly all
mathematical systems is saying something
beyond the formal. It is saying something
more than any other formal result. It is
pointing out that existence may be --
not necessarily so, but possibly so --
greater than that which can be depicted
on paper. And that's maybe the trouble
with bible-readers; people who think
they reach God merely by sticking to
a text and repeating it, over and over,
like hypnosis. If it isn't in the text,
it isn't of God; Goedel is not discussed
in the bible, hence he cannot mean anything,
his results are but formal.
As I see it, the challenge for the religious
aspect of our minds is to find resonances
that go beyond thought, beyond the words.
A great reading may enhance the resonance
but not if the word becomes a jail. One must
sail on words as music and leave them
behind and take in other words, and blend
with one's own silence, with fresh
experience, also sex, also the sensation
of flowers, of beauty; one must forget
oneself in artistic action, and in dance;
and suddenly something religious can
penetrate.
If you have spent some time with
programming, you know that it can cleanse
the mind deeply; it can do something like
a deep massage can.
Programming is abstract like the stuff
that these folks we mentioned worked with,
but more concrete, for the machine lights
up with faithful, predictable responses
when we program it well. It can give a
measure of self-confidence to have a
machine be a slave to you; only we don't
overdo it so we treat people like machines.
And so, we can make programs that do a
bit of analysis, they can have, with luck,
a bit of what we can call first-hand
mentality (FCM), an off-spring of your
mind.
Let's look, then, at what Goedel worked
with, but in more modern terms.
When you think about programs in general,
you are thinking about what in the 1930s
couldn't be spoken of that way – only as
“route procedures” in general. So in our
days we can say something along the lines
Goedel said but in an easier language.
There are various ways of doing this,
some nearer that original work from the
1930s than others. But we can go a long
way with simple means -- keeping in mind
that it is all about twisting a
self-reference to act against itself.
In a while I will try and explain what
that last sentence means – twisting a
self-reference to make it act against
itself. Russell had a metaphor about
shaving, but here it is elevated
somewhat to the more modern grunge
friendly mind into a metaphor about
self-eating zombies.
Take any G15 program and look at it.
Can you tell for sure whether it will
leave a green square in the middle of
the screen, the screen being otherwise
black around it? For simple programs
it is easy and for more complex programs
-- well, it would require work.
Now in the programme of mathematics
(“programme”, with an -e ending it,
doesn't mean computer program, but it
rather means a large-scale long-lasting
project and project description), --
the notion that Mathematics Hath Principles,
involves, then, -- jumping over some
steps and lending a little bit from the
works of both Goedel and his successor
Alan Turing -- that there are principles
in the form of a route procedure to
judge whether any proposition involving
its type of things -- sets of numbers
and sets of sets and such -- is correct
or not.
If the proposition is meaningful, it
is correct or it's negation is correct.
In the case of the green square, clearly,
either all the criteria we listed --
that the square is green and that it
otherwise is black around it -- is
fulfilled, or those criteria are not
fulfilled.
So it's the type of thing so that if
the proposition -- let's call it p1 --
isn't true, then "not p1" is true instead.
If there is, in the programme of
mathematics, even a single proposition p1
which is so that it is true, or "not p1"
is true, without it being possible to go
by means of the standard route procedure
one imagine could exist to prove it, then
we can say that 'mathematics Hath No
Principles'.
We mean by that to say: the programme
which tries to summarize all deduction as
such, all number thinking as such, and
put it into a mechanical scheme, doesn't
work in any complete sense. Bear in mind
that we're simplifying how we state things
now: we are really using the value of
hindsight and summarizing various approaches.
If one stands in the middle of the
thoughts as they were first presented,
it may look rather different. But though
we're summarising now, I think it is
clearly so that it's still precise enough
to capture most of the fullness of the
core insights.
We should also keep vividly in mind that
we do this as a kind of wild dance where
we pretend to go along with the notions
of this programme of mathematics (far
more so, that is to say, than in nearly
all other writings from these hands on
related themes).
This wild dance is undertaken in order
to get to see the illusion implied in it,
and so have greater freedom to work
outside and without that illusion. So, then,
does the mathematical programme work?
Maybe: if it does work, argued Goedel,
we get into a self-reference; and by that,
we'll be able to show very quickly that
there's a self-contradiction.
So eventually we come to this: no procedure
exist which is so that you can feed it with
any proposition and all the definitions
and such that go along with the proposition
and get a clear-cut answer as to whether
it is true.
If no such route procedure exists, then
intuition is a necessity when it comes to
making perceptions over things.
Alan Turing, one of the fore-fathers of
the whole computer notion, worked a lot further
on Godel's thoughts for he feared such a result;
but he achieved only a deepening and strengthening
of the original result by Goedel.
Of course, the by-product of inventing the bulk
computer idea does count for something, but it
wasn't primarily what Mr Turing was after.
He was after a negation of Goedel's result,
and he didn't get it,far from it.
Goedel's work in 1930 goes over many dozens
of highly complicated pages. We have the
benefit of having the computer concept
in our midst, so to speak.
Let me say that the breakdown of the
notion that Mathematics Hath Principles
doesn't mean that there is no such thing
as clear-cut deduction.
What we are talking about is the attempt
to make a recipe out of abstract thinking.
Also, when we work in such as geometry,
and we look at the interplay between such
as the sine and the cosine functions, we
are touching on phenomena in an abstract
form that resonates within us with an
aspect of what we mean by waves.
This is, we can say, a question of abstract
perception -- a perceiving by the mind of
sensations of thought, which resemble
sensations through our bodily senses of
such as the dance of the sunlight on
water waves at a wild beach.
Such perception involves, in supermodel
theory, a nonmechanical open principle
which goes infinitely beyond all such
route procedures as those described
by Bertrand Russell and Alfred North
Whitehead.
This principle, called in supermodel
theory for PMW, for 'a principle of a
tendency of movement towards wholeness',
we say relate to concepts such as contrast,
similarity, and reverberating wholeness
(such as water waves in their meditative
feel).
When we perceive we are not -- repeat not
-- merely thinking of sets of membership,
we are not merely grouping things and
pushing other things outside of that group.
Perception cannot be limited to set theory.
All the talk by Russell & Whitehead about
sets and memberships in sets, overlapping
(intersecting) sets, the combination and
extraction from sets, and so on, it is at
best but part of the much more rarely
refined process of fluid perception of
the living human mind, at all levels.
It is Mind you call on also when you
make a G15 program to solve something.
But it is also this your mind you call
on when you find the perfect dance to
unfold in a certain setting, perfect
for that moment, for the light in
that setting and for how your hair is
that day and what clothes you have on,
what is expected of people in that
setting, and relative to what skills
you have, and so on and so forth. The
formulation of PMW is consciously so
poetically vague that it embraces far
more than merely inclusion or exclusion
of memberships in conceptual sets.
Such inclusion and exclusion is but
part of the perceptive process –and
thus also but part of what we can, today,
long after Goedel, call The Art of
Deduction.
Perceiving what fits and what doesn't
fit, perceiving how something creatively
ought to be involves a freshness that
goes beyond simple route procedures.
That is intuitively perhaps obvious to
you as one who is touching on enlightened
themes. But the fact that it is possible
to show that route procedures are rather
impossible when it comes to thinking
about something as apparently well-defined
as numbers and groups of numbers, may be
considered to be less obvious.
So, if there is a simple or complex
single route procedure that can be started
so as to permute any set of definitions
leading up to a proposition, telling
whether this proposition is true or not,
we would have a systematic mathematics;
but it turns out that this is not just
beyond what has been achieved, but it
is beyond what can be achieved.
And the enlightened reader should be
glad for this!
Put very very simply, then, again:
Mathematics Hath No Principles.
Some would say it's stating it a bit
strong, but I feel it is ultimately
implied if you really think through what
it is all about.
A rather logical further step then is
to say: let the fields of geometry,
arithmetic and so on be kept sweetly
apart rather than grouped grossly
and crudely together. Each must have
its own clarity, we must learn about
each field and use intuition as well
as a general sense of clear thinking
and good deduction in each area.
We can find all sorts of useful route
procedures but there isn't one single
super-predictor machine that can tell
what's what in all these domains.
So, given even just one very general
example is enough to negate the whole
idea that any single route procedure
can do the trick. And the trick to show
it in a general way is to route the
route procedure to itself. Bring in
the self-reference; that's when all
such finite approaches break down.
For any route procedure must be finite.
You can't have a recipe that's infinitely
long if you're in the kitchen and gonna
apply it to brew, say, an unusual cup
of coffee.
So also with computer programs:
they are composed of a finite number
of lines of text, typically with lots
of numbers.
A computer program has input and
it has output. This output can be of
many types. So also with propositions
about numbers and such: they can be
of various types. So we if we want
to have one single route procedure
to judge em'all, it must have a kind
of input that tells it what type of
proposition it is.
By analogy-- and it is an analogy
that most workers in the area up through
the decades since computers began coming
around have accepted -- any route procedure
in mathematics taking the form of a 'proof'
correspond to a program working on input
towards producing output. This type of
comparison can be made more precise in
various ways, but for our purposes, it's
near enough, and plausible enough, and
it never was here the controversies existed
anyway.
The controversy in humanistic philosophy
existed between those who believed in the
notion of the master route procedure –
which can be called, 'artificial intelligence'
-- and those who believed more in the notion
of 'intuition'. The latter won. They won in
an absolute way, but the battle was fought
in so complex ways that even many decades
after Goedel's work, few has any notion at
all of what it was all about.
Normally, this writer is hyper-aware of
the value of asserting ranges, boundaries,
constraints, and not sneak in the notion
of infinity by keeping the boundaries
needlessly loose or arbitrary. In G15
Yoga6dorg language all this is done
consciously and throughout. The G15
language is full of numbers so as to
encourage a first-hand relationship to
data and numbers and also boundaries.
However: to go nearer the way Goedel
worked, we loosen up -- just very temporarily
-- so as to implode the notion of a master
route procedure from within. Having done
so, we congratulate ourselves and then begin
again asserting boundaries as normal.
This approach can be compared to a
setting in a thriller where a spy emulates
the vile ways of the gangsters -- for a
while, so as to resonate with them and
befriend them and fool them, only to take
down the headquarters of them when they
least expect it. That's rather the emotional
effect Kurt Goedel had on the best-brained
of his those of his contemporaries who had
taken a stance in favor of a master
prediction machine or route procedure.
So let's imagine that the programs exist
in a kind of abstract space, although we
still get output on a screen and put input
to them in a suitable way.
In this a bit limitless manner, we see
that some programs -- the program number
0001 at the completion of this essay
(which is there called, using any of
the free warp table numbers above two and
a half million, #3000001) is one such --
some programs produce a green square somewhere
near the middle of the screen keeping the
rest black, -- the center of this showing
a little different depending on the physical
monitor we use in each case.
In contrast, programs such as the one
labeled 0002 (#3000002) produce a black
square inside a much larger green area.
And then we have programs such as 0003
which accepts numbers in local variables
1, 2 and 3, and which produces an answering
number in local variable 4 (confer g15
documentation for why the code is this
way). The number is 1 if the sum of 1, 2
and 3 is bigger than one thousand, otherwise
it is 0. In the case of 0003, it is provided
with numbers adding up to just 999, so the
result there is 0.
In the case of 0004, the numbers adds
handsomely up to ten thousand, which is
patently much greater than one thousand,
and so the answer in local variable 4 in
this case is 1.
So 1 here means, "yes".
If we like, then, we can also read in
this number in another program, calling
on this again, and do something with this
number (this is the interaction between
'gi' – get from local variable into
one of the main variables; and 'pf'
– put to local variable from one of
the main variables). Now, in this area
where programs are floating around,
surely one can imagine that one tries
to make a program that at least in some
cases manage to figure out how another
program works.
Say, in variable 1 and 2 one specifies
where the program exists -- in G15,
that's done by giving a disk, such as
c (3), or d (4), and a number, which is
the 'card number' at that disk -- it can be
e.g. 300,000, just to take a meaningful
enough number.
Then we can say that the value in
local variable number 3 is to be 1 if
the answer sought is to the question,
"does this program produce a green
square rather in the middle of the
screen, with black around it?", and
it can be 2, or 3, or something else
for other relevant questions.
At first sight, surely we could imagine
ways and means of going about the
answering of such a question for some
simple cases. But if we really are going
to have the analogy to a master route
procedure which can answer any question
not just some about eg arithmetic, then
we must have a program that can judge
the result of running any other program.
I repeat, ANY program.
No matter how complex.
Of course, we can ourselves start up
a program and see for ourselves. But we're
now talking about having programs working
on programs. We're not talking off stepping
outside of the domain of programs and
watching the results. We want a master
predictor program, because if we can
show that this is not possible, then
by analogy, we have found a well-defined
clear-cut question that in each
particular case has well-defined
clear-cut answers but not answers
that can be decided upon by any totally
general route procedure in any programme
of mathematics. So that's why we begun by
emphasizing the analogies between the two
fields so strongly.
Note again that we only require ONE
counter-example for the whole paradigm
of completeness about a single route
procedure to come tumbling down. One
counter-example is all we need, if
that counter-example is shaped so
that it applies to the whole range
of POSSIBLE and POTENTIAL route
procedures of the master kind.
So we have our master program
predictor, say: we imagine that we
do, and we'll get into a really tricky
self-reference that way, which leads
us to perceive a kind of self-contradiction,
so that we prove the point of the
nonexistence of the possibility of such
a master program.
We play along, imagining, then, a
program 0005, so that it takes the
specifications we indicated just now --
in local variables 1 and 2 we give the
location of the program we want
evaluated. Program 0001 to 0004 are
real programs that you can compile
by CTR-A and run by CTR-X in G15.
Program 0005 is but the shell of
an imagined probably very large
program indeed, the master detector
program; remember the spy thriller
analogy here. Imagining the presence
of this that we know we're about to
disprove is being a 'spy'.
In variable 3 we indicate '1', because
our question is the deceptively simple
one, namely whether the program produces
a green square or not, with black around
it.
And we want the 0005 program, the master
predictor program (which is analogous
to the master route procedure when we
imagine that Mathematics Hath Principles),
to give a clear-cut answer as 1 or 0 in
local variable 4. Right?
Part of the whole route procedure idea
is that of clear-cut-ness in answers,
not admitting of ambiguous answers.
We want a yes or a no. A 1 or a 0.
We want the answer in a clear-cut,
digital form, not as a vague gesture
in the wind (however immensely
important vague gestures in the wind
may be).
Well, then: does 0005 exist?
There are two possibilities, that
0005 can be made and that 0005 cannot
be made.
If we run into a self-contradiction
by assuming it can be made, we conclude
it cannot, and then we have, at least
informally and with a precision level
that is adequate for our purposes so
as to gain real and actual insight
into the whole spectrum of ideas involved,
shown that Mathematics Hath No Principles
in the sense we have indicated above.
We proceed by thinking in terms of
self-reference. We're going to tie the
program to itself somehow.
Here's the idea -- and it's a simple
one, but it's immensely powerful when it
comes to deduction and digital or boolean
logic of all sorts: we're going to have
a route procedure that acts on the answer
so as to falsify the answer.
No matter what the answer is, we're
going to act so that it becomes false
as soon as it is given.
And in our entirely modern contribution
to this somewhat dusty field of goedelian
thinking, we're going to touch on the
land of xxx, a land of crazy zombies, let's
call them cz folks. These cz folks eat all
the can get their teeth into, – the crazy
zombie part is that they gnaw on
themselves, trying to eat themselves –
eating more and more of themselves until
there's that famous final big quick where
the person swallows himself into nothingness.
Some in the land of xxx has a tendency to
be almost entirely natural and eat only other
cz folks. But some are extremists and are
exclusively self-eaters, and would never
dream of eating anyone else. So we have
two categories here.
But there's a particular cz person, who
has a motto: 'I only eat cz persons who
don't eat themselves.' This crazy zombie
goes after all crazy zombies who aren't
self-eaters. ALL of them.
So far, so good. But does this cz person
eat on himself? Or herself, we better say
(women being a superior race). The cz girl
eats on herself, maybe: but she eats ONLY
and ALL those who don't eat themselves,
so if we say that she eats on herself,
we have to say next that she doesn't.
Try and look at it: it's a self-reference
that literally bites on itself, in a
self-contradicting way. It's part of
the goedelian way of imploding the
principles of mathematics.
Assume, for instance, that she doesn't
eat on herself. Well, then, she doesn't;
but that means that she is herself
including in her target group, for she
goes after absolutely all – without any
exception who doesn't eat on themselves.
So again we come to a contradiction.
These crazy zombies probably therefore
don't go around with such clear-cut
statements, at least not if they want
a decent measure of consistency in their
lives. But see the principle involved:
if action can be modified according to
a statement so as to negate the statement,
then neither one statement nor the other
can be correct.
Now, by analogy, can you make program
0006 so that it contains program 0005 --
which gives the answer about it -- and
so that it acts to negate the result?
Of course you can.
You simply feed the program location of
0006 to itself, having 0005 as its input-part,
its first set of program lines; then you
look at the answer -- is it 1, that a green
square is made in the center? Well then,
be sure to make a black square there and
green around it.
Is it 0, that no green square is made?
Well, oppose that answer in program behavior
by making a large, beautiful green square
there. Start program 0006, and you enter
straight into the sixth dimension.
There simply is no way you both can have
route procedure 0005 and also incorporate
that program within a larger program that
does this simple twist, this twisted
self-reference. But if you cannot incorporate
0005 inside another program, what kind of
program is 0005 anyway? Have you ever
heard of a program that cannot be
incorporated within a larger program?
Neither have I.
So, we DEDUCE, informally or by what
Goedel called meta-reasoning, that the
0005 program doesn't exist, cannot exist,
cannot be made, for any such making
leads to an infinite variation of possible
programs that opposes the answer in action,
impossibilising it.
And this means that a lot of programs
not consciously made so as to
impossibilize it also exist, that befuzzle
and bedazzle the artificial intelligence of
that program.
In short, artificial intelligence doesn't
exist; and Mathematics Hath No Principles.
Be glad, then, you have intuition!
And that's an intuition, therefore, that
can be used both in concrete areas of life,
and in more abstract areas, such as when we
purify our capacities to think deductively
by making smart programs or reasoning along
clear lines.
Having said all this, if you keep on thinking
about it for several days, once in a while,
also while walking, relaxing etc, and you
have understood pretty much of the assumptions
in what we have worked through, it is easy to
get a lot of questions about it.
I'm going to try and give a couple of
more comments, therefore, but rest assure that
the quantity of questions that easily can arise
after thinking about such which goes in the
borders between the notion of the finite and
the infinite can be great indeed, and a
considerable percentage of the logicians who
struggled with such questions in the 20th
century ended up as very peculiar people
indeed.
I advice you, therefore, to take it lightly
with all such questions, and not seek
absolute certainty.
We must work a little bit once in a while
in order to shore up new levels of insight
into our being and existence, but then we
must leave the meditations and let other
waves come and wash in on the beaches;
we mustn't cling to a sense that we can
nail infinities into a set of limited
propositions. But come back to such questions
regularly for the brain is a muscle that
needs exercise; and the brain is part of
your sexuality, dance, everything.
One of the questions that can arise is this:
can't we just have the program which is to
judge other programs allow the first to
run its course, and THEN it applies its
detectors on it, on top?
In other words, can't we just let the
program take a 'wait and see' attitude to
other programs? But we have to remember
the fact of self-reference: this is all
about self-reference. this is an argument
about an abstract world of possible route
procedures where all procedures themselves
can be subject to scrutiny by just these
route procedures.
So if the program takes a 'wait and see'
approach and is tied up to look at itself
that way, the program won't produce an
answer for it is waiting for an external
event that won't happen. But then the
definitive aspect of the route procedure,
viz., that it is to give a definite
answer to any meaningful well-defined
question, vanishes.
What if there are two different computers,
then? This is again an attempt to reduce
self-reference. You can perfectly well
line up any number of computers, each
one can photograph and detect the behavior
of the next, but the whole point of the
argument resides in the fact of sharing,
that programs exists in a kind of abstract
space where they can incorporate each other.
This can be practically realized by as many
computers as you have access to, but as
long as the program is accessible within
the sphere of programs that are generally
available when new programs are made,
then the program can be tied up to itself;
and if it isn't available, then it is not
a proper answer to the question.
What if the answer is produced on paper,
then? But no matter how the answer is delivered
-- whether easily and digitally or in some
cumbersome way which has to be photographed
and scanned and analyzed, we require of the
route procedure that the answer is delivered
in clear-cut terms. The program must give the
answer and we insist that the answer, alongside
the program as a whole, exist in an accessible
sphere -- if that's the phrase I want --
even if that sphere is a bit fuzzy in its
edges. So if it is produced on paper by a
printer, then we must insist that we also can
read in that answer by a program that incorporates
the first, and that larger program can act
to negate the content of the answer.
Finally, isn't it a bit arbitrary to have
a program construed in this manner -- to
negate an answer derived by a finely tuned
analysis of any program by a program? But
what we must take into consideration is
the original area of discourse, the original
zone of thinking: namely, that of the
notion of a Mathematics that Hath Principles,
complete route procedure principles operating
on 'for all members of so-and-so sets there
exists such and such property' -- and without
any neat pleasant limitation. This is a giant,
huge, abstract area, with all sorts of vast
structures possible; and with a vast sloppiness
as to boundaries.
And in such a situation, if there's even one
thread that gets loose, it starts running like
a thread of an old sock, and there's more and
more of the same.
We conclude, therefore, that the project
of such folks as Bertrand Russell and Alfred
North Whitehead is dead; it didn't work;
mathematics was a programme idea, but
a programme idea whose time has passeth.
Welcome instead the notion of the creative
intuitive deductive-loving thinking
programmer – and thinker in general --, who
uses something like a boundary aware G15
programming language and other boundary aware
concepts in ways that are perceived in this
moment's dance to be right. And let's go
beyond logic when we have to.
***SKETCH OF ROUGHLY HOW SUCH PROGRAMS AS
THE ESSAY ABOVE MENTIONS COULD BE MADE
IN G15 YOGA6DORG (such programs, when
included in essays, are not necessarily
performable without possibly much correction,
they are meant as a sketch of the code)
Why all the numbers? Why all the letters?
For the numbers encourage thinking about
psychologically meaningful data structures
which are actually the ones performed
by the underlaying electronics, and the
individual operations are actually what
is done by the electronics. It is like
having a machine with transparent plastic
cover, so you can get a sense of its
workings (there's no problem but little
teaching in making a program generator so
each of these programs can have lengths
of half a line or two lines or so; but
this would be meta-programming and a
second-hand relationship to data).
G15 program 0001, cfr
text above (the + signals that what
was just before it is a comment,
rather than an active part)
^nu #3000001 green_center+
v3 500 x+ v5 300 y+
v4 600 x2+ v2 400 y2+
wp #33000 draw_green_rec+
ret
G15 program 0002, cfr
text above:
^nu #3000002 black_center+
v3 1 x+ v5 1 y+
v4 1000 x2+ v2 600 y2+
wp #33000 draw_large_green_rec+
v3 500 x+ v5 300 y+
v4 600 x2+ v2 400 y2+
wp #34000 draw_black_rec_within+
ret
G15 program 0003, cfr
text above:
^nu #3000003 compare_ex1+
nins :#3050003 go_to_main+
^nu #555 subroutine_evaluate+
v1 1 pf1 8 prelim_result+
gi1 1 gi2 2 ad123 v3v1
gi2 3 ad123 v3v1
v3 1000 v5 :#270
v1gv3n5 branch>1000+
v1 0 pf1 8
;#270
ret
;#3050003 main+
prep #555 call_evaluate+
v1 997 nx 1
v1 1 nx 2 nx 3 warp
v1 @#555 8 w5v1
pf1 4 result+
ret
G15 program 0004, like
0003 but different start, cfr
text above:
^nu #3000004 compare_ex2+
nins :#3050004 go_to_main+
^nu #555 subroutine_evaluate+
v1 1 pf1 8 prelim_result+
gi1 1 gi2 2 ad123 v3v1
gi2 3 ad123 v3v1
v3 1000 v5 :#270
v1gv3n5 branch>1000+
v1 0 pf1 8
;#270
ret
;#3050004 main+
prep #555 call_evaluate+
v1 9998 nx 1
v1 1 nx 2 nx 3 warp
v1 @#555 8 w5v1
pf1 4 result+
ret
Shell around imagined G15
assembly program 0005, cfr
text above:
^nu #3000005 masterpredicter+
nins :#3050005 go_to_main+
^nu #888 detectorsubroutine+
v1 1 pf1 8 prelim_result+
..+
..+
..+
ret
;#3050005 main+
prep #888 call_real_oracle+
gi1 1 nx 1
gi1 2 nx 2
gi1 3 nx 3
warp
v1 @#888 8 w5v1
pf1 4 result+
ret
G15 program 0006,
extending and combining the
earlier examples; calling on
#30000005 (program
example 0005 of an imagined
master predictor or 'oracle'),
and on either #30000001 or
#30000002 acts opposite to
prediction, cfr text above:
^nu #3000006
location_for_prog:h1+
nins :#3000777 jump_to_start+
ret
..program_3000001_here+
..program_3000002_here+
..program_3000005_here+
^nu #3000007
;#3000777 startingpoint+
prep #3000005
v1 8 disk_h+ nx 1
v1 1 card_num_1+ nx 2
v1 1 nx 3 type_diagnosis+
warp
v5 @#3000005 4 w5v1 result+
nv1n :#8 nonzero_means_jump
result_0:action_green_center+
wp #3000001
ret
;#8
result_1:action_black_center+
wp #3000002
ret