Aristo Tacoma

Background. In the 19th century, and with feverent
activity and highly interesting contributions not in the
least from the hollender L E J Brouwer very early in the
20th century, mathematics as field came to be regarded as
something whose foundations are crumpling. There is no
need to mince the words. In the view of some, mathematics
never regained its clarity; though certain ways of
relating to the questions in the foundations of
mathematics came to be more or less mainstream for a
while. In this exploration, we will consider that we can
continue what was begun inside mathematics--including the
exploration of ideas of the infinite--without relying on
any definition of mathematics, whether according to
Brouwer or anyone else. Rather, we can use the type of
formalism we employ when we program computers to explore
not just structures involving finite numbers, but clear
ideas about infinity. But to do this, it can help to
extract some of the features of the various approaches
to the foundations of mathematics in an eclectic way.

It often happens that while a programme may be formulated 
according to idealistic principles with a definite ring 
of truth to them, the follow-up of this programme is far
less convincing. For while great energy may go into the
formulation of the motto or mottos of a new undertaking,
a level of confusion may set in when this is sought to be
put to practise. But when this is the case, it is
important to be able to view dispassionately and
eclectically what went on, so that we can pick out that
which makes sense independent on whatever less sensical
things went on. This seems to be a more fruitful
approach than judging the programme according to the
least attractive of its results.
  And in this spirit, I wish to call on attention to the
greatness in what L E J Brouwer, the dutch logician and
thinker, sought to create--in terms of a programme. As
soon as this programme was put into praxis, there are
features coming in that may not ring as true. We'll
briefly look into something of this, and move beyond, to
suggest what we think is a more coherent research
programme, but learning from both him and a number of
other good thinkers.
  Brouwer suggested that a great deal of clear thinking
can begin with the intuitions we have about numbers and
elementary arithmetic. He then sought to impress upon his
readers and friends the following: every idea we then make
in the wake of this--(let us not now call this 'mathe-
matics', for we are interesting in a somewhat novel
application)--should have a similar feature of intuitive
  In other words: Brouwer sought to call on attention to
clarity in our ideas--and to elevate doubt into something
that can be used to cast away propositions that entail
unclear ideas.
  Of this I fully agree.
  Let us put this into contrast into what has sometimes
been called the 'formalist' point of view, or approach. 
Here, meaning is de-emphasized; and rather permutations 
of abstract symbols according to strict rules are what 
is in focus. We notice that the computer idea, and the 
computer language (such as G15 PMN) indeed involve very 
quick permutations or handlings of signs, which may 
indeed be very abstract. We have, in the 21st century, 
in contrast to very early in the 20th century, machines 
that do the 'formalist' type of activity to some extent. 
We can therefore, with some leisure, make programs in which 
not every bit of them have supreme clarity, and then 'check' 
them on the machine and regard that checking as a meaningful 
degree of mild 'verification' of the validity of what we 
did there.
  In that sense, dealing with computers let us pick out
something--quite eclectically--from the formalist or
attitude, at least when it comes to finite structures, even 
as we can declare full agreement with the aforementioned 
core postulate of Brouwer.
  There is another approach, held by a significant number
of people who have worked hard in fields of thought
related to those Brouwer worked in,--sometimes called
the 'Platonic' or, a bit confusingly, the 'realist'
approach, or point of view. Again, it must be possible to
look for clear ideas also here, and apply a measure of the
eclectic (indeed, thinkers such as Kleene, influenced by
Brouwer, sought to combine Brouwer's "Intuitionism" with
such a "Realism", but not necessarily in ways that
Brouwer would have approved of).
  The idea employed in the Platonic view is that there is
something more to reality than what the senses can pick
up--something mystical and pure--and that can be reflected
in formalisms, also when they deal with infinity. For
instance, a thinker working within this approach may argue
that a perfect circle hasn't been seen anywhere by the
senses (for the circles around us aren't absolute in their
perfection, at least not when studied with better
instruments), but the idea of the circle exists out there
in the platonic realm of pure forms somehow, and our
formalisms can reflect that. Some of them will also say
that our minds can pick up these forms by intuitive
perceptions, AS WELL as represent them in our formalisms.
  This Platonic view then ascribes a reality to abstract
notions, at least of some kinds: and so the word
'realism' has been used to characterise it.
  For someone who is used to look for clear ideas and who
is attentive to unclarities about ideas that MIGHT seem
clear before we look on them further (consult e.g. the
chapter on "Long strings, longer strings, and infinitely
long strings", in the "Learning PMN" document linked to
from, it may at once be pointed out that 
the idea of a circle isn't altogether formally clear. For 
in order for this idea to be formally clear when our 
foundations are thinking about whole numbers and arithmetic, 
it is adamant that the idea of the perfect and infinite-
decimaled number PI, or some suitable related number or 
formula, is also regarded as clear, so that the shape of 
the circle can be easily described by means of only clear 
ideas and the corresponding formalisms. But a number of 
such a kind requires that one fully agrees to the definition 
of PI, and of that kind of numbers in general, and, armed 
by the slogan of only accepting clear ideas, we may not 
get to that point. In other words, while the Platonic 
or 'Realist' point of view may have something to it, we 
cannot at once jump into agreement with a lot of 
assumedly 'easy' and 'obvious' examples.
  Rather, in one of our texts on infinitely long strings 
(referred to just above) it is indicated a way by means 
of an idea that does indeed seem to be clear, unfolded 
by means of applying the G15 PMN formalism on an 
imagined "infinite computer", and suitably extended with 
a number of the "infinite" kind, denoted there "...", 
as three dots, --by this we can come to what may be fairly 
clear ideas of infinite arrays.
  However, these may not correspond to what was typically
described either in the Realist or the Formalist or in any
other school of enquiry into formalisms. We are then at
liberty to consider the proposition that such infinities,
of some sort, can be said to have a more or less "real"
existence, perhaps in some vein vaguely suggested by
the ancient hellene Plato. Note that we do this without
calling into question a core proposition by L E J Brouwer,
namely, that we must only allow clear ideas.
  Also, when we do explore formalisms, we may admit some
value to some formalisms that not at first correspond
to clear ideas when we get machines to check them, and
imagine that if we wanted to, we could invest attention
into the analysis and synthetic understanding of these
formalisms so that also these would show up to be pure
ideas, or else unclear ideas that are discarded. Indeed,
when we speak of First-Hand Relationship to Data, and,
in the field of programming, of First-Hand Programming,
or '1st-hand programming' for short, we are indeed
emphasizing the validity of restricting the use both of
formalisms and computers to that which admit to clear
ideas--to our having a psychological, meaningful relation-
ship in thought with them.
  Is this unduly restrictive? I think not, for we must
ask what type of society, and what type of science we
want: do we want a society and a science that is
increasingly embroiled in complex formalisms and programs
and systems outside the practical possibility of human
understanding? But what factor will then ensure that we
create a meaningful society, and a meaningful science?
What possibilities are there for correction, when we
must use machines to check the activity of machines, but
no longer retain in our minds the capacity to check the
very designs we implement in our machines, or in the
formalisms governing them?
  So 1st-hand programming is an ethical concern. And it
ties in well with a core postulate by L E J Brouwer--
namely, that we must be willing to say that clear ideas,
meaningfulness in what we mentally construct in
connection to the formalisms we evolve around numbers,
must remain the chief criterion as for our acceptance
of them. Those who are fond of -isms {this writer is
however not fond of -isms except in a few well-thought
cases like 'impressionism' in art, and 'small-business
friendly capitalism' in politics} like to try and put 
Brouwer's works under the slogan of the baggy concept 
of 'constructivism', where the idea that it is essential 
that mental ideas are constructed as we go along 
developing formalisms. But this isn't necessarily all
that clear--and clear ideas, and willingness to admit
doubt as to clarity as good enough foundation to reject
a proof, are far more powerful postulates.
  For reasons that to this writer clearly seems to be
tied up to historical pressures of the various schools of
thought with which Brouwer, at his time, was concerned
with, and focussed at, Brouwer ALSO postulated things that
--seen outside of this context--seems to be not at all
necessary given his core emphasis on intuitive clarity,
or intuition.
  In particular, Brouwer became concerned with focussing
on the way in which so-called 'proofs' were made, and he
found a number of cases where the proofs relied on the
power of negation, but without leading to clear ideas.
To summarize a great deal of highly complicated patterns
of reasoning, we might then say that he challenged the 
notion that by showing that something is false, it 
doesn't follow that 'the opposite' is true--and this
especially when engaged in proofs connected to the
infinite. (This has sometimes been expressed as Brouwer's
opposition to 'the law of the excluded middle'.)
  While we regard it as a flash of genius in Brouwer
to engage in this opposite AS A FORM OF DOUBT relative
to some previous (and, we agree, too-quick) works on
infinity, it must also be said that in the opinion of 
the undersigned, once it becomes an opposition to the
use of negation in a broad sense, it doesn't ring
intuitively clear anymore. Brouwer's work in this regard 
doesn't seem to follow from his core postulate (on there
being only intuitively clear ideas) (an emphasis that
he, by the way, shared with the French philosopher Renee
Descartes who sought to employ it on a number of themes,
including geometry and some form of existential
philosophy, centuries earlier).
  While it is clear that in natural language, the denial
of a statement leads a lot of room open as to what, if
anything, is affirmed by this denial, it is far more a
clear idea, it appears to me, that when we have achived
a formal expression which is FULLY understood, we are
also able to fully appreciate the depth of negating this.
And a clear-cut negation of something false IS an
achievement, however small it is: it is indeed an
achievement towards some kind of deduction of the
opposite. Because, indeed, in formal language, the notion
of 'the opposite' do often come forth as a clear idea.
However, when the starting-point is influenced by unclear
ideas of which we may not be entirely aware, a negation
means much less: and only in such more narrow cases, does
it make sense to open up the middle ground 'between the
true and the false' as it were.
  In human activities, arithmetic is needed, in some form
or another, by everyone, from childhood and up, and in
some avenues of life, all the time. The natural extensions
of arithmetic into simple forms of (computer-friendly,
whole-number oriented forms of) trigonometry and such,
as well as suitable calculations in various fields such
as electronics, mechanical pressures, chemistry, and so
on, are entirely natural. These applications involve
rather finite structures and finite measurements.
  When we find that we want the clear-cut conciseness of
a good first-hand formalism to show things by means of
some degree of the dialogue with oneself one can have by
means of computer programming in a language like G15 PMN,
then we can also quickly reach the understanding that
there are possibilities for careful, even scientific
quests into what it would mean to apply certain notions
of formalisms and certain ideas of the finite when
translated, in one way or another, to what we may surmise
as "the infinite".
  Since the work published in 2004 by the undersigned, 
privately {a book available under my pen name Stein von 
Reusch both in the Firth platform, in the main text folder 
there as the document named A.HTM, and in the National 
Library of Norway, at, as well, and the book
title involves the term "physics", for it also introduces
the first public formulations on the so-called super-
model theory or interpretation of physics}, I have 
maintained  that the following proposition has been 
  The REAL exploration of infinity, where clear, coherent
ideas are honored all the way, hasn't yet begun very
  While at the time I wasn't able to go further with the
particular early formulations I had of this at the
University of Oslo (which led me to break contact with
them instead of revising this bit out of a thesis), I am
optimistic that the time will come where clear ideas, in
the spirit also of the core postulate by Brouwer as
mentioned above, will regularly be produced on the nature
of infinity. What with one thing and another, Brouwer did
not feel at ease with features of what Georg Cantor had
done just some years previous to his own first
productions. As the years went by, Brouwer both challenged
some features of what was said about 'real numbers' but
then also accepted some other features of these. He
sought to speak of 'potential infinity' in contrast to
the idea of 'actualised infinity' that he meant that he
saw in the works of Cantor-influenced thinkers. For me, it
is clear that I do not regard that Brouwer went far
enough in his demand on clarity in ideas. The works done
by Cantor can and should be challenged as for clarity in
ideas to a much deeper extent than what has been heard,
in the mainstream 'history of formal thought' as
summarized in the encyclopaedias we find around at present
and in the best-selling books on such themes.
  In other words, this writer feels that there has been
a lack of doubt, and a too liberal attitude as regards
certain proofs, which again has led to what might be
called, perhaps, an "arrogant" relationship to the
infinity concept in later mathematical thinkers. This has
carried over into the computer industry with the easy-
going and superficial repetitions over the (illusory)
theme of "Artificial Intelligence". This may again be
seen as part of a larger but dark stream of human thought
that seeks to reduce the human mind and body to mere
machine-like processes,--although some of the thinkers
are incorporating an openness for something 'platonic'
as possibly 'real' on 'the other side'.
  This development, or lack of development, fits with a
statistical emphasis on computer programming, in which a
language more often is judged according to 'what it can
do'--what it can manipulate of existing over-developed
structures of computers and networks and robots and
computer-related things--rather than according to the much
more significant criterion of whether the language is
esthetically pleasing and reflects clear ideas.
  For certainly, once we admit the importance of the core
postulate of clear ideas--where we do not have to use the
somewhat cumbersome notion "intuitionism" not even the
over-used concept "mathematics"--we must regard it as
adamant, necessary, vital that when we deal with finite
numbers on finite machines, our languages are clear and
easy-going and allow first-hand thinking and a relation-
ship to the data that is direct and meaningful. For only
with this foundation in place, can we venture into the
important (for human well-being) explorations of the
  Each person can, in this regard, by adopting such a
passion for clear ideas, regard herself or himself as a
"researcher". The scientific attitude involves engaging in
research without putting ego first. It involves being
willing to look for perceptively clear notions and
insights into reality regardless of emotional judgements.
In this way, we call on feelings of a deeper and more
true kind than the emotions of prejudice. This type of
research, also into infinity, can then call on words, such
as can be explored in an essay. The resulting essay
category, whether we call it 'philosophy' or 'science',
can then fruitfully also engage formalisms of a suitable
first-hand type, such as run on a digital computer, but
--with great care, and as thought experiments done with
sharp, good attention--applied to the infinite.
  I have evaluated the G15 PMN programming language from
the point of view of having made it in the past: in other
words, I am done with attachments to it, I have done so
many other things since it was complete; and I have
completed other, though related, languages before. I have
had a chance to evaluate whether something else should be
made. I have had several opportunities to use the
language to explore themes in thought, and I have, indeed,
done so. The language G15 PMN does live up to the desires
for such a suitable formalism as this essay calls for. It
is a language that stands forth in a complete form, and
that doesn't need a redoing to serve this purpose. Had 
there been another such language available, I would gladly
have been using it instead of making a new. There is no
other good programming language and no other good formalism
that stands clearly forth as a golden pathway into the
future of all our upcoming explorations. G15 PMN became
a far better vehicle of good thinking around clear ideas
than we had dared hope for. I think that it is an objective
statement that this is infinitely better than any formalism
mathematics has ever had within its realm, when it comes
to shaping new ideas of the finite AND of the infinite. I
suggest, now, that those who seek clarity in their own
minds about also philsophical themes in the wake of a
sense of caring for numbers and elementary arithmetic
as soon as possible realize the unique powerful features
of this formalism and start using it, both on a computer
and in thought experiments allowing a safe leap to
infinity and back, along the lines suggested in the
chapter on infinite strings I referred to earlier.
  Let it be clear that--as we've pointed out in some 
other (relatively complicated, and earlier) texts on
infinity, that the whole area of infinity may be, in a
sense, infinitely more complex than what it first appears.
Each exploration into it must be willing to admit that.
As soon as we admit the notion--in my repeated postulate--
that a coherent, clear-thought process of developing a
notion of the full complete set of all whole numbers
involves a lack of possibility of cutting away a new
and different type of numbers of an infinite kind, we
are led into a whole host of implications and these do
not follow mechanically from the starting-point. Rather,
as Alan Turing also pointed out: some of the works 
we do when we have a starting-point which is rule-based
or mechanical follow mechanically, as it were: but some
other features of it call on fresh attention, fresh
insight, or intuition, or these leaps to new features
won't take place. To call on intuition and a willingness
to distinguish between clear and unclear ideas involves
a sense in which we invest a form of reality to that
which is beyond the sensory--something that, if we wish
to expand it into some grandiose (and complicated 
concept) involves (what I've earlier named as) a form
of 'metaphysical empirics'. 
  When we grant the infinite complexity of exploring
the infinite, even as we begin with what we take to be
clear ideas and a simple starting-point, and add to this
a strong dose of doubt as to ALL historical explorations
of infinity, THEN we have a new form of scientific
research programme in a coherent form. We do not have
to call it anything snappy, we do not have to make a
new -ism or a new -logy or a new -ics. We can simply 
call it a use of a formal language in a research into
clear ideas also about infinity. A long phrase, 
connected to a form of research which we must blend with
many other forms of research and practises, so we do not
overindulge in this form of mind-boggling activity. We
must protect our minds as we engage into a fresh
thinking, where we find out more about the universe and
our minds--the multiverse and our manifold minds--than
ever before. This is our quest. The G15 PMN is the
clear-cut concise tool created after decades of intense
awareness and questioning into all sorts of themes like
these. Let's wish each other luck in using it for future
research, blended with excellent natural language, good
English, and suitable meditations when we explore
the infinite, and new features of the finite in light
of the infinite, in the future.

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