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