Aristo Tacoma [[[ESSAY found at norskesites.org/essay20110913.txt Written talk. Yoga4d.org/cfdl.txt redist license. Consult published works as at yoga6d.org/cgi-bin/news/f3w which are containing key concepts connected to what this writer calls 'neopopperian science' and also 'supermodel theory', which involves also perceptive processes -- concepts effortlessly drawn upon here.]]] ON NUMBERS, INFINITY, LOGIC THOUGHT AND THE WORLD -- And some brief notes on the f3 and golden ratio Now I happen to know that some of you have given time to some forms of science, philosophy or logic, but I do not know exactly what all of you have worked with, or how well you have succeeded, personally, in bringing it all together as a meaningful whole. Does all of logic appear to be nothing but a collection of puzzles to you? Well, in that case you may have hit the mark perhaps better than another, who possibly thinks that logic is nothing but a collection of cut'n'dried simple answers to the full set of big questions. In this informal miniseminar, a bit of higher education in a sense, I will sketch some ideas that to me represent insights, and bring in some bits of the f3 programming language so as to show that it is coherent in thought to do so -- it is a fluid transition, from thinking about these things to sketch this and that and the other bit in the Lisa GJ2 Fic3 formalism. First, let us muse over the question: what is the relationship between language and the world? By language I mean also anything we do in logic, and as a formalism, but also anything we say such as in English, in natural language, and I also mean our togetherness in doing so. By world I mean the wholeness of existence in the most ultimate sense, but also the beaches, the flowers, the long legs and the sex and the food. So what is the relationship between language and the world? First, though it would not be practical, we could imagine an existence in which people didn't talk, but wandered about in silence, eating, making love, bathing, swimming, -- the world still exists. So the world is simply not dependent on language. Are we seeing this together? The world is beyond language. I do not mean that it is easy to say -- that is the world, and this thing over here is language -- for in each particular case, we may be a bit confused unless we have given the question great attention. But as a general statement, the world is the greater, and language is the smaller. Language is within the world, not just possibly but certainly, but unless we are trying to take the point of view of God writing the universe into existence along the lines of a story-writer, then, for all practical purposes, for humanity and for the muses -- you know? -- for all but God, language occurs within the world but the world isn't dependent on language. Naturally, language may engage cause-and-effect and such within the world, but the world is still the greater. This is a fact, and the fact can be called a kind of humility that language must have relative to what is not language. Put in other words, actuality doesn't depend on thought. Thought must have the humility to regard the infinity beyond itself as essentially beyond it, and as the carrier also of what there is of thought. If there is any disagreement about this, let me propose that it may be that those opposing this humility are more interested in expressing themselves -- at best it can be called poetry -- rather than being honest about the understanding of the general features of reality. There are those who try to make a kind of expressive type of language-music in which questions like the ones we just touched on are brought in, and negated whismically. I don't think they REALLY agree, but I do think that they are trying to make themselves interesting and perhaps, for fake reasons, try to make others say that they are genius-folks, when they are simply throwing words around clownishly, without awareness or attention to what they are doing. I would not recommend it, in short. If it is a kind of self-therapy, I would suggest more humble means of self-therapy than throwing about words clumsily concerning the greatest questions of what thought is to the world. No matter, then, what thought portrays to itself as a so-called General or Deep insight or a Pure Deduction, it may be that it has nothing at all to do with the world. It may be that it is but a piece of false play, a false note, having nothing to do with anything. The world is not constructed by thought, but it exists and it is real and actual, and then thought tries to construct something and it may call it 'the world', but what thought constructs may simply be a map which is a map of nothing at all -- the map of a fantasy-land. But what we are suggesting, when we look at the words we just used, is that thought CAN indeed relate to the world beyond it somehow. Some thoughts match and some thoughts do not match. In the case of the weather, it may be simple to work out; in the case of something like reincarnation, it may involve a large number of assumptions and a great deal of intuition to be able to do, as I frankly do -- and claim that reincarnation indeed is real, for all humans, always. But the word-sounds themselves may mean one thing in one religious context and another thing in another religious context, and so we cannot always speak of how well a thought matches reality just like that. We must rather speak of the whole network of thoughts, the net of assumptions, and we can weigh, intuitively, whether we have many or few or very few or possibly none illusory assumptions in a context. Is the network of thought involving CORRECT assumptions? So the notion of a CORRECT assumption, and the notion of an INCORRECT assumption, means that we are allowing ourselves to pass judgements on thoughts of some types. Not every thought is claiming anything much about the world. Those that do claim something about the world, we attend to; and we also attend to the world; and we attend as best we can to the degree of matching, connectedness, -- and from this attention, we pass a judgement. When this goes deeper than the sensory organs, we speak of intuition. When we focus on sensory checking, we may speak of experience, or data, or even research data, and all of the latter we can also call 'empirics', which is a word meaning experience, but experience of a kind that has been paid very clever, analytical, good, honest attention to, whether in a scientific context or not. Note that we have not spoken of whether the thoughts of one person is somehow viewable by another directly or not. In other words, we have not spoken on the nature of mind and minds in this world, whether they are all like petals of a rose or more divided, whether they can be seen by higher beings -- as I in more religious contexts suggest that they can -- and all sorts of things like that. It is also not easy to very quickly, using experience, if you have been deeply and intimated related to particularly mind-sensitive or 'psychic' people, say for sure just how independent human minds are. Let me say that I regard human minds as largely interrelated, but that they are also having more or less private zones to some extent. Now, where does logic and formalisms and such come in? Once one has the essential humility that thought doesn't for sure say anything about the world, one may find that the world tends to have, as shown by objective experience, certain regular features, which may look very similar to some patterns one may model e.g. on a computer, or just think about, in more or less formal terms. When one then, with humility -- always with humility! -- plays with something like a little computer program, in a vague inspiration from experience, also intuitive experiences, to use that phrase, -- one may find that the program, in turn, inspires one to look afresh at the world -- perhaps with a new sense of question. Is it then science? I would be careful to use the word science just like that -- see, if you have time, the list of points that I think one should consider, as a kind of checklist, before considering whether something like an article could be considered a contribution to science. Let us here more humbly say that at present, we are exploring. For instance, have you thought about there being something about long legs relative to a short torso -- something a child may have, it is not about overall height -- which relates in turn to what is often called 'the golden ratio'? The longer legs, given some approach to the measurement, may compare to the torso, the upper body including the head, by a factor of something much like 1.6, a number which is not far from 5 divided at three. It may apply in more ways than one, also relating to the hips accentuated by a g-string (which is a general, generic name and we do not need any such name as "t-string", in the qualified opinion of this beach meditator). Why does such a golden ratio speak to the mind? For the mind, eager to adjust thoughts to fit with reality, must be aware of similarities of contrasts and contrasting similarities of all kinds, not just some kinds, since both thought and the world beyond are so infinitely complex. And there is something about the golden ratio which involves a repetition of similar contrasts within one another, and also spiralling out: in order words, the simple shape may be a visual metaphor of a type of ensnaring infinity; an infinity one may also encounter by interesting angle-similiarities and shape-likenesses between such as lips and jaw, lips and eyes, the high wrist of a girl's foot with the curve of her thigh, and so on. Each girl of beauty is an equation onto herself, unlike all others in a marvel that speaks of infinity. The religious mind is, like the scientific mind, willing to concede that beauty speaks of the highest order of the universe, and as such do not deny the sensual, nor the sexual which is always naturally part of the sensual. So the love of beauty is part of what a true understanding of God must be, in the opinion of this thinker. Let us for the sake of starting with something formal, not because it is exactly the most general question or theme one can touch on, but because it is inspiring for our artistic aspirations, look at the numbers which, when divided pairwise on one another, tend to be more and more accurately the golden ratio -- for numbers which each are within the typical computer range of 32-bit or about plus minus two billion. The series of numbers goes back in the history of number studies to one Fibonacci, and so we speak of 'Fibonacci numbers'. (LET FIBONACCI-NUMBERS BE (( )) (( }* .. }* )) OK) So we are going to make something formal, give some flesh to a structure like this. The formal language is a kind of mantra, it is a set of reminders, or rituals, so as to bring about a contemplation for the type of activity that is resonant with doing formalisms. So it is important to not just look for computational content, or calculations. We must allow the art of the formal structure to arise for a sense of beauty and wholeness in the mind to come about simultaneously. I find personally that doing even small programs once in a while make me open ever-afresh to the beauty of girls. It helps art. It helps paintings. It cleanses the mind. When we program, therefore, we are going to do things which are not always as compressed, if that's the word I want, as it can be. We want to expand a thought by means of the formalism f3. Later on, it can be compressed more, if we have reasons to make a quicker or more compact version of it as part of another program. But it is typically so that when we watch the world, and vaguely appreciate some patterns, these have to be expanded before they are compressed. But in art, in a way, you start with the compressed sense of it, the contracted image within as it were, and then expands it when you are drawing or painting or dancing it. In that way, it is not an imitation. Fibonacci noted that adding two numbers, like 1 and 2, gives a new number which can be put to the right of the two first, so that the process can be repeated; and that when it is repeated just some times, the ratio between the two numbers quickly settle to about 1.618. So 1 and 2 gives 3. Then 2 and 3 gives 5. Then 3 and 5 gives 8. Already, 8 divided at 5 is much the same as 5 divided at 3. And even 3 divided at 2 is a pretty good approximation. ((DATA FIBO-NUMBER-A )) ((DATA FIBO-NUMBER-B )) Just to be very explicit, this time we do not use anonymous variables inside a function, but spell it out. The two numbers defined -- higher up in the program than the function which we have still to give flesh to -- then should have some initial values. Let us say 1 and 2. (( 1 FIBO-NUMBER-A <N1 in the function header. To print a number on the screen, with a space after, we can use the word POPS. This works fine in text mode. In graphics mode, we would use something like B9-POP. But here we assume text mode. Finally, the <>> is what we want to retrieve the value from the variables, as they get higher and higher. The word INTGREATER compares two whole numbers, two integers. The word (MATCHED and the completing word MATCHED) then leads to an action on condition of the comparison working out. We would want something like a GOUPn here, such as GOUP1, where the number 1 refers to a point higher up in the function to return to, because the function is not yet finished. This we indicate by GOLABEL1:. The addition we do by the word ADD. All right, we are getting something like this -- where I put in the extra blanks, the extra (( and )) and the extra => in order to make it easy to look at -- using the conventional correct type-setting of the Lisa GJ2 Fic3 language of mine. Also, I use N10 and not just N1 to store some temporary values, for each function has its own set of eleven such, N1..N11. (LET FIBONACCI-NUMBERS BE (( >N1 )) (( (( FIBO-NUMBER-A >>> => POPS )) (( GOLABEL1: )) (( FIBO-NUMBER-B >>> => >N10 )) (( N10 => POPS )) (( FIBO-NUMBER-A >>> ; N10 => ADD FIBO-NUMBER-B < INTGREATER (MATCHED (( GOUP1 )) MATCHED) )) )) OK) This is all well and good but if we want to run several test-runs of it, we would like the numbers to be reset to their beginning, and perhaps we want a more snappy function name also. Let me put those initial statements giving start values to the variables inside a function with a quick name, which then calls on the above. The completing line, CRLN, puts in a lineshift. This is text mode, so we have to allow the lines to be a bit unformatted -- a number going too far right will be chopped and its completing digits will be all on the left side -- just like political extremists of the right equals much those on the left ;) (LET GOLD BE (( >N1 )) (( (( 1 FIBO-NUMBER-A < FIBONACCI-NUMBERS )) (( CRLN )) )) OK) I put the above in a file called GOLD.TXT, then started F3 by the command F3, and typed :GOLD IN to load it. Then I typed 50 GOLD and the computer displayed 1 2 3 5 8 13 21 34 55 Naturally I couldn't hold myself and put in really large numbers just for fun. It is a good thing, to lean back and have the computer do such tasks, stretching it a little bit, a kind of BDSM whip on it. So is every programmer a sado at heart? Let us note that 34 and 55 looks a bit like 3 and 5. But, clearly, 5:3 and 55:34 are not exactly the same number. It seems that we can get from 5:3 to 50:30 by multiplying with ten, multiplying both factors in a ratio by the same number doesn't change the ratio. But to go from 50 to 55 we would add exactly one-tenth of 50; whereas to go from 30 to 34 we would add more than one-tenth of 30. In other words, the number 55:34 ought to be slightly smaller than 5:3. I go into the F3 again, and use it as decimal calculator. The function \R divides using not just large whole numbers, but also such large numbers with decimals, and R( prints them directly in the text mode of f3: $5 $3 \R R( and get a number like 1.666666 and more digits like that, completing with a 7, as a rounding up. Then I type $55 $34 \R (R and get cirka 1.618, which confirms the little informal calculation. (The $ and the 'R' in the above refer then to such caculations, which in f3 formalism is called "using the rich stack", so R for Rich.) It is often so that the mind has a greater joy in going into something deep, if one does it without too much exhausting detail in each session. But there is always a progress. Next time we explore, we can surf through earlier insights with swiftness and delve right into something new. Then, coming back to earlier insights in a new round, one will have learned something of the overall progress and can see new aspects one didn't pick up the first time. I think that reincarnation is just this: the aptitude for learning is heightened, even if the memories of past lives may be very far from concrete indeed. It is not about ego. But before we complete this little exploration into some great features of the world, I wanted to touch on questions of the infinite, for the human mind needs a little bit of the wilderness of a beach of this type. Not too much at a time, and certainly not all the time, but as a contrast to questions of finiteness. Let us appreciate the truth of a falseness. In other words, let us dwell a little bit on an illusory way of relating to infinity, so as to see more of a right way to relate to, and respect, that awesomeness called 'infinity'. What I have in mind is that some people, on looking at the numbers such as the Fibonacci numbers above, and noting that as they go from a range such as cirka 50 to such as, for instance, cirka a million, they get steadily more decimals in place of a number which tend to level out -- namely this golden ratio -- where the first digits indeed are as noted with 55:34, namely 1.618. You can also switch the ratio, 34:55, and you get much the same number just one less, namely cirka 0.618. So they notice, aha, we go from 50 to a million, but what if we simply let it go 'on and on'? And they try and calculate around this 'on and on' and a large number of people, including such as the logicians Abel and Cantor and Russell, all have had a tendency to regard this as pretty much an easy thing to do. But with what right can one assume that any such thing as addition works once one lets numbers go on and on? It is not obvious that it can work in a context which no longer has a well-described boundary on the numbers. What will an endless, infinite context do to the addition? What will it do to the comparison? What will it do to the division? What will it do to the storage of two numbers? Indeed, what is a number, once the context has to an utterly extreme degree transcended all possibility of being part of the manifest universe, and is a mental fleeting idea of boundlessness? One must be in a great haste, and rather hypnotised, to avoid conceeding that on changing from a context of something such as f3, which has a well-defined range of numbers essentially between minus two and plus two billion and a related range for decimal numbers and for the amount of digits involved in decimals, to a context of presumed endlessness, it is extremely questionable whether there is a well-definedness of absolutely every every other feature of this context. And, indeed, the more one looks into what happens when one crosses from having a range of numbers such as 1..1000, to having a kind of 'et cetera' situation without any upper bounds, the more one will see that this involves a full change of the senses of all operators and even of all numbers. I am not claiming that the infinite is not real, but I am claiming that as human thought grapples to come to terms with a finite calculation, it cannot merely paste the letters "et cetera" or some three dots after a number series with a clear-cut beginning, and assume that it still grapples with the situation. I say the infinite is real, indeed that the infinite in some sense is the foundation (not basis, which means zero, but foundation, or ground) of all and everything else -- including the ground of the finite numbers and the simple forms of addition, division and comparison we can do with these finite numbers. The world IN ITSELF is infinite. Thought, and more generally the mind, may have something about it which is infinite. But then thought must be humble enough to regard the infinite as living and coherent and beyond such simple rules as one may find to apply, as it were temporarily, when we play finite games. Let us round off with a brief reflection on the relationship between the idea of the golden ratio as simlarities of contrasts of interest to art, and what we can rather immediately surmise when we look at the Fibonacci numbers, that finite series. When we watch 1 2 3 5 8 we are watching 1 and 2 together as 3, 2 and 3 together as 5, 3 and 5 together as 8. So there is a sense of containment, one within another -- and, as we have just calculated on, with something of the same relationship or ratio being preserved. The mind is doing its visual calculations spontaneously on all that it apperceives. Thought knows, for instance, how to chop a square out of a rectangle -- subconsciously, we do such things all the time, in order to weigh and ponder and muse on the order of great things. And so, when a shape is about 5 units -- it may be centimeters -- along one side and 3 along another, then a square chopped off would mean 3 times 3 chopped off, leaving a smaller shape about 2 units to 3. But that is again part of the Fibonacci series. So you see there is a kind of containment, or spiralling reflection of the Fibonacci series, in visual shapes having such proportions. Let me be clear that I regard Fibonacci numbers as an EXAMPLE of similar contrasts and contrasting similarities involved in perception, and that beauty is not one but several.