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