T H E O R E M S

B A T C H 1 The newest theorems Written by Aristo Tacoma. Each can be further distributed when copyright license Yoga4d CFDL (as on yoga4d.org/cfdl.txt) is respected. This is the newest batch of theorems. For the foundation batch, click here. Still, the bottom part of this batch #1 is also regarded as fairly foundational -- the line of *'s divided the bottom part, which is stable, from the newest theorems, which is above that asterix line! ************************************************** UNDERNEATH THIS LINE: FAIRLY FOUNDATIONAL THEOREMS AND THUS A STABLE PART OF THIS BATCH #1 Theorem: Sort isn't a first-hand concept 1::B::2013::11::30 Background of terminology: In this section with theorems, we use the words with a precise enough meaning within this context, which is a context informed by an understanding that the infinite is a subtle concept. Given the work on such as essence numbers, it is a context in which it is clear that in a vast perspective, anything said about the so-called finite must be imagined as taking place inside an infinite realm -- rather as a "contact zone" between two forms of pulsating infinities. It follows from this that the infinite is irreducible to finite concepts and that it is out of the question to "build" the concept of the infinite by some attempt at extrapolation from the finite. The infinite can be said to "behave" differently in key ways (and as such, the concepts sought to have been used in e.g. the 20th century -- such as "approximating a limit" -- are found to be inadequate to sum up a coherent clear set of ideas about the infinite). This leads to an understanding of words such as definition, theorem, corollary, deduction and such in which we will normally not assume that it is practically possible to make explicit, at least not in the form of one article or even one book, all the boolean-logic type of strict inferences necessary to fully provide a logical proof of anything such as a general or metaphysical point about infinity-related concepts. And even if we are presented with the whole explicit schema of inferences, these in turn start with definitions and axioms that require a perception into the infinite to make any much sense. Instead, when we use the word 'theorem' here, we intend by it to say that we regard it as within the "landscape" of correct deductions but where the mind of the reader must engage at an intuitive not merely logical level in order to apperceive this reality. Background of the particular theorem: In the works such as in the 20th century connected to algorithms, in particular computer algorithms, a recurrent theme is that of imposing what is called a "sort" on an unorganised list, perhaps a list of names or numbers, which can be very large indeed. Algorithms have been proposed with some sort of subdivision of lists such as "quicksort", which, after many permutations, puts one name after another so that all those e.g. with letter a come first, then the letter b, and so on up to z, and at the second position of letters after the first, the same sequence is imposed also, and so on all the way until the length of the name in each case is reached. Before the era of interactive personal computers, humanity relied to a large extent on printed-out paper catalogues sorted by such a principle in order to retrieve data from large collections. (A computer may mimick such a search by what is called "binary search".) However, there is no necessity of employing this type of organising principle in order to obtain a quick look-up using a personal computer. Nevertheless, the notion of sort of large lists has come to dominate 20th century computer science to an extent where it has been, by many, regarded as part of its essence. In contrast to sorting of large lists, which is a complicated process, the sorting of very small lists is intuitively simple by the completely different algorithm often named "bubblesort" (and similar such), and which form a typical sub-component of the larger and involved sorts. Theorem: Sorting a significantly sized list into such as an alphabetical or numerical sequence isn't in general a core part of first-hand computer programming. Deduction: Our informal deduction of this theorem will then be an attempt to summarise the "milestones" in the "landscape" of definitions, axioms, deductions, sub-theorems and corollaries to these sub-theorems, all needed to erect this theorem as a proven statement. This summary will be extremely brief and it will be readable only in an intuitive sense, but it is proposed that a full deductive analysis COULD be made. First of all, in watching how humans are naturally organising information, it is clear at once that what is grouped and the sequence in which things are grouped relies typically more on meaning, "fields of semantics", rather than on names or numbers. In watching fifty girl names and ten names of machines, the natural groupings are those fifty girl names in contrast to the group of the names of machines, regardless of the characters in those names. Similarly, for people who regularly work with numbers, the patterns of digits inside a number evoke various meanings and the instant recognition of such patterns in the numbers presented then become a natural way to group them. These groups may then loosely have a sequence such as from left to right which may not have much to do with the sizes of the numbers. Similarly, a computer, having a range of algorithms available, and also a set amount of ram addresses with the psychological meaningful 32-bit data range, will not necessarily store incoming data in a sorted way when we speak of large lists. For example, in some contexts it may be more natural for the computer to do a rough grouping -- e.g., it can be derived from a hash number principle. The hashing algorithm -- which also can be said to generate an 'implicate key' from the 'explicate key' or 'explicate data' given to it, -- can be proposed to be a KIND of perceptive or semantic ingredience, to some extent. And it is very clear that the sequence of the hash numbers typically have little to do with the sizes of the incoming numbers, or with the alphabetical sorting if what is hashed over are names. This is not to say that size of numbers, or alphabetical sequence -- two concepts that really, at the level of data, are the same -- is not sometimes having a component of meaning. And meaningful relationship to data is indeed a core part of first-handedness. But the principle of first-handedness is that meaningful relationship to data is put first and foremost, and it is typically not so that for any wide spectre of data, size is adequate measurement of meaning. Even in a beauty contest, where length of legs may be said to be a size of great meaning, it will still only be meaningful to talk of length of legs in such a context given that one also talk of a number of other factors which interplay to create beauty -- including but not limited to length of torso (so that the proportion of length of legs to length of torso is big, rather than the absolute length of the legs), smooth symmetry and sensuality of face, shapeliness of feet, massiveness and shine of hair, glow of skin, firmness of body, shapeliness of muscles, and so on -- all in all factors that lead the child beauty girl and the adult beauty girl to be sexually fairly much the same, as enlightened observers of human sexuality has noted, but which has been combatted in the anti-sexual culture that quasi-religious semi-moralists have purported to be 'healthy attractiveness'. In the totality of meaningfulness in such a context, only when other factors of meaning are the same (which they nearly never are except for very small amounts of data), can size alone take on the chief meaningful organizing principle of the data. More generally, it is by semantic ordering that one in general finds it possible to program computers well, given a finite region of ram. The creation of a sorted list is an imposition that generally must come after the data has been stored in some other way, and then only by relatively heavy use of computational resources -- which, all taken into consideration -- are not well-spent computational resources unless the aim is to created printed catalogues for human manual searching. But it would be more satisfactory -- and, thus, meaningful -- to produce an ordering that comes naturally the first time data enters the computer, suitable for the finite segements of ram set aside for each semantic portion of data -- so that the required data can be reproduced for the human interactor without the artefact of going through a sorting mechanism (whether implemented as batch or in some way directly, such as by 'trees' of data). Are there then no circumstances in which sorting is semantically first-hand when it comes to data? It is proposed that the only circumstances are special cases which lends themselves to simplistic sorting inside a prior, larger organizing principle that doesn't entail use of any sorting algorithm. For instance, a free sequence inside of an alphabetical or numeral ordering can make sense in some contexts. Perhaps the two first digits of a longer number can provide an entry-point for which ram segment to store the longer number in. On some occasions, the numbers within a particular segment is called for, and then in a sorted way: but this has then already been strongly limited in size by the initial approach of categorising these segments, so that any imposition of a complicated sorting algorithm such as "quicksort" should not be necessary in any first-hand data model. Rather, the notion of "bubblesort" can then be used -- and indeed, studies of various ways of doing the "quicksort" approach have found that once the quantity of data is small, bubblesort is actually more efficient than the complicated algorithm as a whole. It is therefore not uncommon that when any large "quicksort"-like algorithm has been implemented, there is within it a check on how many numbers or names are left to sort, and under a certain limit, the algorithm calls on bubblesort as a sub-algorithm. The bubblesort, used within a larger context OTHER than a sorting algorithm, is then a first-hand approach. This other context can be such as an instant decision to use such and such segment of ram given such as a hash or such as the first part of a name or number. This instant decision is not a sorting algorithm, but in some cases it paves the way for a meaningful use of bubblesort, which then also is first-hand. The theorem is, then, intuitively now regarded as established. For completeness of the indication of the concepts here used, let's briefly state the procedure of the bubblesort. The procedure, described in next paragraph, will sort a list like 3 4 1 2 gradually, in several small steps. In the first cycle, 3 4 1 2 becomes 3 1 4 2 then 3 1 2 4. In the second cycle, 3 1 2 4 becomes 1 3 2 4 then 1 2 3 4. Each cycle consists of comparing two and two numbers beginning on the left and proceeding to the right. First, the first and second number is compared. If these are in incorrect sequence, they are swapped. THen the second and third number are compared. Again, if these are in incorrect sequence, they are swapped. A number entirely out of place is like a 'bubble' going through the sequence, especially if read vertically rather than horisontally. When the completing pair of the numbers (or whatever) is compared and, if need be, corrected, one cycle is complete. When a cycle has completed in which no corrections were necessary, the bubblesort algorithm has finished. THEOREM: Sound, video and timing isn't properly part of the philosophically important and clear concept of the Personal Computer. 1::B::2014::07::18 INFORMAL PATHWAY OF DEDUCTION: 1. BACKGROUND A computer, in its pure, philosophical form, is also a limited, manifest instrument where performing an algorithm and looking up data by this algorithm is assumed to take some unspecified amount of time. Not much time, compared to the psychological meaningful unit of time such as a second -- we can speak of thousands even millions but the performance speed has to make psychologically 1st-hand good sense for the Personal Computer to be a coherent good concept. This computer has limited RAM, limited speed, it has a keyboard and a mouse as input, it has a disk for more enduring and larger but slower storage than RAM, it has a monitor like 1024 times 768 and it has a variety of pixels, such as bright green to black in a variation over e.g. about 60 tones. This is suitable for a non-emulative non-imitative reproduction of quality photography so that to the living mind, the SENSE of reality is stimulated within, and the living mind will provide a sense of colors, scents, motion and so forth. This is philosophically an important concept, and as other theorems in this series of intuitively evaluated informal theorems indicate (assumed to be possible to bring forth deductively in a strict form only by masses of books), a far more important concept than any abstract notion of the computer where one imagines that there aren't limitations here and there. For there is a fundamental philosophical difference between a structure which has definite limitations about it of a psychologically meaningful kind, and a structure which undergoes such dubvious 'extensions ad infinitum' as e.g. in the geometry postulates of the ancient hellene Euclid, and followed up in logical consequence, but with incoherent results, by such as Georg Cantor, Bertrand Russell and others laying the foundation for the type of infinity-unaware 'mathematics' as came to be the fall of physics, once it got wedded to this incoherent approach. (The approach taken by here of essence numbers, supermodel theory and the popularisation into q-fields avoid such incoherent ideas connected to a sloppy relationship to infinity.) 2. CONCRETELY Considering the theorem above, it connects to three themes -- timing, sound and video. It is clear that whether or not video is regarded as with sound or without sound, it is a concept which involves a simulation of movement by fooling the eye by providing changes of visual content faster than some 20, 25 times pr second, and doing so over a period of time in a way which depends on timing. Sound, such as the reproduction of music or talk, obviously also depends on timing. We can therefore concern ourselves primarely with the concept of timing, and ask what relationship this concept has to the concept of the Personal Computer as indicated. Let us, before we do this, allow ourselves to bring in the notion of electronics, such as through our Elsketch first-hand electronics approach, or through some form of chip-making or another, eg ASIC, Application-Specific Integrated Circuits, which is nothing but Elsketch on a microscopic form, burned into thin layers of the same type of silicone or such that would otherwise constitute such as the transistors in the Elsketch format. It should be clear that sound recording and replay, as well as whatever visual means it makes sense to produce, can well be done by means of the same type of electronics as also can give rise to a Personal Computer. A Personal Computer with many megabytes of RAM involve a lot of high-speed components, a great quantity of them, but the principle is by and large exactly what is needed also to produce something like a music station. And one can also imagine that the box called a 'Personal Computer' is equipped with extra components beyond what is coherently called for in a Personal Computer, and since the electronics components are of the same type, this should provide no practical problem at all! As for the question of timing, it is clear that in the usual design of a Personal Computer, there is such as a computer clock -- an oscillator -- providing a sequence counter for the instruction performance. It is not thereby clear that doing a particular instruction takes exactly a certain amount of microseconds or the like. It is rather that the computer organises things around the notion of ticks of this computer clock. This is -- although the timing signals can be read of, to some extent, by an instruction -- assumed to go on in the background. It is not the same as to measure what time the performance of each instruction takes, nor is it the same as to promise that the performance of each instruction cannot take place over more such clock ticks. Some CPU instructions are complex, and others are simple; the complex ones obviously take more time. Indeed the whole programming context is such that IN ORDER TO DO GOOD, MEANINGFUL AND COHERENT PROGRAMMING, ONE MUST BE FREE TO LOOK ASIDE FROM THE QUESTION OF PRECISELY HOW MUCH TIME IT TAKES TO DO EACH PROGRAMMING THING, and merely have a sense of the overall duration questions involved with the program relative to the human, living context within which it is going to be used. But this also means that while the program can read off a timer, whether an external timer or the same used by the CPU to sequence the instruction steps, the programming langauge as such -- for the pure philosophical idea of the Personal Computer -- isn't tied in sharply with any form of manifest timing AT ALL. Thus, for instance, it matters not so much exactly when a pixel is drawn on the monitor, -- it can even be a quarter of a second late, if the monitor electronics is doing other things -- as long as the right pixel get up at the right place in the right sequence. The visual aspect of the monitor, even with the device of the mouse and the mouse pointer symbol, is naturally tied up to the computer program. But such freedom in connection with timing isn't the approach that electronics dealing with sound or video or even timing in general ought to take. A variation in when a frequency is produced in the loudspeaker within something like a quarter of a second is enough to make the sound psychologically meaningless. A first-hand computer, programmed in a first-hand way, is an instrument which is then crafted out of a conscious relationship to limitations such that the visual idea of the monitor fits perfectly. There is no way in which the idea of exact timing can fit equally perfectly. The 1st- hand computer isn't supposed to be speeded up beyond the natural speed where 1st-hand programming involves a contact with the sense of duration for the program. But sound-production properly involves another instrument -- which can well be electronical and can be put into the same box as the Personal Computer -- without having the same affinity to the concept, and indeed also the super model, or q-field, of the Personal Computer itself. The same argument goes for simulation of movement by rapid changes of the whole matrix of pixels many dozens of times pr second, as video; and in both cases, it has to do with the importance of letting time be a deliberately somewhat 'indeterminate' variable for the program. ***** ATWLAH