Fractal Foundations of mathematics: Axioms notions and the set FS as a model

[jehovajah]

The spaciometric concepts of mass and density have many contributory sensate notions that will need to be combined in any definition or denotation.

The notion of space has to be realised as a matter or substance, a material that is elemental and exhibiting of phase states and regional attributes that are relativistic motion attributes. Regions of space will have a spaciometric mass and density.

The mass i will denote by the extents of a region's boundary surfaces, the count of internal boundaries, structures,  internal regions, surfaces, forms,and the variation of the auditory signal, colour and opacity and strength of smell within . The region's defining boundary surfaces are not assumed to be continuous or contiguous and in fact may be an imposed or iterative approximation, by axioms 1 to 5 of the set FS. In addition surfaces are equally definable by focal lengths and magnification parameters as well as extension. All surfaces and forms boundaries and structures are computational products of processes within the central nervous system and are fundamentally iterative. Thus spaciometric mass is an iterative product of a computational CNS process.

Spaciometric density is the numerous structural internal relativity of the forms, structures, surfaces and boundaries within the defining boundary surfaces of a region. The numerosity of the relative internal structures is enhanced by the colour and smell variation and the auditory soundscape. In density it is the structural relativity that distinguishes it as a property of mass. Thus for a gaseous medium the relativity is clearly over a greater extent than for a liquid or solid phase. Also the motion of a gaseous form is relativistic and is more apparent and diverse than that of the liquid and solid phases. The structural relativity means that density is always apprehended relative to the region it is appreciated in. However i can recognise congruent or similar regions by colour or smell or texture (variation of colour and structuraland auditory form and kinesthetics) in which the same structural relativity is apprehendable and this will lead on to constructing through equivalences a fractal scale for density and then mass.

[jehovajah]

What i have found for me is that spaciometric mass and density is synesthesiastically reified by the kinesthetic and proprioceptive senses. Thus i apprehend the spaciometric mass or density of a region "magically" by contacting it! Thiness, gaseousness, fluidity of mass is apprehendable by this and indeed comprehendable! It also gives me insight into why "weight" or balance equivalences were considered originally in the definition of mass for the SI units.

While the smell aspect of mass may seem a little odd it is a real contributor to the notion as does the auditory signal, which assists in orientation and mass boundary determination through a sonar sense.

On another point it dawns on me that in spaciometry i ought not to assume that pythagorean forms will be the standard. Rather these will be special forms. Thus my "definition" of conic curves and motions will have to be generalised accordingly. Now i read that einsteinian relativity uses geodesic to classify spacetime curves so i will use spaciodesic.

[jehovajah]

I have to mention one thing in passing, cos i am currently thinking about the "angle" in spaciometry as an imposed cultural norm or abstraction that though an extremely useful tool, has its drawbacks and non sequiturs.

the form a=b^2 has its shape determined by the reference system or rather the reference frame devised for the parameters. Suppose i use an orthogonal set of axes      C(x,y) with x and y being the parameters of the points in this frame; compare the shape with a parameter frame P(ø,r) which has no orthogonal axes but uses a rotation parameter and an extension parameter or B(l₁,l₂,ø), which again has no orthogonal axes , but rather 2 line egments that cross at an angle ø with the parameters measured along tese line segments.

the same form traces out distinctly different shapes:

y=x^2 being a parabola curve, x= y^2 being a rotated parobola:

ø=r^2 being a spiral curve, r=ø^2 being a spiral curve rotating and expanding differently to the first.

l₁=l₂^2 being a quadratic Bezier curve shape varied by ø.

If the reference frame can fundamentally determine the shape i discover i feel that formulae are not of themselves a shape. They are a relationship between the parameters which we can explore by reference frame by algorithmic procedure sequence and most relevantly by iteration.

The iteration setup for a=b^2 is b=0:a=b^2, b=b+1.

There are various implementations in code for this.

The plot function is where the parameter frames can be implemented , the algorithm for b^2 is hidden but can be b*b, b+b+...... b times, b mod(2π)*2 , a taylor maclaurin expansion etc.

The iterator can also be b=b^n+c, b= bmod(n)+cmod(m),z=z^2+c etc including any series expansion form,and any calculus form such as x=d(x)^n/dx+c, or x=dⁿx/ds+c  etc where the parameters have to be in there calculation form not the manipulation proforma which itself is iterative in expansion to the calculation form.

By manipulation profrma i mean the symbolic notational form which is then used as a procedural guide to the calculation form. I will elaborate on this distinction in mathematical notation and symbology at another time, but suffice it to say these static forms lauded for there elegance obscure the dynamic often iterative process of manipulation and oerator action required to arrive at the calculation form which is itself an algorithm enjoining action by the reader or processor. these levels of required action are what make algebra and math so daunting, because they appear to have no referent.

[jehovajah]

Spaciometric mass and density i have outlined as counting procedure to apprehend its "quantity" aspect or appreciation. However, as basic as counting is it is a foretiori to measuring. Measurement is fundamental to sensor arrays of huge numeral dimensions, and measurement is essential comparison of equivalences. Whenever i want to count i set up a region as a standard and measure that region against others which i then count off as they match. The match criteria can be anything from exact congruency to a specific item.

It is this measurement that is the basis for counting and all quantification and this is the connection i will pursue to the comprehension of mechanical mass.

[jehovajah]

I have been working under the apprehension that my primary language response to the set not FS would be naming and numbering; from which i would at a later stage proceed to show how mathematical thinking chiefly in its algebraic form would naturally derive with counting as a specific application of an algebra called in full arithmetic based on notions of natural numbering which is to say the culthural counting iteration +1. This counting iteration is in fact a specialised form of naming, that is numeralisation- the naming of quantity by numeral names, such as "one", "two" etc..

It is the notion of quantity that goes unrecognised in this explanation of the derivation of mathematical thinking. The notion of quantity is a notion of measurement. It is the notion of measurement that is fundamental to the language response to not FS, so it is naming and measuring that seem to be my first response and the measurement is an inate response. 

[jehovajah]

Measurement then i consider as an essential apprehension and comparison of quantitative signal output of a sensory "array", where the "array" is itself a very large quantity of individual sensors each one capable of giving various signal outputs. At this level of detail it does not matter if the output is binary trinary or indeed analogue, because the individual response is subsumed within the overall structural response of the "array" and regions within the "array" .

Thus the structure behaves in a manner as follows:

The individual signal has no meaning, as it is in no context to give it a meaning. Two sensors providing 2 signals are the basis of a comparison, but require a combining sensor to record the comparison. So the immediate sensing level passes signals to a comparison level, and the fundamental structure of this arrangement is 2 sensing to 1 comparing.

Although i can immediately divine a binary structure forming, it is a structural binary system not a sensor that outputs a binary signal.

It iss again clear that the structure has to be iterated up another level to give meaning to the comparator level signal, so a fundamental structure would be 4 to 2 to 1. At this level of comparing the comparators the system is able to hold individual signal information, and comparative information. If i iterate the structure one more level then the system can hold information about regional response at the 2 sensor level and can compare at the 4 sensor level as well a individual sensor information: the structure is 8 to 4 to 2 to 1 .

This structure of sensors can hold information as individual signal as comparative signal out put as regional signal output and as comparative regional signal output, as well as some other combinations, for example comparative comparative signal output. However, without some detailed diagrams this would soon become confusing and i am merely noting a structure as an example.(good news, see below).

It is clear that the structure is an information system, but what is not so clear is that it is a self referential information system, and that it acts by measurement /comparison of raw signal, that is by equivalences. The system clearly has emergent properties at each iteration and as you might guess self reference is not the top emergent property. Self reference occurs at a lower level than the structure goes up to. By this i mean that for  the 8 to 4 to 2 to 1 structure self reference occurs when the system compares the 2 regions below a given level. So in the 2 to 1 comparator level there is potential self reference but it requires the 4 to 2 to 1 structure to have self reference at the 2 to 1 level. This means that the structure holds information about its state but cannot describe its state true state to "itself". However it can precisely describe its constituent states up to the level where the self reference/ comparison is occurring.

Now suppose the structure had 2 comparators, by which i mean the 2 sensors feed their signals via 2 separate structures, say a second comparator that connects directly to the top level of the structure. This would give the whole structure a direct comparison feed which could allow the structure to have an "internal and "external" representation of the same information, thus allowing the system a freedom to be "creative". This means the system could measure the information in the structure an assign for example a " truth" value to it. These kinds of sensory structures are relevant to the description of innate sensate notion forming processes, and thus to the topic of consciousness and of course information processing at the thinking level.

If the description above is valid then it is also valid that my initial response to not FS will be measurement. Thus measurement would come before language, and at the level of language that humans are at would become a forgotten or mis-described foundation to our conscious response to not FS. It would lead me to say that "non verbal communication" is the basis of all language and that this NVC is primarily measurement of sensor responses. Thus my appreciation of my experiential continuum is based on measurement and apprehension of comparative differences in those measurements. This would then lead "naturally" to a naming response to those comparative differences and a numbering response in terms of quantitative differences. The numbering response would be in terms of "instance". "instances", the "one, two, many" response reported in many so called primitive numbering systems. I am using numbering here as a precursor to "counting", and as the process of "naming numerals". I have distinguished between numerals and numbers for reasons of rigour to avoid as much as possible the mystical attachments to numbers. This is not to say that i am against mysticism, but rather to be clear what i am referring to. Thus numbering as a response precedes counting which is the cultural iteration +1 and which is performed by rehearsing the numerals, that is the numeral names given to us through a cultural "numbering" process, which more directly put is just a cultural numeral naming.

It is clear to me that measurement processes of an iterative nature in an iterative structure for sensors have given rise to the impuls to name and number, and from this we have culturally derived a counting iteration which has been the basis of our algebraic apprehension of quantity, which we formulated into various arithmetics. But at the same time we continued to respond to our measurement impulse to extend our appreciation of quantity in terms of  orientation, extension, boundarisation, relation, motion, form and structure, a well as other comparative differences, such as colour, smell, kinesthesia and audition.



I offer this image from the Internet Encyclopedia of Science as an illustration of the sructure discussed above. One ought not to think that this structure is unique to the eye. The sensors are unique to the eye in the body i believe but of course not biologically unique as this is a general vertebrate eye form.

[jehovajah]

As a restatement of some initial discussions earlier in the thread i revisit the notions of quantity and quality and express these notions as products of a measuring and distinction process or set of processes that occur from the sensors to the central nervous system within the organism that inheres "my" experiential continuum.

The sensor "arrays" do have a spaciometric input which is again an information source about the spaciometry of the experiential continuum. Vision contributes to the notions of form, surface, orientation, hue; audition to extension, orientation. pressure, sonar, balnce and rotation; gustatory contributes to regional identification, memory for place, kinesthesia contribute to locality awareness, extension, motion , movement, balance, motion transfer and a grounding "presence" which synesthesiastically combines and comprehends the whole sensory map. This is just some of the result of the measurement and distinction processes that inhere in the relativistic forms that structurally combine to form the organism in which my experiential coninuum inheres.

[jehovajah]

When researching the foundation of calculus i find a catalogue of misconceptions and missed opportunities.
 The situation is not unusual however a s mathematicians a re children of their cultures and time. There are several precurdors to the notion of calculus but the topic is formalised by Leibniz and Newton in 1670's on the basis of a dispute. The nature of Leibniz, a poly math and a high achiever versus Newtons more secretive and mystical nature lead to an unsteady start to the topic with weak foundational axioms and posits. Berkleys  challenge to Newton's calculus sparked off an intense revision of the foundations that supported or supplanted the notions of the founders Leibniz and Newton. In the course of this new notation experimental at the time was tried out and developed and accepted on partisan grounds rather than sound communication principles thus d²y/dx² though familiar is a confusion of signals.

Euler's patient revision of the symbology , methodology, and pedagogy did much to provide an acceptable explanatory schema of the algebra of the developing calculus and enabled mathematicians to revise the foundations in a more coherent way .

I hope to explore the precursors and the founders notions to see why such a powerful working process delivered and delivers the goods despite seeming to be ill founded.

The Greek notion of proof or rather the classical notion of proof is the first culprit, and the religious sensibilities of the time are the second- for if you sincerelu hold that there is a realm of man's enquiry and the rest is god"s perogative then venturing on the infinite and the infinitesimal is a great impertinence.

All things greek though admired were not wholeheartedly accepted, so challenging the god of the roman catholic church was not a culturally encouraged thing, and many of the priest intelligentsia made it their duty to point out errors in all things scientific. For this reason and due to his autism and sexual ambiguity Newton was very private about his ideas. Heknew that roof under the greek system was by trial. Trial by our peers who may have personal axes to grind. Leibniz knew also that logical arguments, as a lawyer are designed to convince, not t prove in the modern sense of some mathematicians. Logic in fact is the study of argumentation to convince or demonstrate the inadequacy of a notion, person or thing. Chiefly to discredit a person was a valid and accepted form of argumentation and proof, on the basis that the gods would vindicate the truth.

It is therefore of paramount importance to apply the dictum of Karl popper that Falsifiability is necessary and sufficient to defend the utility of a notion. Proof by contradiction or the "impossible" proof in fact only supports consistency, it doesnot "prove" a thing. The only sense for the word proof in mathematics that makes any useful contribution is trial to falsify..

Both Newton and Leibniz used a notion of vanishingly small; this was a value so small that when it was squared it became 0. These values came to be called infinitesimals. Now it makes no sense to us who are heirs of the "reai number" system after Dedkind, that squaring a small value would com to  exactly 0, but for Newton and Leibniz the Rational Numbers and logarithms were all they had, and it was possible to see that a very small number vanished when squared. It got so small that according to Newton it behaved as if ir were 0, and logarithms of these numbers produced values that were exactly 0 to 4 or 6 decimal places in a log calculation.

Newton went on to apply this notion to rates of change of velocities as they moved on some polynomial curve, which he called fluxions, but he did not explain what they were as he had no idea which is why he gave it a made up name. Leibniz at least was able to account for his infinitesimal elements.

The notions that guided Newtons thinking was the compound interest process, and its relation to the binomial theorem, the  notions of polynomials, the notions of extrapolation and interpolation.

[jehovajah]

It strikes me at the moment that Newon's laws of motion are validated by him by an intense study of motion, and Leibniz had a deep interest in and study of motion. Both hit upon the use of the binomial coefficients to describe or framework their exploration, based on the then complete model of growth and rates of change afforded by the study of compound interest. Nobody studied the nature or algebra of time so intensely until William Hamilton in his Theory of couples 200 or so years later,
 Therefore i am not suprised that modern theorists like Einstein and his wife have combined elements of the two in their descriiption of motion. In fact Hamilton's quaternions have proven to be a succint and powerful way to describe space time motion along wih newtons analysis of motion.

The combinations of the calculus and hypercomplex algebras has proven very useful especially since Euler"s Formula.

[jehovajah]

My study of calculus foundations is proving fruitful.

So i note from Wikipedia on Ratio"
History and etymology   Look up ratio in Wiktionary, the free dictionary.


It would be impossible to trace the origin of the concept of ratio since the ideas from which it developed would have been familiar to preliterate cultures. For example the idea of one village being twice as large as another or a distance being half that of another are so basic that they would have been understood in prehistoric society.[6] However, it is possible to trace the origin of the word ratio to the Ancient Greek λόγος (logos) appearing in Book V of Euclid's Elements. Early translators rendered this into Latin as ratio, meaning "reason". However a more modern interpretation of Euclid's meaning is more akin to computation or reckoning.[7] Medieval writers used the word proportio ("proportion") to indicate ratio and proportionalitas ("proportionality") for the equality of ratios.[8]

Euclid collected the results appearing the Elements from earlier sources. The Pythagoreans developed a theory of ratio and proportion as applied to numbers.[9] The Pythagoreans' conception of number included only what would today be called rational numbers, casting doubt on the validity of the theory in geometry where, as the Pythagoreans also discovered, incommensurable ratios (corresponding to irrational numbers) exist. The discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables.[10]

The existence of multiple theories seems unnecessarily complex to modern sensibility since ratios are, to a large extent, identified with quotients. This is a comparatively recent development however, as can be seem from the fact that modern geometry textbooks still use distinct terminology and notation for ratios and quotients. The reasons for this are twofold. First, there was the previously mentioned reluctance to accept irrational numbers as true numbers. Second, the lack of a widely used symbolism to replace the already established terminology of ratios delayed the full acceptance of fractions as alternative until the 16th century.[11]
[edit]
Euclid's definitions......

And from the dictionary

com·men·su·ra·ble  (k-mnsr--bl, -shr-)
adj.
1. Measurable by a common standard.
2. Commensurate; proportionate.
3. Mathematics Exactly divisible by the same unit an integral number of times. Used of two quantities.
[Late Latin commnsrbilis : Latin com-, com- + mnsrbilis, measurable (from mnsrre, to measure; see commensurate).]

Their are many notions to draw from just these two quotes alone, but i will confine myself to pointing at the greek apprehension of the measuring response inherent in our neurology is conveyed by the single grapheme logos and the phoneme"logos". As far as i can tell this is a uniquely greek apprehension and thus combines the sensory stimulus of visual , audio, gustatory and kinesthetic sensors. Thus logos is a full response that does not differentiate measurement and distinction processes, "compare and contrast" processes,sensor comparison circuit processes. It is thus ground zero for the greek response to the set not FS, and from it the language (Logos= word) and the measuring(logos= reckoning) response is drawn and subsequently differentiated by application of the word logos to particular contexts. It is interesting to note that comparison the fundamental root activity of measuring is drawn out in the term ratio by earlier translators into latin, and then by later interpreters into the term reckoning, which though a different word is hardly a differnt notion. However it has more potency for thought for non mathematicians. And this of course is the sad thing: Our specialist use of language has divided us as did the mythological event at the tower of Babel. Nevertheless i am content to proceed on the basis that my measuring response is the basis of impulse to distinguish by quantity and name, that is to say by "logos".

I then hope to advance the fundamental nature of iteration in my whole experiential continuum in the development of the "logos" response, and thereby to facilitate a intuitive response in mathematical thinking as well as language study to the fractal products of these iterative processes.

One thing stands out already: the fundamental nature of ratio and its development: proportion, to an intuitive facility with mathematics from spaciometry to abstract symbolic manipulation. In fact the loss of ratio is very much akin to the loss of rationality in mathematics!

Keeping in proportion then, the fundamental nature of Equivalence classes is noted, and the practice of thinking in ratios and proportions is to be encouraged, by which i mean the method of positing or establishing a ratio and then by induction equating ratios to establish a rule or law from which a proportion can be derived for each instance. This encompasses not only exactness where it may be found but more importantly relativity wherever the ratio holds. Thus by ratio exploration it seems we can explore not FS by induction and deduction, and tus we can intuitively interpolate a proportion and even extrapolate a proportion once the inductive rule for a ratio has been established. Hidden within the term induction is of course the iterative nature of exploration of instances from which i may derive a rule and then by test of interpolation accuracy and then extrapolation accuracy determine whether it is a law for such parameters and contexts which i am currently investigating

The falsifiability of this approach is evident and thus satisfies Karl Poppers admonition on proofs.

I say these things after noting that earlier mathematicians had this facility to think Rationally and then to derive proportions as needed. Having had sight of Newton's papers for the first time this is what struck me as most significant.

I venture to add that Newton in his earlier explorations in 1665-1668 developed such a facility for rational thought through training , particularly by John Wallis who directed his studies, That by this method he was able to perceive and develop the generalised binomial series. And from intense consideration of the problems in algebraic geometry at the time he fused this binomial series together with tangent theory under the ubiquitous compound interest methods of Schiller to form geometric calculus. He then rethought this in terms of motion and curves of orbital motion, and happily found that the same rules and methods applied to motion. Because He derived the same rules from fluid motion elements he called these fluents, and the limit tangents fluxions. He was not working with algebraic geometrical curves, but with fluid(continuous) motions broken down by the geometrical method of decrements into small fluents, the limits of which were the tangent of the motion at that point, which by compounding(extrapolating by the binomial series expansion) using the compound interest methodology, enabled him to integrate the tangents into the curve traced out by the motion in its entirety.

This was an insightful moment based on his facility with rational thinking, enabling him to proportion the area under general curves and from considerations of tangents(ratios) to extrapolate(integrate) the curve. Of course he had to note some funny ratios called infinitesimals thereafter, but the binomial series inductively supported that these were tending to 0 and that some even were zero! That i think was more a reflection of the limit of tables and accuracy of his time.

[jehovajah]

I do want to look at polynomials in the light of our decimal system being an instance of a polynomial called a geometric series, because the development of operators and algorithms in transforms have a fundamental shaping effect on the polynomial concept and may have a visible accelerating effect when the decimal and arabic form of numeral formation were adopted widely. In any case it provides possibly a secure foundational theory base for a revision of number theory along iterative lines.

My knowledge of polynomials however will need to be refreshed by exploration.

The binomial series alone in my opinion validates Newton as the discoverer of Calculus with Leibniz in the role of a fellow collaborator rather than plagiarist. However Leibniz ambitions are suspect and he may have published early to beat Newton and for the advantage the recognition gave him in his own country. Leibniz you must remember seemed to survive by patronage, so his needs would affect his ethical considerations.

[jehovajah]

I woke this morning and thought about laser light diffraction gratings which are variable and tuning. This often happens and i feel it is an effect of the computational nature of my experiential continuum. This manipulations at a so called quantum level will play out the same response in all systems tuned to that sympathetic resonance frequency. Some will get a clearer signal than others,and some will be able to act on it while others will assist the manipulations enabling them to persist and extend. If you want to call them ghosts in the machine you can, but essentially they form a kind of software encoding of a continuing distributed universal computation .

The http://en.wikipedia.org/wiki/Virtual_retinal_display and the basic hardware http://en.wikipedia.org/wiki/Grating_light_valve pretty much nail the concept that i had come up with, happily. The notion itself is therefore not originated by me which i do not at this stage in my apprehension of my experiential continuum expect to be the case. Rather i expect there t be some coexisting creative development of an idea that i am perceiving or formulating. This is because as we communicate many seeds are scattered which grow as weeds to some and as beautiful flowers to others, but they grow in a multitude, rather than in isolation. I see this pattern which i have used in the several metaphors and paradigms i have used so far in this post, in the bacterial growth behaviour or in mitosistical cell division.

Because the idea is well established with regard to laser light diffraction gratings i can now advance  a "pixel" for the diffraction gratings. Physically this would look like a diffraction grating in which the diffraction slits are crossing each other, let us say initially orthogonally. The actual slits are what allow an interference pattern to emerge on the other side, showing the wave nature of light. In this case the light source(s) would be white laser light in the form of a beam. Now it occurs that if the beam is rastered then variations in the interference pattern will occur due to the solid angle from the point laser source, and this would have to be compensated for, but essentially a set of parallel slits should produce an interference pattern local to the slit. The emissive radiation would thus be modulated, and perceivable by the eye. I might then see the expected oil-film rainbow patterns. Now add the modulation of the laser beam as per a televisual signal, what would i see?
Now add to this the orthogonal diffraction grating arrangement or some similar arrangement of parallel diffraction gratings and what would i see?

I would expect the "square slit" to have a radially emerging interference pattern locally behind it. This interference hemisphere i would hope would behave like a "pixel", and would with a modulated laser beam raster produce distinguishable images in or local to the diffraction grating.


So now i am going to find out who has done this at any frequency level.This may then provide me with a robust model for quantum interactions at the planck distance, modeling the spaceometric structure as a diffraction grid.My ultimate goal however is to model a relativistc motion grid with diffraction grating properties.

Of course the 3d diffraction grid would have to be explored built up of elements of thin diffraction gratings. Already i can see that the signal attenuation would be marked as the diffracted light moves from layer to layer, but this is consistent with the darkness of the generally.

I also thought about ratio and proportion the day before in terms of the way my neural network of actual axonic nerve impulses records and manipulates my apprehension of the set not FS and the set FS, but i will detail that in another post. The central nervous sysytem i see as a distributed processing neural network with a central coordinating processor, running a base operating system at the nanozoe level with additional software installed by a mentoring process called parenting and socialisation.

[jehovajah]

Looking again at the foundation of tensor analysis and the work of Ricci-Cabastro and Levi-Civita i realised that Einstein like Newton stood on the shoulders of many giants, not to mention David Hilbert.

I have given a lot of thought to Ratio and Proportion in the development of the "logos" response and see the unifying nature of this impulse and in particular a qualitative difference in mathematical thinking before the modern period. I now suspect that many of the innovations in the development of maths derive from the ratio proportion type of response and thinking whereas mathematics today seems to be about remembering formulae and equations and how to manipulate them symbolically, developing axioms and so called proofs that lead to an internal consistency in mathematics but very few new conceptions because it has all become too damn abstract!

"Abstract" means looking at the essential transform, that is algorithm or operation,and working with that as if it were a real entity.

"And YH¶H spoke and it was so!". Our language system has a reification operation embedded within it. It can bring something into a kind of "existence" simply by naming it. The archetypical example is the word "quiz", but what the etymology of this word shows is a reification process for all languaged notions and concepts, which may not be based in a reciprocal apprehension of not FS, but rather a new reciprocal apprehension within the set FS .

The Tensor notion is unsatisfying in how it is communicated because it is couched in abstract language and not interms of everyday things.

However our world has changed by the introduction of cameras and LCD displays and VLSI circuit wafers etc which enable us to give a referent to a tensor which is concrete. In fact  "concrete block" is a tensor model which can be effectively utilised to ground the notion, or even better a "lego brick" block. A matress is another good model of a tensor.

Any way let you know if an other things "spring" to mind!

[jehovajah]

So i am on the way to the dentist and i think: extend the tensor notion to include spaciometric forms which are structured including "arbitrarily" structured forms eventually, but think of a plants cell structure in cross section, to begin with.."

Of course the spaciometric form is invariant under basis transform, that is to say reference frame, and spaciometric mass and spaciometric density can be attributed to the tensor.

Then i heard that because England scored some goals and won a game the economy was estimated to respond in various sectors "memorabilia, electrical items,sports items, FMG... to the tune of about a billion pounds. I saw a "complicated" input oitput structure there which could be represented in a tensor.

Then my dentist actuall y drilled my tooth and i saw a tensor relationship there between the input drill speed and path and the spaciometric result due to chemical and physical bonding energies being exceeded and strain maxima being exceeded being representaable in a tensor which would amalgamate the physical and chemical effect of the action of the dentists drill. The information stored in the tensor being accessed in various ways specific to the required output information.

Tensors as you may be seeing, although a spaciomtric form, may hold relational information, that is relativstic information, in this structured form. The structure of the tensor itself can mimic the structure of the form that it holds the information for or it can just be the "block" form with the correct structural relations adhered.

The fundamental relationship of a tensor is that the form does not alter as the reference frame (basis) alters. Thus if i reflected the components of a tensor in a "mirror" i would fundamentally alter the form of the tensor without realising it as the form looks the same just reflected. This would not be a tensor transformation as currently defined and needs a development in tensor algebra to accommodate it.

I am used to reflections as mathematical transforms but physically they have no counterpart except in the notion of chirality(left and right handedness). Whereas i can relate a left handed version to a right handed version by a mirror reflection i cannot physically do the transformation in the "solid" phase. These transformations will typically be found in chemical reaction products in the liquid,gas or plasma phases.

This relates to a discussion i began with regard to unary operators defined as spaciometric rotations of orientations and how a reference frame with an additional rotational attribute to the axis vector transforms a referenced region on its transformation.

In fact i proposed a more comprehensive vector definition, but now think that a tensor form would more than cover it! ah well such is life!

iimagined that funny effect you can get in those halls of mirrors taking place physically: The left side of me would squeez into the middle and then "pop" out on the right side of me and vice versa, but it would be an anti me that results! Everything would work in the opposite way! Kind of weird science!!

Any way for that reason i say reflection is not a tensor transform even if it is by Ricci and Levi! Except as i noted in the area of reaction products. I suppose i ought to allow seeing what you look like in a mirror for those of us to whom it is a daily necessity. Go on then, put it in but don't say i did not warn you!

My current interest is in trying to define a ralativistic frame of reference, and when i do i will have to make it tensor invariant also.

[jehovajah]

A tensor is a quantitative model of a spaciometric form that is firstly relational and this enables it to be relativistic, secondly differential and this enables it to be mapped to a network of related reference frames and finally quantitative in each differential element of the model of the spaciometric form in a relational way.

Although this is a one line staged explanation it is still not a comprehensible denotation. It is however more apprehendable.

We utilise and view many possible candidates for a tensor in our modern computing environments: any spreadsheet aplication allows me to create a tensor as a spreadsheet for example, where the spreadsheet is the spaciometric form and each cells represents the differentials which are thusly relational  and referenceable relative to a common origin and also relative to each other. Thus a network of reference frames can be created that enables the whole form to be referenced by the origin/common  reference frame or by a relational reference frame. This allows for macro and micro relations to be "describable" and is why i thought that tensors might be a relativistic reference frame initially. They are not, but they possess a relativistic reference frame, which i have just described.

Finally each cell can contain quantitative data which is corresponding to its reference either relativistically or by common reference  or both. And additionally the quantitative data may be related by function/formula to any number of cells within the spreadsheet. It is to be noted that the quantitative data may be measurements or ratios of any quantifiable property of the spaciometric form at that referenced region

Now spreadsheets are a specific instance of a much more general class of applications which are: relational databases. Thus any relational database can be constructed to form a tensor which will have the description already advanced. Some relational database apps do not have all the mathematical functionality of a spreadsheet but this is a design consideration not a fundamental distinction, as this can be easily added to any relational database.

It is of course possible to reference between spreadsheets collected into a book format and across books et al .

Thus a good example of a tensor is a quantitative relational database. Many scientific measurements of experimental results are held in such relational databases, which when constructed appropriately would be exact tensor candidates .