# Nature and spaciometry

Originally posted by author:

A technical point , but of interest to me is the notion of Archimedian  groups and fields. These are fields that specifically exclude infinitesimals and infinity. Since i was clasically trained my notion of these conceptions have always been based on the notions of bounds and limits. In addition i have induced that the concepts of number and numeral are related but distinct and that mathematics or thinking about iterative processes and procedures and operators and algorithms to describe or construct the same do not require the notion of number, but rather the process of counting or numbering . The cultural iteration +1 being a numeral naming scheme is not infinite but is not bounded above, nor are rational numerals bounded below in the naming of any increasingly smaller fraction.

There is a clear disconnect between notions at this point but that is not unusual or indeed worrying in mathematical thinking that is to say iterative exploration. Discontinuity and boundary differences are normal in "fractals" and self similarity is crucial to identifying these discontinuities and boundaries.

In any case Archimedes was interested in quantization which is the basis of SI units, and for this you cannot have "infinitesimal numerals"  as such nor "infinite numerals". I acknowledge only 1 infinite numeral for computation purposes and that is infinity with
the recently declared numeral nullity as its axiomatic inverse.

Infinitesimals have always meant to me the limit process applied to a region in geometrical space as finer and finer scales are applied. In principle this is Archimedian because i practically have to stop when :sufficiently small to produce no appreciable difference in the result. The use of the term infnitesimal here means i have freedom to continue until such a situation arises. This can be well expressed in archimedian axioms and so leads to no difficulties at that level.

We could define all archimedian Quantity groups as clock arithmetics with various non integer  modulo so for example [tex]R_{mod(2pi)}[/tex] would represent the radian principle set of real values.

The question is can i live without [tex]R_infty[/tex] and [tex]R_0[/tex]? In the numeral namespace they clearly have a role so my guess is no. The flexibility it gives algebraically for notation is also not a mean advantage.

The notion of proof by contradiction also deserves a note. This really should be named what it is, that is: a test of consistency with the preferred axioms. Thus every mathematical model built with this test should avoid major additional inconsistencies beyond those introduced in the axioms.

Further it is worth pointing out that the models i build of the set notFS are always going to disconnect from and have a region of applicability to the set notFS, this in itself is iterative and leads to the development of fractal models of notFS.

Originally posted by author:

:rotfl:

I am laughing because in 1973 when i was at uni learning about math and computers nobody thought to tell me that the definitions of the number set R was changing to exclude infinity! In fact a lot of time was spent on Cantor infinities, which i rejected at the time or rather the then extant diagonal "proof". I rather liked Cantors infinities the countable and uncountable ones. The kind of "numbers" we used back then were called real but now they are called hyper-real ˆR. Now it is made explicit that a transfer rule applies from R to ˆR so i can do the things we used to do back then.

These technical distinctions mask the boundary between different models of the iterative processes in notFS and add the additional smokescreen of logic. This is not to say that i do not appreciate logic but rather that it adds nothing in credence to any mathematical statement, but rather tends to abstraction as Bertrand Russel and A N Whitehead  clearly demeonstrated.

Usually mathematicians will start with N the natural numbers assuming that these are trivial and well understood, whereas in fact these are not trivial because they do not exist. What does exist are the process of counting and the naming of each stage/ iteration of the count. What this cultural iteration may be used for is then up for grabs. We often play with it, console ourselves with part of it go to sleep doing it backwards and oh yes  we also Quantize with it! We can Order and rank with it and we can derive an algebra and an arithmetic for it . We can and do often derive mystical significance from the names we give to the stages, and we can and do attribute to it properties we derive later in higher mathematics! A bit circular i know, but what can i say: That is how it is !

So looking at not FS i see Aggregation, disaggregation, translation and rotation and perhaps the most important thing iterative motion. From these things i can construct a model that underlies all my so called mathematics, that is my thinking about all these fundamental iterative processes. I need a language to express all this in and that is why mathematics is a fundamental subset of all languages or means of communication. It therefore is not a subset of logic as Russel Fauningly thought, nor is it a Philosophy or a meta physics. It should be a playful engagement of our senses in contact with notFS and our languaged response to that experience, And how does that differ  to a writer or a poet or an observer of nature? My contention is that it does not except in the style and density of the notation and the utter repetetiveness of the subject matter.

I wonder what names we shall agree to call the infinitesimals when we realise that since 1973 we have had these "numbers" at our command. I bagsy "oneeta" for the least upperbound for the infinitesimals ;D

I have learned something today courtesy of phractal phoam phil: solidity does not inhere in space it is attributed to space.

I explain this anecdotally. When i was learning about science iwas given examples of solid liquid and gas to help form the concepts of solidity liquidity and gaseousness. This of course seemed pretty straightforward and comprehensible. Then i was taught atomic theory. the result of this was that i reviewed my understanding of the liquid and gaseous phase in the following way : solidity inheres in atoms; the liquid and gaseous phase are explained by the kinetic theory of atoms. Thus solidity remained unchanged and in fact spread into the liquid and gaseous phase by attribution of spacial properties via the kinetic theory. Then i was introduced to high energy physics and Niels Bohr's work and solidity was gradually pared back to protons and neutrons and electrons and massless photons, with the particle wave distinction gradually blurring the description of matter. Solidity was replaced by probabilistic description of position and the newer particles of quarks and eventually colour Quantum Dynamics(QCD). By this stage i had simply forgotten about the concept of solidity having reached out for a concept called "energy" which being indistinct and not well founded remained the linking concept for all new results from high energy physics research. I might as well have called it "Bleh!" for all its use, but it sounded /seemed scientific and utilitarian. However it was not falsifiable and so is in fact tantamount to a faith statement. Nothing wrong with faith statements but they do not tend to advance scientific enquiry. The fact that it was an ill formed conception i now realise because i could not understand how "space" could be solid. Even though i have briefly flirted with a pixel description of space unrelated to the planck length i could not clarify how that related to motion in space etc, other than by a switching on and off of pixels. The energy conception of space would have plasmas as exemplars but again solidity is inhered in some particle or other.

What dawns on me now is that solidity does not inhere in a particle it is an attribute of space in that it relates regions of space to one another through relativistic properties and not inherent properties of a particle. Thus space can be in constant motion but the relative motions encode relativistic properties which are the attributes we appreciate as rigidity liquidity and gaseousness. This does not require a substance other than space itself but any substance should exhibit the same properties by fractal entrainment to the fundamental properties of space in motion relativistically.

In the set FS i have set an iterator as a fundamental universal motion which fractally entrains all motions in setFS. This fundamental iterator is a vortex. That means space is spinning universally in some fractal pattern in setFS. The spinning patterns consist in vortices in infinitely various arrangements size and boundarised regions. It is these arrangements sizes and boundary conditions along with fractal entrainment in which all the the attributes of solidity, liquidity and gaseousness inhere along with plasma phase. Energy then becomes synonymous with rate of spin of the vortex and vortices , that is energy is proprtional to ∂2Ω/∂p where Ω is the angle of rotation in radians and p is the period length of 2π radians in metres.

The fractal entrainment and coherence within the vortices is the "medium" through which conic sectional curve motion is transmitted through the vorticular space and arises as tangential motion to conic helical motion within the vortices. Also arising out of the vorticular motion and interaction are loxodromic spiral motions giving rise to spherical and toroidal curve motion within the sururface of a conical helix vortex system.

This is entirely testable within any medium capable of sustaining a vorticular fractal pattern.

Note on computational consciousness. The computational refers to the manipulation of the molecular and cellular structures within an organism or a structured environment . These structures are at least fractal in that there are structures within structures at different levels of investigation and of different level of dcomplexity.

the arrangement at the dna /rna level allows for a manipulation of the molecular structures through electrical, chemical and hydraulic pressure, and gas pressure feedback feed forward interactions.

At the cellular level the arrangements and connections of cells provides a cellular substrate for a circuit board analogue which provides for an electronic cybernetic system to be expressed, while similarly hydraulic and mechanical cybernetic systems are also established through cellular structure and manipulation. Inherent within these structures is also a chemical messaging system with microbiological inputs and outputs which also form a cybernetic system. These feedback feedforward systems interact so that the hydraulic and gas pressures within the structures and cell structures alter electronic properties within the systems which are then fed back to cascade other alterations in a cybernetic way through a mechanical biological structure in tandem with chemical cybernetic systems .

Without complication the nervous system delivers the main identifiable structures for the computational systems but this is in tandem with the other structures within the organism. The senses of visual, auditory,gustatory(smell and taste) and kinesthetic/proprioception are global descriptors of this systems inputs and output systems, which necessarily are complex and iterative and fractal in structure.

Although computational consciousness is a neat phrase it is no different to information processing consciousness or data processing consciousness .

The iterative nature of computational consciousness is clear and the products of that iteration are fractal perception and and construction and concept development. In addition fractal structure is evident in all products of the organism as a whole in particular reproduction.

In order to rethink Geometry i am coining a phrase Spaciometry.

Spaciometry will be the study of the forms, boundaries and surfaces, and structures of and within "objects and structures in space" from the point of view of computing properties of equivalence in and between those objects and structures. It will be foundational to spaciometry that iteration and iterative techniques will be clearly identified and utilised. It is hoped that this may reveal a referent for "fractal geometry" which encapsulates the idea of "roughness" as Benoit Mandelbrot originally had a predilection for. Meanwhile i will start with Riemann's Hypotheses and go from there.

i have been studying the spaciometry of my garden and it is interesting to look at and derive notions from. Topography is naturally apart of spaciometry and the topography of the structures and objects in the garden is a treat in itself! The topogahical arrangement of forms and structures by juxtaposition or sequential patterning or relative patterning and structural arrangement is fascinating . And the arrangement of structures within structures and the asymmetry and asimilarity is refreshing. The relationships of growth in structures to the similarity within structures is interesting and the notions of boundaries as applied to complex structures made up of well defined objects which themselves are structured is very fractal. The notions of surface continuity discontinuity regions and spatially oriented objects which are distinct but connected is also exciting . The notions of space filling and dense yet not solid are also fascinating.

These forms appear by a growth process which is a combination of internal dynamics within the structures modifies by external conditions, very much like the operations within a fractal generator. I can also see the dynamic disaggregation or erosion or degradation of less dynamic forms by dynamic boundary conditions.

It is interesting to note that Euclidian forms simply do not exist because there are no Euclidian lines of points. Even man made objects exhibit a cobbling together of Euclidian ideas which in one explanation of Riemannian geometry makes these object representative of Riemanns hypotheses.

Measurement by equivalence is strongly suggested and twigs and nets of stalks seem natural measuring tools. Orientation is flexible as some structures are rigid and others are motile, and some follow the sun. Notions of twist and rotation and translation are evident and tessellation of a surface is hinted at but indicative of the difference between living and non living or slowly growing forms. Convexity and concavity are exemplified and many polygonal forms are in evidence as well as polyhedral. Truncated conical forms f varying heights are in evidence as are loxodromic folding and other spiral/vorticular forms for plant stems and the like.

There are no planes only surfaces and consequently 2d forms are an abstraction. However there are very thin 3d forms which have a consistent depth over a surface region which will approximate a plain. Cartesian and polar coordinate systems seem a natural mesh like measurement tool with variable orientation mimicking the natural rotations in some structures. I can establish from each of these types of structure an extension measure and a rotation measure. I can also establish a density measure as a measure of iteration or tessellation which is commensurate with notions of surface area. I think that a notion of space filling is easy to establish but volume is less obvious because density is so intrinsic to space filling but volume attempts to exclude density. The common notion of volume it is apparent is a surface area bounded space, which is a generalisation of area which is the edge or boundary enclosed space.

Just found that my notion of conic sectional curve motion is on the track that others have been exploring before me. So i am pleased that i have adopted it for the set FS. It along with my latest axiom on (energy) motion allows me to think of gravity as the motions defined by these curves.

In spaciometry i have a chance to define a spaciometric density, which is an apparent notion and axiomatic to spaciometry, but i have to go back to axiom 1 to define it.

The construction of my experiential continuum utilises inate processes and procedural forms, hence my interest in axiomatising parsing and syntax, which i will get round to. It also uses inate forms in fact form is an inate notion derived from as it turns out all the sensory inputs. Form is a synesthesia. We have an extensive-model of the world which gives location information not only by binocular vision but by bin orificial sensation! So location is given by the visual kinesthetic auditory and gustatory and proproceptive sensual systems through processing algorithms in the brain. Kinestthetic systems properly include proprioceptive but i am distinguishing the external from the internal. Interestingly our kinesthetic includes th whisker effect of the hairs on our skin, which also contribute powerfully to location information .

Thus my primary reference system of orientation extension and rotation relative to the orientation is a conglomeration of the sensory input processed by an evolutionary and possibly revolutionary algorithm process in the brain which the whole organism accepts s a fundamental stabilisation . However alaong with these stimulus response products are also products such as mas, density, surface, form structure etc. The notioins of region and boundary are such products where region uses orientation and a soft focus to give a meaning to its denotation. The soft focus provides a boundary impression without defining a boundary. Boundaries can then be defined or discovered in this region in terns of sharp focus and the particular rods and cones in the eye stimulated by this procedure and aor any other binary type output from the supporting senses.

Mass then refers indiscriminately to the substance being sensed that is space itself. I can now quantify mass by boundarisation and form and structure and surface. Mass can be defined by a numeral representing the count of the structures and forms and surfaces and boundaries identified within a region

On the face of it a smooth surface may count at 1 form and/or 1 structure within a defined region, but i have to be careful not to confuse abstractions such as planes etc with regions. A surface is only usually one face of a form and special forms should be looked at in their actual not ideal format. Currently i can link mass through Avagadros constant to molecular structures in a form. Spaciometry includes these atomic forms in its conception, but this really highlights the fractal nature of all forms and regions and the structure of a region when boundarised may be analysed in this way to determine a mass.

Smooth surfaces are interesting because they highlight another aspect of mass and that is extension. This is why there is always some confusion between volume as it is defined and mass. the notion of extension is utilised as soon as i boundarise a region ie put a boundary corresponding to cone and rod signals from the eye that are binary. Thus i will modify the notion of mass as i explore it more.

I also have to look at the density of surfaces and structures within a boundary to appreciate and determine mass and kinesthetic contributions have to be included as well. Mass as a conception is fundamental to my notion of space and abstracting from that a mechanical definition has i believe divorced me intuitively from space itself!

I have yet to consider gaseous phase matter in terms of appreciation of its mass. Although not strictly gaseous i will begin by looking at clouds.

The profound result for me in developing a spaciometric mass and density is the discovery of magica trigger that gives me a full synesthesia of mass combining the notions to reaify mass.

Considering the gaseous phase of matter and in particular water vapour and clouds i realise that these regions of space have colour, from invisible in the near locality to opacity in tone and rainbow hue. This variation in itself gives me information about mass and density of the gas in the region. If i then include the contribution from smell i begin to realise how pervasive, subtle and powerful the apprehension of mass and density is. simply by smell i can appreciat the mass of a gas in the region i am identifying.

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.

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.

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 proforma 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.

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

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.

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.

http://www.fractalforums.com/gallery/3/410_26_08_10_3_36_15_1.png

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.

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.

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 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.

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.

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.