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

[jehovajah]

Aha ! Classical Hamiltonian Quaternions! Nice!

[jehovajah]

Of Logarithms and modulo/ clock arithmetics.

Aggregation  is usually notated with a + symbol, but this activity is far from simple and far from binary. It is algorithmic in nature and is different to measurement.

Measurement is an activity of apprehension, of response to the environment, of applying or observing the application of unity to a form in plethora, a sensing of the structure and construction of a form by application of the unit measure, or by watching how the unit measure populates a form dynamically.

Aggregation is an activity that we perceive as happening to all sorts of things as they gather or are gathered together.My perception links, yokes, ties things together in a subjective collection. I may even perceive or draw or define  boundary or an envelope for the aggregation that dynamically changes.

Dynamic as these forms structures and aggregates are i still perceive a static form as a resolution, or a completion, a whole an answer a sum etc.

y perception of aggregation and measurement is therefore dynamic and i only "know" when i have an answer when i stop, or it stops being dynamic and becomes stable or equilibrium.

The dynamism allows for rhythm and motion and dance and song and metre and ritual and repetition, iteration and convolution and poetry and rhyme of drawing and painting of surfaces. Sculpture arises out of disaggregation.

So when we look at the Indian decimal system for noting numbers we first have to recognise that it is an aggregation, and then a special rhythmical form of aggregation. It is an aggregation of nested unities, with a gate + algorithm. The gatre+ algorithm controls the addition of the unities on a modulo basis. I could write the gate + as  +modulo(10), meaning only add when this bracket = 10. There is one other thing and that is the power of the modulo, which is its place value power.

So its complete form is +modulo(10^n) where n is 0,1,2,3,4...

I can write the indian rhythm as


   ..........+modulo(10^3)+modulo(10^2) +modulo(10^1)+modulo(10^0)

which i can write as

........a*10^3+b*10^2+c*10^1+d*10^0 wnere ab,c,d are in modulo(10) or mod(9).

There are several ways of notating Napiers Logarithms but this one  makes clear the previous discussions link to them and all calculation

n    =     a*log[((a-1)/a)^n ] where (a-1)/a is the base of the logarithm which is n/a. a is a factor to make the log into an integer.

As previously discussed the logarithm applies to all the values in between the integer value of the log. Therefore each base is portioned  by the log method on a modulo arithmetic scheme: each whole log producing a proportional decrease by the base, so each partial log producing a proportional decrease somewhere in between each base.

Thus  mod(base)*base^ logarithm is a way of representing the naperian notion of antilogarithm as Brigg expressed them in base 10.

Thus Brigg and Napier are really linking the indian rhythm to the dynamic motion of a hypotenuse of a right triangle in the sky, a hypotenuse attached to some moving planet or star, and thus linking its motion to the logarithms base sinπ/2.

Now the indians had already shown themselves masters of calculating sines for their astronomical use so it would appear that the sines were of great use and significance to astronomers of all cultures for calculation, but it was Napier who through dint of hard calculation turned them into a labour saving tool of great utility.

Prosthaparesis thus recommended the trig ratios as a useful source of exploration for quick reliable calculations, and Just Burgi made great use of it and some other methods which turned out to be correspondence relations between series not logarithms

We find that logarithms are a fundamental form of all computational structures since the indian rhythm was adopted.Logarithms are the basis of polynomials, the basis of our p-adic number system, the analysis of geometric and arithmetic series, the fundamental structure of our aggregation schemes, the link between the entities called roots of unity and the yoked units of shunaya, the essential relation between scale and proportion, an essential link in describing motion and relative motion and a describer of growth and gravitational constraints, and definitely more.

[jehovajah]

Of measure and of measuring, arithmoi and arithmetic,numero and numbering calculus and calculating , count and accounting reckon and reckoning proportion and proportioning, manipule and manipulating, putative and computation, ratio rate and rationing, gnomon and trigonometry ,vehicle and vector, magnification and magnitude .

We are in a dynamic space we sense it therefore we are. We have sensations therefore we are conscious. We manipulate sensations therefore we think,compute, and consciously measure and manipulate and language. We respond to shunaya and grow in shunaya because we are of shunaya. Our survival imperative our evolutionary imperative are manipulations and computations within shunaya.

Thus we are, and thus we cease to be absumed into shunayas ceaseless computation  and our thinkig consumed by it all our distinct days.

[jehovajah]

A vector definition as per Hamilton is inadequate, and betrays a misaaprehension of greek apprehensio of the space in which they geometrised. Shunaya has always been dynamic and pregnant for the Hindu, the Buddhist the Confucianist. The greeks branched out into something different, they looked for unity an atom to match the sanskrit "atem". a "non divisible" to match the "spark of self".

All the while their units were in constant motion, like atoms, becoming plethora, being aggregated, disaggregating and dividing and dissolving.

Thus to the greeks their units were dynamic magnitudes.

    A unit is a dynamic magnitude

They watched the units aggregating or being aggregated; or how they aggregated and manipulated them themselves. From this study they learned that motion and dynamism of a unit can and is revealed by using smaller units. As the units decrease in magnitude the detail of motion and structure and relation and relative internal motion as well as external_to_the_unit motion and dynamism of the unit is revealed. This is like magnification of detail, and the greater the magnification the smaller the unit size being used to measure.

The units measure magnitude, relative motion, rotation, twist and stretch and skew, internal relative motion and structure and plethora interaction.

Thus a unit is a dynamic magnitude used to measure and through a measure to display and describe relation, motion, etc.

The measurement tool is therefore fundamental to measuring, describing, and displaying and verifying our exploration of shunaya. Such a measuring tool aggregates units and relations and motions into a handy tools and holds in a static form a representation of a dynamic magnitude or representation of structure of dynamic magnitudes.

When i look at a tool i almost immediately discount the activity i engage in to use the tool. This activity is as important as the tool as it restores dynamism to a static form or structure of our conceived units.

[jehovajah]

When looking at aggregation rhythms and the +gates of various algorithmic structures, i have looked at a fundamental logarithmic structure.  The logarithmic structure using various bases ties together nearly all rhythmic aggregations.

However their are aggregation rhythms tha are not simply logarithmi:  n! means n*n-1*....*3*2*1.

So an aggregation rhythm based on n! would look like

                            n!+n-1!+*n-2!........+6!+5!+4!+3!+2! +1!

These types of aggregations include
1/1!+1/2!+1/3!+1/4!+......1/n-1!+1/n!+

There are other aggregate based around the permutation and combination formulae.

I think this aggregarion can be written in terms of the gamma function and their is an analytical treatment of function relations  i do not understand called Verblen functions which may be related.


The Γ function

The function Γ enumerates the ordinals α such that φα(0) = α. Γ0 is the Feferman-Schutte ordinal, i.e. it is the smallest α such that φα(0) = α.

For Γ0 =Γ(1)
 Γβ+1= Γ(β+2)

 the aggregate would be
 
  Γβ+1+ Γβ+.......+ Γ0

where the +gate would be +mod(Γβ+1/Γβ).

I am not certain my notation is correct, but i think the general idea is clear.

[jehovajah]

David Attenborough illustrated very simply and powerfully that the size of the mammalian brain correlates with social group size, not intelligence.

This is of importance because it gives a diagnostic indicator of autism.

On reflection one might conclude that all whose brain size is below a norm for a social group will be relatively autistic within that group. Thus individuality would enable a new group dynamic to form gradually through evolution of  a family group structure . Such groups would display distinct characteristics called tribal traits, which in fact we could rename relative to a larger societal group as autistic traits.

In addition we would suspect brain dysfunction within a normal brain size that displayed autistic traits.

We know that sexual evolutionary pressure is a major diversifier of group characteristics, and it is of course entirely reasonable to expect brain size to similarly diversify a social population into gangs, tribes , clubs etc, with social order within such groups correlating to brain size within the group.

Now the male-female division has been under investigation for some time and a result has been announced making it clear that the brain has a functional role in sex differentiation over evolution. This differentiation of course is statistical and not genetically set in stone. Therefore the operation of 2 genes are in evidence and the variation through this combination is sufficient to explain the characteristics of male-female gender sexual preference and sexuality, and why some are male inside a female phenotype and female in a male phenotype.

Of course these descriptions are subjective, but a combinatorial study should perhaps show the various population percentages expected within each identified possibility. This would help to demonstrate if this gene effect is real or  a figment of imagination.

[jehovajah]

I have toyed with the idea of using "recipe" for the idea of function, not only to make it more accessible but to alert those who everyday use recipes to the essential mathematical activity of their labours!

Today many are labouring over recipes traditional and modern in order to celebrate Christmas! Merry Christmas to all you mathematicians who are making enjoyable use of your inate mathematical abilities today, especially you mothers, whose formulations and activities will result in a successful christmas day for all your families.

Shalom

[jehovajah]

I had thought to look at an aggregation rhythm based on the  logarithms to different bases but realised that these make little sense as aggregates because they are not a rhythmical depiction og plethora or aggregation. They are a signal of a stage in a calculation algorithm

The Greek notion of plethora which taken together with unity forms our basic notion of scalar and scale is a useful one. It covers dynamic increase in the units whether by multiplication, scaling the rhythm of the musical bar while counting, or by division of unit size, setting up sub scales of rhythm within a larger initial unit. Thus a plethora is a dynamic description of unit motion whether observed in a living organism with cell division or a mass production or replication of the unit through some specified agency.
dynamic
Replication of the unit is the macro notion of plethora, whereas plethora through a division process of unit size reduction is the micro notion of plethora, but it is always dynamic.

Aggregation is the apprehension of gathering items together, both physically and psychologically, mentally. An aggregation is a more general structure than form, and is an appreciation of gatheredness. It is not as dynamic and self organising as plethora, but of course could be made of many plethora!

The factorial number system, or as I would draw attention to it, the factorial rhythm aggregation has been explored in relation to probability distributions, and a link to logarithmic aggregate rhythms has been shown, but I would observe the underlying structure of any aggregation purporting to use or relate to logarithms would have to be an antilogarithm aggregate scheme, whereby the base of the contiguos unity in the arrangement will increase by one:

+ mod(n)*n^logarithm+ mod(n-1)*(n-1)^logarithm+......+ mod(2)*2^logarithm+ mod(1)*1^logarithm+mod(0)*0^logarithm

The + gate is not the +mod()  gate we have encountered earlier. The factorial rhythm has a +mod(n!/n-1!)
 + gate. Because of this I cannot make an aggregate rhythm and therefore this type of aggregation is different.

Since this represents an aggregation of rhythmical aggregations each strand of the aggregation is a complete system by itself,so there is no need to move to a different strand to change scale, the logarithm does that , so this aggregation is a ranking or quality aggregation,allowing qualities or ranks to be aggregated.

The +gate thus is a quality yoke and does not allow summation except on an arbitrary imputed assignment. The transform between strands is a ratio of logarithms
Logb(1)/logb(b+1)

When that ratio exists between the strand members those members can be summed at the logarithm level and we know that represents a product at the antilog level.
Apart from rank or quality measurement there may be other uses for such an aggregation mesh of rhythms.

The +gate has become an equivalence relation.

[jehovajah]

If asked to represent in any artform or mixed media the notion of curved motion, how many would use  stereophonic soundscapes, tactile intensity maps, or gustatory intensity maps?

I have mentioned before the fundamental nature of the Logos Response to apprehending notFS, particularly in relation to proprioception maps. However this is only one aspect of the synaesthesia we use in apprehending shunaya, notFS. 

[jehovajah]

Spin field effect transistor announced, using the effect of a spin helix field.

And i realise thar polar equations make spirals easy and plot in spacetime.us.

Jakob Steiner is my next post.

[jehovajah]

Jakob Steiner  gives me hope that the geometrical/spaciometric basis of Algebra is evidently fundamental.

[jehovajah]

Of the mesh of logarithmic aggregates and the dance rhythms of street dancers and Break dancing.

Of the factorial rhythm aggregate and the factorially fractal structure of infinite possibility space, as it condenses   stochastic probability clouds in regional spaces , containing statistically significant propensities to computational rearrangement: some of which condense into dense regions of relativistic motion space.

Of discontinuous attributable space, to which regionalised properties  are attributed subjectively, and which regions are dependent on the measuring instrument adopted subjectively.

of a dynamic density function that regionalises space into condensing and evaporating potentials, and which is dynamically in equilibrium

[jehovajah]

3d polar coordinate geometry is quite hard to find a very general treatment of it which is not fully wrapped in tensor or vector notation.


Take a unit pole and a variable rod, and attach the rod o the poke so that it is joined in a corner which is free to rotate. This measring device will be the basis of some measurements of regions, and works by pointing the pole at one point on the boundary of the region and moving the rod around the region boundary noting the scalar length of the rod as it moves round and the radian angle measure between the rod and the pole as it moves in this way. I thus record but clearly not the rotation of the polar axis( the pole) as the rod traces round the region boundary. For this purpose i have an orthogonal unit that is attached to the pole, thus the pole becomes a right triangle, a Bombelli vector, and i have a third marker of unit length orthogonal to the pole but free to rotate in a plane orthogonal the pole . This marker is also fixed on a point on the boundary of the region if possible or some point relative to which the region is fixed.

Now i can measure pole's   axial rotation in radians. This is exactly like using a pair of compasses with the region being traced out by the pencil. Thus we record .

We can then write equations that have to distinguish and and therefore these require +gates and the cos and other trig functions.

When we look at the role of the roots of unity we find that they do one thing, they rotate the unit magnitude in space. So + and - are π rotations of the unit magnitude. The functions that control this rotation and its rate and nature are the trig functions. The + and the - are +gate modifiers, like mod() and control how units are aggregated. and what quantity.


for example.

[jehovajah]

Standing in the middle of the bank holiday sale motions of people , discerning the aggregation rhythms of their motions, the dance of their hurried predilections.
Suddenly the logarithmic rhythm of their dance gradually emerges.

I think of disaggregation and am immediately surprised by the difference in rhythm. The modulo minus gate -mod() starts with a bang! The rule that you only sum units that are the same in any aggregation applies to disaggregation, but the dissolution of a unit into a smaller nested unit is a shattering surprise! One sees a whole unit dissolve before ones eyes into smaller units that disaggregate and disperse in unexpected ways!

Really the imposed order of aggregation lulls one into a false sense of an orderly build up. But the unfamiliarity of disaggregation means that anything is possible!

So A unit kind of dissolves into smaller units enabling smaller units to be disaggregated. Whole units can obviously be moved away as a whole, or roll away in blocks, etc, but smaller units require  unit to dissolve or  shatter into smaller units.

Therefore the shattering of a plate when it drops on the floor, the shearing off of a precipice of ice from a glacier, a disintegration of a piece of wood into shards and splinters are all examples of the logarithmic rhythms of disaggregation.

So now the aggregation of things has a moment of unification of a larger unit which is somehow less dramatic than the shattering of a unit, but in fact it is more interesting than that because we see these unifications in everyday situations. Traffic jams, where a collection of rhythms forms  solid looking block, that dissipates as the rhythms play out. The formation of planets where a cloud of smaller units surrounds and moves with the larger units. Standing observing a crowd one sees clumping and dispersing of groups of people, larger groups forming and splitting , moving and whirling away and together, finally splitting into single units.

The formation of waves in a dynamic ocean as mounds of water surrounded by smaller dissipating ripples, amonst the coagulating mounds.

This is a great cycle of motion in a motion field experiencing aggregation and disaggregation at the same time, leading to a density distribution that is dynamic and polarised around intense density and extreme rarefaction.

Thus it became a syncopated thought that the logarithmic rhythms of aggregation and disaggregation play their part in the dance of cosmogeny.

[jehovajah]

What we attribute is what we tend to find.

Aggregation and disintegration images