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

[jehovajah]

Sensors and sensor meshes are the basis of The Logos Response. Therefore how they depend on the underlying relativistic motion field is of interest and of importance.

At the level of molecular biology research is being done that supports a electro mechanical reductionism of sensor function.

This means that amongst other thing we can look at cell groupings and structures in terms of electronic circuitry.

I would go further and view any molecular structure as a proto electric circuit.

I remember developing two characters, Thermo and Electro to dramatise what happens at this atomic level, so i can see Electro building circuits and Thermo trying to destroy them, but all he does is help them evolve!

I guess i can give them partners: Therma and Elektra! No i am not going to give them babies today! They have only just met!!!

[jehovajah]

So when i was positing a motion field as an axiom for the set FS, that was stated in terms of relativistic motion and relativistic motion transfer, to which i would now add relativistic equilibrium or relativistic motion equilibrium. This to me would be the basis of all mechanics and dynamics including QCD.

This may take a bit of thought and exploration to see how the latest work on phase change models my ideas but i feel that it is worth it. This is not to say my idea are cool, just to say that this is what i am talking about, willis!

[jehovajah]

Yea Leibniz, yea Leibniz, go Leibniz!

[jehovajah]

There is a a direct line from Huygens through Leibniz to the modern notion of Energy. .

Leibniz vis viva was influenced by Huygens formulation of mechanical sytems in which mv^2 featured significantly. In particular the derivation of centripetal motion kinematics reminiscent of Kepler's law, mv^2 was highlighted in closed systems suggesting that it was conserved. Leibniz took a logical step toward identifying this with  the inherent god given "force" that keeps things moving, in distinction to Descartes who deduced an impinging force from god always acting making things move but in no particular direction, and according to a law which kept mv constant. Newton while still upholding Descartes mv conservation empiricaaly found that god's force appeared in accelerating bodies and resisted initial impression to move then swung round and accelerated the body.(An equilibrium system).

So there were 3 notions or uses of the term force which have been retained to this day under different names.

Only Spinoza posited eternal motion unaided.A review of his philosophy

[jehovajah]

Man the thinker
man the handler
The footler
the tailer
with every part of him
she thinks
With his penis he thinks
with her womb she thinks
with their heart
in his heart
she beats out every breathing moment
of their intertwined life
apprehending
comprehending
prehensile
in every taut
and tautological thought
with which he measures the world
that she manipulates
As toys and tools and do s and don'ts.
Holding on to notions taught
As if they lorded over all
Why not let them go
And cease the day
Why not?
Let us play..
for we discover more that way.

[jehovajah]

Albrecht DÚrer an artist mathematician who so symbolises what people at Fractalforums are doing.

The incredible melancolia

[jehovajah]

I once loved a woman,
  a child i am told,
       i gave her my heart,but
She wanted my soul..
No, don't think twice cos it's alright!

Bob Dylan

I   love my wife of 22 years more than ever. Happy Anniversary Mate!

Just in passing a force IS a change in velocity, precisely, The notion of force that takes form after Descartes actually is confused, although of course Newton describes it clearly. I prefer pressure to force because it does not require me to think of some insubstantial person pushing or pulling, which is the essential but non empirical referent of force. The genernal force is like the wind or an expanding or moving gas, and by usisng Boyles Analysis i have a sufficient and robust notion of force related to pressure, and need only to observe that this is resusltant in a change in velocity. The directional nature of force is not to be simplified as Newton did but to be derived by quaternion vector addition over the surface of pressure.

[jehovajah]

a wolf spider in my wifes garden said to me,
"the world is as it is because of radials and trochoids.
in this world, such as it is, if you want a flat plane
join precisely 3 straight lines into a triangle".
and then through his neusis he spiraled away.
i was left wondering and in wonder...

[jehovajah]

So for me this is the final part of the puzzle about polynomials.

The history of polynomials starts in proportions and equations from proportions. The application of proportions particularly to geometrical concerns naturally leads up to the cubics and intimates the quartics and beyond. But the difficulty that arose through this research was the development of generalisation. Generalisation is called algebra after the arabic meaning restoration, restoring the greek and indian methods of calculating in general.

Along with arithmetic i can derive the notion of algorithm and the 2 ideas about sum up the dimensions of algebra: Rules and methods of calculation specifying what where when and how and what is acceptable as a result. Some time between the greeks and the Arabs influence number took precedence over proportion and measure.

So when the equations started to not have solutions that were numbers, mathematicians were slowly forced to creep up on measure and proportion again. At the same time the chinese and Japanese were developing solutions to equations that used a table or matrix form. They were developing an algebra of matrices for linear equations.

Before any advancement to polynomials could be made the issue of negative surds had to be resolved. Negative numbers had been introduced in all forms of mathematics worldwide but negative surds had not received a serious acceptance until Bombelli. Once Bombelli had produced his "Algebra the major part of the arithmetic",  other mathematicians were able to synthesise a theory of polynomials, and to think in terms of generalised polynomials, and finally to begin to use the taxonomic name polynomial to describe the field of study. And as polynomials developed so did matrix theory and determinants as it already had in the far east.

Along with the theory of polynomials the development of algebra moved eventually in several directions:Calculus, Cartesian geometry, Advanced Algebra including extension to Quaternions and ring  and group theory, function theory,Tensors etc and through algorithms and symbolic algebras computational science and computers developed.

Despite this fascinating development from Bombelli's impetus, mathematicians still have not acceded to the vital role of geometry/ topology in the foundation of mathematics.

[jehovajah]

With my grandchildren i would start in kindegarten/nursery with Fractals. These would be solid and real entities as in fractals in nature, ephemeral and fleeting as in fractals in soundscapes and as thrilling and as exciting as fractals in rhythm and motion.

From this great experiential base including others i have not mentioned i would by degrees construct the fundamental notions of boundary  form and shape , surface and density and viscosity and pressure.

From these i would define spaciometry and study topology and set theory to distinguish by notation and language form how we proceed to construct an algebra. Having introduced the young mind to such experiences i would drill down into historical geometries and attributes and develop the notion of proportion and quantity and ratio.
I would then delineate fractals and their use in scales and measurement, and reiterate the fundamental nature of iteration. only then would i introduce the historical development of number and a critique of the cultural iteration of numbering or counting.

[jehovajah]

Space the final frontier of mathematics, or rather spaciometry!(both readings).

I have come to an appreciation of notFS that i did not start with. NotFS is perceivable as space, but not an arid Newtonian space but a dynamic form that everywhere impinges at different scales through relativistic motion, relativistic motion transfer and relativisric equilibrium. And where it impinges on me is precisely where i interact with it on all sensory levels and beyond those by scale to levels of interaction i call quantum and universal .

These mysterious levels of interaction are truely beyond my comprehension, forcing or more exactly energising me to accept them as essentially existing but unknowable, but excitingly and surprisingly apprehendable: something i can devote my feeble and mortal life to with the fullest of satisfaction.

That i see iteration at all levels is a symphony of the essential harmony between the processes that govern me and my form and the inherent processes within notFS. My response , the Logos Response, reveals to me the apriori informationinherent within notFs which  i uniquely both discover, uncover, and recover as well as create and transform cause to exist in a certain form and relativity and initiate into my consciousness  and thereby uniquely store and retain it as a transformation of notFS. These transformations i call the set FS, what i know and have modelled from interacting with notFS.

Therefore randomness and complexity are not attributes of notFS, but of the set FS. This is not to say notFS is not random and complex, but that i cannot know if it is, i have only reacted to it at certain stages in my life and development by calling it so.

The better reaction i feel is to think of it as containing all possibilities, and thereby to approach it positively and exuberantly, all the days of my vain and futile life! 
In my approach to notFS i intuitively utilise iteration(unavoidable) and iterations of signals convoluting them into a format  called by me measure. These measures proportion notFs and spatialise it giving perspective and ratio and pressure maps in all sensory systems, utilising signal interference and thus inhering a notion, an intuition of iterative pattern making on the basis of an equilibrium pattern of forms called sensor meshes. These evolved forms of sensor meshes represent a transformation from biological molecular conformation networks, responding to basic quantum electromagnetohydrodynamic motions in notFS.

Thus the logos response generates my spaciometry and i distinguish my spaciometry by language forms which provide me with the basic mathematics of my computationally constructed spaciometry. Mathematics is language development and language development is mathematics.

Therfore i sought to rechristen mathematics as manipumæ.

[jehovajah]

I guess we just do not remember. Few of us di recall the first experience of consciousness because it is in the womb. But the first thing we iteratively measure through our sensory mesh is the dynamic motion of pressure and sound. And then when we are born we are ovewhelmed with the dynamic motion of vision and taste and smell and gravitational pressure.

In some animates the sensory mesh has evolved to provide apriori motor and spatial ability through "instinct", which of course is a genetic neural programme or operating system enhancing following and copying behaviour through chemical and sensory stimulation. Even plants follow the sun!

So we use our sensory mesh instinctively in and iterative way to capture the dynamic nature of notFS. How we organise this flux of information is the proper study of computation, but mathematically,by manipue we use relativity based on radials and meshes and we sense equilibria int hese meshes by rotational, translational, orthogonal and proproceptive sensors.

[jehovajah]

It is probably a longshot, but i think there is something about Chladni patterns that remind me of Trochoids.

So far i have found that few people are properly studying [urlhttp://www.cymatics.co.uk/=]cymatics[/url]
but fluid dynamics and high energy plasma physics have already linked trochoids to standing wave phenomena at distinct energy levels. So i will watch the developments with interest.

One question i have asked myself recently is how is light refracted around an edge in a thin slit interferometer, and i suspect it has something to do with trochoidal motion in 3d when light is polarized.

[jehovajah]

@ Einstein.

How ironic i must say
if god turns out with dice to play
or that we shall come to see
 that god is us , that's you and me
 and that is not to say
 that we shall live forever and a day
 but that the world does come about
by how are processing turns things out!

[jehovajah]

As usual lots of things to research on the significnce of Hamilton. Time and of course Leibniz views on it with regard to motion, randomness in which Benoit makes a welcome appearance, Quaternions through which i hope to place in my mind Hamilton's mileu and how he presented his discovery in different fields of maths and physics, starting with his battle with cartesian coordinates and the introduction of versors,and his application to generalized coordinates,

After that Hamilton is subsumed in the general Lagrangian mechanics and distinguished as a Extension field in a more general study of classical mechanics and lighting the way to quantum mechanics and Feynman path integral mechanics. Hamiltonian mechanics can now be approached as a "worked over" reformulation of Hamiltons notations and inventions, generalising them into other fields with dry and self important names which totally disconnect the general reader!
We need to look at the Legendre transform, but after that it becomes all greek to me!

An overview of vectors is instructive given the history discussed before in an earlier post. It is interesting to note that Newton's contribution to vectors is little recognised, eing overshadowed by his Principea.
the impact of compound interest in the development of Calculus i have noted before, and hidden in that is Newton's contribution to vector application, through Descartes diferrentials and tangents.

i need also to look at the Laplacian of scalars and vectors and spherical harmonics which i want to explore in relation to trochoids in 3d.