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

[jehovajah]

Found what i wanted 3d trochoids! Roulettes have not got this far yet.

[jehovajah]

I woke up this morning and started thinking about matter and its cognate mass, related through the attribute of density to space. Ended up thinking that certainly in the west if we want a theory of everything we will need to veer away from 2 unacknowledged principles: space is nothingness and space consequently is static. Similarly any notions of the void in eastern culture must have the following two principles: the void is somethingness and that somethingness is dynamic.

With these principles i can see that trochoids and more recently roulettes provide a spaciometric topology of the motion field or dynamic somethingness of space that account for both classical and quantum Physics.

Usually one then says i have not done the math yet to demonstrate this, however i want to point out that the math only corroborates if you believe and understand the math, and the math in this case is topology or rather spaciometry. Anybody who can think geometrically can corroborate it for themselves. Whether i or they can convince others is not my concern. To each their own i say.

Vector's post here on this page
http://www.fractalforums.com/3d-fractal-generation/truerer-true-3d-mandelbrot-fractal-(search-for-the-holy-grail-continues)/60/

may be very significant in this regard. In any case it is illuminating . I found it so then and even more so now.

All Vector's posts are found here
http://www.fractalforums.com/index.php?action=profile;u=1056;sa=showPosts

And i hope he will let me know if he minds but i have taken the liberty of uploading his work into my gallery.

"When i was a child i thought like a child and did childish things....  " this quote from st. Paul is significant in that we all have to embrace change and revolution in our thinking as we mature.

[jehovajah]

buckyballs and Buckminster Fuller a real example of geodesic curvature see the glossary.

Notice that a parameter is a choice of measurement technique, a way of turning geometry into numerals to utilize the properties of a set of geometrical spaces homomorphic to the set of numerals.

What we do is we standardise a geometry, we standardise a measurement fractal on a part of that geometry. we develop the algebra of the fractal on that geometry and then we apply the algebra to what seems to us as similar cases . When we do this the standardised fractal is called a parameter, because it is used alongside the geometry we are investigating. But it goes deeper. We actually dimension the geometry we are looking at using the standardised geometry so that we can use he parameter.

The dimension is how we cut up or boundarise the geometry and forms in the geometry. The parameter is how we measure the cut up pieces. So i can cut up 3d space into extended strips or areas  or volumes, and i can parameterise these by using a very thin strip as an extension measure; fractalise the strip for greater paramtric distinctions and dimension area and volume by radiating orthogonal strips of this fractal parameter. Each of these cuts (dimensions) is then measured by the parameter.

Now suppose the extended strip is chosen as a "straight" strip, we get a cartesian dimension of space, but if it is chosen as a curve we get a Riemannian dimension of space. Further if the curve is a closed loop then we get a "polar" or "spherical " dimensioning of space and the Lie Algebras that go with it .

We are more familiar with a mixture( straight and circular) of dimensioning curves when we deal with spherical coordinates, but this is not the only case. And hitherto i have been seeking a spiral dimensioning curve, but now realise that a trochoid set is probably the only uniquely parameterizable set of curves, at least up to the cardioid.

[jehovajah]

The Logos Response is a subject of intense investigation by scientists.  and the cold and the heat affect everything.

Enzymes as batteries

[jehovajah]

Truth like beauty is in the eye of the beholder.

[jehovajah]

 The eye  of the beholder.

Feeling blue? Do not listen to the blues!!.

Of mice and men

The Logos Response clearly has an emotional contingent as well, which makes blue skies very attractive and orienting.

[jehovajah]

Just a note about some spaciometric issues.

In general spaciometry deals with regions and boundaries of regions. Regions may be forms and boundaries may be surfaces.

The issue of a boundary is where or what is it?  I have repeatedly stated that it is a computed entity, and therefore iterative, which in my book makes any boundary a product of iteration and therefore a fractal.My  issue is the intersection of boundaries.

In spaciometry i would propose that a boundary crosses another spatially, in which case it is like to sticks crossing, or metaphorically, in which case the so called boundaries are actual regions that originate a a common source region.

Now the common source region is naturally a candidate for the term origin, and personally if the region can be bounded within a decreasing spiral that tend to the "centre" of the region, so that at each whorl, or properly trochoid, the origin of the two or more boundaries is evidenced i would be prepared to call that, under those conditions a point.

It is not so much that the limit exists but that i can arbitrarily define an approximation good enough for the purposes at hand. Being able to do this i would suggest is the intention behind the word "fractal" and encompasses for this notion of point the attribute of scale invariance and self similarity.

Although i have considered these points as origin, the limit process relies on them also being "sinks", that is a region from which or to which the boundaries are , in motion terms, moving.The spiral motion bounding such a region is crucial to the notion of the point originationg or capturing all so called boundaries if closed, or radials if not.

In a similar way the spiral or vortex motion is crucial to defining an axis, and therefore a notion of a straight boundary or radial.

I note as i write that i am preferring to use radial for any line originating from a point or sinking into a point.

I also note that the crossing of radials can only be approximated by drawing pencil lines and assuming that these lines are in fact origins or sinks, otherwise we are approximating crossing sticks or any 3d object crossing, and a thickness attribute is necessary.
Similarly if i roll up a boundary or radial around its sink or origin a thickness attribute is necessary. Therefore even in the point notion above the thickness of the origin has to be adjusted by negative magnification to apply the "rule" and that is why self similarity is inherent.

In spaciometry therefore scale magnification is also adjusted in limit processes, and is a determiner of when the limit process has a natural end.

[jehovajah]

Descartes laws of motion compared with Newton's, but rather an example of a long line of evolving thought on motion and motion fields.

Laws of conservation therefore have there philosophical basis in a mystical notion of the immutability of deity. Laws of conservation have proven extremely useful, and that is their main ground for retention, i.e. they seem to work. However they have no intrinsic validity if they are not derived from an immutable "something". and immutability is hard to find nowadays.

It is also of interest to me that Descartes proposed that all other motions could be reduced to his conception of motion, and we find this simplification in the reducing of motion to straight line events today. Kepler it is mentioned thought otherwise, and accepted motion as intrinsically curvaceous. The consequence of this is to upset the Newtonian model slightly and to precursor the Einsteinian model greatly.

[jehovajah]

Newton and Descartes on  Force and inertia is an interesting insight as is Newton on Matter.

Newton's notion of matter is the more confusing and unnatural. In my opinion also unnecessary as Density is a separate attribute of mass and could have been clearly defined as such at that time. However Given that Descartes had no conception of the role of density in force and motion transfer it has to be seen as a step in the right direction.

Newton's notion of force is new. Descartes notion goes back to the greeks and though more obvious it is not based on observation in detail, or analysis of Galileo. Newton's notion is more mysterious: Force only appears in the action and application of it, and acceleration is proportional to the applied force, and inheres in he applied force. Plus force resists force initially!

What Newton was describing was Equilibrium both static and dynamic. Because it was not understood as an accelerator until Newton, Equilibrium and stability took some time to be appreciated.

It is important to realise that along with a motion field there is a concommitant equilibrium field, called an inertial field.This is the viscosity of space and the density of the region in which we have our being, and is a relativistic phenomena.

[jehovajah]

The motion field which we exist in has not been identified yet in the same terms as other field theories.
It will have to account for inertia, friction, acceleration, equilibrium, both static and dynamic and complex motion of "bodies" in various phase states. The behaviour of motion should encompass both wave and particle motion and spin at all scales.

[jehovajah]

When considering the notion of motion one has to acknowledge a rational framework for it and for Descartes this was a god, as it was for Newton and particularly so for Leibniz. The idea was therefore to correctly describe the workings of this god in the substantive realm, there being , as assumed by all , a non substantive realm. Descartes by minimal observation and maximal deduction from religious principles derived laws for the substance and motion in this substantial realm, but in ignoring observation committed "fallacies" to which Leibniz drew attention, and in particular created a problem with the insubstantial interacting with the substantial. Leibniz attempted to correct for this by invoking a kind of life force in motile bodies, derived from god and not intrinsically related to matter in the cartesian way.

Newton felt inspired to see this force as being intimately involved with motion and proportional not to the mass but to the acceleration, thus extricating it from Descartes  matter notions and Leibniz unclear proportional relationship, based on Huygens and his desire for immutability to support his God cause.

Newton was Equally motivated by divine inspiration to set out accurately what his God knew to be the working of his motion field of matter, but this time based on acute observation. Thus the mysteries of the acceleration and inertia revealed themselves to his gaze  but through a 17th century prism that made force an interference from god or beings uknown who nevertheless obeyed god given laws.

Force was not defined because it was axiomatic that it was a "divine" push, transferred and maintained by bodily interaction.
Newton demonstrated that this was through change of acceleration,and noted resistance to acceleration in his laws of motion as inhering force. This subtle difference enhanced the role of density in his thinking, and the maleability of mass within a magnitude.

For Newton a body was a mass within a magnitude of space,but he could not seem to clarify this thought and therefore treated of point masses.

From Descartes to all following the substantial nature of space was dual : it contained matter and nothingness: God was in an entirely different untouchable realm. Thus matter devoid of its cause and in a non impinging space called nothingness is the root of the difficulty in comprehending motion and our motion field in particular

[jehovajah]

Spinoza attemped to make motion inherent in space as an infinite attribute. While Spinoza can be a difficult read because of the subtlety of the ideas and the comparison with Descartes overwhelming the contrast, it is clear on reflection that the Cartesian Axiom of Extension and motion being an inherent unity is what causes the differences in opinions.

Thus, correcting an earlier statement i posted Descartes and Spinoza And Leibniz delineate a chain of thinking starting with a substantive whole differentiated solely by motion to a dualistic whole which is intrinsically differentiated requiring insubstantial attributes such as god derived forces and immediate infinite action to maintain observed motion behaviour. Newton in his philosphy adopted a truely absolute spatial "nothingness" as a non impinging backdrop to his discussions, and thus firmly fixed the duality in modern scientific thought, by default!

I cannot think that Newton intended to decide the issue through his philosophy, but rather to provide the most accurate information and description of how motion actually worked that he could, admittedly as a divine revelation, but not as a declaration of war on Cartesian thinking.In fact Newtonians and Cartesians jointly attacked Leibniz school of thought, not each other. The popularity of Newton's mathematical mechanics and dynamics meant his working axioms were absorbed by osmosis.

As Newton did not set out to Debunk Descartes philosophy, there is no clear working out of how his working axioms, assumptions affect the Cartesian Economy, and indeed Einstein and others have paid little attention to the effects of the philosophical underpinnings, regarding the metaphysics as beyond them, and dismissing it with a few pithy sound bite comments.

The philosophy of relativity was overwhelmed in the existential philosophy of the early 20th century, firmly placing the Cartesian Description on the back burners of history.

The relevance of Descartes is the unwinding of confused modalities in modern metaphysics of Physics for example; and for illuminating the notions we use everyday confusedly, despite having hard referents for them.

Spinoza by making motion immediate and infinite in his philosophy lifted the causal bond Descartes places in his philosophy between gods and motion, but without any causal replacement he has to make it infinite and eternal, ie he did  not think it necessary to have a impinging non substantial cause forever at work, for his own reasons, just as Descartes had his reasons for deducing such an impingement.

Leibniz for his own reasons wanted an inherent impingement, as an internal and eternal property of matter attempting to deal with the insubstantial /substantial problem of cause. Thus the problem derived from the insubstantial realm and its action of impingement on a substantial realm. Newton sidestepped this problem by dealing with the two realms separately in his work, hoping to reveal the connection through his work rather than to assume it.

Thus Newton's description of force and inertia are in context the beginning of his contribution to how the insubstantial impinges on the substantial in observation. That his method has supported along with Darwinism a general decline in belief in the agency of gods i am sure would be of concern to him, but that is the nature of cultural reworking of important mores, any idea that gives grist to the agitator's mill is used to promote change. We call ir spin doctoring nowadays.

I find after engaging in this study that i am reworking Cartesian notions in the light of Einsteinian Relativity theory,and Feynman QCD theory. It is about time i think that somebody attempts this.

[jehovajah]

Hermann Weyl  

Spacetime

Philosophy of science.

I have found Kant to be the one who advanced the slow unravelling of Descartes conception up to Newton, by advancing Newton's model, without his Philosophy. Therefore Kant provided a way forward into 20th century philosophy and scientific thinking.

[jehovajah]

I find in Descartes the kind of thinker who typifies the Great philosophers of the ancient ancient world. Thus Descartes could be Chinese or Indian or Buddhist or Babylonian, Islamic or a magi from the far eastern regions of Europe, or of  the natives or indigenous peoples of many islands and coasts.

What Descartes philosophy reflects is the gathering of knowledge afforded by Empire, and the industrious aggregation and compilation and processing of knowledge by the Islamic Arabic empires with its seat of learning at Baghdad.

FGrom this perspective Descartes is possibly the last of this old school philosophy where perfection is the organizing principle and the absolute is the self justifying basis and goal of thought.

Although Chinese philosophy encapsulate the no absolutes dynamic nature of "reality", it does do perfectly! Therefore they avoid the stagnation and inconsistencies of perfection within their philosophical model but have no falsifiable limit to it. tus they are supreme analogical thinkers, with a system of philosophy that will encompass any change or new observation by rearrangement of the emphasis on the elements of the old and by attachment of referents to existing "yi".

This makes chinese philosophy moribund, confusing and taciturn, requiring years of study to master even but a portion of its applicability. It is a blunt instrument for analysis but a marvelous repository of knowledge and information and categorisation.

Reform of the chinese philosophy by Lai Zhide for example reveals how new scientific insights can be clothed in ancient symbols and practices, giving a false sense, but a very chinese ideal of respect for the wisdom of the elders. This leads to a slow plodding approach to innovation.

Descartes was the tipping point for a period of ceaseless innovation in the west. Because Descartes was absolutist, he was vulnerable to attack, and indeed Spinoza reformed Descartes conception with his own absolutist ideas, and so on.

However Galileo and the great Leonardo Da Vinci were observationalists, called empiricalists. They were hounded because they were not absolutists, therefore they were against god and deceived by the antagonist to god, a very zoroastrian notion.

Some how what the magisterium feared and tried to prevent happened and the seed of empiricalism grew and finally toppled the absolutist paradigm, and Newton and Leibniz were crucial to the unraveling of the absolutist philosophy of Descartes, although only Spinoza at the time proposed an infinite substantial property not requiring any agency as an axiomatic possibility.

Today we still have absolutists contending with empiricalists and all shades in between! We could eventually resolve this in the style of chinese philosophy, but to me that is to miss a trick and an opportunity.

Benoit Mandelbrot provides me with a Type or paradigm of geometry which it is possible to philosophise into a system that i hope can resolve these issues in a modern, fresh and innovative way. Moreover it provides a substantial rethinking of the roots of knowledge (epistemology) the fundamentals of motion( metaphysics) the aesthetic response and the ethics of animates.

The Essential idea of Mandelbrot's paradigm is Roughness and i will address this idea in what i think are its essential components: iteration and approximation and computation and manipulation and scale/magnification dependence (self similarity).

[jehovajah]

   the question is can empiricalism deliver a grand unified philosophy? Furthermore should it even strive to?

I think any unifying idea or ideal is not an empirical product, but a computational one arising out of a pattern seeking algorithm, which is perhaps typified by iterative function systems.

This area of pattern analysis is probably best suited for constructing an overstructure for empiricalism.

The question of striving i think is an over statement as i think the algorithm organises in this way whether we are conscious of it, attending to it or not.