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

[jehovajah]

I hope that others will be suitably inspired to contribute to the thread. Having has a rather harrowing day, and being reminded of the fate of the late Henry J S Smith, i feel it is a good time to go on and extended holiday.

In many many ways i have personally gone beyond what i could ever have imagined on setting out. Happily there i no end of things to explore: for example reciprocals, Kujonai operators, Tim Golden's generalised coordinates, the parsing function we as measurers use to establish a read of a 3D dynamic magnitude and its relation to electronic video cameras and very fast camera chips/sensors, and image and signal processing systems of a convoluted nature. etc. etc. etc.

Fascinating stuff which i hope someone out there will contribute to the thread.

```i am having a break before my "heart gets broken and my brain gets eaten" , some lyrics from Lady Gaga!

[jehovajah]

Suffice it to say that their are many great mathematicians and many small, bur few who define their area of study well.

Manipume is the study and theory of ratio and proportion in all aspects of reality.

Eudoxus is credited with the foundation of this description, and thus may well be the founder of Mathematics.

[jehovajah]

So when i did not understand i went all over the place. Now i can save you time by linking you to this video.

<a href="http://www.youtube.com/v/YAdEfQsIGt8&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/YAdEfQsIGt8&rel=1&fs=1&hd=1</a>

[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 to the pole 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 <Quoted Image Removed> 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 <Quoted Image Removed> .

We can then write equations that have to distinguish <Quoted Image Removed> and <Quoted Image Removed> 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.


<Quoted Image Removed> for example.

Divine proportion may help.

[jehovajah]

I have to say that having read just now Descarte's Geometry in translation, i find that Descartes, While not acknowledging Bombelli fully concurs with Bombelli's use of Neusis and the use of the gnomon or carpenters rule.

`in fact he identifies elegantly the source of this method from the greek fathers, and also how to employ it in circular or semicircular constructions in which relevant measures are set orthogonally.

There is a lot of neusis in his explanations and proofs and the whole subject is accordingly very dynamic. Not once do the standard cartesian coordinates that i was taught, rigidly fixed in the page and to which every curve or line must conform, appear. instead the form took precedence and the "coordinates" were constructed as and when and where needed by means of a circle construction!

This may not seem of importance to you, but i can assure you that this is very greek, very natural, and immediately apprehendable by any child who has been taught the rudiments of Euclidean construction. Thus amazingly as if appearing out of nowhere Descartes by proportion is able to sensibly write down proportions and equations necessary to the finding of measurements on many forms.

It is also apparent within his terms of useage that he neither intended to slur or in any way denigrate Bombelli's codification of the "roots of minus one". Within his method these roots were imaginary as they existed only by rotation, translation etc of the form, in short by neusis. In addition some roots existed in multiple form and so needed to be distinguished from those that existed by moving the form hence his terms imaginary and real.

His explanation is full, gentle and above all reasonable and accomodating.

`i of course withdraw any comments made hitherto to his intentions in calling complex numbers imaginary. Indeed if the translation is literal, it seems he used the term complex liberally when dealing with equations which may give rise to these kinds of roots.

He finishes...
" I hope that posterity will judge me kindly not only as to the things i have explained, but also as to those which  i have intentionally omitted so as to leave to others the pleasure of discovery.."

[jehovajah]

As I begin my exploration of Greek geometry and Euclid in particular I find many assumptions of my youth hiding curious facts about my world.

Angles for example come from the idea of bending to form a hook in all languages and etymologies the idea of corner therefore is too static!

The gnomon is a carpenters or artisans measurement tool enshrining orthogonality and was so well known and useful that it was sufficient to represent hold or support a rectanngle area or a rectilinear form. The artisans and builders hodder for carrying bricks and tiles was a type of gnomon holding solid forms. The gnomon could be used to cast shadows and Mark off shadow lines, and so was useful for celestial and terrestrial measurement.

The dynamic circle is a curious creature! It hides an infinite number of relations in it's perfect form and links all measurement in some relation one to the other. The drawing of a circle which seems so simple reveals a curious fact about time differentiation which is as intuitive to grasp as Einsteins curious time distortion as speed approaches light speed. For to draw a circle in it's full circumference takes a time t but as the radius of the circle increases the time taken to draw the circle increases. Thus at infinity it takes infinity in time to complete one circle. Thus to the stationary observer the time to complete one revolution appears to slow down.

On the other hand, should the time appear to remain the same then the speed of traversing the circle appears to tend to infinite speed!

Such curiosities do not reside only in these apparent measurements, but commensurately the length of arc diminishes as the radius increases in the sense that should two circles be tangential, that is kissing, then the same distance travelled along each arc to each circle results in a different rotational experience to the travelling observer. Therefore the larger the arc one traverses the smaller the sense of rotation one experiences. This is of course a suitable notion of curvature measure.

It seems also fairly clear now that angle measure was not corner angle but arc distance along a circle or inside of a sphere. Each arc is indivisibly associated with a chord, a proper tie that in many ways linked circle and sphere to the triangle or rather gnomon. Thus the gnomon is a convenient and versatile measuring tool for all fields of measurement especially when linked to the sphere or circle.

These fundamental relations form the basis of all our systems of measurement by hand and eye, and enable our logos response to standardise a spaciometric measurement response to a constant form with a constant set of relations.

That we have over time been able to extend these measures to define evens gustatory and all sensory signal measures through the modular arithmetic models is truly amazing.

I am struck by Descartes geometry which at the last he characterises by the circle, the gnomon and the form these are used to measure. Thus Bombelli's vector as well as his operator took full pride of place in Descarte's geometry, being used to distinguish plane geometry from solid geometry and as Descartes hoped geometries of more exotic descriptions .

Plane geometry is characterised by a straight line intersecting with a circle, solid geometry by a conic curve intersecting with a circle, and I daresay other numerous geometries may be described by how their standard form of curved surface or curve intersects with a circle or sphere.

[jehovajah]

The nature of all possible geometries and all future geometries will be found in the geometrical objects generated by the relative  interaction of at least 2 dynamic spheres. These said geometrical objects will be found to concord with every generalisation or designation of the set of geometrical curves that are now called the Roulettes, but which formerly were named the Trochoids including the cycloid. The concord may one day be shown to be a congruence which after the suggestion of Descartes in his Geometry serves to categorise every geometry we will ever invent.

" ...i have found a proof of this that is most wonderful! However the margins of the Forum are too small to hold it....."

[jehovajah]

Here , dear reader is the origin of logarithms in mathematics.


I have to say, that like Bombelli Euclid aimed at a more popular audience of Artisans, that is artists engineers, bui;ding contractors, land surveyors etc. We have been at the mercy of classical scholars who have made his work more high brow han it is!

It still requires the genius of Napier to bring this and Ptolemy's work to the practicalities of logarithms, but it is clear nither would have been able to without this definition by Eudoxus.

I also have to adjust my understanding of ratio and proportion in line with Eudoxus.

[jehovajah]

I am looking at time dilation.

It occurs to me that this Lorentzian transformation has more to do with the rotational attributes of the circle than The limit of light speed. The ratio appears to be the relation between the tangent and the circle, the tangent ratio relative to the circle arc.

[jehovajah]

You are looking at Roger Cotes Harmonium Mensuraram. For Millenia scientists have measured the arc after the Babylonians. After Robert Cotes scientists measured the arc after π, but it was Cotes who set the unit which later became known as the Radian.

Cotes found out , under the noses of Newton, Wallis and De Moivre that this one standard unified all measurement.

Of course like Newton he stood on the shoulders of giants to see a little further. When Cotes died Newton expressed genuine and heartfelt regret at what he may have brought to science and astronomy.

This i believe is what he saw.

This i think is something like how he thought

[jehovajah]

Anyone care to post info about how trigonometric ratios encouraged the development of Fractions? Spherical trig nad Regiomontanus had something to do with it.

In the meantime a ratio called the versine was more significant in navigation and surveying than the the sine, as was the haversine.

The versine was ≡ to 1-cosø and the coversine1+cosø.

In the unit circle these are magnitudes the diameter is sectioned into, therefore the geometric mean of the products is the sine as the product (1-cosø)*(1+cosø) is  sin<sup>2</sup>ø.

This enabled a geometrical tool and a calculation for finding the square roots of values whose factors formed the diameter of a circle.

In addition the surprising omplexity of the trig idenits is revealed especially in the regime of directed numbers. Simply changing the signs and applying the bombelli operatoe lead to varying but relate results depending on the internal relations!

Thus was revealed by Wallis et al the strange an complex relations in measurement.

Today we begin to realise that these complex relations are not artifacts but "revealings" of a richer and more dynamic relationship in "what" we measure, and how we measure.

[jehovajah]

Of the work of Turing, and the computer, computer programming and computer science as a model of communication, a paradigm of cybernetic interactivity in complex structures in equilibrium or dynamic equilibrium.

Of the recursive formula for Theodorus spiral in polar coordinated, and how a visually straightforward geometric process requires an intense trigonometric convolution to describe and calculate, revealinf a mandelbrot type application in a linear dimension:f(r^2)=f(r^2)+d^2(initial f(0^2),d=1)
 f(ø)=arctan(f(ø))+arctan(d/√f(r^2)) these 2 recursions being coupled so as to have the radius drive the angle.

Of the relation of the mandelbrot to thr polar spiral r(t)=t^2, ø(t)=2t in terms of circles intrsecting the spiral giving reference coordinates to???

Of how configuration for purpose is a solid explanation of both computer and human behaviour and efficiency of behaviour, to whit: why humans procrastinate on some things but overexcel on others.

[jehovajah]

Of the nature of light and angular momentum: light beeing the extension of angular momentum axially through space; precisely accounting for its 3 main characteristics: massless, straight lined and frequency defined propagation.

As a torquing of space angular momentum establishes axes within the "grain" of the flow of space, thus bent space bends angular momentum axes with it, and consequently light curves round condensing objects.

The structure of complexity exists in 2 states : complex equilibrium structures in  a static relation, "melded" or contiguous, and complex dynamic flows of spatial structures in a dynamic equilibrium relation.

These states are the self organising, self assembling states which arise out of the complex motion of space naturally.

There  is a third state of complexity which i might describe as "explosive" or "explosion"; implosive states inevitably leading to equilibrium states as previously described. Explosive states may be stochastically explosive in that they give rise to a new arrangement of order; or non stochastic which i may then define as true Chaos or random.

There is a thermal/electrical gradient relation to motion field processes and their products: extreme motions are indefinables as to product,temperature and electrical flux all of which are logarithmic or index measures of the internal motion convolutions within space.

however currently the less than extreme produce a temperature and elctromagnetic flux known as plasma. Below that at lower temperatures and electromagnetic flux individual atoms and molecules are produced.

Again below that at noticeably cooler temperatures and less violent flux compounds and various chemicals in solid, liquid gas states are produced.

Below this we find the production of viruses.

There is a marked distinction for us here because after the gradient reaches viruses it is a long complex interaction that produces the thermophile bacteria , and time becomes significantly involved with the temperature gradient in the production then of bacteria and finally more complex life forms.

[jehovajah]

Suppose as i do that angular momentum is the source of the attribute called electromagnetism, then what is the source of "heat"?

Heat is the contact sensation of translational wave motion, which arises due to internal equilibrium adjustments to a disturbed equilibrium state in a static equilibrium form.

There is a frequency range for a translational wave to be felt as heat. Above that range heat is not felt, but material damage is sustained.

However if the "logarithmic" index we call temperature is utilised we can see a proportion of the translational wave is used to transfer translational motion to a measure we calla temperature gauge.

Thus the kinetic theory which describes heat as motion of regions is modified to describe it as relative motion of regions in a wave like form.

The effect howeer is now linked o the angular momentum of regions giving a combined translational angular momentum wave description of thermo electro magnetism,.

Howevr i can go further and include gravity as a consequence of torqued space.

Thus the combination of angular momentums leads to long range gravitational effects short range electric and magnetic effects ie shorter range than gravity so called and even shorter range weak and strong nuclear forces.

All of these attributes derive simply from the angular momentum of regions and the relative translational wave forms in arrangements of regions. However it is torque which combines all together in its gravitational embrace.

When angular momemtums align it produces a strong magnetic flux field  with a translational field effect in a wire called an electric current flux,orthogonal to it.

[jehovajah]

Over the centuries the notions of mass and density have become confused and remain confused till today.

Mass naturallyrefers to magnitude of volume, volume being a dimensioned region of space. We dimension space by deciding what kind of measuring device we are going to use and "cutting" a form out of space in alignment with that measuring device. The form naturally is described in more general terms asa region of space.

Usually we dimension space using orthogonal cuts, but I think it more natural to use a circular cut and a radial cut as an envelope for all "dimensions", or more precisely a radial and orthogonal arcs.

So mass is fairly straightforward.

Density is equally straightforward but we need a measuring device to measure it. The historical device for measuring "density" has been q pair of balance scales.


This is where the confusion has arisen , I think. In all cultures the balance scale has been used to measure out "amounts" of material. Since they were used it has always been recognised that a standard quantity or magnitude has to be employed to allow for a fair measurement of magnitude or amount. It became essential for trade that this was upheld within a social institution with power to enforce fair weights and measures. "Buyer beware" stems from these days as a scale though universally recognised and excepted has no basic fundamental unit magnitude just as is the case for all measurement.

There are many unit quantities that societies have employed but the weights of a material were the most important for trade in materials that were tedious to count or measure the length of. So weight was used to describe an amount of material a magnitude of material and the mass of a quantity of material.

The problem of mass and density arises naturally here, due to a confusion in measuring devices. A quantity of grain occupies a volume, therefore it could be "measured by volume", and this would be important for working out how many transporters one would need to transport it to a different location. But it then becomes obvious that some grain is "lighter" than others although it occupies the same or greater volume. This is a confusing set of affairs and the confusion has remained until today.

To avoid feeling robbed a new measure called density was employed and defined as the balance weight of the grain divided by the volume, that is the weight divided by the volume.

So two processes are involved in determining this historical density a weighing or balancing process and a volume  measuring process. The results of each process are then arithmetically divided to produce a scalar called density. Thus density was an abstracted concept from mass that is volume, attributing a notion of something that came to be called "weight" to volume.

Weight before this had been synonymous with volume or mass before this, so a confusion that has remained unresolved was set up from the outset. The confusion is not a real confusion, it is the result of using and combining two different processes based on two different perceptual neural networks. "Weight" is a proprioceptive network output, volume is a visual network output. They refer to precisely the same form.

Now density is actually perceivable, so to use this traditional measure of it requires some clarification, and this is what has been so confusing. Most pedagogues respond to the confusion by in effect shouting at you! They re emphasise the definition of density over and over as if you were unable to understand it. It is an attempt to hide their lack of understanding by coercing you to accept as they did the confusion which has persisted for centuries.

The simplicity is this: we take a volume of a common substance like water because everybody can get hold of it. We take a standard fractal scale to measure the volume in length dimensions as most people can get hold of a standard measuring tape. We construct a standard volume to contain a standard amount or mass or quantity or magnitude etc of water. Thus the volume of water is the standard mass and mass is exactly a volume.

Now we weigh this mass of water this volume of water against itself, and it balances. What we are weighing is the comparative density, the relative density.

So all our weighing scales etc are not measuring mass but relative density.

Now the confusion of mass and density has been transported down tomthe atomic and quantum level. When we measure any chemical substance we do a relative density nowadays against an isotope of carbon. Traditionally we have explained these relative densities in terms of neutrons positrons and electrons and their nuclear binding "forces".

Simply let us compare the relative density of water with a lump of iron. To do so we take the same mass, that is volume of iron and water. We then weigh them on a balance and find that iron is " heavier" than water ! What we are observing is the relative density displayed by a glance that is out of dynamic equilibrium! We call this "being heavier than".

So to restore dynamic equilibrium which we eventually rename static equilibrium when it stops oscillating, we add mor volumes of, that is mass of water. The mass of water is now the standard measure of the relative density of iron. However instead of realising that we makema simpler bu t confusing relation calling the mass of water the mass of the iron!

So when later we come to think about mass it is in fact shrouded by this simple confusion between relative density and mass.

So returning to the molecular or atomic case we assume that because the relative density of hydrogen and carbon is 12 that there are 12 lots of hydrogen particles in carbon. However since they do not, clearly, have the same volume they must be compacted or fused together in some way that warps and bends space with immense "forces" called nuclear binding energies! This is a direct consequence of our confusion of mass and density.

This may in fact be the case, but what I am saying is that it ain't necessarily so. We may in fact not need to have a particulate description of relative density at all. If not I would replace it with a gas density model based on Boyle's law.

Of course this confusion impacts on the notion of force, but I have discussed that elsewhere and my view is that pressure is in fact the more natural and useful description of the equilibrium restoring motions we observe in static and dynamic systems.

In short mass is everywhere equivalent to volume and density ior relative density is what we routinely measure using scales etc and pressure arises in a disturbed equilibrium system in exactly the vector quantities required to restore equilibrium, wheter static or dynamic.