Mathematics derives from the french mathematique which derives back to the indo European roots through Latin and Greek and the Sanskrit: man.
So in every sense mathematics is essential to the defining of man or what it means to be a human. This does not imply arrogance ut rather a time when animals became self aware enough to describe there learning and thinking ability as "man". From this one distinction flows every human notion of difference to other animals.
However the Root of interest for the Logos Response is: me. Now this distinction is what our CNS and PNS do prior to and during man (that is learning and thinking) and that is MEAsuring and comparing.
Thus me and man are possibly the 2 oldest "human" notions, but are not unique to humans as all animals and even plants and fungi do this. It is a fundamental response of life (living things).
From this root analysis no human is excluded from mathematics as it is part of ones definition of being human! And every human not only thinks and learns but also measures and compares. Thus the problem is the distinctions that are drawn, or posited and adopted. These are what set up and establish barriers as well as boundaries between the myriad human activities that would otherwise qualify to be called mathematical!
So be proud and loud you craftsmen and craftswomen, for everything your hand produces from your mind is mathematical through and through, especially if done with love:
Mind Mind (m[imac]nd), n. [AS. mynd, gemynd; akin to OHG.
minna memory, love, G. minne love, Dan. minde mind, memory,
remembrance, consent, vote, Sw. minne memory, Icel. minni,
Goth. gamunds, L. mens, mentis, mind, Gr. me`nos, Skr. manas
mind, man to think. [root]104, 278. Cf. Comment, Man,
Mean, v., 3d Mental, Mignonette, Minion, Mnemonic,
From Free dictionary.
From this it is but a short step to……..play!
The establishment of distinctions is not only a language response, but also drives communication, explanation, education and differentiation. It is a powerful aspect of the Logos Response which accounts for many communication motivations and road blocks.
However just as thinking and learning measuring and comparing are at the heart of being human, so my friends is doing math, or rather playing with the things around one and thinking and learning about what that reveals.
Experiment in proprioception 1
Slowly, very slowly turn, or twist with eyes closed.
There is a quantum size to sensors action potential. So i know i am moving by 1 intention, 2 lateral differences between proprioceptors, 3 mesh of proprioceptors self referencing, 4 exterioceptors (open eyes) audiometric and gustatory differentials 5, memory comparisons that is perception comparisons.
Each one in this list requires an increased potential to activate.
When i generalise the notion of a coordinate system to have n ordinates/ axis not necessarily mutually orthogonal, which in any case is a non sequitur beyond 3, i find myself tracing a path through all the ordinates ending on the point i am referencing. Thus it seems to me that a generalisation of a coordinate bracket would be a path bracket, based on a sequence of visiting the ordinates.
Thus it seems fair to think of a peano curve as the basis of a sequence for visiting each ordinate on an extremely dense ordinate system.
Experiment in proprioception 2
Pick something you know how to do well. Do this as fast as you can. Repeat and attempt to go faster. If you can repeat faster still.
Your consciousness of doing this task is altered. You will not know how each movement flows into the next, because your attentional focus is limited.
You routinely do more through unconscious proprioception than you can distinguish.
@ lycium Hey man if you want weed, you can smoke weed, but if you want what i am on you need to spin out, dude–right to the edge and beyond!
Originally posted by author:
So now the spiral gives up another secret.
The circle and the spiral are in competition in my mind. Well not really as i think that the spiral is prior to the circle, and in fact i have gone a long way to generalising the boundary properties for generating spirals. So it is really a bit of penile envy. Why is the circle so damn easy to use to explain spiral motion in general?
The circle is a very very special spiral form, and i have yet to rigorously determine it as an intersection form for the general spiral boundary. It can't be because the general spiral form is chaotic, so why do circles introduce an order to that chaos? Why are they so facile as a starting point for anything spiral?
A sphere or a circle can enclose any shape, and then the boundary of the shape shrunk to, thus providing an increasingly accurate map based on the radial adjustments to a circle or a sphere. This means i can develop a theory of interactions based on circles or spheres and then apply it to real objects by using a radial transform map.
Is the theory based on the circle or sphere more general than the one based on the actual shape of things? Yes, but that makes it an approximation which has to be made more applicable by minute adjustments.
Well i have heard of this type of limit process before, and it in fact underlies the notions of differential geometries. So even in our most general thinking the circle plays a powerful role, and there is no obvious sign of the ubiquitous spiral or torus forms, In fact we tend to explore these form using the circle!
And i have done the same.
I was thinking about trochoids and cardioids in their relation to the Lai Zhide Torus based on the I Ching algorithm. I wondered what the path of points on an unrolling spiral would look like. It turns out to be trivial in one sense but non trivia in another as the programmes that can generate an "unrolling carpet" animation are non trivial.
It is trivial because the answer is that they form a set of trochoids originating on the line along which the spiral is rolling. It is non trivial in that the trochoids dimensions are radially decreasing to zero for unrolling, and increasing for rolling.
Thus i can see the straight line and the trochoids are geared to the spiral and the circle is inextricably enmeshed. I can of course transform to a different boundary say a spirangle but that only emphasises the general applicability of the circle in solving these types of problems.
However if only for the Quote at the end i like this exploration into thr problems of locomotion on roads.
I always enjoy a good blow in 3 dimensions! but for me the simple bouncing of a ball before it comes to rest hopefully before smashing that window is illustration enough of the trochoidal nature of movement and motion and the interlinking of spiral and circle, of vortex and torus.
The circle is a very special form of spiral as the torus is a very special form of vortex. I do not think spheres actually are real objects,but ovoids are which are a form of torus approaching zero central radius.
The power of the circle or sphere lies in its unique abstraction from every vorticular form, and that is the sense in which i think it is an intersection of vorticular or spiral forms unique to animate consciousness.
Ah yes, i forgot to add that i think the mandelbrot set is intimately related to the trochoids of an unravelling system of spirals in the polar coordinate plane, and consequently the mandelbulb is intimately related to the three dimensional trochoid shells unravelling from a system of tori (toruses!) cut off, and ultimately shaped by the circular or spherical boundary condition.
Proprioception experiment 3:
Go into a familiar room and stand in a familiar space. Now reach behind you to grab, or make contact with something.
Notice how proprioception is tied in with visualisation, and how you use the visualisation and the audiometric senses to locate and position your hand and body to ensure maximum success.
If you are struggling, try to smell out the object or surface or use your Whiskers (small hairs on your body) to sense how close you are. Maybe a temperature feedback might help, but these are all exterioceptors, so realise how exterioception assists and coordinates with proprioception.
And you thought geometry was just Euclid!
Originally posted by author:
i have commented on the notion of dimension. in trems of the characteristic 3 d and 2d nomenclature. So essentially the idea is that a 4th orthogonal dimension exists, etc. Of course a 3 d consciousness cannot perceive a 4d spatiality, so the rubric goes.
Well i will be clear and perhaps reactionary in saying i do not think an mutually orthogonal direction beyond 3 exists.
I think that there are lots of dimensions, but orthogonality is a unique characteristic to 3d. That is to say to flatlanders that they would not perceive orthogonality in their flat world, only us 3d types can see it in their world and in ours.
Now flatlanders is a "teaching myth", and thus it is not an analogy that i would care to continue beyond 3d,because in 3d we have all the dimensions we wish.
So why is orthogonality important? Orthogonality is in fact a sensor driven notion, and we have an orthogonal sensor within the vestibular system . That for me locates it in what we call 3d. Now if we had a 4d orthogonal sensor i would concur, but we do not.
What we do have is a proprioceptive mesh network which drives our motion of geometry,and our imagination. Thus proprioceptively we can imagine another dimension, but in the sense of a "degree of freedom" . We do not like to be stuck and 3d is so often defined in that sense of a limit to our motion. Of course mechanics and engineers know that there are degrees of freedom and the rigidity implied by 3d is a difficult thing to achieve without taking these into account.
So while it is a nice story and all that the 4d fiction is actually masking what is going on. We are actually dimensioning in terms of degrees of freedom.
Thus if i build a so called hypercube "shadow" i am actually building a shape that with an extra degree of freedom providing it is an elastic or compressive degree of freedom can deform while preserving local relationships. As a consequence like any of my real world clothes seen as mesh networks i can turn them inside out continually by rotating in the correct way. I can also twist them and deform them in other ways while maintaining relative relationships.
Some of the more complex web like structures can still do this but in a complex internal radiating in/out pattern which in fact may be good models for growth and development within a 3d cellular structure.
So are my clothes shadows of 4d objects? You know when you get that feeling you are not lone…. :dink:
Originally posted by author:
Just a note to myself. Trochoids and torii are related in some way. Maybe that torus is a 3d trochoid mesh, and this links to helical wraps of torii, maybe they are types of trochoid in 3d. Now i have seen one description of the lie group as the symmetry of 248 circles arrange in an orthogonal twisted like pattern so maybe trochoids relate to Lie groups in some fundamental way.
Originally posted by author:
Benfords Law, Eudoxus and Archimedes are related i think. We typically do not use the full range of any aggregate number system, preferring to rescale. This kind of hides the infinite fractal nature of things, but also relates to computational boundaries being a scale issue. That means that my PNS and CNS based massively parallel distributed computing system computes surfaces based on scale information. Therefore a smooth surface at one scale becomes a rocky mountainous terrain at a high magnification because the scale information falls within the computational accuracy limits.
In one sense computational error is what i live with and in everyday, so notFS may be a scale issue as well as a sensor issue. Who knows what i am missing because of scale or not having the right sensors?!
It's a bit scary really, cos i am blind and deaf and insensate in so many areas! Or its exciting news- plenty to explore and find out for everyone!
I want to extend the conic section curves to include loxodromes and trochoids. The way i would do it is by projecting the loxodromes onto the cone that is enclosing and tangential to the sphere. Similarly i would project the trochoids onto the cone.
All these additional conic section curves would then be the defining set of curved motion for gravity.
Additionally i would project the trochoids and loxodromes on a torus onto the cone. The torus would again be enclosed and tangential to the cone.
With this thought in mind i would project the curves of a lie group onto a cone and search for matches with the conic section curves. For the lie group to model gravity it would have to match precisely with all the conic curves including the extensions.
Topology, i say, is the analysis of the the properties of the space that inheres within a form. And when i say propeties i mean attributes to that space.
So i attribute a property to a space and maintain that attribute by some means and explore the consequences of that attribution to that space by any specific or general action.
This is as general as i want to get because i feel that the mathematical definitions of topology are way to abstract to be graspable. I have used form property and agent to define topology so that i can locate the subject in reality. One of the reasons why i escaped to spaciometry was so that i could get clear of all the abstractions and words. It seems that mathematical definitions are more about words than anything tangible, whereas i know every mathematical idea comes from a tangible source, but some would want to make out otherwise.
Topology naturally includes any geometry and is an interface between geometries in the abstract and the material arts and sciences.
So a concrete example: an elastic band inheres a space to which i attribute elasticity as a property. I manitain that property by playing with loads of elastic bands and referring to a cultural database on elastic bands, which informs my exploration and apprehension f the property of elasticity.
One of my explorations for example may be what happens to to marks on the elastic band as i act on it by stretching twisting folding. A question arises? what happens when the band breaks? This is of course explored in material sciences and thus the tw studies mutually inform each other.
So topology is not an abstract set of words and notations but an exploration of real spaces.
With this in mind i look at the property of "tangential". In reality this does not exist.
A surface of one form grazing another surface does exist. In practice we deal with tangentiality by using a variable area of contact that can be made arbitrarily small.
In fact it is a variable volume of contact, and the arbitrary measure of closeness has to be suitably defined. The tangent is then defined as the limit of a process that makes this volume tend to zero. Although this is defined as a point in practice it is never achieved as the materials breaks free of each other before this singleton point is ever reached. The reason is quite simple: a singleton point is a fiction that does not exist. The consequence of this is that tangents do not exist as defined normally.
This leads to a point about points in Euclidean geometries. They do not exist. Instead intersection of lines is and always has been used. The introduction of the notion of point into Euclidean geometry is a later addittion to try to axiomatize the system. Since it has no practical geometrical purpose that intersection does nor fulfill, point has not been analysed that much. However i say that it does not bear scrutiny and its legacy in mathematics has always been troublesome.