Axioms for the setFS

[jehovajah]

I have actually revisited these axioms in many different guises in my blog, but have not set them out in a list fashion.

As i have, i feel, come to a natural conclusion in a theory of space, i feel inclined to slowly and carefully sift and revise the axioms.

It is more inviting, and it makes more sense to start afresh in a new thread, and that i propose to do. However, it is likely to be very terse and a bit like a contents table, hence my reluctance to do it. Therefore i will retain this thread as a commentary on the progress.

I have to say, that the axiomatic approach has been a kind of philosophers stone. As a springboard it has been useful but in reality i have written as the mood and insight took me. Therefore i may find inconsistencies and contradictions as i do this, and that in itself will be interesting!

[jehovajah]

I have now selected 3 guides to help me in my revision: Euclid, Hermann Grassmann, and sir William Rowan Hamilton.

I apologise to any one who has been following my ramblings, as i got the preeminency of Hamilton over Grassmann slightly wrong. The correct relations are discussed in my blog.

[jehovajah]

As usual, further research leads to revision of concepts. I have come upon the source of all mathematics and science in the western world as is named mathematics and Philosophical science; that is Pythagoras. Consequently  have added his work to the body of advisors i shall need to review my axioms.

What are axioms anyway? When i started 3 years ago i was under the impression they were greek in origin and established by Euclid. But i am reading Euclid in the greek and have yet to find the word. I will ponder mor e on this as i shape the matrial.

As luck would have it, my axioms have been very essential to the unraveling of this whole fractal structure, and so i am encouraged to pursue a more accurate version.

[jehovajah]

I have posted the revision to axiom1 or axiom! as i like it. It is hard and meant to be rigorous and obscure! This is not deliberate except in the sense that it invites any one to deliberate the axiom. The terms and notions are fully derived, as far as possible in my blog. You will have to search the archives there but that is not a bad thing  .

Hey, do not be afraid to reply guys and gals, Damen und Herren, Mesdames et Messieurs. Take a bite. I promise i won't bite back, but those who have followed my meanderings know that already. However, the moderators may moderate any vitriol so be considerate

[jehovajah]

The second axiom is quite detailed, but gives a potential dynamic process that underpins the first axiom. At some stage i want to support the notion of self and then i as a fractal product of iterative processes, and i think i need axiom 2 and 1 before i can posit that.

Axiom by the way is derived from the same root as axis, and that is a kind of direction to travel in.

[jehovajah]

Axiom 3 is a draft. "What a mighty work is man...... in intellect how like a god!", and all that jazz..

"What a piece of work is man!
how noble in reason!
how infinite in faculty!
in form and moving how express and admirable!
in action how like an angel!
in apprehension how like a god!
the beauty of the world,
the paragon of animals! ". - (Hamlet Act II, Scene II).

Pretty cool guy,Shakespeare.

"Time is not a mensurable quantity" I think Lagrange may have said that

Axiom 3 in fact applies to all self aware animates and biological entities from microbe to mammoth, from bacteria to Blue whale, which evolve over time.

The virus is not excluded but is a special cse of horizontal gene transfer as opposed to vertical gene transfer, Without the sensorially dense boundary only simple interactions can take place with a virus. A bacteria however with a more sensorially dense boundary can exhibit more behaviours and interactions over time.

[jehovajah]

Axiom 4 is more related to the old axiom 3.

The philosophical axioms i originally proposed have served me well, but new insights have clarified things in my mind. I know the new axioms are even more weird and wonderful, but so are fractals! Hang in there and see where they take you!

[jehovajah]

Less is more on many an occasion. Although i have conducted research into the basis of the axiomatic system, and know it to be  a Euclidean teaching style, based on the best Rhetorical practices of the day, that same research indicates that Kairos, or proportionality is required. Thus, to over complicate the presentation is in fact to go against the rhetorical rules established millennia ago and championed by Pythagoras.

The Axiomatic approach of course has a history, but the Euclidean and indeed the Newtonian examples of it are models of appropriateness and accessibility. One cannot help being the self that the neural interaction has established, but one can review and modify expressions to make them plainer or more accessible.

The seemingly pernicious goal of mathematical abstruseness is a difficult habit to eradicate, seeing as all so called mathematicians are spoon fed it as pabulum. However when one matures it is hoped that one will put aside childish things in favour of portehouse, that is good convivial communication.

Quite often, when this is done the triteness of the topic is revealed, and indeed we may fall prey to thinking in terms of trivialities. However Fractal geometry comes to the rescue, for it is from such trivialities that the most beautiful and profound effects are constructed via iteration. Thus a true democracy ensues, because anyone can contribute, and watch the unfolding fractal patterns that iteration produces.

[jehovajah]

So 2300 years old and still packing a punch! While reading the greek of Euclids Stoikeioon i found out that The so called postulations are in fact demands! The word postulate does in fact mean to demand or petition, and not as i thought to propose! I guess the clue is in the use of 2 different words

In any case the master of axiomatic style is no more than a petulant teacher demanding of us that we accept certain things as so!
Thus my lofty ingrained ideals of the merit of such an approach are brought crashing down into the mud of human frailty!

I do not demand of you that you accept these axioms, i beg you to critique them.

I have no confidence in the systematic logic  per se, because the conclusions depend entirely on the quality of the premises.

Therefore, Axiom this or axiom that is not to be taken in its highbrow meaning in this thread. It is merely the ramblings of the fevered mind of some old geezer in the uk!

However, Euclid's ramblings have had quite a powerful effect on modern intellectual thought, due entirely to their Pythagorean content. So make of it what you will, but my intention is not to browbeat anyone, but to welcome all 

[jehovajah]

I have recovered slightly from my shock, and have researched the terms axiom and axis in my blog. Plus also, Plato is the kingpin here in the west. He redacted Pythagoras and employed Euclid in his Academy. Thus his demands are not petulant, but driven by a need to establish academic standards of construction and rigour. Later pedagogues may have acted petulantly.

The so called axiomatic system is also a late 19th century invention. Euclid and Plato relied upon Duality and veinal systematics, not axiom. There are no axioms in Euclid, only Aitema, that is "Items demanded". and Ennoia, that is "common notions".
The axiomatic system i will thus modify accordingly, retaining what is useful.

[Apophyster]

I am a recent (FF) member and sparse contributor.  Until about the time you relocated some of your writing, I was reading with interest.  I was not comprehending all of it, and cannot personally contribute to you lines of reasoning.
While I was musing around this morning I did have a curious thought and I wondered what you might think about it.

You are interested, I think, in general, of "other" sets.  I get the notion you are presenting is that the recognized sets in mathematics are incomplete.

What do you think of the concept of "impossible sets".  heheh...  Something besides "infinite sets", or "complex sets".  I don't do well at those classifications.  Actually I sort of thought that "impossible sets" were what you might be pointing toward with your descriptions and analysis of the setFS.
For me as a human being, impossible sets are valve to escape compulsive trying to solve for the construction of things constructed.  
We aren't really using the "sets" we know of now very well as a species.
Or something like that.

Could I also intrude upon your interest in fractals to introduce my main interest, which is a script I (continue to) write that animates fractal flame parameters.  (I promise I will be impolite only this one time intruding into the development of your thought here.  But I would like you to see. please. I don't have time to be here at FF myself: [edit 120717: I didn't fully realize the link would put the actual animation here, which I think is inappropriate.  So here's the link separated a bit to keep the anim from loading:] http:/ /www.youtube .com/watch?v= uAJ-WASvX30  )

I hope I am able to get back into reading your setFS and otherwise interesting postings.

Fred's my name.  and on the web of course, others a few.
pleased to virtually meet you, if I may.

[jehovajah]

Hiya Fred!

Am i allowed to say OMG? Well i have said it anyway!
I am Sam, Fred and pleased you took time out to just chat a little.

Set Theory was developed particularly by Cantor and Venn, and i forget who else, though Boole made a significant contribution. It was particularly Bertrand Russel who highlighted the paradoxes of the theory as written, ie the predicate paradoxes. In so doing he motivated a whole generation to a form of rigourous mathematics that has turned into rigour mortis! It is so abstract it just about kills the subject for most people!

It just about killed it for me too, until i came to FF as you call it, and got caught by the wonderful surge toward the mandelbulb.
When i first came here i assumed everyone would be mathematicians!. Lickily for me they were Artists and artisans in the main. Lucky in 2 ways: the mathematics forum was under used; my meandering and sometimes obtuse style was not vigorously challenged, so i did not have to spend time in argumentation etc. I could philosophise.

Impossible became a word i rarely used because i thought we had a handle on infinity(which we don't by the way),so the strongest i could commit to would be improbable(because i thought we had a handle on probability, which again we kind of do not).

So set theory as a means of describing a collection of anything is  a bit of "mathematical window dressing". i picked up the lingo when at school and it was established in me at university, but what it means other than a Summetria i have not got a clue. It sounds mathematical though.

I shall give your impossible sets some thinking time, and look at your interest too.

[Apophyster]

Hi Sam!

It's good to be able to use a shorter name. Thank you.  Although I've seen your user name in several similar forms - I like your spelling choice.
Using or saying OMG is not something I find offensive.
There are so many I'd like to chat with more substantially.  Oh the time...

I have so enjoyed your survey of earlier mathematicians.

I was seen to be good in math when I was in my teens and twenties.  But I didn't want to be a teacher.  What was left to interest me?
Hahaha... I didn't know, but it would be fractals that rekindled my interest.  Oh the time...

I follow your obtuse writing ok, I think, except for its mathematic depth.
I've read considerable material from the 19th century and lacking media writers of the era tending to compose very long sentences.
It's good really... reading long complex sentences helps hone one's concentration - at least in some cases including mine.

I think I presented you with some kind of paradox really.
By impossible I more or less meant "humanly unreachable" - or for those who take such "other realms" to be existences of fact, I meant these "impossible sets" to also be unreachable by or through any beings capable of consciousness.
So anyway, an impossible set is impossible of definition. 
I suppose in that, I've left mathematics entirely and ventured into philosophy an area which I try not to tread over.
I'm a nitwit when it comes to philosophy! ;-/

So... just some bits more of my fractal flame interest.  The most difficult math I use is spline interpolation.  Not much really, but once I learned that splines were used once to frame boat hulls - to make the necessary smooth curves - I began to see how widespread their use is in our modern world.  Some of the math I encountered was developed by a person working for "Peugeot".  The math was so developed so that engineers had an easier way to describe the smooth curves of a car's body and I suppose other features.
When I was a kid, the shapes of cars were very square. Now cars are all bubbles.  I think the car mfgrs got somewhat carried away with their curves.  :-(
But the curves are everywhere.  I think nearly everything that's made in some kind of mold uses some type of spline-like interpolation math to design and manufacture the shapes.
As Artie Shaw used to say:  "Interesting, but not funny".
I mean, a 1957 Ford Thunderbird is a very interesting car to look at.  About any late model car I look at today *might* be a late model thunderbird...

But wait... there's more.  The windows on cars at one time were flat.  One would only see the reflection of the sun bouncing off those windows if one were in some particular spot.  Now, it doesn't matter where a person might be.  ALL cars on a sunny day reflect light at a person as long as there is not obstruction between the person and the rays from the cars.
My feeling is that all of these inescapable rays of sunlight on streets where there are many cars are very very bad for our eyes.   

Popping on spline-designed, darkly shaded sun-glasses is not a very good solution. 

Fred

[jehovajah]

Fractal "splines", Fred are related to trochoids and Roulettes, so you got my interest there. Trochoids are what i once called the manufacturers dirty little secret, because these smooth polished curved surfaces rely on them for manufacture and finish. and of course, trochoids are at the heart of the Mandelbrot set.

Before computers were used, numerical calculations were done by hand. Functions such as the step function were used but polynomials were generally preferred. With the advent of computers, splines first replaced polynomials in interpolation, and then served in construction of smooth and flexible shapes in computer graphics.[5]

It is commonly accepted that the first mathematical reference to splines is the 1946 paper by Schoenberg,[6] which is probably the first place that the word "spline" is used in connection with smooth, piecewise polynomial approximation. However, the ideas have their roots in the aircraft and shipbuilding industries. In the foreword to (Bartels et al., 1987),[7] Robin Forrest describes "lofting", a technique used in the British aircraft industry during World War II to construct templates for airplanes by passing thin wooden strips (called "splines") through points laid out on the floor of a large design loft, a technique borrowed from ship-hull design. For years the practice of ship design had employed models to design in the small. The successful design was then plotted on graph paper and the key points of the plot were re-plotted on larger graph paper to full size. The thin wooden strips provided an interpolation of the key points into smooth curves. The strips would be held in place at discrete points (called "ducks" by Forrest; Schoenberg used "dogs" or "rats") and between these points would assume shapes of minimum strain energy. According to Forrest, one possible impetus for a mathematical model for this process was the potential loss of the critical design components for an entire aircraft should the loft be hit by an enemy bomb. This gave rise to "conic lofting", which used conic sections to model the position of the curve between the ducks. Conic lofting was replaced by what we would call splines in the early 1960s based on work by J. C. Ferguson[8] at Boeing and (somewhat later) by M.A. Sabin at British Aircraft Corporation.

The word "spline" was originally an East Anglian dialect word.[9]

http://en.wikipedia.org/wiki/Spline_(mathematics)

[Apophyster]

Thanks very much for that excerpt Sam.
I've been walking and looking at some of the "Victorian" vintage architecture lately.

I've been wondering how some parts of some of the structures were built.
I cannot quickly find a good example, but I believe some similar technique would be used in the construction shown here:
http://leadingleadworkservices.com/ESW/Images/Cupola.jpg?xcache=2874

Haven't time to look further now, but I will try to snap a photo of some very particularly interesting "domes" I've seen to show you. 

Ummm... is all of this ok in the setFS thread here?

I so appreciate your considerate reply, once again!
Fred