Axioms for the setFS

[Apophyster]

Hi Sam

I haven't figured out yet how to use a digital camera I recently borrowed.  But I will try to get a picture of that architecture I mentioned in my last reply when I understand using the camera.

Some clarity about the way the notion of an "Impossible Set" occurred to me has returned.
I was simply thinking about the way "imaginary numbers" are in fact, not "imaginary" at all.

So my notion was that an "Impossible Set" was some set not genuinely "impossible".

Maybe related somehow was a "koan" I've been wrestling with lately.  I suggested in another topic to those who have pondered the idea "I think therefore I am" to do the thought experiment: "I am not".
Or in other words, what is the experience of "I think 'I am not'"? 
And what about those who can be "I am" with a stilled mind, absent of the arising and fading away of thought constructions?
(Maybe such yogis and monks would understand "impossible sets".)

The connections between these two notions are still in a formative state in my mind.

Fred

[jehovajah]

Only just recently learned the term koan in some research i was doing on the pythagorean scholarly material. Pythagoras may have been mythologised who knows, but some of his harder sayings seem to fit the description of a koan.

While you may think you are off topic in this thread, you are not. The axioms are meant to be the most tautologically obvious statement of an attribute. So imagine my shock when i find that axioms are not even from Euclid, but a late 19th century school of thought about logic and logical systems. Aitema and Ennoia and Oroi are all that Euclid apparently laid out as the entry requirements to his course on Philosophy. The rest he builds as he goes along using these foundational principles.

Now splines may be foundational principles that need to be added, particularly if we want to describe human engineered products within the set. Trochoids for me are fundamental in a way i can not fully explain. I derive much benefit from meditating on them, especially as Laz Plath presents them.

Your particular koan i have tackled in the other thread on the Axioms, which is slowly being developed and refined.

[Apophyster]

I have an orientation inclined to the East (but have lived on the West coast of the USA about 4/5 of my life).  It is so interesting to me Sam, with your wide range of knowledge of historic mathematicians (East and West) that you should be also interested (it seems) in the mystical(?) representations you've mentioned.

I pulled myself back from "metaphysical" speculations some time ago.  I had a terrible time when I overindulged my belief in some of the presentations.
Once all that outer shell was removed, I started understanding (I think) some of the firmer bases of practices and the thinking behind them that I've read in scriptures of all the religions. 
I must say though, that I recognize only superficially, many of the names you note.  And many names I recognize not at all, whether the names represent ideas or people.

Din't someone, at sometime, say something like "East is East and West is West, and never the Twain shall meet"?
(I cheated and looked: it was Rudyard Kipling.)

Perhaps both the author (or speaker) of those original words and myself both need some further study time!

Cheers!
Fred

[jehovajah]

I am not sure if i wrote this in my blog or here, but one of the things i found out when i got older, was just how many "fairy" tales i was told as a kid to keep me interested! Now i have come to see how human our knowledge is, and thus dependent on individuals and their belief systems.

We arrogantly were taught to pooh pooh those belief systems, while we used the results of that way of thinking. Very pragmatic, but ultimately creatively sterile.

I do not say that one has to believe a belief system to use a method derived from it, but i do say the belief system deserves equal merit of mention when expositing a method, otherwise the heuristics is lost, and creativity is harmed. This is why i think empiricism is so important. By being empirical we can judge one method against another as regards its underlying principles, and determine which principles are empirically more "congruent".

Unfortunately you do hurt peoples"feelings" in doing this, but the individual's beliefs, though judged, are not hopefully otherwise denigrated. And every belief system i know of has that big area called "the mysteries" from which devotees may garner further and more compelling insights.

[Apophyster]

Howdy,

I mentioned around reply 44 that I would try to snap a photo of a structure that I believe was constructed with the help of some kind of "spline" technique.  It's the top piece of the cylindrical section of the building pictured in the attached photo.
If you turn the photo upside down you can see that the profile of that "dome" has some resemblance to the profile of a ship's hull.
I know that some kind of spline mechanisms were used in constructing ships but I don't have a very good grasp of how that was done.  I think the method used to construct the dome must have been similar.

The building is a relatively modest structure compared to others in this area which I think may have been built around the same time.
But this one's dome shape is more unusual than the others I've seen that have similarly curved or rounded structural features.  Most of the others have simpler shapes like cones or multi-gons.
(My architectural vocabulary is miniscule.)

As far as I know there was no mathematical spline at the time these building were constructed.
Even blueprints would have been drawn with the assistance of stiff metal guides that were bent and kept in place by pins.

Fred

[jehovajah]

The house looks modern to me, judging by the facade. Is it really that old?
The mathematics of splines developed from the artisans pragmatic approach. Many pagodas and other Chinese roofs will fit a spline model in some aspect.

A lot of architects perhaps forget the considerable wisdoms in basket weaving, beadwork and knitting. These do not look like hi tec solutions, but in fact they are way beyond hi tec in terms of material technologies. When an earth process crumples a layer of the ocean sea floor like a piece of cloth, one should look beyond just the analogy to the scales of forces involved.

Fractals as a paradigm provide some vey intuitive insights and expectations.

[jehovajah]

Since I discovered that axioms are not what I was taught to believe they were, I have not been so inclined to axiomatise the thread. Instead, I am thinking carefully about indexing it under common or fundamental ideas/forms.

I cannot deny that the axioms I started with, no matter how much they were not what I thought they were, or what they aspired to be, have been pretty crucial to holding my meandering thought processes together!

I do not know how useful this thread is as a resource , whatever, but I do know that if I had not moved to my blog site, it would have been about ten times longer! So some indexing is in order!

[jehovajah]

Currently i am working around the development of the axiomatic system in a direct Historical line from Euclid to Wallis to Newton to Kant to Schelling and Ficht and to Justus Grassmann. This is revealing the historical dynamic to the axiomatic school of thought and of course many fundamental mistakes and misdirections and misinterpretations! Its a wonder we get anything right at all!!

[jehovajah]

Only just recently learned the term koan in some research i was doing on the pythagorean scholarly material. Pythagoras may have been mythologised who knows, but some of his harder sayings seem to fit the description of a koan.

While you may think you are off topic in this thread, you are not. The axioms are meant to be the most tautologically obvious statement of an attribute. So imagine my shock when i find that axioms are not even from Euclid, but a late 19th century school of thought about logic and logical systems. Aitema and Ennoia and Oroi are all that Euclid apparently laid out as the entry requirements to his course on Philosophy. The rest he builds as he goes along using these foundational principles.

Now splines may be foundational principles that need to be added, particularly if we want to describe human engineered products within the set. Trochoids for me are fundamental in a way i can not fully explain. I derive much benefit from meditating on them, especially as Laz Plath presents them.

Your particular koan i have tackled in the other thread on the Axioms, which is slowly being developed and refined.

<a href="http://www.youtube.com/v/ct_uGOSPtok&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/ct_uGOSPtok&rel=1&fs=1&hd=1</a>
Norman gives this lovely insight into the algebraic constructions of splines, and indeed they are fundamental!

[jehovajah]

Recently certain insights have transpired to lluminate my path Ahead. I have come to see that my research is close enough to that o Normans for me to usefully spark off his research and motives.

I just want to recommend his universal Hyperbolic geometry as the best set of "Axioms" you could currently have from a geometrical point of view.

The Axioms i have historically laid out here are more `philosophical viewpoints in that regard, a bit like Newton's `principles of Philosophy. The newer Thread on the Axioms Exemplifies this and i will tackle them on that basis rather than as so called axioms.

[jehovajah]

If i have not said it before i am saying it now:

The dynamics of the sphere underpin all of our Astrological, so called mathematical, notions , and the spaciometry of that sphere, its so called geometry, derives from Mechanical Philosophy.

The Stoikeia of Euclid are not mathematical textbooks but a 2 or 3 year introductory course into Platonic Socratic Theory of Form/ Ideas in which the Pythagorean School of Thought is most purely transmitted, with Aristotle's version being a variation which introduces some logical inconsistencies, due to his incomplete graduation.

The Euclidean outpost of the Academy recognised the Graduated status of Eudoxus as mathematikos, and passed on his introductory teachings on the Logos Analogos method of Rhetorical justification and measurement.

Beyond this introductory course in Philosophy there were many other rhetorical works dealing with Mechanical , Astrological and Social, metaphysics which the Seniors of the Platonic Academy engaged in Discourse.

That Newton, particularly was sensitive to the profound teachings and investigations of the Pythagorean schools, and penetrated their heart, goes without question, and that he laid out such a groundwork for philosophical reasoning that we might always conclude aright, is commended by many if not all.

To  this i might add the thoughts of Hermann Grassmann and Sir William Rowan hamilton as able commentators and developers on the Newtonian Principles derived from the Pythagorean schools of great Antiquity.