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

[jehovajah]

I can highly recommend Normans courses but in particular his Maths foundation series.
http://m.youtube.com/playlist?list=PL5A714C94D40392AB

This forms a fantastic introductory course into the foundational issues of so called mathematic.

Norman is a good presenter and clearer than I am. I have learned a lot though his presentations?

In particular his resynthesis of mathematics demonstrates the power of fundamental and detailed structural analysis in making a really useful tool.

[jehovajah]

This is the Arithmoi

<a href="http://www.youtube.com/v/0QwESq4hCZQ&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/0QwESq4hCZQ&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/IUAVqWMybxU&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/IUAVqWMybxU&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/1mSk3J3GlA8&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/1mSk3J3GlA8&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/cP50PQ&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/cP50PQ&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/h0Woqc&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/h0Woqc&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/9txb9FgLDNY&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/9txb9FgLDNY&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/k4z2Y8Y&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/k4z2Y8Y&rel=1&fs=1&hd=1</a>

And this is the Kabbalah!
This is the Gematria and the Numerology that is the Pythagorean geometry..
 This is that Algebra of the Kabbalah , the twisting of the mind, the torsion of the brain the contortion of the thought that is characteristic of the Kabbalah.
Nothing is as it seems and everything is representing something else! What you thought you knew, now comes into question, for what is encoded in the letters and the words of the Kabbalah?

Do not be deceived. For the simple find simple things in the Quabbalah, and the wise wise things. But those whose mind is twisted beyond what they can stand are in peril of losing their reason! Such is the power of the Al Jabr m'Quabbalah!

I have given it a fair chance in my lifetime, and my conclusion is Algebra is not for the faint hearted, nor those who are easily confused. It's twists and turns are not for those who become dizzy in thought or who cannot sustain meditation . It is not for those who hate rhetoric and have no love of words that stand for concepts, nor have any conceits of another alternative interpretation.

Those who want certainty turn away from Algebra. There is nothing certain in it. But those who wish to engage themselves in a struggle, who do not mind meeting themselves going backwards, who can hold on to one reality while exploring another, let them do algebra, let them revel in the Kabballah. They will find wisdom in nonsense! They will bend, but not be broken, they will twist in the wind and return with knowledge!

[jehovajah]

Some comparisons of Normans rational trigonometry assertions can be viewed here
http://web.maths.unsw.edu.au/~norman/papers/TrigComparison.pdf
http://web.maths.unsw.edu.au/~norman/papers/RationalTrig.pdf
Normans opinion on angles I have always questioned. But the generalisation of the 5 laws to higher geometries is very convincing. In higher geometries the use of angles is very problematic. However it does not follow that we should make the use of angles problematic in everyday use.

Normans aim was to prepare the student to think consistently about key concepts, and angle is not a general enough concept as usually presented.

While this method cn produce some painful looking computations, the fact that general calculators can cope with most of them makes it less daunting. The use of cosine or sine tables speeds things up on the back of centuries of labour by earlier mathematicians. We generally do not realise the kinds of computations that went into the making of these tables, or the algorithms in calculators.

Nevertheless, for ease of use, any calculation algorithm ought to use a calculator nowadays. However for understanding what you are doing, I have to agree that Normans approach is more fundamental.

Norman tries to make some distinction between spreads and rotations. But a simpler counting system is possible in which the number of quarter turns valued at 1 are aggregated. For a triangle it is known that the three angles should sum to 2. While this constant relationship for spreads does not work because of their quadratic nature, summing to a right triangle of 1 does. And in rotation this can be incorporated as the value of each quarter turn. The flipping about of the spreads, ie essentially using the cosine in certain quadrants may seem abnormal, but in fact this is what is done under the hood all the time in trigonometry, and it is called the CAST rules.

Prior Greek systems used the chord to radius ratio, that was precisely sin 2ø while Indians used sinø. There is little evidence they used spread. However Norman defines spread to make a consistent case in all geometries. There are several other options which he could use including one which relates quadrance to the Quadrea of a sector.

Also, once a quarter turn has been reached the new spread could be combined with the line point to identify  which spread is acute or obtuse by identifying negative correspondence . Although there is a test for acuteness. Vectors also help distinguish.

There are some details to iron out, but they are minor. The major benefit is in not having to learn many many arcane trig relatioships. However that gives you more time to solve the extensive computations that may arise!

The following shows the powerful analysis behind Normans thinking in rational trigonometry
http://web.maths.unsw.edu.au/~norman/papers/Chapter27.pdf

In light of this , not being able to measure an obtuse "angle" or rather a spread greater than 1, or in the other quadrants seems very minor.

There are many signals that guide the appropriate interpretation of results, so to make invidious comparisons is to miss an opportunity to really simplify and make congruent ones metrical experience. For example the constant relation of angles in a triangle summing to is replaced by an invariSnt Formula! This is a subtle but powerful observation. Constancy may not be the numerical sum but rather the formal expression!

[jehovajah]

Polynumbers are  a fundamental redefinition of Algebra. Norman explains it in the Maths foundation videos, but the theory behind it I have expounded  in this link
http://my.opera.com/jehovajah/blog/2013/11/30/the-polynumbers

Because it is a foundational coding technique it is clearly demonstrably correct. But what Norman has done is made it his mathematical foundation. It is a fundamentally valuable recasting of the whole of mathematics into one logically consistent format. It gets my vote.

Normans exposition is eloquent but he does need to clarify some of his presentation. I hope by reading at the link it will help you grasp how awesome this reformulations is.
<a href="http://www.youtube.com/v/DAHBgcDJQjw&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/DAHBgcDJQjw&rel=1&fs=1&hd=1</a>

[jehovajah]

I have given rational trigonometry some thought.

http://web.maths.unsw.edu.au/~norman/papers/TrigComparison.pdf
This paper provides a good framework for comparing the two approaches. It reminds me of the old comparison between a computer doing a calculation and a human using an abacus. We used to love these comparisons, because in our brave new technological world it showed that humans were superior to machines– just! However, those days have long since passed.

You still my hear individuals protesting that machines are simply following programmed in instructions, and they do not think, nor are they alive! I am afraid we are deluding ourselves if we adopt that point of view. It may be too extreme just yet to say a modern technological robot is a living being, but equally it is extreme to say it is just a hunk of " metal". Our boundary lines, drawn in a less technological less electronic age where chemistry was still in its infancy are no longer sustainable .

So, in my opinion Norman's assertions were meant to be provocative. Rising to the bait Gilsford confirms Normsns Analysis Classical theory is not the best way to teach or understnd trigonometry. Nomans method, flawed as it is has much merit . One of its aims was to be understandable by high school students. Gilsford amply demonstrates that this is the case. The claim to be faster or more accurate is not upheld, but in practice a high school student could solve a problem that requires considerable sophisticated classical knowledge and set up, by applying the 5 main laws of rational trigonometry, and quadratic algebra.  While it may take a little longer, it can be started sooner and processed in a known solution path.

High school students would be better equipped by learning rational trigonometry.

Normans arguments about quadrance and angles just need to be revised a little to be less polemical, and that would make them useful jumping off points for further research by graduate level students. Metric theory is vital for all philoshers to apprehend and the distinction between measuring using different magnitudes understood. Quadrance is dependent on distance both in formulation and practical application, but this is because it is designed to spread itself unevenly I might add, over two distinct but related magnitudes. Then it is used to define a third distinct magnitude! This may seem very convenient for calculation, but for apprehension it represents that torsion of mind so characteristic of Algebra!

[jehovajah]

Norman tackles a fundamental concept. That of limits. In preparation he defines the notion of sequence in some specific cases. You will have to look at the videos prior to this one
<a href="http://www.youtube.com/v/K4eAyn&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/K4eAyn&rel=1&fs=1&hd=1</a>
If you are interested.

Sequences and series are fundmental undefined but usually exemplified notions. Google" Jehovajah sequences" to find out more. In my opinion a sequence and a series are conjugates in subjective processing. When a sequence is perceived one actually tautologically perceives one's own processing strategy, structure or behaviour. One is experiencing one's individual subjective processing patterns as they impose order on the undefined stream of conscious awareness or perception.

Consensus sequences are absolutely fundmental to social and group processes including communication, but idiosyncratic sequence utilisation is what encodes the experience of self and self awareness and my experiential continuum.

That being said, it is fundamentally important to apprehend ones limitations.

[jehovajah]

<a href="http://www.youtube.com/v/fEWj93XjON0&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/fEWj93XjON0&rel=1&fs=1&hd=1</a>
Way to go Euler?

[jehovajah]

I have to post this video here because my work on Grassmann and the arguments of Norman have coincided with the basic conception of this thread.
http://youtu.be/xYPw2gY_3PI
http://www.youtube.com/watch?v=xYPw2gY_3PI

[jehovajah]

This video promotes the misleading idea that Euclid couid not trisect an angle! This false idea is still not corrected . The game that is played restricts the artisan to specific "naff" or unfit for purpose tools!. You are not allowed to improve on the tools in the game! Origami relies on Neusis. This was known to Euclid and any other Greek artisan. In fact Archimedes solved this game challenge, using the tools, by neusis. But his proof was frowned on by the game masters!

http://youtu.be/SL2lYcggGpc

http://www.youtube.com/watch?v=SL2lYcggGpc
Cubics and Conics were regularly solved by Neusis in Euclids day.
http://youtu.be/6Lm9EHhbJAY

[jehovajah]

This is the Euclidean like version:

http://youtu.be/vTsR1Q8S3Oo
http://www.youtube.com/watch?v=vTsR1Q8S3Oo

[jehovajah]

This is the Archimedian method where the Neusis is reliant upon good judgement

http://youtu.be/wgQayWMxHvw
http://www.youtube.com/watch?v=wgQayWMxHvw

Archimedes had several more advanced methods of trisecting any acute angle , or any angle. He constructed the trisectrix which basically is the radian curve for circles wher the point on the trisectrix is a function of the radius and the radian measure. As the radius of a circle increased its radian measure is increased in step. Thus got a radian measure of the radius has length . Thus an angle which is a half circle is associated to a radius of length ? Thus Archimedes had a very good representation of and a very good value for the circumference or perimeter of a circle to construct a trisectrix.

The importance of circles and proportions is fundamental to apprehending these constructions. Rather than making it impossible to trisect an angle or any other proportion of sn angle circle proportions made it possible to accurately estimate or approximate the required position in space.

To minimise the requirement for excellent judgement, tables of chords or dines were and are constructed that allow any level of precision required . These tables are constructed from accurate diagrams , many thousands of them, making the Neusis very accurate indeed, and allowing interpolative formulae to be designed for better in between estimates.

[jehovajah]

This is the most accurate approximator to the trisection of any angle. Here circle proportion is used at its most basic and direct level.

We are misdirected to the vertex of an angle even while measuring with a protractor along an arc!  All the basic information about angles in a triangle refer to arcs of a circle. Thus the angles in a triangle sum to?   Stop and think for one minute. Using a protractor you are no way measuring in a triangle! You are measuring along a standard circle. So in fact what we are doing is finding the arcs for each corner of the triangle as the limbs cut a standard circle. What we then find is those arcs always add up to a semicirclular arc of that standard circle .


In the same way we can find the proportion of arc we require by using proportional circles. A circle of 1/3 rd radius or better diameter will roll its perimeter 3 times into the perimeter of a circle with the full diameter. Thus by concentric circles we can find the proportional arc. However the process does not then involve marking off 3 circles on the larger perimeter because that in fact cuts that perimeter according to the chord not the arc. The process is a little more involved,but simply it is the neusis of rolling the circle along the larger perimeter without slipping.

It is this kind of Neusis that was frowned upon as an accurate or reliable solution. It is however, like any skill as accurate as the master craftsman . There are some guiding lines and circles and triangles that can make that rolling even more secure.

Study this method and free your mind.

http://youtu.be/eNYgDfi5PdY

http://www.youtube.com/watch?v=eNYgDfi5PdY

[jehovajah]

The simplest innovation is to replace the pencil tip in a pair of compasses with a gear wheel of a known radius. Such a tool wold mark out as precisely as skillful artisan ship would allow, the coordinate system on every perimeter so that direct proportional comparisons can be made, and any angle subdivided at will!

[jehovajah]

One of my abiding interests is how to construct the motion of a rolling circle.

I recall in technical drawing learning how to transform a circle onto a parallelogram at say 60° angle to the square. This was done by creating sections and mapping sections to the transformed image point by point, then drawing the curve freehand.

Similarly I divided the circle into any number of sectors and then translated the heights by parallel lines ont a rectangular grid with the same number of horizontal intervals , to frm the sine curve, but the actual rolling of a circle by drawing I did not do.

Out of interest I carefully sectored a circle and stepped off those sectors along a straight line to estimate the perimeter of a circle, but of course it was not possible to get an accurate value of this way.

Several things remained unexplained: how was the protractor itself so accurately divided? How did the radius come to precisely divide the circle into 6 sectors and yet was not a rational number?

Eventually I perceived that the circle and it's proportions we're an older more pervasive geometry than Planar geometry. We had somehow lost connection with its proportional simplicity. If I wished to divide a circle perimeter into 6 I merely made a disc 1/6th the diameter( or a gear wheel) and rolled it round the perimeter of the larger disc.  From these pragmatic artisan like beginnings the formal empirical findings were discovered and set. The radius does not section the perimeter into 6 but it is as near as dammit! Thus we formally defined that relationship because pragmatically we could not tell the difference.

That a square has 4 equal sides is again a formal definition, based on the formal definition of a circle diameter. 4 equal circles can be so folded that their diameters form a square , and the rotation of the quarter arc stands out.

O how to draw a circle moving , or rather a cycloid locus by construction  is my interest. I see computers doing it regularly, but how can I do it with a straight edge a pencil and a pair of compasses with a gear attachments!

[jehovajah]

I have finished my analysis of the trisector method, and think it is a good approximation. Whether it extends to obtuse Nigel's equally well I doubt.

The proof at the end is to support the assertion that the ratio 1:3:4 remains constant. However, as you can see the 2 sides tht lie within the secondary and concentric arcs do not match. They are very close and this is what makes it a useful approximation, but only for acute angles I suspect.

The fact of the matter is the sine tables represent a centuries long measurement and interpolation exercise to completely characterise the sines or chords for equal sectors of the circle. The attempt to do this simply and accurately generally is a great exercise but can only give approximate results. The pragmatic way is to mark off equal steps on a fisc using a rotary gear.

All of our measurements are based on these pragmatic approximations, and that is not a demeaning thing. Rather, idealists hoped to establish by reason lone invariant truths and all attempts do far have relied on assertions such as the one made in the Trusectir method.