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Creepy Sleepy

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A Re-orientation of Your Mindset
« on: July 11, 2021, 11:48:51 pm »
"This is probably the most important mathematics video that you are likely to see."

(not hyperbole)

Rational trigonometry: an overview  (presented in a simple and elegant fashion)

"If you absorb the lessons of this video, you will be re-oriented in your approach, not only to geometry, but also mathematics more generally; and you will see that there is a huge revolution in mathematics education which is looming ahead of us - and no amount of orthodoxy or inertia is really going to end up being able to prevent it."

"If you are a high school student,  undergraduate student, [or even a] graduate student, you are going to learn very important things.  Make sure you understand everything that I am saying in this video.  If necessary, watch it several times."   (sounds like Schopenhauer)

"If you are a professional mathematician, don't be dismayed by the elementary aspect of this video.  You are going to learn something very, very important for YOU.  You may not LIKE it, because it represents a re-orientation of your mindset.  To be honest, we've been indoctrinated as pure mathematicians, into thinking, ultimately, in the wrong direction."

https://www.youtube.com/watch?v=dVk3CpjHR4Y

reference:  see University of New South Wales (NJ Wildberger YouTube Videos)


Chapter 1 of Divine Proportions

Note that I had hesitated to explore this text in detail here so as not to lead Holden astray, but now I am thinking that I can study Wildberger's Rational Trigonometry alongside my exhaustive going-through-the-exercises of the Dolciani classic, Modern Introductory Analysis as well as the 1988 edition, "Introductory Analysis".   Supposedly, Rational Trigonometry deals only with rational numbers.  Pythagoras would be quite pleased!   The foundations would then not require analysis or transcendental functions, I suppose.

This student/researcher now may be prepared to consider Wildberger's controversial book in a more serious manner, albeit, comparing it to the rigorous classical (and so-called "modern") trigonometry.

Like Holden, I find the best time to allow for such meanderings and intellectual wanderings is waking so early in the morning that it is still the middle of the night. 

Just becoming engrossed in such research might decrease the anxiety caused by this sense of lack (being broke all the time, observing just how unattainable some desires can be).  There is something to be said for this kind of study.  When truly engrossed and interested, the practice could spare me from a great deal of unnecessary heartache that comes from emotional entanglements and romantic delusions.
« Last Edit: July 12, 2021, 09:56:28 am by "No No Bad Dog!" »
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Holden

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Quantum Mechanical Patience
« Reply #1 on: July 12, 2021, 10:08:09 am »
Thanks for the links. I reread them and especially liked the lengthy essay by you. While I did notice a number of times that you often mention Wildberger,I did not explicitly refer to him as I was not familiar with his ideas and did not know what to make of him.

I think now I understand a bit as to what you have been saying about him, and what he says himself about mathematics.

Correct me if I am wrong, but I think, what he is saying is that we do not need infinite sets and real numbers, for one. That these are not genuine mathematical entities.

I finished reading Men of Mathematics, and in there, I came across a man called Kronecker who liked finitism and did not think highly of the set theory.
Wildberger  sounds quite similar to a man called Brouwer too. It might be fair to say that what all three of them have in common is called the constructivist approach to mathematics.

Very interesting. I like such ideas. Even a year back, I would not have been able to write this message to you.

Mr.Wildberger is asking us to show him 5+7i oranges:)

I feel like I have just started to grasp the soul of mathematics and yet, I also very clearly remember ,that in Die Welt als Wille und Vorstellung ,Schopenhauer writes that even if a thousand once-in-a-millenium  geniuses should die just before they bring their project to fruition, the Earth merely laughs at their deaths. And I am no genius.
According to Quantum Mechanics,it is quite possible that a cup of ice-cream might materialise on your table and then turn into a Cuban cigar.It would not be contrary to any laws of physics ,because the world would have changed in a compensatory way.

This is not a mystical interpretation or an extreme interpretation of Quantum Mechanics by me.Provided one has enough time, the unorthodox interpretation would be that such an event would not take place.True, I am talking about a period of time greater than the life of known universe.
But given enough time..it would happen…

So,if we are stuck in a everlasting torture chamber, then, maybe there is hope for me ,after all, that I ( whatever entity that might be) would manage learn some mathematics after all:) I just need to have Quantum Mechanical Patience while I continue to exert myself:)

If wanted to ask you, if I might, what in your view,would someone like Godel make of  Wildberger?

I continue to study down to earth mathematics ,and I refer to the thinkers ,mostly for their “approach” to mathematics and not for the content. I agree with Mr. Wildberger that the way they teach mathematics is all wrong. But then again, most teachers in my no-name provincial school were there only for the money.
 
Their teaching approach  towards mathematics was suffused with the typical dreariness that is always present when there is a total lack of reflection of greater issues.
I hope your foot heals fast.
 While there have been many well- read men on my mother’s side of family,my father’s side of family is mostly made up of peasants who are   scared stiff of mathematics( and death).
Maybe by truly embracing mathematics ( and death in a Schopenhauerian fashion), I would bring the millennia long turmoil the lineage has experienced to an honourable end.
Get well soon.
« Last Edit: July 12, 2021, 10:15:07 am by Holden »
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Creepy Sleepy

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a misunderstanding of geometry
« Reply #2 on: July 12, 2021, 06:57:51 pm »
Quote from: Spy der Mann
Dr. Norman Wildberger, of the South Wales University, has redefined trigonometry without the use of sines, cosines, or tangents. In his book about Rational Trigonometry (sample PDF chapter), he explains that by replacing distance and angles with new concepts: quadrance, and spread, one can express trigonometric problems with simple algebra and fractional numbers. Is this the beginning of a new era for math?

From New trigonometry is a sign of the times:

Quote
Mathematics students have cause to celebrate. A University of New South Wales academic, Dr Norman Wildberger, has rewritten the arcane rules of trigonometry and eliminated sines, cosines and tangents from the trigonometric toolkit.

What's more, his simple new framework means calculations can be done without trigonometric tables or calculators, yet often with greater accuracy.

Established by the ancient Greeks and Romans, trigonometry is used in surveying, navigation, engineering, construction and the sciences to calculate the relationships between the sides and vertices of triangles.

"Generations of students have struggled with classical trigonometry because the framework is wrong," says Wildberger, whose book is titled Divine Proportions: Rational Trigonometry to Universal Geometry (Wild Egg books).

Dr Wildberger has replaced traditional ideas of angles and distance with new concepts called "spread" and "quadrance".

These new concepts mean that trigonometric problems can be done with algebra," says Wildberger, an associate professor of mathematics at UNSW.

"Rational trigonometry replaces sines, cosines, tangents and a host of other trigonometric functions with elementary arithmetic."

"For the past two thousand years we have relied on the false assumptions that distance is the best way to measure the separation of two points, and that angle is the best way to measure the separation of two lines.

"So teachers have resigned themselves to teaching students about circles and pi and complicated trigonometric functions that relate circular arc lengths to x and y projections – all in order to analyse triangles. No wonder students are left scratching their heads," he says.

"But with no alternative to the classical framework, each year millions of students memorise the formulas, pass or fail the tests, and then promptly forget the unpleasant experience.

"And we mathematicians wonder why so many people view our beautiful subject with distaste bordering on hostility.

"Now there is a better way. Once you learn the five main rules of rational trigonometry and how to simply apply them, you realise that classical trigonometry represents a misunderstanding of geometry."
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

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Creepy Sleepy

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A Purely Rational Theory
« Reply #3 on: July 12, 2021, 11:23:26 pm »
Is it not particularly enchanting that when Wildberger explains the Pythagorean theorem in terms of quadrance, the diagram on the board - the triangle with sides squared (the side squared being quadrance (NEW [yet ancient] CONCEPT)) - is the very image projected in the Embodied Animal Mind as Schopenhauer explains how we "perceive" the qualitas occulta of the triangle, the relationship between separation of points and lines, and how the areas of the two shorter sides squared equals the longer side squared when right triangle?

here: https://youtu.be/dVk3CpjHR4Y?t=360


WWR:V1p72

The theorem of Pythagoras teaches us a qualitas occulta of the right angled triangle; the stilted, and indeed subtle, proof of Euclid forsakes us at the why, and the accompanying simple figure, already known to us, gives at a glance far more insight into the matter, and firm inner conviction of that necessity, and of the dependence of that property on the right angle, than is given by his proof.

Now, the following image is not from Schopenhauer's work, but he is talking about intuitive knowing - directly through perception.

Rather than thinking a^2 + b^2 = c^2, we imagine

The sides are squared.  You gain insight at a glance.

So what does Schopenhauer mean when he says "Pythagoras teaches us a qualitas occulta of the right-angled triangle"? Pythagoras teaches us the hidden quality of right-angled triangles?  Pythagoras teaches us the ultimate cause or inner nature of right triangles?
____________________________________________________________

Quadrance is distance squared.
  (Distance is the square root of quadrance)

Holden, I sense that this is one of those academics we can trust.  It would be a radical delight to analyze triangles without recourse to calculators or tables.   Imagine the delight for the philosophical mathematician in chains, in the dungeon, gulag, or bedlam!   It would revolutionize the way people apply trigonometric theory. [surveying on the back of a napkin with a telescoping device]

Note:  Wildberger unknowingly gives a nod to Schopenhauer when he points out the logical weaknesses in Euclid's ELEMENTS:  https://youtu.be/dVk3CpjHR4Y?t=588
« Last Edit: July 13, 2021, 07:48:59 am by "No No Bad Dog!" »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Holden

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Re: A Re-orientation of Your Mindset
« Reply #4 on: July 13, 2021, 10:10:33 am »

That is very interesting and I see what he is trying to say.He also says somewhere that mathematics does not need  axioms. Almost the opposite of what Frege and Russell were trying to do at one point of time.

I hope that your foot is healing .I read somewhere that watching horror movies or reading horror books might actually help some people to get a handle on their anxiety.

As regards ,mathematics, like I have said before, I am a tortoise and while I walk slowly, but I have been walking almost continually for the last three years now.I have been reading a bit about Frege and Quine.

I hope that you continue with your mathematical and programming studies and your body get healed at the earliest.It is twilight and when the Sun is dying, I feel quite strange. There might after all be something to the thesis that sentient life is just a one-time (very painful) accident.It is much better than the alternative.

I am sure Schopenhauer would have found life very strange too( I mean on a day-to-day basis) I think that if he were forced to work for a living all his life then he might have turned far more bitter(justifiably) than he was.

Sometimes, like now, I feel that I just wont be able to carry on, so tired that I cannot be bothered with the task of drawing another breath. But then the great tide of Life pushes me along and takes me once more, to its nefarious shores.

Get well soon.


https://youtu.be/mHD08tI0T30
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Creepy Sleepy

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The Will to Understand
« Reply #5 on: July 13, 2021, 11:02:11 am »
"Fuck logic." ~ AH from video above.

Funny.

He might also have exclaimed, "Fuck the square root of negative one!"

You know I have made the Herculean effort of returning to the challenging texts of my high school days, actually in the spirit of "self-ordained" post-graduate RESEARCH.   I wanted to virtually master the classical representation of trigonometry, with the complicated definitions with unit circles, arcs, projections, etc ..., BEFORE becoming a serious "disciple" of Dr. Norman Wildberger.

Now I see that there may be a method to my madness, for I did not just say, "Fuck set theory and the so-called real numbers," but have used every bit of diligence I could muster simply to deepen my UNDERSTANDING, or at least become more intimate with the subject as it is presented by actual mathematicians, and not mere "educators.".   I am going to continue to try to deepen my understanding of the fundamentals.  It's just that not all that is at the foundations of set-theoretical notation and real number decimal representations are FUNDAMENTAL. 

Quadrance is more fundamental than distance, since distance is the square root of quadrance.   I am more enthusiastic now about including rational trigonometry in my personal curriculum, for it would make sense of my life in a Joseph Campbell way.  This animal creature that "I am" is no longer an overwhelmed teenager, but an aged beast who has developed just enough confidence to become a great student of Wildberger.   I would not become the evangelist for his theory until I study/master how to apply it myself first.

Hence, in my own little world narrative inside my head, this is what I may have been so intensely preparing my mind for over these past five or six years while "regrouping my mindset."    I may now be prepared for a genuine re-orientation.

Wildberger makes some very strong statements, such as the following, where he claims to be presenting the two most important theorems in mathematics:

https://youtu.be/dVk3CpjHR4Y?t=939

I would say that mathematics students have cause to celebrate, and that mathematicians have an opportunity to defy the false authority of their "academic masters."

Note:  The least expensive used copy out there is at Abe Books for less than $50.

I will go through the PDF, using small notebook computer, and printing pages when necessary, taking notes from it as religiously as if I were a True Disciple of Finitism (although I am, like Sweden, quite neutral; that is, I had been indoctrinated into set-theoretical notation and the Field Axioms.   It has not been a waste of time struggling to find a way to deepen my understanding.  If anything, this has just made me that much more ripe to appreciate a purely rational theory.

Do you not feel the holy ghost of Pythagoras grinning?

You do realize that my first real computer programming project was "experienced" in a community college C++ course in 1998 or so after losing my job with the NJ State Park service for, Jupiter forbid, "eluding the police."   That project was a Fraction class that has evolved over the last two decades.   I mention the instructor's influence on me in the Generic Programming thread and in the The One Who Uses Parrhesia thread.  My subconscious mind is already daydreaming of the elegant code I could write to accompany my exploration of this rational approach to trigonometric problems.

Call me crazy, but, Holden, we may be mere instruments of the Will-to-Understand.

___________________________________________________-
Be forewarned that I am of two minds (at the very least).  That is, I have been religiously indoctrinated into the set theoretical view, and I understand how infinite series are used to define the decimal representation of the so-called Real number system.   1 = 0.9999999...

 It is ALL so very interesting to me, personally.   Still, I do not want to be one of those who dismiss Wildberger as a crank.   I want to consider his theory and be open to the re-orientation of my mindset.   

The bottom line is that the Will to Understand is strong in this embodied animal mind.  This is what has cemented our communications since 2014 - a mutual will to understand.

Anyway, you will find Wildberger under attack by the conventional academics, posturing to be taken seriously by their "masters."    He is surely poised for crucifixion.

Case in point:
Quote from: Houshalter
(2017: Hacker News [Don’t Fall for Babylonian Trigonometry Hype ])
Wildberger is a recurring subject on /r/badmathematics including this recent story. He's almost a crank and has some very very strange views on mathematics. He strongly objects to the use of infinity and proofs and mathematics that involve it, which includes real numbers.

If I understand correctly, he goes even farther and claims even really big but finite numbers don't "exist". And through this he claims to have "resolved" the Goldbach conjecture and other strange things.

Quote
Even regular finitists are kind of absurd, denying the existence of pi or the square root of 2.

... the Math Wars commence ... ::)

Quote from: roenxi
(Aug 30, 2017)

We've entered fuzzy-definition territory; but I'd say it is a stretch of the language to imply Wildberger is a crank. Which you have now done twice with no references.

He is unorthadox in his pushing of rational numbers, sure, but he gets through an extremely large amount of maths correctly. It is well within his job domain to reflect on odd academic questions about what happens outside the world of human measurement.

Amen.  It is my intention to honor Wildberger's unorthodox stance, and to open my mind to understanding the framework of rational trigonometry.

Quote from: bergoid
(on Aug 30, 2017)

In mathematics education, I always disliked the way infinity is shoehorned into the different sets of numbers and treated as "just another number", so that you can write shorthands like: 1/∞=0, instead of the more accurate ∀ε>0, ∃c>0: x>c ⇒ 1/x<ε

Infinity is not a number. It's a limit.

Treating it as a number only confuses real understanding of it.

Quote from: jacobolus
(Aug 30, 2017)
It makes sense to treat ∞ ≡ 1:0 as a “number” in the context of projective geometry, the Riemann sphere, etc.

The quarrel Wildberger has is not with the proportion 1:0, but with the concept of a limit.

Quote from:  chroem
(Aug 29, 2017)

This article (Don't Fall for Babylonian Trigonometry Hype) seems to boil down to "stop liking what I don't like." Is Wildberger's distaste for real numbers eccentric? Yes. Is there anything fundamentally wrong with what he is proposing? Absolutely not.

As long as rational trigonometry is logically consistent then you really can't argue against it except on aesthetic grounds. I don't see why popsci outlets like Scientific American should feel the need to rally the public against rational trigonometry on emotional grounds.

Quote from: akyu
(Aug 30, 2017)

Glad to see Wildberger getting some recognition, even if some of it is mixed. His rational trigonometry is an entirely rigorous and interesting piece of mathematics even if you don't think it's particularly pragmatic.

https://www.youtube.com/watch?v=77nB_9uIcN4
« Last Edit: July 14, 2021, 12:33:00 am by "No No Bad Dog!" »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Holden

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Re: A Re-orientation of Your Mindset
« Reply #6 on: July 14, 2021, 10:06:20 am »
It is very interesting,no doubt.I hope your foot is not in pain anymore and maybe you would be able to write finitism oriented codes. I checked out the links you left behind.Most interesting.

I would say I understand about 80% of things you and Wildberger are saying. Maybe someday, I would be able to write finitism oriented code too but that is far in the distant future, I guess, if I still continue to breathe.

At the moment,I am in the “nose to the grindstone phase” ,which I have started to enjoy a little bit. But I did like exploring finitist mathematical ideas and understood about 80% of what all I read.I get up early in the morning, and before I go to office, I study mathematics, because that is the time when I am most (mentally) energetic. I do not know if I would ever even be able to reach the stage of coding,because first I wish to be quite confident with rudimentary mathematics. I do not like to think about the future a lot because I cannot see it in a bright light.

Only different versions of dystopia. There is no doubt,however, that I am getting better at comprehending mathematics.Maybe in a few years ,if I am still around, I might be able to understand these ideas better.

I do not know what horrors life has in store for me, but, I am doing my best to pick up a little mathematics( for its own sake). Ageism is even more rampant here than it is in the U.S. and unlike the professors of mathematics who are akin to the clergymen, we are like, as you rightly say, defrocked  friars.

Get well soon.
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
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Holden

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Re: A Re-orientation of Your Mindset
« Reply #7 on: July 15, 2021, 10:13:08 am »
For close to two thousand years everyone believed that  through a point, parallel to a given line, only one line could be drawn. Then, one day one man tried to find out if he would get any contradictions if he drew many lines through the point and ended up discovering a new geometry.No one could have imagined it. This was something unheard of.

Then another man came along who assumed that no line could be drawn through the point and he also ended up discovering any entirely new geometry too.

I think Wildberger is doing something similar with Rational Trigonometry. Sometimes when we challenge an axiom then really interesting things might happen.

I hope you would keep you posted about your study of his ideas.

Get well soon.
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
-van Gogh.

Creepy Sleepy

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Re: A Re-orientation of Your Mindset
« Reply #8 on: July 22, 2021, 12:43:19 am »
I am currently able to walk around without a cane or crutch, although I do utilize them if and when I find myself exhausted (or simply feeling vulnerable), which has been a tremendous help in doing some research and actual Study/Practice of Rational Trigonometry.  I opted for this before "Concrete Mathematics" as I feel this reorientation of my mindset might benefit more "future disciples" [of The Weird Truth] than a haphazard voyage through the more "elitist" number theory presented by Knuth and Company - although, were I to live to be very old, that text, Concrete Mathematics: A Foundation for Computer Science, is also alarmingly tempting.

Here is a Fellow Traveler who might assist us on this intellectual adventure:  Dr. Gennady Arshad Notowidigdo, PhD (Solutions to Divine Proportions).

Off of this PhD's site, I just discovered that he has something called "Concrete Mathematics Forum," and I am curious to see if it is (and how) related to Concrete Mathematics: A Foundation for Computer Science.

From the more recent instructional videos created by Dr. Wildberger, he appears to be aging, as is par for the course.  It is great that he took the time to create the instructional videos he has.     Maybe my years of searching for "math(s) worth learning" might not be in vain if it saves you some decades of banging your head against the wall.   

I will have to memorize the spelling of this name:  Notowidigdo

Noto wi dig do

He may turn out to be a precious guide in this intellectual/spiritual dark age Matrixtan we find ourselves BREATHING AS.   May I be blessed with a Beginner's Mind in the spirit of The Dancing Wu Li Masters.

https://www.youtube.com/watch?v=08h0IVs4RKQ
« Last Edit: July 22, 2021, 01:36:31 am by Kaspar Hauser »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

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Creepy Sleepy

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Re: A Re-orientation of Your Mindset
« Reply #9 on: July 23, 2021, 11:36:52 am »
Quote from: Holden
It is very interesting, no doubt.  I hope your foot is not in pain anymore and maybe you would be able to write finitism oriented codes. I checked out the links you left behind. Most interesting.

Quote from: Holden
I think Wildberger is doing something similar with Rational Trigonometry. Sometimes when we challenge an axiom then really interesting things might happen.

I hope you would keep you posted about your study of his ideas.

While going through section 1.2: Laws of rational trigonometry (page 10), I found the word quadrea (of a triangle), which author Dr. Wildberger declares is the single most important number associated to a triangle.   With humility I confess I have never heard this term, so I did some research.

This brought me to a page off of Euler Math Toolbox in Germany: Rational Trigonometry (off of sourceforge DOT net): 
Rational Trigonometry

Quote
Inspired by a talk N.J.Wildberger gave at our university, I expanded the Euler file for geometry. In his book "Divine Proportions", Wildberger proposes to replace the classical notions of distance and angle by quadrance and spread. Using these, it is indeed possible to avoid trigonometric functions in many examples, and to stay "rational".

In the following, I introduce the concepts, and solve a few problems. I am using symbolic computations Maxima here, which hides the main advantage of rational trigonometry that the computations can be performed by paper and pencil only. You are invited to check the results without a computer.

The point is that symbolic rational computations often yield simple results. In contrast, classical trigonometry yields complicated trigonometric results, which evaluate to numerical approximations only.

It is there I see that the quadrea of a triangle is its area squared.

Quote
We first compute the spread at B for a triangle with sides a, b, and c. Then we compute the area squared (the "quadrea"?), factor it with Maxima, and we get the famous formula of Heron.

Admittedly, this is tough to do with pencil and paper.

By the way, I imagine that the computations performed with Euler might also be done using SymPy or Sage. We shall see.   The main advantage of rational trigonometry is that the computations can be performed by paper and pencil only, but it is always comforting to check our work with a down and dirty computer algebra system.   :)

Myself, since I have this homegrown class (fraction.h, fraction.cpp) which has been evolving over the past two decades, I would most likely opt to create my own command-line programs in C++.   I even have a special "Big Integer" Calculator I managed to construct, which I mention in the Modified Calculator Program in Stroustrup Textbook C++PPP (Bjarne Straustrup's Programming Principles and Practice Using C++) thread.

I have already been using "isympy" (improved Interactive Symbolic Python) to check my algebraic manipulations [WORK] when "proving" (to myself) the laws of the theory.
« Last Edit: August 07, 2021, 01:02:55 pm by Kaspar Hauser »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

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Creepy Sleepy

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Little if any research has been conducted looking into educational benefits of rational trigonometry.

How do mathematics majors approaches to solving problems pertaining to triangles change after studying rational trigonometry?


Student responses to instruction in rational trigonometry
« Last Edit: August 07, 2021, 01:05:08 pm by Kaspar Hauser »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

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Re: A Re-orientation of Your Mindset
« Reply #11 on: August 07, 2021, 02:03:08 pm »
From the end of the "Overview" [Chapter 1] of Divine Proportions: Rational Trigonometry to Universal Geometry, Dr. Wildberger discusses "Modern ambiguities," which I think Holden would find exciting, to say the least.   We do not need to engage in philosophy to do geometry.
_______________________________________________________

1.7 Modern ambiguities


In the nineteenth century, some of the difficulties and limitations with Euclid’s geometry became more recognized, and for this reason and others, the twentieth century saw a slow but steady decline in the importance of Euclid as the basis of education in geometry. Unfortunately, there was no successful attempt to put into place an alternative framework.

One of those who realized the importance of the task was D. Hilbert, whose Foundations of Geometry [Hilbert] was an attempt to shore up the logical deficiencies of Euclid. This approach was not universally accepted, and the thought arose that perhaps the task was intrinsically impossible, and that a certain amount of ambiguity could not be removed from the subject.

Twentieth century mathematics largely resigned itself to an acceptance of vagueness in the foundations of mathematics. It became accepted wisdom that any treatment of Euclidean geometry had to incorporate the philosophical aspects of the ‘construction of the continuum’ and Cantor’s theory of ‘infinite sets’. Distance, angle and the trigonometric functions required the usual framework of modern analysis, along with its logical deficiencies.

A common feeling was that the best strategy was to adopt an axiomatic system, involving formal manipulations of underlying concepts that are not given meaning. It was often supposed that this is what Euclid had in mind. 

But this is not what Euclid had in mind. When he stated his ‘axioms’, Euclid wanted only to clarify which facts he was going to regard as obvious, before deriving all other facts using deduction. The meaning of these basic facts was never in question. Euclid’s work in this sense is quite different from formalist systems which came to be modelled on it.

When we begin a study of universal geometry, these preconceived twentieth century notions must be put aside. It is possible to start from the beginning, and to aspire to give a complete and precise account of geometry, without any missing gaps. The logical framework should be clearly visible at all times. The basic concepts should be easy to state to someone beginning the subject. Pulling in theorems from the outside is not allowed. The ‘continuum’ need not be understood. ‘Infinite set theory’ and its attendant logical difficulties – deliberately swept under the carpet by modern mathematics – should be avoided.

With such an approach, the logical coherence and beauty of mathematics become easier to verify and appreciate directly, without appeals to authority and convention. To build up mathematics properly, axioms are not necessary. You do not have to engage in philosophy to do geometry, nor require sophisticated modern logic to understand foundations. Clear thinking, careful definitions and an interest in applications should suffice.

So let’s begin the story, starting with an informal review of the basic assumed knowledge from algebra and arithmetic. Then let’s build up trigonometry and geometry, one step at a time.
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I thought you might want to stick around, continue to feed your Animal Body enough food to keep it as healthy as possible, and see if we might get past these mathematical ambiguities together, discovering a universal geometry independent of visualizations in spatial fields.

My interest in this and how it ties into the Foundations and Fundamentals of Mathematics may be what sustains me spiritually and psychologically.  I do not know what else could rejuvenate my mind and inject some passion for intellectual adventure in my day-to-day life.

My goal has always been UNDERSTANDING.  My sense of obligation to present something unambiguous to you has led me to the conclusion that these foundations [of set-theoretical modern analysis) are indeed ambiguous since the circle (arc, angle, etc) is used in the definition of distance between two lines.   It's a shame that the kernel of truth walks such a razor's edge.  That is, I wish we could form our own kind of mathematics monestary up in the mountains somewhere ... Pythagoras Rebel Math Monks ...

If not for my interest in mathematical truth and the elegance of the manner that Dr. Wildberger presents his theory, I might very well be tempted to drink myself into oblivion first thing in the morning.  I can honestly say that I am interested in this, interested enough that I would invest in a new copy to be mailed into whatever dungeon, hospital, prison facility, or group home this tired Animal Body ends up "residing" in.   For now I am using the PDF file off a notebook computer.

Often I print several pages so as to be free to work outdoors, take notes, add comentary.

Many dramas play themselves out in the local monkey sphere, but I try to protect my spirit and not allow my ATTENTION to be drained by the tyranny of public opinion or the unpleasantries of creaturely existence.

https://www.youtube.com/watch?v=3GU9mGyxz04

Wildberger's blog:  https://njwildberger.com/

​​New Paths for Mathematics Education with​ N J Wildberger: https://www.wildegg.com/

Research Gate: https://www.researchgate.net/profile/Norman-Wildberger
« Last Edit: August 07, 2021, 02:41:32 pm by Kaspar Hauser »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Holden

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Re: A Re-orientation of Your Mindset
« Reply #12 on: August 07, 2021, 05:06:37 pm »
My sense of obligation to present something unambiguous to you has led me to the conclusion that these foundations [of set-theoretical modern analysis] are indeed ambiguous since the circle (arc, angle, etc) is used in the definition of distance between two lines.-Herr Hauser

Thank you for that. I hope your foot is better now. Every single day for the last few years I have been studying mathematics and I think I might be beginning to get the hang of it. I can see why this rather unconventional approach to mathematics might interest you. I also can see where Dr. Wildberger is coming from.

I can ,I think see now, that the approach adopted to  teach maths when I was a  schoolboy  was deeply flawed because the teachers probably did not themselves possess a profound understanding and my father is mathphobic.But I have been rebuilding the foundations, brick by brick.As they say, Rome wasn’t build in a day,but after all it was built.

You want to know how they trap folks like me? With the lure of promotions and marriage. Only, they don’t know and won’t understand even if they did know that I drank  deeply from the wells of Schopenhauer. That I think was the turning point. 
                                                                                                                                                                                                       
As regards our social difficulties, I think it is by design. The society, American or Indian, was never built with the likes of us in mind. We just get in the way. Most people,back in the day, never understood Schopenhauer. They still do not.Now, I might never reach the really advanced topics in mathematics,but I would very much like to have built rock solid foundations as regards the discipline, before I call it a day. I think, thanks to  you,I have been acquainted with the last word in philosophy, and on the day I could honestly say to myself that I grasp, at least, the foundations and the fundamentals of mathematics, I think ,then I would have nothing more to do in this wretched world.

However, it goes without saying that on the days when the panic attacks are particularly severe, I just lose my head.

The gorts WANT all sorts of responsibilities. They WANT to have kids. All of it is not social conditioning. There is something deeper, and virulent at play here.

Get well soon.
« Last Edit: August 11, 2021, 01:47:14 am by Kaspar Hauser »
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
-van Gogh.

Creepy Sleepy

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Love and Respect
« Reply #13 on: August 11, 2021, 02:15:31 am »
Quote from: Holden
The gorts WANT all sorts of responsibilities. They WANT to have kids. All of it is not social conditioning. There is something deeper, and virulent, at play here.

This word you use, virulent (bitchy, catty, cruel, hateful, malevolent, malicious, malign, malignant, mean, nasty, spiteful, vicious  (even my own mother admits that many women can be extremely catty)), reminds me of Nietzsche's concept of ressentiment (the French term).

actively poisonous; intensely noxious:

malignant or deadly

violently or spitefully hostile

intensely bitter, spiteful, or malicious: a virulent attack
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I left an essay over at the wordpress blog:  https://xhentric.wordpress.com/2012/12/02/love-and-respect/


It is mostly notes from a book by sociologist Zygmunt Bauman.

I also mention this in Why Take Comedy Seriously? thread:

Quote
Evidently much of what we take to be humor or comedy contains resentments, ill-will, the things I mentioned in Love and Respect, an essay inspired by my reading of Zygmunt Bauman about the three types of relationships he believes are particularly prone to produce “ressentiment” (rancor, malignancy, acrimony, grudges, spite, repugnance):

1. humiliation (denial of dignity)
2. rivalry (status competition)
3. fearful ambivalence

Used wrongly humor can be cruel or distancing.  The kind of humor I am looking for is a kind that would allow me to laugh at myself, especially when it comes to my expectations of myself.

Notice that it was around 2012.   Uncanny?  We did not begin correspondence until 2014 when I was residing in a studio apartment next to a library down by the ocean.  I wish we might Teach Respect like some kind of Ligottian Corporate Horror ...

https://www.youtube.com/watch?v=dBOQjl2tYTA

https://www.youtube.com/watch?v=TkzVQeaHQpg

The full essay:

LOVE and RESPECT

Before getting into this stream of notes, I want to list the 3 types of relationships Zygmunt Bauman believes are particularly prone to produce “ressentiment” (rancor, malignancy, acrinomy, grudges, spite, repugnance):

1. humiliation (denial of dignity)

2. rivalry (status competition)

3. fearful ambivalence

Civilization is based on an irresolvable contradiction.  How can I “love my neighbor as myself” when my neighbor may not show the slightest consideration, or, whom, when it suits him, would not hesitate to injure me, jeer at me, or slander me?

The injunction to love one’s neighbor as oneself is less likely to be obeyed than any other norm.  To love one’s neighbor would be setting oneself up against Nature, turning into an unnatural being too unlike the beasts in the wild.  (Bauman 2008).

We need to love ourselves before we can love others, but we need others to love us in order to love ourselves.  Say what?  Do we even need to love ourselves?  What else could it mean to love oneself but to try to stay alive for better or for worse?  Actually, survival, animal survival, bodily survival, can do without self-love.  Sometimes survival does better without self-love.  According to Zygmunt Bauman in Does Ethics Have A Chance in a World of Consumers?, “self-love may rebel against the continuation of life if we find that life hateful.  Self-love may prod us to reject survival if our life is not up to love’s standards and therefore not worth living.”

Bauman goes on to suggest that what we love when we “love ourselves” is a self fit to be loved.  In order to have self-love, we have to be loved or have the hope of being loved.  Refusal of love, a snub, a rejection, denial of the status of a love-worthy object … all this breeds self-hatred.

We believe we are worthy of love when we are talked to and listened to;  when we are listened to ATTENTIVELY, with an interest that signals the listeners readiness to respond.  We gather then that we are respected.  It is from this “state of being respected by others” that we derive the conclusion that what we think, do, or intend to do matters, that our staying alive makes a difference, that we are worthy of being cared for.  Is it possible for us to value each other’s uniqueness?  Are we able to “value each other for our differences, which enrich the world we jointly inhabit and make it a more fascinating and enjoyable place?”

Bauman admits that this is the bright side of the presence of the Other.  Each of us is the Other’s Other.  What are the obstacles we face in teaching respect?  What are the obstacles to experiencing authentic respect (and not merely coerced respect, known as fear or “learned helplessness”)?

Nietzsche, Scheler, and Bauman himself point to “ressentiment” as a major obstacle to loving the Other as thyself.  Bauman’s work is fascinating, and he wastes no time in exposing us to this term, ressentiment.  While both Nietzsche and Schiler wrote in German, they both used the French term ressentiment, which has a complex meaning not conveyed by the English word, resentment.  When reading the word ressentiment, think of rancor, repugnance, acrimony, grudge, spite, malignancy – or a combination of all these.

rancor – bitter, long-lasting resentment; deep seated ill will

acrimony – bitter or sharp animosity, especially in speech or in writing

malignancy – malevolent (wishing to harm others)

For Nietzsche, ressentiment is what the downcast, the deprived, the discriminated against, and the humiliated feel against their so-called “betters” (the self-proclaimed betters and self-established betters): the wealthy, the powerful, THOSE WHO CLAIM THE RIGHT TO BE RESPECTED TOGETHER WITH THE RIGHT TO DENY (or refute) THEIR (so-called) “INFERIORS” RIGHT TO DIGNITY.

Ressentiment, for Nietzsche, was a mixture of acrimony, envy, and spite.  The deepest cause of ressentiment is the agony of cognitive dissonance, that irresolvable ambivalence:  approving of qualities one does not possess involves disapprobation.   Ressentiment leads not to freedom but to alleviating the pain of one’s own indignity by pulling others down.  Max Scheler’s concept of ressentiment and the role it plays in society is opposite to Nietzsche’s.  It is a feeling that appears among equals.  The middle-class competes to promote themselves and demote others.  Ressentiment results in competition and also, what Thorstein Veblen called ostentatious consumption (that shameless display of one’s own opulence and wealth to humiliate others who don’t have the same resources …) —- (Bauman 2008)

Zygmunt Bauman adds a third instance of ressentiment, the timeless kind, the most unstoppable obstacle to “loving thy neighbor:”  the resentment toward strangers, outsiders – vivid tangible embodiments of the resented and feared fluidity of the real world.  Many of us have served as natural props in the exorcism rituals against evil spirits threatening the orderly lives of the pious.

TV Land broadcasts the mantra, “Life is a hard game for hard people.  In the game of survival, trust, compassion, and mercy are suicidal.  If you are not tougher and less scrupulous than all the others, then you will be done in by them.”

I choose to resist being influenced by the cola-selling, car-culture-promoting TV as well as the big screen.  Instead, I go directly to my own nervous system for the news of the world!  How can I ignore this “sense” that there are some kind of invisible forces at work in my willy-nilly coming across Zygmunt Bauman’s recently published (2008) work?  Early in the text, after quickly bringing to the foreground the concept of ressentiment as an obstacle to loving our neighbors, Zyggie mentions that, in Emmanual Levinas’s first publication in 1930 (a prize winning essay on the role assigned to intuition in Edmund Husserl’s work) was dedicated to the “exegesis and interpretation of the teachings of the founder of modern phenomenology” [Husserl] – his philosophy teacher.

On the authority of a procedure conceived, practiced, and legitimized by Husserl – the phenomenological reduction – Levinas endorsed putting ethics before ontology.  Deploying tools like epoche (detachment, ask yourself, “Am I dreaming this?”) led Levinas not to transcendental subjectivity but to the “impenetrable transcendental otherness of the Other.”  (Bauman 2008)

Is societal coercion necessary to prevent “war of all against all?”  – like in H.G. Wells’s The Island of Dr. Moreau, when each “god-beast” was toting an automatic machine gun, coming down to an **** of bodies dropping?  Would the cessation of social coercion render humans unable to resist the morbid pleasures of their own essentially antisocial instincts?  Look at the way the cities of the liquid-modern consumers are designed.  See the obsession with security?  Do you notice how the poor and hungry serve and even protect the hallucinations of the well-connected?  Civilization is unthinkable without coercion.

Freud presented social coercion as the very essence of civilization:  since the “pleasure principle” (desire for orgasm, or the inborn inclination to laziness) would guide [in Freud’s view, “misguide”] individual conduct toward “the wasteland of asociality or sociopathy,” unless it were constrained and counterbalanced by “power-aided, authority-operated ‘reality principle,’ civilization could not function.”

Now, at the threshold of the modern era, Nature was viewed as the major source of uncertainty that haunted human life.  The ill-will, malice, and uncouth conduct of the neighbors next door [me], or the next street, or beyond the river [the inner-cities], that made people fear and tremble, were classified on the side of Nature, as distinct from the man-made part of the world.   Giorgo Agamben suggests that the constitutive feature of the sovereign state was the “relation of exception,” through which “something is included solely through its exclusion.”

The modern state was about managing human affairs through the exclusion of everything unmanageable and thereby undesirable. Uncertainty and all that caused it – all that was resistant to management, all that evaded categorization, all that was under-defined, all that was category-crossing, all that was ambiguous, and all that was ambivalent – was the major, most toxic pollution of the would-be man-made order that had to be excluded.

Uncertainty had to be excluded!  Honest souls most certainly pose a problem to the managers of the social machinery.  Am I a philosopher-in-chains?

In our society of consumers, where so many seem to act on an urge to replicate the lifestyles currently recommended by the markets and praised by the markets’ hired and voluntary spokespersons, management has ceased to be associated with external coercion.  This urge to replicate social norms is perceived as a manifestation and proof of personal freedom!

Only if one tries to opt out and retreat from the chase – or if one is blackballed and chased away from the chase (a truly horrifying scenario) or refused admission a priori – will one learn just how powerful are the forces that manage the racetrack, guard the entries, and keep the runners running – and only then will one find out how severe is the punishment meted out to the helpless and insubordinate.

Those who opt out of the rat race may easily slip into the ranks of the “collateral victims” of consumerism – the excluded, the outcasts, the underclass.  Most gorts are terrified of slipping through the cracks of their bubble-worlds and “falling by the wayside” or “going down the tubes,” and so they fall prey to the market forces, consuming the products which will grant them approval, thereby avoiding exclusion, abandonment, and loneliness in the Taker Prison.  Those of the silent majority most terrified of the fluidity of the real Natural World, which we are all still the presence of, enclose themselves in elaborate bubbles, always in some kind of bubble, whether it has wheels or flies or floats on the water or is stationary.

Those who get by without such bubbles, without anything but basic shelter and a life of eating from day to day, are flawed consumers and so also become social outcasts.  What enormous psycho-socio-economic coercion is required to promote and instill such “civilizing choices” as the work ethic!  And yet, this variety of the civilizing process arouses little if any dissent, resistance, or rebellion as it represents the obligation to choose as freedom of choice.  Individuals are expected (obliged) to choose to consume more and more, to reinvent their identities according to what they can “afford” or how much debt they are willing to commit themselves to climbing out of.

This is the culture that has farmed us, mis-educated us, and continues to drive us insane.

Since it is the United States that defines the meaning of the new planetary order, designs its shape, and manages, monitors, equips and polices its implementation, the real challenge to Europe is that the sole superpower of the planet fails abominably to lead the planet toward peaceful coexistence and away from imminent disaster.  This superpower may become a prime cause of disasters’ not being averted.   The US military behaves like a fearful giant striking about wildly, further intensifying peoples’ already bitter resentment of the callousness and arrogance with which their needs and ambitions are treated by the high and mighty princes and princesses of the planet. Conflicts and antagonisms multiply and the chances for peaceful cohabitation become ever more remote.

How does one hold onto respect in a world of consumers?  Are we able to respect such depth of thought that Bauman displays in environments contaminated with the mantra of TV-land spreading fear and toughness – where respect is reserved for only brute-militaristic-force or obscene wealth?  Is respect in our world reserved for external power?    While the managers and engineers work on ways to get around why human beings are so reluctant to love their neighbors, or even reluctant to love themselves, I wonder why the concept of neighbors is restricted to human beings, or even human beings around us?  What if we love our bird neighbors and tree neighbors and cloud beings more than we even love ourselves or our bloody human neighbors?  What if we know quite well what side we are on in this monotheistic-God versus Nature battle, in this Scientific-humanistic-State versus Nature battle.  I shake my rattle.  I’m not like trained cattle.  Somebody’s gonna have to kill me before I am farmed into chattel.

Your mind for a job.

Your mind for a hair-dryer.

Your mind for a car.

Your mind for consumption.

What’s YOUR function?

Nature does indeed belong to itself, and something that the managers of the liquid-modern societies of consumers seem to always overlook is the fact that Nature is within us and IS every mineral in our bodies.   You don’t have to love the poisonous snake.  Just don’t tread on it.  Respect it.  Don’t try to subdue it or decide it is too aggressive and hostile and reptilian to deserve your respect.

We shouldn’t confuse respect with fear.

Ressentiment, for Nietzsche, was a mixture of acrimony, envy, and spite. The deepest cause of ressentiment is the agony of cognitive dissonance, that irresolvable ambivalence: approving of qualities one does not possess involves disapprobation.

disapprobation – the state of being disapproved

It has to do with our dignity.

We are supposed to admire qualities we hate?  We’re supposed to admire greed and stupidity?

My lack of interest in most all of the products promoted by the liquid-modern society causes me to be a social outcast.  Perhaps I was even chased out of the race.  Maybe people like me have been wildly resistant to being managed, controlled, or manufactured.  So, what does it mean to be unmanageable?

management := to cause (humans, animals, etc) to submit to one’s control

To manage is to manipulate probabilities.  To manage means to limit the freedom of the managed.

There is a neat division between the managers and the managed, the powerful and the submissive, those behind locked doors “engineering” and those being engineered upon.   The managers-and-managed is intrinsically agonistic; the two sides pursue two opposite purposes and are able to cohabit solely in a conflic-ridden, suspicion-infected, and battle-ready mode.

It is so frustrating to realize that the toughness and hardness of those who would claim to have been made this way “by the streets” may actually be the result of television viewing and TV-Land’s enculturation to HATE.   The world today conspires against trust.  The city in ancient times may have kept danger out, but in our world, the city itself breeds the danger.  The city is a death sentence.

As far as what Nietzsche calls “ressentiment” (see above), the “old” Natives of North America knew how to handle this.  They just got rid of anything anybody wanted.  They didn’t own property and they dressed in rags.  They laid low and let the aristocrats, egalitarians, sycophamts, and assassins all look on them as worthless.  Pirsig examines this in his second and last book, LILA.

It seems that I can only love “my neighbors” in the abstract, only at a distance.  I am finally reading Dostoevsky’s final novel, The Brothers Karamazov; and I’m not ashamed to confess that, reading some sections aloud, I could not keep from crying tears.  There is gut-wrenching scene about the pain, bitterness and anger of a nine year-old boy defending his shattered impoverished father’s “honor” against a heartless pack of mocking teenagers. But that’s not the passage that applies to this thread.  It is a passage from Part Two Book Five Chapter Four … Ivan speaking to his brother Alyosha, who is living in a monastary but is not a monk:

“I must make you one confession,” Ivan said.

“I could never understand how one can love one’s neighbors.  It’s just one’s neighbors, to my mind, that one can’t love, though one might love those at a distance.”

“Beggars should especially never show themselves, but to ask for charity through the newspapers.  One can love one’s neighbors in the abstract, or even at a distance, but at close quarters it’s almost impossible.”

“I think if the devil doesn’t exist, but man has created him, he has created him in his own image and likeness.”

https://www.youtube.com/watch?v=hnABBXAOPOw

https://www.youtube.com/watch?v=bZQcQlOrwAo&t=443s

« Last Edit: August 13, 2021, 01:08:51 pm by Kaspar Hauser »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Holden

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Re: A Re-orientation of Your Mindset
« Reply #14 on: August 12, 2021, 11:28:19 pm »
Not feeling well so could not respond. Hopefully would respond soon.
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
-van Gogh.