Philosophers in Rags :: We dive deeper, and come up muddier.

OK, yes, I do love calculators and computers, but ...

I also do plenty of sketching, scratching my head until it bleeds, feeling bewildered.

One thing Holden and I definitely have in common is that in our youth we were kind of shocked by the idea of the square root of negative one.  I still find that a bit disturbing. 

In the 1500's some Italian mathematician used these to solve a problem, and since they cancelled out, and were not part of the answer, he figure it was quite valid.

Sorry, it's after midnight.  I will try to stay focused here.  If you are lurking about, Raul, you do not have to skip this post just because it mentions "trigonometry".  This can also be considered social commentary on the state of mass education and our reluctance to question authority when it comes to mathematical matters.

Some background to what has me typing on this obscure message board after midnight.    I wrote up some small program for converting rectangular coordinates to polar coordinates and visa versa.  After all the annoying syntax errors were hunted down, and the infinite loops removed, when it compiled and ran and spit out one result after another, I was left feeling disappointed.  Emptiness ... the kind of dissatisfaction Schopenhauer predicts I will experience looking for deeper insight behind algebra and geometry and trigonometry - calculations and computations ... empty ... spent ... disappointed in the lack of deeper insight into what it is the numerical approximations represent.

Numerical approximations ... that's all ... Although I added the knowledge of which quadrant the point was in, this still involved finding the arctangent ...

So how do the calculators and computers come up with the arctangent?  They aren't magic oracles.  The calculators "compute approximations using finite numbers of infinite sums" behind the scenes ... Where is the intuitive grasp in that?!

This has nothing to do with intuitive geometric understanding! 

So, something else to consider.  Another trail:

See How do you find the sine, cosine, and tangent inverse without a calculator?

If you make it to the video, yes, Divine Proportions is available at Library Genesis.

We're going to need an eternity to get anywhere, so why bother?

Well, if you download the book, on page 11 I notice the very wall I have repeatedly hit is discussed.

The section title sounds Schopenhauerian:  "1.3: Why classical trigonometry is hard"

For centuries students have struggled to master angles, trigonometric functions and their many intricate relations. Those who learn how to apply the formulas correctly often don’t know why they are true. Such difficulties are to an extent the natural reflection of an underlying ambiguity at the heart of classical trigonometry. This manifests itself in a number of ways, but can be boiled down to the single critical question:

What precisely is an angle??

Hooo Ha!     Great question!

That sounds like something Holden would say!  He does not want a standard textbook definition. 

He goes on to say that the problem is that defining an angle correctly requires calculus.  This is precisely the "angle" taken [pun intended] by the young graduate student author of BURN MATH CLASS (Jason Wilkes).  Wilkes had suggested we learn calculus before leaning about the tangent function since it turns out to be the slope ... but ... around and around ...

Let’s clarify the point with a simple example. The rectangle ABCD in Figure 1.5 [see Library Genesis] has side lengths |A, B| = 2 and |B, C| = 1. What is the angle θ between the lines AB and AC in degrees to four decimal places??

And this is how I know this Wildberger is a gortbuster.   He is pointing out that elementary as well as advanced geometry texts are reluctant to make this supposedly basic idea clear.  Then the student is made to feel "stupid" or "lazy" for not having a grasp of it ... so we resort to memorizing SOCATOHA and all these algorithms ... and, of course, the calculator and tables ... and numerical approximations.

Without tables, a calculator or calculus, a student has difficulty in answering this question, because the usual definition of an angle is not precise enough to show how to calculate it. But how can one claim understanding of a mathematical concept without being able to compute it in simple situations? If the notion of an angle θ cannot be made completely clear from the beginning, it cannot be fundamental.

He goes on to say, "If the foundations of a building are askew, the entire structure is compromised."

Quite Holdenesque!

In the introduction he states, "Mathematics is a conservative discipline, and it is not easy to acknowledge that traditional thinking might involve elements of misunderstanding."

And, of course, eventually we have to sleep.  We can only focus on so much at a time.   

At least here is someone facing the abyss.  It is not just me who has difficulty with this.  It is a universal phenomenon.  What is an angle, anyway?  Why is it so difficult to get the arctangent without calculators - and why do the calculators resort to approximations using infinite sums?

Students are constantly given examples that deal essentially with 90 ◦ /60 ◦ /30 ◦ or 90 ◦ /45 ◦ /45 ◦ triangles, since these are largely the only ones for which they can make unassisted calculations.

Exactly, those are so crystal clear!  But stray from these and one is left feeling mentally impotent ... or, worse, like a FRAUD!   

Small wonder that the trigonometric functions cos θ, sin θ and tan θ and their inverse functions cause students such difficulties. Although pictures of unit circles and ratios of lengths are used to ‘define’ these in elementary courses, it is difficult to understand them correctly without calculus.

And then he introduces "Rational Trigonometry"  ....

1.4 Why rational trigonometry is easier

I sure don't want to get sidetracked, but it really does irk me that we must rely on approximating infinite sums (which is what the calculators do) or even that our grandfathers relied on tables ...

How is it we lack clarity on something supposedly so fundamental as this angle that, as Schopenhuer would say, is clearly apprehended in every way by our faculties of perception?  When it comes to expressing it in terms of numbers and measurement, wow ... there seems to be no exact representation, but only approximations.   Oh well, the truth is out of the bag.  Math is not an exact science after all, but an approximate one. 

Maybe it is not so fundamental after all ... maybe what is presented as elementary depends on much more advanced mathematics that are left out of the explanation because it is not known (and taken on faith) ...

Hence, those who are most intellectually honest will call the bluff and admit there is mass confusion throughout the entire edifice in universities around the globe.

The most honest students, like Holden, will be mocked and accused of not studying (memorizing) enough, and yet he has realized that teacher may be lacking in understanding himself and that there is a great deal of monkey business having to do with "acting as if" everything were crystal clear, when in fact, some of the so-called basic ideas, like angles, are glossed over.

Have you ever noticed that when you set a calculator's mode to EXACT, as opposed to APPROXIMATE, and enter arctangent(-1/3) or most any other value, it spits out the very question you asked?

It is unable to give an exact answer except in terms of the expression, arctangent(y/x) ...

It can only approximate.  We can only approximate.

The trains have to run on time.  Most do not have the leisure to demand more understanding.

So, what is the point of our educational institutions if there is no time to understand things?

All the questions involve 30-60-90 angles or 45-45-90 angles.  It makes us feel smart.

We are deluded ... Only those with intellectual honesty allow themselves to face this confusion squarely. 

Confucius said, "Rest in confusion".

Time for some math nightmares.   

I am not the only one who thinks about such things.  The Internet sure does have many benefits if you can hide from the advertz.

I had a situation yesterday that I would have solved by whipping out the DM15, but I had forgotten it, so manual calculation was the only option. I must though humbly admit that I utterly failed. I wonder how I should have done.

We had a BBQ at a friend, who was constructing a new shed. The roof was not yet finished and we starting talking about roofing materials, but then came into that depending on the roof angle, there is a need for an underlayer. And then we wanted to know the roof angle...

We got as far as seeing it as a right angled triangle were the base is 3 and the height is 1, the side thus sqrt(10). Thus we only needed to calculate arctan(1/3) to know the roof angle. (I later put this to my neighbour who was a carpenter, he knew by heart several angles that you have when triangle height is 1 and base is an integer).

No calculators at the house. No smartphones. Nobody wanted to go in to boot up a computer.

Everyone pondered over the evening about how to solve this (series expansions: but who remembered the formula, integrals: something with 1 over a square root of something).

In the end, several calculations leading to nowhere, we had to give up and ring one of the kids who opened a Wolfram Alpha tab (what a computation overkill).

I don't know the arc trig or the regular trig expansions by heart, was there a way this could be solved logically with pre-calculus math?

How ironic, a search for "arctangent without a calculator" led me to the freakin ...
Museum of HP Calculators !

Am I using the word ironic correctly? 

Anyway, my apologies, but I do tend to use this board as a memory bank to compensate for my lack of memory ... I may forget when I wake up feeling groggy ... and shot out.   So, a few more links I want to check out.  No need to investigate at this point ... unless you you want to, of course.  I mean, you probably will want to skip it.  I really want to look at it when I am less shot out tired.  I think this may help me break through a mental block that I've been frustrated about.

I would like to be able to figure certain things out even were I prevented access to electronic devices.  It would give me great satisfaction.  When I say without calculator, I also include tables of values, protractors, rulers, etc ... Just pencil and paper.   (or sand?)   

From Solving the ArcTan of an angle (Radians) by hand?

Finding the exact arctangent of other values would be much more complicated, though you ought to be able to estimate the arctangent by picturing it. For example, it's easy to estimate that arctan(1/3) should be about 15 or 20 degrees, just by picturing a line with slope 1/3.

Edit: By the way, if you really want to compute arctangents by hand, one possible method is to use the identity arctan(x)=2*arctan(x/(1+√(1+x^2))),  which follows from the double-angle formula for tangent. The quantity in parentheses on the right is less than x/2, so you can iterate this identity to find a sequence of smaller and smaller angles whose arctangents you want to figure out. (Note that you need to be able to compute square roots by hand.)

Once your angle gets small enough, the approximation arctan(x)≈x

Its 288.I used LCM.Its the only method I know.
The question may have appeared as baby stuff to you,but bear with me-noblesse oblige.

What was the deja vous about,if I may ask?

It was a feeling of remembering this very question ... and finally having a use for least common multiple.  LCM sometimes comes up when writing code for the Chinese Remainder Theorem. 

Maybe I tend to make things more complicated than necessary.  Initially I wanted to apply something from number theory, and I found it was best to use brute force sense, just thinking about your question without any algorithms .... You know, in a freestyle arithmetic manner.

Yes, the deja vous was the feeling of remembering that moment as having been lived before ... maybe I had gone without sleep for too long.

As for the Mystic River, I will get back to that.  I intend to proceed with this inquiry into finding arctangents without a calculator BEFORE I eat any kind of food ...

____________________________________________________________
Do you remember posting the following?

Mathematical reasoning, Schopenhauer argued, is fundamentally different from ordinary logical or syllogistic reasoning in being based on intuition or construction, not on deduction from premises to conclusion; and accordingly Schopenhauer advocated the revision of Euclid, who, he believed, mixes the genuinely geometrical with the spurious logical proof. Schopenhauer even offered specimens of the right kind of proof. While the idea is interesting, I feel daunted by complexity of the problem S. has raised.Do you think his work on the logical foundations of mathematics has any value to-day?

Well, this Norman Wildberger (if interested, he has a series of videos on foundations mathematics, WildTrig series) says we have been "following pictures" which can lead to flawed reasoning.  He says Euclid's geometry is full of this kind of thing.  Isn't this what Schopenhauer was stating in no uncertain terms, even as "the artist's philosopher"?

Well, I do intend to  give Wildberger some of my attention. 

In the current framework, the notions of distance and angle are not algebraic.  They are "transcendental", meaning they go beyond algebraic methods.  Norman Wildberger's approch is to stick with Descarte's reformulation of Euclidean geometry. 

At the end of video zero in the series, he has written on a board:  For a purely algebraic approach, we need to replace distance and angle with new concepts.

He insists that everything be done completely 100% algebraically.

Restating all of trigonometry ... reformulating all of geometry.

This new approach aims to unify arithmetic, geometry, and algebra.  He wants to bring these subjects together in Rational Trigonometry.

It does sound quite interesting.  I only stumbled upon it trying to figure out how to find the arctangent without a calculator.

Do you think this is something to keep an eye on, Holden?

Señor Hentrich,
Let me express my gratitude to you and Holden for taking the time to read my comments. You cover so much and many deep topics in your blog. I apologize for my language, but sometimes when I am sitting on the toilet relieving myself and expelling that smelly stuff, I ask myself why the Big Bang, only for human beings to ****,take a ****, go to bed, brush my teeth, drink water or whatever and swallow food? Why do we exist anyway? We are born or made to be born into this hell and much worse people bring more babies to suffer this nightmare. Do you think that people who do not procreate live longer? If this is the case the punishment is much bloodier. And inside our our heads hell is much more horrible. I am starting to feel less empathy for my fellow human beings.I feel a blunting. Life is a lethal weapon used against us. Here sometimes when they ask me if I believe in God, I tell people that if there is a God, that selfish guy in the sky must have washed his/her/its hands of humanity. You see, here in Paraguay every December 8, Catholic pilgrims go to the Caacupé, 50 kms from Asuncion to worship the Virgin Mary and pay their promises.Of course the President and his cronies are there to raise their hands and utter empty words infront of the bishops and priests. It is a good show, nice entertainment. The Stockholm Syndrome in full at broadaylight. Thank you again and please be safe. Raúl 
 

Thanks for your response senor Raul.I think you would find the following of some use-The sphere of consciousness shrinks in action; no one who acts can lay claim to the universal, for to act is to cling to the properties of being at the expense of being itself, to a form of reality to reality’s detriment. The degree of our liberation is measured by the quantity of undertakings undertakings from which we are emancipated, as by our capacity to convert any object into a non-object. But it is meaningless to speak of liberation apropos of a hurried humanity which has forgotten that we cannot reconquer life nor revel in it without having first abolished it.
   We breathe too fast to be able to grasp things in themselves or to expose their fragility. Our panting postulates and distorts them, creates and disfigures them, and binds us to them. I bestir myself, therefore I emit a world as suspect as my speculation which justifies it; I espouse movement, which changes me into a generator of being, into an artisan of fictions, while my cosmogonie verve makes me forget that, led on by the whirlwind of acts, I am nothing but an acolyte of time, an agent of decrepit universes.
   Gorged on sensations and on their corollary—becoming, we are “undelivered” by inclination and by principle, sentenced by choice, stricken by the fever of the visible, rummaging in surface enigmas of a piece with our bewilderment and our trepidation.
   If we would regain our freedom, we must shake off the burden of sensation, no longer react to the world by our senses, break our bonds.
Cioran

WARNING:  Please feel free to ignore this.  It is just a curiosity.
____________________________________________________________________________

OK, so, if I am going back to the start, I can't assume that I will be able to stay entirely focused on my self-imposed curriculum.  Since I will eventually be re-studying Geometry, a subject that I do not particularly enjoy, it's not going to do me any harm to keep the work of Wildberger (Rational Trigonometry, c. 2005) in mind.  It is not my intention to cause myself or anyone else confusion.  We might gain insight into what Schopenhauer was talking about when he was critical of Euclid's methods of proof.  I suspect that Wildberger has taken on a project that Schopenhauer might have pursued if he were more mathematically inclined, since he seems to have hinted at this.

Rational Trigonometry provides the mathematical foundation for a dynamic and elegant new approach to teaching trigonometry and geometry.  Divine Proportions revolutionizes trigonometry, re-evaluates and expands Euclidean geometry, and gives a simpler and more natural approach to many practical geometric problems.   This new theory unites the three core areas of mathematics–geometry, number theory and algebra–and expels analysis and infinite processes from the foundations of the subject.

This book finally addresses the failure of modern geometry to win the minds of young people , and provides the mathematical foundation for a dynamic and elegant new approach to teaching trigonometry and geometry.  [Wildberger 2005]

I am not trying to bypass a formal review of Geometry (Euclidean).  I might be getting to this over the summer.   Still, it would be interesting for me to consider Wildberger's ideas during this "going back to the start" process.  It is possible to see things both ways so as to better understand what Wildberger is actually replacing.  He says he can make the measurement of angles  unnecessary, that Rational Trigonometry can algebraize geometry as it is not based on the distance of the sides of a right triangle, but on the squares of those distances.


Here I will place a link to  the Rational Trigonometry Site which has links to preface, contents, introduction, Chapter 1, and Chapter 27.

Or, you can find Divine Proportions: Rational Trigonometry to Universal Geometry.

I have been searching for a hardback copy, but they are just too expensive.  At least we have the pdf version or else I think I would not be able to resist.

Take a deep breath.  This could be a ground-breaking summer.  I will be going over classical geometry and classical trigonometry anyway, so I want to definitely get through Wildberger's book as well (over the summer) during this "re-education campaign".

I am wondering if you (Holden) might want to see if you can help me articulate precisely what Schopenhauer's criticism of Euclid's methods were, and to see if perhaps Wildberger's Rational Trigonometry might serve as a clue to help us make some headway in answering your questions:

Mathematical reasoning, Schopenhauer argued, is fundamentally different from ordinary logical or syllogistic reasoning in being based on intuition or construction, not on deduction from premises to conclusion; and accordingly Schopenhauer advocated the revision of Euclid, who, he believed, mixes the genuinely geometrical with the spurious logical proof. Schopenhauer even offered specimens of the right kind of proof. While the idea is interesting, I feel daunted by complexity of the problem S. has raised.Do you think his work on the logical foundations of mathematics has any value to-day?

I think that Wildberger has answered the call, as far as a revision goes.   I am interested in this, and I think Wildberger is worth looking into alongside a thorough study of the classical methods.

Since Rational Trigonometry is a revision of the classical methods handed down to us by Euclid, I would think it is best to become intimate with what is being revised.

I am not sure what order to suggest.  Everything is rather cyclic.   They teach classical trigonometry before calculus, but classical trigonometry uses transcendentally defined functions (which require calculus to create the tables as well as program the calculators to come up with the angles).   So, maybe it is best I take another look at classical geometry and classical trigonometry before focusing too much on Wildberger's revisions, which may never be formally implemented on a mass scale in high schools or colleges.

See A brief introduction to Rational Trigonometry


The first video of Wildberger's WildTrig series is Invitation to Geometry

A little after 3 minutes into the video, which is very interesting, by the way, Wildberger says that Euclid's methods have been in use for over 2000 years, but that a little over 100 years ago, people started noticing some flaws, some small cracks ... Did not Schopenhauer see these flaws?   Maybe he just wasn't able to articulate on this particular matter as well as Wildberger has.  It's a controversial and radical re-evaluation of Euclidean geometry, and i suspect Schopenhauer wanted to concentrate his energies on mroality and matters of the heart. 

 By 4 minutes into the video Wildberger explains the problems you can get into by relying on pictures.  This is why he wants geometry and trigonometry based totally in algebra.  It's kind of exciting, and I do sense that Schopenhauer would approve.  What do you think, Holden?

Note: see WWRv1: references to Euclid (p. 52, 55, 63, 438)
References to Mathematics WWRv1 (50, 54, 81, 85, 95-6, 121, 144, 189, 222, 247, 342, 346, 431, 449, 465, 469, 480)

I am not so sure where I stand in this controversy over the foundations of mathematics, but I do lean strongly in the direction of a set theory approach.   Still, it doesn't hurt to keep Wildberger's "Rational Trigonometry" in mind as I revisit algebra, geometry, and trigonometry and the way it was presented in the Dolciani series during the set-theory based "new math" experiment which was later overthrown in the math wars reform.

Most of the problems with the foundational aspects arise from mathematicians' erroneous belief that they properly understand the content of public school and high school mathematics, and that further clarification and codification is largely unnecessary. Most (but not all) of the difficulties of Set Theory arise from the insistence that there exist `infinite sets', and that it is the job of mathematics to study them and use them.

Set theory as presented to young people simply doesn't make sense, and the resultant approach to real numbers is in fact a joke! You heard it correctly - and I will try to explain shortly. The point here is that these logically dubious topics are slipped into the curriculum in an off-hand way when students are already overworked and awed by all the other material before them. There is not the time to ruminate and discuss the uncertainties of generations gone by. With a slick enough presentation, the whole thing goes down just like any other of the subjects they are struggling to learn. From then on till their retirement years, mathematicians have a busy schedule ahead of them, ensuring that few get around to critically examining the subject matter of their student days.

[italics and bold-faced mine, of course]

This is worth repeating: From then on till their retirement years, mathematicians have a busy schedule ahead of them, ensuring that few get around to critically examining the subject matter of their student days.

Holden, do you remember when you suggested that my mental and emotional breakdown at age 17 might have been some kind of weird blessing?   You concluded that, if my academic life proceeded without that catastrophic nervous breakdown (and eventual suicide attempt, incarceration, happy days of humble employment with the State Park service), I might have gone on to be one of these professors with the busy schedule who would never get around to critically examining the subject matter of my school days.

You show great insight into the great paradox between appearance and reality.

This insight of your gives me a totally different perspective on the "narrative" of how my life has unfolded, where I actually treasure the days of having "nothing to do, nowhere to go, and no one to know" rather than becoming distraught over being such a "nobody" or "nonentity".

You have also suggested that there is the possibility that mathematicians do not understand math because what they think they understand is based on a fundamental falsehoods.  These are controversial thoughts to be entertaining in your head.

Maybe there is not such a problem with set theory, but just that there's a limit to formal reasoning.

Do those students learning `infinite set theory' for the first time wade through The Principia? Of course not, that would be too much work for them and their teachers, and would dull that pleasant sense of superiority they feel from having finally `understood the infinite'.

it would have dulled that pleasant sense of superiority they feel from having finally `understood the infinite'

funny   

This experiment I am engaging in - of revisiting the algebra, geometry, trigonometry, and analysis, presented to me in my school days (with set-theoretical formality) - can certainly be considered as evidence that I have no such "pleasant sense of superiority".

I'm just trying to get a better grasp of the fundamentals before proceeding to get back into anything more advanced than calculus or linear algebra.

And why am I at liberty to do this?  Because I have been blessed with undesirable qualities.  I am not desired by any woman or any employer.   This curse is now my blessing!

Let's flip the script on academia and actually care about the subject matter as opposed to building up a gort-pleasing resume!

For a critique on Wildberger's radical attempt to revision Euclid, see Dirty Rotten Infinite Sets and the Foundations of Math

You know, as fascinating as the methods of Wildberger's Rational Trigonometry are (and I appreciate how it eliminates the need to find angles using the transcendentals sine, cosine, tangent), I think I'll focus my attention on solving problems in algebra, classical geometry, and classical trigonometry since I prefer not to totally out on a limb.  I mean, it's one thing for an established mathematician to create such a revision in hopes of eliminating the dependency of trigonometry on transcendentals (infinite series, calculus) and classical geometry, but for us outcasts of society, I think we might be better off taking a more traditional approach to critically examining the subject matter of our school days.

Just for the record, Herr Wildberger is at it again, this time with "Algebraic Calculus" (coming soon to a ZooTube theater near you in 2018!)

If you are curious ...

algebraic calculus

Another one for you -I call it the Mystic River:
A girl leaves home with x flowers,goes to the bank of a river.On the bank,there are 4 temples in a row.She dips all the x flowers into river,the number of flowers double.Then she enters the 1st place of worship,offers y flowers to the deity.She dips the remaining flowers into the river &again the flowers double.She goes to the second temple& offers y flowers again.She dips the remaining flowers into the river and again number of flowers doubles.She goes to the 3rd temple &offers y flowers to the deity.She dips the remaining flowers into the river &again flowers double.She goes to the fourth temple and offers y flowers to the deity.

Now she is left with no flowers in hand.

If she leaves home with 30 flowers,the number of flowers she offers each deity is?

1.  Start with x flowers.  She dips all the x flowers into the river. The number of flowers doubles.

---> 2*x

2. Then she enters the first temple, offers y flowers to the deity.

---> 2*x - y

3. She dips the remaining flowers into the river, and again the number of flowers doubles.

---> 2*(2*x - y) = 4*x - 2*y

4. She goes to the second temple, offers y flowers to the deity.

---> 4*x - 2*y - y = 4*x - 3*y

5. She dips the remaining flowers into the river, and again the number of flowers doubles.
---> 2*(4*x - 3*y) = 8*x - 6*y

6. She goes to the third temple, offers y flowers to the deity.

---> 8*x - 6*y - y = 8*x - 7*y

7.  She dips the remaining flowers into the river, and again the number of flowers doubles.

---> 2*(8*x - 7*y) = 16*x - 14*y

8. She goes to the fourth temple, offers y flowers to the deity.

---> 16*x - 14*y - y = 16*x - 15*y

9.  Now she is left with no flowers in hand.

---> 16*x - 15*y = 0

------------------------------------------------------

If she leaves home with 30 flowers,the number of flowers she offers each deity is?

16*(30) - 15*y = 0
480 = 15*y
y = 480/15 = 32