{∅, {∅}, {∅, {∅}}} : Rage Against the Meat Grinder

If, therefore, we want to have abstract knowledge of space-relations, we must first translate them into time-relations, that is, numbers. For this reason, arithmetic alone, and not geometry, is the universal theory of quantity, and geometry must be translated into arithmetic if it is to be communicable, precisely definite, and applicable in practice.   It is true that a spatial relation as such may also be thought in the abstract, for example "The sine increases with the angle," but if the quantity of this relation is to be stated, number is required. This necessity for space with its three dimensions to be translated into time with only one dimension, if we wish to have an abstract knowledge (i.e., a rational knowledge, and no mere intuition or perception) of space-relations--this necessity it is that makes mathematics so difficult.

Holden, remember when you asked me if I look at a curve in a book before drawing it?  I will explain the general scenario.  What I quoted from Schopenhauer [WWRv1] is very relevant.

Suppose I am asked to find the area of the inner loop of the polar curve r = 1 + 2*sin(theta), where theta is the angle in polar coordinate system.

What I generally do first is use a computer algebra system such as Sage or even gnuplot (or a graphing calculator) to graph the polar curve from theta = 0 to theta = 2*pi.

What I have found helpful is to display a table evaluating r as a function of theta, and then using the table to plot the points on a graph I draw by hand, a polar grid where the circles represent distance from the pole.  I plot the points and connect the dots.

Were I to attempt to just sketch the curve on the fly, this is when the curve looks skewed.  This has everything to do with what Schopenhauer wrote nearly 200 years ago.  As he was fond of saying, we are living in the self-same world.  Even with the assistance of technology, we are still dealing with the fundamental nature of the relationship between time and space, which is manifest in the relation between algebra/arithmetic and geometry, the relationship between numbers and geometric objects.

"geometry must be translated into arithmetic if it is to be communicable" ~ S

In other words, I can't just mechanically apply the formula for finding the area of a polar curve, but have to first analyze the graph to decide upon the appropriate interval, the limits of integration.

It's kind of fascinating to witness the exact nature of what Schopenhauer is talking about.  You know, for many years I did not pay much attention to his statements about arithmetic and geometry, and their direct relation to his statements about time and space.  I have been mostly influenced by his thoughts on the denial of the will, the wickedness of optimism, etc.

Schopenhauer has been called "the artist's philosopher," but what he wrote about mathematics is quite profound.  I am currently quite enthusiastic about this bridge between algebra and geometry that used to be called "analytic geometry".

Where I used to race into the mechanical operations involving calculus, I am finally beginning to take my time and really analyze the situation slowly and calmly, creating graphs, tables, taking advantage of computer algebra systems graphing capabilities, but also jotting down the table of values next to a hand sketched graph.  Maybe I have finally matured enough to take my time with each exercise I engage with.

PS:  Even though seeing the computer generated graph and finding the angle theta where r is zero, I could also skip all the visualization and find this by algebraic means, what Schopenhauer calls "with arithmetic".

In fact, I prefer analyzing it this way with pencil and paper before proceeding with the visualization process (geometric representation).

You just ask yourself, "When is r = 0?"

r = 0 when 1 + 2*sin(theta) = 0

... when 2*sin(theta) = -1

... when sin(theta) = -1/2

theta = arcsin(-1/2)

This occurs when theta = pi + pi/3 = 7*pi/6

Also when theta = -pi/6

Note that 2*pi + 7*pi/6 = 4*pi/6 + 7*pi/6 = 11*pi/6, which is the same angle as -pi/6

At this point, as is the scenario frequently, one could find oneself drawing a graph of sin on a rectangular coordinate system to note where sin(x) = -1/2

The depth and the degree of analysis depends on just how much one really wants to grasp the described situation that the curve represents.

Remember that all this analysis takes place even before applying any calculus to determine the area of the "inner loop".  When working in haste, this stage might be done abruptly, whereas as once the definite integral is set up, one considers the calculus involved as the "real part" of the problem solving activity.   I confess that this is how my brain attacked such problems for years.

For whatever reasons I am a little calmer now and take some pleasure in the analysis stage that appears to be boundless, limited only by something nudging you to "make progress", "forge ahead."

God forbid one should go off on a tangent!

How does one balance this tension?  I mean, on the one hand, you want to understand things deeply, and on the other hand, there is a kind of pressure to not spend "too much time" on the task at hand.

So, when looking at r = 1 + 2*sin(theta), it helps to look at this with both polar AND rectangular coordinates, where r(theta) = 1+2*sin(theta) translates to f(x) = 1 + 2*sin(x).  This shows visually what I initially found algebraically.

It seems to be a matter of "extracting intelligence" by whatever means necessary, and there are no laws dictating precisely how to go about extracting such intelligence.  Maybe I was the type of student who wanted a fool-proof step by step procedural algorithm so I could compute and calculate in a mechanical manner.  Now that I am no longer in an environment (i.e., "school") where memorization and stressful fast calculations were demanded (exams), and since I no longer despair about "transforming from a janitor into a 'scientist' ", I can at long last enjoy the process of gaining intelligence through exploration, inspection, and in depth analysis.

At this juncture I come to realize that I do not have to choose between polar or rectangular coordinates, but am free to use both simultaneously since each gives its own valuable information in the "intelligence gathering" phase of the "problem solving" process.

It only took me over 20 years to realize this, although, 20 years ago, I actually was not even stretching my mind enough to consider polar coordinates and rectangular coordinates being used in harmony.  I told you I was a slow learner. It's as though not even a great teacher from a distant {exotic) land could have taught me this.  I had to learn it in the spirit of exploration, in a state of mind attained through what the industrious refer to as "goofing off."

Speaking of exotic lands, with all the commotion over politics and religion all over the world, it is kind of revealing that like-minded individuals from diverse cultures can discuss mathematical truths as autonomous subjects, thereby transcending ego and ethnicity.

Might a shared fascination with the art of mathematics demolish the socially constructed masks dr-ape-d over our subjectivities? 

Contemplation of sine and cosine occur in an inner realm that transcends the boundaries of states and nations ... and is truly "apolitical".

Returning to the very beginnings of The WWRv1 in search of anything the great genius Schopenhauer had to say about mathematics has had an effect on my approach.  No longer do I feel like an old Steppenwolf struggling with school-boy mathematics exercises, but embrace each set of problems and each textbook I engage with in the spirit of the ancient scholars who who studied with in an exploratory manner, and not simply to have gold stars to put on their job resume.

I am not a man who prays to any gods, but I find myself saying as like Sioux medicine man Black Elk prayed, "Make me to walk in a sacred manner."

I want to explore mathematics in a sacred manner.  Schopenhauer pointed out exactly why mathematics can be difficult, and so we might embrace these difficulties as inherent in the limits we face in translating geometrical representations into algebraic representations.  May we proceed with humility and patience rather than arrogance and frustration.