This semester I took trigonometry, and really learned nothing. I mean, I did learn about things like the Law of Sines

or that the magnitude of a vector is equal to the square root of secondth-power of the sum of it’s coordinate points

, but I didn’t actually learn what they are.

Towards the final exam, I decided to drop the exam review

as I felt it to be useless for actually learning the material. It wasn’t so much of just a grind of repeating problems, but my own skill level made some of them a bit to far removed from their original context for timely research. Instead, I decided to read the trigonometry section of Basic Mathematics by Serge Lang as to be a starting point of my more conceptual understanding of the subject: and how so much more lucid! Right away I’m told that we’re starting on a plane constructed by a area of a disc denoted by \(\pi)\) and having radius 1, where a angle shares a vertex at P with the disk making them stacked on top of each other. Ah yes the unit cicle you’d reflexivly say back to me but yet I don’t recall it being introduced as a disk, or anything else as intuitive as such: the “Unit Circle”, a too complex name!

Needless to say the exam went well,

but I could’ve done better: to be frank, I only did well by finding that with three hours and twenty five questions, I only have seven minutes to complete a problem (which after accounting for reading the problem, it’s like six). This meant I could not preform how I usually do to solve problems mathematically,

This problem has me preforming operations between vectors, where $y=<-3,-4>, u=<3,-4>$.

   For addition, the sum of two vectors on the plane will be the the tail of the second vector in the sum order starting on the end point of the first vector. (NO CHANGE TO MAG OR DIR)
   ,#+begin_src latex
   \if y=<-4,-4>\and u=<3,-4> \then u-v=<3-4,3-4>=<-1,-8>
   ,#+end_src export
   note that this sum isn't just adding the values together but transforming the vector space into an new one which will have the same magnitude and direction of the original vectors: viz, it's the line perpendicular to the inner angle that is u+v.

   For subtraction, the difference between two vectors on the plane will be the tail of the second vector in the difference order starting on the endpoint of the first but going in the opposite direction of it's original(alone). The vector u-v will be the line perpendicular to the inner angle of the opposite direction.
   ,#+begin_src latex
     \if y=<-4,-4>\and u=<3,-4> \then u-y=<3-(-4),-4-(-4)>=<7,0>
   ,#+end_src latex

   this time in a different order, which would actually make a different vector but symmetrical to the first order example.
    ,#+begin_src latex
     \if y=<-4,-4>\and u=<3,-4> \then y-v=<-4-3, -4-(-4)>=<-7,0>
   ,#+end_src latex

   and finally multiplication, which I've done in the first problem. As stated before, the new vector by the scaler is placed on the tail of multiplicand vector.
   ,#+begin_src latex
   \if u=<3,-4> \then 2*u=<3*2,-4*2>=<6,-8>
   ,#+end_src latex

and instead I had to look through my notes to quickly replicate only the mathematical notation framework for a quick computation; lucky I had ample repetitions of the problems throughout my notes, each in a detailed way, so as to not exceed the time limit. It was actually at this point did I realize what I was actually doing: I was being timed on how fast I could repeat the notional framework to solve a individual problem form a set of problems that I’ve done before repeatably.

Is this just me? Am I the one which fails to see the understanding behind it all? Apparently not as illustrated by Paul Lockhart’s A Mathematician’s Lament. I won’t go into it here but I’d recommend it if you feel the same way about how mathematics (usually lower level) is taught.

So I started to review Trig over the summer

using mainly Serge Lang’s book and others like Gelfand’s Trigonometry & Geometry and my course’s textbook. It’s been really fun due to org-mode within Emacs finally having the org-latex-preview module now, so if I ever get lost in the latex sauce, I can just visually see what’s going on as if I wrote it.

Recently I’ve finished Pragmatism by Williams James.

It’s a small but dense book which in my experience as having a intermediate knowledge of philosophy tends towards needing a second read: although I’d love to I don’t have the time for that unfortunately (only as much to review section two as you’ll see later), so YouTube it is! Thankfully Dr. Sadler exist and is so vast in knowledge that he has a playlist devoted to going over it! So that’s my reading hour for now.

But anyways, while I was watching the actual lecture two of the Pragmatic Theory of Truth lecture by Dr. Sadler, I’ve realized a connection between how I came upon a method of solving problems and the meaning behind Pragmatism.

While doing a problem which put me in a very directed space

requiring me to actually learn the concept behind the problem to solve it, I realized a way to conceptualize the situation of not knowing what you don’t know. Leading with example, if I want to solve this problem but yet I lack the skill to do so, how would I put myself in a position as to know intuitively how to solve the problem? In ratio with what Charles Peirce said in How to Make our Ideas Clear where it’s quoted in William Jame’s pragmatism as an explanation of what Pragmatism is, “To attain perfect clearness in our thoughts of an object, then, we need only consider what conceivable effects of a practical kind the object may involve—what sensations we are to expect from it, and what reactions we must prepare.”, it can be seen that the point of understanding intuitively how to solve a problem would mean knowing the practical effects of what makes it ‘is’ and the kind of contextual reaction it might produce due to it being in some use. I think this type of viewing it a problem does require maturity as you would have to have enough background knowledge to figure the practical reasons for it’s existence in relation to other objects or ’truths’ in it’s domain.

I’ve also just been viewing mathematics much more visually as of late,

which has helped me in practicing a more reading & writing approach to mathematics: and that’s another thing! Mathematics is much then just writing out computations; it’s reading other works and writing out your own as you introduce your objects for their effects, arrangements, etc.

In totally different news, I finally got to take our bikes out!

I seriously forgot how fun a bike is. It made me remember being a kid and being almost always on mine during the summer when there wasn’t anything else to do.