Motivation!

So, for the past few days, I've been trying to overcome a combination of repeated interruptions by various errands and chores, and a strong sense of "Pervasive Drive for Autonomy" where apparently I cannot do something because someone is putting a demand on me, and mentally, I'm not prepared to fulfill people's demands -- even if that person making the demands is me, and the demands are work on one of those things you know you want to work on, darn it!

This post isn't about my motivation to do things, though.  This morning, as I was settling in to finally start a project, deciding to watch a couple of videos before actually starting it, thinking I'm not even sure if I'm going to write a blogpost (or even just post something) -- because I'm afraid that a blogpost takes away precious energy needed to work on projects -- and besides, I'm not addicted to videos, I can stop any time! -- I came across The Problem With Math Textbooks on Youtube Shorts that resonated greatly with me.

The TL;DW (too long, didn't watch -- wait, isn't this a short video? -- well, maybe it won't be there by the time you internet archaeologists get to this post) summary:

Pure Math textbooks delve right into the axioms, which is a problem, because students are left thinking that we could just pluck axioms from thin air, giving us infinite possibilities.  Where do these axioms come from?  We need to describe the motivation that led to these axioms!

This is, indeed, an approach I've been wanting to take with mathematics for years.  When I took a "Physics for Scientists and Engineers" class as an undergrad, my room mate was explaining that he was taking the Physics class that didn't use calculus -- and thus, the math was significantly harder! -- and this led me to the conclusion that both physics and calculus would benefit if they're taught as physics gives birth to calculus -- or, perhaps, rather, as both are given life as twins!

But I was initially at a loss as to how to find motivation for everything else -- when I realized I had answered this for myself years ago too!  The motivation comes from the history of mathematics.

• Euclid's Geometry was motivated by an attempt to standardize the measurement of the Earth (hence the geo of geometry!) -- and its alternatives were motivated by attempts to prove that Euclid's axioms were the only alternative, only later to be discovered that they have their own physical analogs.
• Calculus was motivated by physics, and each refinement to the idea by mathematicians like Euler, Riemann, Gauss, and Lebesgue, were done to address philosophical concerns, and to refine the techniques.
• Modern Abstract Algebra was motivated by solving the classical Greek problems of trisecting the circle, doubling the cube, and squaring the circle, using only a straight-edge and compass.

As for everything else?  Well ... I'm not sure if I can tell you ... because I'm not as confident on the history as I'd like to be.  The problem the field of Mathematics has is that, as the math becomes more refined and purified, the older techniques are jettisoned -- and little to no effort is put into understanding the history!  A good example of this is in Calculus itself -- everyone who goes through the mathematics classes know about epsilon and delta proofs (My blog's nom de plume "Epsilon Given" is taken from this!) -- but far fewer know about Newton's fluxions or Euler's infinite and infinitismal numbers, among other approaches to the subject.

Indeed, my own understanding of the history of mathematics is a mixture of "The History and Philosophy of Math" class I took in my first year of college, and being self-taught.

To this end, I have spent some time trying to collect older mathematical works by early mathematicians, with the hope of exploring the more "intuitive yet unrefined" approaches to mathematics.  I have Euler's Elements, a work or two by Archimedes and another mathematician I can't remember, Euler's "Introduction to Algebra", a book of Leibnitz's works on calculus, and Sir Isaac Newton's work on calculus, A Treatise of the Method of Fluxions and Infinite Series, With its Application to the Geometry of Curve Lines -- wait, shouldn't that have been Principia?  Well, that was Newton's physics book, published in Latin, but using complex, difficult, and sometimes incorrect "simple" math, because Newton was highly jealous of calculus during his lifetime -- and thus, A Treatise of the Method of Fluxions was published posthumously in English (albeit translated from Latin).

Sometimes the motivation is simply "I don't know.  It seemed like an interesting problem at the time!"  And that, too, is good, because it's a reminder that sometimes we just have to play, and see where are games take us!