How not to be Wrong - Part I - Linearity

This one word tends to cloud our judgements most of the time. Before I take you on to this demystifaction ride of interpreting too much I suggest that some of us need to know this “jargon”. Know means “really know”.

Line and Curve in Chapter One - Less Like Sweden

Jordan has been very kind to make us understand Lines and Curves with the help of examples and plots. He shares his observations on debates and differing perceptions on the matter of taxation.

We can wear our respective hats and derive the meaning out of a line or a curve. Fixating our views on lines might not result in satisfaction because the real world is non linear. Numbers and plots if presented without critical analysis might result in unintended consequences too. Mathematics is really about critical thinking, questioning, making changes to our inferences and communicating it with the help of diagrams.

I am not new to Lines and curves,nevertheless I have shown those in images below. The things that I want to compare are to be quantified (using real numbers) and to be plotted against each other with the help of something we call axes. i.e. Horizontal axis is called X and the vertical one is called Y.

X and Y are called with some more names such as dimensions, features, columns etc. There is nothing to get confused or scared with these names as long as we can sucessfully communicate our point.

However, I need to remind myself that I can compare only 02 variables at a time with the help of a line/curve in a two dimensinal space to find out

where I am and decide where to go

Where I am means “The Present” and where to go means “Future”. If I follow a line it will take me in one direction but if at any point in time I feel that there is too much or too little in case of Y then I can safely use Curve, as it has the highest point.

Ok It was easy to understand X and Y then what is next? I am wondering what to chose as X and what to chose as Y. X is something that you have control on and Y is something that you expect to change because you are changing X. If I could plot Present and Future in terms of X and Ythen I will keep Present as X and future as Y.

Note: I have intentionally kept capital X and capital Y as notations

I read this chapter of the book to know about taxation and how mathematics helps officials, leaders, politicians and economists in taking decisions.

Well just because I mentioned it above, it does not mean that a line or a curve is enough to uncover insights and decision making.

Digging deeper.

Calculus in Chapter 2 - Straight Locally and Curved Globally

In this chapter, Jordan has communicated that thinking of curve as a line is not that bad afterall. Our perceptions are limited to how much we see.

Let us say I go for a walk on the street after dinner to burn some calories and plot the path on a piece of paper to analyse the route; I will probably not get a straigt line. However, if I plot only one step on the paper then I will get a straight line(in any direction).

X is latitude and Y is longitude here.

Now if I want to plot the relationship between no of calories I burn in a day and the weight over a period of one week then I might not get a straight line. However, if I zoom in my view and observe relationship between burnt calories and weight for every minute of my activity hour on a particular day, I might see a line as shown below:

#TODO:

On any day looking at the line over an hour of activity is enough to deduce that burning calories is good because it results in weight loss, burning more of calories is even better. We tend to think linearly and in a way it is fine to say that all curves are straight lines.

In the book, Jordan helps me in understanding the difference/similarity by sharing the story of Archimedes, who took an approach called the method of exhaution, to calculate the area of a circle(Curve) using inscribed and circumscribed polygons with 65,536 sides(lines). Why at that time Archimedes took this approach ? Because it was doable with the methods available to Archimedes.

#TODO: image of circumscribed and inscribed ploygons

Before getting to know about the section on calculus, I would like to quote another example from the book.

An ant on the circle, aware of his own tiny immediate surroundings, would think he was on a straight line, just as a person on the surface of the earth (unless he/she is clever enough to watch objects crest the horizon as they approach from afar) feels like he/she is standing on a plane.

Jordan teaches calculus, in one page, by giving the example of the a fired missile’s arc. He asks to take a conceptual leap and reduce our field of view until it’s infinitesimal - so small that it cannot be named but not zero on the arc and voila! the arc becomes a line.

#TODO: image of the missile arc

The slope of this line what Newton called the fluxion, and what we would now call the derivative.

I read the chapter to know about fluxions, the evanescent increments and to know why they are called the ghosts of departed quantities.

I too am going to mention the derivatives and their use when I post about how they are useful in calculating gradient descent while training a neural network.

Calculus works and a real life example from the book below:

*If you swing a rock in a loop around your head and suddenly release it, it’ll shoot off along a linear trajectory at constant speed, exactly in the direction that calculus says the rock is moving at the precise moment you let go.

These evanescent increments and associated unnecessary perplexities are explained with the help of Zeno’s paradox on going to an ice cream parlour, but never reaching it, and Grandi’s series. I could understand 0.9999.... riddle. Going further I got to know that there’s a whole field of mathematics that specializes in contemplating numbers of this kind, called nonstandard analysis. Cauchy’s contribution to make Newton’s calculus completely rigorous is commendable and was convincing enough to understand that we need to make choices in mathematics too. There are times when we have to settle the conflict between our intuitions.

Coming back to Linearity in general and heading towards the next chapter.

To be continued….