How to Science

Let's say a mechanism to compute the area of a circle (given its radius) is needed. Do we throw bunch of "data" of radius/area pairs to a parrot learner and hope it spits back something approaching real answer? No.

Here is what you do; mentally take out a piece of the circle,

Figure 1

.. and other pieces like that, flip every other one place them side by side

Figure 2

Again, mental exercise, we think 'if I made the pieces smaller does that approach a known calculation?' Yes it does,

Figure 3

It approaches the area of a rectangle. We already know how to compute that, height times width, $C/2 r$, $C$ is circumference, $C = 2 \pi r$, area is $\pi r^2$. Boom.

We just used mental transformations (in complex problems the algebra is more complex but still symbolic) to reach something simpler. That final formula is the end-result, the fruit of science. The end-result is fast to compute, and accurate. Simple manipulation w algebra, geometry saved me from countless computations. This is how science is done, through representations, mental transformations.