Editing The Road to Reality Study Notes (section)

===2.1 The Pythagorean theorem===

To explore the process of pursuing mathematical truth, Penrose outlines a few proofs of the [https://en.wikipedia.org/wiki/Pythagorean_theorem Pythagorean theorem]. The theorem can be stated as such, "For any right-angled triangle, the squared length of the hypotenuse <math>c</math> is the sum of the squared lengths of the other two sides <math>a</math> and <math>b</math> or in mathematical notation <math> a^2 + b^2 = c^2. </math>

There are hundreds of proofs of the Pythagorean theorem but Penrose chooses to focus on two. The first involves filling up a plane with squares of two different sizes. Then adding a second pattern on top of tiled squares connecting the centers of the larger original squares. By translating the tilted pattern to the corner of the large square and observing the areas covered by the pattern you can show that the square on the hypotenuse is equal to the sum of the squares on the other two sides. While the outlined proof appears reasonable there are some implicit assumptions made. For instance what do you mean when we say ''square''? What is a ''right angle''?