One of the universal laws of nature is the law of the least rate of energy expenditure. Thus, for example, the light emitted from point A chooses the path that requires the least amount of energy among the infinite number of routes to go to point B, which is often also the fastest.

If points A and B are in the air, the light follows the AB line because the AB line is the way light will reach point B as quickly as possible, with the least amount of energy expended. However, if point A is in the air and…

There’s a mathematical order inherent in our universe. Let’s start with rivers. If we measure the length of a river and divide it by the direct route from the start to the end, we’ll get its sinuosity or bendiness.

Mathematicians found that the average sinuosity of every single river in the world is pi. The same mathematical constant used to calculate the mass of an electron and the gentle breathing of a baby helps define how bendy all rivers are. …

The moving sofa problem is inspired by the real-life issue of moving furniture around. A mathematician was probably carrying his sofa down a corridor. He had to navigate some obstacles and asked himself that question; what is the sofa of the largest area to move around the corner?

Consider these two examples:

** “This statement is false,”** &

- If the statement is true, then it’s wrong because a statement is false.
- If the statement is false, then that the statement must be true since the statement is false.

Thus a paradox results in a puzzling problem that cannot be resolved. With paradoxes, logic is sometimes defied. Other times it involves circular reasoning. And occasionally, the paradox results from an invalid argument. …

How do you explain the deep interaction between philosophers and mathematicians throughout history?

In the history of human thought development, there has been constant interaction between philosophical and mathematical aspects. Many intellectual movements have sought their support, inspiration, and even their model in the style and way of mathematics. What could be the reasons for this approach?

Philosophers try to understand and unravel the many enigmas that the real world proposes to them. But the reality is too chaotic to try to address it as it is. …

One of the best-known features of Jupiter is the great red spot. It’s been there for hundreds of years, and it’s a big storm, a vast hurricane on Jupiter. And when the Voyager spacecraft got a look at that, it noticed you get complicated, turbulent vortices spinning off. What we see on Jupiter is turbulence. Big vortices, smaller vortices, tiny vortices, and if we could look closely — smaller and smaller and smaller ones.

Mathematics is something extraordinarily precise. Moreover, it is more accurate in different areas than in some areas we know less about it. People often find mathematics very abstract. But mathematics can describe reality as we understand it.

When we think about solid stuff like an umbrella scientifically, we picture molecules in our heads. We know molecules are made of atoms. Atoms are made out of nuclei and electrons going around. But what about the nucleus? What is an electron? At that stage, the best we can do is to describe some mathematical structure.

Mathematics describes things in the ordinary physical…

What is reality? Does reality exist? Is reality independent of the mind? How does a person find reality? What is “truth”? What does it mean to understand something truly? How does the brain come to understand something? How do we know if we understand something? What do we mean by “thinking”? How to can we get different conclusions from the same output? What is the proof?

If you don’t ask these and similar questions and are incapable of pondering the answers, you can not be considered an intellectual.

I am not saying that *all* intellectuals have to find reasonable answers…

We know that Kurt Gödel (1906–1978) worked intensively with Leibniz (1646–1716) towards the end of his life. Gödel was obsessed with Leibniz. When some people had destroyed some of Leibniz’s writings, Menger asked Gödel, “*Who could have an interest in destroying Leibniz’s writings?”* Gödel would say, *“Naturally, those people who do not want men to become more intelligent!”* (Menger, 1994). When his friends advised him to focus on his studies, he would ignore them rather than studying and reading Leibniz’s works. Eventually, the expected happened, and Gödel continued to follow Leibniz’s footsteps and gave God’s ontological proof like Leibniz.

Mathematics is a vast and diverse field, and so many good mathematicians can share their knowledge with humans in a beautiful way.

Although this year, 2020, has ruined our lives, it helped me read many math books this year. And now, I am eager to read some new good math books.

I have chosen 17 books for lifelong learners. I have read some of them this year and also I added the books that I am planning to read in 2021.

I hope you enjoy them.

If infinity has always been a bit of an issue for you, this book…