Here are some of the subjects that I think are the most valuable to study, along with some recommended books in each category.

## Mathematics

Study mathematics! Not only will its methods and results empower you to solve difficult problems, it will also improve your thinking in a variety of ways. Here are the first few that come to mind.

1. Logic: It is a paragon of logical reasoning, requiring absolutely sound arguments. The mathematics community demands careful definitions and does not allow for fuzzy leaps in logic.

2. Abstraction: This is hard to describe briefly, but there’s a nice little Wikipedia article on it.

3. Creativity: Laypeople often think there is little difference between what a mathematician does and what a calculator does, but in fact creative thinking is central to higher mathematics. Anyone who doubts this need only learn about some of the astoundingly creative proofs in foundational mathematics.

This is the book that showed me how deep and fascinating mathematics is.

• Godel, Escher, Bach: An Eternal Golden Braid - Douglas Hofstadter

But it only scratches the surface. It’s almost impossible to appreciate the level of clarity and rigor in foundational mathematical thinking without years of deep study. Along those lines, here is a textbook that I think is great for starting the transition from Freshman math to real mathematics.

• Discrete Mathematics and Its Applications - Kenneth Rosen

Finally, I can’t imagine a better book connecting mathemtatics to real-world reasoning than this next one.

• Probability Theory: The Logic of Science - E. T. Jaynes

## Randomness

Let me define a few concepts. A belief system is a set of propositions that a person believes to be true. Two belief systems $A$ and $B$ are contradictory if there exists a proposition $p$ such that $p \in A$ and $\sim p \in B$. (There’s a richer version of these definitions in which each belief system assigns a probability of truth to its propositions. For now, though, let’s just stick to this simpler formulation.)

Even very smart people are very often wrong. In fact, if I have a set of N people with pairwise contradictory belief systems, then at least N-1 of them are wrong (i.e. they believe at least one false proposition). The world contains plenty of very smart people, but it also home to a plethora of different belief systems. Typically, these belief systems contradict each other, even when they have much in common. My conclusion is that the vast majority of people have imperfect belief systems.

How do false propositions sneak their way into people’s belief systems? The main problem, it seems, is that we humans are bad at dealing with randomness. We live in a noisy world. It’s not hard to find “evidence” for pretty much any hypothesis you can dream up. The secret to maintaining an accurate understanding of reality is thinking careful about the implications of this randomness. To that end, I recommend reading anything you can find on understanding randomness or on cognitive biases. Some of my favorites are listed below.

Keep in mind that these authors aren’t perfect. Some parts of their books are more speculative than others. Try to maintain a keen awareness of this as you read.

• Fooled by Randomness - Nassim Taleb
• The Signal and the Noise - Nate Silver
• Uncontrolled (particularly parts I and II) - Jim Manzi
• Don’t Believe Everything You Think - Thomas Kida
• Damned Lies and Statistics - Joel Best
• More Damned Lies and Statistics - Joel Best
• How to Lie with Statistics - Darrell Huff

Also, let me recommend against something. Avoid news/politics like the plague! Seriously.

## Science

Before I studied probability and statistics, I used to read a lot of books praising science. Indeed, science is enormously underappreciated by the average person. But I fear that science is overappreciated by parts of the pro-science community. Science has weaknesses that should be taken seriously, and the insights of the books listed above bring many of these weaknesses to light. Some of those weaknesses are listed below.

1. Scientists are only human and are therefore subject to the same cognitive mistakes as the rest of us. The scientific process (experimentation, peer review, etc.) is supposed to keep them on track, and it does in fields where the signal to noise ratio is good enough. However, in many cases (especially social sciences), research is weakened by a combination of a very poor signal to noise ratio (in part, due to the impracticality of experimentation), an underappreciation of randomness, and often strong ideology. Anyone who has done enough data analysis knows how easy it is to over-explore a dataset and almost always find a significant result that supports whatever hypothesis you want to show. In other words, science is an incredible tool for discovering truth (or at least for gaining increasing control over our universe), but not all branches of science are created equal. I’m not trying to trash social scientists here; they do interesting and useful work. But the complexity and noisiness of their objects of study make real knowledge hard to come by. See the book Uncontrolled (mentioned above) for more on this.

2. In the complex system of human society, there can be implicit knowledge in our culture and our institutions. Because of the inherent problems of studying these systems scientifically, the implicit knowledge may rival or surpass the explicit knowledge that science produces. Again, see Uncontrolled for more.

3. To me they seem overly dismissive of religion. It has been my experience that many people in this community will not take seriously the idea that religion could have important personal or social benefits, regardless of whether or not it’s true. They say that it’s false, they list the problems it causes, and they think that settles the issue. But religion has been a central part of humanity across the globe for tens of thousands of years; to me that warrants a more honest consideration of its benefits.

Although the pro-science community seems to have developed its own brand of dogmatism, its core ideas are worthwhile. A classic that I thoroughly enjoyed is

• The Demon-Haunted World - Carl Sagan

Study the basics of the hard sciences: Physics, Chemistry, Biology. The fundamentals in an introductory textbook provide a superb approximation to reality.

Some people contradict scientific knowledge without any evidence. Others present scientific sounding language without having any of the key ingredients of science underneath. For instance, academia provides some examples of the peculiar views that can result from this, as described in

• Higher Superstition - Paul Gross and Norman Levitt