7. (unless you work entirely in theoretical spaces -- then it's ok because there's no difference between theory and practice)This. You cannot frame a painting without knowing its dimensions. Many foundational computer scientists are or were mathematicians for good reason. Smullyan's "Knights and Knave" style puzzles often touch very deep ideas in mathematical logic.

Perhaps the best way to improve abstract thinking skills is to increase your mental endurance and start thinking about things more.

He won the USA Mathematical Olympiad (USAMO) and was a member of the first American team to participate in the International Mathematical Olympiad (IMO) in 1974. It probably isn't a coincidence that a vast number of high level practitioners in science were also artists in some form or another (Einstein, Newton, Feynman, etc).

Not a knock on them, just not everyone has the opportunity to use their mental faculties 8hrs a day at their day job.While I agree with most of your comment, this caught my eye, because that's exactly what "Starting Strength" by Mark Rippetoe (an HN recommendation, I might add) does. Others argue either for it or against it, refining the argument until they feel they have a better understanding of a given topic.For actual programming mindset, I think this a learnable skill that I'd love to see a good article on. Philosophers have honed their abstract thinking skills to the max :-)Or Abstract Algebra as whole including ring and field theory.I don't know a lot about Bayes theorem but you caught my attention. You'll study sets of operations on groups -- if you are able to identify Collection X as a group, you immediately know you can apply theorems a, b, c, etc. I would definitely do the trial period though because it may not be for everyone.

).This comment is strikingly myopic and follows a strain of insular opinion that seems to have become more popular in recent years in tech circles ("Most of them are also not necessary").You might want to look up the definition of necessary!How sad life would be if we only learned what was necessary!It's true that HN is mostly for devs/engineers but aren't we that awesome that we can be interested in everything else too?

And science.Abstract Algebra and Linear Algebra are adjacent fields with a lot of clean abstractions that I believe exercise the same muscles as programming (even if a lot of it isn't directly applicable)i.e.

My more resourceful colleagues on the other hand managed to ship yucky but working solutions -- which eventually got less yucky.I learned that sometimes you have to let go of the ground-up mathematical-derivation type of thinking, and just release practical yet inelegant solutions into the wild, collect data, and then iterate. I have mostly used it to review material that I basically should already know so I don’t know how difficult it would be for someone approaching it at a perspective of not knowing much about the subjects.

The three that made the biggest difference for me were:Set theory: Covered proof construction in a deeper fashion than high school geometry and was the first proof-heavy math course I had taken in college (others used them and expected regurgitation, but did not expect construction; this was my first math-major only course).Later I took both Abstract Algebra and Linear Algebra (high level) together. If you've been through high school, books like that are self-contained. Abstract thinking is very powerful and you can train it.IQ can be higher and lower during a lifetime, surprisingly, as long as you maintain brain plasticity.It is not, and by thinking so you are limiting yourself.Accumulate more concrete knowledge. (I want to add category theory, but I don't feel I can as I only have the barest grasp of the fundamentals myself...) As with all pure math courses, it will quickly move beyond the depth/level you can actively use in programming, but the mind-expanding it does is really great at encouraging the sort of abstract thinking the OP's post is about. While I agree that you should continue down the path of Math, there's other ways to practice abstracted thinking. After all, that high precision is difficult to express and comes with a lot of assumptions.Lastly, another pure math method: study abstract algebra.I can really see a difference here with my peers in the sense that it takes them a lot more to understand each other's lines of reasoning both in verbal and written speech, and it takes them a bit more to string together complex ideas when they are too far away from what they were taught (this last bit even affects the way We must be mindful of irrational fear causing a false sense of urgency.


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