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Ask HN: Resources on how to improve abstract thinking skills?
195 points by mezod on Nov 20, 2017 | hide | past | favorite | 81 comments

This is not quite related, but I learned something yesterday. My wife tells me that she wants to be smarter so while we were walking I was trying to teach her mental chess so we can play chess while we walk. And she didn't want to because she was mentally exhausted (perhaps from all my talking, which is totally okay).

Some of the smartest people I know don't actively seek out to improve their thinking skills. Its simply a by-product of thinking more about anything and everything. When it comes to thinking, depth first is the way to go. Start asking questions about everything you see (why is Louis Vuitton always on the first floor in a shopping mall? If I spilled my drink, how long does it take before it gets cleaned up?)

Perhaps the best way to improve abstract thinking skills is to increase your mental endurance and start thinking about things more. Eventually, you are going to start connecting things and the abstraction will start to happen.

Also, perhaps increasing your creativity might also help. Every morning I shuffle a deck of cards and I memorize the order (timed. I started this year at 15 mins, now it takes about 1min 30s). To memorize a deck of cards, you must create a story out of your ass and it helps with your creativity.

You memorize the order of an entire deck of cards every day?

Although I'm not as intentional about it as the parent commenter, I try to play with my memory palaces when I get the chance. I'll get a list of 12 random words and memorize them with the technique. This helps me get better and quicker at the process, and it feels good to impress myself every so often.

How do you check you didn't forget some of them?

Shoulder surfing and spitting back bitcoin wallet words would be a neat trick.

I am a bridge player, so I memorise 24 boards easily.

That's pretty cuil.

"Time spent thinking" is actually a great metric for getting smarter. However playing mental chess is probably way too difficult for her and the majority of the population.

A great way to understand how to get mentally fit is to look at physical fitness. For most unfit people, exercising muscles that they don't use is unpleasant. What they need is simple exercises that are short, easy to understand, repeatable, cover a range of muscles, and have variations that provide increasing difficulty for advancement. You don't want to start someone off doing power lifts.

If you've been to any gym classes, you'll realize that there's no such thing as an objective scale for exercise difficulty. Squats may be trivial to someone but impossible for someone else. Finding a person's level of fitness and figuring out a good set of exercises that are appropriate for their progression is the key to a good long term fitness plan that actually works.

How about starting her off with something simple like sudoku? I know some may balk at the idea of sudoku being a mental challenge, but it's the mental exercise equivalent of yoga. The reality is that some people who haven't used their brain in a long time may be at a very, very low level. Not a knock on them, just not everyone has the opportunity to use their mental faculties 8hrs a day at their day job.

> You don't want to start someone off doing power lifts.

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. Granted, Rippetoe points out, multiple times, to start with the empty bar. It's also probably not a bad idea to have your form checked, although I lifted for a year before doing so, and the only time I've injured myself was a bodyweight exercise (situps, which aren't part of "Starting Strength").

Mental chess is a bit tough of a challenge. Why not try mental four in a row? Still a bit of memory work required, but not to the extent that chess or blindfolded go require.

I wrote an app to let you play, starting with playing with one colour, before moving on to invisible.

https://simonduff.net/invisilink to play in your browser, and a link to the Android app. Appreciate any feedback :)

A bit (more) offtopic, but regarding your deck of cards, it's an old technique that magicians use (used?) to display "amazing memory".

The usual play is to get people from the audience say random words while you assign those words to something you do everyday (the most common tool is the way to work/school) in order.

With training (as you are doing), you can remember long lists of seemingly unrelated things (and the order they are in).

As I said, a bit offtopic, but I found it funny for that old trick to come up here :)

Ah yes the method of loci[1], the book Moonwalking with Einstein explains this and other techniques very well

[1] https://en.wikipedia.org/wiki/Method_of_loci

>why is Louis Vuitton always on the first floor in a shopping mall?

Tried googling it to no luck. Any idea?

Some hypotheses:

1. First floor space is the most expensive (a testable assumption), and only luxury brands like LV can afford it.

2. Brand name on the first floor attracts people walking by into the mall.

I feel like iterative deepening is a more optimal strategy in this context.

Study higher level math! Category theory is a brainfuck that exercises that skill well. This book https://mitpress.mit.edu/books/category-theory-sciences doesn't have any real mathy prereqs but is fun.

Added benefit from studying math is that its a foundation for a whole bunch of other cool stuff.

Yes, exactly this. Since I've started studying category theory I am finding a lot of changes in my thinking. For example, I used to struggle a lot with mental overload when trying to accomplish business goals because of the classic "so much to do don't know where to start" type of thinking that I think is common with a lot of people. It's hard to explain but I find myself becoming better able to mentally organize my thoughts and break things down into units, etc to start work on specific things even though the end goal is overwhelming. I still have a lot to learn about category theory but I think it's very beneficial.

I was going to recommend “How to Prove It”, if someone doesn’t know proofs already.

Very good suggestion. Here's is a similar but free book: http://people.uleth.ca/~dave.morris/books/proofs+concepts.ht...

Conceptual Mathematics and Sets for Mathematics from Lawvere are really good for this purpose too. Challenging but doable with the mathematical maturity that most programmers have after extensively working with functions and types.

Suspension of judgment was the very first step I discovered, I clearly remember, from when I was quite little. Perhaps I started out especially daft, but I kept making discoveries ever since.

Something which served me well a long time was a mental game of following logic to extreme conclusions and then asking myself "what if this is true, and my premise was wrong?"

One of the most powerful methods I use now when approaching a new subject is to create my own narrative to the best of my understanding. Then as needed I can refine my narrative. - Apparently a lot of people have a thing where they become deeply and personally invested in the first narrative they discover, and it does not serve them well. - Don't be dogmatic.

Coming up with counter-examples gives you great analytical abilities. It can also earn you enemies, so don't be too hard on the friends you have chosen.

When you encounter something seemingly paradoxical, you have an opportunity to find a new perspective which resolves the paradox. - Here you have further opportunities to observe yourself as you are making observations. Once you are able to do this, you should be self-sufficient in improving your intelligence.

Don't spend all your time pondering! All this work lets you see and experience life more deeply, so it is a waste if you don't choose to have a good time.

I prefer your style of thinking over the "brute force" method in the top comment. If thinking was an olympic sport, I feel it is more akin to gymnastics than power-lifting. I believe that if you are rational, disciplined, and take your time you can have more accurate beliefs and make better decisions than someone with greater raw intellectual power.

I also agree that premature conclusion is the most common error of thinking. We must be mindful of irrational fear causing a false sense of urgency. Thinking compulsively without forming a conclusion can be a problem too, but it's probably a better problem to have.

Thank you, this is very relevant to me right now. It is easy to become overly invested in your idea of what a narrative should be.

Could you please elaborate and give an example of these techniques.

I can try. Suspension of judgment: say a provocative crime has been committed and a news outlet reports on a suspect. - You should be emotionally prepared that new information could arise which explains even if not forgiving the actions of the suspect. e.g. A parent kidnapping their own child, and then it turns out that the other parent was abusing the child.

For taking stuff to the Xtreme, taxes is an approachable subject. Say if you have 100% tax rate, you might be looking at communism, but you might also look at a pirate ship where the captain divvies up the booty. Another example would be the mental exercise of trying to figure out that maximum possible current through a conductor. Even superconductors suffer from magnetic pinching, but if you have ever seen a plasma globe at a science fair you might discover the magnetohydrodynamic phenomena of Alfven waves. - The taxes example is more direct than these about current, but I'd argue the physics discoveries are more interesting.

I very often see dogmatic beliefs in science. The mistake of 'taking the map for the terrain' is common. People will for example pick a favorite interpretation of quantum mechanics and then with complete conviction deny that effects predicted by other interpretations are worth looking for, because their interpretation excludes it. - Why try to travel around the globe? You will fall off the edge! Figuring out that someone is making this mistake can be frustrating and time-consuming.

Counter-example takes the most creativity, and can be constructed to be too impenetrable. Recently I argued that if the cost of healthcare in the US would best represent the optimum quality, then countries with just as good service and universal healthcare too would be bankrupt.

For paradoxes, you can apparently find a very long list on Wikipedia. I don't remember ever having gone out specifically looking for paradoxes. I have encountered them during sessions of deep pondering as apparent contradictions where I expected none, so to me they are nameless.

Hope this helps! I might turn these two posts into an article... :o)

please make sure I hear about such article at mezood @ gmail :)

A lot of the way experts think is actually very domain-specific. So if you want to be good at thinking about a specific domain, might be worth reading one of Gary Klein's books about naturalistic decision making. https://www.amazon.com/Power-Intuition-Feelings-Better-Decis... is the most practical of the ones I've read.

More generically: learn how to write. Most hard problems won't fit in your head. Most of the ways we think are too vague and fuzzy. Writing things down and working through something in writing can help find the limits of your thinking process, and help you find solutions. I review a book that explains this very well here: https://codewithoutrules.com/2016/06/15/writing-book/

(You could alternatively take a good academic writing class at your local university.)


Here are two mental frameworks that have served me well throughout my career:

I am process oriented and find that my “abstract thinking” improved significantly when I applied a somewhat consistent mental framework / process to questions, situations, challenges or projects (which are often just a series of questions)

#1: the three to five “whys”

Super simple, ask why or how three to five (or as many as necessary) times to try to unravel the root of a problem or issue. “Why did this happen, well why did that happen, how do we know that is why that happened, etc”

#2 break problems or complex issues down into decision trees. Sales are down X%. Well, there are probably 3-5 core events that drive sales (inbound, outbound, retention, competition etc). Looking at the outbound tree, there are 3-5 core activities that drive outbound (phone calls, emails, in-person meetings, conferences, etc). The more you do this on paper, the easier you do this in your mind which is what I believe improves what others see as abstract thinking.

Final thought: I don’t think improved abstract thinking has much to do with memory. You encounter a problem, employ a consistent process and get to a result. The better the process the better the result over time.

Happy to elaborate on any of this further. I was putting together examples but it took away from the core message.

I get it though, thank you!

Eno and Schmidt’s Oblique Strategies is a classic. It’s based on the idea that creativity is based on constraints, so artificially adding new boundaries to a problem helps the mind find new solutions. https://en.wikipedia.org/wiki/Oblique_Strategies

Read up on “lateral thinking,” it’s sort of the gospel of graphic design. https://en.wikipedia.org/wiki/Lateral_thinking

Interesting - there are two different takes on this question in this thread. This comment is about generating new ideas and thinking creatively, whereas itamarst answered in the sense of organizing thoughts in order to solve problems and communicate them.

one of the things I love about asking random questions (I care about) on HN is that answers never get constrained in the ways I had devised or expected :)

If it's about programming the book How to Design Programs[1], Chapter 3 is about Abstraction

[edit] And is more general as some other use is suggestion philosophy is great.

I'd recommend you a podcast titled "In our Time: Philosophy" from BBC Radio 4.[2]

[1] http://www.ccs.neu.edu/home/matthias/HtDP2e/part_three.html

[2] http://www.bbc.co.uk/programmes/p01f0vzr/episodes/downloads


I'm just being a snarky smartass, but you could probably improve your abstract thinking skills by trying to find the answer to this question yourself. :-)

Yeah there's no one fix other than struggling to contextualize the skills being built here.

The way people think and make decisions is prone to different types of cognitive errors. Learn, recognize, and apply ways to overcome those classes of errors. Then learn, recognize, and apply mental models to improve your decision making.

Why do you want to improve your abstract thinking skills?

I would be surprised if there is anyway to improve these skills, g(aka IQ) is notoriously hard to improve.

But I imagine you want better abstract thinking skills for a specific reason, and it'll probably be much more effective to skip directly to the reason you want to improve these skills.

To be honest, there's no definite reason other than personally growing and being able to understand more concepts. I don't know if IQ can be "improved", but I do know that people who have battled through difficult problems develop ways of thinking that make it easier for further difficult concepts to be understood. 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.

This works even better for people who think of intelligence as something improvable, like physical skill. You might not start with great ability, but with practice you'll get better.

It is not, and by thinking so you are limiting yourself.

Citation please

Accumulate more concrete knowledge. Our brains develop abstractions when we identify patterns in disparate phenomena. Can't identify those patterns if you don't study the concrete phenomena themselves.

As you learn more you'll build up a toolbox of abstractions that will make thinking and learning easier.

I've always understood abstract thinking to be of broad nature, generalities, non-specific, a vastness.

The idea that abstract thinking is devoid of specifics has never struck me as a prohibition that you can't get to abstract thinking by starting with examples and metrics. Looking for patterns, mind mapping, and tabletop exercises all seem like ways to take concrete thinking elements and bridge them to the abstract.

In doing so, I think there are some specific areas to watch for, namely the influence of cognitive bias. I'm a big fan of the Cognitive bias cheat sheet which has been covered on HN in the past, https://betterhumans.coach.me/cognitive-bias-cheat-sheet-55a...

I also believe that the concept of multidisciplinary approaches can be one of abstract thinking -- when you begin your exploration of a topic from the viewpoint of a discipline you have not mastered, your mind is more likely to be able to explore concepts and solutions which are not bound by fact -- you simply don't know the facts and principles of these foreign disciplines intimately.

And just as disciplinary viewpoint can provide interesting triggers in abstract thinking, so can applying empathy and imagination. The best books on things like this are often written for children. Try "The Book of Think: Or how to solve a problem twice your size" (Burns, 1976).

thank you

So I'm laughing at all the examples that have made the rounds on HN innumerable times. How about this.

"Possible Worlds: An Introduction to Logic and Its Philosophy" https://www.amazon.com/Possible-Worlds-Introduction-Philosop...

Analytic philosophy has produced a methodical approach to thinking about things that we don't seem to teach as methods. Instead it trickles down into pop-science takes on thinking which tells you a lot /about a domain/ but not how to /use/ things in the domain.

Thinking in terms of "possible worlds" is a nice little hack that unifies a lot of our intuitions in how we determine what is true. It also implies a lot of nice methods that we already use implicitly, like making thought experiments, or conceptual analysis, or testing arguments with ad absurdums, or working with modalities (like treating what we should do differently from what we could do). This book teaches possible worlds at an introductory level, when typically it's considered a graduate-level topic. It also gives exercises, which are ultimately the important bit. I don't think I've seen too many books that teach philosophy /as a method/ beyond argument construction.

I also like this guide: https://www.av8n.com/physics/thinking.htm "Learning, Remembering, and Thinking". I recommend checking out his other work for a model of working through problems coming from physicists.

One more thing. Oftentimes the key step to thinking is figuring out what you're questions are, and questions are always determined by what uncertainties you have in a domain, as specifically relevant as you can make them.

I'm gonna quote Venkat Rao (of Breaking Smart and Ribbonfarm fame) from an article he deleted years ago:

> Real questions, useful questions, questions with promising attacks, are always motivated by the specific situation at hand. They are often about situational anomalies and unusual patterns in data that you cannot explain based on your current mental model of the situation… Real questions frame things in a way that creates a restless tension, by highlighting the potentially important stuff that you don’t know. You cannot frame a painting without knowing its dimensions. You cannot frame a problem without knowing something about it. Frames must contain situational information. There are two types of questions. Formulaic questions and insight questions. …. Formulaic questions can be asked without knowing much. If they can be answered at all, they can be answered via a formulaic process. …. Insight questions can only be asked after you develop situation awareness. They are necessarily local and unique to the situation.

The world is /extremely/ information rich to the point of absurdity, and what fails is not the richness of our input data but rather our awareness of how we ought to use it. George Polya tried to teach his students how to problem solve in mathematics by means of getting people to ask questions. By verbalizing his thought process he hoped to convey these principles, as well as giving them a standard template to prompt their cycle of questions. But to adhere to a strict plan like that is to defeat the point. The real point is to maintain a conversation with yourself, giving yourself and refining your own questions until insight develops, and keeping yourself talking.

Ultimately I like to take an information-theoretic approach as the basis of my philosophy here. /Some/ information is /always/ going to be contained in /any/ comparison that I can make between two phenomena in the world. Most of this "information" would be considered noise relative to most reference frames. But it is always possible to extract /something/ from a situation by creating these tensions between yourself and your uncertainties in the world.

You can muddle around questioning things for awhile, but gradually things come up. The key is to let your uncertainty start off however it is and keep pruning away at it until your solution is sculpted from the clay. It can and will happen.

If you've ever tried doing Fermi Estimates (like those prescribed in https://www.amazon.com/Street-Fighting-Mathematics-Educated-... , https://www.amazon.com/Art-Insight-Science-Engineering-Compl... , https://web.archive.org/web/20160309161649/http://www.its.ca... , https://www.amazon.com/How-Measure-Anything-Intangibles-Busi...), then you'll be able to perceive the mindset that has significant transfer to many problems that have even just approximate answers.

thank you

There are three rules for writing the novel. Unfortunately, nobody knows what they are. - 'W. Somerset Maugham on enterprise architecture', Peter Hilton

In practice, designing seems to proceed by oscillating between sub-solution and sub-problem areas, as well as by decomposing the problem and combining sub-solutions. - Nigel Cross

Firmitas, utilitas, venusitas. ("Firmness, utility, delight") - Marcus Vitruvius, De Architectura (22BCE)

Less tongue in cheek, there are quite a few decent, mature systems encouraging thinking from different perspectives. Examples include De Bono's hats system, IDEO's mantras for innovation, Skillman's englightened trial and error, 99% perspiration (ie. just stay focused), design for data, partitioning the problem, prioritizing simplicity and clarity, prototyping, Alan Kay's dream while you are awake, etc.

Culled from http://github.com/globalcitizen/taoup .. there are more there :)

Learn and do things outside of your domain, for example learn to play an instrument. The mental gymnastic that music requires (which needs both intuition and rational thinking) will teach you to be more creative and to analyze things in a different way.

This is akin to telling an arts major to learn to program.

Personally, I believe it made my thinking much more visual and geometric (or maybe it was all that Legos I played with)

One of the books I still reread. Its not so much about "abstract thinking" but its very good at teaching you to find the real problem & make a decisions. https://www.amazon.com/Thinkers-Toolkit-Powerful-Techniques-...

Read https://www.farnamstreetblog.com. They have a lot useful content on the subject of thinking, mental models...

Thanks for the pointer. That site looks very interesting.

indeed great blog!

Read literature and/or philosophy (the latter is good for understanding abstractions, the former is good for understanding others' abstractions), do lots of math.

I do an exercise where I'll visualize geometries. Starting with a point, then a line, then a triangle, a square, a tetrahedron, and so on. At each stage once you really have control over it (you're able to translate/scale/rotate your visualization) add another point to the object.

I started studying economics in my free time because abstracting how humans behave with math fascinates me and you start to understand what a complex system really is.

Markets are very complicated and unpredictable in the short term, but you can understand the atomic unit of the market just fine; a transaction. It’s the same with all complex systems; try to work from the bottom and up, always begin with what you can wrap your head around.

After a while, you see that the methodology of building abstract models that are good enough is very similiar in most fields. You discover that it’s never easy and you should always be careful trying to recycle a model by mapping it to an entirely new system.

I've been working on a project that is something like "a framework of abstractions". I've found that studying and developing its concepts has been a great exercise in abstract thinking. Check it out and see if it's the type of thing you might be interested in. https://github.com/perspectivesonatheme/patterns

I'd also recommend learning some Haskell and category theory. It's difficult stuff and I won't pretend to have a solid handle on either, but what I have learned has been eye-opening.

My personal recommendation just based on personal experience would be to study philosophy. It doesn't much matter which philosophies, but a variety that differ fairly substantially would probably be best. When you read a very different viewpoint you can occasionally almost feel your brain shift sideways when you 'get' how the other viewpoint is seeing things. That leads to questions about the exact points where your viewpoint and their actually diverge, and considering a wider scenario where both exist or develop is inherently a move in the abstract direction.

Write, read challenging works of literature, learn an instrument and play music, listen to art music, learn to paint or draw, or take an acting class.

Or any other art, such as woodworking or pottery.

The question was posed on a site dedicated to discussion of programming and related subjects. It therefore should be assumed that the author was requesting advice for developing abstract reasoning skills such as are employed in mathematics or software engineering. Your suggestions require explanation as to why they would help this sort of abstract reasoning. As a matter of fact, many people who are proficient in the skills you mention are hopeless at the type of abstract thought used in math and computer programming, so the skills you list are certainly not sufficient. Most of them are also not necessary (all apart from writing and reading challenging literature?).

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").

Music and the arts have long been documented as wonderful ways to enhance abstract thinking. And you might become a well rounded person as an added bonus!

You might want to look up the definition of necessary!

> Music and the arts have long been documented as wonderful ways to enhance abstract thinking.

That is entirely consistent with them not being necessary to develop abstract thinking :)

I was just being logical, and trying to provide some subtle humor by giving the dry reply of a caricatured technophile rationalist to OP's evident enthusiasm for the arts.

How sad life would be if we only learned what was necessary!

Oftentimes abstract thinking isn't about learning to create programmatic mental models of things, but about learning to see a problem from various perspectives. Doing so gives one's understanding more depth. These recommendations, the arts as a whole, are extremely valuable at teaching different perspectives for understanding things.

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? :P My original question was broad on purpose, and I appreciate every single answer, it allows for a healthy discussion!

Euclid's Elements teaches geometric concepts that you're probably familiar with, but with interesting proofs: https://en.wikipedia.org/wiki/Euclid%27s_Elements

It's free, it's time tested, and probably just the right amount of rigor to get your proof abilities up. I think the ability to prove theorems and the ability to construct abstractions go hand-in-hand.

There's a really great mobile game called euclidea. It's basically Euclid's Elements - The Game.

My answer to your question is math. Learn to read and write proofs. Any intro to proofs will do: those employed in discrete math, the ones in analysis, the diagram chasing ones, whatever...Working with math proofs will definitely straighten out your thinking and whip your mind into shape.

Some suggestions to get you started:

Book of Proof by Richard Hammack: https://www.people.vcu.edu/~rhammack/BookOfProof/

Discrete Math by Susanna Epp: https://www.amazon.com/Discrete-Mathematics-Applications-Sus...

Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al: https://www.amazon.com/Mathematical-Proofs-Transition-Advanc...

How to Think About Analysis by Lara Alcock: https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0...

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers: https://www.amazon.com/Learning-Reason-Introduction-Logic-Re...

Mathematics: A Discrete Introduction by Edward Scheinerman: https://www.amazon.com/Mathematics-Discrete-Introduction-Edw...

The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by Rafi Grinberg: https://www.amazon.com/Real-Analysis-Lifesaver-Understand-Pr...

Linear Algebra: Step by Step by Kuldeep Singh: https://www.amazon.com/Linear-Algebra-Step-Kuldeep-Singh/dp/...

Abstract Algebra: A Student-Friendly Approach by the Dos Reis: https://www.amazon.com/Abstract-Algebra-Student-Friendly-Lau...

That's probably plenty for a start.

A more graduate-level book but one I found pleasing:

A Logical Approach to Discrete Mathematics: https://www.amazon.com/Logical-Approach-Discrete-Monographs-...

And a more pragmatic approach to the same material (with a lot of cross-over in terms of proof-style, etc):

Programming in the 1990s: http://www.springer.com/gp/book/9780387973821

But one I particularly enjoyed early on was written for liberal-arts level students of maths (who might've been traumatized by maths in the past):

Introduction to Graph Theory: https://www.amazon.com/Introduction-Graph-Theory-Dover-Mathe...

It will actually get you into writing proofs in set theory within the first couple of chapters.

Oh gosh the equational logic rabbit hole.

To add to the fire: http://mathmeth.com/

great, thank you!

Do math exercises.

I would recommend Elements of Programming by Alexander Stepanov and Paul McJones.

What evidence do you have that it's your general abstract thinking skills that are lacking? You'll probably improve faster at the tasks you're interested in if you just practice those tasks.

LSD according to Steve Jobs.

Cubensis are also highly effective.

Play d and d


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