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Ask HN: How to improve my abstract thinking?
264 points by max_ on June 30, 2020 | hide | past | favorite | 104 comments
A few days ago I got into Mathematical Logic[0] and learned how to reason about problems through using various branches of mathematical ideas like proof theory, model theory e.t.c.

I found this abstract way of thinking about problems clear, & organised. "Mathematical Logic" is diffrent from the kind of Math I was taught, which was a top down approach to solving problems.

"Mathematical Logic" seems to be able to derive solutions to problems in a ground up fashion where a solution can somtimes elegantly present its self as long as you apply correct mathematical properties.

What other techniques do you hackers use to improve your abstract thinking?

[0]: https://en.wikipedia.org/wiki/Mathematical_logic

Three things I recommend:

1 - Read lots of things unrelated to your field of study. Read about flowers, or art history, or music theory.

2 - Take long walks, get away from the screens and distractions. Walk some place that stimulates thought, like in nature, or in a city, or by a river. Skip the treadmill or places like malls that demand your attention. You need a place to allow your mind to wonder and to process/organize the info you read about.

3 - Dream, in a very literal sense. Do you ever remember those moments right before you fell asleep where your mind gets a bit too creative. Harness that. It has been said that Edison would take naps in a chair with heavy ball bearings in his hand. Right when he was just about to be asleep his hands naturally relaxed and dropped the bearings on the floor waking him. Quickly, he would pop up and recollect on dreams/ideas he just had. Use a similar technique to your advantage, see what randomness your mind designs.

> 1 - Read lots of things unrelated to your field of study. Read about flowers, or art history, or music theory.

I think reading is the most consistent way to get high quality information on any subject, and a lot of it. Don't get me wrong, there's a lot of other great methods of learning, but reading tends to just be very high quality, specifically books. There are always a handful of must read books for any given subject, whether that be compilers or growing mushrooms, and they achieve that status through years of agreement. Also, it seems to me that the average book tends to be higher quality that the average YouTube video or Medium article. I'm assuming it's due to a higher barrier for entry, but maybe there are other factors.

I wonder the origin of that story, I've heard a version of a painter who holds a brush and pallette, falls asleep, drops them, wakes up and paints.

And I do love my lucid dreams, at some points of my life they would ruin my sleep but I found them a good trade off.

You are thinking of Salvador Dali. He would hold a spoon in his hand and rest it above a plate. When he fell asleep the spoon would fall and make a clang with the plate. Jolting him awake again, he would try to recall whatever was happening in his subconscious right before being startled awake and try to express it on canvas.

> He would hold a spoon in his hand and rest it above a plate

I do a variation of that too. And my plate has cereal in it ... The moment I wake up I go: "oh hello, what's that ... breakfast!" - and instantly I'm in a good mood. It's like a form of meditation I call "bowlfullness"

I've heard it specifically about Salvador Dali, except he would drop a key.

I'm not sure, the attribute to Edison is apocryphal at best.

> Dream, in a very literal sense. Do you ever remember those moments right before you fell asleep where your mind gets a bit too creative. Harness that.

This sounds similar to Rich Hickey’s “Hammock Driven Development”. He gives some tips on how to harness it in his talk: https://youtu.be/f84n5oFoZBc

Thank you for the insights.

Lately, I have also been thinking about different subjects such as flowers, art history, music theory, about animals (how insects, or bats work).

My question, how do you choose books if you intend to learn such subjects as non-expert? And if you can recommend any good books?

I would love to recommend The Peregrine by J.A. Baker if you've never read it before. It describes in fantastic and vivid detail every tiny nuance of how a peregrine lives, flies and hunts. The book is poetic in its prose and i've taken to looking at birds when I walk outside to try and see what Baker saw when he looked at the sky.

Thank you, looks great. I bought it.

If there are any other books like that, please let me know.


I have a sister with a degree in biology. I borrowed some of her books, like "introduction to zoology". If you've been through high school, books like that are self-contained. For instance, it had an appendix on organic chemistry.

Highly recommended, those books taught me much more about the world we live in than the various books on physics I've read on my own. And science.

some even suggested that horizontal makes your brain work differently

I think such mathematics is probably the best way to improve abstract thinking, so you're on the right track.

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)

Related comment: https://news.ycombinator.com/item?id=23152152

i.e. basically linear algebra for engineers is not abstract. Linear algebra for mathematicians is quite abstract, e.g. a 400+ page textbook that doesn't refer to any matrices.


On the flip side, I think mixing abstract thinking with testing/debugging is the ideal combo for programming.

It is a skill to write useful test cases. To explore the state space efficiently.

If your code isn't grounded in real examples, then it may become overly general and not RUN run very well. It's a big danger to abstract before you have enough examples.

And probably the hardest thing that programmers do is debug OTHER people's code, as opposed to your own code.

(I guess you didn't ask about programming, but I'm sort of assuming you are programming by asking on this site :) )

I was going to comment on this as well.

I added a math degree to my CS degree (which also added a year, but I was going to be a year late graduating anyways). So I took a lot of courses in math my last two years. 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. I wasn't struggling in either, but in my mind they were two separate courses. One day we were doing a proof in Linear and I realized I'd already done it in Abstract, only we had been dealing with (I don't recall what) some other objects than matrices or vectors. What I realized then was that we were dealing (in both classes) with a class of objects and operations on them that were the same in the abstract, but different in the concrete (if you wanted to actually apply the math to solve a problem). Given the right perspective, I could apply the proofs of one to the other so long as the objects had the same properties the proofs relied on. Both classes became a breeze after that because, coincidentally, the order the material was covered in meant that every other week one class had been largely covered by the other when viewed in this fashion.

I totally feel that last bit w/ Linear. I took it with Data Structures/Algos, which was fascinating! In one class, we had a guided lab that was (basically) implementing PageRank, while in Linear we were approaching that problem _entirely_ from the theoretical perspective.

How would one get into abstract algebra having flunked High School math very early on? I've been going through Khan Academy pre-algebra and find it incredibly disheartening how little I know. I did manage to get a CS degree but the math taught there is different.

This is several days late, but I had this tab open along with many others and I'm finally getting to it.

What is it that you find challenging about math? If it's the actual computation/calculation -- the part where you are finding a numeric answer -- don't worry, that's nowhere to be found in abstract algebra (or any higher level math). Pure math (of which abstract algebra is a part) is about the study of patterns more than anything that has to do with numbers.

In fact, the name "abstract" refers to the fact that it's concerned with abstract collections of things -- for instance, groups. 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. to it. For these reasons, the sort of things you are likely learning in pre-alg on Khan Academy don't have much direct applicability.

I think it's an enormously beneficial subject for programmers to study, maybe the only math course beyond the standard discrete math that I think should be required. (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. It has strong relations to generics, interfaces, polymorphism, etc.

As for how to get into it, I used this book: https://www.amazon.com/Book-Abstract-Algebra-Second-Mathemat....

I'm 20 years out of school, so unfortunately not very qualified to answer that question. I took a look at the 3Blue1Brown videos but it seems he focuses on subjects other than algebra.

I will say that high school math is very different than abstract algebra, and in some ways abstract algebra is simpler.

I heard a long time ago that in France they teach some abstract algebra to 10 year olds??? They are simple concepts that don't require much background knowledge.

I actually got a little bit of it in 6th grade. I remember doing some optional assignment that introduced the idea of functions as things you can operate on, rather than things you just apply to concrete data. In other words, higher order functions.

So everybody knows you have numbers and operators:

    1 + 2*3 = 7
Well functions are also algebraic objects, with operators, e.g. 'o' is the function composition operator, so if you have functions f and g, f o g is also a function.

I think 20 years ago, programmers didn't think like this. These days, many programmers do, whether formally or not. JS/Python are more flexible than C or BASIC in this regard (and Haskell, Lisp, etc. even more so). So this is directly applicable to programming and it's worth a little bit of study. You probably know it, but seeing the mathematical view will make the reasoning more fluid.

It's not just functional programming either -- I don't use functional languages currently, but I use the ideas. I think that most people use inheritance poorly and I suspect that knowing algebra helps you "factor" code correctly.

Math is sort of the end of the road for abstraction. If you go any further you drive off a cliff. Take a trip to the cliff and look around for a while.

So-called mathematical maturity lets you think about many domains without considering concrete reality, and this lets you solve a different class of problems. Many foundational computer scientists are or were mathematicians for good reason. Think von Neumann, Knuth, and Turing.

There is a magnitude of difference in abstract thinking between a mathematically mature scientist and a mathematically immature one. Mathematical maturity seems to allow an easy transition from abstract theory to empirical experimentation. The converse does not appear to be as true.

My advice would be to study math directly.

As I got further in my Mathematics degree what really made it all "click" was classes on Philosophy. The logic and approach you forms in Philosophy give a greater understanding of how mathematical proofs are formed and how to approach problem solving. A series of lectures from a Philosophy 101 course would do a lot to help with the mindset you speak of so look to any of the MIT, Stanford, etc online courses.

As someone who studied mathematics as an undergrad and then intellectual history (with an emphasis on philosophy) postgrad, I second this. I would add too that learning how to write effective philosophy papers is an excellent way to improve your writing, and argumentation, and analytical ability more generally. (I often share the Jim Pryor's guidelines [1] with students.)

[1] http://www.jimpryor.net/teaching/guidelines/writing.html

this is interesting. thanks for sharing! these guidelines seem like they would apply well to science papers, or really, any writing where the objective is to prove an argument or perspective.

A friend of mine once said that if he could do college over again he'd major in math and philosophy. He said something to the effect of philosophy teaches you to argue from a set of premises, math teaches you to argue with a given conclusion.

I don't think math is about arguing, and neither is good philosophy. Both are about truth.

You could view math as arguing, if you studied it by endless theorems and proofs, but a better way to learn math is by trying to solve problems whose answers you don't know (and picking up theories as needed). That way you learn to be curious for the truth out there, not just a convincing argument. That's the best approach to philosophy as well.

I think the way we're taught math in school might lead us to believe that it's not but it is. you're right that the best way to learn math, or anything really, is to take a problem or project or whatever that's just outside your current skillset and work towards achieving it, getting help where necessary. But when you are seeking new knowledge, the best tool you have for verifying truth is argument with others. Math researchers and really anyone doing research of any kind spend a lot of time arguing methods and results - this is how you seek truth. Philosophy is the same way - someone makes a claim and uses evidence to support that claim. 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. I see a lot of intermediate programmers try to "abstract" by kitchen-sinking things, such that they end up with a perfect machine that is fairly inscrutable. And then when a change is needed, the machine falls apart and needs to be rewritten entirely.

I think abstraction is about recognizing patterns, but also about recognizing what elements are more likely to change - you don't want to abstract away the changing aspects. In that sense, it becomes similar to creating a model that depends on parameters.

Also, generally, when you are implementing a solution, that solution exists within a context of a problem. If we are simply told to implement a solution, there is a temptation to just trust that the solution will solve the problem - and sometimes we are told to trust that. But if we take a step back in abstraction and fully understand the problem ourselves, we can derive our way to the correct solution, freeing us from having to "trust" that the solution is correct.

Abstraction is just continuing from there. Understand the larger context of the problem, why it is a problem. Maybe you'll discover it isn't, and that the solution isn't actually needed. And so on, if you ask another Why you can discover that maybe another problem is more important to solve, which would make this problem irrelevant.

A dirty trick that is used all the time but goes unnoticed is concretizing abstract thoughts. The entirety of abstract maths is just that. Let me provide an example:

You've got some abstract concepts of relationships. How do you go around thinking about relationships? You make them concrete. You do so by first representing them as perhaps, two points on a piece of paper and then drawing an arrow between them. This act of representation is an act of concretizing an abstract thought.

Now you start adding rules to your concrete representation. If you are rigorous about following your rules you almost inevitably end up when Saunders MacLane ended up with Category Theory.

Another example:

Think about computers and what they do. Now try to represent them on paper. Depending on how your mind works, you might end up with something that looks like a Turing Machine or cellular automata. I've not come across anyone who thought long enough and came up with something like lambda calculus, so I suppose Church was an alien.

I didn't use any mathematical logic examples because those examples typically involve historical battles over symbols.

I would strongly recommend the book, "The Art and Craft of Problem Solving" - https://www.abebooks.com/book-search/isbn/0471135712/

It teaches you how to solve [math] problems, but not the kind you've seen in school. The author was a coach for the Mathematical Olympiad, and the kinds of problems he presents are creative, often requiring creative abstract lateral thinking to solve.

> Paul Zeitz studied history at Harvard and received a Ph.D. in mathematics from the University of California, Berkeley. He currently is an associate professor at the University of San Francisco. 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. Since 1985, he has composed and edited problems for several national math contests, including the USAMO and helped train several American IMO teams, most notably the 1994 "Dream Team" which, for the first time in history, achieved a perfect score. In 2003, he received the Deborah Tepper Haimo award, a national teaching award for college and university math, given by the Math Association of America.

The most important part of abstract thinking, at least after you get the hang of it, is to then step back and make it concrete and pull at the threads to see how a theory comes apart. The more layers of abstraction you add to a situation the more opportunities for a tiny, tiny error to grow into a larger problem.

If you follow along carefully with the arguments in most classical philosophy in great detail, you notice this more and more: everything has a logical error or assumption eventually and the author typically does not catch their own error.

> How to improve my abstract thinking?

The simple answer, is practice. I know this is absurd sounding, but it really is the goal here.

While I agree that you should continue down the path of Math, there's other ways to practice abstracted thinking. Pick up some kind of artistic hobby. Painting, music, poetry, story telling, etc. These are all just things that force you to think creatively. Whatever you find interesting. 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). Abstract thinking is often associated with connecting ideas from different subjects (abstract: disassociated from any specific instance ).

Additionally, focus a lot on creating analogies for things you are studying. This has 2 major benefits (to you). #1, it helps you remember. #2, you have to abstract a concept to create an analogy. Remember that the point of an analogy isn't to be precise, but rather to create an elegant means to abstract a concept (this also is why they are good for communication). An analogy primes you to remember a concept, it doesn't describe it exactly. After all, that high precision is difficult to express and comes with a lot of assumptions.

Lastly, another pure math method: study abstract algebra.

Agreed. I think the more you push yourself to learn and pursue things that seem abstract to you, the better. As you do so in various fields that interest you (the arts, maths, whatever) you start to draw a lot of patterns around seemingly unrelated things, and I think it teaches you a lot about yourself.

I agree with everyone here that maths is the frontier of abstraction but here's my two cents: delving into a humanities or social studies discipline you are interested in. I'm a self taught programmer and studied a history major. I was pretty good at history and got to learn a lot in there, but the most important skill I learned there was to manage abstract thoughts and complexity through reading and writing history essays.

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 code is written).

I guess any discipline that cares for truthfulness or sound reasoning is useful for this. The most important thing is you enjoy studying it and it conveys complex thoughts. Some other areas besides maths and history I can think of are philosophy, political science, economy and even architecture.

Of course this is N=1, so I'd take this with a grain of salt.

Deliberate practice in abstraction. Forcing yourself to solve abstraction problems at the edge of your current ability. This should be hard and frustrating. If it’s not, you aren’t solving hard enough problems. Do this daily and after a year you’ll notice real progress.

+1. Deliberate practice—working at the edge of your ability, where you fail at least as much as you succeed—is nearly the gold standard for continual improvement at any practice.

I would amend that do this daily and you'll see progress after months if not weeks.

I would also add that puzzles are a fun way to improve your thinking, IMO. My favorite source is https://www.ocf.berkeley.edu/~wwu/riddles/intro.shtml.

If you're really interested in the problem, it can be hard without being that frustrating. The best people can always keep going when tackling difficult problems because they genuinely enjoy the problem space.

Many people have mentioned mathematics, but I'd like to add philosophy to that list. Things like logic and set theory very much overlap with philosophy (and many mathematicians can be considered philosophers). The main reason, however, is that philosophy is thinking about thinking. So if you'd like to think abstractly, while also covering a broad range of topics, philosophy is great!

Good luck.

+1. Read some Saussure, Derrida, Foucault.

And if you're going to read Derrida, make sure you know your Husserl first...

Semi on-topic mindset related note: be careful about going too far in the direction of elegant "mathematical" thinking. It's good to have certain mental techniques available to you, but real world problems require a certain suppleness/flexibility of mind and willingness to deal with exceptions that don't fit nicely into logical molds. (unless you work entirely in theoretical spaces -- then it's ok because there's no difference between theory and practice)

As someone who was for many years enamored with theory and mathematical elegance (I drank from the wells of lambda calculus and category theory in hopes of discovering something that would set me apart), I had to unlearn much of it to actually make progress in my work. When faced with new problems, I found myself trying too hard to find an elegant solution, and when I couldn't, I was paralyzed. 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. As one iterates, some solutions will tend toward elegance and others will not -- some problem spaces are just naturally messy and the solution needs to reflect that. If you've ever worked with an ERP, you'll understand how hard it is to unify competing concepts, yet that's what an ERP does with varying degrees of success. (everybody hates ERPs, but different people hate it for different reasons, and on the whole they kind of work)

Take something like Category Theory for instance: it seems like it should lead to amazing elegant solutions, but in practice it's rarely used -- and Haskellers might disagree with me here -- to design solutions (except in rare cases like LINQ). Instead, it's often deployed as a post-hoc gloss to explain solutions that have emerged by trial and error (like SQL perhaps). Its utility is often usually retrospective, i.e. either to verify correctness or to add rigor to existing solutions.

p.s. don't underestimate the value of experience + good taste in producing good thinking. People who design good abstractions are not always deeply mathematical people, but instead are people who have good intuition, like Anders Hejlsberg (architect of Turbo Pascal, Delphi, C# and Typescript), Rich Hickey of Clojure fame (who actually trained as a musician), etc. Guido van Rossum (Python) once said he probably couldn't have designed Python when he was 17 because at that age he didn't have enough experience and good taste in programming languages -- which is why most programming language designers tend to be over 35.

Speaking of Category Theory: what can be more concrete than points and arrows between them. (That's the thing about "abstract" math - it is in fact no more abstract, and is often simpler, than the "concrete" math, especially applied math. That's why someone [don't remember who] said that less talented people should stick to working in pure mathematics.)

This. It's easy to get lost in abstractions and start thinking that the real world is just getting in the way of solving your beautiful abstract problem.

If you are looking for other branches of mathematics, I'd recommend topology and group theory; both can be pretty abstract, and work quite differently than the math you tend to be taught at school.

Otherwise, I recommend learning and using "extreme" programming languages, languages that go very far off your standard language. Examples:

* Lisp: everything is a list

* TCL: everything is a string

* Haskell: everything is pure and strongly types, and you use monads and other crazy stuff

* Rust: on the extreme end of safety through zero-cost (at runtime) abstractions

* Regexes: Just how much you can do with a powerful regex engine (declarative programming)

* Prolog

Finally: studying philosophy. Philosophers have honed their abstract thinking skills to the max :-)

>group theory

Or Abstract Algebra as whole including ring and field theory.

Lesswrong.com (a starting point is readthesequences.com) is probably worth a look. It’s usually not math-heavy (though there are plenty of those as well), but about actually applying the truths we do know to think better. E.g., lots of people know the Bayes theorem (I was taught that in K12), but almost no one actually uses that insight in real life.

> almost no one actually uses that insight in real life

I don't know a lot about Bayes theorem but you caught my attention. Why is it the case that no one uses that in real life?

Bayes theorem is a result in conditional probability, but the real life problems that lend themselves to being reliably quantified and solved with probability theory may be relatively few.

Good point. Thanks.

It may not be exactly what you had in mind, but poetry and metaphor.

I was a CS major with a Creative Writing minor. I picked the minor as an escape from CS but over the course of my career as a developer it has been some of my most useful time spent. When I was trying to write about a thing in terms of another thing (or reading other people's much better attempts) I would look at the lower level similarities and mess around trying to make the metaphor as tight as possible. Along with that, both writing code and creatively are mostly about constant revision. So both interests kind of played off each other reinforcing that process.

A different take -- drugs. Especially marijuana and hallucinogens. Helped me better understand how to think about abstract computations vs the nitty gritty details like pointers. Not for everyone, but helped me.

Similar direction but different:

Spirituality, mysticism, religion, "alternate" non-sciency stuff, but also history. When not approached with sceptic mindset but one that tries to understand it can be very enlightening.

I think it trains to think with vague, incomplete and also contradictory thoughts (its bit similar to simulated annealing in contrast to deriving a solution analytical). On a side note, its damn interesting what our heritage has to offer.

Regularly revisit it with some scepticism so you don't get lost.

I suppose.. when logic is located in the left brain hemispere, this other stuff is located in the right hemisphere. Don't fixate on only the one side. Boost it with the help of the other one.

Disclaimer, I am in no way affiliated with brilliant.org but am a happy user. Do yourself a favor and invest in a brilliant.org membership (there should be a trial period, but tell yourself if you like it and stick with it you’ll consider buying the lifetime membership). Then do all the Logic Courses. You could work your way through their Computer Science stuff after that. They are actually pretty fun and making mistakes is all part of the process and viewing the answers will often correct what you missed. The way it works is you are given multiple choice or fill in the blank questions usually with only one right answer. The problem statements are very exact and taking in all the relevant information is necessary. Then you kind of have to deduce for yourself the answers. Brilliant.org is literally the best investment I’ve ever made to increase my mathematical ability second only to University where I took up to discrete math. I would definitely do the trial period though because it may not be for everyone. 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. But, Brilliant setups up prerequisites for each course so you should know atleast what you could work on before certain ones. Best of luck and I hope someone tries it after this recommendation because I truly do love the platform.

@johnsonjo: asking this late --- which university did you do your online distance education at? I'm super interested to know! [https://news.ycombinator.com/item?id=22176989]

Hey no problem with asking late though sometimes people may not respond or they may take a while though I have hacker news replies setup [1], so I sometimes have to get through my 300+ emails per week (mostly marketing emails [just saying I’m not that popular haha]). I actually was doing it at Utah State University (USU) and I was on Campus my distance-ed program was across different campuses for a University that has multiple campuses across Utah. So, I’m not sure if they have many people that do off campus distance ed classes.

I loved my University though. They were accredited for their CS program (not sure how important that is honestly in fact I only learned about that because I had to do an exit interview). Though there are many arguably better Universities in Utah such as BYU and U of U to name a fewer, I’ve come to believe that pretty much any University will likely give you a world class education. Also Utah State is a research University which is an added bonus, because it helps to have teachers that research in their field. I felt I got a really good grasp on the theoretical principles there. My favorite course was probably Discrete Mathematics at USU. It helped me get a good grasp on abstract thinking (might as well bring that up since this thread is about that). I really recommend anyone take something similar to Discrete Math by starting to go through the readings for the Mathematics for Computer Science course from MIT Open Course Ware and possibly the videos too (OCW) (e.g. [2] or [3]). Reading the beginning of the Math for Computer Science course manual prepared me for my Discrete Math class so well (In fact I ended getting over a 100% final grade because the teacher had to move the grading curve, because likely too many were likely not doing so good). Usually before you take a class like that you need to take Calculus, but really all you need to know from Calculus to complete that class is likely sums and series. Any ways best of luck with your studies and if you plan on doing online education I’m not sure the best University to do it through. I was really glad that I was able to attend University on campus though, because though I can be disciplined I feel like I lack that discipline whenever I don’t have enough pressure to keep me focused and not get distracted on other things. Again best of luck in studies!

[1]: http://www.hnreplies.com/ [2]: https://ocw.mit.edu/courses/electrical-engineering-and-compu... [3]: https://ocw.mit.edu/courses/electrical-engineering-and-compu...

thanks for your super awesome reply! :)

I used to struggle a lot when I first started studying mathematics seriously as part of my Computer Science degree. I could never understood how people ever came up with proofs for problems that seemed completely abstract and foreign to me. Even the very notion of proving a theorum or time complexity itself seemed alien.

It was a painful journey, but in the end I found that it comes through the following:

Practice, a lot. I don't believe that anyone pulls solutions out of thin air. When tackling an abstract problem, people draw on the similar things that they do understand and then leap to the solution they're chasing.

Study the field you're trying to understand. Read over as many examples of solutions to abstract problems, and really analyze why they work. Try to solve your own problems, using what you've observed in those ones. Hop between these two until you start to close the gap more.

In reality, you're always going to make lots of mistakes and the practicing is never going to end. Its all part of the learning, and that's a journey that shouldn't stop.

If you're interested in this field of Mathematical Logic, I suggest you listen and read about it as much as possible, that you try to pursue problems you are interested in, and that you collaborate and discuss things with others. This constant doing, in small quantities but without getting up, will eventually help you see the abstract problems for something that is much more relateable.

Systems thinking and system dynamics. Run simulations. Short intro https://www.bm-support.org/pdfdocs/BMS_BusinessDynamics.pdf

Abstraction is like a multi story building. The concrete at the bottom is unrefined unprocessed raw sensory present moment experience. Building up, a floor is an abstraction built on top of that concrete floor. The concept of addition could be seen as a first floor. As we know, math builds on to of itself. The farther up the tower you go, the more abstract.

Abstractions require a solid foundation, be it a floor below, or concrete. So, when I struggle with an abstraction, I look at its base parts, identifying what is required to understand the abstraction. What I find is I often do not understand a prerequisite as well as I think I do. Sometimes I have to go multiple levels down to hash something out, but once the abstraction clicks, it becomes effortless and as difficult to think about as any other word I use in English to talk and think.

Getting a solid foundation can be a time consuming process. If you go slow and relax, you'll find the missing pieces and everything will come together.

A mental image of a concrete building somehow messes with my attempts to think in abstract terms.

Ironic given the metaphor is an abstraction.

A more accurate representation is a mind map.

I listen to interviews with thinkers that I'd admire from all sorts of spheres: business, music, film, etc and try to stitch together how they got where they are (make a point to read the books they mention or research the people that inspire them).

It really helps your brain to form pathways to seeing connections that may not be obvious on the surface.

I do the same, glad to know there are others!

I don't read as much as I used to anymore, but I listen to podcasts. Pick a subject of even mild interest to me, add a dozen podcasts on the topic. Listen to a few episodes from each podcast and delete the ones that I don't like (for whatever reason). Over a period of time, one ends up with a few podcasts that are thoroughly enjoyable.

There are people who read/listen to extreme views on topics (raw food vs meat, left vs right wing politics, yoga vs weights etc).

Read up on topics of interest to you ... join online forums on these topics ... get immersed in the unsolved challenges ... work to solve problems which take time to resolve ... progressively solve ever harder challenges ... talk to people who listen and ask hard questions

Above will get you started ... engaging in abstract thought takes time and focused attention ... coding software can provide a medium to express yourself so can expository long form writing ... before bedtime bring to mind an unsolved question then upon awakening harvest solutions ... that habit can provide feedstock for ongoing evolutionary jumps

over time increase the complexity of these long form projects ... grow them so their course may stretch for days to weeks to months to ...

nurture friendships with interesting folks ... travel wide ... gain inspiration give guidance cherish the moment

When I was taking graduate-level mathematics courses, my professor told me that abstract concepts become much easier to wrestle with if we make them real through focused realization and imagination. This has served me well.

Would you mind expounding a bit on what "focused realization" and "imagination" mean in this context?

We understand the world through our senses and our body. The best way to understand an abstract concept is to turn it into something that we can visualize or somehow "feel" in a tactile way.

For example, sometimes functions feel like a dance to me. The movement of the dance represents the transformation of the domain to the image of the function.

I think mathematical logic is potentially a bit of a red herring. I could imagine it's easy to think, "I want to practice thinking logically, surely a good starting place is... logic?" But actually I don't think necessarily involves any more creative thinking or problem solving than any other area of mathematics, and is perhaps hard to appreciate until you've practiced mathematical logic less formally.

I'd suggest something like linear algebra or analysis (in the mathematical sense of formal proofs in calculus). Those are both a bit higher level than logic, but still very abstract.

You should do a course in Automata Theory. Check out cs154.stanford.edu

I used the book "Language, Proof, and Logic" which was also excellent.

These courses will formalize concepts and allow you to apply them much better than random anecdotes.

By forgetting. Abstraction is the process of focussing on what two (or more) things have in common by forgetting all the concrete details which make them different.

pick up a functional programming language and learn its [concept] library, it's a great way to experiment with mathematical concepts while also having something concrete to play with. I suggest Haskell.

P.S. challenge yourself to solve problems with as few lines as possible, that way you're forced to find better (combinations of) abstractions

Polya’s book “How To Solve It” is the canonical introduction to this topic and applies far beyond mathematics.

beat you to it ;)

Roughly in order of user friendliness and accessibility:

Puzzles can help introduce very powerful ideas without any baggage like mathematical notation. Smullyan's "Knights and Knave" style puzzles often touch very deep ideas in mathematical logic.[1] To Mock a Mockingbird[2] is probably his most famous book.

Godel, Escher, Bach has very clear, fun, and memorable descriptions of formal systems and their fascinating properties. After reading that it will be easier to view real world systems as formal systems and to understand the implications of that.[3]

Most of object-oriented programming and entity-attribute-value models can be found in the writings of Plato and Aristotle. For the purposes of abstract thinking, Plato's theory of forms[4] and Aristole's Organon[5], especially its Prior and Posterior Analytics which describe syllogistic reasoning, are probably the most important. For roughly 2000 years, this was logic. The Theaetetus[6] is also a very good introduction to epistemology and the deductive method of philosophy. In a practical sense, there is very little that programmers do in terms of modeling data or systems that does not derive more or sense directly from these two thinkers.

It's only been in the last two centuries that we've improved on Greek logic. Boole and De Morgan for propositional calculus[7], Frege and Pierce for quantification[8], which combine to create first order predicate logic[9]. From their you can either go to second-order logic or to set theory in order to begin talking about collections of things. Naive Set Theory[10] is a good introductory book, although you can jump straight in to ZFC set theory[11] for an axiomatic approach.

Relational algebra, which will be familiar in a loose sense to anyone who has ever worked with a relational database, is a formal theory that can be studied in the abstract[12]. I find the terminology (like "theta join") to be useful for thinking about advanced SQL statements. It's also very interesting to contrast relational algebra with ZFC set theory - many of the axioms are similar, but there are also crucial differences.

Lately, in the last century or so, abstract algebra[13] has proven very useful in modelling all kinds of real-world phenomena. For example, Lie groups in physics, or finite fields in cryptography. Abstract algebra basically strips down numbers to their most basic axioms and generalizes them. In group theory we study structures that have a single operation (say addition) then "rings" allow a second operation (say multiplication) and "fields" allow this second operation to be inverted. It is incredibly fruitful to model your real-world system as an abstract algebra and then to add axioms that fit your system (do your operations commute? Are the associative? Can they be reversed?) because you can then leverage a huge number of appropriate theorems.

The mother of all "abstract thinking" has to be category theory[14] which is so abstract I can hardly even describe it. Nevertheless many people find it a useful framework, with commutative diagrams[15] showing up all kinds of papers.

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

[2]: https://en.wikipedia.org/wiki/To_Mock_a_Mockingbird

[3]: https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach

[4]: https://en.wikipedia.org/wiki/Theory_of_forms

[5]: https://en.wikipedia.org/wiki/Organon

[6]: https://plato.stanford.edu/entries/plato-theaetetus/

[7]: https://en.wikipedia.org/wiki/Propositional_calculus

[8]: https://en.wikipedia.org/wiki/Quantifier_(logic)

[9]: https://en.wikipedia.org/wiki/First-order_logic

[10]: https://en.wikipedia.org/wiki/Naive_Set_Theory_(book)

[11]: https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...

[12]: https://en.wikipedia.org/wiki/Relational_algebra

[13]: https://en.wikipedia.org/wiki/Abstract_algebra

[14]: https://en.wikipedia.org/wiki/Category_theory

[15]: https://en.wikipedia.org/wiki/Commutative_diagram

Fantastic recommendations. Thank you!

Breadth in what you think about and active attempts to connect seemingly disparate pieces of information. IMO, "abstract" refers to something that is a pattern which applies to many things. As opposed to "concrete", which refers to specifics about a specific domain.

And psychedelics tbh. There is a propensity of science that says that among other things, psychedelics improve your ability to connect what you know. https://www.sciencedirect.com/science/article/pii/S105381192...

If you enjoyed logic, you might enjoy reading/working through this text[0]. I learned from that book when it was just written, and it remains one of my favorite books/classes. As you say, it was so very different from previous math classes I'd taken, and it opened up a whole new way of thinking for me. Although it was used for a college course, the prerequisites are minimal -- maybe at most elementary algebra?

[0]: https://www.whitman.edu/mathematics/higher_math_online/

You should read about different thinking frameworks:

- https://untools.co/

- https://mindfold.co/

For me, abstract thought comes best when doing abstract things. Particularly reading, camping, cycling, playing/listening to music. As somebody who spends way to much time in front of a computer monitor, I find that anything that takes me away from screens for a few days is really beneficial for mentally tackling big problems.

A great test of how well you understand a (mathematical) concept, is if you can visualise and reason about it when you are away from a desk and doing something else. That said- you can never know all the stuff you want to know, so dont stress about it :)

This may be a bit too "abstract" but using mathematics to think about mathematics (ie metamathematics, philosophy of mathematics ...which is just more mathematics) is definitely the way to go.

Just learn more math and you'll get all the abstract thinking "workout" you can handle and then some.

But what is your goal? Is this just as a form of exercise?

If so, I recommend going through (really understanding) Godel's Incompleteness Theorem(s). In my opinion, it's very hard but it is something that's doable without an insane level of prerequisites depending on how you approach it. It's also quite interesting and one of capstones of the 20th century, but don't bring it up at cocktail parties.

The course Paradox and Infinity recently started on edX. If you start now, I think it's still possible to get a good enough grade for a certificate.


Listen to audio books in bed when you're going to sleep. Get an Audible subscription. Listen to Treasure Island or Of Mice and Men. It doesn't matter of you're really listening to the story. Just listen to it while you're falling asleep.

Bonus: it will really help you to go to sleep.

I did a Maths degree before becoming a software engineer, and honestly I think its really changed the way I think, just in general. There's something about being given a problem, or theorem to prove, and grappling with it until you really start to get a deeper understanding. After spending hours on a single problem, sitting there trying various ideas, getting flashes of inspiration, hitting dead ends, grabbing a drink, coming back and doing it all again. Finally actually getting to the point where it all just suddenly clicks and you realise that actually if you just think about it in these ways the solution is just, well, obvious! It's really intensely satisfying; just a three line proof of "without loss of generality we can assume X, which implies Y, and so clearly Z is true". So satisfying! (Then you realise you still have another nine problems to try to do before tomorrow, oh god...)


To me, it really taught how to tackle Hard Problems, where you do just sit there making seemingly no process for hours/days/weeks. When you first start tackling such problems it can feel really frustrating, but actually with experience you realise that progress is being made when you slowly manage to map out the problem space and get a better intuitive understanding what's going on. I kinda do imagine it as stumbling round in the dark in an unfamiliar place, slowing groping around, hitting dead ends, then slowly but surely getting a mental model of what's around you and how it all interconnects. Once you have that understanding and intuition the problem is often, kinda, easy? Or obviously impossible and you'll need to make some trade offs.

Changing the way you think about progress to be less goal oriented and more about expanding your understanding is really quite crucial to tackling such problems I think. Both just to keep you motivated through the process and stop you from getting discouraged, but also helps you realise when you've stopped making any progress and should take a bit of a break and come back with a fresh mind.

Most of the time this skill is entirely useless, but sometimes it really is quite powerful. I guess working on Matrix is a bit of a special case, but I would never have been able to sit down and spend weeks trying to come up with a new state resolution algorithm, to pick one example, without that sort of experience. I just wouldn't know where to start, and I'd become demotivated by the end of the first day and likely give up (knowing me).

All of this rambling is to say: I think Maths is really something you have to do. Reading books about it is interesting and great and all, but if you really want a deeper understanding you have to get stuck, get your hands dirty and try to solve problems. I don't mean problems where you take that cool theorem you just learnt and apply it or figure out how to apply, but problems where you actually have to come with ideas and theories on your own. (Now, I have no idea how feasible that is outside a formal setting and without supervisors, but that's really the dream).

I hope that in some ways helps, even if its probably entirely devoid of practical advice :)

Just wanted to say this was a great answer. As an adult just getting back into math it is very encouraging and motivating.

I find algebraic geometry really interesting and helps me with my abstract thinking too.


Sure, but you may need to learn commutative algebra first...

Conversely, I seem to have the opposite problem and not being able to work on anything in the concrete. Always just expanding onto different perspectives and revelations about the systems I am supposed to work with.

Any advice for that?

Write down & summarise everything you read. It doesn't have to be thorough and you don't even have to read it again, but somehow putting it down in words clears up any unknown confusions for me.

You might like Teach Yourself Logic, https://www.logicmatters.net/tyl/

I've enjoyed working through https://regexcrossword.com/

Algebra - group theory. I think that it kind of similar to abstract thinking in programming, through I have hard time to express exactly why.

Depends on what you are thinking. Mathematical thinking without emotions - may yield better algorithms/analysis - consumable by machines - not sure about humans. Humans need emotions attached to their thinking. Suppose if we are solving a human problem or building a product consumable by humans - the end goals are different and attached to emotions - delight/happiness/experience etc., Hopefully, I didn't misunderstand your question :)

Draw on paper with a pencil.


Read novels.


Play Go

Most competitive programmers suggest Polya's "How to Solve It". His "Mathematics and Plausible Reasoning" is also good, but longer and more dense.

I'm guessing you're not actually Linus Torvalds?

I'm not, and frankly I think I need to choose a new name as I keep getting downvoted and I believe it's due to the username lol...

Yes, seems likely. It makes it look like you're trolling.

Read books.

It seems to me that the original poster is already following this advice.

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