Anyone trying to learn mathematics on his/her own has had the experience of "going down the Math Rabbit Hole".

For example, suppose you come across the novel term vector space, and want to learn more about it. You look up various definitions, and they all refer to something called a field. So now you're off to learn what a field is, but it's the same story all over again: all the definitions you find refer to something called a group. Off to learn about what a group is. Ad infinitum. That's what I'm calling here "to go down the Math Rabbit Hole."

Upon first encountering the situation described above one may think: "well, if that's what it takes to learn about vector spaces, then I'll have to toughen up, and do it." I picked this particular example, however, because I'm sure that the course of action it envisions is one that is not just arduous: it is in fact utterly misguided.

I can say so with some confidence, for this particular case, thanks to some serendipitous personal experience. It turns out that, luckily for me, some kind calculus professor in college gave me the tip to take a course in linear algebra (something that I would have never thought of on my own), and therefore I had the luxury of learning about vector spaces without having to venture into the dreaded MRH. I did well in this class, and got a good intuitive grasp of vector spaces, but even after I had studied for my final exams (let alone the first day of class), I couldn't have said what a field was. Therefore, from my experience, and that of pretty much all my fellow students in that class, I know that one does not need to know a whole lot about fields to get the hang of vector spaces. All one needs is a familiarity with some field ($\mathbb{R}$, say).

Now, it's hard to pin down more precisely what this familiarity amounts to. The only thing that I can say about it is that it is a state somewhere between, and quite distinct from, (a) the state right after reading and understanding the definition of whatever it is one wants to learn about (say, "vector spaces"), and (b) the state right after (say) acing a graduate-level pure math course in that topic.

Even harder than defining this familiarity is coming up with an efficient way to attain it...

I'd like to ask all the math autodidacts reading this: how do you avoid falling into the Math Rabbit Hole? And more specifically, how do you efficiently attain enough familiarity with pre-requisite concepts to move on to the topics that you want to learn about?

PS: John von Neumann allegedly once said "Young man, in mathematics you don't understand things. You just get used to them." I think that this "getting used to things" is much of what I'm calling familiarity above. The problem of learning mathematics efficiently then becomes the problem of "getting used to things" quickly.

EDIT: Several answers and comments have suggested to use textbooks rather than, e.g., the Wikipedia, to learn math. But textbooks usually have the same problem. There are exceptions, like Gilbert Strang's books, which generally avoid technicalities and instead focus on the big picture, and they are indeed ideal introductions to a subject, but they are exceedingly rare. For example, as I already mentioned in one comment, I've been looking for an intro book on homotopy theory that focuses on the big picture, to no avail; all the books I've found bristle with technicalities from the get go: Hausdorff this, locally compact that, yadda yadda...

I'm sure that when one mathematician asks another for an introduction to some branch of math, the latter does not start spewing all these formal technicalities, but instead gives a big-picture account, based on simple examples. I wish authors of mathematics books sometimes wrote books in such an informal vein. Note that I'm not talking here about books written for math-phobes (in fact I detest it when a math book adopts a condescending "for-dummies", "let's-not-fry-our-little-brains-now" tone). Informal does not mean "dumbed down". There's a huge gap in the mathematics literature (at least in English), and I can't figure out why.

(BTW, I'm glad that MJD brought up Strang's Linear Algebra book, because it's a concrete example that shows it's not impossible to write a successful math textbook that stays on the big picture, and doesn't fuss over technicalities. It goes without saying that I'm not advocating that all math books be written this way. Attention to such technical details, precision, and rigor are all essential to doing mathematics, but they can easily overwhelm an introductory exposition.)

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    Sometimes you just have to swallow the red pill. – Dan Rust Dec 24 '13 at 22:11
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    Try reading books that are called an introduction to _____. – Jorge Fernández Dec 24 '13 at 22:13
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    @DanielRust: the choice of linear algebra for my example is better than I thought. – kjo Dec 24 '13 at 22:14
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    Read a book. Don't learn through Wikipedia. Not that there's anything wrong with Wikipedia (browsing wiki is a great way to learn), but it should be for exploration or reference. Once you're ready to sit down and learn a subject, you find a good book. – Jack M Dec 25 '13 at 11:59
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    Don't go down the rabbit hole too far. You might end at Bourbaki Vol. 1. – Hagen von Eitzen Dec 26 '13 at 18:56

21 Answers 21

up vote 156 down vote accepted

Your example makes me think of graphs.

Imagine some nice, helpful fellow came along, and made a big graph of every math concept ever, where each concept is one node and related concepts are connected by edges. Now you can take a copy of this graph, and color every node green based on whether you "know" that concept (unknowns can be grey).

How to define "know"? In this case, when somebody mentions that concept while talking about something, do you immediately feel confused and get the urge to look the concept up? If no, then you know it (funnily enough, you may be deluding yourself into thinking you know something that you completely misunderstand, and it would be classed as "knowing" based on this rule - but that's fine and I'll explain why in a bit). For purposes of determining whether you "know" it, try to assume that the particular thing the person is talking about isn't some intricate argument that hinges on obscure details of the concept or bizarre interpretations - it's just mentioned matter-of-factly, as a tangential remark.

When you are studying a topic, you are basically picking one grey node and trying to color it green. But you may discover that to do this, you must color some adjacent grey nodes first. So the moment you discover a prerequisite node, you go to color it right away, and put your original topic on hold. But this node also has prerequisites, so you put it on hold, and... What you are doing is known as a depth first search. It's natural for it to feel like a rabbit hole - you are trying to go as deep as possible. The hope is that sooner or later you will run into a wall of greens, which is when your long, arduous search will have born fruit, and you will get to feel that unique rush of climbing back up the stack with your little jewel of recursion terminating return value.

Then you get back to coloring your original node and find out about the other prerequisite, so now you can do it all over again.

DFS is suited for some applications, but it is bad for others. If your goal is to color the whole graph (ie. learn all of math), any strategy will have you visit the same number of nodes, so it doesn't matter as much. But if you are not seriously attempting to learn everything right now, DFS is not the best choice.

So, the solution to your problem is straightforward - use a more appropriate search algorithm!

Immediately obvious is breadth-first search. This means, when reading an article (or page, or book chapter), don't rush off to look up every new term as soon as you see it. Circle it or make a note of it on a separate paper, but force yourself to finish your text even if its completely incomprehensible to you without knowing the new term. You will now have a list of prerequisite nodes, and can deal with them in a more organized manner.

Compared to your DFS, this already makes it much easier to avoid straying too far from your original area of interest. It also has another benefit which is not common in actual graph problems: Often in math, and in general, understanding is cooperative. If you have a concept A which has prerequisite concept B and C, you may find that B is very difficult to understand (it leads down a deep rabbit hole), but only if you don't yet know the very easy topic C, which if you do, make B very easy to "get" because you quickly figure out the salient and relevant points (or it may be turn out that knowing either B or C is sufficient to learn A). In this case, you really don't want to have a learning strategy which will not make sure you do C before B!

BFS not only allows you to exploit cooperativities, but it also allows you to manage your time better. After your first pass, let's say you ended up with a list of 30 topics you need to learn first. They won't all be equally hard. Maybe 10 will take you 5 minutes of skimming wikipedia to figure out. Maybe another 10 are so simple, that the first Google Image diagram explains everything. Then there will be 1 or 2 which will take days or even months of work. You don't want to get tripped up on the big ones while you have the small ones to take care of. After all, it may turn out that the big topic is not essential, but the small topic is. If that's the case, you would feel very silly if you tried to tackle the big topic first! But if the small one proves useless, you haven't really lost much energy or time.

Once you're doing BFS, you might as well benefit from the other, very nice and clever twists on it, such as Dijkstra or A*. When you have the list of topics, can you order them by how promising they seem? Chances are you can, and chances are, your intuition will be right. Another thing to do - since ultimately, your aim is to link up with some green nodes, why not try to prioritize topics which seem like they would be getting closer to things you do know? The beauty of A* is that these heuristics don't even have to be very correct - even "wrong" or "unrealistic" heuristics may end up making your search faster.

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    You mean something like this? – Erel Segal-Halevi Dec 25 '13 at 16:06
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    That is exactly the approach that Khan Academy is taking. I am not saying that KA is the greatest thing ever, but I find, we should all try to re-structure our knowledge in non-linear DAGs, rather than the wanna-be linear crap that it is right now. Look at every advanced math text book and be amazed by how they assume that in chapter 12, you still know what "Corollary 4.32" is. – Domi Dec 26 '13 at 5:43
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    +1 to the question and the answer. I'm not much of a math person so to speak, but this question and it's answer apply probably to any area of study. I have run into the rabbit hole problem many times trying to learn some new thing about a programming language or how to administer a Linux server. I never have the patience to reach that recursive return you mentioned. I look forward to trying your BFS strategy. – TecBrat Dec 26 '13 at 12:37
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    An elegantly succinct and informative answer using analogies from computer science! Kudos to you! – Paul Dec 26 '13 at 17:20
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    So what is a graph and depth-first-search? opening wikipedia – k.stm Mar 21 '14 at 20:10

You don't learn what a vector space is by swallowing a definition that says

A vector space $\langle V, S\rangle$ is a set $V$ and a field $S$ that satisfy the following 8 axioms: …

Or at least I don't, and from the sound of things that isn't working for you either. That definition is for someone who not only already knows what a field is, but who also already knows what a vector space is, and for whom the formal statement may illuminate what they already know.

Instead, if you want to learn what a vector space is, you pick up an elementary textbook on linear algebra and you start reading it. I picked up Linear Algebra and its Applications (G. Strang, 1988) from next to the bed just now, and I find that "vector space" isn't even defined. The first page of chapter 2 (“Vector Spaces and Linear Equations”) introduces the idea informally, leaning heavily on the example of $\Bbb R^n$, which was already introduced in Chapter 1, and then emphasizes the crucial property: “We can add any two vectors, and we can multiply vectors by scalars.” The next page reiterates this idea: “a real vector space is a set of ‘vectors’ together with rules for vector addition and multiplication by real numbers.” Then there follow three examples that are different from the $\Bbb R^n$ examples.

A good textbook will do this: it will reduce those 8 axioms to a brief statement of what the axioms are actually about, and provide a set of illuminating examples. In the case of the vector space, the brief statement I quoted, boldface in the original, was it: we can add any two vectors, and we can multiply vectors by scalars.

You don't need to know what a field is to understand any of this, because it's restricted to real vector spaces, rather than to vector spaces over arbitrary fields. But it sets you up to understand the idea in its full generality once you do find out what a field is: “Just like the vector spaces you're used to, except instead of the scalars being real numbers, they can be elements of any field.”

If you find yourself chasing an endless series of definitions, that's because you're trying to learn mathematics from a mathematical encyclopedia. Well, it's worth a try; it worked for Ramanujan. But if you find that you're not Ramanujan, you might try what the rest of us non-Ramanujans do, and try reading a textbook instead. And if the textbook starts off by saying something like:

A vector space $\langle V, S\rangle$ is a set $V$ and a field $S$ that satisfy the following 8 axioms: …

then that means you have mistakenly gotten hold of a textbook that was written for people who already know what a vector space is, and you need to put it aside and get another one. (This is not a joke; there are many such books.)

The Strang book is really good, by the way. I recommend it.

One last note: It's not usually enough to read the book; you have to do a bunch of the exercises also.

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    "Just like the vector spaces you're used to, except instead of the scalars being real numbers, they can be elements of any field.” -- replace "real numbers" with "field" and "field" by "ring" and that's the first definition of a module I was given. :) – tomasz Dec 25 '13 at 0:14
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    I too have had difficulty getting any traction on new areas from sources such as Wikipedia, and moreover am frustrated to see areas I do know explained in ways that will not help a beginner. For an autodidact, sources of free online textbooks are valuable; is one place to start. – half-integer fan Dec 25 '13 at 3:54
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    @kjo: Dear kjo, One point about homotopy theory is that it is at a higher level of sophistication than basic linear algebra. The reason is that to formalize the intuitive ideas (of continuity, spaces and shapes, deformations, etc.) takes a bunch of technical machinery (the language of topological spaces, continuous maps between them, product topologies, etc.). It is not true that all (or even most) books on homotopy theory will presume you already know the definitions, but they will all presume a facility at absorbing definitions and joining the dots between formal definitions and ... – Matt E Dec 25 '13 at 3:56
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    ... underlying intuitions. It sounds like you're still at the stage of learning how to do this, and so possibly homotopy theory isn't quite the right thing for you to be reading right now (even if that's your ultimate goal). There are good avenues for learning more foundational material in topology, building (for example) on what you know about linear algebra. Perhaps this would be a good avenue for you to pursue? Regards, – Matt E Dec 25 '13 at 3:57
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    Amusingly enough, most lecturers do begin with a formal definition, since they know what they're talking about, so this dreadful definition comes very natural to them. Teaching people is much harder than one might think... – Dunno Dec 25 '13 at 19:26

A very well known mathematician showed me how he avoids the rabbit hole. I copied his method, and now I can stay out of it most of the time.

I had private weekly seminars with him. Every week, he would research a topic he knew nothing about (that was our deal and that's what was in it for him). I would name the topic (examples: Bloom Filters, Knuth-Bendix Theorem, Linear Logic), and the following week he would give a zero-frills Power-Point presentation of what he found out. The presentations had a uniform pattern:

Motivating Example
Lemmas and Theorems

By beginning with the motivating example, we never got lost in the thicket of technicalities, and the Applications section would circle back and explain the Motivating Example (and maybe some others if time allowed) in terms of the technicalities.

This is how he taught himself a topic without going down the MRH.

Limit your rabbit-hole time (one week)
your presentation must be one hour long
Focus on a Motivating Example
do just enough technicalities to explain the example and optional variations

I have since copied this style. When I teach myself a new topic, I make a slide presentation like that, and then I present it to others in a weekly reading group.

  • Wow, thanks for posting this. It's an amazing approach. The reading group in particular; it's a resource I'd not considered at all. Of course, it's not a trivial matter to form such a group and keep it going, but I can see how it would be invaluable. Also, let me say that I'm sooo envious: how I'd love to have a knowledgeable mathematician (even one who was not well-known) as my math tutor, or even just as my math "sensei"... (If I had the $$ I'd hire such a person, of course.) – kjo Dec 26 '13 at 0:17
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    Yes, I was very lucky to have this opportunity. The reading-group concept works quite well. Just circulate the responsibility for next week's presentation amongst the participants. It helps if you can provide lunch. I'm doing it now with about five of us going through category theory. Forcing the motivating example, no matter how weedy, keeps us out of the rabbit hole. It really works! – Reb.Cabin Dec 26 '13 at 1:12
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    This approach embodies some basic learning-cycle wisdom. When I was training math/science teachers to function in a developing-country environment for the Peace Corps, we used something called the 4-Mat method to get the basics to subject-matter experts who didn't have (much) teaching experience. 4-Mat has 4 stages that closely mirror yours: – Faust Dec 27 '13 at 11:48

I think that sometimes, you don't really need to know exactly what every term used means, not right away, anyway. Most of the time, a vague idea is enough to get you started.

Check out the definition (without necessarily understanding it at first -- ploughing through a huge mess of a formal definition is not always helpful at this point, but it helps to see its general structure), then see some examples, tinker a little, see how it works. If I told you everything about horse riding for a month, you probably wouldn't be as good at horse riding as you would be if you had instead tried practised horse riding for a week (and not just because I don't know a thing about horse riding ;) ).

As you get deeper into the subject matter, it might help to understand the details of the definitions, as well as the auxiliary objects. What are they for? What do they really mean? But at first, you shouldn't expect to understand everything, especially when studying more in-depth stuff which (unlike vector spaces) can get you really deep into... rabbit hole.

Familiarity comes with experience. There is no other way.

As a side comment about your vector spaces example: I don't think you can really understand linear algebra if you restrict yourself to reals. They have characteristic zero, are not algebraically closed, they are naturally ordered... this can be very misleading. It's good for starters, but I wouldn't say you understand vector spaces if you just understand real vector spaces.

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    +1 for the final paragraph. You don’t understand VS if your (only) intuition is just $k^n$ for some field $k$, and your understanding is extremely limited if you only understand real vector spaces. – Christopher Creutzig Dec 25 '13 at 10:30
  • Same +1 from me. I actually understood what vector spaces are even for when we were doing various things with matrices (kernel, image, rank etc) – Dunno Dec 25 '13 at 19:30

It is a good idea to learn about vector spaces first in the context of real scalars rather than general fields. But afterward, it is worthwhile to observe that, in most of what you learned (everything short of inner product spaces, in the usual presentations of the subject), you never used the fact that the real numbers come with an ordering; you never needed to consider whether numbers were positive or negative. And for some purposes, like eigenvalues and eigenvectors, it's actually helpful to allow complex numbers into your picture. In fact, all you needed about the real numbers was that you can add, subtract, multiply, and divide them (except of course that you can't divide by $0$) and you can manipulate equations as you learned in elementary algebra. That's why it's safe to allow complex numbers into your picture --- they share all those essential (for linear algebra) properties of the real numbers. And at this point, you know what a field is, even if you've never seen the definition or even the word, because a field is just a collection of things that resemble numbers to the extent that you can add, subtract, multiply, and divide them (except of course that you can't divide by $0$) and you can manipulate equations as you learned in elementary algebra. The formal axioms that define "field" are just the result of the observation that all those algebraic rules you learned are consequences of just a few of the rules; i.e., most of them are redundant. So "field" can be defined by giving just the necessary rules, not all the redundant ones. Of course, that makes it easier to check that something is a field, because you have far fewer rules to verify, and it also makes it easier to write the definition of "field" in a book, because it's shorter than it would otherwise be. But the true idea of "field" remains that all the usual manipulations of equations are valid.

  • +1, but this prescription seems to leave out a modern view point what MacLane described as the protean nature of math: sume and product means different things in different forms. Eg non-negative and cone-preserving matrices and their link to combinatorial geometry. In which case the order properties of the field greatly extend the expressiveness of vector space. Looking at math chronology, quaternions, octanions and one more level could also be substituted in the vector space. – alancalvitti Dec 5 '14 at 23:42

I have fallen into this vortex with a lot of my studying. The only way I think you can get out of it is to first start off by reading soft math books that don't focus in the details/proofs but try to convey what the general goal of the topic is.

I'm referring here to such books like Garrity's "All the Mathematics You Missed: But Need to Know for Graduate School" which outlines the different disciplines of mathematics, what their roles are and how they relate.

I think anybody who self-studies (and if you're learning math I think you have no choice) has figured out that there are two kinds of math books: those that explain things to you so that you understand them and those that assume you already know the basics and put a new perspective on things.

For me a recent example is combinatorics. A lot of people on here suggested Peter Cameron's book - which is great if you already know a lot of stuff he neglects to mention. If you don't, you're in hell trying to figure out where he is coming up with stuff. And then there is there is Brualdi's combinatorics book which explains things so that you understand them and is a joy to read. Now to properly understand and appreciate combinatorics, I think you need both books (Brualdi first and Cameron second). But I would have saved myself a lot of grief if I had started off with Brualdi first.

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    Thanks for the pointer to Garrity's book. I confess that I saw it some time ago on my way out from a bookstore, and snap-judged it (basically "by its cover") right into the "for-dummies" category. Now I see that this quick assessment was quite off-base. I'll give it a more serious look. Also, thanks for the pointer to Brualdi's book. BTW, I see that you are a Don Q fan. Teaching oneself math does sometimes strike as a rather quixotic project. – kjo Dec 25 '13 at 17:05
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    Garrity's title is misleading, in my opinion. The book is not a math book, but a book about the fields of mathematics. Its role is to build the graph/nodes of topics other people mentioned here. Other books I have found in the same vein are Paul Sally's "Tools of the Trade" and Louis Lyons's "All You Wanted to Know About Mathematics But Were Afraid to Ask". – rocinante Dec 25 '13 at 17:52
  • Thanks once more for the additional titles. Those two are completely new to me. – kjo Dec 25 '13 at 18:08

Learning in an organized manner (as opposed to "on your own") may help. Plans of what order the topics should be studied in have been worked out by teachers and textbook authors over the years. Trying to start in the middle (with "Vector Space", for example) probably will be difficult.

Metacademy is a community driven, open source web platform that tries to solve this exact problem. In their own words:

When you try to learn a given concept, Metacademy can show you the full prerequisite structure of the concept and provide a custom learning plan for you to learn the concept as efficiently as possible, complete with curated learning resources as well as discussions on the concept's relationship to other relevant subjects

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    Thank you for that link. It's nuggets like this that make me really really glad the Internet exists. – rocinante Dec 31 '13 at 2:59

how do you avoid falling into the Math Rabbit Hole? And more specifically, how do you efficiently attain enough familiarity with pre-requisite concepts to move on to the topics that you want to learn about?

the "rabbit hole" you describe exists in all scientific fields and in many ways is part of the difficulty of its highly specialized nature in the modern age in which it can take many years of study to get to the frontiers of modern research. yet, it is somewhat unacknowledged by experts and considered unavoidable/inevitable. (its great to get some visibility on this prominent issue with your question here.)

in many ways it is unavoidable, yet here are a few ways/strategies around it.

  • toy problems. there are sometimes simple problems in an advanced theory that are accessible with a few key definitions in that theory. that doesnt mean they are solvable with those pieces only that they are expressible. this can be psychological leverage to delve deeper in the field.

  • textbooks instead of/vs papers. textbooks often are more organized than papers, have a better sense of the overall map of the theory, and are written with beginners in mind sometimes with very careful description and order of introduction of concepts (eg calculus). there can be a lot of variation in coverage/style in textbooks on the same subject. try to find the best ones and then pick a textbook that suits your style.

  • "survey papers". these are papers that dont try to prove anything new but "survey" the field. if the field is significant then typically these papers exist. they are not always easy to locate. "insiders" know about them.

  • brilliant teacher-writers. often a field has a few writers who are known for their expository rather than research skills, and it pays to focus on their conceptualizations of the field. in some rare cases this can overlap eg Feynman comes to mind. (and conversely there may be few hard-core high-prestige researchers who clearly have little interest in making it all comprehensible/accessible to neophytes or disregard the problem entirely, understanding their primary agenda can help avoid frustration.)

  • software/algorithms. approaching mathematical subjects from the pov of algorithms that compute the various entities can be a useful pedagogical approach and has increasing relevance as some key advanced areas of math/TCS are starting to overlap. (eg combinatorics field).

  • seek discrete versions of continuous problems. sometimes it seems that math gets most abstract with continuous aspects. discrete versions of the same problem exist and can be simpler to understand/conceptualize. for example the Riemann hypothesis has a lot of very advanced continous math associated with the Zeta function, but there is a discrete version of the conjecture that doesnt even mention the Zeta function.

  • possibly approach the field from the problems it addresses/solves (aka "applications") rather than its overarching theories. more generally, there are often equivalent formulations of problems, one simpler to conceptualize for beginners than another.

  • look for interesting "bridge theorems" between fields that you know about, and fields that you are interested in. this is a theorem that shows something like an "uncanny correspondence" that seems to have hints it can or will be be developed further.

  • wikipedia can be useful but note it often is unnecessarily complex/technical on difficult areas of math/science. at times on some topics it reads as if its written by experts for other experts. raid the references at the end of the article.

  • blogs written by experts are increasingly more common and contain remarkably sophisticated expositions, some even surpassing what can be found in textbooks. it will be hard to find specific pages on subjects but once found can be gems. use google and search only on a specific blog, use their topic organization links, etc.

  • visual approaches to mathematics. many concepts can be graphed or have visual representations that may be more accessable or intuitive to beginners.

  • there are sometimes articles written on "common misconceptions in this field". they can be useful for beginners to avoid common mistakes.

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    also—focus on the statements/assertions/definitions of (key) theorems and not so much the technical proofs. math has a pyramidal structure and one can study it top-down looking for "important theorems" 1st. – vzn Dec 27 '13 at 1:50

In favor of the rabbit hole

Begin at the beginning, and go on till you come to the end: then stop.

My answer is somewhat contrarian, but I do believe it strongly: You have to fall into the rabbit hole, and you have to go as deep as possible at all times.

Math isn't science. To understand why a frog croaks we might study its respiratory system, to understand this we might get into how cells divide, then the chemical processes in living organisms, organic molecules in general, then the physics of atoms, the sub-atomic particles, etc. By now we have certainly gone too far. The best way of understanding how a frog breathes is to make simplifying assumptions at much higher scales. So we assume that cells are little machines that do a certain thing, or at least that atoms are small balls that bounce off each other and stick together.

To understand vector spaces, there is no point trying to 'gloss over' fields. Best is to know fields inside out. If you can't learn them inside out, you should know the basics as well as possible. The rabbit hole doesn't go on forever - in all cases you will reach either basic definitions, or things that we all agree on intuitively (like the counting numbers).

Sentence first, verdict afterwards

I have adopted a discipline of studying mathematics, where I never go past a single word that I don't know what it means, and I never skip over a statement that I can't understand, or justify, or understand the justification of. As soon as I have a question of the type "but why wouldn't that work if that condition wasn't true", I stop and think about it until I understand it. If I need to go back earlier in the book or to another book, I do so, even if it means I end up reading books backwards.

I don't always work like this, and there could be places where it is unnecessary, but your example definitely isn't one of them. Suppose that you are trying to learn about vector spaces. Frankly you aren't doing yourself any favors if you believe deep down that all fields are $\mathbb{R}$ and all vector spaces are $\mathbb{R}^n$ for some small natural $n$. Each time you justify something or try to picture something using this model you are storing up problems for yourself when you encounter infinite dimensional spaces, finite fields, or spaces over $\mathbb{C}$ and your mental models don't work anymore. Not to mention the huge problems you would have encountered if you had some misconception about a field (a field is a special case of a group, or a field is an ordered set with some other properties...) and had continued studying vector spaces for a while with this wrong picture in your mind.

Obviously you wouldn't adopt this method to read a newspaper, or a history book. If you don't know what an arquebuse is, but you have an idea that it's some kind of weapon, you might as well assume it's a type of sword. When you find out it's a kind of gun, you can simply slot this knowledge into the understanding of whatever you were reading about arquebuses. Similarly if you think Wisconsin is in Canada. Either it doesn't matter to the story which country it is in, or it does matter, and you find out soon enough where it is, with little effort wasted in reinterpreting other parts of the story.

Curiouser and curiouser

Now suppose that you do follow my method. You start to research fields. The information about fields you need is this:

  • You need to know the axioms that define a field. These easily fit on a sheet of notepaper in large caps written in sharpie. All of them are pretty much self-explanatory. Knowing the names (commutativity etc.) will be invaluable. Arguable any pure mathematician should know them.
  • You should know several examples of fields, eg $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, $\mathbb{F_p}$. If the p-adics or other more advanced fields are accessible to you, then all the better, but if not you can live without them. Again, there is nothing here that is not worth knowing to anyone in any branch of pure maths (or in other words, this material is something all undergraduate mathematicians have to learn before they specialize).
  • You should know some examples of things that are a bit like fields, but not fields. Eg $\mathbb{N}$, $\mathbb{Z}$, residue sets modulo a composite, reduced residue sets, $\mathbb{R}^n$, boolean rings. There is nothing extra to learn here, you only need to be able to check that various objects you have already heard of don't satisfy the definition of a field. If you haven't heard of some space, you don't need to know that it's not a field. In each case you just figure out one line of the definition that you can point to and say 'This fails, therefore not a field'.
  • You should be aware of some theorems about fields. In particular, you should know which properties of 'numbers' apply to all fields. This is again just a case of chasing the definition of a field through some knowledge you already have. Is the quadratic formula applicable in any field? No because square roots are not part of the definition. Is the formula for the solution of a linear equation applicable in any field? Yes, because the multiplicative inverse must exist because...
  • You should be aware of some differences between fields. Which fields are finite? Which fields are complete (if you already know what complete means)? What is the characteristic of a field (a one line definition)?
  • You should understand how a field has two groups embedded in it.

Now, in all of this there are really only three things which could cause you to get pulled further down the rabbit hole. $\mathbb{C}$, $\mathbb{F}_p$, and the definition of a group. All of these are things that everyone should know. In the case of the first two, all you need to know is the how the field structure works, not any other properties. This can be worked out or taught in an hour (how to add, how to multiply, how to divide, in $\mathbb{C}$ or $\mathbb{F}_p$). As for groups, you need to know the equivalent of the list above for fields. But you would honestly not be wasting your time if you were to read a whole introductory book on groups, studying every proof in detail, even if your goal is vector spaces. A vector space is also a group, as are half of the next thousand spaces/objects you are going learn about. Furthermore, as you learn about groups you are learning:

  • how abstract algebra works: you don't assume inverses, commutativity, etc unless the axioms say you can.
  • how to do logical proofs, eg how to prove that two sets are equal
  • notation in abstract algebra, eg writing binary operations as sums, as products, or in some other way.
  • some examples of groups that you can later think about when you are trying to consider vector spaces or fields in generality (for instance, why can you add a multiplicative structure to some additive groups but not others).

All of these are going to help you massively. Even if you NEVER get to vector spaces (you will), it will have been worth you while. Someone who understands groups is a better mathematician than someone who thinks they know what a vector space is, but doesn't understand groups.

Now, instead of thinking of $\mathbb{R}$ everytime you read a fact about a vector space, you can think of vector spaces over all the fields you know. Rather than limiting your imagination, you are challenging it to understand the statements you read in as close to full generality as possible.

Did I load my example? If you are studying something more specialized, then the 'rabbit hole' subjects are not things everyone needs to know, but they are things you need to know. If you are studying Brownian motion, any fact about elementary probability you come across is something you should be able to master.

Everything's got a moral, if only you can find it

When we speak to professors and other experienced mathematicians about difficult mathematical concepts, they are often able to summarize them with beautiful and compelling generalizations ('Harmonic analysis is about dualities between smooth behavior at small scales and convergent behavior near infinity'). This often makes us worry, because we aren't able to figure these things out ourselves, and they don't appear in our elementary textbooks.

The person who can make that generalization isn't fluent at doing manipulations and solving problems about Fourier transforms because she has understood 'what harmonic analysis is about'. She is fluent, AND she can come out with nice generalizations, because of all the work she has done on the mechanics of harmonic analysis, solving problems, reading slowly through proofs etc. To emulate this person, we shouldn't try to be able to summarize a topic in a neat way. We should try and acquire their detailed knowledge of the mechanics of their subject. Once we've done this, the flashes of insight will come by themselves.

You should also know that some harmonic analysts might think that this 'motivation' for harmonic analysis is wrong, irrelevant or trivial. However all kinds of harmonic analysts would be able to follow each others' work by giving the precise definitions they are working with, starting with the things they all agree with.

Seeing the big picture is nice, but you can often afford to miss it. Miss the small picture, and you're not doing mathematics anymore, just reading about it.

  • 2
    This is all very well in theory but completely inapplicable in practice. There is no "beginning" and "ending" in mathematics - unless you're talking about learning things in the historical/chronological way when they were developed. And if you do that, you've just traded your rabbit hole for all the circles of hell in Dante's Inferno. – rocinante Dec 31 '13 at 2:43
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    @rocicante: Firstly, as I said, this is how I work 'in practice'. Secondly, of course there is a beginning in mathematics - every definition refers directly or indirectly to undefined terms, every theorem follows directly or indirectly from axioms. – jwg Jan 1 '14 at 23:56
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    "Seeing the big picture is nice, but you can often afford to miss it. Miss the small picture, and you're not doing mathematics anymore, just reading about it." This is true in my case. Thanks for providing this view. – Gabriel Fair Jan 3 '14 at 19:26

My 2 cents: break the text into chunks you can swallow by allowing yourself multiple passes, where you accept less and less on faith and intuition each time through. I dimly recall Terry Tao giving similar advice on his blog somewhere; hopefully someone can give a link.

For elementary topics, a lot of us are able to "absorb" everything at once, all the definitions, examples, theorems, proofs, unspoken intuitions, etc. At some point it just becomes too much, so do the only thing mathematicians know how to do: break the problem into pieces. Get a high level overview on the first pass, get a sense for what's important on the second pass, figure out what you care to understand on the third pass, and break up those parts you care about iteratively until you reach something manageable.

To take a hypothetical, suppose you're reading a chapter with 50 propositions, lemmas, theorems, and examples. On the first pass you might read the introduction and very quickly skim the rest. After you have a very hazy idea of what the chapter is about, re-skim enough to identify the important theorems and examples. Here I sometimes write a paragraph or two summarizing the chapter. (It'll be incomplete and might be badly wrong, which is fine. It's kind of funny to read my own hazy ideas after I actually understand a topic.)

Now figure out what you actually want to understand; at some point there just isn't time to understand every detail, so you'll have to pick and choose anyway. Perhaps the first half of the chapter proves Theorem M and the second half Theorem Y, which you don't care about yet. Great, no need to slog through Lemmas P, Q, and R, since those are only used in the second half. Maybe Theorem M roughly says Example A is the "only" example; that really says you should get comfortable with Example A, maybe play around with it a little on scratch paper before starting the journey to Theorem M. If it's a long journey, break it up into multiple passes too--maybe start by reading its proof so you know which lemmas are important, and make another rough summary. Perhaps make the path easier by assuming an abstract object is something particular, eg. that a general field is just $\mathbb{R}$. It's fine to read non-linearly.

Eventually you'll have to actually "reach the bottom" and get your hands dirty with technical details, but at least for me it's very helpful to have a high level overview of where things fit into each other before getting down and dirty. I have no hope of retaining much of a complicated topic otherwise. Oh, sometimes I write myself a technical summary, which can be helpful--perhaps rigorous definitions and statements of the ingredients to Theorem M together with a proof of Theorem M in my own words using those ingredients. If I really want to understand something, I add "proof ideas" to a technical summary, where (at the time) I can reconstruct rigorous proofs from my summary. Exercises can also help; sometimes I look for relevant exercises after the first few pages, to get me used to basic definitions that are fundamental to the rest of the chapter.

This works better with some sources than others. If you can't even begin to give a hazy summary (maybe you have no idea what any of the words mean), you're probably reading something written for experts, which doesn't yet include you, so a gentler introduction might be in order. Always be kind to yourself: written math is typically the product of years (or decades, or centuries) of labor by brilliant people. It's no wonder it often takes a long time to absorb.

Finally, I'd like to note that I wrote this post roughly in the style I advocate. The first paragraph is a sketchy overview of the main idea, the second fills in some details, and the third through fifth "get down and dirty" with actionable advice.

  • "write a paragraph or two summarizing the chapter. " This is critical into formally learning the topic into a form that can be recalled easier. – Gabriel Fair Jan 3 '14 at 20:17

What you are talking about is the process of learning and curiosity.

When researching, you have to keep in mind your goal. Why do you need to study rings when you are just trying to implement an FFT? You don't. So you have to know when to cut off curiosity in exchange for working towards your set out goals.

Now if you don't have defined goal in your study of math, then your goal was to wander about in the rabbit hole. And there's nothing wrong with that, you'll become a stronger analytical thinker the more you study mathematics.

  • bobo: Sorry, I can't stop my curiosity, and I think no one should. In this case, shouldn't I read scifi or fantasy because I live in a real world. This is a sad life... – Ernesto Iglesias Feb 17 '17 at 15:08

The real question from the OP:

How do you avoid falling into the Math Rabbit Hole?

MJD's answer hints the the answer to this question: Read a textbook. Textbooks are designed to introduce the material in the correct order, without overburdening or going into too much detail. They are designed to help you progress.

Knowing which textbook to read is a problem unto itself, and for that you should really ask a professional (See: asking a doctor instead of self-diagnosis). To get an idea if the textbook is suitable for you, skim through the first chapter only. If you feel that you grasp or could grasp the content of that chapter, then you're golden. If not, then go find the textbook which will bring you up to speed for that first chapter.

  • 3
    I think that's a really simple (and potentially hazardous) way of going about things. I think the OP knows to read a textbook instead of Wikipedia; the sticking point is to figure out which textbook is appropriate. The accessibility of the first chapter is often misleading. – rocinante Dec 25 '13 at 15:13
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    @rocinante: That is why I mention that he should ask which textbook to read. – dotancohen Dec 25 '13 at 16:16

It's possible to visualise a 2D or 3D spatial object with some structure in your imagination, which is made of abstract stuff and behaves in a certain way. For example, for a vector space, I visualise something like a couple of arrows from a point (the origin), pointing in general direction (in the actual imagined instance, the directions are quite specific), and a part of a plane stretched between and a bit beyond them. And all these are in a state of frozen animation.

As one gets used to vector spaces (for example), this sort of thing helps. The reason is that a lot of questions about all vector spaces could be answered by considering just one non-trivial example that you are familiar with. This speeds up the search through the list of things you know, by using extensive spatial reasoning and memory capabilities of the brain.

In general, the behaviour of, say, a vector space, is infinitely complex. The rabbit hole is infinite! (And splits into exponentially many burrows as a function of depth). That is to say, while the definitions are small and finite, the set of all possible behaviours is infinitely complex, for objects such as groups, fields, vector spaces.

If someone spent a lifetime working with vector spaces, they would probably have in their mind a museum of representative visualised objects, and some efficient ways to choose which to use in order to answer some question or other. Some people, I'm pretty sure, also have at least one other way of thinking that is much faster than this kind of spatial visualisation but consistently yields intuitive answers. Some students in the International Mathematical Olympiad can solve previously unseen problems ten times faster than the average professional mathematician, and I'd like to know how. I've also read that some (most?) chess grandmasters maintain their level of performance when they are asked to continuously solve simple sums while playing chess. So perhaps there is another form of familiarity.

Anyway, to answer the question:

  1. It's very useful to explore the nearby reaches of the rabbit hole. With a short Ariadne's thread. It's easy to miss interesting areas even there, so a solid set of exercises from a classic set of lecture notes is invaluable to get good coverage rapidly, and train whatever construct is your understanding.

  2. If you have some goal, that's even better, and while some pure mathematicians find their nirvana in a random walk through relatively scary reaches of the rabbit hole (which gets rapidly more horrendous and useless with depth; though I'm sure there are elusive veins of gold and astonishing shortcuts etc.), you can't go wrong with a useful goal from the real world.

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    I object to the notion that the rabbit hole is infinite. But perhaps we should start acting like it is :) – Eric Stucky Dec 25 '13 at 16:51
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    @EricStucky Hmm, Lewis Carroll also didn't seem to think so... – Evgeni Sergeev Dec 25 '13 at 23:08

I read a lot of math on my own as a young person. I would go sit in the stacks, and sift through the books on what I wanted to learn about till I found one that "spoke to me". Part of that test was if I could work some exercises. Once I found such a book, I went through it, working as many exercises as I could. If I couldn't find such a book, I just spun my wheels for a bit till my interests shifted.

Avoid swallowing definitions and going from there. While they give you the illusion you are learning, you are potentially missing insights and masking problems.

What I'd suggest is reading some university textbooks aimed at first and second year mathematic-students. In my experience (limited) they start from the ground up and go from there.

However, if you don't want to spend your time working through books, most universities offer non-degree programs. You can select the classes you want and learn what you want thoroughly. This has the advantage that you'll have access to cheap summaries, exercises and somebody willing to explain it to you.

For vector spaces specifically, every introductory book in linear algebra should do.

Here's my short naive answer, not always possible but really good when it is: ask someone who knows and whose pedagogical skills you respect. Interrupt when you need less (or more) technical precision in his/her response.

  • I wish more american students were more assertive in the classroom on when to ask for less and more precision. But after learning about student participation in the classroom in Asian and Indian countries, Americans don't seem so bad. – Gabriel Fair Jan 3 '14 at 20:35

This question has much broader implication. "Going down the Math Rabbit Hole" simply reveals the nature of being any kind of academic, not just a mathematician. This kind of no ending thing is good for some people who enjoy having things they do not know so that they can keep doing things (whether the things they are doing are worthwhile is a completely different question). This no ending thing, however, really is bad for other people since it indeed is frustrating to know that you can never finish and you can never have a closure. As an actuarial graduate who tries to make up sufficient mathematics, I find it is helpful to take some fundamental subjects like calculus and algebra without which it simply is too difficult to do anything else. Also I suggest a healthy attitude towards study. Nowadays, inter-discipline work is the norm. One could never acquire all the knowledge they need for their work. For my own purpose, I only need to know enough so that I can communicate with my colleagues who are mathematicians, which is the ideal balance in my opinion.

  • And this is becoming more common. Academic fields can get so specialized that working together with colleagues to research a subject is all but mandatory. – Gabriel Fair Jan 3 '14 at 20:41

I'm not good at algebra but as far i can tell you is that when you need it, you will get it. "deep stuff" exists only as answers to deeper questions, and this kind of learning appears to be systematic in every age. Is tempting to dig on Wikipedia, but previously said is like you here acting like a graph visitor algorithm: visiting adjacent nodes and stacking previous. Note also that Wikipedia is turning a Monster - contributors are more concerned about the completeness than the audience. And this completeness in a mathematical sense is ... formality and verbosity!!! Thats why we need books and exercises. After a lot of exercises your brain will ask "why i'm not organizing or relating those mathematical objects in that in someway"... and then you go to Wikipedia looking for answers to find out that another set of brains did that. And then you acknowledge that you understand 5 or 6 hyperlinks and you say "thanks wikipedia! without you wouldnt be possible" :P

The thing is, you don't have to know what a field is in order to work with vector spaces. The field "stuff" relates vector spaces to a generalization which connects them to other spaces. Skip the rabbit hole, read the rest of the chapter, and do some exercises.

You don't have to understand the word "beverage" in order to swallow the Kool-Aid.

As one of many "Alices" in the MRH called stackexchange I rather think that studying math with out structure is like waking up in free fall. You see everything around you, you may even come to understand a few things that wiz by your head but you can't shake the feeling that you are falling. More to the point it is like reading a book in a foreign language with a translation book near by. Its a difficult read but doable

  • Interesting to point out how much of human information is being assembled and dissected within online forums like SE. I believe that in the next few years we will see crawl bots that parse the internet, fall into the MRH and will invent applicable algorithms on how to escape from the MRH. – Gabriel Fair Jan 3 '14 at 20:47

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