# What is “the intuition behind a result”?

A search for "What is the intuition behind" on the main site yields loads of questions that use this phrase, many in the title. Seeing it used a lot in recently posted questions prompted me to try to understand what it means.

It seems that people use "What is the intuition behind ...?" roughly synonymously with "Could you please explain ... to me?". But I find it intriguing that they add this extra level of reification. They're not directly asking you to explain something to them, but to explain something lying behind that – an "intuition" which presumably is the source of an understanding of the thing itself. But what could possibly be the difference between understanding the thing itself and understanding "the intuition behind it"?

Perhaps someone who uses this figure of speech themselves or has a better understanding of it can throw some light on this? Or perhaps I should be asking what is the intuition behind asking what is the intuition behind something?

• Are you asking for the intuition behind understanding what is the intuition behind a result even means? :-) – Asaf Karagila Apr 21 '16 at 10:37
• They are essentially asking for a general strategy to understand (and hence able to deal with) not a single but a whole class of similar problems. Explaining/solving a problem is equivalent to catching a fish for them. Some people prefer to know how to fish themselves. – achille hui Apr 21 '16 at 10:58
• You know, understanding is inferior, it's too "verkopft" (washes and disinfects keyboard). Intuition is superior because it's "ganzheitlich" (takes keyboard to the Sondermülldeponie and gets a new one). – Daniel Fischer Apr 21 '16 at 11:49
• As a simple example, the intuition behind a proof that a series converges might be that the terms go to $0$ as fast or faster than a "good" series. The actual result involves filling in the details. – André Nicolas Apr 22 '16 at 0:40
• It can mean many things, but in this particular context it often is simply, "Explain this math concept to me without using any math." – anomaly Apr 22 '16 at 7:16
• I often come across such questions with regard to seemingly unmotivated definitions. Here the question is: why is this the right thing to do. – Mathmo123 Apr 22 '16 at 15:52
• For example, I don't "understand" this short, elementary proof (at the bottom) that a Jacobi matrix is self-adjoint if its off-diagonal elements grow no faster than $n$. – Keith McClary Apr 22 '16 at 17:14
• This should be re-asked on, or migrated to, the main site. Nothing of meta relevance. – zyx Apr 26 '16 at 17:33
• @zyx It may not be quite on-topic here on meta, but it would definitely be way off-topic on main. – Daniel Fischer Apr 26 '16 at 18:22
• @DanielFischer, how so? The first two paragraphs are a well-formed question about mathematical terminology: can phrases like "intuition behind" something mean anything different from "explanation" of the thing. – zyx Apr 26 '16 at 18:24
• @zyx "Intuition" is not mathematical terminology. Psychological maybe. – Daniel Fischer Apr 26 '16 at 18:25
• It is a commonly used term in mathematics publications, and some aspects of its meaning and use can be defined precisely. Which might be the outcome of a main-site thread. @DanielFischer – zyx Apr 26 '16 at 18:27
• Thank you for asking this question which has been haunting me as well for quite some time. I – Hagen von Eitzen Apr 29 '16 at 15:27
• Terence Tao's blog post describing three stages of mathematical understanding is relevant. – littleO May 1 '16 at 0:06
• @zyx: I agree: this is a question about (a specific element of) the language that mathematicians use to talk about mathematics. It may verge on being meta-mathematical, but it certainly isn’t meta-MSE-ical. – Brian M. Scott May 2 '16 at 9:37

I posted this question regarding intuition of denseness not too long ago, so maybe I can shed some light on your question by explaining this from my point of view.

If I wanted a definition of denseness, for example, I could read my analysis text or go on Wikipedia and be slapped with a bunch of jargon I might not know a priori. And if you read the question, I do have an idea regarding the definition of denseness.

But definitions aren't the most useful things when it comes to real problem-solving. Problem solving requires intuition, which is literally defined as "being able to solve something without conscious reasoning."

So, for example, let's take the line integral over a scalar field. You can go up to your average multivariable calculus student and give them a line integral problem and they'll bash out some algebra and calculus and give you an answer. But if you ask them, for example, "Can you quickly give me a nontrivial case where the line integral equates to zero," they might fumble with it. It's not because they're not thinking of cases quickly enough, but because they don't have intuition for what a line integral means.

Then, show them this animation of how to calculate a line integral. To be completely honest, this animation doesn't really help me solve any single line integral problem - that's what the definition and simplification formulas are for. But what this animation does do is show me what exactly is a line integral.

If you ask them the same "line integral equating to zero" question after showing them the animation, I think they'll be able to answer it much more easily.

tl;dr: giving someone intuition for a topic is explaining the topic to them in a way that doesn't help solve any single problem, but helps develop reasoning for how to solve problems. I think this might be a subset of "can you explain ___ to me," but I also think it's a really important specification.

Just as a side anecdote: I was once asked if you have a central circle $A$ of radius $r$, how many other nonoverlapping circles of radius $r$ can you place such that they are each tangent to $A$? To visualize it, I pulled out a box of Cocoa Puffs and picked a few up with my spoon, and quickly realized that the answer had to be $6$. I gained intuition for the problem by doing this, and then rigorously proved it not too long after.

• I have to admit that this Wikipedia animation makes no sense to me. But then again, I do not possess any reasonable geometric intuition. But I do have some idea about what is a line integral. At least because I understand the formal definition. – Asaf Karagila Apr 24 '16 at 4:23
• Do you find visual and physical intuition of the deepest kind instead of (stupid) symbolic manipulation ? – user185498 May 1 '16 at 5:03
• This is almost like a behavioral psychology-type definition of what "mathematical intuition" is. Well put. – Jack M May 3 '16 at 23:46
• @AsafKaragila I had no idea what the line integral represented (although I could work with it) until that animation in particular. Being able to work with something and being able to understand it, although pragmatically equivalent, are usually different things. – user123641 Nov 13 '17 at 19:13

(I once asked a question about the intuition behind the dual problem in optimization.)

As a math learner I've frequently had the experience of reading a proof and being convinced by it, and yet having no idea how someone would have thought of the proof or why someone would have expected this result to be true. Then, after thinking about the theorem a lot and reading other sources, I eventually find a way to look at it that makes it seem easy and obvious. That's what I mean by "intuition" -- an explanation for why the result is secretly easy or obvious, even though it might appear daunting upon first sight.

A good example is the proof of the multivariable chain rule. The proof seemed quite difficult to me when I first read the proof in baby Rudin. (Try looking at Rudin's proof on p. 214 and pretending you don't already know the intuition for this result.) But compare Rudin's proof with the explanation given by Terence Tao in his book Analysis II, on p. 553:

Intuitively, one can think of the several variable chain rule as follows. Let $x$ be close to $x_0$. Then Newton's approximation asserts that $$f(x) - f(x_0) \approx f'(x_0)(x - x_0)$$ and in particular $f(x)$ is close to $f(x_0)$. Since $g$ is differentiable at $f(x_0)$, we see from Newton's approximation again that $$g(f(x)) - g(f(x_0)) \approx g'(f(x_0))(f(x) - f(x_0)).$$ Combining the two, we obtain $$g \circ f(x) - g \circ f(x_0) \approx g'(f(x_0))f'(x_0)(x - x_0)$$ which then should give $(g \circ f)'(x_0) = g'(f(x_0)) f'(x_0)$.

Once I learned to look at the chain rule this way, I realized that it's one of the easiest / most obvious results in math. This intuitive argument can be made into a rigorous proof just by keeping track of the errors in the approximations. That's what's "really going on" in baby Rudin's proof. With this new understanding, I began to love the chain rule. It's hard for me to understand why people would write about it in textbooks without mentioning this intuitive viewpoint. (Perhaps they forgot that it isn't already obvious to new readers.)

Edit: By the way, Terence Tao wrote an interesting blog post describing three stages of mathematical understanding.

1. The "post-rigorous" stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory. (For instance, in this stage one would be able to quickly and accurately perform computations in vector calculus by using analogies with scalar calculus, or informal and semi-rigorous use of infinitesimals, big-O notation, and so forth, and be able to convert all such calculations into a rigorous argument whenever required.) The emphasis is now on applications, intuition, and the "big picture". This stage usually occupies the late graduate years and beyond.

The transition from the first stage to the second is well known to be rather traumatic, with the dreaded "proof-type questions" being the bane of many a maths undergraduate. (See also "There’s more to maths than grades and exams and methods".) But the transition from the second to the third is equally important, and should not be forgotten.

• Rudin is proving the multivariable chain rule; Tao is explaining the single-variable chain rule. – Gerry Myerson Apr 23 '16 at 23:43
• @GerryMyerson No, that's not correct, on p. 553 of Analysis II Tao is discussing the multivariable chain rule. At the beginning of the passage I quoted he states, "one can think of the several variable chain rule as follows". – littleO Apr 23 '16 at 23:47
• What does $f'(x_0)(x-x_0)$ mean, in a multivariable context? – Gerry Myerson Apr 23 '16 at 23:49
• Assuming $f:\mathbb R^n \to \mathbb R^m$ is differentiable at $x_0$, then $f'(x_0)$ is the unique linear transformation from $\mathbb R^n \to \mathbb R^m$ such that $\lim_{x \to x_0,x \neq x_0} \|f(x) - (f(x_0) + f'(x_0)(x-x_0))\|/\|x-x_0\| = 0$. So $f'(x_0)(x-x_0)$ is what you get when you plug the vector $x-x_0$ into the linear transformation $f'(x_0)$. – littleO Apr 23 '16 at 23:53
• In some other books (like Calculus on Manifolds by Spivak) $f'(x_0)$ is an $m \times n$ matrix, so then $f'(x_0)(x - x_0)$ would be the product of an $m \times n$ matrix with an $n \times 1$ column vector. – littleO Apr 24 '16 at 0:01
• I think this is a great example. If someone asks for the intuition behind a proof (as opposed to a definition/theorem), they often just want to get the overall structure (or the main idea) behind the proof - without getting bogged down in $\varepsilon-\delta$ (or other) details, for they don't see the forest with all those trees in sight. It is also quite common for me that I can follow a formal proof step by step but in the end, I ask myself "what did we really do, how can one come up with this proof", etc. – PhoemueX Apr 24 '16 at 15:32
• A more extreme example of this kind is multivariable change of variables. If you look at it from the correct point of view (and support it by some kind of argument "determinant describes the distortion of volumes") it seems nearly obvious that it holds, while the proof is extremely tedious piece of analysis. – Wojowu May 4 '16 at 18:49

A really simple example that came up in the class I teach today... In probability you can define independence in the following way:

"Two event $A$ and $B$ are independent if $\text{Pr}(A\cap B)=\text{Pr}(A)\cdot\text{Pr}(B)$"

Then you can prove that if two sets $A$ and $B$ are independent, then $\text{Pr}(A\mid B)=\text{Pr}(A)$.

You can prove this pretty easily:

$$\text{Pr}(A\mid B)=\frac{\text{Pr}(A\cap B)}{\text{Pr}(B)}=\frac{\text{Pr}(A)\cdot\text{Pr}(B)}{\text{Pr}(B)}=\text{Pr}(A)$$

If someone wanted the intuition for why that formula is true though, you would say something more like:

"If two events are independent, knowing that one happened won't affect the chances of the other happening because they have no connection to one another."

This doesn't give a mathematical proof, but builds an intuition for what independent sets are.

• I disagree with this example. IMHO you gave the wrong definition of independence. The correct definition is $Pr(A|B)=Pr(A)$ which, in words, is quite intuitive and can be illustrated with simple examples. I agree that the equivalent statement $P(A\cap B)=P(A)P(B)$ is not intuitive, but happens to be very computationally useful. – David Hill Apr 26 '16 at 17:17
• @DavidHill I see what you are saying, but it is hard to say what is the "wrong" definition. It all depends on how the material is presented. For example, I could motivate this definition with an example as follows: "Say the chances it is raining in London are 30% and the chances I am in a car are 10%. Imagine if we chose 100 random times and check if it is raining and if I am in a car. We expect around 30 of those times to have it be raining and out of those 30, 3 to have me in a car since me being in a car has nothing to do with it raining in London(I'm from the US)." – Sean English Apr 26 '16 at 17:29
• @DavidHill Also it might not hurt to mention that the textbook my school uses for the class I teach has the definition that way, so it isn't me just randomly choosing to present the definition like that. I agree that the characteristic is more intuitive though, and if I wrote my own text I would probably present it as the definition. – Sean English Apr 26 '16 at 17:38
• Well, as I said, the statements are equivalent. One tells you what independence means and the other tells you how to compute with it. It is up to you as an instructor to decide the most constructive way to present the material to your class. I think I have made my personal opinion clear. – David Hill Apr 26 '16 at 20:15
• @DavidHill Sure, you have, but I don't see what your complaint with my example is. You can have a problem with which one is a "better" definition, but that doesn't affect the example trying to illustrate the idea of intuition. – Sean English Apr 26 '16 at 20:20
• @DavidHill $P(A \cap B) = P(A) P(B)$ makes sense even if $P(B) = 0$, whereas (as far as I know), $P(A|B)$ is not defined in that case. So that is why the definition is $P(A \cap B) = P(A)P(B)$ instead of $P(A|B) = P(A)$. – Bungo Apr 26 '16 at 22:00
• @Bungo: Usually it is convenient to define $P(A \mid B) = P(A)$ whenever $P(B) = 0$. So that's not a good reason. – user21820 Apr 29 '16 at 13:17
• @user21820 One reason why it is convenient to define $P(A\mid B)=P(A)$ when $P(B)=0$ is so that it matches the independence definition of $P(A\cap B)=P(A)\cdot P(B)$. Since this definition motivates the convention, I would say it is a good reason personally. – Sean English Apr 29 '16 at 13:23
• @SeanEnglish: Okay alright I concede. =) – user21820 Apr 29 '16 at 16:14

When people say intuition, they mean two different things, which are often treated as the same:

1. Heuristics. A continuous map of topological spaces is one such that the inverse image of an open is open (confusing) but heuristically is one such that nearby points go to nearby points (not confusing); however, sitting down and making the heuristic precise ends up forcing the original definition on you.

2. A certain mathematical ability to make guesses that we can't put into language quite yet but that turn out to be right once we think them through, which arises both as a natural talent and out of experience of working intensely with the same mathematical objects for a long time.

I am in the minority but I rather dislike the use of the word intution in the mathematical community; I especially don't like "give me the intuition behind the result" which smacks of magical thinking (of course someone who says this means it in the sense of definition 1, i.e., "can you please give me a heuristic for understanding this instead of these details," but it often comes across as a naive belief that there is a shortcut into definition 2 of intuition).

• sed s/'in the mathematical community'// ;) Intuition is just a pretentious word for "enough experience to know how it works without consciously thinking about it". – Daniel Fischer Apr 26 '16 at 12:07

An example. In set theory describing sets intuitively means leaving some terms of a definition undefined and thereby relying on common sense so as to its meaning. Several paradoxes came from this.

In order to avoid these paradoxes an axiomatic approach to set theory arose where attempt was made at greater formalization. Whilst being more precise it is more terse and so not as 'intuitive' and therefore more difficult to grasp without knowing the "local context".

Thereby when someone asks for a an intuitive understanding, they want an answer which is grounded in common sense.

• Are you sure? My intuition about set theory says otherwise. – Asaf Karagila Apr 22 '16 at 6:48
• Hence the importance of shared context for intuitive explanations! – Will Apr 22 '16 at 17:01

I think that questions of the form

What is the intuition behind...

are inherently I'll posed.

Some time ago I dared to ask a similar question on some intuition on another SE site and got a comment like

Your intuition may be different from mine...

Intuition is not something that is behind some result or proof or such. It's something that is in one's mind. And minds differ. Some people find geometric arguments intuitive, others can't visualize more than two dimensions. Some prefer analogies to ecological or even psychological situations (think of all the riddles involving hats, prisoners and stuff), others get distracted by such analogies.

So I think any answer should state "My intuition is..." or "I explain this to me like...".

• A mathematics question without the word "intuition" can have multiple valid answers posted by different people. This happens rather frequently. So I do not see why this is a reason to treat "intuition" differently than other words. – zyx Apr 26 '16 at 20:14
• @zyx So you would welcome the question "What is the solution of $x^2=4$"? Come on, we do math and care if there is the answer, an answer or probably no answer. – Dirk Apr 26 '16 at 20:37
• I think you are confusing solving a problem (which involves things like writing and explaining an answer, giving a proof, or posting on the internet) with the mathematical idea of solution set of an equation. If the question is "solve x^2 = 4" there are different valid answers but that does not make the term "solve" ambiguous or personal in a problematic way, unless one is writing software to respond to such questions on a computer. – zyx Apr 26 '16 at 20:47
• -1 This answer gets completely side-tracked on formulation and does not address the why of intuition questions at all. – Lord_Farin Apr 26 '16 at 20:56

The first two paragraphs of this question are a reasonable and answerable question for the main site, but not the meta. "Intuition" has all kinds of straightforward meanings that do not involve strange reification as this question puts it, but the place to explain that is the main site. It is off topic for meta.MSE.

A search for "What is the intuition behind" on the main site yields loads of questions that use this phrase, many in the title.

The search also yields loads of questions that define the phrase in some way, such as a request for visual images or examples to keep in mind when using a definition. I take that as further evidence that the question can be productively discussed on the main site.

https://math.stackexchange.com/search?q=%22What+is+the+intuition+behind%22

I guess the only meta relevant item here is: if intuition-as-distinct-from-proof is part of a MSE question, the tag [intuition] can be added so as to make that searchable.

• Why not put your money where your mouth is, and explain it on main. – quid Apr 26 '16 at 18:09
• Is there a question to explain it at? It seems it is joriki's task to post that if he is interested. The word "intuition" seems clear enough for many others including myself, that I have no reason to post that question from my account, and even less reason to do so as a self-answered post. If it's only a matter of cleaning up the meta, then this thread can be abandoned, migrated, or OP can request a deletion. @quid – zyx Apr 26 '16 at 18:20
• To me, and likely also OP, the question seems off-topic on main. You claim otherwise. It seems fair the onus is on you to test the topicality. – quid Apr 26 '16 at 19:09
• Instead of the topicality of hypothetical questions that you are volunteering others to write, why not explain how the actual posted question on meta is on-topic for the meta site. It does not discuss the site software, or how the [intuition] tag should be used, or anything like that. – zyx Apr 26 '16 at 19:14
• The edit may help you. If the OP is not satisfied with the notions of intuition stated in the questions asking for it, he could be more specific about his dissatisfaction. @quid – zyx Apr 26 '16 at 19:22
• The question asks for clarification of the intended meaning of a type of formulation used on the main site that OP seems to consider (and I'd tend to agree) as somewhat site specific jargon. This is a question about an aspect of posts on the main, and as such a meta question. That you seem to be of the opinion that this is not site specific jargon, but common terminology does not alter this (conveying this perception might qualify as answer though). The fact that case studying the usage on the main site is possible does not corroborate it not being a meta question, rather the converse. – quid Apr 26 '16 at 19:34
• There are over 230000 search hits for "intuition behind" proof excluding Stackexchange, which is almost 1000 times the number of posts with the words "intuition behind" on the MSE main site. There are over 700 hits on Mathoverflow, suggesting that professionals also are comfortable with this phrase. The idea that it is "somewhat site specific jargon" for MSE is hard to reconcile with those statistics. @quid . [refs: google.com/… google.com/#q=%22intuition+behind%22+++site:mathoverflow.net ] – zyx Apr 26 '16 at 19:46
• I am of the firm opinion it is somewhat site specific jargon for MathOverflow. – quid Apr 26 '16 at 19:49
• The hits on (MO + MSE) are less than 0.005 of the total, 1 part in 200. Excluding scraper sites we are still talking about 1-2 orders of magnitude below the total. How is this an MO/MSE localism? – zyx Apr 26 '16 at 19:51
• 8600 hits for "intuition behind" at arxiv.org. google.com/#q=%22intuition+behind%22+site:arxiv.org . How is it a local jargon again?. @quid – zyx Apr 26 '16 at 19:57
• The word "hint" is quite frequently used elsewhere too, still the usage on this site has some specifics to it that can lead to meta questions about it. It is at least plausible the same applies to "intuition." As explained, the fact that you do not think this is the case does not make the question off-topic; instead it might qualify as answer. – quid Apr 26 '16 at 19:57
• All discussion of "hint" on meta has been about actually on-topic matters such as how to respond to "hint wanted" postings, the merits of hint answers, or adding a [hints] tag. At no point did anyone suggest that the commonly used word "hint" required clarification or involves mysterious processes of reification. (Provide a link if you have seen otherwise.) In this question we have none of those things that might create meta topicality; the only such thing is the mention of [intuition] tag in this answer. @quid – zyx Apr 26 '16 at 20:05
• @zyx I must say that it surprises me that you use the answer box for a close vote, while there are none of those on this question. Besides that, I think your over-arguing your case to the point where none of your arguments come across any longer due to their number is adversely affecting its reception. – Lord_Farin Apr 26 '16 at 21:02
• There was no reason to cast a close vote as I do not want to prevent additional answers to the question. I do think the question should be posted at MSE or migrated there, which also would allow answers. Closing is for spam and the like. If answering the comment invasion is "over-arguing", well, yes, comment invasions are like that, and attempts to reduce the rate of invasion have not worked. @Lord_Farin – zyx Apr 26 '16 at 21:07
• @zyx I guess our criteria for close-voting are quite different, then, the extent of which I find somewhat surprising. Also, I just tried to make clear how things came across, irrespective of the motivation and if it was any good. – Lord_Farin Apr 26 '16 at 21:20