John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them." This was a response to Smith's fear about the method of characteristics.
Did he mean that with experience and practice, one obtains understanding?
This is quite an old post, but I choose to answer, because I feel that I offer a completely different understanding of this quote.
I am very surprised to find out my understanding of it is different from others, since when I first read it, I thought to myself: exactly!
To me, this quote is how I feel all throughout studying mathematics. New concepts enter my mind, I learn about their properties and uses, I use them myself, prove theorems with them, yet in the time in between the first sight of the definition and the time when I am fully comfortable with using the concept, there was no aha! moment, when I'd finally understand it.
Take the example of the concept of infinity. You learn about $\lim_{x\to\infty}$, understand Zeno's paradox, $\lim_{x\to c} \frac {f(x)-f(c)}{x-c}$, understand that $\bigcup^{\infty}_{n=1} (0,1-\frac 1n) = (0,1)$, and keep seeing infinity again and again. One has many small epiphanies, but none of them could be considered the moment when one finally understands infinity. And yet there is some kind of road beginning at the first moment of utter confusion as to what infinity actually is and resulting in the feeling of infinity not being all that mysterious at all.
Thus, in this sense, the quote is full of hope. It gives me the reassurance that I don't need to push myself to try to grasp infinity in one evening, there is no piece of information I need to understand in order to say "I got it". Instead, I will gradually get used to its oddness until it becomes a very familiar object.
This process is much better described as getting used to rather than understanding, and thus I understood Neumann's quote in this way and it's been on my mind every time I encounter a new mathematical object.
In my opinion, what many people mean by the word "understand" simply isn't practical or relevant to mathematics. For example, it often carries the connotation that understanding something means reducing it to something obvious (e.g. something the speaker can "picture"), or that understanding is about what something "is" rather than about how you can use it. And, of course, Einstein's quote
You never truly understand something until you can explain it to your grandmother.
I would interpret von Neumann's quote as rejecting these notions of understanding, and stating that what's truly important in learning and practicing mathematics is actually using it, getting used to how it works and how you can use it to derive things.
One thing he might have been referring to is that in mathematics you often have to learn to apply a method without actually understanding what it is all about.
Take for example matrix multiplication. You could (and many students do) beat themselves up about why it is so "weird" in comparison to say multiplication of the reals. But it turns out that yes it has those weird properties because it is perfect for representing a linear transform, amongst other things.
In general I have found it counter productive to try and understand every aspect of something before moving on to the next thing. I just accept that that's the way it works, trust that one day it will have some sort of application, be useful or otherwise "make sense".
Note that the history of mathematics is full of branches of mathematics that didn't even have this sort of utility when they were initially created and explored, but have later turned out to be enormously important. Take for example Boolean algebra and knot theory.
Another important point is that mathematics is the study of abstract logical systems, including wholly invented ones. Therefore it can be pretty fruitless to understand some of the deeper meanings of a mathematical concept, because they might not even exist. Sure there might be deep connections or generalisations to other mathematical concepts, and applications might be found, but trying to say that the application is "the true form" of the mathematical concept is putting the cart before the horse.
One word frequently missing from this discussion is "joke". Von Neumann joked around a lot. I'd guess he was 50% joking when he said this. But it's one of those jokes that's funny because it has an element of truth to it. We all know the feeling of "getting used to" ideas that once seemed strange to us, so that now it is hard to remember how unfamiliar the ideas once were.
And, there is something encouraging about this psychological phenomenon. Even if new ideas seem very strange now, a year from now you will be used to them and they will seem much easier.
I interpret the quote in a very different way. In general, we understand new ideas based on old ones. In math we can't always do this. I came into math from a applied math background, and when I started to learn math outside of the "plug and chug" engineering math I knew, I had difficulties associating the new concepts I was learning with what I "understood". Jack Quine, who was one of the first mathematicians I really got to know, used to tell me I had to "liberate my mind". What I took away from his advice was that math has its own logic and its own set of rules which do not necessarily correspond to anything one really understands well (maybe it is a bit like quantum mechanics in this sense). Sometimes, it is just a matter of believing it until you get enough experience and finally in a higher level course, the structure and logic become apparent. For people very good at abstract thinking, the structure of certain parts of math may be easier to "understand" in this sense. However, I suspect everybody, at some point, comes to questions in developing areas of math where the structure is not laid out nicely, and they have to use tools that they don't have such a good understanding of. This is not such a big leap of faith as some would make it out to be though: after all, when calculus was being developed, Newton could not really defend his use of infinitesimals, although people still used them b/c they worked.
I would just like to add a reformulation, an isomorphism if you will, of von Neumann’s quote, in the shape of a story/anecdote, by a rather perspicacious fella that goes by the name of John Cleese, who, from what I have read so far of his autobiography, is either a professional thief, a bible salesman, a professional soccer player, or one of the founders of a legendary comedic troupe who's gone totally insane... but, I digress... Here's the anecdote in question:
“[…] I encountered the teacher who made the greatest impression on me: Mr. Bartlett. He became my maths master, and during the first term he taught me, I have to confess that I understood next to nothing. But when he taught me the same things next term, I grasped them instantly: they had become self-evident. So I was moved up a form, where Mr. Bartlett introduced me to new mathematical ideas, all of them incomprehensible—until the following term when they became blindingly obvious, and I assimilated them effortlessly. Promotion, in other words, was followed by bewilderment, and the next term, by full comprehension. Mr. Bartlett was a very good teacher.”
Cleese, John. So, Anyway... (p. 47). Crown/Archetype. Kindle Edition.
He meant that when you think you understand math, you have in fact fallen under the sway of an illusion. This illusion of understanding is brought on by familiarity ("getting used to it") coupled with intellectual laziness. People get used to doing math and having it work and they forget that no one knows why it works, they forget that it is fundamentally mysterious.
If you read this and object, let me ask you,
What is number?
I hope Von Neumann was one of the very few of us who realized that remembering and familiarity are not at all the same as understanding. Understanding means that we see the need, the origin, the purpose of definitions, the REASON a definition is made. The a priori purpose. And it means that when we encounter a "theorem" that reveals a property of a mathematical object, that it is identified as such. Mathematics is NOT an abstraction of reality. It is very MUCH a PART of reality. Because it is created and used to model reality does not make it out of this world. There IS no other world that is not fictitious. Mathematics is a thing in itself. Telling us that it's "abstract" and about "reasoning" is terribly misinforming. For example, geometry is ABOUT the existence and quantitative properties of plane and solid closed figures, usually bounded by straight line segments; but not necessarily. Circular and other shaped sides are perfectly possible and just as real. The emphasis on proof is obscuring and debilitating in all areas of mathematics. Proof is unique to mathematics but it is NOT what mathematics is ABOUT. It is interesting to wonder WHY proof is possible in mathematics.
I actually reject most claims on this page. Coming from an (astro)physical background, pure mathematics was a blessing for me. It took only a few months for me to cry out to all my physics friends that this pure maths thing, as opposed to physics, is something "that can actually be understood!"
Why? In physics, there will always remain another question. In every matter. "But what is an electric field? What is the wave function?" Etc. So I never had the experience of understanding something from A to Z. Not even the mathematics I was taught, because the story of its construction was always curtailed to save time, so ad hoc and heuristic arguments remained. In pure mathematics it is actually possible to follow an argument entirely down to the axioms, leaving no statement unproven in between. Here the questioning stops, as it is silly to ask "but why these axioms?" That's just because that's the game the we are playing. No one forces you to play this game, you could've played any game you wanted. That isn't to say, and this is important to note, that being able to follow a proof down to the axioms comprises "complete understanding of everything the situation entails". There are always ways to look at it that you haven't seen yet. Ideally one would see all these simultaneously, and if you demand that in order to say you understand it, we must indeed conclude that no one understands anything.
Concerning Von Neumann's quote, I don't believe he would disagree with me on this. Like some other answers I think he was at least half joking, and alluding to the well-known fact that we slowly get used to certain concepts 'playwise' (Dutch: spelenderwijs; there doesn't seem to be an English translation), going from hard and opaque to easy and transparent. The same thing happens in physics too, but the difference is that in the math case, you can justify every assertion. Now if that doesn't exhibit understanding, I don't know what does.
My 2 interpretations:
To the arrogant, don't think you know that much. You don't know what you don't know. As Nero aka Nassim Nicholas Taleb said 'If you don't feel that you haven't read enough, you haven't read enough'
To the doubtful, don't feel bad for not understanding/not having understood. This is how maths is.
"Getting used to it" is memorization that doesn't feel like memorization, more like speaking a language fluently, or recognizing your grandmother, or knowing how your furniture is arranged.
Knowing axioms and some comsequences of those axioms is a similar memory, but it's not understanding.
When you become fluent in a language, it's not that you understand it, but that you can use it. Understanding is different - you might see some patterns, know the historical origins of some words, there might even be computational efficiency arguments for grammar structure, and some might be contingent on the human language system. But this is not full understanding.
Axioms can't really be "understood"; they are given. And Godel and Turing seem to show that we can't even predict what will be true - let alone understand it!
Poincare once wrote that all of mathematics is one plus one equals two, and a lot of definitions. Speaking as a physicist, I believe the most important thing to understand about mathematics is whether those definitions are useful in describing reality (i.e., observations). I don't believe mathematicians are similarly concerned, though certainly von Neumann himself is a most prominent counter-example.
Just to be provocative, I have found Newton's calculus to be sufficient for my needs and have never understood \delta -- \epsilon proofs. There seems to be no place for "infinity" in physics. On the other hand, nonstandard analysis has proven quite useful to me.
I became aware of this quote just now and am enjoying the conversation :-) My first impression about the quote was that we relate 'understanding' to something we experience in the physical world. But, when it comes to mathematics, it is an algebraic system beginning with some axioms and then we develop a theory around it. That theory helps mimic the real world phenomena. Sometimes that relationship to the real world is not obvious, or is hidden. We still continue using the theory because it is useful, and we get used to it and we start believing it for real.
