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?

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    No, he means that with experience and practice, one obtains experience and practice. – Qiaochu Yuan Nov 21 '10 at 18:49
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    Anyway, I have always thought that this is unnecessarily pessimistic. I don't know why people are so fond of quoting it. – Qiaochu Yuan Nov 21 '10 at 19:16
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    There is some truth in it. Of course, it's not a mathematical truth. More like wisdom. But everybody who has been a teacher and had trouble explaining things to students who were genuinely trying to understand must have somehow felt the meaning of the dictum. – Raskolnikov Nov 21 '10 at 20:26
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    A more constructive interpretation of von Neumann's cute rhetorical flourish is that experience and practice change_one, so that what may have once seemed alien becomes familiar. This psychological aspect of "learning" is very often grossly under-estimated, I think. So, yes, as far as I can tell, after I've seen a dozen examples of a phenomenon that at first seemed bizarre and counter-intuitive, and had some months to let it percolate into my head, I "understand it" in the sense that it now seems reasonable, and I can "act" with it as part of my reality. – paul garrett Jul 8 '11 at 20:47
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    Young man, with von Neumann you don't interpret quotes, you just get used to them. – Meni Rosenfeld Nov 22 '16 at 12:13

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 seing 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.

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    Wow! You have penned down my thoughts here. I too think that this is a very positive quote, full of hope, instead of a pessimistic quote as some people mention in the comments. – shivams Jul 1 '16 at 2:13
  • Shouldn't that be $\bigcap^{\infty}_{n=1} (0,1-\frac 1n) = (0,1)$? – Mateen Ulhaq Nov 24 '16 at 1:23
  • @MateenUlhaq nope – arkeet Nov 24 '16 at 2:37
  • @Mateen Ulhaq What you wrote actually equals $\varnothing$ – J. C. Nov 24 '16 at 17:11
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    @MateenUlhaq they are intervals with rational endpoints, not rational tuples. – Dahn Jahn Nov 24 '16 at 22:17

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.

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    Frankly,I love that quote of Einstein's a whole lot more then Von Nuemann's, which always struck me as just a snarky bit of sarcasm from the Master. – Mathemagician1234 Dec 8 '11 at 7:18
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    I think this is a reasonable interpretation. One often hears questions from beginners of the type "But what is the square root of minus one?", and it's hard to explain, because they are coming at it from the completely wrong direction: what matters isn't what is is, but what it does. – MJD Sep 20 '12 at 23:20
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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.

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.

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 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.

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?

My two interpretations:

  1. 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'

  2. To the doubtful, don't feel bad for not understanding/not having understood. This is how maths is.

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.

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