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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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I remember being dumbfounded when math students (I took it as minor) juggled around with integrals without being able to explain to me what they were doing and why the steps were valid. Fubini's theorem was among the more prominent things. –  Raphael Nov 21 '10 at 20:43
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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

3 Answers 3

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.

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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.

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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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