Young man, in mathematics you don’t understand things. You just get used to them.

John von Neumann, to Felix Smith

Go read about John von Neumann. He was a brilliant person who I wished would have had more time on Earth. You owe him, among other things, for the very computer you are using at this moment. If you’re studying mathematics you must find out more, and if you’re a regular person you should find out more.

Now, the topic of this post of course is the quote at the head of the post. I could be taking it too seriously, because von Neumann was known to make jokes like this. However, the problem is that this quote is often used to give the impression that advanced mathematics is non-intuitive or abstruse even to those who are experts in it.

That is what I want to dispute, in the context of cognitive science today.

Here are the problems I see with it.

It suggests things are never understood

“… you just get used to them. So don’t even try to understand!” That’s the most pessimistic interpretation, which I hope everyone agrees is clearly not true. Understanding some concepts can be a very deep experience, one which may not ever be fully complete. But there are clear signs you understand a topic: if you can converse about it intelligibly with other experts, if you can teach it, if you can solve problems using it, and make new discoveries about it.

Unless you have a very narrow definition of what “understand” means, I think most people would agree that mastery follows lots of work to understand something. It may take time, but time is not really a factor, which leads me into my next point…

It suggests there is a dichotomy between comprehension and familiarity

“Get used to it” suggests that a long time passes while you acclimate, perhaps unwillingly, to the idea. The quote sort of insinuates that “understanding” only had a chance to happen before that, but failed to happen, and can’t thereafter, so the only thing left is “getting used to it.” Ugh!

Understanding something is not always something that has to happen the instant you hear a new idea. It can be a long and ongoing process. Cognitive science says that when you’re encountering something genuinely new to you, it’s going to take some time for the new “chunk” to crystalize. But given time you understand it and you get used to it. At some point, you like your mental construct enough to say that you’re comfortable with it, and that is itself a form of understanding.

Your brain needs time to physically form connections, so “getting used to” a topic should not be separated completely from understanding a topic.

Finally, notice nothing I said is tied to mathematics in any real way, so my next point is…

It suggests this only happens in mathematics

Nope. This happens while learning any advanced concept. Not immediately understanding something isn’t alarming: we’re all bound to run up against things which trip us up someday.

All that’s needed is patience and thought, and hopefully some good teachers or classmates, and you too can build understanding and get used to the thing you are studying.

Conclusion

It’s not an idiotic quote, but it’s a little awkward because it’s not used in flattering ways.

If you’d like to see other intelligent opinions on von Neumann’s quote, you should take a look at this mathematics.stackexchange discussion.