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I know that everyone here is familiar with the quote of von Neumann which is:"Young man, in mathematics you don't understand things. You just get used to them".

I read his quote and it got me thinking how did he say something like this or what did he mean by (don't understand things) ?

According to my experience in mathematics, it's all about understanding what the definition really mean for example:one can't understand the limit properties without understanding the $ \epsilon-\delta$ definition or the darboux sums without understanding the upper sum and the lower sum

So it's all about understanding what the definition want to say and derive from it another theorem

Mans
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    People understood limits before the $\epsilon-\delta$ definition, they just didn't have a rigorous understanding. Mathematicians had the idea of limits for centuries before they defined them rigorously. – Thomas Andrews Mar 29 '23 at 00:29
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    @ThomasAndrews , Newton's idea of calculus represented by the infinitesimals was not accepted because it didn't fit with the nature of real number line until the work of Cauchy and the introduction of $\epsilon -\delta$ definition, besides the understanding of limits before the $\epsilon -\delta$ was about saying "what value the function is tending to when the argument of function is tending to $a$" and we know that rigours understanding of limits is far more complicated than this primitive idea – amin Mar 29 '23 at 00:51
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    I'm not sure your point. People were using calculus for two centuries before it was made rigorous, and people pretty clearly understood it to the extent they needed to do calculus. @amin The real question for people learning the $\epsilon-\delta$ definition is to match the intuitive understanding with the definition. That's a different kind of understanding, and, for the people doing calculus in the real world, it is a kind of understanding that is nearly completely unnecessary. – Thomas Andrews Mar 29 '23 at 01:00
  • Another example: Bertrand's Postulate. I've seen it proven, but I've never dived into it. Do I understand it, really? I certainly couldn't prove it, myself, but I can give some hand-wavy argument about why one might expect it to be true. I do know it well enough to use it. I've tried to read the proof that exponentiation is diophantine many times, but on some level, I'm still not sure it is true. The proof is such a tangle of algebra, and piecing it together never feels convincing. Yet I know how important it is, and how it helps solve Hilbert's Tenth Problem., – Thomas Andrews Mar 29 '23 at 01:11
  • @ThomasAndrews , according to my experience in studying calculus the introductry definition of limits was ambiguous to me and made contradiction with some functions like $f(x)=c$ or $f(x)=xsin(1/x)$ and I didn't really understand limits until I learned $\epsilon-\delta$ definition, matching the intuition to rigours definition is a must in mathematics to remove the ambiguities ,the use of people of calculus before $\epsilon-\delta$ was founded on a concept that most of the mathematicians and philosophers didn't accept at that time – amin Mar 29 '23 at 01:16
  • This has already been discussed in some form here: https://math.stackexchange.com/questions/11267/what-are-some-interpretations-of-von-neumanns-quote.

    My interpretation of his (somewhat joking most likely) comment is that sometimes after working in an area of math for a while, one simply "absorbs" a lot of the intuitions and ideas by osmosis. The process doesn't seem to feel like an "a-ha!" moment, but rather a slow acclimation to the area. My personal experience is that this is to some extent true.

     – Alekos Robotis Mar 29 '23 at 02:20

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