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It seems to me that as one goes higher up in mathematics, the proof of theorems get more involved and convoluted, that at some point one must postpone (or even give up?) understanding all prior theorems, and take them as "Black Boxes" instead? Is this true?

Personally, in an ideal situation, I would certainly like to understand and know (and remember) the proof of the theorems, but am slowly finding it starting to be an uphill (and near impossible) task.

In high school, it is still quite possible for the conscientious student to learn and understand the proofs of Pythagoras' Theorem, limits proof of calculus, etc, even though it is not being taught in school. At the undergraduate level, things get harder, but I suppose it is still possible, to know all the necessary theorems proofs like Intermediate Value Theorem, Mean Value Theorem, etc.

As things go higher, the key theorems like Egorov's Theorem, Lebesgue's Differentiation Theorem, etc, have rather non-trivial proofs that span pages. I am sure there are more difficult theorems than these which have even more difficult proofs.

If I were to learn the full proof of the theorems first, before proceeding to learning other stuff, my progress would be painfully slow.

My question is: To do mathematics at a sufficiently high level, what are some tips to judge which theorems to learn in detail, and which to just take it like a "black box"? Any experiences to share?

Thanks.

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    $\begingroup$ One remembers the key points and hopefully recalls details. But mainly it is useful to have a crude outline of the ideas and tools used. As a simple example, one reads "now, by a standard spectral sequence argument" a lot, but one doesn't really care about the nifty details. For example, the proof I know of the Lebesgue differentiation theorem involves the Hardy Littlewood maximal inequality and a density argument, and that really is what I need to pinpoint more details. $\endgroup$ – Pedro Tamaroff Nov 16 '16 at 3:59
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    $\begingroup$ No new research would ever get done if the researcher tried to intimately understand the entire path from Pythagoras to the state of the art. $\endgroup$ – OJFord Nov 16 '16 at 5:03
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    $\begingroup$ It's not economic to read every proof, but please be reminded that sometimes a published proof may be wrong, leaving the theorem statement unproven. Even worse, sometimes a published "theorem" may be wrong too. There is always an issue of faith/trust. $\endgroup$ – user1551 Nov 16 '16 at 9:01
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    $\begingroup$ I finished my Maths degree and got a job as a Data Scientist which involves working with black-box machine learning algorithms. When trying to understand these better I have to open the black-box using some of the maths I learnt at university. Turns out this black box is just filled with black-boxes. The saying should be that "it's just black-boxes all the way down" not turtles. $\endgroup$ – josh Nov 16 '16 at 17:26
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I think it's important to remember the key ideas in long proofs, as these may serve you later. Trying to remember every single equation in a long proof is a loss of time and energy. However, it is a good idea to memorize the statements of theorems in order to be able to recall them without having to reread them every single time. The thing is, the more you know by heart, the more you will be able to make links in your knowledge and recognize patterns.

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    $\begingroup$ Besides knowing the general ideas of a proof, I also find it useful to know where to find a spelled-out version of the proof if I see a similar question somewhere else. Sort of a balance between memorizing the proof and remembering just the theorem's statement. $\endgroup$ – Bob Jones Nov 16 '16 at 3:51
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    $\begingroup$ @BobJones Sure, I agree. A physics teacher of mine has said once that what's important is to know that something exists and know where to find it. However, I believe one must be careful with such thinking. It is certainly not enough to just know that something exists. If you want to be an efficient mathematician, it's better to have great knowledge off the top of your head, otherwise it's like you always keep stepping backwards. $\endgroup$ – Guest Nov 16 '16 at 3:57
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I think one thing to bear in mind is that there is a big gap between "black box" and "learn in detail".

In my experience, people use the phrase "black box" for big pieces of machinery or big theorems, often from outside of their field, whose statements or conclusions they want to use but which they have little or no understanding of whatsoever.

Let's say "Learning in detail" means learning in a way that you could write out the details from memory, e.g. in an exam.

In between, let's say you are an analyst and you want to use some theorem about PDE or measure theory that you did not already know. Most of the time you don't do either of the two aforementioned things. You would look up the theorem in a textbook or paper and read enough of it to get the main ideas so that a) You could cite it precisely if someone asked and b) You could roughly explain the idea so that another practitioner in your field would ultimately find it very plausible.

In summary, this answer is similar to that of Guest

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The question asked is refined --- albeit not definitively answered --- by Vladimir Voevodsky in recent works that include, in particular, the lectures "How I became interested in foundations of mathematics" (2014) and "UniMath" (2016).

Voevodsky's web pages provide links to both the lecture-slides (respectively, here and here) and the lecture-videos (respectively, here and here). These lectures are suited particularly to young mathematicians.

In a nutshell, Voevodsky argues in his "UniMath" lecture of 2016 that:

Today we face a problem that involves two difficult to satisfy conditions.

On the one hand we have to find a way for computer assisted verification of mathematical proofs. This is necessary, first of all, because we have to stop the dissolution of the concept of proof in mathematics.

On the other hand we have to preserve the intimate connection between mathematics and the world of human intuition. This connection is what moves mathematics forward and what we often experience as the beauty of mathematics.

The purpose of this answer is not to argue that Voevodsky's views are right or wrong in any absolute sense, but rather to suggest to anyone interested in these questions---students in particular---that Voevodsky's recent lectures and writings will amply reward careful study.

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I think technical theorems can be used as a black box. Generally the more advanced a subject goes the more technical it becomes.

P.s. if you want to use theorems as black box be always really sure you got all the details of the statements. It's often really easy to overlook a word which is crucial for your need.

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  • $\begingroup$ What defines something as technical? Is there a measure or index of "technicality"? Would that be similar to level of abstraction? $\endgroup$ – minmax Sep 22 '18 at 21:22
  • $\begingroup$ It doesn't have a relation with abstraction, it's simply a term to indicate a theorem that have a proof that has "ad hoc" definitions, terms and procedure that are not particularly meaningful outside the context of proving that specific theorem. so at the end of the study of a "technical theorem" you just end to knowing that the theorem was indeed proved but you don't have any other meaningful insight $\endgroup$ – Dac0 Sep 23 '18 at 16:35

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