When I look at first time to a theorem and I try to understand it or when I try to memorise a useful theorem I always have difficulties (I am not the only one. For example: I read a question: I always have trouble memorizing theorems. Does anybody have any good tips? )

However, I have learn from the theorems that I am able to prove, that what I am doing is not following the theorem itself but a rather very clear non too abstract idea (usually a concrete example) I have on mind preconceived somehow.

I wonder to know if there is any sources of pictures that big mathematicians has drawn when proving an important theorem. I would like to see what was the concrete idea (usually rather simple) that they have on mind. It is not difficult some time to write a nice representation of the theorem after is conceived, as usually everybody does, but I am interesting in other way around.

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    $\begingroup$ This is a good point. For example, in topology (IMHO, of course), it is considered mauvais ton to write the proofs the way they were conceived: instead of a clear geometric idea (and maybe a simple picture) one is supposed to write a bunch of exact sequences, just to make it sure that no one can understand the proof. $\endgroup$ – Alex Degtyarev Mar 10 '14 at 12:34
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    $\begingroup$ @AlexDegtyarev For an interesting counterexample, see Goresky and MacPherson's Stratified Morse Theory. They mention the tension between geometric intuition and rigor, and how they were led to introduce new techniques ("moving the wall") to help lay bare the intuitive content of technical proofs. $\endgroup$ – Todd Trimble Mar 10 '14 at 12:40
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    $\begingroup$ @ToddTrimble Yes, of course, some people are still trying to fight the trend :) Unfortunately, the trend is so strong that many young mathematicians (e.g., homotopy theorists) are forgetting that topology is geometry. Needless to say that in algebraic geometry the word geometry was forgotten long ago... $\endgroup$ – Alex Degtyarev Mar 10 '14 at 12:43
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    $\begingroup$ While there is plenty to sympathize with in user39115's post, I am not entirely sure what the question is. Is there a particular theorem you would like to know the intuition behind? Or are you searching for a big list of theorems and intuitions ... ? The latter would be much too broad in my opinion. Perhaps what you are after overlaps to some extent with this earlier question? mathoverflow.net/questions/38639/thinking-and-explaining $\endgroup$ – j.c. Mar 10 '14 at 14:43
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    $\begingroup$ @AlexDegtyarev: do people really discriminate against proofs based around clear geometric ideas? Or is it more that "intuitively clear" geometric ideas are not so easy to turn into watertight proofs? $\endgroup$ – Tom Leinster Mar 10 '14 at 16:21

Short answer: I believe Paul Cohen's piece on his own development of forcing provides an answer to your question. I will elaborate further below the break, but you can find his article written up as:

Cohen, P. (2002). The discovery of forcing. Rocky Mountain Journal of Mathematics, 32(4).

Elaboration: A brief idea of the "big picture" seen by Cohen (here I assume, perhaps incorrectly, that you mean "big idea" rather than a literal drawing) is as follows:

$1.$ Believe that philosophical arguments can be applied to questions in Number Theory;

$2.$ Believe there are many different models of mathematics (cf. the Löwenheim–Skolem theorem); and

$3.$ Believe in the existence of some sort of "decision procedure" that simplifies complex statements until they are decidable.

These "big picture" ideas are what evolved and interacted with one another in leading Cohen to his proof of the independence of the Continuum Hypothesis. It might be worth remarking that another important belief, more related to affect but also mathematically relevant, held by Cohen was that:

$4.$ The Continuum Hypothesis was the sort of problem Cohen would like to tackle, and he was determined not to lose the chance to resolve such an outstanding question.

Here are some quotations by Cohen to support items $1$ to $4$, respectively:


In set theory when dealing with fundamental questions, one often has a kind of philosophical basis or conviction, rooted in intuition, which will suggest the technical development of theorems.

$ $

Basically [the Continuum Hypothesis] was not really an enormously involved combinatorial problem; it was a philosophical idea.


We will never speak about proofs but only about models.

and, expatiating further,

The existence of many possible models of mathematics is difficult to accept upon first encounter, so that a possible reaction may very well be that somehow axiomatic set theory does not correspond to an intuitive picture of the mathematical universe, and that these results are not really part of normal mathematics . . . . I can assure you that, even in my own work, one of the most difficult parts of proving independence results was to overcome the psychological fear of thinking about the existence of various models of set theory as being natural objects in mathematics about which one could use natural mathematical intuition.

On the relation of $1$ and $2$, Cohen remarks:

Of course, in the final form, it is very difficult to separate what is theoretic and what is syntactical. As I struggled to make these ideas precise, I vacillated between two approaches: the model theoretic, which I regarded as roughly more mathematical, and the syntactical-­forcing, which I thought as more philosophical.


If you actually wrote down the rules of deduction—why couldn’t you in principle get a decision procedure? I had in mind a kind of procedure which would gradually reduce statements to simpler and simpler statements.


Because of my interest in number theory, however, I did become spontaneously interested in the idea of finding a decision procedure for certain identities... I saw that the first problem would be to develop some kind of formal system and then make an inductive analysis of the complexity of statements. In a remarkable twist this crude idea was to resurface in the method of ‘forcing’ that I invented in my proof of the independence of the continuum hypothesis.

$4.$ This concerns, in particular, Friedberg's solution to Post's Problem using the priority method, and parallels between this and Cohen's work on ~CH using forcing. See the earlier MO question here, and, e.g., Cohen's remark:

It was exactly the kind of thing I would like to have done. I mentally resolved that I would not let an opportunity like that pass me again.

You can find the points made above in a slightly more (or less?) organized write-up here.

As a final, general remark: Having a big picture in mind when proving something occurs, of course, outside of mathematics as well. The best exposition of this (of which I am aware) is Howard Gruber's (1974) book Darwin on Man, which explores how Charles Darwin developed his theory of evolution. Though Origin of Species is structured as a presentation of overwhelming evidence that simply led Darwin to his final theory, the reality is that he was constantly theorizing along the way. Gruber explores this by going through previously unread notebooks of Darwin's, and includes a discussion of a literal picture that turns up repeatedly - in different forms - of a branching tree. Although this books concerns work outside of mathematics, I think reading it can be both informative about "creative thinking" in general, and can also provide a way for those who are mathematically-inclined to try and write up the biography of a mathematical theorem and the big ideas/big picture behind it.

  • $\begingroup$ Cohen's reminiscences may not be so reliable. He described parts of the paper as "pseudo-history", commented that Godel gave him contradictory reminiscences in 1966 and 1967, and took an oversimplified view of Cantor. Specifically, he says Cantor "probably regarded his theorems as correct theorems exactly in the same spirit as other results", neglecting that Cantor proved the theorem that "two aggregates have the same cardinal number if, and only if, they are equivalent" using a definition of cardinal numbers referring to "our active faculty of thought" (unseres activen Denkvermögens). $\endgroup$ – Matt F. Mar 21 '19 at 18:14

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