How important is building up intuition for a theorem before trying to prove it?

For example, consider trying to prove that:

If $A$ is a set and $F \subset P(A)$, then the relation $R := \{(a, b) \in A \times A$ such that for every $X \subset A - \{a, b\}$, if $X \cup \{a\} \in F$ then $X \cup \{b\} \in F\}$ is transitive.

Keep in mind that I don't want you to prove this for me. I only give it as an example, in order for you to understand the question better. This theorem is not obvious to me: I can't envision it, so to say. Am I approaching mathematics in a wrong way, by trying to envision every theorem and build an intuition on it, or is this not possible when the theorems themselves are too complex?

Secondly, if building an intuition is often not possible, how do mathematicians come up with new conjectures, ideas, etc?

Edit: What I am saying is that I can prove some theorems by simply applying previously learned techniques, algorithmically, although I don't understand completely these theorems before I attempt to prove them. The question is:

Is it important to try to first understand the theorem completely, and then attempt to prove it ?

• Few clarifications:(1) are you just starting proofs or familiar with their mechanics? (2) I assume R is defining a relation, correct? – Chris C Mar 23 '15 at 19:13
• Yes this is a relation and yes i am new to proof writing. However i know the basic mechanincs. What I am saying is that i can prove some theorems using these mechanincs altough i dont understand the theorem itself. I feel this is not the way it is supposed to be. – Anton Mar 23 '15 at 19:17
• Somewhat related was MESE 1373 (though its title was not very descriptive). – Benjamin Dickman Mar 23 '15 at 19:26

3 Answers

(Include von Neumann quote about not understanding things in mathematics so much as getting used to them.)

If the goal is for your understanding, then you need to break it down as you can to aid in your understanding. I would ask why you need the relation to be transitive. Are you going to form an equivalence relation eventually? Do you have some idea of how you are going to use R and its presumed transitivity?

You have to build up your intuition about some abstract constructs, usually by relating them to things you know. ("It behaves like this other thing, see?") Without any intuition, you have little or no chance of conveying the meaning, importance, or even correctness of a theorem to someone else, including your later self. Yes, it would help if you understood R. At the beginning though, you just work with it to develop your intuition and understanding.

If you want to try a meta-experiment, observe and record your thoughts and states of understanding along the way as you work with R. You may find and identify certain blocks to understanding and ways to overcome them, which can be quite useful to yourself in other contexts and to the person after you who has to work with R.

Gerhard "Depends On Your Particular Goal" Paseman, 2015.03.23

• Thanks for the answer. The reason why i need to prove that R is transitive is because i am learning about relations and this is an exercise in the book i am reading. – Anton Mar 23 '15 at 19:54
• Spoiler alert: OK. One of the things you will use to develop your intuition for transitivity is to note that (roughly) "transitive relations follow from transitive concepts". In this case, the notion of "if-then" is like a transitive concept. Building such patterns is part of building such intuition. Gerhard "Wants Not To Spoil Things" Paseman, 2015.03.23 – Gerhard Paseman Mar 23 '15 at 20:21
• I have a proof of this theorem. The point is that I dont think I am able to come up with such a theorem myself. I think the problem is that I cannot imagine it. It is hard to explain really. I am having trouble visualizing it, that is all. However when it comes to logic i can prove this. – Anton Mar 23 '15 at 20:49
• As pointed out in the other answers, you aren't expected at the beginning of your studies to assume near mastery. Being able to craft theorems is a skill that is acquired; crafting them well is an art, and one should try it early only to check one's progress and gain practice. If you don't feel ready for it, it's ok to put it off to tomorrow. Many of us take years to develop (and continue developing) such visualization skills. Gerhard "Explaining To Someone Often Helps" Paseman, 2015.03.24 – Gerhard Paseman Mar 24 '15 at 20:10

I think there are (at least) two different skills involved in what you are asking. One is being able to work with formal definitions, to be able to prove things from the theorems you know by working with the basic structure, even without understanding particularly what is going on. The second is having the understanding and intuition to picture how the proof should go. I think both these skills are useful, and different bits of maths are suited to these different approaches. There are some bits of algebra when it's best to just multiply out the terms and cancel down. I've often had student who are overwhelmed by a problem because they try to see the whole proof, when working one step at a time will lead you through. When dealing with functions of functions of functions... I find it helps to work through the formalities. On the other hand, there are many places where having no intuition will make it very hard to make any progress, especially in the sorts of proofs where you need to pluck the right object essentially out of thin air, before proving it has the properties you want.

In short, I think doing either one has a value, but never doing the other would be more of a problem.

Edit after reading your comment on the other answer:

Being able to come up with the result in the first place is a third skill, somewhat unrelated to the other two. If you are at the stage of trying to understand enough to prove the given result, you are not yet at the point where you should be worrying about coming up with the result (in my opinion). That will come later. In particular, this specific example of a transitive relation is not something you are likely to stumble upon out of nowhere. It is likely whoever came up with it had some other reason to be thinking about that particular relation, or else was actively looking for more unusual transitive relations to use for teaching.

It would seem to me that you're not yet expected to come up with such things yourself. You just have to prove a bunch of statements provided in a sort of "black-box", that is, just accepting that there is a motivation for this that might not be clear to you at the time.

There are a few ideas behind why an instructor would ask you to prove such a thing:

• To reinforce a proof strategy or concept that is critical. It could be that such a statement is of no particular use, but the methodology used in the proof is key to later success in the course. In fact, I know that using equivalency classes is critical in many later courses (such as Topology or Algebra).
• To prove a lemma/theorem that will be used in a larger theorem later. This happens more often than not in the exercises in books. Early in the text, it will list a bunch of exercises for you to prove without much motivation other than to get a foothold in the subject. Then later, the fact will appear in a few proofs in the text or even is needed in later exercises.

Thinking back to when I was first learning to prove things, there were many such statements that didn't make sense as to why they were formulated a certain way. It was not until much later that I saw that many were technical theorems/lemmas that were used in a more larger concept. It is not usually until much later, after proving and going over many statements, that you will have the intuition built to write such statements yourself. In fact, I find they're usually from asking questions about relations between the topics you are learning or a common proof strategy.