What is the Line Between Intuition and Insight?

[MagiOmni]

There's no doubt mathematics in public education is filled with half baked explanations, subpar instruction and most worriedly the rejection of any question of why something is the way it is beyond whats necessary to traverse the path between the question and the answer. This has given rise to a culture of "intuition" which tries to supplement this lackluster education with metaphors and visuals which, while potentially interesting and beautiful connections, never give rise to anything rigourous. The problem with this, of course, is that those on the receiving end never understand how one would discover such things; they never learn to think like a mathematician.

On the other hand, mathematics isn't discovered in a black box of implications and axioms. Rather mathematics was built up on visual and symbolic reasoning which map between real world experience and phenomena. Much of this reasoning shares striking similarities to intuition, but is backed by something rigorous. This "rigourous intuition" (insight) is clearly key to understanding mathematics on a fundamental level, and getting a grasp on what it is (and what it is not) could prove useful in mathematical education, both in the classroom and in one's own ventures. Hence, I pose the question posed in the subject line, where should the line be drawn between intuition and insight?

[Alfred Cromwell]

I think the distinction you're looking for is that intuition is psychological, whereas insight is mathematical.

An intuition is simply a feeling that something should be true, or a mental picture that makes a result seem plausible. Insight, by contrast, is an understanding of the structural reasons a result is true. The key difference is that insight can, at least in principle, be unpacked into a rigorous argument, whereas intuition cannot necessarily be.

From a rigorous perspective, intuition has no evidentiary value. A diagram, analogy, or visual metaphor may suggest a theorem, but it does not establish it. Mathematics is full of examples where seemingly obvious intuitions turned out to be false or misleading once pushed beyond their original domain. Intuition is useful as a heuristic, not as a justification.

What experienced mathematicians often call "intuition" is usually something closer to what you're calling "rigorous intuition." It is not merely a feeling that a statement is true; it is a compressed understanding of the proof structure. When an analyst says they "see" why a theorem holds, or a geometer draws a picture, they are typically relying on a network of definitions, lemmas, and prior results that could be expanded into a formal argument if necessary. The novice sees a picture; the expert sees a proof strategy.

So I would draw the line as follows:

Intuition tells you that something is probably true.
Insight tells you why it is true.
Proof demonstrates that it must be true.

Insight sits between intuition and proof. It is not itself a proof, but unlike mere intuition it remains accountable to proof. If you keep asking "why?" an insight eventually bottoms out in definitions and rigorous arguments rather than in "it just seems obvious."

In that sense, insight is not a substitute for rigor but a compression of rigor. It is valuable precisely because it can be unfolded back into mathematics when needed.

[StudentDriver]

It seems like insight only falls out after reflection on the proof that is pretty fleshed out. Intuition can be misleading, but its purpose is to guide your questions. Here Professor Cromwell seemed to have define insight as--- explains why something is true. And also stated that insight in principle can be unpacked. Well.... here we go.......

- Before we ask the question "what is the line between intuition and insight?" on the pure mathematics channel, there is a video called "mathematicians are near divine" and I think asking questions about self-evident axioms is necessary before answering that question.

Professor Cromwell talked about "self-evident axioms" in this video.

And here I claim in the comment section: self-Evident axioms???? <-- reminds me of the need of intuition ... though a lot more refined!
But PROFESSOR Cromwell! I found some really interesting "Axioms"/statements that are based on a very LOCAL perspective (without using the Sacred Words), that really make the set theoretical axioms a lot more personal (aka make the set theoretical axioms ALOT MORE SELF EVIDENT!)....

Though I did not have the lang of Professor Cromwell, here I would replace intuition with insight. "reminds me of the need of insight"

-- before I list out and perhaps get destroyed in this discussion, I want to see if anyone is interested in this very simple Local perspective hehe.

I'll give a hint... the statements start with the word "I".

[nyquistfrequency]

It seems like insight only falls out after reflection on the proof that is pretty fleshed out. Intuition can be misleading, but its purpose is to guide your questions. Here Professor Cromwell seemed to have define insight as--- explains why something is true. And also stated that insight in principle can be unpacked. Well.... here we go.......

- Before we ask the question "what is the line between intuition and insight?" on the pure mathematics channel, there is a video called "mathematicians are near divine" and I think asking questions about self-evident axioms is necessary before answering that question.

Professor Cromwell talked about "self-evident axioms" in this video.

And here I claim in the comment section: self-Evident axioms???? <-- reminds me of the need of intuition ... though a lot more refined!
But PROFESSOR Cromwell! I found some really interesting "Axioms"/statements that are based on a very LOCAL perspective (without using the Sacred Words), that really make the set theoretical axioms a lot more personal (aka make the set theoretical axioms ALOT MORE SELF EVIDENT!)....

Though I did not have the lang of Professor Cromwell, here I would replace intuition with insight. "reminds me of the need of insight"

-- before I list out and perhaps get destroyed in this discussion, I want to see if anyone is interested in this very simple Local perspective hehe.

I'll give a hint... the statements start with the word "I".

I don't exactly understand what you are writing which is probably why you haven't gotten a reply.
But there's only one way an axiom proves itself, if it is shown that it cannot be otherwise,
in other words all other alternative has been ruled out because it denies it's own possibility of being true. For example, in order for the statement "there is not truth" to be true there must be truth, so it denies its own possibility of being true. This is how axioms prove themselves, this is how axioms are self-evident, they aren't true because feel they are true, they are true because they cannot be otherwise.

[StudentDriver]

My bad! Lmao I am Sorry people, for having to read that.
Damn I am a real crackpot. xD. I mean, Im like on the edge. Will try to explain myself better. Anyway, what I say is very simple.

Here is what I meant in the comment:

-I was just simply trying to apply the definition of insight that Professor Cromwell gave.

Insight: explains why the proof is true. In principle insight can be unfolded into a rigorous argument. Doesn't necessarily mean that all insights can be unfolded back into rigorous argument.

Yes axioms are simply true, because that's how you engage in mathematics or philosophy. (Especially in mathematics, axioms are irreducible statements. These handful of statements are the basis to derive the rest of that specific body of mathematics). But I'm talking about how we arrive to those axioms. That requires insight outside of those axioms. (They are much more grounded/ self-evident quite literally). And the philosophy that explains where these axioms come from, have statements that start with the word "I" (Not the letter I THE WORD I) (the hint) ... for example, if you think about Rene Descartes "I think therefore I am". well it's similar to that, but Im applying it for (naive) set theory.