Why is the unique factorization of natural numbers nonobvious? I get that everything has to be proven, but why is the unique factorization of natural numbers (or Fundamental Theorem of Arithmetic) nonobvious and nonintuitive? I thought I understood why it's nonobvious but then after some time I go around in circles again and think wait, why is it nonobvious?
Whenever I see answers attempted at this (e.g. in this blog entry), some more "complicated" number system is introduced to show non-unique factorization (e.g. even numbers, algebraic numbers), and I roll my eyes and think of course some properties will no longer hold if you add more restrictions to (or change) the structure/system. We are no longer talking about the natural numbers so I feel it's an unfair comparison.
Or they would come up with a large number and say we don't know a priori that there is only one way to factorize it. Yeah, okay sure, but I am not able to convince others (or sometimes myself - other than we need proof) that this is a convincing argument against the obvious nature of unique factorization.
I still get that mathematics is all about proof and you still need to be able to write things down clearly, but that doesn't really explain the supposedly nonobvious nature of the unique factorization of the natural numbers.
Should we just ditch the idea of "obvious" or "intuition" in mathematics altogether?
I'd be grateful if someone could explain to me about this kind of conundrum I am in. Preferably explained in layman terms because I am not a math (or math heavy) major. Just really interested in elementary number theory.
 A: In a domain with unique factorization, irreducible elements – those which can not be factored into smaller pieces – are the prime elements, and they can be thought of as "fundamental multiplicative building blocks", analogous to atoms. Just as the compound water has a specific chemical formula: $H_2O$, so too the number $12$ has a specific factorization $2^2\cdot 3$.
Since the numbers that we usually work with, the natural numbers, have this property, we tend to learn about prime numbers as if "prime" = "fundamental building block". When we do this, we are building in a misleading intuition, specifically that "prime" and "irreducible" mean the same thing.
In any domain where "prime" and "irreducible" are synonyms, we have unique factorization, and it has a sort of "obvious" feel to it. For it to feel non-obvious, we would need two things:

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*A better definition of "prime", distinct from "irreducible".

*An experience-based feel for why those two don't just mean the same thing.

With these in place, unique factorization would not seem obvious, as we would have more refined intuitions about multiplication. Obtaining that experience-based feel for the difference between "prime" and "irreducible" would come from playing with various domains, where the two notions sometimes coincide, and sometimes don't.
When we adjoin $\sqrt{-2}$ to the rationals, we get "integers" where every irreducible is prime. When we adjoin $\sqrt{-5}$ to the rationals, we get "integers" that include non-prime irreducibles. Adjoin $\sqrt{5}$ instead: every irreducble is prime. Adjoin $\sqrt{10}$ instead: not every irreducible is prime!
Seeing these examples, one is led to ask what makes some of these domains special, so that they have unique factorization. From there, it's not much of a stretch to think that we should verify that good old $\mathbb{Z}$, the familiar rational integers, is one of the nice kind, where every irreducible is prime.
Someone without experience of non-prime irreducibles would not think to verify such an "obvious" fact.

What does this tell us about intuition, and about things being "obvious"?
Whether something seems obvious is relative to one's intuitions, and those intuitions grow and change with exposure to diverse examples. With such exposure, our ideas about what's "obvious" change in a interesting way:

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*A property seems obvious, because it's there in every example we've seen. Our intuitions are trained to see it as a background fact that's always present.

*The property becomes non-obvious, because we see a variety of contexts where it can be present, or not. Our intuitions have now been trained to see it as a variable property, one that can "come and go". That's what we really mean by "non-obvious".

*The property becomes obvious again, relative to context. This happens after we've proven that it holds in some context, and established necessary and sufficient conditions for it to hold when the context changes. Our intuitions have been trained to connect the property with details that had previously gone unnoticed.

What does this process look like for unique factorization? Maybe something like this:

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*Of course factorization into primes is unique, why on Earth wouldn't it be? If you break something down into fundamental pieces, you obtain the pieces that the thing was made of, obviously. Irreducible elements are prime, right?

*The familiar integers are just one of many domains where we can play factorization games. Some of those domains have unique factorization, and some do not. For a specific domain, you have to check to find out which kind it is. Irreducible elements need not be prime!

*Sure, domains can behave in different ways, but once you have a Euclidean algorithm in a domain, it will obviously be a Principal Ideal Domain, which is obviously a UFD as well. We have a Euclidean algorithm in $\mathbb{Z}$, which is obvious because of some basic geometric intuitions about line segments. So obviously, $\mathbb{Z}$ has to be one of the domains that does have unique factorization. Irreducible elements have to be prime in a PID, and every Euclidean domain is a PID.

Let's link back to your first question: Why is unique factorization non-obvious? It's only non-obvious for someone at step 2 of the progression we just outlined. Someone at step 3 could probably also allow that it's non-obvious along the way.

Your second question is much more general: Should we just ditch the idea of "obvious" or "intuition" in mathematics altogether?
I would say no. In fact, I would say that throwing away such concepts is not only impossible, but undesirable as well. The process, by which "obvious" things start to seem non-obvious (and then eventually turn out to be either obvious again for better reasons, or turn out to be false) is precisely the process of discovery. The really great "what if?" questions often challenge the obviousness of something that has been taken for granted.

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*"What if we were to allow square roots of negative numbers in our calculations?", asked Cardano, and this allowed him to develop general formulas for solving cubic polynomials. This is how we started understanding complex numbers.

*"What if, through a point not on a given line, more than one line could pass without intersecting the given line?", asked Lobachevsky, and this allowed him to develop hyperbolic geometry. This is how we started understanding ideas that led to differential geometry.

Although I can't speak from experience, I don't believe that this same dynamic could occur for mathematicians who somehow lack all intuition, or notion of obviousness. Seeing some things as clear facts, and seeing other things as conundrums, provides us with a richness of feature, a landscape where a clever explorer can make informed choices. Such a mathematician doesn't throw away intuition, buy they recognize it as being simultaneously a helpful guide, and a work-in-progress. They take advantage of obviousness, in not having to constantly verify every clear fact, but they're also willing to challenge it at the right times.
I hope I have addressed your question adequately. If you feel that anything is missing here, please let me know.
