Note that the question is not "is one a prime" but rather, "should one be a prime". It is important to recognize the difference.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
I see your point. We could change the definition (and we would need to change lots of consequent things as well) but the result may still be consistent and maybe even neater.
The prime factors of, say, 10 would become 1, 2 and 5 (seems a little unnecessary here).
And the prime number list would be 1, 2, 3, 5, 7, 11, etc ... (looks neater)
And the definition could be simplified to "divisible only by 1 or itself" (1 would qualify on both accounts!)
But would we actually *break* anything? Or could it simply be fixed by rewriting?
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
Interestingly, any number would always be divisible by itself. And any number would also be divisible by 1. But in the case of 1, both are one and the same. For every prime number, there are atleast two factors, one and itself. But does 1 pass this test? It has only one factor. Hence, it neither qualifies as a prime nor a non-prime.
Further, but for summing the factors for check of perfect numbers, amicable numbers, abundant numbers and deficient numbers, 1 is never considered to be a prime factor at any point of time.
The case is different with 2. 2 is divisible by both itself and 1, the pre-requisite of a prime number. Hence 2 is treated as a prime number and rightly so.
It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.