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Zack_M_Davis comments on Bizarre Illusions - Less Wrong
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I think you're really failing to grasp the content of the unique factorization theorem here. Firstly we don't think about factored numbers as products of primes up to permutation, we think of them as products of distinct prime powers (up to permutation, I suppose - but it's probably better here to just take a commutative viewpoint and not regard "up to permutation" as worth specifying). But more importantly, you need to take a multiary view of multiplication here, not a binary one. 1 is the empty product, so in particular, it is the product of no primes, or the product of each prime to the 0th power. That is its unique prime factorization. To take 1 as a prime would be like having bases for vector spaces include 0. Almost exactly like it - if we take the Z-module of positive rationals under multiplication, the set of primes forms a free basis; 1 is the zero element.
Information and expertise like this is why hanging out at Less Wrong is worth the time. I estimate that I value the information in your comment at about $35, meaning my present self would advise my former self to pay up to $35 to read it.
So, I get it. My brain is more wired for analysis than algebra; so this isn't the first time that linear algebra has been a useful bridge for me. I see that we could have a 'vector space' of infinite-dimensional vectors where each vector (a1, a2, ..., an, ...) represents a number N where N = (P1^a1)(P2^a2)...(Pn^an)... and Pi are the ordered primes. Clearly 1 is the zero element and would never be a basis element.
I should admit here that my background in algebra is weak and I have no idea how you would need to modify the notion of 'vector space' to make certain things line up. But I can already speculate on how the choice of the "scalar field" for specifying the a_i would have interesting consequences:
I'd like to read more. What sub-field of mathematics is this?
Oh! And orthogonal vectors are relatively prime!
I'm not sure that the idea of orthogonality is defined for modules, is it? Is there a standard definition of an inner product for a Z-module?
Yes; the same definition works. See here.
Yay! I actually got something right!