4 ms·

My guess is part of the issue is people remembering that 0 is neither positive nor negative and neither prime nor composite and naturally extending it to even v

My guess is part of the issue is people remembering that 0 is neither positive nor negative and neither prime nor composite and naturally extending it to even vs. odd. But even here, one has to be careful about what definition of "even" you're employing. = 0 (mod 2)? Has two as a factor? One less than an odd number? !(x % 2)? I remember my confusion when I found out that the answer to "is 0 a natural number" was "it depends" and I still run into people who insist 1 is prime, which I think is insane. A lot of these things depend on the norms of the mathematical community discussing them.

derefr 6y ago
To me, the “is 1 prime” question gets at a very subtle distinction people don’t realize they’re making. Some people conceptualize the prime numbers as a mathematical sequence definitionally — akin to the Fibonacci sequence. And mathematical sequences have both inductive-case members — that is, members created by the inductive-step “algorithm” for the sequence — and also base-case members — members by fiat, that bootstrap the sequence into being. In the mathematical inductive sequence of the prime numbers, you clearly need 1 there in order to “get the sequence going.” Any parsimonious formula you could define for a prime-sieve induction step, needs to start off with a 1 already “in the sieve” in order to generate the sequence you want it to generate. Any special-case that excludes 1 (or rather, unit values) from “being prime”, can be reformulated into a syllogism for 1 (/unit values) being base-case members of the primes sequence. And yet, it turns out that the definition of “prime” isn’t actually “a member of the generative sequence of the prime numbers”, but rather more specifically “an output of the generative process of the prime numbers.” This is unusual for a mathematical property — usually we use an adjective “Foo x” to refer to an x that shows up as a member in a mathematical object Foo. But prime-ness and membership-in-the-primes-sequence aren’t equivalent. Primality as a property, was created as a mathematical tool to reason about perfect numbers; and you don’t get perfect numbers (or at least, not the same set of perfect numbers) if you label 1 as prime. But this, obviously, has nothing to do with the primes sequence! It’s as if we had a concept of a “Fibonacci number” that didn’t include the base-case members of 0 and 1. It would feels kind of strange and arbitrary to laymen, who probably know about the Fibonacci sequence, but who wouldn’t realize that there might be something interesting about the Fibonacci-ness property of the inductive members, that doesn’t apply to the base-case members.
- ajarmst 6y ago
  That makes sense. For me, including 1 breaks the fundamental theorem of arithmetic (for values of "breaks" that include requiring an ugly "except for one" in the definition).
keeglin 6y ago
My first comment on HN; I have strong feelings on this topic. I was specifically taught as a child that zero is neither even nor odd, because zero has no value. I don't think it was confusion or misunderstanding. Understanding the value of "nothing" or "the opposite of something" is what separates us from most other brained-animals, correct? I understand there are patterns that can be applied. However, I never once was misserved by that understanding, either through academics or physical life. Ultimately, it's more upsetting today to try to reset that thought process in my mind. So to me, zero is neither even or odd, and I'm happy with that.
- ajarmst 6y ago
  A reasonable case can be made that (if we employ the "is evenly divisible by 2/has 2 as a factor" definition) zero is not composite, so has no factors and therefore can't be even. I happen to disagree, but it's not clearly right or wrong. This is why when mathematicians from different subdisciplines/communities spend a fair bit of time on "when you use the term ... what do you mean?" Outside a shared definition of exactly what "even" means, the question is meaningless, and within one, it's trivial---the least interesting possible questions in mathematics, but we seem to spend a lot of time on them.

derefr 6y ago

To me, the “is 1 prime” question gets at a very subtle distinction people don’t realize they’re making. Some people conceptualize the prime numbers as a mathematical sequence definitionally — akin to the Fibonacci sequence. And mathematical sequences have both inductive-case members — that is, members created by the inductive-step “algorithm” for the sequence — and also base-case members — members by fiat, that bootstrap the sequence into being. In the mathematical inductive sequence of the prime numbers, you clearly need 1 there in order to “get the sequence going.” Any parsimonious formula you could define for a prime-sieve induction step, needs to start off with a 1 already “in the sieve” in order to generate the sequence you want it to generate. Any special-case that excludes 1 (or rather, unit values) from “being prime”, can be reformulated into a syllogism for 1 (/unit values) being base-case members of the primes sequence. And yet, it turns out that the definition of “prime” isn’t actually “a member of the generative sequence of the prime numbers”, but rather more specifically “an output of the generative process of the prime numbers.” This is unusual for a mathematical property — usually we use an adjective “Foo x” to refer to an x that shows up as a member in a mathematical object Foo. But prime-ness and membership-in-the-primes-sequence aren’t equivalent. Primality as a property, was created as a mathematical tool to reason about perfect numbers; and you don’t get perfect numbers (or at least, not the same set of perfect numbers) if you label 1 as prime. But this, obviously, has nothing to do with the primes sequence! It’s as if we had a concept of a “Fibonacci number” that didn’t include the base-case members of 0 and 1. It would feels kind of strange and arbitrary to laymen, who probably know about the Fibonacci sequence, but who wouldn’t realize that there might be something interesting about the Fibonacci-ness property of the inductive members, that doesn’t apply to the base-case members.

ajarmst 6y ago
That makes sense. For me, including 1 breaks the fundamental theorem of arithmetic (for values of "breaks" that include requiring an ugly "except for one" in the definition).

keeglin 6y ago

My first comment on HN; I have strong feelings on this topic. I was specifically taught as a child that zero is neither even nor odd, because zero has no value. I don't think it was confusion or misunderstanding. Understanding the value of "nothing" or "the opposite of something" is what separates us from most other brained-animals, correct? I understand there are patterns that can be applied. However, I never once was misserved by that understanding, either through academics or physical life. Ultimately, it's more upsetting today to try to reset that thought process in my mind. So to me, zero is neither even or odd, and I'm happy with that.

ajarmst 6y ago
A reasonable case can be made that (if we employ the "is evenly divisible by 2/has 2 as a factor" definition) zero is not composite, so has no factors and therefore can't be even. I happen to disagree, but it's not clearly right or wrong. This is why when mathematicians from different subdisciplines/communities spend a fair bit of time on "when you use the term ... what do you mean?" Outside a shared definition of exactly what "even" means, the question is meaningless, and within one, it's trivial---the least interesting possible questions in mathematics, but we seem to spend a lot of time on them.