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Sometimes the definition is wrong (which should be a big hint that the definition is just pointing at something). Like the definition of prime that I was taugh

by juped 7y ago

Sometimes the definition is wrong (which should be a big hint that the definition is just pointing at something). Like the definition of prime that I was taught in primary school, "a prime is a positive integer whose only divisors are itself and 1". The right definition is "a prime is a positive integer with exactly two divisors", because otherwise all interesting statements about primes would have to be confusingly rewritten as statements about primes greater than 1. (Sometimes we have theorems about "odd primes", i.e., not 2, but the important set, primes, definitely includes 2.) I don't get the monad joke fwiw; I've done a ton of Haskell and a ton of math, but the Haskell wasn't deep abstraction and the math was differential geometry and the vicinity.

gjm11 7y ago
To make matters worse, it turns out that there are actually two different concepts commonly labelled by the word "prime", and in some highbrow contexts they are different -- so mathematicians actually define "prime" in an entirely different way: p is prime iff p|ab => (p|a or p|b). That is, a prime is something that can't divide the product of two numbers without dividing at least one of the numbers. The thing about having no nontrivial divisors is called being "irreducible". (When are they different? When you're working with some kind of number other than the ordinary integers. For instance, suppose you decide to treat sqrt(-5) as an integer, so your numbers are now everything of the form a + b sqrt(-5) where a,b are ordinary integers. In this system of numbers, 3 is irreducible, but it isn't prime because (1+sqrt(-5)) (1-sqrt(-5)) = 6 which is a multiple of 3 -- but neither of those factors is a multiple of 3 even in this extended system of numbers.)
eximius 7y ago
> primes greater than 1 I mean, many theorems about primes do not apply to 1. 1 is very often not considered a prime. And, as another commenter noted, there is not just one definition of prime.

by juped 7y ago

gjm11 7y ago

To make matters worse, it turns out that there are actually two different concepts commonly labelled by the word "prime", and in some highbrow contexts they are different -- so mathematicians actually define "prime" in an entirely different way: p is prime iff p|ab => (p|a or p|b). That is, a prime is something that can't divide the product of two numbers without dividing at least one of the numbers. The thing about having no nontrivial divisors is called being "irreducible". (When are they different? When you're working with some kind of number other than the ordinary integers. For instance, suppose you decide to treat sqrt(-5) as an integer, so your numbers are now everything of the form a + b sqrt(-5) where a,b are ordinary integers. In this system of numbers, 3 is irreducible, but it isn't prime because (1+sqrt(-5)) (1-sqrt(-5)) = 6 which is a multiple of 3 -- but neither of those factors is a multiple of 3 even in this extended system of numbers.)

eximius 7y ago

> primes greater than 1 I mean, many theorems about primes do not apply to 1. 1 is very often not considered a prime. And, as another commenter noted, there is not just one definition of prime.