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Obvious is different than easy to prove. The concepts of multiplication, division, prime number and "divisible by" are much older than formal proofs and arbitra

Obvious is different than easy to prove. The concepts of multiplication, division, prime number and "divisible by" are much older than formal proofs and arbitrary sets of axioms. Let's say I only now what multiplication is and that AxB = BxA and that prime number can't be written as AxB unless A or B are 1. Now it's obvious that there are factorings of a number: you just divide it by smallest possible prime divider until you reach 1, that process obviously terminates every time and produces a finite factoring. Now let's assume that our number A has two factorings F1 and F2. Let's sort them from the smallest to the biggest divider. Is it possible that F1 and F2 are different at the first position? It isn't as that would mean the same number has different smallest prime divider. We therefore divide our number by the prime divider at that first position and continue the process proving that dividers at all the positions must be the same or that F1 F2 are factorings of a different number. It is in fact obvious to someone who understands multiplication. As to some points from the article: >>it is obvious that 23\times 1759 is not the same number as 53\times 769. It is, we sort them from the smallest to bigger prime in the factorization. If they were the same number that would mean the same number's smallest prime diviser is 23 and 53 at the same time. You may want to argue that it's not obvious that to be divisible by a prime number it must appear in a factorization - here I just assume you understand it's obvious for anyone who knows what multiplication is and that the fact that it's not obvious to your system with your axioms is your problem altogether. >>If it’s so obvious that every number has a unique factorization, then why is the corresponding statement false in a similar context? It's not a similar context, at least for non-mathematician. You are introducing some weird objects in the form of a + b*sqrt(-5). I may not even understand the concept of a complex number but I still can understand FTA. Btw, for that FTA isn't obvious because there is no order for those objects, it's not clear what is bigger or smaller than something else and which position given object is in if we sort them from the smallest. I get it: FTA is hard to prove in your nice formal system with your nifty arbitrary chosen axioms and formal deduction rules. It's bloody obvious to the caveman who can do multiplication by putting stones in a rectangle though.

ColinWright 10y ago
> Now let's assume that our number A has > two factorings F1 and F2. Let's sort > them from the smallest to the biggest > divider. OK, I've done that. > Is it possible that F1 and F2 are > different at the first position? It > isn't as that would mean the same > number has different smallest prime > divider. So why is this false in Z[ sqrt(-5) ] ?? There we have: 6 = 2 x 3 6 = (1 - sqrt(-5)) x ((1 + sqrt(-5)) Now 6 has a "smallest" factor of 2, and a "smallest" factor of (1 - sqrt(-5)). > It is in fact obvious to someone who > understands multiplication. So you are claiming that the author of the linked article, Prof Sir Tim Gowers, winner of the Fields Medal, Fellow of the Royal Society, doesn't understand multiplication? Might I instead suggest that you don't understand the things that might go wrong, and the subtleties that lurk underneath.
- bluecalm 10y ago
  >>So why is this false in Z[ sqrt(-5) ] I don't know, I don't know what sqrt is, let alone sqrt for a negative number. It's like asking someone who made a nice geometric proof of Pitagoras theorem why it doesn't work on a 4 dimensional sphere for stuff which is kinda like triangles. It's different multiplication you are mentioning here. I can't do (1-sqrt(-5) x (1 + sqrt(-5) by putting some stones in a rectangle and counting them. The concepts of primes, multiplication, divisor don't instantly make sense for those objects and I am not sure why you ask me to extend them. I am just claiming FTA is obvious for natural numbers and straightforward multiplication. >> 6 = 2 x 3 >> 6 = (1 - sqrt(-5)) x ((1 + sqrt(-5)) I mentioned sorting from the smallest, can't do that with your sqrt thing. Another obvious thing with multiplication is that the more you multiply the more you get which isn't true for a + sqrt(-5). >>So you are claiming that the author of the linked article, Prof Sir Tim Gowers, winner of the Fields Medal, Fellow of the Royal Society, doesn't understand multiplication? No, I haven't claimed that. I think FTA is quite obvious to him but it's not obvious if you try formally define arithmetic using the smallest sensible subset of axioms. It is obvious is you just understand multiplication and can perform it by forming rectangles and then rectangles from rectangles (in case of 3 terms to multiply). >>ight I instead suggest that you don't understand the things that might go wrong, and the subtleties that lurk underneath. There are many subtleties in defining and proving things formally. That's a different point altogether.
  - ColinWright 10y ago
    So what you are saying is that if you don't know what can go wrong then it's "obviously true." Let's try some other things. * If you draw a distorted circle in the plane then it's obviously true that it has an inside and an outside. * The inside is obviously contractable to a point, and the outside is obviously contractable to a plane with a hole in it. * In three dimensions if you have a distorted sphere then it obviously divides space into an inside and an outside. * The inside is obviously contractable to a point, and the outside is obviously contractable to 3D space with a hole in it. All obvious, right? Now, pick the statement (or statements) from the above that are in fact false.
    - 10y ago

ColinWright 10y ago

> Now let's assume that our number A has > two factorings F1 and F2. Let's sort > them from the smallest to the biggest > divider. OK, I've done that. > Is it possible that F1 and F2 are > different at the first position? It > isn't as that would mean the same > number has different smallest prime > divider. So why is this false in Z[ sqrt(-5) ] ?? There we have: 6 = 2 x 3 6 = (1 - sqrt(-5)) x ((1 + sqrt(-5)) Now 6 has a "smallest" factor of 2, and a "smallest" factor of (1 - sqrt(-5)). > It is in fact obvious to someone who > understands multiplication. So you are claiming that the author of the linked article, Prof Sir Tim Gowers, winner of the Fields Medal, Fellow of the Royal Society, doesn't understand multiplication? Might I instead suggest that you don't understand the things that might go wrong, and the subtleties that lurk underneath.

bluecalm 10y ago
>>So why is this false in Z[ sqrt(-5) ] I don't know, I don't know what sqrt is, let alone sqrt for a negative number. It's like asking someone who made a nice geometric proof of Pitagoras theorem why it doesn't work on a 4 dimensional sphere for stuff which is kinda like triangles. It's different multiplication you are mentioning here. I can't do (1-sqrt(-5) x (1 + sqrt(-5) by putting some stones in a rectangle and counting them. The concepts of primes, multiplication, divisor don't instantly make sense for those objects and I am not sure why you ask me to extend them. I am just claiming FTA is obvious for natural numbers and straightforward multiplication. >> 6 = 2 x 3 >> 6 = (1 - sqrt(-5)) x ((1 + sqrt(-5)) I mentioned sorting from the smallest, can't do that with your sqrt thing. Another obvious thing with multiplication is that the more you multiply the more you get which isn't true for a + sqrt(-5). >>So you are claiming that the author of the linked article, Prof Sir Tim Gowers, winner of the Fields Medal, Fellow of the Royal Society, doesn't understand multiplication? No, I haven't claimed that. I think FTA is quite obvious to him but it's not obvious if you try formally define arithmetic using the smallest sensible subset of axioms. It is obvious is you just understand multiplication and can perform it by forming rectangles and then rectangles from rectangles (in case of 3 terms to multiply). >>ight I instead suggest that you don't understand the things that might go wrong, and the subtleties that lurk underneath. There are many subtleties in defining and proving things formally. That's a different point altogether.
- ColinWright 10y ago
  So what you are saying is that if you don't know what can go wrong then it's "obviously true." Let's try some other things. * If you draw a distorted circle in the plane then it's obviously true that it has an inside and an outside. * The inside is obviously contractable to a point, and the outside is obviously contractable to a plane with a hole in it. * In three dimensions if you have a distorted sphere then it obviously divides space into an inside and an outside. * The inside is obviously contractable to a point, and the outside is obviously contractable to 3D space with a hole in it. All obvious, right? Now, pick the statement (or statements) from the above that are in fact false.
  - 10y ago