> 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.

 I really appreciate your patient engagement with the doubters in this thread, but I think that your determination to make your point has led you astray here.> Now 6 has a "smallest" factor of 2, and a "smallest" factor of (1 - sqrt(-5)).As [nilkn](https://news.ycombinator.com/item?id=11955341) mentioned upthread, if you consider well ordering part of the intuition of the positive integers, then you have (depending on what else you consider obvious) practically pinned down the integers already. This reference to the 'smallest' factor is not just a throwaway, and it sinks this example (why is 1 - sqrt(-5) smaller than 1 + sqrt(-5), for example?).> 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?This gives the impression that math is subject to an appeal to authority, which I think is a shame. The newest student can find the error in the work of the Fields Medallist—though he or she probably won't, and the error he or she seems to have found is more likely to be a concealed subtlety—and to suggest avoiding dissenting on the grounds of eminence gives entirely the wrong idea of how mathematical argument should proceed.
 >>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.
 You're approaching this with a hostile attitude, which is preventing you from understanding and/or addressing what other people are saying, and substituting (light) mockery for attempts to understand what others are saying. You're not going to learn anything or convince anybody of anything this way.The point of using the ring with the sqrt in it is to conveniently demonstrate that the FTA is non-obvious. Since it is only being used for demonstration, and not as part of a proof, faking ignorance of sqrts and imaginary numbers is not helpful to you. There are many things in mathematics where the subtleties only came out later; heck, that's basically the entire history of set theory. Sets are also trivial if you come at them with an attitude of artificial ignorance like that.
 But he clearly does understand, and some gentle mockery at the other side is called for when they play dumb.I'm sure the fundamental theorem of arithmetic has a non-obvious proof. And perhaps that's precisely what a mathematician means every time they say "non-obvious". If that were all just made explicit to this general interest site in the first place, perhaps we'd have nothing to discuss.But if we are trying to play coy here, 1+1=2 also requires a non-obvious proof to anyone not versed in formal methods. I looked a proof up:http://mathforum.org/library/drmath/view/51551.htmlI don't know how long it would take me, working alone, to come up with that proof. It's non-obvious because it took humans probably 100,000 years to come up with it, even though we've had the IQ to do it for a long time. I don't think we could agree on what constitutes a proof of it without some social aspect and convention, so non-obvious by means of proof.But, we've been using and making predictions about the world using 1+1=2 for very much longer. That seems like a pretty worthwhile definition of obvious.
 >>You're not going to learn anything or convince anybody of anything this way.The tone of the original article is light mockery as well. That's why I am imitating it. I think the author is wrong, it's quite condescending for you to say "you are not going to learn". What about pointing the non-obvious step in the simple reasoning I gave?>>The point of using the ring with the sqrt in it is to conveniently demonstrate that the FTA is non-obvious.This is not a correct way to point something is non-obvious. I illustrated why already. It's just not a correct way of arguing.>>Since it is only being used for demonstration, and not as part of a proof, faking ignorance of sqrts and imaginary numbers is not helpful to you.It is. It points out why the argument made in the blog post is not correct way to argue something is non-obvious.>>There are many things in mathematics where the subtleties only came out later; heck, that's basically the entire history of set theory. Sets are also trivial if you come at them with an attitude of artificial ignorance like that.Some things in set theory are trivial, some aren't. I don't understand your point here. I am claiming FTA is obvious when it comes to natural numbers not that something similar to FTA on some other objects is obvious.
 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.
 >>So what you are saying is that if you don't know what can go wrong then it's "obviously true."I never claimed that. I only claimed that FTA is obvious and that pointing out that something similar to FTA on some complex objects (a + sqrt(-5) things) doesn't work isn't a correct way to argue the FTA itself isn't obvious.>>If you draw a distorted circle in the plane then it's obviously true that it has an inside and an outside.Yes (as long as definition of "distorted" doesn't contain any surprises, I am assuming you mean not exactly a circle but something like it).>>The inside is obviously contractable to a point, and the outside is obviously contractable to a plane with a hole in it.Those things already aren't obvious. Go ask someone a bright kid in high school what "contractable to a plane" is. It's not obvious in any way to non-mathematician.I am claiming FTA is obvious for a bright person who understand multiplication (or for a caveman who can do multiplication by putting rectangles together, then making rectangles from those rectangles etc.)>>In three dimensions if you have a distorted sphere then it obviously divides space into an inside and an outside.Yes, obvious.>>The inside is obviously contractable to a point, and the outside is obviously contractable to 3D space with a hole in itAgain, those things are very far from obvious. What "contractable to 3D space with a hole" means is very far away from "obvious" by any reasonable definition of the word.