4 ms·

Yes, but the only way in which that ordering compels the integers to be a unique factorization domain is through the fact that it allows us the Euclidean algori

by Chinjut 10y ago

Yes, but the only way in which that ordering compels the integers to be a unique factorization domain is through the fact that it allows us the Euclidean algorithm; put another way, through the fact that, for any bunch of integers, their smallest positive combination (in the sense of adding or subtracting various multiples of them together) divides all of them. But this fact about the integers actually doesn't seem to be obvious to many people! If I were to ask "What's the smallest positive value of the form 98X - 60Y?", how many people would say "Psh, obviously it's 2"? If I were to pick three particular primes p, q, and r, and ask what the smallest positive value I could get by adding and subtracting copies of p * q, p * r, and q * r together was, how many people would see right off the bat "Oh, it has to be possible to get 1 itself, as that is their only common factor"? Not many, I suspect. And if your magic integers-with-< intuition doesn't even give you that, then I don't think it's in any sound way giving you uniqueness of prime factorization. I think you're just back-rationalizing your sense that uniqueness of prime factorization must be obvious because you've never seen it fail and heard people talk about it a lot and so on, instead of having access to genuine grounds for certainty.

nilkn 10y ago
> I think you're just back-rationalizing your sense that uniqueness of prime factorization must be obvious because you've never seen it fail and heard people talk about it a lot and so on, instead of having access to genuine grounds for certainty. I don't think it's fair for you to make assumptions like this about me. I have a degree in mathematics and have a fair amount of experience with this stuff. I certainly do not have the same level of experience as Gowers, but the odds are very high that you don't either. > Yes, but the only way in which that ordering compels the integers to be a unique factorization domain is through the fact that it allows us the Euclidean algorithm; put another way, through the fact that, for any bunch of integers, their smallest positive combination (in the sense of adding or subtracting various multiples of them together) divides all of them. What do you mean by "only"? You'll need to use something roughly equivalent to this, but you certainly don't have to use it stated in this precise form. For instance, a proof has already been given in this very comment thread which does not require the Euclidean algorithm in the form in which you've stated it. It requires only a much more intuitive statement about modular arithmetic, which can be proven relatively easily using the properties I've mentioned (as was done by makomk). If you're still not satisfied, I encourage you to read this thread: http://math.stackexchange.com/questions/385967/origin-of-well-ordering-proof-of-uniqueness-in-the-ftoarithmetic http://math.stackexchange.com/questions/385967/origin-of-wel... --- The other questions you've posed seem irrelevant to me. Any mathematical fact can be spun so as to make it seem difficult or counter-intuitive. For instance, I might tell you that a very simple and elementary proof of the infinitude of primes makes it obvious that n^8+4n^7+8n^6+10n^5+9n^4+6n^3+3n^2+n has at least three prime factors for all positive integers n. But how obvious is that to you?
- Chinjut 10y ago
  Oh, don't worry; the "you" in "I think you're..." was generic (for anyone who claims complete and utter obviousness of unique prime factorization without ability to explicate the reasoning behind it or even grasp that reasoning might be required) and not targeted at YOU per se; I'm not accusing you of lacking mathematical sophistication. It was clear from your other comments that you have significant background in mathematics. I'm just accusing you of being misguided when claiming that humans are able to access uniqueness of prime factorization as obvious because of special inbuilt integers-with-< intuition. I acknowledge that you clearly know perfectly well the sort of reasoning Gowers and I would claim is necessary to soundly claim understanding of UPF, but you seem to feel this reasoning is something people are actually implicitly aware of when considering UPF obvious, and I think that is erroneous. That is: Sure, all you need is that, in arithmetic modulo a prime, nonzero values are closed under multiplication. But this fact is not obvious! The only reason to believe this is via, implicitly or explicitly, the reasoning that tells us how to compute GCDs as linear combinations (what I have referred to as "the Euclidean algorithm" for shorthand, though I do not mean to fixate on any particular presentation of such reasoning). Here is makomk's argument: "Brief outline of an argument: suppose nm = 0 mod p (p prime) and m != 0 mod p. We can write this as n(ap+b) = 0 mod p for some integers a, 0<b<p, and trivially nb = 0 mod p. So there is some smallest b', 0<b'<p for which nb' = 0 mod p. We can also see nb'r = 0 mod p for all integers r. Find the smallest r for which b'r >= p. If b' != 1, then since p is prime b'r != p and we get a smaller b'' := b'r-p for which n*b'' = 0 mod p, a contradiction. Therefore b' = 1 and n = 0 mod p." That is good, that is fine, that is the sort of argument one needs. But I would not consider that to be at all "obvious" in the way laypeople often claim UPF to be. It takes some insight or work to see the possibility of that argument! There's no reason to consider that sort of argument as deep-in-our-bones obvious, implicit to laypeople by basic intuition about integers-with-<. There are myriad quite closely related facts about integers-with-< that laypeople have no awareness of, so why should we consider the reasoning in this particular case to be built-in? Especially when calls to make such reasoning explicit fail entirely. (FWIW, makomk's argument is also closely related to the following natural algorithm for computing the gcd of A and B: starting with an initial guess x which is any combination of A and B, keep replacing x with any nonzero value among either A % x or B % x (up to negation, if you like), till eventually one cannot continue reducing x in this way; at all times, x is a combination of A and B, and one must stop by finding such a combination that divides both A and B (and is therefore their GCD). This isn't quite the Euclidean algorithm as usually presented, but serves just as well as an algorithm computing GCDS-as-combinations recursively, and makomk's argument essentially follows the structure of computing the GCD of m and p in this way) But nevermind the full generality of GCD computation. That may be a distraction from the point I intended to make. (I phrased things in terms of this because it was the easiest wording, but that's not quite what I'm concerned about). Even just the special case of this reasoning used by makomk is not something "obvious". I do not believe the layperson who claims UPF to be obvious has an argument like makomk's latent in their head. I think they are just making unjustified leaps. I think they are, as I said, just presuming UPF obvious because they've never seen it fail and have heard it discussed so many times and so on, to the point that they can't even conceptualize that any further reasoning could be called for. As for "using the properties I've mentioned", presumably in reference to the integers being a well-ordered ring: sure, being a unique factorization domain follows from being a well-ordered commutative ring where the ordering interfaces with the ring structure in the expected ways... because the ONLY well-ordered ring is the integers, and the integers are a UFD. But I still don't think the average person, when looking at uniqueness of factorization as "obvious", is accessing the reasoning that goes into this.

by Chinjut 10y ago

nilkn 10y ago

> I think you're just back-rationalizing your sense that uniqueness of prime factorization must be obvious because you've never seen it fail and heard people talk about it a lot and so on, instead of having access to genuine grounds for certainty. I don't think it's fair for you to make assumptions like this about me. I have a degree in mathematics and have a fair amount of experience with this stuff. I certainly do not have the same level of experience as Gowers, but the odds are very high that you don't either. > Yes, but the only way in which that ordering compels the integers to be a unique factorization domain is through the fact that it allows us the Euclidean algorithm; put another way, through the fact that, for any bunch of integers, their smallest positive combination (in the sense of adding or subtracting various multiples of them together) divides all of them. What do you mean by "only"? You'll need to use something roughly equivalent to this, but you certainly don't have to use it stated in this precise form. For instance, a proof has already been given in this very comment thread which does not require the Euclidean algorithm in the form in which you've stated it. It requires only a much more intuitive statement about modular arithmetic, which can be proven relatively easily using the properties I've mentioned (as was done by makomk). If you're still not satisfied, I encourage you to read this thread: http://math.stackexchange.com/questions/385967/origin-of-well-ordering-proof-of-uniqueness-in-the-ftoarithmetic http://math.stackexchange.com/questions/385967/origin-of-wel... --- The other questions you've posed seem irrelevant to me. Any mathematical fact can be spun so as to make it seem difficult or counter-intuitive. For instance, I might tell you that a very simple and elementary proof of the infinitude of primes makes it obvious that n^8+4n^7+8n^6+10n^5+9n^4+6n^3+3n^2+n has at least three prime factors for all positive integers n. But how obvious is that to you?

Chinjut 10y ago
Oh, don't worry; the "you" in "I think you're..." was generic (for anyone who claims complete and utter obviousness of unique prime factorization without ability to explicate the reasoning behind it or even grasp that reasoning might be required) and not targeted at YOU per se; I'm not accusing you of lacking mathematical sophistication. It was clear from your other comments that you have significant background in mathematics. I'm just accusing you of being misguided when claiming that humans are able to access uniqueness of prime factorization as obvious because of special inbuilt integers-with-< intuition. I acknowledge that you clearly know perfectly well the sort of reasoning Gowers and I would claim is necessary to soundly claim understanding of UPF, but you seem to feel this reasoning is something people are actually implicitly aware of when considering UPF obvious, and I think that is erroneous. That is: Sure, all you need is that, in arithmetic modulo a prime, nonzero values are closed under multiplication. But this fact is not obvious! The only reason to believe this is via, implicitly or explicitly, the reasoning that tells us how to compute GCDs as linear combinations (what I have referred to as "the Euclidean algorithm" for shorthand, though I do not mean to fixate on any particular presentation of such reasoning). Here is makomk's argument: "Brief outline of an argument: suppose nm = 0 mod p (p prime) and m != 0 mod p. We can write this as n(ap+b) = 0 mod p for some integers a, 0<b<p, and trivially nb = 0 mod p. So there is some smallest b', 0<b'<p for which nb' = 0 mod p. We can also see nb'r = 0 mod p for all integers r. Find the smallest r for which b'r >= p. If b' != 1, then since p is prime b'r != p and we get a smaller b'' := b'r-p for which n*b'' = 0 mod p, a contradiction. Therefore b' = 1 and n = 0 mod p." That is good, that is fine, that is the sort of argument one needs. But I would not consider that to be at all "obvious" in the way laypeople often claim UPF to be. It takes some insight or work to see the possibility of that argument! There's no reason to consider that sort of argument as deep-in-our-bones obvious, implicit to laypeople by basic intuition about integers-with-<. There are myriad quite closely related facts about integers-with-< that laypeople have no awareness of, so why should we consider the reasoning in this particular case to be built-in? Especially when calls to make such reasoning explicit fail entirely. (FWIW, makomk's argument is also closely related to the following natural algorithm for computing the gcd of A and B: starting with an initial guess x which is any combination of A and B, keep replacing x with any nonzero value among either A % x or B % x (up to negation, if you like), till eventually one cannot continue reducing x in this way; at all times, x is a combination of A and B, and one must stop by finding such a combination that divides both A and B (and is therefore their GCD). This isn't quite the Euclidean algorithm as usually presented, but serves just as well as an algorithm computing GCDS-as-combinations recursively, and makomk's argument essentially follows the structure of computing the GCD of m and p in this way) But nevermind the full generality of GCD computation. That may be a distraction from the point I intended to make. (I phrased things in terms of this because it was the easiest wording, but that's not quite what I'm concerned about). Even just the special case of this reasoning used by makomk is not something "obvious". I do not believe the layperson who claims UPF to be obvious has an argument like makomk's latent in their head. I think they are just making unjustified leaps. I think they are, as I said, just presuming UPF obvious because they've never seen it fail and have heard it discussed so many times and so on, to the point that they can't even conceptualize that any further reasoning could be called for. As for "using the properties I've mentioned", presumably in reference to the integers being a well-ordered ring: sure, being a unique factorization domain follows from being a well-ordered commutative ring where the ordering interfaces with the ring structure in the expected ways... because the ONLY well-ordered ring is the integers, and the integers are a UFD. But I still don't think the average person, when looking at uniqueness of factorization as "obvious", is accessing the reasoning that goes into this.