Why isn’t the fundamental theorem of arithmetic obvious? (2011) • Hacker News Board

Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)

#201

Post by JadeNB » Thu, Jun 23, 2016, 3:25 PM UTC

kazinator wrote: post #52

curiousgal wrote: post #21

Earlier quoted context omitted.

I didn't claim that FTA is the most practical way to perform Gödel numbering. I just said that it's my favorite application of it, in a general sense, not that I'd use it to do so. Preferences are subjective.

"Application" in mathematics usually means that something is used as a necessary component of a solution in another area. If Gödel numbering is an "application" of the FTA, that strongly suggests that you're saying that FTA somehow enables the possibility of Gödel numbering, or else that it is exploited somehow to endow that numbering with convenient properties (without loss of generality). Is that true?

> "Application" in mathematics usually means that something is used as a necessary component of a solution in another area.

I think that sentence is entirely true if you delete the word 'necessary', and otherwise entirely false. I think that almost everyone would agree that at the heart of modern cryptography is an application par excellence of modular arithmetic, but I think that no-one would claim that cryptography cannot be done without modular arithmetic.

Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)

#202

Post by fenomas » Thu, Jun 23, 2016, 4:16 PM UTC

psyklic wrote: post #192

Someone wrote: post #150

Let me try to explain with an example outside of mathematics: All swans are white. For centuries (possibly millennia, as Juvenal thought it, too), that was obvious (in western Europe) to anyone studying nature. Then, Willem de Vlamingh returns from a journey to Australia with some dead black swans. Now, there are various options. Some of them are: - You can drop your claim that all swans are white. - You state these…

Gowers argues that even without considering generalizations of the reals, it is not "obvious". He argues that if the Theorem were "obvious", we should quickly be able to say that 23 x 1759 != 53 x 769, without multiplying them out.

Perhaps I'm dim but that seems like a ludicrous argument. The inequality is obvious, if you know that all the numbers are prime, and it's not if you don't. Ergo, what's non-obvious is that the big numbers are prime, not that fundamental theorem is true.

Shouldn't the standard of proof should be something like whether it's obvious that:

    23 * 7 * 11  =/=  17 * 7 * 13

? In this case the inequality is obvious, to the extent that you can convince yourself of it without multiplying anything. Doesn't this imply that therefore the theorem is obvious?

Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)

#203

Post by psyklic » Thu, Jun 23, 2016, 4:55 PM UTC

fenomas wrote: post #202

psyklic wrote: post #192

Earlier quoted context omitted.

Gowers argues that even without considering generalizations of the reals, it is not "obvious". He argues that if the Theorem were "obvious", we should quickly be able to say that 23 x 1759 != 53 x 769, without multiplying them out.

Perhaps I'm dim but that seems like a ludicrous argument. The inequality is obvious, if you know that all the numbers are prime, and it's not if you don't. Ergo, what's non-obvious is that the big numbers are prime, not that fundamental theorem is true. Shouldn't the standard of proof should be something like whether it's obvious that: 23 * 7 * 11 =/= 17 * 7 * 13 ? In this case the inequality is obvious, to the exten…

It is only obvious if you take the theorem as a given (as mathematicians typically do). However, I bet if you are given a slightly different system, it would be difficult to say whether this is true: For some natural number N, do there exist n primes p1,..,pn such that p1..pn = N and no other m primes q1,..,qm where q1..qm = N ? As an example, Gowers gives the complex numbers, where it may intuitively appear to be true ... but it is not.

Gowers mentions that, to him, an obvious argument would be: 23 x 22 != 21 x 25, since 2 divides 23x22, but 2 does not divide 21x25.

Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)

#204

Post by bluecalm » Thu, Jun 23, 2016, 11:03 PM UTC

jerf wrote: post #114

bluecalm wrote: post #83

Earlier quoted context omitted.

>>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…

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…

>>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.

Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)

#205

Post by bluecalm » Thu, Jun 23, 2016, 11:09 PM UTC

ColinWright wrote: post #86

bluecalm wrote: post #83

Earlier quoted context omitted.

>>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…

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 obv…

>>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 it

Again, 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.

Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)

#206

Post by fenomas » Fri, Jun 24, 2016, 2:32 AM UTC

psyklic wrote: post #203

fenomas wrote: post #202

Earlier quoted context omitted.

Perhaps I'm dim but that seems like a ludicrous argument. The inequality is obvious, if you know that all the numbers are prime, and it's not if you don't. Ergo, what's non-obvious is that the big numbers are prime, not that fundamental theorem is true. Shouldn't the standard of proof should be something like whether it's obvious that: 23 * 7 * 11 =/= 17 * 7 * 13 ? In this case the inequality is obvious, to the exten…

It is only obvious if you take the theorem as a given (as mathematicians typically do). However, I bet if you are given a slightly different system, it would be difficult to say whether this is true: For some natural number N, do there exist n primes p1,..,pn such that p1..pn = N and no other m primes q1,..,qm where q1..qm = N ? As an example, Gowers gives the complex numbers, where it may intuitively appear to be tr…

> Gowers mentions that, to him, an obvious argument would be: 23 x 22 != 21 x 25, since 2 divides 23x22, but 2 does not divide 21x25.

I don't follow the distinction - isn't the same true of my example, just swapping "2" for "11" or "23"?

I mean, given Gowers' standing I'm prepared to assume his premise is true, and I didn't follow the last argument (by comparison to other kinds of rings) at all so I'm guessing that it's the "real" reason why the theorem isn't obvious. But the preceding arguments made no sense to me at all.

Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)

#207

Post by ellyagg » Sat, Jun 25, 2016, 1:03 AM UTC

Someone wrote: post #150

Let me try to explain with an example outside of mathematics: All swans are white. For centuries (possibly millennia, as Juvenal thought it, too), that was obvious (in western Europe) to anyone studying nature. Then, Willem de Vlamingh returns from a journey to Australia with some dead black swans. Now, there are various options. Some of them are: - You can drop your claim that all swans are white. - You state these…

Many people would consider the set of numbers the fundamental theorem of arithmetic applies to be more different from other sets of objects that don't abide by the fundamental theorem of arithmetic than white swans differ from black swans. I'm afraid this isn't the last word on the debate.

Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)

#208

Post by psyklic » Sat, Jun 25, 2016, 1:09 AM UTC

fenomas wrote: post #206

psyklic wrote: post #203

Earlier quoted context omitted.

It is only obvious if you take the theorem as a given (as mathematicians typically do). However, I bet if you are given a slightly different system, it would be difficult to say whether this is true: For some natural number N, do there exist n primes p1,..,pn such that p1..pn = N and no other m primes q1,..,qm where q1..qm = N ? As an example, Gowers gives the complex numbers, where it may intuitively appear to be tr…

> Gowers mentions that, to him, an obvious argument would be: 23 x 22 != 21 x 25, since 2 divides 23x22, but 2 does not divide 21x25. I don't follow the distinction - isn't the same true of my example, just swapping "2" for "11" or "23"? I mean, given Gowers' standing I'm prepared to assume his premise is true, and I didn't follow the last argument (by comparison to other kinds of rings) at all so I'm guessing that i…

The difference is that divisibility by 2 can be tested almost instantly. (Only the least significant digits of each factor matter.) However, testing each factor for divisibility is often as tedious as the original multiplication!

I think we're all at a loss for a good definition of "obvious".

Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)

#209

Post by ellyagg » Sat, Jun 25, 2016, 1:24 AM UTC

Someone wrote: post #150

Let me try to explain with an example outside of mathematics: All swans are white. For centuries (possibly millennia, as Juvenal thought it, too), that was obvious (in western Europe) to anyone studying nature. Then, Willem de Vlamingh returns from a journey to Australia with some dead black swans. Now, there are various options. Some of them are: - You can drop your claim that all swans are white. - You state these…

Oh dear. So you're taking the much stronger stance that it's not even just non-obvious to prove, it's non-obvious to intuit!

If you are being honest, the set of numbers the fundamental theorem of arithmetic applies to is much more different from sets of objects it does not apply to than black swans are from white swans.

Obviousness is probabilistic, so proving one seemingly obvious thing non-obvious doesn't mean we have to stop calling everything that seems obvious non-obvious, which is the implication of your analogy.

Sets of objects that don't follow the fundamental theorem of arithmetic are already super non-obvious themselves, so it seems strange to use their features as an excuse to invalidate an obvious feature of the set of integers.

Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)

#210

Post by ellyagg » Sat, Jun 25, 2016, 8:50 PM UTC

ellyagg wrote: post #209

Someone wrote: post #150

Let me try to explain with an example outside of mathematics: All swans are white. For centuries (possibly millennia, as Juvenal thought it, too), that was obvious (in western Europe) to anyone studying nature. Then, Willem de Vlamingh returns from a journey to Australia with some dead black swans. Now, there are various options. Some of them are: - You can drop your claim that all swans are white. - You state these…

Oh dear. So you're taking the much stronger stance that it's not even just non-obvious to prove, it's non-obvious to intuit! If you are being honest, the set of numbers the fundamental theorem of arithmetic applies to is much more different from sets of objects it does not apply to than black swans are from white swans. Obviousness is probabilistic, so proving one seemingly obvious thing non-obvious doesn't mean we h…

Not only that, but there are other even more obvious properties of the set of integers that don't inhere to other sets of objects since discovered. So your claim is essentially that there are no obvious properties of integers, which leaves one feeling a little flat.