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

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

#191

Post by [deleted] »

pavelrub wrote: post #189
mcguire wrote: post #188

Earlier quoted context omitted.

Well, according to the Wikipedia page on Russell and Whitehead's Principia Mathematica [1], " 54.43: "From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2." —Volume I, 1st edition, page 379. (The proof is actually completed in Volume II, 1st edition, page 86, accompanied by the comment, "The above proposition is occasionally useful." - they go on to say "It is used at leas…

> if you can prove 1+1=2 at all, then the generalization to n+m=l (where l is the "intuitive" value of n+m) should be easy enough to use directly But the proof of 1+1=2 is itself trivial in PA. So this really doesn't mean much. In fact it seems like you are agreeing with me that having explicit proofs of m+n=l for m,n>1 is meaningless. > But my point is that, if you are constructing a proof in formal mathematics and…

[deleted]

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

#192

Post by psyklic »

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.

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

#193

Post by raincom »

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…

Or explain why swans are black in one environment, white in another environment. Here is where knowledge increases. Otherwise, we would be counting all generalized claims, stereotypes as scientific hypotheses.

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

#194

Post by Someone »

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.

But that's a weak argument. By that token, checking whether two numbers add up to a third or whether two numbers are equal or even whether two numbers have the same number of digits aren't obvious, if you make the numbers large enough.

"Can be answered quickly" is not a good indicator of "is obvious".

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

#195

Post by lostlogin »

jdc wrote: post #190
delazeur wrote: post #174

Earlier quoted context omitted.

> I've seen that on LessWrong too, really smart math/CS people talking about biology can get demolished by a second year biology student. IMO, LessWrong and other communities based around critical thinking tend to either foster a sense of intellectual arrogance or attract people who already have that quality. > Sometimes I even feel that math/CS education has damaged me in some ways, made me too arrogant, though obvi…

I generally agree, however I would like to know for what cases you think the skill of solving unfamiliar problems is dangerous.

Watch what happens when you let the post doc PhD "help" set up a scan in the MR scanner. It's amazing/terrifying what a very clever person will do in an unfamiliar environment with different rules to the real world.

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

#196

Post by contravariant »

Someone wrote: post #194
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.

But that's a weak argument. By that token, checking whether two numbers add up to a third or whether two numbers are equal or even whether two numbers have the same number of digits aren't obvious, if you make the numbers large enough. "Can be answered quickly" is not a good indicator of "is obvious".

Well, yeah. Having the same sum, or having the same number of digits aren't really obvious properties of numbers.

Similarly, both being prime and having the same product aren't obvious properties either. Making the fundamental theorem of arithmetic non-obvious (or at least individual cases of it, but if the general case were obvious then the individual cases should also be obvious).

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

#197

Post by delazeur »

jdc wrote: post #190
delazeur wrote: post #174

Earlier quoted context omitted.

> I've seen that on LessWrong too, really smart math/CS people talking about biology can get demolished by a second year biology student. IMO, LessWrong and other communities based around critical thinking tend to either foster a sense of intellectual arrogance or attract people who already have that quality. > Sometimes I even feel that math/CS education has damaged me in some ways, made me too arrogant, though obvi…

I generally agree, however I would like to know for what cases you think the skill of solving unfamiliar problems is dangerous.

It's dangerous when it leads a person to choose "I'm going to figure this out on my own" over "I'm going to find someone who can solve this" in cases where they really should have known to pick the latter.

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

#198

Post by playerqq »

Someone wrote: post #184
playerqq wrote: post #179

I haven't looked at any proofs for the FTA. Could somebody point out if I made any mistakes on the one I've arrived at? [1] Proof by induction that if a positive integer has a prime factorization, then it is unique. We're inducting over Z_N, where Z_N is the set of all positive integers with at least one known prime factorization using exactly N number of primes. Call this factorization Pn = p_1 * p_2 * ... p_n For e…

"If Fn has the same number of factors as Pn, divide both sides by p_i. Since Fn / p_i must be an integer, Fn must contain p_i, or else one of its factors f_i actually isn't prime by Euclid's Lemma." You would have to prove that first. https://en.m.wikipedia.org/wiki/Euclid%27s_lemma : "This property is the key[4] in the proof of the fundamental theorem of arithmetic [4] In general, to show that a domain is a unique f…

Got it. I was trying to do a proof while avoiding abstract algebra as much as possible, given how rusty I am with it.

Thanks for the feedback, and additional things to take a look at!

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

#199

Post by JadeNB »

Arnavion wrote: post #136
scott_s wrote: post #120

Earlier quoted context omitted.

Please eventually provide the answer, because I'm curious!

The correct spelling "contractible" yields useful Google results, namely that spheres in 3 dimensions (point 4) are not contractible to a single point. Edit: ... and maybe circles (point 2) too? Not a mathematician.

> The correct spelling "contractible" yields useful Google results, namely that spheres in 3 dimensions (point 4) are not contractible to a single point.

The claim is about the interior, not the sphere itself (which certainly is not contractible, as you say).

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

#200

Post by JadeNB »

ColinWright wrote: post #81
bluecalm wrote: post #75

Earlier quoted context omitted.

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 u…

> 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 h…

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.

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