1310: Goldbach Conjectures

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Re: 1310: Goldbach Conjectures

Postby gmalivuk » Mon Dec 30, 2013 9:15 pm UTC

In particular, that one looks like it's a parody of an anti-evolution "argument" about how no new information or speciation or whatever could happen just from the sum of existing things.
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Re: 1310: Goldbach Conjectures

Postby xtifr » Mon Dec 30, 2013 9:17 pm UTC

Flumble wrote:
garaden wrote:Alice: "Given that you can create any number over 7, they keep going."

I can't read "numbers keep going" in any way other than "there is a successor to every number",

You're doing fine so far.

which means 7 has a successor which is over 7.


Highlighted your error. You're stuck in Euclidean thinking. If the parallel postulate is incorrect and parallel lines meet at 4 and 0, then the successor to 7 is 0.

(Someone else presented a much simpler way of explaining this, earlier, but I like to make my students work. :) )
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Re: All odd numbers are prime.

Postby Klear » Mon Dec 30, 2013 9:25 pm UTC

cellocgw wrote:somehow I fail to see how including the words "liberal" , "expert" , or "indoctrination" add to this semijoke. Besides which, I rather doubt Einstein would ever have said anything so moronic. Consider, for example, adding NaOH to HCl .


Here.

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Re: All odd numbers are prime.

Postby jc » Mon Dec 30, 2013 9:48 pm UTC

Flumble wrote:
jc wrote:There were a lot more, but I've forgotten most of them. I wonder if they're collected somewhere on the Web ...

A quick search shows numerous collections, for example http://rationalwiki.org/wiki/Fun:Proof_that_all_odd_numbers_are_prime.

Yeah; if you feed google ""all odd numbers are prime" (with the quote), that's the first hit. As someone mentions below, there are only a few there that are really good examples of the joke, mostly because they don't bring in anything at all mathematical. Most of the rest are just silly.

But I did like the right-wing chain letter, and it's a not-very-mathematical one. And part of the fun of it is that we already have a case of Poe's Law [q.v.] for this joke in this discussion.

I've also seen the proof of the inverse: All prime numbers are odd. This consists of first noting that for all primes > 2, this is trivial, since all the even numbers are a multiple of 2, so those > 2 can't be prime. For 2, we observe that being even, it's an extremely odd prime, which completes the proof. (I actually came up with this one myself, but quickly discovered that there were at least N others who had discovered it before me. I didn't feel bad about this.)

There is, of course, the special case of 1, which generally falls into the "It depends on how you define prime" bucket. But we've had someone here refer to calling 1 a prime as blasphemy. Probably the best answer to this is to observe that mathematicians have historically taken such insults and turned them into technical terms. Thus, when numbers less than zero were introduced, they were denounced as an absurd attempt to count things that don't exist, so they can't exist either. This led to them being called "negative" numbers, a clear admission that, useful as they are, they don't exist. But no numbers have any actual physical existence, so this doesn't much matter. "If you don't like them, don't use them."

Then people demonstrated there were fractional number that weren't the ratio of integers, and those people were denounced as "irrational", so of course that became the name for those numbers. Later, some people started theorizing about numbers whose squares were negative, and this was soundly denounced as talking about something that can't possibly exist. So the response was to call these new numbers "imaginary", as opposed to numbers with non-negative squares, which were called "real". Again, we have to note that no numbers actually have any physical reality, so these terms are another case of mathematicians adopting insults as their new technical terms.

Maybe someday soon, we'll be able to read esoteric mathematical treatises on the concept of "blasphemous numbers", of which the prime number 1 will be the first member of the set. I wonder what the others will be like? Maybe aleph-0.5 will be among them ...

(And this time, I did google "blasphemous number", out of curiosity. None of the hits seem to be at all mathematical. I'm disappointed. ;-)

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Re: 1310: Goldbach Conjectures

Postby cellocgw » Mon Dec 30, 2013 10:36 pm UTC

gmalivuk wrote:In particular, that one looks like it's a parody of an anti-evolution "argument" about how no new information or speciation or whatever could happen just from the sum of existing things.


Aha -- looks like I got whooshed by that one :oops:
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Re: 1310: Goldbach Conjectures

Postby bondsbw » Mon Dec 30, 2013 11:36 pm UTC

ManaUser wrote:
bondsbw wrote:
  • If there are no numbers greater than 7, then every odd ("1", "3", "5", "7") would be prime.
  • if every odd number were prime, then every even number n over 2 would be the sum of two primes, namely (n-1) and "1".
  • If every even number over 2 is the sum of two primes, then every odd number n over 5 is the sum of "3" and any even number (which is already the sum of 2 primes).
  • Given all those odds over 5 and evens over 2, you can create any number over 7.
  • Given that you can create any number over 7, they keep going.

And yet, if we accept that each step here works, that must also mean "there are no numbers above 7" proves "numbers just keep going", which can't possibly be right.


I forgot to mention that we were suspending transitivity of relations.

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Re: 1310: Goldbach Conjectures

Postby Plutarch » Tue Dec 31, 2013 11:02 am UTC

As a non-mathematician, when I come across anything mathematical on XKCD, I investigate it on Wikipedia, which is about the full extent of my investigative powers. Sadly, this rarely enlightens me.

I'm wondering about one. Used to be regarded as a prime number, but not any more. Did this make a difference in maths? Were there things that didn't work if one was counted as a prime number?

Edit: Are mathematicians fed up with people asking about one being a prime number?

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Re: 1310: Goldbach Conjectures

Postby Yuan » Tue Dec 31, 2013 11:14 am UTC

Plutarch wrote:Were there things that didn't work if one was counted as a prime number?


I know that for example prime factorization becomes pretty stupid, because you can now tack on *1 on any factorization, but they are supposed to be unique.

Code: Select all

12 = 3*2*2 = 3*2*2*1 = 3*2*2*1*1 = ...

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Re: 1310: Goldbach Conjectures

Postby Klear » Tue Dec 31, 2013 11:31 am UTC

Yuan wrote:
Plutarch wrote:Were there things that didn't work if one was counted as a prime number?


I know that for example prime factorization becomes pretty stupid, because you can now tack on *1 on any factorization, but they are supposed to be unique.

Code: Select all

12 = 3*2*2 = 3*2*2*1 = 3*2*2*1*1 = ...


Isn't that a bit like how you can tack ".0" to any integer? Like 5.0?

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Re: 1310: Goldbach Conjectures

Postby Kit. » Tue Dec 31, 2013 12:17 pm UTC

We are getting too close to the topic not welcomed here.

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Re: 1310: Goldbach Conjectures

Postby Klear » Tue Dec 31, 2013 1:17 pm UTC

Kit. wrote:We are getting too close to the topic not welcomed here.


I don't see what this thread has to do with planes on a threadmill =P

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Re: 1310: Goldbach Conjectures

Postby Flumble » Tue Dec 31, 2013 2:53 pm UTC

xtifr wrote:
Flumble wrote:
garaden wrote:Alice: "Given that you can create any number over 7, they keep going."

I can't read "numbers keep going" in any way other than "there is a successor to every number",

You're doing fine so far.

which means 7 has a successor which is over 7.


Highlighted your error. You're stuck in Euclidean thinking. If the parallel postulate is incorrect and parallel lines meet at 4 and 0, then the successor to 7 is 0.

In my case I'm stuck in thinking that "greater than" and "less than" imply a partial order in which case 0≤7 (obvious because of successorship and transitivity) and 7≤0 (stated) require that 7=0 (and all the other numbers must be equal to eachother).

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Re: 1310: Goldbach Conjectures

Postby Quizatzhaderac » Tue Dec 31, 2013 3:41 pm UTC

The successor function is defined as S(x) = x+1
in mod 8 arithmetic S(7) = 0 and it happens to be that 7 + 1 < 7

For intuition purposes, it helps to remember that definitions are just what we choose, not some platonic truths.
The successor function above defines a successor to have "next-ness" but not "greater-ness"

Likewise, modular arithmetic has it's own set of definitions, which may or may not be useful. The canonical example of modular arithmetic is hour of the day.
Modular arithmetic is concerned with the question "Is 10 pm plus four hours later or earlier in the day than 10 pm?". You seem to be hung up onbeing later in absolute terms, which is not what modular arithmetic addresses.
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Re: 1310: Goldbach Conjectures

Postby shokoshu » Tue Dec 31, 2013 5:03 pm UTC

Why has nobody yet mentioned the [Expletive Deleted] Weak Goldbach Conjecture: Numbers exist.

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Re: 1310: Goldbach Conjectures

Postby Flumble » Tue Dec 31, 2013 5:44 pm UTC

Quizatzhaderac wrote:The successor function above defines a successor to have "next-ness" but not "greater-ness"

I didn't know you could split those properties. :o

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Re: 1310: Goldbach Conjectures

Postby TV4Fun » Tue Dec 31, 2013 7:19 pm UTC

The Russell twin prime conjecture states that the Russell twin prime conjecture is false.
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Re: 1310: Goldbach Conjectures

Postby gmalivuk » Tue Dec 31, 2013 7:28 pm UTC

Reals and rationals already have greater-ness but not next-ness.

And even if we interpret "keep going" as there being infinitely many of them, that doesn't require any of that infinitude to be greater than 7. We'd just need to use a different order relationship than the one that is implied by the successor function.

For example:
Put the rationals in [0,1] in a sequence, such as <0, 1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6, ...> (increase the denominator by one, list all the fractions with that denominator in order, skipping ones that have already shown up, and repeat).
Adjust this sequence so that the 7th element is 1: <0, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 1, ...> or <0, 1/2, 1/3, 2/3, 1/4, 3/4, 1, 1/5, ...> (depending on whether you start with the 0th element or the 1st).
Define a new greater-than relation on the natural numbers such that m GT n iff rm>rn (with rk being the kth element of our sequence of rationals, and > being the usual order relation on the rationals).

Now we have all the usual natural numbers, but 0 LT 1 GT 2 LT 3 GT 4 LT 5 GT 6 LT 7 GT 8 LT 9 LT 10 and so on, but now all naturals are GT 0 and LT 7, so there are none of them "above" 7.
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Re: All odd numbers are prime.

Postby rmsgrey » Wed Jan 01, 2014 11:16 pm UTC

jc wrote:Then people demonstrated there were fractional number that weren't the ratio of integers, and those people were denounced as "irrational", so of course that became the name for those numbers.


Actually, "irrational numbers" are so called because they're not ratios, and no-one could stomach calling them "non-ratio-esque numbers"...

Imaginary numbers really were named derogatively.

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Re: 1310: Goldbach Conjectures

Postby Trickster » Wed Jan 01, 2014 11:53 pm UTC

I see I'm not the first to mention this, but I noticed two errors.

The big one: the Extremely Weak conjecture is not implied by any of the others (in fact, the opposite is true).

The small one: the Extremely Strong conjecture does not imply the Strong conjecture as written (this could be fixed if Strong says every odd number other than 1).

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Re: All odd numbers are prime.

Postby Trickster » Wed Jan 01, 2014 11:56 pm UTC

rmsgrey wrote:Actually, "irrational numbers" are so called because they're not ratios, and no-one could stomach calling them "non-ratio-esque numbers"...

Imaginary numbers really were named derogatively.

They still had it easy compared to the stupid-as-fuck numbers, which were fortunately renamed "quaternions" shortly thereafter.

(I'd mention what they used to call octonions, but I'd get perma-banned if I posted it here.)

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Re: All odd numbers are prime.

Postby Trickster » Thu Jan 02, 2014 12:01 am UTC

Klear wrote:The name of that student: Albert Einstein.

What's sad is the number of people who actually buy made-up crap like this wholesale.

Thank your deity of choice, or lack thereof, for Snopes. I hope someone has that whole site backed up because we're screwed if it ever goes down. :P

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Re: 1310: Goldbach Conjectures

Postby Steve the Pocket » Thu Jan 02, 2014 2:51 am UTC

I was on Snopes recently. Holy gosh do they need to get someone to redesign their site. It looks like something you'd expect to see pushing penis-enhancement pills. The fact that they're just about the only reputable site left that still has pop-up ads (uComics being the only other one I know of) most certainly does not help.
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Re: 1310: Goldbach Conjectures

Postby Kit. » Thu Jan 02, 2014 10:27 pm UTC

ManaUser wrote:And yet, if we accept that each step here works, that must also mean "there are no numbers above 7" proves "numbers just keep going", which can't possibly be right.

They just keep going till 7.

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Re: 1310: Goldbach Conjectures

Postby Klear » Thu Jan 02, 2014 11:35 pm UTC

Kit. wrote:
ManaUser wrote:And yet, if we accept that each step here works, that must also mean "there are no numbers above 7" proves "numbers just keep going", which can't possibly be right.

They just keep going till 7.


There's a big difference between "They just keep going" and "They just keep going".

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Re: 1310: Goldbach Conjectures

Postby shadowthrust » Sun Jan 05, 2014 11:33 pm UTC

Hey everyone, I figured this strip would be a good opportunity to have some fun with interactive theorem proving. Unfortunately, the spam filter won't let me post urls, so you'll have to copy/paste this into your address bar yourself.

home.in.tum.de/~brunnerj/goldbach.png

More information on the used software (Isabelle), can be found at isabelle.in.tum.de

You can of course ask me questions, too ;).

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Re: 1310: Goldbach Conjectures

Postby orthogon » Tue Jan 07, 2014 4:38 pm UTC

Flumble wrote:
Quizatzhaderac wrote:The successor function above defines a successor to have "next-ness" but not "greater-ness"

I didn't know you could split those properties. :o

Sure you can: consider for example kings, presidents or prime ministers. ;)

shokoshu wrote:Why has nobody yet mentioned the [Expletive Deleted] Weak Goldbach Conjecture: Numbers exist.

That statement may be weaker than the others in a sense, but it is nevertheless highly debatable.
xtifr wrote:... and orthogon merely sounds undecided.


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