Is minus 7 prime?

[jack]

My name:
Yes. Obviously the domain in question is the ring of integers, and the definition of prime for non-natural-number rings includes -7 in the integers.
No. Unless epxlicitely specified, the domain is always the natural numbers, and negative numbers and fractions are no more prime than american presidents or colours.
Moofle. Mu. Your question is malformed. Without specifying the domain, you can't correctly say either.
Even in the integers, -7 isn't prime, even though most mathematicians use that definition.
Even in the integers, -7 isn't prime, and everyone knows that.
Other. I will explain in comments.

This question actually came up in an online multiple choice test. However, another option was a positive prime, so it was clear what they meant. (Foruntately there wasn't an "other" option. Have another vote for "tests where you might know everything about the subject and popular conceptions and misconceptions thereof, but you have to second-guess the level of sophistication of the examiner to get full marks suck!" :))

But mathematical pedantry aside, it illustrates how much context can be important to the meaning of a sentence. This seems about perfectly balanced about what context you assume, but there's a continuum. At one end, the sentence is without context, you interpret it literally.

At the other, you have something like:

Q. Were you telling the truth?
A. Yes!

Where "yes" isn't the answer to *that* question, but a delayed answer to something someone else asked the other day. I think everyone could agree that's a lie. But there are intermediate stages.

If you say a sentence which would be literally true out of context, but in response to a question in a situation where it sounds like a natural reply meaning one way, when the opposite is true, and you intended that, I think that's a lie. But some people think lying is always wrong, but this doesn't count as lying.

OTOH, if you have some reason not to tell the whole truth, and don't volunteer something important, that can be acceptible. It can be misleading, and may or may not be wrong, but isn't necessarily lying.

It seems like people have seized on an overly restrictive notion of the meaning of language, and don't accept how we actually use it.

Comments

[aldabra]

If the context involves -7 being prime then there's an implicit domain in which it makes sense. Usually the domain for this is the positive integers but in this case it implicitly isn't.

(Though if K asked I'd say no, by definition negative numbers aren't prime. In a room full of mathmos, I'd shut up and see where the conversation was headed.)

[cartesiandaemon.livejournal.com]

Thank you. Yes -- you do need to tailor the answer to the audience. I guess I'd say something like "Only the natural numbers have primes" which is true enough -- that definition of prime is enough for her for the next 10 years, and as far as she's concerned primality just doesn't apply to negative numbers. The other is there only if she needs it.

[ewx.livejournal.com]

On reflection I find myself believing both "-7 is prime", and "if someone says this works for all primes they only mean positive ones" are reasonable things to claim as true in the absence of context.

-7 plainly isn't composite, for instance. If you believe it is then I suspect you risk ending up talking yourself into -1 being prime which is plainly ridiculuous. From another angle, it is a non-unit that satisfies p|ab => p|a or p|b.

Still, if people talk about all primes, they generally do mean the +ve ones.

Obviously what's going on is that the implicit context induced by the two statements is different.

Obviously what's going on is that the implicit context induced by the two statements is different.

[cartesiandaemon.livejournal.com]

Exactly! Thank you.

Without context, primes refer to the normal, positive integer primes.

With context that is nonsense if you don't allow negative numbers, you use the obvious field.

But here, it could be either, the context is ambiguous between the two.

Re: Obviously what's going on is that the implicit context induced by the two statements is differen

[ewx.livejournal.com]

Not a field. Fields don't have primes, because everything but 0 is a unit, and 0 doesn't divide anything l-)

Re: Obviously what's going on is that the implicit context induced by the two statements is differen

[cartesiandaemon.livejournal.com]

Ring! Ring! Sorry.

[douglas-reay.livejournal.com]

I was in a maths lesson today where a 14 year old asked about negative numbers and primeness. The teacher looked stumped for a while and then came up with the following:

7 is prime, because it has exactly two factors: 1, 7
4 is not prime, because it does not have exactly two factors: 1, 2, 4
2 is prime, because it has exactly two factors: 1, 2
1 is not prime, because it does not have exactly two factors: 1
-1 is prime, because it has exactly two factors: 1, -1
other negative numbers are not prime because they do not have exactly 2 factors. eg: -7 has three factors: -1, 1, 7

[cartesiandaemon.livejournal.com]

Meefle! I see where it comes from. But I don't think claiming -7 isn't a factor of -7, or -1 isn't a factor of 7, works. Admittedly, I don't know if I could have explained it better.

[beckyc.livejournal.com]

other negative numbers are not prime because they do not have exactly 2 factors. eg: -7 has three factors: -1, 1, 7

And did the 14 year olds then say "Ah, but 7 is also the product of -1 and -7) or similar?

1=-1*-1

[ewx.livejournal.com]

I hope you pointed out (perhaps privately) that they were talking nonsense...

Re: 1=-1*-1

[cartesiandaemon.livejournal.com]

Of course, maybe he knew that :(

[beckyc.livejournal.com]

But some people think lying is always wrong, but this doesn't count as lying.

Doing a degree in computation led me to favour even more the idea that lying, or at least any morally bad parts, is more in the act of intending to deceive than in making a false statement. Otherwise we'd be a bit stuffed for boolean algebra: "No, I'm sorry, I can't evaluate that, it's immoral".

[cartesiandaemon.livejournal.com]

I think deceiving is the right definition.

OTOH, I've always thought there was something sinister about boolean algebra ;)

[gileswl.livejournal.com]

On lying: http://www.newadvent.org/fathers/1312.htm

On bullshit: http://web.archive.org/web/20040722082643/http://www.jelks.nu/misc/articles/bs.html

G K Chesterton puts his oar in: http://www.online-literature.com/chesterton/tremendous-trifles/10/

[cartesiandaemon.livejournal.com]

Thank you!

[ewx.livejournal.com]

Incidentally this article provoked me into correcting the ring-theoretic definition of prime in [[Prime number]], which curiously also uses -7 as its example of a negative prime in Z.

[cartesiandaemon.livejournal.com]

LOL. Yeah, I noticed that. I guess 7 is always the "random" number chosen for this sort of thing :)