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Is 1 Prime, and Does It Matter?

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Re: Is 1 Prime, and Does It Matter?

#121

Post by dullcrisp »

JadeNB wrote: post #114
dullcrisp wrote: post #113

Earlier quoted context omitted.

And if we treat zero as not a number, it would make division much easier to define. I wrote that sentence as a joke but now I wonder if maybe it’s true. Does addition really need to have an identity? Maybe we just saw that multiplication has an identity and got a bit carried away. I’m not too sure about this negative number business while we’re at it. Could be that we just took a wrong turn somewhere.

> And if we treat zero as not a number, it would make division much easier to define. I wrote that sentence as a joke but now I wonder if maybe it’s true. Does addition really need to have an identity? It probably doesn't, but, if you want to allow negative numbers, then addition is partial unless you have 0. It's perfectly reasonable to disallow negative numbers—historically, negative numbers had to be explicitly al…

Well you did say you were okay with set intersection being partial (or I guess also set difference for the more direct analogy). Maybe not everything needs a solution. (Plus we’ve just gone from division being partial to subtraction being partial…but when I say that I begin to suspect that this argument has been made a lot before and we decided that the negative numbers get to stay. I don’t have anything against them personally but they’re probably less natural than the empty set being a set.)

I might be reading too much into what you’re saying about the empty set though and you just mean we could use the word “set” to mean “non-empty set” and then say something like “set-theoretic set” to mean what we now mean when we say “set.” But that sounds like a mouthful.

Re: Is 1 Prime, and Does It Matter?

#122

Post by amavect »

Some other definition fun: Should we define 0 both positive and negative, or neither positive and negative? Does monotonically increasing mean x f(y) f(x)≤f(y)? Should we deny the law of excluded middle and use constructive math? Does infinity exist? If infinity exists, is it actual (as an object) or potential (as a function)? Is the axiom of choice true? Or, is the axiom of determinacy true?

Should we use a space-time manifold, or separate space and time dimensions? Do future objects exist, and do past objects exist? Do statements about the future have a definite truth value? Does Searle's Chinese Room think? Which Ship of Theseus is the original: the slowly replaced ship, or the ship rebuilt from the original parts?

I find that so many philosophy debates actually argue over definitions rather than practical matters, because definitions do matter. Well, add your own fun definition questions!

Re: Is 1 Prime, and Does It Matter?

#123

Post by JadeNB »

dullcrisp wrote: post #121
JadeNB wrote: post #114

Earlier quoted context omitted.

> And if we treat zero as not a number, it would make division much easier to define. I wrote that sentence as a joke but now I wonder if maybe it’s true. Does addition really need to have an identity? It probably doesn't, but, if you want to allow negative numbers, then addition is partial unless you have 0. It's perfectly reasonable to disallow negative numbers—historically, negative numbers had to be explicitly al…

Well you did say you were okay with set intersection being partial (or I guess also set difference for the more direct analogy). Maybe not everything needs a solution. (Plus we’ve just gone from division being partial to subtraction being partial…but when I say that I begin to suspect that this argument has been made a lot before and we decided that the negative numbers get to stay. I don’t have anything against them…

> Well you did say you were okay with set intersection being partial (or I guess also set difference for the more direct analogy).

Good point!

> I don’t have anything against them personally but they’re probably less natural than the empty set being a set.

An interesting idea, which history supports: 0 was considered as a number before negative numbers were, and we still usually consider only "natural sets" and not "negative sets" (except for Schanuel: https://doi.org/10.1007/BFb0084232).

> I might be reading too much into what you’re saying about the empty set though and you just mean we could use the word “set” to mean non-empty set and then say something like “set-theoretic set” to mean what we now mean when we say “set.”

Right, or a different word entirely, just like we refer to 1 only as a number that's not prime, not as a "number-theoretic prime." But, anyway, the analogy was just the first one that sprang to mind; it doubtless has many infelicities that could be improved by a better analogy, if it's not just a worthless idea overall.

Re: Is 1 Prime, and Does It Matter?

#124

Post by dullcrisp »

JadeNB wrote: post #123
dullcrisp wrote: post #121

Earlier quoted context omitted.

Well you did say you were okay with set intersection being partial (or I guess also set difference for the more direct analogy). Maybe not everything needs a solution. (Plus we’ve just gone from division being partial to subtraction being partial…but when I say that I begin to suspect that this argument has been made a lot before and we decided that the negative numbers get to stay. I don’t have anything against them…

> Well you did say you were okay with set intersection being partial (or I guess also set difference for the more direct analogy). Good point! > I don’t have anything against them personally but they’re probably less natural than the empty set being a set. An interesting idea, which history supports: 0 was considered as a number before negative numbers were, and we still usually consider only "natural sets" and not "…

Yeah I guess what I got stuck on is that we don’t currently have a word for “a set that’s not a set” (I guess a class?) like we do for a number that’s not a prime but I think I was just lacking linguistic imagination.

Re: Is 1 Prime, and Does It Matter?

#125

Post by thehappypm »

bluepnume wrote: post #102
jordigh wrote: post #50

Earlier quoted context omitted.

To be fair, 2 is also a very odd prime because it's even. So many theorems have to say, "for every odd prime..." https://math.stackexchange.com/questions/1177104/what-is-an-...

It's hardly odd. "Even" just means "divisible by 2" "2 is the only prime that is divisible by 2" "3 is the only prime that is divisible by 3" "5 is the only prime that is divisible by 5" ... "N is the only prime that is divisible by N"

Exactly, we could also have a word for multiple of three or multiple of five

Re: Is 1 Prime, and Does It Matter?

#126

Post by [deleted] »

amavect wrote: post #122

Some other definition fun: Should we define 0 both positive and negative, or neither positive and negative? Does monotonically increasing mean x f(y) f(x)≤f(y)? Should we deny the law of excluded middle and use constructive math? Does infinity exist? If infinity exists, is it actual (as an object) or potential (as a function)? Is the axiom of choice true? Or, is the axiom of determinacy true? Should we use a space-ti…

[deleted]

Re: Is 1 Prime, and Does It Matter?

#128

Post by LudwigNagasena »

Maxatar wrote: post #38
gerdesj wrote: post #27

Earlier quoted context omitted.

I'm no expert but: "...ignoring the trivial case of 1 being an obvious factor of every integer." I remember quite a big chunk of GEB formally defining how integers are really not trivial! The main problem seems to be is that you soon end up with circular reasoning if you are not razor sharp with your definitions. That's just in an explainer book 8) Then you have to define what factor means ...

Correct, it's impossible to specifically and formally define the natural numbers so that addition and multiplication work. Any definition of the natural numbers will also define things that look very similar to natural numbers but are not actually natural numbers.

Second-order arithmetic formalizes both natural and real numbers. Though it has a host of issues with inference.

Re: Is 1 Prime, and Does It Matter?

#130

Post by JadeNB »

tux3 wrote: post #94
JadeNB wrote: post #86

Earlier quoted context omitted.

> That universe would be deprived from the bottomless wellspring of dryness that is the set theoretic foundations of mathematics. Unthinkable! "Wellspring of dryness" is quite a metaphor, and I take it from that metaphor that this outcome wouldn't much bother you. I'll put in a personal defense for set theory, but only an appeal to my personal taste, since I have no expert, and barely even an amateurish, knowledge of…

I should apologize if I came off too colorful, I only meant it as a friendly jab - but my bias is showing :) Appreciate the defense of set theory, I can't find a problem with it!

No apology needed! It's all in fun, and we might as well enjoy the discussion.
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