Since 1 is the multiplicative identity (x * 1 = x for any x in the set) and any definition of "prime" needs to use multiplication then one way or another 1 is going to be special when talking about primes whether it is included in the set of prime numbers or not. You can't avoid 1 being "special"
Is 1 Prime, and Does It Matter?
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Re: Is 1 Prime, and Does It Matter?
#82Since 2 is prime 1, wouldn't it be more symmetric if 1 was prime 2?
What makes you think two is prime? Not everyone would agree with that statement as the artical points out.
The article states that historically Nicomachus of Gerasa didn't consider 2 a prime, in like 100 AD.
Nowadays 2 is considered prime. Seems silly to question why someone is claiming 2 is prime if that is how it is defined in modern day.
> What makes you think two is prime
The current mathematical definition of a prime number
Re: Is 1 Prime, and Does It Matter?
#83Earlier quoted context omitted.
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.
Can you elaborate on this? My understanding is you can specifically and formally define the natural numbers with addition and multiplication, although multiplication means the language is no longer decidable. You can define natural numbers with just addition ( Presburger arithmetic ) and it’s decidable. Im not sure how undecidable “will define things that are similar to natural numbers but are not” but maybe I am mis…
Yeah for sure.
If a sentence S is undecidable from your axioms for the natural numbers then there are two models A and B satisfying those axioms where A satisfies S and B satisfies not S. So which one is the standard natural numbers, is it A or B?
Either A or B will be an example of something that satisfies your definition of natural numbers and yet is not the natural numbers.
Re: Is 1 Prime, and Does It Matter?
#84Earlier 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-...
The concept of “one” holds a dual role. It represents a countable unit: something you can put in a bowl and also stands for indivisibility itself. When you divide any quantity by an indivisible unit, you’re simply counting how many of those indivisibles fit within it. Then comes 2: the first number that is divisible, but only by itself and the indivisible one. That’s what makes it prime. A prime is a number divisible…
> Then comes 2: the first number that is divisible, but only by itself and the indivisible one.
This does hold in the ring Z. In the ring Z[i], 2 = (1+i)*(1-i), and the two factors are prime elements.
Re: Is 1 Prime, and Does It Matter?
#85Earlier quoted context omitted.
When I was younger I had a period I often was thinking about prime numbers (before I got old and started thinking about the Roman Empire). I noticed the same as you, and IIRC the (some?) ancient greeks actually had an idea about 1 as not a number, but the unit that numbers were made of. So in a different class. 2 and 3 are also different, or rather all other primes from 5 and up are neighbours to a multiple of 6, (th…
If you work not only the primes, but also the modulus function value of each non-prime, things get even more interesting than thinking of base changes! To me, it reveals much more.
Also, rearrangements.
In two dimensions is easier.
I cannot rearrange one pebble.
I can rearrange two or three pebbles equidistant from each other in just one distinct way (inverting the position of a neighbouring pebble).
And so on...
There are many ways to think of natural numbers without actual numbers.
Re: Is 1 Prime, and Does It Matter?
#86Earlier quoted context omitted.
> One reason that 1 is often excluded from the prime numbers is that if it was included, it would complicate the theorems, proofs, and exposition by the endless repetition of "not equal to 1". This is true and compelling as things developed, but I think it's an explanation of where history brought us, rather than a logical inevitability. For example, I can easily imagine, in a different universe, teachers patiently e…
That's an interesting thought, but I think that'd break the usual trick of building up objects from the empty set, a set containing the empty set, then the set containing both of those and so forth. That universe would be deprived from the bottomless wellspring of dryness that is the set theoretic foundations of mathematics. Unthinkable!
> 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 set theory beyond the elementary; but I'll also acknowledge that set-theoretic foundations are not to everyone's taste, and that someone who has an alternate foundational system that appeals to them is doing no harm to themselves or to me.
> That's an interesting thought, but I think that'd break the usual trick of building up objects from the empty set, a set containing the empty set, then the set containing both of those and so forth.
In this alternate universe, the ZF or ZFC axioms (where C becomes, of course, "the product of sets is a set") would certainly involve, not the axiom of the empty set, but rather some sort of "axioms of sets", declaring that there exists a set. Because it's not empty, this set has at least one element, which we may extract and use to make a one-element set. Now observe that all one-element sets are set-theoretically the same, and so may indifferently be denoted by *; and then charge ahead with the construction, using not Ø, Ø ∪ {Ø}, Ø ∪ {Ø} ∪ {Ø ∪ {Ø}}, etc. but *, * ∪ {*}, * ∪ {*} ∪ {* ∪ {*}}, etc. Then all that would be left would be to decide whether our natural numbers started at the cardinality 1 of *, or if we wanted natural numbers to count quantities 1 less than the cardinality of a set.
Re: Is 1 Prime, and Does It Matter?
#87Earlier 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.
>Any definition of the natural numbers will also define things that look very similar to natural numbers but are not actually natural numbers
This isn't correct. This is only true for first-order theories of the natural numbers using the axiom schema of induction. Second-order Peano arithmetic with the full axiom of induction has the natural numbers as its only model. This property is called "categoricity" and you can find the proof here [1] if you're interested
[1]: https://builds.openlogicproject.org/content/second-order-log...
Re: Is 1 Prime, and Does It Matter?
#88Earlier quoted context omitted.
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.
>Any definition of the natural numbers will also define things that look very similar to natural numbers but are not actually natural numbers This isn't correct. This is only true for first-order theories of the natural numbers using the axiom schema of induction. Second-order Peano arithmetic with the full axiom of induction has the natural numbers as its only model. This property is called "categoricity" and you ca…
This isn't correct. While it's true that in second order logic the natural numbers admit categoricity, second order logic lacks axiomatic semantics. So yes, there is a single set which can be called the natural numbers in second order logic (namely the intersection of all sets that satisfy Peano's axioms), but this set has no interpretation.
You can adopt Henkin semantics to give the naturals an interpretation, which is still second order logic, but then you're back to lacking a categorical model of the naturals.
Re: Is 1 Prime, and Does It Matter?
#89Earlier quoted context omitted.
A good example of this is the natural numbers. Algebraists usually consider zero to be a natural number because otherwise, it's not a monoid and set theorists want zero because it's the size of the empty set. My number theory textbook defined natural numbers as positive integers, but I'm not entirely sure why.
> My number theory textbook defined natural numbers as positive integers, but I'm not entirely sure why. Since both the inclusion and exclusion of zero are accepted definitions depending on who’s asking, books usually just pick one or define two sets (commonly denoted as N_0 and N_1). Different topics benefit from using one set over the other, as well as having to deal with division by zero, etc. Number theory tends…
> commonly denoted as N_0 and N_1
Oh my, it had never occurred to me that one could disagree, not just about whether the natural numbers include 0 or don't, but also about how to denote "natural numbers with 0" and "natural numbers without." Personally, I'm a fan of Z_{\ge 0} and Z_{> 0}, which are a little ugly but which any mathematician, regardless of their preferred conventions, can read and understand without further explanation.
Re: Is 1 Prime, and Does It Matter?
#90"Only divisible by itself and 1" is a darn elegant definition. 1, 2 and 3 are kind of special to me. In prime distribution studies, I discovered that they are special. It gets easier for some things if you consider primes only higher or equal to 5. Explaining distribution gets easier, some proofs become more obvious if you do that (tiny example: draw a ulam-like spiral around the numbers of an analog clock. 2 and 3 w…
It's not entirely clear if that definition includes 1. On one hand 1 is certainly divisible by both itself and 1, but on the other hand they are the same number, so maybe it shouldn't count for "both", because the word "both" vaguely implies two distinct things. The usual "natural number with exactly two integer divisors" definition may not be as elegant but I think it is harder to misinterpret.