- Aug 7, 2006--- Shi Huang <shuangtheman@...> wrote:
> Following a list of primes that starts with 2, the

It does not.

> author Derbyshire wrote: At this point, someone

> usually objects that 1 is not included in this or any

> other list of primes. It fits the definition, doesnt

> it? Well, yes, strictly speaking, it does, and if you

> want to be a barrack-rood lawyer about it, you can

> write in a 1 at the start of the list for your own

> satisfacion. Including 1 in the primes, however, is a

> major nuisance, and modern mathematicians just dont,

> by common agreement. (The last mathematician of any

> importance who did seems to have been Henri Lebesgue,

> in 1899.) Even including 2 is a nuisance, actually.

> Countless theorems begin with, Let p be any odd

> prime . However, 2 pays its way on balance; 1

> doesnt, so we just leave it out.

>

> from Derbyshire, J. Prime Obsession: Bernhard Riemann

> and the Greatest Unsolved Problem in Mathematics. New

> York: Penguin, 2004. Page 33.

>

> So the above quote proves my point that 2 is treated

> as a prime today the same way as 1 is not, by human

> agreement rather than objective truth.

You're right that it's human agreement, but any selection

of axioms (and postulates) and inference rules, for example

is a human agreement. To think otherwise shows a complete

disregard for how the foundations of modern mathematics are

defined.

However, there are fundamental differences between the issues

that '1' causes, and the issues that '2' causes which make them

not comparable.

> 2 as prime

I assume that should read "doesn't". But it's still a gross

> serves us better and so lets call it a prime. 1 as a

> prime does serve us as well so lets ignore it.

misrepresentation of the truth. Having a unit as a prime messes

up /almost everything/.

> Clearly serving us is not the same as serving God.

Obviously. Ockham's razor indicates that there's no need to have

brought up the latter at all, and persuades us that the simplest

set of rules is usually the better one. Having a unit as a prime

complicates almost every otherwise simple rule we have, and

therefore is unwarranted, and unwanted.

It appears that you don't understand _why_ 2 causes the problems

that it does in the situations where one needs to say "an odd prime".

It's usually not its primeness that causes the problem, but it's

_size_. It is the only prime for which x == -x (mod p) for all x,

which messes up assumptions about orders (see carmichael's lambda,

for example). Lack of divisibility by 2 messes things up too,

but lack of divisibility by 3 messes things up in elliptic curves

over GF(3^n), and there's no temptation to not call 3 a prime -

it wasn't the _primeness_ of 2 that was the problem, merely the fact

that 2 occured as a multiplier that one wanted to invert.

Coupled to this, it appears that you don't understand why having

1 causes the problems that it does too. The fact that it is a unit

messes up practically everything it touches.

The reason why units have been isolated in their own special

category is for a very simple reason - they behave fudamentally

differently from the other members of the multiplicative group.

To group them all together and then to have to separate them again

almost everywhere provides the mathematician with no perceptable

gain, and plenty of pain.

I notice that precisely _no_ sources for primes to concretely be

defined by a mathematician to exclude 2 have been cited yet, which

reinforces my previous post on the subject.

Phil

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