Look: I am eager to learn stuff I don't know--which requires actively courting and posting smart disagreement.

But as you will understand, I don't like to post things that mischaracterize and are aimed to mislead.

-- Brad Delong

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Everything that appears on this blog is the copyrighted property of somebody. Often, but not always, that somebody is me. For things that are not mine, I either have obtained permission, or claim fair use. Feel free to quote me, but attribute, please. My photos and poetry are dear to my heart, and may not be used without permission. Ditto, my other intellectual property, such as charts and graphs. I'm probably willing to share. Let's talk. Violators will be damned for all eternity to the circle of hell populated by Rosanne Barr, Mrs Miller [look her up], and trombonists who are unable play in tune. You cannot possibly imagine the agony. If you have a question, email me: jazzbumpa@gmail.com. I'll answer when I feel like it. Cheers!

Friday, July 23, 2010

What the Hell? Friday -- On the Primacy of the Number 1

Or not.

In another part of the universe, I was corrected by a math teacher for calling the number "one" a prime.

After a bit of research, I retorted thusly:

I read the info on 1 not a prime and am deeply underwhelmed. Unless I'm missing something big, the reason that 1 is not a prime is that mathematicians have conspired, in a completely arbitrary and discriminatory fashion, to deprive 1 of primehood, by including in the prime definition "greater than 1." If you take a prime to be a number divisible only by 1 and itself, 1 clearly qualifies. Granted it's the degenerate case (chemists will understand what this means) but that really makes no difference.

Tongue in cheek - sure.  But I'm not above a bit of japing.  Anyway, I was advised to have a look at this link.  Which I did.  Here is my new and improved response, re: the four reasons found there (which see.)

Answer 1: 
My point exactly: by definition, with no serious reasoning.  It's a totally unconvincing word game, and totally lame.

Answer 2:
Re: the fundamental theorem - the word "uniquely" is the kicker here.  But that is still just word play.  Tacking on any number of "x1" factors is clearly redundant, and could be eliminated by a more elaborate and linguistically clumsy definition - which I will not attempt at the moment.  BTW, this also illustrates what I meant by a degenerate case.  All the infinite number of potential "x1"s collapse into a single "x1."

Actually, I'll go further and say that the fact that you can write the product non-uniquely is irrelevant to the fact that you can still write it with a single "x1" in the formula, and that THAT is a unique answer.

So we still do not have a really sound reason.

On the other hand, though, lets take another look at the fundamental theorem, "Every positive integer greater than one can be written
uniquely as a product of primes."  Ignoring the problematic "uniquely" for the nonce, clearly, if 1 is excluded from the primes, then this theorem crashes and burns.  What are the factors of any prime, say, 5?  They are 5 and . . . and 1!  How can this simple fact not put a Q.E.D. on the primacy of 1?

Answer 3:
By the logic expressed there, I cannot be both a trombonist and a grandfather, though each is a subset of the set  "real humans"  -  being a grandfather is sooo much more important.  Really?  A thing cannot belong to two sets at once?  Five is both a prime and a Fibonacci number.  This reasoning is embarrassingly fatuous.

Answer 4:
This seems to be just an elaborated restatement of definition 3.

Now it's possible that I'm missing something, but here is my verdict.

Reasons for excluding 1 from the primes boil down to, "It's not because I say it's not."   I think that claim has no merit, and can be summarily rejected.

Reason for including 1 in the primes is that without it, the fundamental theorem turns out to be false.   This is not really a big deal for me, but mathematicians will probably find it disconcerting. 

Ergo, 1 is a prime.


Where did I go astray?  Help me out, somebody.


J said...

A prime requires two divisors, One and itself. Yet one and itself aren't two. Why is that the rule? Because. Sort of like asking, why do knights move the way they do in chess? 'Cuz that's how the game's played, for centuries (that doesn't mean that the chess rules...or natural numbers were written in stone...an abacus shows the origins of mathematics as well as greek philosophers do, ie bean counting) . Everything works out that way for the numbers' game. Holy Sieve of Erastothenes batman.

You'd be surprised at how many academic mathematicians and logicians still believe numbers float in some platonic abode--like next to Truth, Justice and Euclidian axioms.

Jazzbumpa said...

Yet one and itself aren't two.

That is why i call this a degenerate case. Both conditions: itself and one collapse to a single condition. This happens with electron energy levels in molecules, as confirmed by spectroscopy. I really don't see this a problem.

The definition strikes me as being arbitrary, if not downright capricious. And the fundamental theorem fails if 1 is not a prime.

When it's me against the entire edifice of mathematics, it's hard even for me to believe I'm right. But I can't find a valid argument to convince that I'm wrong.

"It is because we say it is" is totally lame, and no science should work that way. I want something grounded in mathematical logic.


J said...

Again, I think it's done for...simplicity's sake--if the eggheads did allow 1 as prime then....any number could be factored in an infinite number of ways, via 1 and exponent: 5 = 5 x 1^to the 10th, and 5 = 5 x 1^ to the millionth. Messy bookkeeping.

The Fundamental Theorem-- “each number has a unique factorization into primes"-- actually depends on one not being prime (see the wiki on primality).

Erastothenes's ancient algorithm left one out.

But analysis, schnalysis. Blogland wants....lifestyles of Rachel Maddow & friends, jzb

Jazzbumpa said...

I had no time to dig deeper today, and won't tomorrow either.

I do want to get to the bottom of this.

I'll check the wiki link when I have more time.