|
|
Re: Non-Euclidean Arithmetic
Posted:
Sep 9, 2012 3:26 PM
|
|
On Sun, Sep 9, 2012 at 11:30 AM, Joe Niederberger <niederberger@comcast.net> wrote:
>>His position is only that we should teach that
> although these two operations of multiplication and repeated addition get the same results, that does not mean therefore that they are the same operation.
>
> The action that the term "repeated addition" refers to, is a computable procedure. Multiplication on integers is typically defined by such a procedure. In that sense they most assuredly *are* the same. If you want to take things from certain axiomatic viewpoints, and avoid "defining" multiplication from more primitive notions, then mult. is just a function, a mapping from pairs of integers to integers. In that view it has no "action" at all, it just is. That's fine too - but its misleading even in that view to compare an abstract function against a computable procedure as if they were comparable. If one wanted to be clear about these matters, they would lay out the options, not go (to paraphrase Kirby) "kicking down other peoples sandcastles, while insisting their own was inviolable."
>
> No, Devlin's writing is clearly misleading - he wants people to think "repeated addition" is one kind of action, and "multiplication" is a comparable that is another kind of "action" (stretching a rubber band). Its all plain to see. Stretching is fine as mental imagery, but if you want to turn that mental imagery into a computable procedure, you will need to rely on iteration or recursion one way or another.
>
> Joe N
Is the multiplication of two irrationals, especially two non-algebraic irrationals like the product e(pi), a computable procedure?
Computation is fine when you actually do it, but when you can't, you will need a definition of multiplication that holds up on all of the reals, not just on almost none of them.
This lack of computability on almost all of a set of numbers that we blithely teach to every kid starting sometimes in middle school or possibly even earlier and on through all of such as algebra, calculus, and differential equations and much, much more that they could study on through college in math, science, and engineering is precisely part of what I think Devlin was trying to get people to see as to why there needs to at be at least an inclusion of another view of multiplication that holds up on the set of all reals including any subset of it like the naturals.
In my view, this idea that if it's not computable then it's no big deal and so we can ignore is what I was talking about in my prior post
http://mathforum.org/kb/message.jspa?messageID=7884356
in this thread. I reproduce some of it here, followed by some more comments on this:
Quote:
"> I'm saying the "repeated addition" meme can be extended to Q and from
> Q to R.
The extension cannot be made at all to R when we take into full account what R actually is: Almost all of R is made up of non-algebraic irrationals. This "generalization" to multiplication in R cannot be said to be made at all if the only way to make this "generalization" is to restrict the domain of one of the two factor variables to Q, which has a cardinality of at least an entire order lower than R.
And my points about cardinality are not irrelevant, my trying to change the subject as you say. The fact that the cardinal numbers for Q and R are different exposes how fundamentally different rationals and irrationals - especially a non-algebraic irrationals - are. The very nature of the proofs that Q is countable is such that we cannot apply the same reasoning to the irrationals we cannot do this because of the nature of each irrational - especially each transcendental irrational - is just not the same animal at all when we compare it to any rational. And it's the irrationals that make R.
The crux of all this is that those who think that "multiplication is repeated addition" is just fine for R seem to treat the irrationals as not all that important, just a side curiosity, just "those numbers over there" that "don't really count" for what is really important (which seems to be finite computation for those who think this way about multiplication), and all that. But again, it's the irrationals and especially the non-algebraic irrationals that make up almost all of R, and is the basis of calculus and much of everything else higher up in analysis and algebra and topology. (This "movement" of a tiny handful of people to ban irrationals is going nowhere, since it is not an answer to anything - it bans so much of analysis and algebra and topology. And so we see the "solution" for the repeated addition folks to the fact that repeated addition just cannot apply to two irrational factors is to, well, kill the messenger - ban those pesky irrationals causing the problem.)"
This disrespect for the irrationals and especially the transcendental (or non-algebraic) irrationals is something I think bothered Devlin enough to write what he wrote. There is no disrespect for the irrationals here by the "multiplication *is* repeated addition" crowd that "multiplication is one and the same as repeated addition" and never agrees with the idea that we should at least include an extra view of multiplication that accommodates all of the reals not just almost none of them? Consider:
What this crowd doesn't get is this:
To ban or at least shove off to the side as unimportant those pesky irrationals and especially those pesky transcendental (or non-algebraic) irrationals is to ban or at least shove off to the side as unimportant all of mathematics based on continuous functions, which is almost all of algebra, calculus, and differential equations and much, much more that they could study on through college in math, science, and engineering.
Great idea, huh?
|
|
