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Re: Non-Euclidean Arithmetic
Posted:
Sep 11, 2012 6:52 PM
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On Tue, Sep 11, 2012 at 5:03 PM, kirby urner <kirby.urner@gmail.com> wrote:
> On Tue, Sep 11, 2012 at 11:27 AM, Paul Tanner <upprho@gmail.com> wrote:
>
> << snip >>
>
>> This doesn't hold up, since almost all irrationals are not computable
>> numbers, according to the standard definition of computable numbers.
>>
>
> Uh oh, sounds like a security breach. We've gone from a strictly
> professed segregationist stance, putting a fence between Q and R, and
> now some of the irrationals are getting through, as definitely
> computed. Pi to billions of digits. Sure, more digits could be
> crunched but to call what's probably a record-holder for computing
> time devoted thereto, "non-computable" is just daft.
Nonsense.
You are just trying to resurrect the attempt to rescue repeated addition with this "roughly" talk I debunked in my post in this thread
http://mathforum.org/kb/message.jspa?messageID=7884356
in which I said:
"That you have to introduce a "roughly" part demonstrates the truth of what I'm saying when both factors are irrationals.
When both factors are irrationals, now we have an even further stretch beyond the original meaning [of repeated addition]: Not only can we not use one of the original factors as the repeated addend, we cannot even talk about things precisely any more."
Each of this merely countable set of irrationals that are computable is of course computable to exact precision only with an infinite number of steps.
And this is true for infinitely many rationals if we view a fraction as the same type of finite representation as sqrt(2) or e, where computability really means what can be done in terms of digital expansion like a decimal form. But if being able to compute an exact answer allows for fractional forms of integers, then of course we're still OK with being able to compute an exact answer without having to complete an infinite number of steps.
And this ""non-computable" is just daft" remark causes me to again reiterate everything in my post in this thread
http://mathforum.org/kb/message.jspa?messageID=7887042
and especially what I said here:
"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.)"
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