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From: Darren New
Subject: Re: I haven't read the entire paper yet, but the analogies are ratherapt
Date: 13 Dec 2010 13:23:27
Message: <4d06649f$1@news.povray.org>
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Invisible wrote:
> Logically, this is a perfectly valid argument. But intuitively, it
> doesn't make much sense. "You can't write this ratio because if you did,
> it would be *evil*!" Um... OK? So how come you can't do it then?
Proof by contradiction isn't reducto ad absurdum.
You say "Assume that X exists." Then you prove that from there you can
logically derive the fact that X doesn't exist. That's a contradiction.
Since you only made one assumption, the contradiction must be logically
derived from the assumption you made, namely that X exists. (X in this case
being the ratio that expresses sqrt(2).)
> On the other hand, any positive integer can be represented as the
> product of unique prime numbers, each raised to a positive integral
> power. For example, 99 = 3^2 * 11^1. And when you square a number, you
> simply double all the exponents. Hence, 99^2 = 9801 = 3^4 * 11^2.
That's a lot more to prove before you can prove the sqrt(2) is irrational.
> Now A/B = Sqrt(2) is equivalent to A^2/B^2 = 2. And in order for the
> ratio to be 2, A^2 and B^2 must have identical factors, except for A^2
> having an exponent for 2 which is exactly one higher than the exponent
> for 2 in B^2. But, as just established, the exponents of all prime
> factors of a square number are always even, so this condition is
> completely impossible to satisfy.
That's a very similar proof.
> This doesn't *prove* anything, but it does explain why you can't solve
> the equation, in a way which intuitively makes sense.
Actually, that *is* a valid proof. I'm not sure why you think it's not a proof.
--
Darren New, San Diego CA, USA (PST)
Serving Suggestion:
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From: Darren New
Subject: Re: I haven't read the entire paper yet, but the analogies are ratherapt
Date: 13 Dec 2010 13:24:53
Message: <4d0664f5$1@news.povray.org>
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Invisible wrote:
> Isn't Godel's theorem the one that says "mathematics is almost
> guaranteed to be logically self-contradictory, so you're basically
> wasting your time, guys"?
No?
> I presume you mean "prove anything that is true", since a system that
> can prove *anything* is a system that's inconsistent...
You can prove anything that you could prove with a more complex mathematical
system covering the same axioms.
And no, inconsistent isn't always bad.
http://en.wikipedia.org/wiki/Paraconsistent
--
Darren New, San Diego CA, USA (PST)
Serving Suggestion:
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From: scott
Subject: Re: I haven't read the entire paper yet, but the analogies are ratherapt
Date: 14 Dec 2010 04:38:33
Message: <4d073b19$1@news.povray.org>
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>> Although it did leave me wondering for a moment: Am I actually "good at
>> math"? Or am I just good at following directions?
>
> Perhaps it could be compared to computer algorithms: If you are given
> a computational problem for which you need to create an efficient algorithm,
> how good are you at coming up with such an algorithm?
ie How many Project Euler problems have you solved? (and by "solved" I
mean properly, so that you get the answer in less than 1 minute of
computer time, which IIRC is the target).
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From: Neeum Zawan
Subject: Re: I haven't read the entire paper yet, but the analogies are ratherapt
Date: 15 Dec 2010 10:41:07
Message: <87mxo782oh.fsf@fester.com>
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Darren New <dne### [at] san rr com> writes:
> FWIW, I learned virtually nothing in music class and nothing at all in
> art class. Art class, such as it was, consisted of the teacher trying to
> come up with rules about what you drew to ensure it occupied the entire
> 45-minute class to draw it.
Yeah, years ago I sent that same essay to a friend, and he said his
music class was exactly the one being parodied.
> And while I took and understood a great deal of math, it wasn't until
> physics class that I suddenly said "Oh, *that* is what an integral is
> for!"
I dunno. My textbook made it clear it was about summing infinitesimal
quantities.
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From: Darren New
Subject: Re: I haven't read the entire paper yet, but the analogies are ratherapt
Date: 15 Dec 2010 10:48:43
Message: <4d08e35b$1@news.povray.org>
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Neeum Zawan wrote:
>> And while I took and understood a great deal of math, it wasn't until
>> physics class that I suddenly said "Oh, *that* is what an integral is
>> for!"
>
> I dunno. My textbook made it clear it was about summing infinitesimal
> quantities.
Yeah, but what is it *for*? When, outside of calc class, would you ever do
such a thing? That's what I hadn't figured out.
--
Darren New, San Diego CA, USA (PST)
Serving Suggestion:
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From: Neeum Zawan
Subject: Re: I haven't read the entire paper yet, but the analogies are ratherapt
Date: 15 Dec 2010 16:14:23
Message: <87ei9i91tk.fsf@fester.com>
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Darren New <dne### [at] san rr com> writes:
> Neeum Zawan wrote:
>> I dunno. My textbook made it clear it was about summing infinitesimal
>> quantities.
>
> Yeah, but what is it *for*? When, outside of calc class, would you ever
> do such a thing? That's what I hadn't figured out.
They had the usual examples, like calculating work by integrating the
force.
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From: Darren New
Subject: Re: I haven't read the entire paper yet, but the analogies are ratherapt
Date: 15 Dec 2010 16:19:29
Message: <4d0930e1$1@news.povray.org>
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Neeum Zawan wrote:
> Darren New <dne### [at] san rr com> writes:
>
>> Neeum Zawan wrote:
>>> I dunno. My textbook made it clear it was about summing infinitesimal
>>> quantities.
>> Yeah, but what is it *for*? When, outside of calc class, would you ever
>> do such a thing? That's what I hadn't figured out.
>
> They had the usual examples, like calculating work by integrating the
> force.
Right. That's the physics I was talking about. I never saw examples in our
textbooks about *why* you want to do this thing. Just curves, and how to
figure out the area underneath. Not where the curves came from.
--
Darren New, San Diego CA, USA (PST)
Serving Suggestion:
"Don't serve this any more. It's awful."
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From: scott
Subject: Re: I haven't read the entire paper yet, but the analogies are ratherapt
Date: 16 Dec 2010 03:48:32
Message: <4d09d260@news.povray.org>
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>> They had the usual examples, like calculating work by integrating the
>> force.
>
> Right. That's the physics I was talking about. I never saw examples in
> our textbooks about *why* you want to do this thing. Just curves, and
> how to figure out the area underneath. Not where the curves came from.
You just needed the right text book. Our maths was split into pure and
applied, with separate text books. Pure had all the theoretical stuff
for calculus etc, applied had carefully contrived examples of how you
applied it IRL, things like analysing rockets taking off and various
mechanisms.
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From: Orchid XP v8
Subject: Re: I haven't read the entire paper yet, but the analogies are ratherapt
Date: 16 Dec 2010 07:13:56
Message: <4d0a0284$1@news.povray.org>
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On 13/12/2010 06:23 PM, Darren New wrote:
> Proof by contradiction isn't reducto ad absurdum.
What is then?
>> This doesn't *prove* anything, but it does explain why you can't solve
>> the equation, in a way which intuitively makes sense.
>
> Actually, that *is* a valid proof. I'm not sure why you think it's not a
> proof.
Presumably since I haven't proven the Fundamental Theorem of Arithmetic,
this doesn't count as a valid proof.
It does, however, make a lot more sense to somebody who is willing to
take it on faith. (I gather that actually *proving* the matter in this
direction is drastically more complex...)
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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From: Darren New
Subject: Re: I haven't read the entire paper yet, but the analogies are ratherapt
Date: 16 Dec 2010 11:05:03
Message: <4d0a38af$1@news.povray.org>
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Orchid XP v8 wrote:
> On 13/12/2010 06:23 PM, Darren New wrote:
>
>> Proof by contradiction isn't reducto ad absurdum.
>
> What is then?
Reducto ad absurdum is reducto ad absurdum.
> Presumably since I haven't proven the Fundamental Theorem of Arithmetic,
> this doesn't count as a valid proof.
That's an implicit lemma of most proofs.
For example, your "on the other hand" paragraph might need proof, might not,
depending who you're trying to prove it to.
> It does, however, make a lot more sense to somebody who is willing to
> take it on faith. (I gather that actually *proving* the matter in this
> direction is drastically more complex...)
No, it's one of the first known mathematical proofs. It's not drastically
more complex.
--
Darren New, San Diego CA, USA (PST)
Serving Suggestion:
"Don't serve this any more. It's awful."
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