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)
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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:
     "Don't serve this any more. It's awful."

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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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Darren New <dne### [at] sanrrcom> 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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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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Darren New <dne### [at] sanrrcom> 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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Neeum Zawan wrote:
> Darren New <dne### [at] sanrrcom> 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:
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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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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...)

-- 
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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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