>> extremely hard to find (especially if you don't now the proper
>> terminology),
>
> *Especially* before google.

Or if, like me, you utterly suck at research...

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>> Thinking about it... No, I'm not very good at math. I never bother
>> proving things.
>
> You don't need unshakable solid proof for conjectures, so maybe you
> should stick to those. In my eyes that's mathematical enough.

A proof is usually a highly abstract sequence of logical steps which 
demonstrates that something /is/ true. Myself, I'm usually more 
interested in trying to get an intuitive feeling for /why/ it is true.

-- 
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http://www.zazzle.com/MathematicalOrchid*

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On 12/12/2010 06:33 PM, bart wrote:

> - it is not that the set theory is boring,
> rather the way it is presented in most of textbooks is.

Possibly. Or maybe it's just that different areas of mathematics 
interest different people?

Let's face it, mathematics is FREAKING HUGE. There must surely be 
something in there somewhere that interests just about anyone...

-- 
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http://www.zazzle.com/MathematicalOrchid*

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>> but I've always thought of mathematics looking at
>> interesting systems and discovering their properties, out of simple
>> human curiosity. Certainly that's why *I* explore mathematics; it's the
>> desire to know everything about everything.
>
> ... despite there existing proof that this desire is insatisifiable

Well, sure. Perfection is known to be impossible (for at least a dozen 
reasons). But we can sure have some fun trying! ;-)

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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On 12/12/2010 10:01 PM, Orchid XP v8 wrote:
> On 12/12/2010 06:33 PM, bart wrote:
>
>> - it is not that the set theory is boring,
>> rather the way it is presented in most of textbooks is.
>
> Possibly. Or maybe it's just that different areas of mathematics
> interest different people?

or at different time of life, yes.

> Let's face it, mathematics is FREAKING HUGE. There must surely be
> something in there somewhere that interests just about anyone...

Absolutely.

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clipka wrote:
> ... despite there existing proof that this desire is insatisifiable 

Diaspora - Greg Egan.

-- 
Darren New, San Diego CA, USA (PST)
   Serving Suggestion:
     "Don't serve this any more. It's awful."

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> Having just said that, as best as I can tell, taking the derivative
> /usually/ makes something simpler, and taking the integral therefore
> /usually/ makes something more complicated.

Derivatives are easy if you have systematic brain that can follow 
logical methods.  You learn a few simple rules and then you are able to 
pretty much differentiate any function, no matter how complex (it just 
takes up more sheets of paper).

Integration on the other hand requires you to have a different skill, 
one to figure out what function might differentiate to give you back 
your original function.  There are a few common procedures and rules of 
thumb, but faced with an unfamiliar form often requires a bit or trial 
and error or luck...  Luckily at school we had a table of common 
functions and their integral, and I suppose the questions were 
specifically designed to make use of these.  In the real world you seem 
to get stuck very quickly though and just rely on a computer :-)

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On 13/12/2010 08:36 AM, scott wrote:
>> Having just said that, as best as I can tell, taking the derivative
>> /usually/ makes something simpler, and taking the integral therefore
>> /usually/ makes something more complicated.
>
> Derivatives are easy if you have systematic brain that can follow
> logical methods. You learn a few simple rules and then you are able to
> pretty much differentiate any function, no matter how complex (it just
> takes up more sheets of paper).
>
> Integration on the other hand requires you to have a different skill,
> one to figure out what function might differentiate to give you back
> your original function.

Indeed, there appears to be a *method* for figuring out a derivative. 
The only method for finding an integral seems to be trying expressions 
at random until one yields the derivative you want. There doesn't seem 
to be any more direct method.

> There are a few common procedures and rules of
> thumb, but faced with an unfamiliar form often requires a bit or trial
> and error or luck... Luckily at school we had a table of common
> functions and their integral, and I suppose the questions were
> specifically designed to make use of these. In the real world you seem
> to get stuck very quickly though and just rely on a computer :-)

Like I say, it appears that "in the real world", the integral usually 
doesn't have a closed form at all.

For example, Wolfram Alpha claims that there is no closed-form integral 
for tanh(x^2). And... well, check this out:

http://www.wolframalpha.com/input/?i=Integrate[x+Sin[x]+Cos[x^2]%2C+x]

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>> I didn't say that set theory is not *important*. I said it is not
>> *interesting*.
>
> It's interesting if you want to know whether arithmetic is complete and
> consistent, for example. Godel's theorem certainly is an interesting
> insight.

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"?

> And *that* is what makes it interesting! Set theory is like the lambda
> calculus - it's tiny and trivial, but you can prove anything with it.

I presume you mean "prove anything that is true", since a system that 
can prove *anything* is a system that's inconsistent...

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On 12/12/2010 09:59 PM, Orchid XP v8 wrote:

> A proof is usually a highly abstract sequence of logical steps which
> demonstrates that something /is/ true. Myself, I'm usually more
> interested in trying to get an intuitive feeling for /why/ it is true.

For example... There is a rather famous proof that asserts that Sqrt(2) 
is irrational. It proves this by demonstrating that any ratio equal to 
Sqrt(2) can be cancelled down an infinite number of times, which is 
absurd. Hence, such a ratio cannot exist.

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?

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.

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.

This doesn't *prove* anything, but it does explain why you can't solve 
the equation, in a way which intuitively makes sense.

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