nemesis wrote:
> clipka wrote:
>> Am 11.12.2010 13:14, schrieb Orchid XP v8:
>>
>>> By contrast, this paper seems to assert that mathematics is about
>>> constructing abstractions and building theories out of them as a matter
>>> of creativity. Obviously I've never seen any cutting-edge mathematics
>>> (and I never will), 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 
>> (which I find an interesting thing to know, too) - and not only in 
>> this world, but also in any imaginable one (another interesting thing) 
>> except for pretty boring worlds.
> 


or perhaps of Turing himself? :p

Post a reply to this message

nemesis wrote:
> suppose you have a set of 2 things and a set of 3 things.  How many things you
> have in total?  Is that not how you even learn basic arithmetic?

Yeah, except you don't start out with "things". That's why it's like that 
1-function variant Andrew always talks about. The S or the K or whatever. 
The only "thing" you have is a set. And sets don't have order. So how do you 
even *represent* "three" using only sets, with nothing in the sets but other 
sets?

> functions?  a formula mapping things from a set to things from another set?

Except, again, you have to do it with sets.

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

Post a reply to this message

Am 12.12.2010 06:35, schrieb Darren New:

> Yeah, except you don't start out with "things". That's why it's like
> that 1-function variant Andrew always talks about. The S or the K or
> whatever. The only "thing" you have is a set. And sets don't have order.
> So how do you even *represent* "three" using only sets, with nothing in
> the sets but other sets?

Oh, that's pretty simple:

First, we start with a contrived unary operation on arbitrary sets we 
call "increment" some obscure reason:

   A++ := A U {A}

(The notation for the operation is, as you may guess, not common in the 
mathematical world ;-))

Next, we define the inverse operation:

   (A++)-- := A

Next, we define a set we call "zero"; we could use an arbitrary 
definition, but as we have no other sets in our little world yet, we use 
the empty set:

   0 := {}

Next, we recursively define a set of objects we call "natural numbers":

   N  := { n | n-- is element of N U 0}
   N° := N U 0

Now we can assign symbols to the natural numbers:

   3  := { {}, {{}}, {{}, {{}}} }

There.

 From here on, we can extend our little world of arithmetics by defining 
an operation on two arbitrary natural numbers called "addition", define 
the inverse operation called "subtraction", define another operation 
called "multiplication", and so forth...

And it all lives within the realms of set theory.

Post a reply to this message

Am 12.12.2010 04:51, schrieb nemesis:

>> ... despite there existing proof that this desire is insatisifiable
>> (which I find an interesting thing to know, too) - and not only in
>> this world, but also in any imaginable one (another interesting thing)
>> except for pretty boring worlds.
>



differently :-P

Post a reply to this message

Am 12.12.2010 04:48, schrieb nemesis:

>> In what sense is calculus "tough"?
>
> are you talking about that calculus? the one with integrals, derivations
> etc? Yeah, pretty hard stuff. I gave up on integrals in the complex
> space...

I preferred to already give up on integrals in the real space, unless I 
had a formula handy for that specific function...

Derivatives are fun though, I find they play along pretty nicely.

Post a reply to this message

clipka wrote:
> Am 12.12.2010 06:35, schrieb Darren New:
> 
>> Yeah, except you don't start out with "things". That's why it's like
>> that 1-function variant Andrew always talks about. The S or the K or
>> whatever. The only "thing" you have is a set. And sets don't have order.
>> So how do you even *represent* "three" using only sets, with nothing in
>> the sets but other sets?
> 
> Oh, that's pretty simple:

Once you figure it out, it's pretty simple, yes. I wasn't really asking. I 
as more saying "think about this problem, isn't it an interesting problem? 
That's why set theory is interesting."

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

Post a reply to this message

>There are plenty of fields of mathematics that are so utterly boring
 >that you have to wonder why anybody studies them
 > - set theory, number theory, category theory, etc.)

And this exactly supports the author's ideas in
 >http://www.maa.org/devlin/LockhartsLament.pdf

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

 >I didn't say that set theory is not *important*. I said it is not
 >*interesting*. Certainly not to me, anyway.
 >The obvious stuff is obvious, and the non-obvious stuff is all tedious
 >riddle-like splitting of hairs rather than interesting insights.

 > If there *is* some kind of connection between set theory and 
arithmetic, it is highly non-obvious.

Check out Knuth's book of "How Two Ex-Students Turned on to Pure 
Mathematics and Found Total Happiness". Unfortunately, it's not so easy 
to find it in UK.

Post a reply to this message

>> In what sense is calculus "tough"?
>
> are you talking about that calculus? the one with integrals, derivations
> etc? Yeah, pretty hard stuff. I gave up on integrals in the complex
> space...
>
> If you don't believe so, you're clearly gifted...

To me, the concepts seem pretty simple. Sure, actually applying the 
concepts in a tricky situation is... tricky. But that applies to just 
about anything.

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. It's easy enough in the 
simple case, but - according to Mathematica - for almost every function 
I'm interested in, no closed form exists for its integral. In other 
words, analytic integration is impossible most of the time. (And I guess 
if I didn't have Mathematica, I'd spend a reeeeeely long time trying to 
figure out how to do the integration...)

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

Post a reply to this message

>> 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...

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

Post a reply to this message

>> 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.

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

Post a reply to this message