>    Or perhaps more like: Programming has been a good incentive to 
learn math?

Pretty much. I don't think it's coincidence that I got a lot better at 
math a couple of years after my family got our first computer. POV-Ray, 
Visual Basic, and JavaScript all pushed my math skills, because there 
was a real, immediate payoff to every thing I learned.

  - Slime

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On 10/12/2010 05:55 PM, nemesis wrote:

> You should really have gone to a special school of another kind. Your
> parents didn't realize your math brain was running faster than those of
> other kids of your age...

My parents followed the advice of the experts - experts who are supposed 
to know what they're talking about. And hey, I can now read and write, 
which is kind of crucial...

> calculus is tough for anyone. I wouldn't ever be able to grasp it while
> a teen...

In what sense is calculus "tough"?

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

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>> I understand that people need to know how to do division. I have no idea
>> why they need to practise double-sized sheets with 40 quotients per side
>> featuring 6-figure numbers. Hell, even NASA used a slide rule instead of
>> pencil and paper! WTF?
>
> Hollywood lied to you about the moment in the Apollo 13 mission when the
> astronauts asked the ground grew to double-check their maths: They did
> /not/ use slide rules in that situation, but indeed pen & paper, because
> slide rules don't help with simple addition of numbers.

OK, fair enough.

> Back then, computer was an actual job, requiring highly trained people.

I gather that this is the original meaning of the word "computer", yes.

> I'd also say it was good for our generation to know how to add,
> subtract, multiply and divide without a pocket calculator

I'd say it would be good for *every* generation to know how to do this. 
Given that it's not exactly complicated, it still astonishes me how many 
people cannot do this. I mean, hell, I haven't memorised the 
multiplication table, but I know the *algorithm* for all of these 
operations.

(I do not, however, know of any way of computing a square root that 
isn't horrifyingly torturous...)

>> JKR didn't get to where she is today by using punctuation
>> correctly. She got there by convincing Warner Brothers to make her books
>> into big-budget films. Oh, wait...
>
> Not really. The books were amazingly popular before the movies.

Yeah, I know that. I was just trying to inject a little humour.

In spite of the insane popularity of her books, there are apparently a 
significant number of people who assert that they aren't very good. (And 
not just the radical Christian sects who assert that any magic without 
reference to God is the work of Satan...)

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

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> This paper seems like a pretty accurate description of everything that
> is wrong with mathematical teaching, and society at large.

Although it did leave me wondering for a moment: Am I actually "good at 
math"? Or am I just good at following directions?

> People seem
> to think that mathematics is about moving symbols around on a sheet of
> paper according to a set of complex and technical rules.

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.

(Well, "everything" is an exaggeration. 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.)

The paper, on the other hand, likes it sound like doing mathematics is 
like modern art - an essentially empty and pointless exercise designed 
to impress art critics and other artists, in the style of the Emperor's 
Magical Suit. And here I was thinking it was about furthering mankind's 
knowledge...

Seriously. I enjoy exploring mathematical ideas just for the hell of it. 
But you just never know when this stuff will end up being useful.

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

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Orchid XP v8 <voi### [at] devnull> wrote:
> > This paper seems like a pretty accurate description of everything that
> > is wrong with mathematical teaching, and society at large.

> 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? (Of course this
requires quite a bit of both knowledge and experience in algorithmical
programming. Eg. you need to have knowledge about computational complexities,
have developed a kind of "computational complexity instinct", have experience
on different problems and their solutions, have the experience and talent to
come up with either modified or brand new solutions to the given problem,
and so on.)

  If you routinely can devise efficient algorithms for given computational
problems, then one could say that you are good at algorithmic programming.
Likewise I'd say that if you routinely can solve mathematical problems,
devise mathematical formulas to describe things and so on, you are good
at math.

  Anybody can calculate 2+2 or even solve an equation, in the same way
as anybody can write a "hello world" program or a program which calculates
the sum of a list of numbers, but that doesn't make that someone *good* at
math/programming.

-- 
                                                          - Warp

Post a reply to this message

>> 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?

To be honest, I've yet to discover a computational problem which hasn't 
already been solved 25 different ways (usually before I was even born). 
The art of writing efficient computer programs appears to be the art of 
figuring out where the **** all these algorithms are written down. 
Because, let's face it, any algorithm that I can invent will be several 
complexity classes worse than what the greatest minds of the 20th 
century could think up.

>    If you routinely can devise efficient algorithms for given computational
> problems, then one could say that you are good at algorithmic programming.
> Likewise I'd say that if you routinely can solve mathematical problems,
> devise mathematical formulas to describe things and so on, you are good
> at math.

Well, when I was a teenager I wrote my first ray tracer. Which doesn't 
sound that impressive, until you realise that I have to figure out how a 
ray tracer actually *works* in the first place, using nothing other than 
the documentation for POV-Ray. (This is *before* Warp added the SDL ray 
tracer, I should point out... That would have made it significantly 
easier, obviously.)

In short, I had to deduce that you perform ray intersection tests by 
solving simultaneous equations. (Usually if you want to draw a shape, 
you want a parametric representation of it. But POV-Ray seems to talk a 
lot about equations rather than formulas, which was a big tip-off.) And 
then I had to work out how you actually solve simeltaneous (non-linear) 
equations. (No, I didn't know how to do that.) And then I had to shift a 
bunch of algebra around to make it happen. (That, arguably, is just 
blindly applying rules. Mathematica(tm) could have done it just as 
capably as me.)

FWIW, I was utterly astonished when I ran the program and it actually 
freaking *worked*!

>    Anybody can calculate 2+2 or even solve an equation, in the same way
> as anybody can write a "hello world" program or a program which calculates
> the sum of a list of numbers, but that doesn't make that someone *good* at
> math/programming.

Actually, I suck at arithmetic. (Fortunately, I have a computer to do 
that for me...) What I consider myself to be modestly good at is 
mathematical reasoning.

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

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On 11/12/2010 12:14 PM, Orchid XP v8 wrote:

> 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.

Thinking about it... No, I'm not very good at math. I never bother 
proving things.

According to the paper, the proof is far more important than the result 
(at least as far as mathematicians are concerned). I always considered 
the proof to be the most boring part.

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

Post a reply to this message

Orchid XP v8 <voi### [at] devnull> wrote:
> >> 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?

> To be honest, I've yet to discover a computational problem which hasn't 
> already been solved 25 different ways (usually before I was even born). 
> The art of writing efficient computer programs appears to be the art of 
> figuring out where the **** all these algorithms are written down. 

  There is an infinite number of possible variations to any given problem,
so you often can't just use a pre-defined algorithm as-is to solve the
problem, but you have to adapt it. Also, you can't know every single
algorithm in existence, and many of the most obscure algorithms may be
extremely hard to find (especially if you don't now the proper terminology),
and hence you have to come up with the solution yourself.

> Because, let's face it, any algorithm that I can invent will be several 
> complexity classes worse than what the greatest minds of the 20th 
> century could think up.

  Not true. There are many, many algorithms where you can *know* how fast
it can be solved (in terms of computational complexity) and you can come
up with an algorithm that solves it that fast. The difference is in details
(basically, speaking in terms of computation complexity, the constant
factor of the complexity function).

  Also, in many cases even if you know that a problem can be solved faster
(in terms of computational complexity), but the optimal solution is extremely
complicated and hard to implement, a less-than-optimal solution may still be
perfectly suitable for the problem at hand (eg. a completely naive solution
would take 10 minutes on a given input, your not-completely-optimal but
good-enough solution takes 1 second, and the known-to-be-optimal would
take 0.9 seconds, it doesn't really matter all that much). If the
less-than-optimal solution is suitable for the problem, and its
implementation is considerably simpler than the most optimal solution,
it might even be better to use the simpler solution from a programming
development point of view (simpler source code, easier to understand, less
chances for bugs).

  If, however, you indeed can only come up with a solution that is
"several complexity classes worse" than the currently known (or proven)
optimal solution, then I'd say you are not all that good at programming
and need more experience.

-- 
                                                          - Warp

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Orchid XP v8 wrote:
> In spite of the insane popularity of her books, there are apparently a 
> significant number of people who assert that they aren't very good. 

They aren't. They're fairly trite, at least the first three or four are 
fairly obvious, the magic is lame and without rules, the stereotyping is 
stereotyped, and it's all fairly predictable.  That's part of what makes 
them popular, tho.

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

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Warp wrote:
> extremely hard to find (especially if you don't now the proper terminology),

*Especially* before google.

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

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Orchid XP v8 wrote:
> anybody studies them - set theory, number theory, category theory, etc.)

Set theory is how you prove arithmetic works. Prove 2+3=5.

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

Post a reply to this message

On 11/12/2010 03:08 PM, Darren New wrote:
> Orchid XP v8 wrote:
>> anybody studies them - set theory, number theory, category theory, etc.)
>
> Set theory is how you prove arithmetic works. Prove 2+3=5.

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.

Also, several sources state that set theory is the fundamental basis of 
all of modern mathematics, but not one single one makes any attempt to 
explain this sweeping statement. Your assertion that set theory somehow 
has something vaguely to do with arithmetic is typical of this type of 
statement. If there *is* some kind of connection between set theory and 
arithmetic, it is highly non-obvious.

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

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Orchid XP v8 wrote:
> 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.

> Also, several sources state that set theory is the fundamental basis of 
> all of modern mathematics, but not one single one makes any attempt to 
> explain this sweeping statement. Your assertion that set theory somehow 
> has something vaguely to do with arithmetic is typical of this type of 
> statement. If there *is* some kind of connection between set theory and 
> arithmetic, it is highly non-obvious.

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.

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

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Am 11.12.2010 12:26, schrieb Orchid XP v8:

> In spite of the insane popularity of her books, there are apparently a
> significant number of people who assert that they aren't very good. (And
> not just the radical Christian sects who assert that any magic without
> reference to God is the work of Satan...)

I've never read them (nor did I see any of the movies). I too heard tell 
that the books were rather bad, but my father insists that while the 
German translation sucks, the English original is actually quite good 
(he's a retired English teacher, and none too easily impressed by any 
"hype", so he may be a comparatively credible witness after all).

Post a reply to this message

On 12/9/2010 1:04 PM, Mike Raiford wrote:
>
>
> http://www.maa.org/devlin/LockhartsLament.pdf

It's an interesting picture of an establishment.  You get accepted as an 
expert by agreeing with the experts.

Regards,
John

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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.

Post a reply to this message

Am 11.12.2010 14:46, schrieb Orchid XP v8:

> Well, when I was a teenager I wrote my first ray tracer. Which doesn't
> sound that impressive, until you realise that I have to figure out how a
> ray tracer actually *works* in the first place, using nothing other than
> the documentation for POV-Ray. (This is *before* Warp added the SDL ray
> tracer, I should point out... That would have made it significantly
> easier, obviously.)

Hehe - when I was a teenager, I wrote a raytracer, too - without even 
knowing POV-Ray; all I had was a few printed b/w chrome spheres on 
checkered planes, the desire to do the same, Snell's law, and a 
classmate who took over the task of developing UI components for the 
interface (we didn't have a library for that either) :-P

Needless to say that the thing was awfully simple, and did not even work 
properly - it got the normals of non-uniformly scaled spheres wrong - 
but hey, you can't do everything right all the time, can you ;-)

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Am 11.12.2010 14:48, schrieb Orchid XP v8:
> On 11/12/2010 12:14 PM, Orchid XP v8 wrote:
>
>> 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.
>
> Thinking about it... No, I'm not very good at math. I never bother
> proving things.
>
> According to the paper, the proof is far more important than the result
> (at least as far as mathematicians are concerned). I always considered
> the proof to be the most boring part.

You don't need unshakable solid proof for conjectures, so maybe you 
should stick to those. In my eyes that's mathematical enough.

I guess more often than not, proving a good conjecture boils down to the 
mere applying of rules again. Sure, there are famous conjectures out 
there that seem to defy any attempt to prove them, but you don't hear 
about the conjectures that are easy to prove. And after all, /every/ 
proof starts off as a conjecture.

Post a reply to this message

Am 11.12.2010 06:37, schrieb Slime:
>  > Geometry, trigonometry and in some cases even calculus has been quite
>  > useful in graphical-heavy and game programming.
>
> Graphical and game programming has been quite useful in learning
> geometry, trigonometry, and in some cases even calculus.

Yup ;-)

Post a reply to this message

Orchid XP v8 <voi### [at] devnull> wrote:
> On 11/12/2010 03:08 PM, Darren New wrote:
> > Orchid XP v8 wrote:
> >> anybody studies them - set theory, number theory, category theory, etc.)
> >
> > Set theory is how you prove arithmetic works. Prove 2+3=5.
>
> 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.
>
> Also, several sources state that set theory is the fundamental basis of
> all of modern mathematics, but not one single one makes any attempt to
> explain this sweeping statement. Your assertion that set theory somehow
> has something vaguely to do with arithmetic is typical of this type of
> statement. If there *is* some kind of connection between set theory and
> arithmetic, it is highly non-obvious.

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?

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

yeah, set theory is fucking huge.

Post a reply to this message

Orchid XP v8 wrote:
> On 10/12/2010 05:55 PM, nemesis wrote:
> 
>> You should really have gone to a special school of another kind. Your
>> parents didn't realize your math brain was running faster than those of
>> other kids of your age...
> 
> My parents followed the advice of the experts - experts who are supposed 
> to know what they're talking about. And hey, I can now read and write, 
> which is kind of crucial...
> 
>> calculus is tough for anyone. I wouldn't ever be able to grasp it while
>> a teen...
> 
> 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... although I'm sure 
povray newsgroups may host a lot of math lovers who put my average math 
skills to shame...

Post a reply to this message

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.

you're talking about that Godël dude, aren't you?

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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.
> 
> you're talking about that Godël dude, aren't you?

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."

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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.

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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.
>
> you're talking about that Godël dude, aren't you?

Almost. I'm taling about that Gödel dude, who placed his Umlaut dots 
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."

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>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*

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>> 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*

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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.

-- 
http://blog.orphi.me.uk/
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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>> 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! ;-)

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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.

-- 
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   Serving Suggestion:
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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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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.

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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

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   Serving Suggestion:
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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.

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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.

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

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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.

-- 
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   Serving Suggestion:
     "Don't serve this any more. It's awful."

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