[extropy-chat] Maths ability

[Adrian Tymes]

--- ben <benboc at lineone.net> wrote:
> Adrian Tymes wrote:
> > (Likewise, x + (-1) = x - 1, because you've three lines, an odd 
> > number.)
> 
> OK, i see what you're doing. But why should the geometry of the
> symbols
> used have any explanatory power for mathematics? I understand that
> this
> is an attempt to make it easy to memorise, but it doesn't help
> understanding. x + (-1) doesn't equal x -1 BECAUSE there are three
> lines
> in the symbols, does it. Nor is it because "a plus and a minus make a
> minus", although i think that's a lot closer to the actual reason.
> Well,
> maybe it IS the actual reason, but put in a way that isn't very clear
> to
> me.

Ah.  True.  I did memorize it that way to memorize "a plus and a minus
make a minus".

As was pointed out, a closer-to-the-pure-truth version is that, in
general, x - x = 0, no matter what x is.  Understand that first - do
not confuse yourself by assigning a given value to x, just come to
grips with the fact that x - x = 0.  (Or if you must have actual values
for x: imagine it for 0, first, to appreciate the absurd, yet true,
simplicity of 0 - 0 = 0.  Then imagine it for positive numbers - say,
1 - 1 = 0.  And so forth, until you're okay with x - x = 0 for *any*
x.)  Once you have that, then substitute -1 for x.

> > But here's a little trick <snip> (hail Google ;) )
> 
> Indeed.
> 
> Thanks for that. I never thought of taking that particular approach,
> even though i use Google all the time. I'll try it.

*nods*  Far and away one of the most useful information coping tools
available today!  ^_^

> (this gives me a headache! - What IS it, how do you SEE
> these
> things? I mean there's no such thing as "-6 oranges", is there?).

Well...if you stretch the concept of antimatter, you can think of an
anti-orange as the equivalent of -1 orange.

It helps that some video games these days actually have icons to show
you negative quantities.  (Like, shortfalls of food in your cities in
Civilization.)  1 normal icon plus 1 negative icon cancel out, and the
game shows you no icon.

> + is: <-<-0->-> and - is: ->->0<-<-
> or
> + is: ->-> and - is: <-<-
> 
> It's neither of those is it? >:(

Well...perhaps it might help to think of negative numbers as
"strange/perverse".  As in, you know what *normally* happens, with zero
and positive numbers; also know that negative numbers cause the
opposite to happen (in certain cases, like addition and subtraction; in
in others, like square roots, they just get wierd).

Another possibility, if you're visualizing it kind of like a conveyor
belt anyway: each -, be that the - of subtraction or the - of a
negative number, reverses the direction of the conveyor.  + doesn't do
anything; it's kind of like empty air, or a null operator, or a
different kind of 0.

In other words...

+ (no reversals) is: ->->
- (one reversal) is: <-<-
+ - (one reversal) is: <-<-
- - (two reversals) is: ->->
- - - (three reversals) is: <-<-

(Yes, this is just visualization, not explanation.  But that's what
you're asking for here.)

> > Hate to break it to you, but sometimes it *is* the tricks.
> 
> Hmm. I know that maths is not a thing in it's own right, like rocks
> are,
> but an invented tool to help us understand the world. But all the
> same,
> it has an underlying unity (it does, doesn't it?!?)

It does...but that's the thing.  Thanks to the underlying unity, you
can start from the tricks - if correctly applied - and get to most of
the other concepts in math...eventually.  (I recall a certain
multi-page proof, starting from certain axioms far removed from basic
addition and subtraction, that 1 - 1 = 0.)  Knowing more tricks gives
you more starting points to choose from...but if you're trying to trace
a path to the "root"/"core" of mathematics, you can in theory start
from any of them.

> I want to understand the concepts. A collection of tricks is
> unsatisfying, and i feel that approach doesn't do justice to
> mathematics
> as a product of human ingenuity. And anyway, you don't have to worry
> about memorising something if you understand it. You can reinvent it
> if
> necessary.

...that's what a lot of people thought.  But some of these tricks took
thousands of years to invent, and are not easy to rediscover.
Sometimes, the tricks themselves have to serve as bedrocks for
understanding in practice, even if in theory there is a deeper
understanding available.

> --- Lee Corbin <lcorbin at tsoft.com> wrote:
> >> Damn right.  Me, every since I was a little kid, I had a "math
> >> line" that quickly, visually, and easily came to me that told me
> >> the answer to many problems.  See "The Math Gene" by Keith Devlin.
> 
> What?!
> I'm not sure if i understand what you mean. Are you saying that you
> (and
> Adrian) have some kind of intuition about maths, and answers spring
> out
> at you in the way that, say, spelling mistakes in a sentence can?

Yup, that's-a right.

> If so, i find that idea deeply wierd.

Imagine, if you will, a stereotype hillbilly redneck American.  Bubba
learned well, for the schooling that was available, but has had
absolutely no exposure to foreign languages whatsoever.  Bubba decides
to move to Quebec...and finds that, while the official language may
still be English, learning French (specifically, the Quebecois dialect)
is a practical necessity.  To Bubba, French is this strange, foreign,
*alien* tongue.  (Fortunately for Bubba's culture-shock-sensitive
heart, Bubba's never heard any actual Chinese or Afrikaans.)

Bubba makes friends with some locals, who grew up bilingual.  These
friends instinctively spot the spelling mistakes in Bubba's written
French.  Bubba, still getting used to this alien culture with its
alien tongue, finds this deeply wierd: intellectually, Bubba can
understand people growing up with it, but after so many years
surrounded by nothing but English speakers, the whole concept of
instinctive French almost feels just plain wrong.

Which is not to say you're Bubba in any way.  Just that this makes for
a good analogy for people like Lee and myself: we've been steeped in
math since we were young, and we understood it...so of course it's
instinctive to us now.  (Genetics might have helped us acquire an
instinctive understanding at a young age.)  Likewise, you have not yet
had a deep understanding of math for long enough for it to become
instinctive.  (Indeed, as you say, you do not yet have a deep
understanding of math at all.)

Which is not to say that my intuition is 100%.  I can and do still make
simple errors from time to time.  (I recall a certain post to this list
about antpower - the horsepower a typical ant could put out.  Someone
copied it over to Wikipedia, before they had their "no original
research" rule that disqualified articles on things like that - which
lead to the article's eventual removal, though you can still find it on
a bunch of Wikipedia's mirrors.  The Wikipedia editors quickly
corrected my calculations.  I *did* disclaim that I wasn't very certain
about my math in that case, but still...)

> Anyway, this is getting a little bit off track. My point was not to
> ask 
> for help with maths (!) but to point out that 'try harder' is not
> really 
> much use to someone who has difficulty with it. It's like telling a 
> depressed person to just pull themselves together.

In this, you are correct.  If one does not know how to even approach a
problem - as in your case with math, or a depressed person with
overcoming depression - then, no matter how intuitively obvious the
solution may seem to outsiders (and thus, no matter how little
explanation said outsiders may think is warranted - one usually does
not waste energy telling a fully functional adult how to walk, for
example), the afflicted person genuinely needs to learn how to perform
the desired task, and a fuller explanation than just "try harder" is
usually required for that.

> Actually, it's quite possible that having people that are inherently 
> innumerate could be a good thing. Diversity and all that.

Uhh...no.  Diversity for diversity's sake alone can be used to justify
all sorts of evils as well as for good.

For example, diversity of atmosphere in a metropolis: we've got the
standard oxy/nitro, but of course we can't let that dominate the whole
city, so we'll keep it over here in the rich peoples' neighborhood.
The commercial district has extra-heavy carbon dioxide - good for fire
suppression and plant growth, dontcha know?  We've got to put the
methane and smog *somewhere*, so we'll let it stay in the working class
residential districts.  And the industrial sector makes its own
atmosphere...just so long as it keeps it over there...

Besides, we're transhumanists.  We're all about overcoming such natural
limitations, aren't we?