Re: Thomas Covenant series

on 17/05/2006 1:37 pm, Jonathan L Cunningham at spam@xxxxxxxxxxxxxxxxxxxx
wrote:

Tim S <Tim@xxxxxxxxxxxxxxxxxxxxxxxx> wrote:

on 16/05/2006 11:46 am, Jonathan L Cunningham at spam@xxxxxxxxxxxxxxxxxxxx
wrote:

"Eyes glazed over" is a strange metaphor. I presume it's derived
from the expression "glassy eyed".

I presume they both derive the same underlying image, of eyes as lifeless
and unmoving as little glass balls.

(Or is that too visual for you?) (That's a semi-serious question.)

Since it's a question, let me counter with a counter-question: why do
you ask if the ideas of lifelessness and unmoving little glass balls
are a *visual* image?

Because they are for me!

Also, because it's not so much the ideas separately, as the connection among
them all. I (think that) I connect actual glass balls with glazed eyes
because they look similar.


More introspection: I think of a differential coefficient as a
ratio, I think of integration as (a kind of) sum.

So, perhaps, the intuitive problem boils down to explaining why
the inverse of a ratio is a sum: it feels like: "the inverse of
two thirds is five: inverse(2/3) = 2+3"

That's very interesting.

I don't really expect to be able to help you with this, I'm just curious
about how you think (and shamelessly milking you for strange new ways to
think about things) but just in case...

That's only why I'm discussing it, too. Your questions help me to
introspect about it. How much ends up as pure invention (like false
memories)

Yes, I sometimes find myself confabulating fluently like this. :-(

is difficult to determine, but it's all potentially useful
for designing alien psychologies, I hope.

(Introspection is a *very* unreliable way to do psychology.)

It depends how carefully you do it, though. Some people are better at it
than others.


Notice that before we add up, we multiply the each of the fractions dy/dx by
their denominator dx : Integral (f(x) dx) matching Sum(dy/dx . dx)

<Tim writhes in agony at the lies he is having to tell>

So we're just adding up the numerators. The ratio in differentiation is
chopping y up into pieces, each of size dx, and integration is putting them
back together again.

<Tim faints from pain>

So you should! Here, have a bucket of cold water (without the bucket).
<whoosh>

If you don't mind indulging me a little more, how do you intuit slope, and
the relation between slope and differentiation? Likewise with integration
and area?

It's very difficult to separate what I *know* from what doesn't *feel*
intuitive.

It's just that, when I think about slopes and areas, I'm always visualising,
so I was wondering what you do.

As I said before, I can make lots of triangles out of any
three of the four sides, I can even make a tetrahedron, but making just
one triangle with four sides feels difficult.

And so it should...

So a better question might be "why does it feel like it ought to be
a triagle? Why not a square?"

As Brian has hinted, elsethread, it might[*] be because I was taught the
things separately. If you're taught that integration is the reverse of
differentiation, then using it to find areas is one thing. If you are
taught that a good way to find areas is to use integration, then finding
that it is the inverse of differentiation is a different thing.

That doesn't entirely make sense to me. I have a lot of different ways to
think about integration and differentiation, with lots of different
generalisations and contexts. Depending on which ways I use, the connection
between them may seem tautological, non-tautological but obvious,
non-obvious but straightforward to discover, or completely opaque and
baffling.


Let me come up with a different example (I've talked about this before,
and Brian has commented on it before, but it might shed light on this
new question).

Ok, you know the simple IQ questions of the form:
A is to B, as C is to which of D1, D2, D3?
Example:
man is to woman as boy is to: [baby, woman, girl]?
Why do you get the same answer if you pose the question in the form:
A is to C, as B is to which of D1, D2, D3?
Same example:
man is to boy as woman is to: [baby, woman, girl]?



<-sex->
^ A --- B
|
a | |
g | |
e | |
|
v C --- D

Questions:
Is it obvious that the ABC pattern (always?) leads to the same choice
of D as the ACB pattern? If so, why?

Because Z_2 x Z_2 is a commutative group. :-)

Can you come up with an example where the ABC pattern doesn't lead
to the same obvious D as the ACB pattern?


If you even attempt the second question, I think that any answer "yes"
to the first must be dubious.

Metaphors about onion skins, oysters, bits of grit and pearls
become appropriate at this point.

Onion skins?

Same as pearls: it's a problem in integration. To find the volume
of an onion, you peel off successive shells. Each shell has
a volume of 4*pi*r^2*dr, where dr is the thickness of an onion skin :-)

Oh, yeah, OK.


[*] Not that anyone ever tried to teach me calculus at five, but
that patterns of thought persist, and get overlaid. More metaphors
about pearls, plus new metaphors about sedimentary rocks and
fossils become appropriate.

I tend to think of it as tools in a toolbox. And machine tools in a
workshop.

And do you throw away your hammer when you buy a screwdriver?

No, obviously not. And I have several different sizes of hammer and
screwdriver, also screwdriver bits for my drill.

What about your plastic scissors, when you buy a new pair of kitchen scissors?

Er...not sure what this is a metaphor for. Probably not, but I guess I don't
use the plastic scissors very often?


Jonathan
[*] I don't even know if that's true: I can't remember how I was
taught integration. Perhaps I was simply born with the knowledge?
Or perhaps, one day, I was simply enlightened??

I learnt calculus originally from a book called something like 'Calculus
Refresher for Practical Men' that must have belonged to my Dad, which he
discovered when he was clearing out a cupboard, and gave to me. By the time
we came to do elementary calculus at school, I had already learnt vector
calculus from a book on electromagnetism, and was trying to understand
tensor calculus on curved pseudo-Riemannian manifolds. I learnt almost all
of my calculus from books. Whether this was a good thing or not, I don't
know.

Tim

.

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