Re: Thomas Covenant series

Tim S <Tim@xxxxxxxxxxxxxxxxxxxxxxxx> wrote:

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

Tim S <Tim@xxxxxxxxxxxxxxxxxxxxxxxx> wrote:

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.

Well, I visualise too, but it's not an image. "I see what you mean,"
said the blind man.

I mean, I know what a slope is: the steeper the slope, the harder it is
to walk up! Or it's the angle which the tangent to the curve makes with
the x axis. Or it's ...

Similarly, the area under the curve is proportional to the weight of
a uniform lamina cut into that shape. What else could it mean? You could
measure it (approximately) by tracing the curve onto a piece of
cardboard and cutting out the shape (and weighing it). That's sort of
an operational definition of area, rather than a visual one. All this
calculus stuff is just a way to guess the answer if you can't find your
scissors.

I'm semi-serious here: I can't really think what else "area" could mean,
in intuitive terms. Maybe it's how much paint you need to buy, to paint
that shape on a wall ... ?

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.

That's much how I feel, except that I feel there ought to be some
over-arching absraction which encompasses them all -- and which is
"obvious" both in the mathematical sense and to my intuition.

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.

In case it wasn't clear, I was really referring to layering: accretion
(in the case of a pearl) and successive shells in an onion.

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

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

It's not safe to let very young children (e.g. 5) play with sharp metal
objects. So very young children get plastic scissors for cutting paper.

When they are older, they can use real scissors to cut paper. Do you
retain, and use, plastic scissors?

The metaphor is in relation to mental tools: do you retain the (flimsy)
tools of a five year old, or do you abandon them. I'm saying that,
though I may have abandoned them, there is retained some echo of their
former existence.

I'm reminded of an ObSF: _The Unpleasant Profession of Jonathan Hoag_

Jonathan
.

Relevant Pages

  • Re: Thomas Covenant series
    ... I mean, I know what a slope is: the steeper the slope, the harder it is ... taught that a good way to find areas is to use integration, ... "obvious" both in the mathematical sense and to my intuition. ... So very young children get plastic scissors for cutting paper. ...
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  • Re: Thomas Covenant series
    ... That was my reaction too ... ... I think of integration as sum. ... The ratio in differentiation is ... Metaphors about onion skins, oysters, bits of grit and pearls ...
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