Re: Ruler and Compass in Mathematics

Mitch Harris wrote:
On Nov 29, 4:23 pm, John Jones <jonescard...@xxxxxxx> wrote:
Mitch Harris wrote:
On Nov 28, 5:46 pm, John Jones <jonescard...@xxxxxxx> wrote:
A curve emerges as an intuitive (in the imagination), pictorial,
non-quantifiable, leap from a selection of such points.
How is a line different from a curve?
That's what I would like to know. I think there is a fundamental
incompatibility between line and curve, as I suggested in my posts on pi.

incompatibility or not, there's a similiarity in that, for all the
inscrutability and unanalyzability that you ascribe to curves, it
equally applies to lines (I think they are different, but not in the
ways that you've mentioned).

All that aside, I am saying that the standard practice of joining up
points by circles or lines amounts to an intuitive, non-quantitative act

OK...sounds like something...

which cannot be described by a mathematics.

...uh...now you lost me. Just look around. Open your eyes. Or rather
get out of the darkness. Mathematics does 'pretty well' in describing
lines, curves, pi, sqrt(2), approximations. You may have different
ideas of what 'pretty well' means, but bridges ain't falling around us
because of the math.


It is something we do for
ourselves when we have to momentarily, as it were, leave mathematics behind.

Why isn't there the same leap
when interpolating lines from points? You're not OK with pi but you
are with sqrt(2)?
I haven't looked at sq.rt.2. but I know that there is, historically,
something very fishy about it. The fact that we invented an 'i' to deal
with it isn't really getting to grips with it.

just for fun, the historical problem about sqrt(2) is that it cannot
be represented as a ratio of two whole numbers. How tht particular
problem was resolved is by not caring, mathematicians just accepted it
(and chose a poor name for similar numbers: "irrational").

'i' on the other hand was an invention (as much as sqrt(2) is or for
that matter 0, or 1, or 10, or even 2). 'i' was 'invented' to allow
solutions to things like

x^2 + 1 = 0

and relevantly, sqrt(2) is nice for solutions to

x^2 - 2 = 0

and negative numbers for

x + 2 = 0

and zero for

x + 2 = 2

and 1 for

x*2 = 2

Usually, these things, historically, if they really bugged you a lot,
you just get over it.

Mitch

Yes, I got mixed up with sqrt-1 - that's 'i', and is what I meant. The
answer of course is that there's no square involved, its an addition.
The two things the same that make -1 are -0.5 and -0.5.

It isn't maths that makes us join up the dots on a graph. There's
nothing on the graph or in y=x^2 that mathematically joins up the dots. We do that for ourselves when we want a curve.
.

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