Re: Ruler and Compass in Mathematics

On Nov 30, 6:55 pm, John Jones <jonescard...@xxxxxxx> wrote:
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.

Some of the things you say are just way out there. This is one of
them. You're very adept at making non-sequiturs.


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.

What hz said. To take this in a different direction... why do you feel
the need to do any joining up of dots? Like in the curve y = x^2,
can't you just be satisfied with 3 goes to 9, 7 goes to 49, etc? and
mathematically joining up of these integer values, well, big deal, who
cares, the integer values are enough to work with? (lots of other
people have been successfully 'joining up the dots' in a very
consistent and efficient and useful manner for many years now, with no
problems. What is your complaint about that process? You haven't
actually articulated what your complaint is beyond something like 'I
don't get approximation'.

Mitch
Mitch
.

Relevant Pages

  • Re: Calculus XOR Probability
    ... which actually curve, such as a section of circular arc. ... directions and lengths of the segments. ... Except that there are no such things as infinitesimals in the reals, ... I feel like I'm already inventing half of mathematics. ...
    (sci.math)
  • Re: Ruler and Compass in Mathematics
    ... mathematically joining up of these integer values, well, big deal, who ... people have been successfully 'joining up the dots' in a very ... contnuous curve or straight line between the facts that are given. ...
    (sci.logic)
  • Re: Calculus XOR Probability
    ... the locations of the staircase in the limit are indistinguishable ... mathematical requirement for a curve to have a length is the the lengths ... it cannot be used as a measure of a curve. ... The mathematics involved is quite straightforward and pedestrian. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... What is the slope of the staircase at x=0. ... infinite, as opposed to the slope of the diagonal. ... As long as those locations are ON the curve, ... The mathematics involved is quite straightforward and pedestrian. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... In standard mathematics, the order in which limit processes are applied ... The limit cannot possibly be a"sequence" of line segments as any ... That was to get a measure on the curve. ...
    (sci.math)
Loading