Re: Ruler and Compass in Mathematics

On Dec 3, 6:15 pm, John Jones <jonescard...@xxxxxxx> wrote:
Mitch Harris wrote:
On Dec 2, 4:13 pm, John Jones <jonescard...@xxxxxxx> wrote:
Mitch Harris wrote:
On Dec 2, 2:37 pm, John Jones <jonescard...@xxxxxxx> wrote:
Mitch Harris wrote:
  y=x^2 isn't continuous, its discrete. If you want to make it
continuous then you employ a non-mathematical manoevure and draw a
contnuous curve or straight line between the facts that are given.
Drawing a curve or line is useful. It's not mathematical though.
Excellent! Progress! Then you agree then that whatever the intention
of your complaints against curves is, what you've said also allows the
same complaint about straight lines.
But I am confused by the first thing you said "y=x^2 isn't continuous,
its discrete". if x is continuous then y will be continuous
(multiplication being well-defined on a continuous domain).
Maybe your complaint is that continuity is not well-defined. It usre
is more problematic than discrete things (arithmetic has a much less
complicated foundation than analysis).
But saying that 'drawing a line is not mathematical' is just being
weird. There are non-mathematical aspects of it, and mathematical ones
(and the mathematical aspects are non-negligeable).
Mitch
Maybe the complaints are a little different. My complaint, or rather
observation, concerning the mathematical treatment of curves is that it
treats them as straight lines in order to keep quantification.

x and y in y=x^2 are discrete, not continuous. Any value of y or x is a
single value. That value can be represented by a point, line or curve,
but these are also discrete and not continuous. If they are represented
as a point which is, after all, a pictorial artifice, then there is no
mathematical reason forcing us to consider it as a line or curve.

OK. I think I don't know what you mean by discrete or continuous. Does
discrete mean for you a single object?

Yes, by 'discrete' I mean a value, or some other unidimensional item,
like a constant, a variable, etc. All items in maths are discrete and
not continuous.

OK. That's a very different use of 'discrete' and 'continuous' than
avirtually anybody else who uses those terms (even non-mathematicians)
but I'll play along.

So by your statement -nothing- that is continuous is mathematical?
(just to confirm)

So a number is discrete, be it 1, 2, -3, i, pi, any variety of
infinity, etc, etc?

A finite set of numbers if discrete?

A straight line is discrete? a curved line?

A function from an infinite set to another set (finite or infinite)?


Is there anything that you
could describe as continuous? (you realize that both terms have their
technical mathematical meaning which seem kind of different from how
you're using them)

I have argued that there is nothing in mathematics or out of mathematics
that can quantify a continuity, eg a curve.

Now I'm confused. Is a curve continuous or discrete?

So if a curve is not mathematical, then what is it? biological? maybe
you have a definition of mathematical that is not what I am used to
either.


To quantify a curve we
always have to change it to a straight line or series of lines.

We -have- to? Says who? Let's just call the length of a circle 'pi',
and move on. To approximate the value, yes, one way to do it is via a
series of lines but you don't -have- to do it that way. Just because
you -see- only seven planets doesn't mean there aren't more.

Thus we
approximate a circle to a line. We do not approximate a curves value,
however, as we have no value to approximate to. Same goes for anything
that is continuous, like space, for example.

So curves are inscrutable. OK, yeah, they're hard to figure out.

Mitch

.

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