Re: Ruler and Compass in Mathematics

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.

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

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