## Continuity## Analytical Geometry |
This page is being developed, and is at an early stage.
## Projective Geometry |

Analytical geometry conflates number, size and equality,with geometric elements. Analytical geometry takes lines to be distributed according to ordinates (which are numbers of at least infinitely-dense
sets of pointsin some “co-ordinate frame”,
so that line-types are distinguished equal units) only by their ordinate-styles,
i.e., by their equations. (A is thought of as the distribution of the actual points, “written” as a track
- i.e., graphed - in actual space, as
directed by the ordinate-style).graphThat is to say, they are distinguished numerically,
geometrically, on the (usually tacit) not that to
every point, without exception, in a distribution, there corresponds a number, or
numbers, as an ordinate, or ordinates, all referred to a co-ordinate frame.
assumptionThis , infinitely-dense association of values with points is
deemed to confer
“continuity” on distributions.
assumedA so-called ‘function’, often written in some abstract form such as y = f(x),defines these ordinates, and can provide a selection of them in a list as ordinates for plotting purposes (i.e., "displaying" or graphing the distribution to some extent in actual space). The inverse procedure is usually supposed valid, especially in the sciences; that is, it is supposed that a limited selection of plottable numbers can successfully imply the full function, using methods roughly equivalent to the, “joining of dots”, sometimes found in children's' colouring books. Accordingly, a “straight” line is merely a
set of points distributed by a certain ordinate-style, equation or “function”, and a “curved” line is merely a set
of points distributed by another, different ordinate-style, equation or function.For Analytical geometry,
lines can either be curved or straight.Indeed, a curve is said to be
straight if it has “zero curvature”. For projective geometry, none of this is the case.An account ^{*} of continuity as it is defined by analytical geometry
appears below. Note that its opening sentence is projectively meaningless.
## Rates of ChangeAnalytical geometry is often concerned with – that is, with how rates an ordinate of a distribution rapidly relative to another ordinate in the same distribution – and thenchanges. the continuity of a distribution is entirely crucial for the quantitative determination of this rapidity is first expressed The rate approximately as a ratio of two quantities, each of which is an arithmetical difference between two adjacent corresponding ordinates of the distribution — If of a distribution described by x and _{1}x are adjacent ordinates_{2}, (these ordinates are expected to graph spatially as the end-points of a ‘geometric’ interval), then the corresponding adjacent ordinates of the distribution arey = f(x),y and _{1} = f(x_{1})y_{2} = f(x_{2})and then the changes of these ordinates are, respectively,δ and
x = x_{2} - x_{1},
δy = y_{2} - y_{1}where the symbol is an 'operator' (not a term), that denotes that we are evaluating a δchange, or increment, of its associated ordinate.
Then the rate of change of the with respect to the y-ordinate-ordinate near x is
said to be x_{1}approximately evaluated as a ratio, (‘approximately’, because the ratio represents the rate of change of a δy/δxchord of the graph, not of the curve itself), on the assumption that the approximation will be improved by reducing δx.
We have, abstractly, y = f(x)if we change, it should produce a corresponding change in x. yy + δy = f(x + δx)Subtracting from both sides, to preserve equality, we getyδy = f(x +δx) - f(x).
Dividing both sides (again to keep the equality) by , we obtain the ratioδxδy/δx = (f(x +δx) - f(x)) / δx,and we hope that this ratio does not disappear when, on test, we take towards zero for some particular, non-abstract distribution,δxbut hope that it instead settles on a particular value,when both and δx
actually δybecome zero,for this should be our rate of change,exact[putatively that of the tangent to the curve at ‘point’ ],(x_{1}, y_{1})not an approximation to it represented by a chord.In other words, we hope that tends to a so-called limitδy/δxwith the reduction of .δxBy way of an example, consider.y = f(x) = x^{2}=> δy/δx = ((x + δx)^{2} - x^{2}) / δx=> .δy/δx = 2x + δxAssuming that this distribution is numerically continuous,then, because as 2x clearly will not change dwindles,δx must δy/δx (that is, without drop-out, break or deviation)continuouslyapproach , as 2x approaches zero,δxso, because of this assumed continuity,we also assume that, “in the limit”, δy/δx = dy/dx = 2x, exactlyThus, the numerical continuity of the distribution is clearly crucial:if a value for is y found for a value of not, xthen for that the rate of change cannot be evaluated.xNow, in an important sense, irrational instances of can be said not to exist, because, by definition, ythere are no units available to sum to them. They must represent discontinuities in distributions that feature them – if, for example, some of the distribution happens to be the square root of 2 (which may be the most famous irrational of all!), y could not be plotted at all.yThis should alert us to the possibility that algebras, which manipulate in the first instance, quantitiesnot lines points and planes, may not in fact conflate properly with Geometry, even of the Euclidean sort. |
Projective geometry does conflate number, size and equality,notwith geometric elements. Projective geometric elements, simply because they are elements, are not composed of anything: for example, a line
is not made of points—nor, as to that, is it made of
line. A line is extension-as-such, pure and simple. It is not of
extension: it made extension.isThus a curve is not a line, because a curve is
a plurality; it is compound.. Because it is composite, it . It is too rich.
cannot be a geometric elementIt is rotation of a line
around a point,as that point translates on that line. Projective geometry has no concept of equality, and
hence none of inequality, so cannot handle approximation or arrive at averages: it
is, for example, meaningless to ask which of n lines is the largest, or
least, or the same length, or which line is the average, or median, of
n lines, because lines do not
have length (size) at all.Most importantly, projective
geometry cannot determine
whether one
interval is numerically absolutely the same as, or different, from, another,
when
those intervals are distinct.A purely projective geometric means to render either the rotation
or the translation continuous may not exist.For detailed discussions in respect of this perhaps startling assertion, seeLimits, Absolutes(3), and the Circling Measure pages. |