The orthodox limit formalism is a challenge-response protocol in which any positive tolerance in one thing must be shown to be attainable by imposing some (typically other) positive tolerance on another; a function is continuous at a specific input if, for any challenge tolerance on changes in the output, all inputs within a given response tolerance of the specific input give outputs within the challenge tolerance of the specific input's output. Likewise, a series converges if, for some given tolerance on variation between its entries, there is some point in the sequence beyond which entries differ by at most the given tolerance; and a function is differentiable at some given input, with a specified derivative, if every positive challenge tolerance – on how close to the specified derivative the gradients of chords of the function must be – can be met by limiting the chords to be between points within some response tolerance of the given input.
Reasoning about such challenge-response protocols typically
requires use of reductio ad absurdum arguments, which I dislike; so I
try to keep my use of them to a minimum. This area of my writings may be
considered an implementation detail (that I dislike and wish I could replace)
for things used by other areas; and shall likely grow side-discussions of how
much I can get away with using more constructive methods of reasoning. In
particular, for continuity and differentiation, I have a hope that one can
replace the for every challenge there exists a response
formulation
(which leads to arguments invoking the existence of a response without
necessarily showing how to construct one) with some here is a polynomial
bound on the response that suffices to meet the challenge
; I'm not sure how
constructive the result shall be, but hope that it may at least point the way to
developing some constructive mechanism that suffices for all the cases I care
about.
Existing fragments:
I think of things for which orthodoxy uses the limit machinery as being
cases of what I term asymptotic truth
, applicable to
any continuum context. By this I mean that some condition is almost true
in some quantifiable way that can be made arbitrarily close to true
(for
the same way of quantifying it) by narrowing attention to all sufficiently
close
approximations to some boundary case, in which the condition's truth
might be either fatuous (so uninformative, where the asymptotic truth says
something meaningful) or inapplicable (making the asymptotic truth as good a
surrogate for it as we can hope for). A common boundary case is everywhere
sufficiencly close to
somewhere (the limit at a point, or near a line or
surface); but another that does coem up is everywhere sufficiently far
from
somewhere (a limit at infinity). Other boundary cases are possible,
but these are the two that most commonly get attention.
What I want from the limit formalism is really some way to reason about such
asymptotic truth, ideally without being too firmly tied down to the details of
how it is specified. It should be possible to reason about, for example, a
smooth manifold with Minkovskian metric having, in a neighbourhood of any point,
a family of covector fields, that is a basis at every point in the neighbourhood
and asymptotically diagonalises (to the usual Minkowskian standard form) our
metric near that point, with various strengths of how close to diagonal it gets
as we approach the point. The coarsest case merely requires the co-ordinates to
be diagonal at the point and continuous near it; but we may also be interested
in the departure from diagonal form rising no faster than some power of
departure from the point. Having differentiable co-ordinates at the point
should hopefully correspond to (at most) linear growth; having derivative zero
at the point and continuous near it should likewise correspond to quadratic
growth. Reasoning about such growth should not be tied down to a particular way
of reducing differences between quadratic forms to a positive real (or zero when
the forms are equal), or of reducing differences of position to a positive real,
in order to show that any requested upper bound on the former can be attained by
imposing a suitable upper bound on the latter. Orthodoxy answers that need by
using abstract topology, which abstracts away the positive real measurements
behond the concept of open neighbourhoods
, but gets somewhat problematic
to reason about when those can be pathologically weird in form.
My approach to this thus far has been to, at least in vector spaces over the
reals, effectively use voluminous
(i.e. non-degenerate) simplicces. The
nature of simplices gives a natural meaning to their interior
for which
the interior of any voluminous simplex does indeed match the interior in the
orthodox topological sense; while the simplices (including boundary) are
conveniently compact, in the orthodox topological sense, so this is merely a
specialisation of the orthodox approach; but it's one I feel more comfortable
thinking about, even though I still haven't escaped from reductio. I
have at least been able to recast the challenge-response protocol in terms of
taking intersctions of simplices satisfying particular conditions.
For example, for my Minkowskian local basis, the co-ordinates
of the metric with respect to that basis lie in some space of lists of lists of
reals, within which I can define simplices; in the smooth manifold, I have
charts that map patches of the manifold to real vector spaces, within which I
have simplices. So I can take any simplex within the vector space image of a
chart, pull it back to the manifold and, if my point of interest lies within the
result, take the collection of co-ordinates of the metric, with respect to the
given local basis, and take an interest in every simplex, in my space of lists
of lists of reals, that subsumes that collection of co-ordinates. The
intersection of all such simplices in co-ordinate space is just the collection
whose single member whose first diagonal entry is 1, the rest being −1,
since the co-ordinates are continuous and that's their value at the point of
interest; I can presumably concoct some similar construction to specify stronger
conditions of being almost diagonal
in a neighbourhood. For example,
back to the simplex within a chart's vector space, in whose interior the chart's
image of the point of interest lies, there is a natural scaling towards the
point's image, by positive scale factors, that always has the point's image in
the scaled simplex's interior; when the factor is less than 1, the scaled image
lies within the original simplex. For each scaling of the simplex, we get a set
of co-ordinates of the metric within the thus-scaled simplex's pull-back to the
manifold; so we can limit our interest in simplices in co-ordinate space to
those which, when scaled towards the standard Minkowskian diagonal matrix, by
the square of any factor t < 1, contains all co-ordinates of the metric
within the chart's pull-back to the manifold of its simplex scaled by factor t
towards the image of the point of interest. If such simplices in co-ordinate
space (exist and) all intersect in a collection whose sole member is the
standard matrix, we've got a suitable sense of the local basis making the
metric's co-ordinates quadratically close
to diagonal near
the
point.
This whole topic is part of the branch of mathematics
called analysis
, which includes topology
; all of which is mighty
interesting, but raddled with reductio. So I haven't written much about
it, but others have. One example is (the later portion of) David Morgan-Marr's
annotation
for one of
Bilbert's puns.