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Lester Zick wrote:
> herb z wrote:
> >Lester Zick wrote:
> >>
[...]
> >Part of the problem here, no doubt, is that I am not grokking you in
> >fullness with regard to "mechanical".
>
> Perhaps not but on general note I'd also like to comment that my basic
> idea of mechanics is one in which an exhaustively mechanical system
> not only specifies properties for the subject at hand but also
> specifies in terms of the same properties what is not the subject at
> hand. In other words let's suppose a system specifies its subject as A
> and that A has certain derivative properties such as B, C, D, etc. so
> we can analyze A in terms of its properties in various combinations.
> However then a mechanically exhaustive version of A would also have to
> specify what its subject is not but not just by saying "not A" but
> showing what "not A" amounts to through the operation of B, C, D, etc.
>
> This is a general epistemological consequence of exhaustively
> mechanical systems of all kinds and is not peculiar to math or
> physics. The idea of a generally mechanical system is one in which all
> definitions, properties, and predicates are reciprocally integrated
> with all others in terms of cause and effect and has nothing to do
> with whether the subjects studied are physical interactions or math.
[...]
> It's all just one hugely complex
> problem of sorting out all possible predicates in mechanical relation
> to one another without arbitrary assumptions that one predicate or
> "model" such as "circle" is mathematical but another such as "mass" is
> "physical".They're all abstractions and the question is not whether
> such and such predicate or property is mathematical or physical but
> whether and how they all fit together in strict mechanical terms. It's
> a problem I like to describe as a "finite tautological regression to
> self contradictory alternatives"
Well, this sheds a lot of light on what you mean by "mechanical".
It seems that a lot of what you wish to do is to make your
initial definitions, and hence the derived definitions, all have
the character of being what is in logic traditionally called
necessary truths.
> >> I think it
> >> would instead be preferable to define "infinitesimals" as those things
> >> having a definite integral according to the rules for the calculus.
Not sure I'm getting it. For example, the definite integral of 1
from 0 to 10 is 10. Where's the "infinitesmal" here?
> >As you may know, beside the classical Newtonian (or Riemannian) notion of
> >integration, there are other kinds of integration that might be more
> >accurate reflections of your intuitions. See
> >
> > http://en.wikipedia.org/wiki/Integral#Definitions_of_the_integral
> >
> >Also, the idea of infinitesmals has been rehabilitated in the non-standard
> >analysis of Abraham Robinson. See
> >
> > en.wikipedia.org/wiki/Non-standard_analysis">http://en.wikipedia.org/wiki/Non-standard_analysis
>
> Well as things stand I'm pretty well satisfied with Newtonian
> infinitesimals and other forms of infinitesimals which satisfy
> Newtonian rules for definite integration.
Well, you might not want to dismiss these other notions so quickly.
It might be worth your time to take a couple of hours and browse
around a bit. It couldn't hurt, might help.
> >> ... so we're left with the
> >> issue of how to define curves in terms of straight line segments, that
> >> is unless we just assume curves as the limits of approximations which
> >> they in fact are but those limits aren't congruent with straight lines
> >> even though their approximations are
> >
> >Right.
> >
> >> and they [curves?] aren't defined exactly
> >> between points [?] which is why there is no real number line [?].
> >
> >Not sure what is meant here.
>
> Well many if not most students of math believe there is a real number
> line. However the limits for curves cannot lie together with straight
> line segments defined between points even though their approximations
> do. Approximations for curves do lie together with irrationals and
> rationals on a straight line but they don't specify where on straight
> lines transcendentals exist but only how close transcendentals come to
> any straight line segment. Even Bob Kolker conceded that there is no
> "real number line" in formal terms.
Well the part above my comment "right" makes sense to me, but none
of the rest does. Does it matter a lot at this stage of discussion?
We've agreed to define curves as velocity with transverse acceleration,
and forget about point-sets, so does it really matter?
> >> A couple of the more interesting implications are that angular
> >> momentum for bodies in circular rotation at constant velocity is
> >> actually not constant but constantly changing instead
> >
> >The velocity of a body following a curved path is certainly not constant,
> >as the directional component of velocity is constantly changing. I'll
> >have to go look at my physics book again, though.
>
> Well if you're seriously agreeing there is a centripetally directed
> finite velocity in circular rotation you'll find very few agree with
> you.
Having consulted my book, I would say that, as I thought, for a body
following a circular path, at any instant there is an an angular
acceleration that can be resolved into component a_tan tangential
to the circular path, and a component a_rad centripetally directed.
In particular, if the speed (not velocity) of the body is constant,
then a_tan = 0 and a = a_rad.
Since the component a_rad is an acceleration, it would seem there
must be a velocity v_rad involved, but I'm not sure what that would
be, offhand.
--
hz
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