Newton's Principia and Inverse-Square Orbits in a Resisting
Medium: A Spiral of Twisted Logic.
Historia Mathematica, Vol. 25 (1998), No. 3, 281-289
by Robert Weinstock, I have some further remarks. In the discussion below I
have simplified the situation, but I have not (I hope) thrown away anything
that impinges on the conclusions.
Weinstock's critcism of Erlichson's interpretation of Newton is based on
the following logic. He (Weinstock) claims that one cannot make the
inference
[E => A] -> [A => E]
where the -> symbol is to be read "implies, in at least one circumstance".
This is what Erlichson does, since he notes that Newton assumes an
equiangular spiral orbit and concludes that the density of the medium
varies like 1/r. In other words, Erlichson says, an equiangular spiral is a
possible orbit in a medium whose density varies like 1/r.
Weinstock shows that the inference cannot be made by constructing an
example for which E => A, but for which A never implies E. Weinstock's
example is over elaborate. A simpler version containing the same essential
features is
E = [x is real and x^2=-1]
A = [x=1 or x=-1]
We can show that E => A as follows:
[x is real and x^2=-1] => [x is real and x^4=1] => [x=1 or x=-1],
but of course A doesn't imply E.
This example (in which E is always false) is, however, not relevant to
Erlichson's observations since in his case E is the statement that the
particle follows an equiangular spiral, something which clearly is a
physical possibility. Erlichson's remarks would be better interpreted as
being of the form
[E => A and E is possible] -> [A => E]
and this inference is (I think, I am no logician) correct. Even though
Erlichson doesn't say (according to Weinstock) that E is possible, it is
certainly implicit in what he writes.
In short, I think that Weinstock's critcism of Erlichson is misplaced.
Chris Linton
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* Department of Mathematical Sciences * Fax: 01509-223969 *
* Loughborough University * E-mail: *
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