🚀 go-pugleaf

On Dec 18, 12:09 pm, Yanick Toutain <yanicktout...@hotmail.com> wrote:
> The earthmoving, for Isaac Newton was indeed a displacement compared
> to the vacuum itself. An absolute displacement.

Newton had no choice but to proceed the way he did, because the ideas
required to fully incorporate Galilean relativity (affine bundles)
would not come into being until Cartan.

The problem he faced is as follows. Galileo's principle of relativity
means that "0" no longer has any objective meaning for velocity. This,
then, would mean that velocities could not be regarded as vectors,
since a vector space has a 0. Rather, it has to be represented as a
point in an *affine* space. For, indeed, that's the very feature that
distinguishes affine spaces from vector spaces -- there is no
distinguished "0" in the former.

But, then, that's the problem. The differences and differentials of
affine points ARE vectors. This makes it impossible to model the
spatial geometry of positions as an affine geometry!

The Galilean principle of relativity, you see, also entails an even
greater relativity for spatial positions -- the relativity of
Genidentity. Genidentity is the term logicians use to refer to the
concept of a point at one time being the "same place" as a point at
another time. Thus, most peoples' notion of genidentity is such that
New York in 2001 is the "same place" as New York in 2008.

But if motion is relative, then the "sameness" vs. "differentness" of
place is too! It would be just as valid, for instance, to say that New
York in 2001 is the "same place" as some place in the middle of outer
space today where the Earth "used to be" before it "moved away" from
that place.

Thus, there is no longer any cohesive concept of a point, in the sense
of a position that endures in time. The very genidentity of points
(the prerequisite to defining spatial geometry) is lost.

This means, among other things, that "point" can no longer be taken as
a primitive in geometry, if you believe Galileo's principle. Rather,
one must go one level deeper and take a point-at-an-instant (or what
we now call an "event") as primitive.

Thus, as an aside, we also see that it was GALILEO and Galileo's
principle of relativity that made the concept of spacetime a
necessity, and a necessary replacement for the lost cohesiveness of
the concept of point; not Minkowski or Einstein or Lorentz; and not
special relativity!

If velocities are, themselves, affine points then that would that make
positions? That was the dilemma Newton faced. While he could answer
the question: "what is the differential of an affine point?" (for,
that's what his bucket experiment was meant to demonstrate: the
differential of the (affine) velocities is a (vector) acceleration),
he could not answer the reverse question: "what is the ANTI-
differential of an affine point?"

And that's because the concepts required to formulate the answer to
that question were not in the vocabulary of anyone prior to the early
20th century and the time of Cartan. The answer, of course, is that
velocities lie in what's known as the "tangent space" of whatever the
underlying geometry is. The geometry whose tangent spaces are affine
spaces is called an affine bundle.

So, Newton (possibly even unbeknownst to himself) did not formulate
the principle of absolute space so much as to assert a new foundation
for physics as he did simply as a convenient expedient to get out of
the jam that he (correctly) saw would arise from taking serious
Galileo's principle of relativity. He asserted the absoluteness of
space, because he had nothing better at the time: he didn't know about
affine bundles (nor spacetime). Had he had these concepts in his
vocabulary at the time, he would have used them instead.

Lack of opportunity, in this case, does not equate to lack of will.