FOM: Analogy between spacetime and formalism?
Jeffrey Ketland
ketland at ketland.fsnet.co.uk
Sun Sep 3 00:18:58 EDT 2000
Dear Vladimir
In describing your formalist position (03 September 2000 02:25: Re: first
order and second order logic: once more), you write:
>There is a good (and of course, not complete) analogy with the
>views on absolute vs. relativistic space-time. Formal systems
>are like coordinate systems with the help of which only we
>can work out (mathematically or rigorously) some approaches
>to the nature or our ideas and intuitions, and between which
>there is a possibility of formal translations / relative
>interpretations.
You've shot yourself in the foot here. In this case, there is always a
manifold M, with an atlas of co-ordinate charts that covers M. The
co-ordinates thus describe something *external to the co-ordinates* - namely
the points in the manifold, their subsets and certain relations (c.f.,
formal systems are descriptions of mathematical structures, which we do not
create). In fact, it's important to distinguish between a point p in M, the
co-ordinate map phi : M -> R^n, and it's value (x1, ..., xn) = phi(p). Given
a pair of (overlapping) co-ordinate maps phi and psi, then the co-ordinate
transformations R^n -> R^n are compositions of the form phi o psi^(-1),
which bounce down to the manifold M and then back up again.
Einstein's (e.g., see the debate between Popper and Einstein in Popper's
"The Open Universe: The Case for Indeterminism", pp. 89-92 - "A Conversation
with Parmenides") and Minkowski's (e.g., see Minkowski paper and tiny quote
below) notion of spacetime is quite absolute.
See,
[1] H. Minkowski 1908: "Space and Time" (in "The Principle of Relativity: A
Collection of Original Papers on the Special and General Theory of
Relativity, by Einstein, Lorentz, Weyl and Minkowski", Notes by A.
Sommerfeld. Dover 1952).
(Minkowski calls his description of space-time "the postulate of the
absolute world" p. 83).
[2] Michael Friedman 1983: "Foundations of Space-Time Theories", Princeton
UP
(especially Chapter 2, Section 3, "Absoluteness and Space-Time Structure",
pp. 62-70).
[3] Robert M. Wald 1984: "General Relativity", Chicago UP
(especially Chapter 1: Introduction).
There is a separate conceptual debate, concerning positions known as
"substantivalism" (roughly, Newton) and "relationism" (roughly, Leibniz),
but I don't think you're referring to that.
Best wishes - Jeff
~~~~~~~~~~~ Jeffrey Ketland ~~~~~~~~~
Dept of Philosophy, University of Nottingham
Nottingham NG7 2RD United Kingdom
Tel: 0115 951 5843
Home: 0115 922 3978
E-mail: jeffrey.ketland at nottingham.ac.uk
Home: ketland at ketland.fsnet.co.uk
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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