arguments (inferences) in physics

[Istvan Nemeti]

Dear All,

> In axiomatic theories such as thermodynamics or relativity, a
deductive
> step consists in obtaining a conclusion from a number of premises.
> Logicians call the respective procedure argument, inference etc.
Authors
> of axiomatic theories claim that they have strarted from a small
number
> of axioms (in the case of relativity only two axioms) and then have,
> step by step, obtained breathtaking results. Yet these authors have
> never found it suitable to put the deductive steps on a list so that
> critics can check their validity, starting with the steps close to the
> axioms and finishing with those producing the breathtaking
conclusions.
...
> Pentcho Valev

We find this a useful question. A complete positive answer can be given
to this question as follows. There are works in the literature,
motivated by exactly this question, which study/reconstruct e.g.
relativity theory in the rigorous framework of mathematical logic, using
the standard deductive rules of classical logic, in a thorough manner.

Such works can be found on the internet address
http://www.math-inst.hu/pub/algebraic-logic/Contents.html (especially
the works of Andreka, Madarasz, Nemeti, Szekely therein). In the papers
and books displayed in this address we take variants and versions/parts
of relativity theory, formalize (and polish) its axioms in classical
first-order logic and then prove (i) that the so obtained axiom system
is consistent and (ii) prove the main predictions of the theory from
these axioms via using the standard deductive rules of classical logic. 

Let e.g. Th1  be the usual Einsteinean version of the kinematics of
special relativity. We postulate 5 simple first-order logic axioms the
meaning of each of which is clear and logically transparent. Then we
prove that these 5 axioms are consistent and after that we prove the
usual predictions of Th1 from our 5 axioms in an explicit, checkable
way. 

In another paper we start out from a richer theory Th2 which is an
extension of special relativity with accelerated observers. Following
Einstein's Equivalence principle we use acceleration for simulating
gravity. Then we proceed as in the case of Th1 above, but now we can
prove some effects of gravity e.g. on clocks proving the so called Tower
Paradox claiming that "gravity causes time to run slow". A similar
approach is applied to other parts of relativity theory (besides Th1,
Th2 mentioned above).

This area (where axioms are carefully polished, proofs are laid out in a
logically rigorous manner, consistency etc are studied) is known as
"logical analysis of relativity theories" or "logical analysis of
space-time" or "logical foundation of space-time". A survey of the
literature of this research direction is presented in
http://www.math-inst.hu/pub/algebraic-logic/lstsamples.pdf (section 5).

A possible sequence of reading the relevant material on
http://www.math-inst.hu/pub/algebraic-logic/Contents.html is: (1)
lstsamples.pdf, then (2) loc-mnt02.pdf, then as a background material
for details of proofs (3) olsort.html.

Istvan Nemeti

  

 

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