[FOM] Re: On the deductive inconsistency of a fundamental physical theory

[Istvan Nemeti]

In his FOM posting of 6/10/2004, Peter Apostoli claims that the special
theory of relativity STR is inconsistent as a logical theory. The same claim
is made in the related paper by Peter Apostoli and Akira Kanda.

(1) We believe that the quoted claims add evidence to the importance of
Harvey Friedman's project (joint with ourselves) for elaborating the logical
foundations of SRT and in particular for building up SRT as a theory in the
sense of logic. (There are relevant postings on the FOM list in January and
February 2004.)

(2) In Harvey's postings as well as in several works quoted from e.g.
[1],[2], the kinematic part SRK of SRT has been axiomatized in first-order
logic (FOL) and has been proved consistent. Hence the claims of Apostoli and
Kanda must refer to the remaining part of SRT not included in SRK.

Even in this restricted form, the claims of inconsistency seem to be
unfounded for the reasons below.

(3) The paper of Apostoli and Kanda pretends that a certain equation is an
axiom of SRT and then derives a contradiction by the unrestricted use of the
equation. One of the problems is that the quoted equation is not an axiom of
SRT. Namely, they derive a contradiction from the equation

(*) m = m0/sqrt(1 - v^2/c^2)

where  m  is the mass of a certain particle  b  and  m0  is the rest-mass of
b . Further,  v  is the velocity of  b  as observed by observer  K  who is
also responsible for observing the mass  m.

Now,  (*) is not an axiom of SRT. Instead, the relevant tentative postulate
of SRT is

(**)  m0 = sqrt(1 - v^2/c^2).m   .

The reason why we call (**) only a tentative postulate of SRT is the fact
that SRT distinguishes two kinds of concepts: observational ones (these are
regarded as important) and tehoretical ones. Postulates like (**) discussing
theoretical concepts like  m0  are regarded as conventions, i.e. matters of
convenience. Should a contradiction be derivable (which is not the case)
from a postulate regarding a theoretical concept, then one would simply
remove that convention from the theory without essentially changing the
theory itself.

However, we emphasize that no contradiction can be derived form (**) in
place of (*) by the methods of the paper of Apostoli and Kanda.

The reason why SRT in the sense of the works quoted in [1] uses (**) instead
of (*) is that  m0  is not an observational concept, the observational side
is  m (indeed observed by  K) and the theoretical notion of the rest mass
m0  is *derived* from  m . As the logic of SRT is explained e.g. in the
works [1], [2], we start out from the observational concept  m  as being
primary, and then we show or define how the theoretical concept  m0  is
derivable (or computable) from the primary, observational one,  m.

Summing up,  SRT  has not been proved inconsistent.

All the same, we can derive two useful consequences:

(i) As we said in item (1), these considerations underlie the importance of
Harvey's and related work on logical foundation of relativity. Indeed, if we
do not point our finger at a concrete theory in the sense of logic, i.e. a
theory with a fixed vocabulary and a fixed set of axioms as formulas of some
kind of formal logic, then it is meaningless to debate whether a thing
(which we call a theory) is logically consistent or not.  Further

(ii) If a physics book happens to use the form (*) instead of the correct
(**), then that book (but not SRT in general) is probably flirting with
logical inconsistency. (The point is that if v=c is permitted, then (*) can
involve division with zero. Therefore one has to make it clear that  v is
different from  c  if and when one wants to use the form (*).)

References

[1] Andreka-Madarasz-Nemeti: Logical axiomatizations of space-time. Samples
from the literature.
http://www.math-inst.hu/pub/algebraic-logic/lstsamples.pdf

[2] Andreka-Madarasz-Nemeti: On the logical structure of relativity
theories. http://www.math-inst.hu/pub/algebraic-logic/PartI.pdf