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What is the meaning of axioms for physics?
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The axiomatic tradition started with David Hilbert, who wrote in 1924
the first (and very influential) textbook on mathematical physics.
http://en.wikipedia.org/wiki/Methods_of_Mathematical_Physics
He defined in his famous 1900 address in the context of the sixth
problem what an axiomatization of physics should mean:
''6. Mathematical Treatment of the Axioms of Physics.
The investigations on the foundations of geometry suggest the problem:
To treat in the same manner, by means of axioms, those physical
sciences in which already today mathematics plays an important part;
in the first rank are the theory of probabilities and mechanics.''
(http://en.wikipedia.org/wiki/Hilbert's_sixth_problem )
This is a quest for giving axioms for probability (done by Kolmogorov
in 1933) and mechanics (at that time only classical mechanics, but
today it means quantum mechanics) that share the clarity and
interpretation independence of the axioms for geometry.
So let us consider a simple example, the axiomatization of projective
planes (assuming that we know already what sets are):
The points form a set P.
The lines form a set L.
There is an incidence relation I subset P x L.
For ''(x,l) in I'' we also say ''x in l'' or ''l contains x''.
Any two distinct points are in a unique line.
Any two distinct lines contain a unique point.
That's all.
The axioms say everything needed to work with projective planes -
although no explanation is given of the meaning of the concepts of
''point'', ''line'', ''incidence''.
The latter is not part of the _axioms_ but part of their
_interpretation_ in real life. And indeed, here all the philosophical
problems appear....
The purpose of an axiom system is precisely to separate the stuff that
is problematic but peripheral from the stuff that is essential and
allows rational deductions. This enables getting universal agreement
about the essence of a theory, without being entanglesd with
controversies about their interpretation.
The axioms for quantum mechanics given in the entry ''Postulates for
the formal core of quantum mechanics'' of this FAQ give such a clear
separation. (But the entry also explains some additional things about
their interpretation.)
Axioms specify in unambiguous terms all properties that are ascribed
to the concepts used, while interpretation rules tell informally
(and in detail often debatably) how these concepts are applied as
models of the real world. Thus the axioms precisely define what the
theory is about, and the interpretation rules uses the concepts defined
by the theory to loosely define how the theory applies to reality.
For example, the interpretation rules for the theory of projective
planes defined by the axioms above are ambiguous and approximate, of
the kind:
-- A point is what has no parts.
-- A point is an object without extension.
-- A point is a mark on paper.
These are already three different, mutually incompatible but common
interpretation rules for the projective point (and doesn't yet
incorporate the interpretation of the points at infinity).
Writing interpretation rules for a projective line is much more
complicated and controversial.
This sort of observations prompted Hilbert to promote the
axiomatization of theories as a means for making the content of a
theory as precise as possible, separating the objective substance
from the controversial philosophy.
Hilbert was a very good physicist - co-discoverer of the laws of
general relativity, creator of the Hilbert space on which all quantum
mechanics today is based, and very productive in using the equations
of physics to extract information tat can be compared with experiment.
It is no accident that today's quantum mechanics is based on Hilbert
spaces rather than wave functions and Born's rule!
In anything more complex than 19th century science, the concepts
(the main ingredient that makes physics differ from Nature) are
_defined_ by the axioms and the subsequent formal theory. They are
then _interpreted_ by rules that usually assume both the concepts and
some social conventions about how experiments are done.
The axioms of physics in the published volume on Hilbert's problems
Mathematical Developments Arising from Hilbert Problems
Proc. Symp. Pure Math.,
Northern Illinois Univ., De Kalb, Ill., 1974,
Amer. Math. Soc., Providence, RI, 1976
are taken to be the Wightman axioms, not as the Born rule.
Note that one cannot separate the mathematics from the physics.
A mathematical theory _is_ a theory of physics once its concepts agree
with those of a branch of physics, and its assumptions and conclusions
can be brought into correspondence with physical reality, no matter
how informal (or even unspoken) the interpretation rules are.
Let me give a concrete example. To define what it means to measure
time, we cannot proceed without first having a definition of the unit
of time in which to make the measurement. The official definition
(found, e.g., at http://physics.nist.gov/cuu/Units/second.html ) is:
''The second is the duration of 9 192 631 770 periods of the radiation
corresponding to the transition between the two hyperfine levels of
the ground state of the cesium 133 atom.''
To be able to make sense of this interpretation rule, one needs to
assume both a lot about the theory of quantum mechanics (the formal
mathematical theory of what can be deduced from axioms that make no
reference to reality) and some additional informal rules that tell
how to measure transitions between two levels, and how to prepare a
cesium 133 atom so that the quantity described can be measured.
To understand the latter, one needs more results from quantum
mechanics of the formal, mathematical kind, and more informal rules
that tell how these results are interpreted in an experiment.
To understand these, one needs.... Etc..
One ends up with a whole book on measurement theory instead of a
simple axiom system. While it is not unreasonable to have such a book,
it _is_ unreasonable to have a book-sized axiom system.
This book would also have to tell how one recognizes a Cesium 133 atom.
The correct answer is: By verifying that it behaves like the
theoretical model of a Cesium 133 atom. (More formal theory....)
This is the only criterion - if an atom does not behave like that, we
conclude with certainty that it is not a Cesium 133 atom.
The situation is here not different from the thermodynamical situation
characterized by H.B. Callen in his famous textbook
H.B. Callen.
Thermodynamics and an introduction to thermostatistics,
2nd. ed. Wiley, New York, 1985.
He writes on p.15: ''Operationally, a system is in an equilibrium
state if its properties are consistently described by thermodynamic
theory.''
This quote can also be found at the end of Section 2 of the article
http://www.polyphys.mat.ethz.ch/education/lec_thermo/callen_article.pdf
In summary, foundations should be concise, unambiguous, and simple.
The only way to get sound, unambiguous foundations of a theory of
physics is to give clear, fully precise axioms for the formal,
mathematical part, then describe its consequences, and finally, with
the conceptual apparatus created by the theory (of course with lots of
hindsight, arrived at through prior, less rigorous stages) to specify
the conditions when it applies to reality in a more informal way, but
still attempting to preserve as much clarity as possible.