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Can Physics be Axiomized?

  1. Aug 14, 2004 #1
    "Can Physics be Axiomized?"

    The question "Can physics Be axiomized?" is one of Hilbert's 23 unsolved problems : http://www.andrews.edu/~calkins/math/biograph/199899/tophilpr.htm [Broken]

    What are your ideas on this particular one?
    Last edited by a moderator: May 1, 2017
  2. jcsd
  3. Aug 14, 2004 #2


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    1) Relativity and relativistic Em have been axiomatized.
    2) Discrete particle quantum mechanics, including Dirac's relativistic one have been axiomatized.
    3) There exists a set of axioms, the Wightman axioms, for Quantum Field Theory, but AFAIK it has not been successfully applied to gauge field theory, such as Yang-Mills theory, which is the type of theory that supports the Standard Model. Axiomatizing Yang-Mills theory would settle the questions about it for which a Clay prize has been offered.
    4) Thermodynamics is an axiomatic system.
    5) Work goes on to axiomatize string theory and quantum gravity.
    6) We don't have a "final theory" yet, so the question of axiomatizing it is moot.
  4. Aug 16, 2004 #3
    Unlike mathematics physics is concerned also with the truth about natural facts,being guided by empiricism,namely the correspondence theory of truth.Thus physicists seek first the correspondence between their enunciations about the world and the observed facts to consider them objective knowledge.Unfortunately there is no way to begin from a (known with certitude) true set of [premises] and,step by step,to deduce true theories,larger and larger in scope,about the world;there is no way to find some first,true, principles,the dream of Aristotle.This is why science has to reside on some basic assumptions,self evident,accepted without demonstration,in order to assure its self consistency in the most economic way (as indicated by the principle of sufficient reason,the base of human rationality).The big difference from mathematics,since physics account for natural facts,is that,in doing so,we do not consider them less fallible,that is if some new data contradicted them then they would be discarded.

    Apart from those basic assumptions axiomatization in science is possible (as a matter of fact it is very often applied in practice in different sciences) in the form of principles from which,in conjunction with other premises (enunciations accepted as representing objective knowledge,not falsified yet,or even other principles) we can deduce novel predictions,corroborated later,about the world.For example,as others have already pointed out,Relativity rest on such principles,there are many other examples.But again it should be clearly stressed that they are considered fallible,if Relativity in the current form would be falsified then the principles should at least be altered (or perhaps other auxiliary assumptions should be introduced) if not dropped altoghether.Thus we do not deal with a fixed scheme buidling over some [more] basic levels,not even the basic assumptions are exempted from being discarded.

    [PS]Though Godel's incompleteness theorems put serious problems for science (since artihmetics is included in the set of enunciations accepted by science) this does not mean that a 'theory of everything' cannot be complete.Indeed there are variants of arithmetics which are complete and currently there is no reason to think that the complete set of Peano axioms [is] needed to obtain such a theory.
    Last edited: Aug 16, 2004
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