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If we construct axiomatic system of physical laws that are independent one another as in axioms in mathematics, what should they be? Can there be such a finite system of physical laws that can explain every physical phenomenon? Or is it impossible to have such finite axiomatic system in physical laws?

Qmechanic
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Henry
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  • Possible duplicates: http://math.stackexchange.com/q/549839/11127 (now migrated to http://physics.stackexchange.com/q/87239/2451), http://physics.stackexchange.com/q/44196/2451 and links therein. – Qmechanic Nov 07 '13 at 12:17
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    The system of standard foundational axioms investigated in mathematics isn't finite in a practical sense. Even KP set theory, which is more on the computable side (more physical, if you will), involves axiom schemata, e.g. replacement. – Nikolaj-K Nov 07 '13 at 12:40
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    Mathematics is a creation of human mind. In a physical theory we try to recreate something using our own language mathematics, its the only language we know. It doesn't mean that mathematics is the foundation of nature. Physics need not be rigorous mathematically. – Self-Made Man Nov 07 '13 at 13:01
  • Could you point your opinion clearly? @NickKidman – Self-Made Man Nov 07 '13 at 13:09
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    The question of "axiomatization of physics" was also posed as Hilbert's sixth problem, see e.g. http://en.wikipedia.org/wiki/Hilbert%27s_sixth_problem – Martin Nov 07 '13 at 13:11
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    @KaziarafatAhmed: Opinion about what? OP wrote "such a finite system", in comparing the physical axioms he searches for with base mathematical axoms. And I pointed out that these are, in fact, not particularly finite. – Nikolaj-K Nov 07 '13 at 13:45
  • Yes, it's called mathematical physics. e.g. wightman, and so on. – Abhimanyu Pallavi Sudhir Nov 07 '13 at 13:57

3 Answers3

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Yes it is possible. As J. Bell eloquently wrote, Quantum Mechanics, together with a finite cut-off QED, explains all of chemistry and nearly everything in Physics. It was axiomatised by Weyl and Dirac by 1930.

There are only six axioms, which is certainly a finite number. Five would be better still..since most physicists no longer believe in the literal truth of the sixth axiom.

There are notorious problems with this axiomatisation, but they can certainly be fixed, although physicsists are not in agreement on how to fix them. The problem was analysed most logically by Wigner and, later, by J.S. Bell, in his "Against Measurement", I have posted a copyright-free copy at http://www.chicuadro.es/BellAgainstMeasurement.pdf. That is, the first three axioms apply to all physical systems, the second three axioms apply only to measurments, but surely measurements are carried out by measurement apparatuses which are physical....unfortunately the answers given by the first three axioms applied to the interaction of a microscopic system with a measurement apparatus are different from the results given by applying the second three axioms to the same physical setup. Not contradictory, but so different that there has been no agreement on how to compare them.

Most physicists now feel that the measurement axioms are only approximations, and ought to be derivable from the first three axioms as approximations. H.S. Green, under (I think) Schroedinger's influence at Dublin, published an extremely important paper analysing the physics of the measurement process as a phase transition, and there has been more recent work as well. See my own http://arxiv.org/abs/quant-ph/0507017, for example.

The only remaining difficulty is to either define the concept of ''probability'' as it occurs in these axioms, or to formulate a few more axioms to connect it with the other axioms. For the quantum case this was done in the paper referred to, and something similar can be done in the Classical Case.

joseph f. johnson
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This is an experimentalist's answer:

I do believe that an axiomatic model , note "model", of nature can be found, but as an experimentalist I am wary of claims that "we have now wrapped up physics and only details have to be mopped up" which was the claim before quantum mechanics rocked the science in the beginning of the twentieth century.

One should be open to the possibility that as we delve further and further into experiments with new technologies, and understand more and more of the cosmos, the axioms might have to be changed. Otherwise physics will become fossilized.

anna v
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Not a complete one.

Kurt Gödel proved this was not possible by proving his "Incompleteness theorem". It turns out that in any axiomatic system (whether or not these axioms were to do with physical laws) we must select either consistency, or completeness, but not both.

Basically the "Incompleteness theorem" says that any 'computable axiomatic system' will have the following properties (such as one containing physical laws):

  1. If the system is complete it cannot be consistent.
  2. The consistency of the axioms cannot be proven within the system

As a corollary of 1, any axiomatic system that is consistent cannot be complete (the system you describe would hopefully be consistent). So if you want your physical laws to be internally consistent you have to accept that there will be true (observable) physical laws that cannot be proven.

With respect to infinite systems, there are two types, countable and uncountable. The Incompleteness theorem has also been proven true for countable infinite sets. In the case of infinite axiomatic systems dealing with physical laws, they would be countable since each axiomatic law could map into the set of natural numbers. Even here Gödel's conclusions hold; either this system would be consistent but incomplete, or complete but inconsistent.

Apparently, we cannot get around the fact there are unprovable truths. Gödel provided a simple example.

Let S be the statement "This statement is unprovable."

If S is true, we cannot prove it since it is unprovable. However, if we can prove S true, the statement is self contradicting, so inconsistent.

Notice from the answer above J. Bell eloquent quote, "Quantum Mechanics, together with a finite cut-off QED, explains all of chemistry and NEARLY EVERYTHING in Physics." Unfortunately for Bell, Gödel has shown that as long as Quantum Mechanics seeks to be internally consistent, it will only ever be "NEARLY EVERYTHING" and not actually "EVERYTHING". If Quantum mechanics does actually achieve the ability to explain everything, Gödel shows us we have good cause to look for its self-contradictions (inconsistencies)

user34445
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  • with all due respect to Godel (I know him from set theory, "the set of all sets is open") the "this statement is unprovable" goes back thousands of years to the Cretan paradox : "A Cretan said all Cretans are liars".http://en.wikipedia.org/wiki/Epimenides_paradox . Paradoxes in this day and age are resolved by meta levels, and I do not think Godel's theorem can be resolved by a meta level. – anna v Nov 24 '13 at 14:35
  • Could be. The Apostle Paul was also apparently aware of that quote [Titus 1:12] – user34445 Nov 24 '13 at 14:44
  • I should also add that no one's proven the Incompleteness theorem cannot be solved by a meta level. Notwithstanding that, the mere fact that it it appears this way suggest that the theorem itself may not be a paradox but a true property of axiomatic systems. – user34445 Nov 24 '13 at 14:52
  • @Joseph f.johnson - no one argued Maths was inconsistent. It is consistent. However it is not complete as per Gödel. Gödel's theorem has 2 possible types of systems; consistent but incomplete ones, or complete but inconsistent ones. Since Maths continues to have new axioms added to it all the time, it is clearly incomplete, and so can also be consistent. It appears you do not understand Gödel's theorem. – user34445 Nov 24 '13 at 15:28
  • Ok, with respect to Gödel's theorem and its relationship to QM the question reframed is what constitutes a computable system. Briefly, a computable system can be represented as a finite automata (operators and operands). QM can be represented as a finite automata and so is a computable system. (In order to do QM calculations one needs to employ grammar and syntax, operators and operands clearly. So QM is axiomatic and computable, so the Incompleteness theorem applies. – user34445 Nov 24 '13 at 15:33
  • @Joseph f.johnson, did you read the question?He asked if an axiomatic system could be constructed that could "explain every type of phenomenon. The answer provided denied only that such a system would be complete. – user34445 Nov 24 '13 at 16:04
  • Gentzen's consistency proof deals only with a simplified subset of maths, specifically the part of Maths that cannot be proved. Gentzen’s proof, for example, also uses transfinite induction which is certainly not using arithmetic itself (as per Gödel's claim no system could prove itself), and was not regarded as persuasive by Tarski Angus Macintyre (“Mathematical significance of proof theory”, Phil. Trans. Royal. Soc. A 363 (2005), 2419–2435, p. 2426). Besides (says MacIntyre) in Gentzen’s work, consistency was not really the main issue at all. – user34445 Nov 25 '13 at 17:22
  • So what you're doing here is setting up a straw-man argument about my use of Gödel and arguing against it. What Gödel proved was that if formalized arithmetic is consistent, a particular numerical encoding of that fact is expressible, but not provable, within that theory itself. His theorem did not rule out persuasive proofs of the consistency of arithmetic that employ means not formalizable within arithmetic.With respect to Gentzen's contribution, the consistency of arithmetic is still not provable from within arithmetic itself. Your criticism do not fairly or objectively recognize this. – user34445 Nov 25 '13 at 17:24
  • I should also add that Gentzen's work is generally seen to be a proof of the Gödel's theorem. By providing a direct proof of the unprovability of the principle of transfinite induction, used in his 1936 proof of consistency, Gentzen provided Gödel with the means of discovering an unprovable formula of arithmetic showing arithmetic to be incomplete. Gödel took Gentzen's proof and used a coding procedure to construct an unprovable arithmetic formula. – user34445 Nov 25 '13 at 17:42
  • I do not understand what you mean by saying «QM ... is a computable system». Do you mean the same thing as prof. Shor, in his answer http://physics.stackexchange.com/a/73366/6432 to a related question, i.e., that every physical phenomenon can be computed to arbitrary precision? (By simulating the relevant physical model). – joseph f. johnson Nov 26 '13 at 02:53
  • Sorry, I should have defined that more clearly. By 'computable system' I'm using the Church–Turing thesis which states that any function that is computable by an algorithm is a computable function. Therefore any system consisting of computable functions is a computable system. Although initially skeptical Gödel argued in favour of the Church-Turing thesis by about 1946! (to see how QM qualifies as a computable system see Werner Depauli-Schimanovich, Eckehart Köhler and Friedrich Stadler's 'Computability in Quantum Mechanics'. Basically, I mean it the same way Gödel (and Turing & Church) did. – user34445 Nov 26 '13 at 04:16
  • I assume that you would agree that there does not seem to be any regular predictions of quantum mechanics which are not computable, given computable data? – user34445 Nov 26 '13 at 04:24
  • There is a subtle difference between computable as used in maths, and computable as used in Physics. If I understand the difference, it is that in maths, the same algorithm must be capable of giving the answer to any desired degree of approximation. But in Physics, we do not care whether a different algorithm would be needed for each different case. This is like the difference between omega-consistency and consistency. There might not be an algorithm within the system for producing the new and different algorithm needed for each case. So, I am not sure how to answer your question. – joseph f. johnson Nov 28 '13 at 00:47
  • In a properly formulated Theory of Everything, questions like «Do I exist?» should be impossible to formulate. IN fact, the pronoun «I» should have no translation into Physics, nor should the verb «exist». (Kant: existence is not a predicate.) It's not an accident that their use in textbooks is rather rare. – joseph f. johnson Nov 28 '13 at 00:49
  • Your clarification still leaves me in the dark as to your meaning. An axiom is not a function. An axiomatic system consists of primitive concepts and axioms. The axioms are certainly not functions. The primitive concepts might be. But in, say, Hilbert's axiomatisation of Euclidean Geometry, I am not sure I remember any functions. And, to go to the opposit extreme, Principia Mathematica has functions but surely they are all computable? negation, etc.? so what this has to do with consistency is still a mystery to me. I'd have thought computable systems are simpler than uncomputable ones.... – joseph f. johnson Nov 28 '13 at 00:57
  • In the three fundamental axioms of QM, the only function that occurs is the exponential of the Hamiltonian. If the Hilbert Space is finite dimensional, then this is obviously computable in either sense. So, for the sake of discussion, let us assume that infinite dimensions are not a problem. In other comments and answers you have been clear that you think it is necessary to include the axioms of logic, as well, so you throw in PM, all the functions are still computable... and now what? – joseph f. johnson Nov 28 '13 at 01:26
  • It's not clear what you mean by 'an axiom is not a function'. Also we're not talking specifically about QM but about the axiomatic system that underpins it. QM must presuppose certain things to be able to work, and it is 'those certain things' that show up in its axioms. Axioms can be derived (i.e. also conclusions) but axioms are not only the axioms of logic. They are specific to the formal system being used. Mathematics has axioms (i.e. 1+0=1 is always true, you can't divide by 0, etc). The axioms of physics are nearly always mathematical, nearly always equations (F=MA). What is 'PM'? – user34445 Nov 28 '13 at 03:18
  • "It turns out that in any axiomatic system (whether or not these axioms were to do with physical laws) we must select either consistency, or completeness, but not both." The real closed field is complete and consistent so either it's not an "axiomatic system" or what you say is wrong. – jinawee May 16 '16 at 15:18