Can Physics Be Made Fully Axiomatic?

[ehrenfest]

Recently it has been really bothering me that physics is so unaxiomatic. See https://www.physicsforums.com/archive/index.php/t-223771.html. Do you think it is possible to construct a completely axiomatic approach to physics and by physics I really mean the heart of physics i.e. quantum field theory and quantum mechanics?

 

[Moridin]

You can derive much of the laws of physics a priori, if you start with point-of-view invariance.

 

[robphy]

Recently it has been really bothering me that physics is so unaxiomatic. See https://www.physicsforums.com/archive/index.php/t-223771.html. Do you think it is possible to construct a completely axiomatic approach to physics and by physics I really mean the heart of physics i.e. quantum field theory and quantum mechanics?

"complete" might be asking for too much [at least right now, when there are lots of loose threads]. There are formulations of many physical theories that try to start axiomatically. Can you be more specific about what you want?

 

[ehrenfest]

Can you be more specific about what you want?

I want to put the theory of general relativity and quantum field theory into a computer, like http://en.wikipedia.org/wiki/Mizar_system. You're right though that they need to be unified first.

 

[tgt]

Any physical theory doesn't give exact experimental results so there is no point in making them more axiomatic - in the name of physics. Maybe important in the name of mathematics, but not physics.

 

[Poop-Loops]

*sigh* what does "axiomatic" mean?

 

[robphy]

I want to put the theory of general relativity and quantum field theory into a computer, like http://en.wikipedia.org/wiki/Mizar_system. You're right though that they need to be unified first.

You might enjoy this seminar
http://pirsa.org/08030001/
Computation and Physics
Robert Geroch - University of Chicago

 

[ehrenfest]

Any physical theory doesn't give exact experimental results so there is no point in making them more axiomatic - in the name of physics. Maybe important in the name of mathematics, but not physics.

In my opinion, the point of physics is to construct theories that are consistent in two meanings: self-consistent and consistent with experiment. You're right the that making theories more self-consistent will not make them more consistent with experiment but still the first type of consistency is just as important as the second. Improving self-consistency will improve the "total" consistency and thus make our theories better. So, I strongly disagree that physics has nothing to gain by being more axiomatic.

 

[tgt]

In my opinion, the point of physics is to construct theories that are consistent in two meanings: self-consistent and consistent with experiment. You're right the that making theories more self-consistent will not make them more consistent with experiment but still the first type of consistency is just as important as the second. Improving self-consistency will improve the "total" consistency and thus make our theories better. So, I strongly disagree that physics has nothing to gain by being more axiomatic.

Mathematical physicsts share your concerns. But for the genuine physicsts like Einstein and Feynman, not so. Not to mention any experimentalists.

 

[Astronuc]

*sigh* what does "axiomatic" mean?

of or pertaining to the use of axioms.

Axiom (from Merriam-Webster)

1 : a maxim widely accepted on its intrinsic merit
2 : a statement accepted as true as the basis for argument or inference : postulate 1
3 : an established rule or principle or a self-evident truth

Well, Newtons Laws of Motion are axioms.

1. An object will remain at rest or in uniform motion in a straight line unless acted upon by an external force.
http://hyperphysics.phy-astr.gsu.edu/hbase/newt.html#nt1

2. External force is proportional the product of mass and acceleration.
http://hyperphysics.phy-astr.gsu.edu/hbase/newt.html#fma

3. Newton's third law: All forces in the universe occur in equal but oppositely directed pairs. There are no isolated forces; for every external force that acts on an object there is a force of equal magnitude but opposite direction which acts back on the object which exerted that external force.
http://hyperphysics.phy-astr.gsu.edu/hbase/newt.html#nt3

http://www.grc.nasa.gov/WWW/K-12/airplane/Newton.html

http://www-istp.gsfc.nasa.gov/stargaze/SNewton.htm


See conservation laws - http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html

 

[ehrenfest]

Mathematical physicsts share your concerns. But for the genuine physicsts like Einstein and Feynman, not so. Not to mention any experimentalists.

Feynmann and Einstein developed the groundwork of modern physics. When they were alive, the state of physics was not really advanced enough to admit an axiomatic approach. I think that now or in the near future, we will have enough experimental knowledge and mathematical tools to make an axiomatic approach possible. If they were still alive, I think they would work things out more rigorously.

 

[Jimmy Snyder]

Do you think it is possible to construct a completely axiomatic approach to physics?

Do you think it is possible to construct a completely axiomatic approach to mathematics?

 

[ehrenfest]

Do you think it is possible to construct a completely axiomatic approach to mathematics?

Yes.

 

[Moridin]

Yes.

http://mathworld.wolfram.com/GoedelsIncompletenessTheorem.html

 

[ehrenfest]

Um--there is a difference between completely axiomatic and being complete if you read that carefully. Mathematics is almost by definition completely axiomatic. The existence of undecidable propositions is not even related to what this thread is about.

 

[ZapperZ]

Feynmann and Einstein developed the groundwork of modern physics. When they were alive, the state of physics was not really advanced enough to admit an axiomatic approach. I think that now or in the near future, we will have enough experimental knowledge and mathematical tools to make an axiomatic approach possible. If they were still alive, I think they would work things out more rigorously.

But you don't know that for sure. So it seems that your deduction here isn't based on any "axiomatic approach" either.

This thread is rather puzzling. If you buy Popper's assertion that experiments cannot prove a theory, but rather can only falsify it, then there is no way that you can come up with any "axiom" to physics. While we accept conservation laws to be valid, we have no derivation to show that they are always true, which is why we continue to do experiments to look for when such-and-such conservation laws are falsified. Have you ever considered such a thing?

Zz.

 

[tgt]

Feynmann and Einstein developed the groundwork of modern physics. When they were alive, the state of physics was not really advanced enough to admit an axiomatic approach. I think that now or in the near future, we will have enough experimental knowledge and mathematical tools to make an axiomatic approach possible. If they were still alive, I think they would work things out more rigorously.

In Feynman's Lectures, Feynman was always distinguishing or separating mathematicians from physicsts. I think he had a very good reason to do that. If he was still alive, he would be furthering physics. And there is actually more physics now then there was during his time so even less time for rigorous mathematics.

 

[Jimmy Snyder]

Yes.

Mathematics is not completely axiomatic. There are insurmountable problems related to set theory. You can find information about this here: http://en.wikipedia.org/wiki/Foundations_of_mathematics" in the section entitled "Foundational crisis".

I quote a small excerpt:

In a sense, the crisis has not been resolved, but faded away: most mathematicians either do not work from axiomatic systems, or if they do, do not doubt the consistency of ZFC, generally their preferred axiomatic system.

Take your choice, don't work from axiomatic systems, or bury your doubts. Mathematicians call it proof by intimidation.

 

[ehrenfest]

But you don't know that for sure. So it seems that your deduction here isn't based on any "axiomatic approach" either.

This thread is rather puzzling. If you buy Popper's assertion that experiments cannot prove a theory, but rather can only falsify it, then there is no way that you can come up with any "axiom" to physics. While we accept conservation laws to be valid, we have no derivation to show that they are always true, which is why we continue to do experiments to look for when such-and-such conservation laws are falsified. Have you ever considered such a thing?

Zz.

You're right that experiments cannot prove theories, but it does not follow you cannot come up with any axioms. In fact the conservation laws would make good axioms since they form the logical basis for a lot of other theory. I am not saying that we should use some abstract mathematical principle like set theory to "prove" physics! I am saying physics should be based on a strict set of ground rules for itself. I think we should take concepts in physics that are most consistent with experiment and most important and call them axioms and formally derive the rest of physics on top of them. There is probably even some axiom that could precede the conservation laws.

You may think physics has already done this but in that case I challenge you to give me a set of axioms that works. It is harder than you might think because you really should not assume anything in an axiomatic approach. You would have to define things such as mass and spacetime that you normally take for granted. Obviously we would borrow a lot from mathematics but there would still be a lot left.

I will try to make this argument more coherent when school ends in two weeks.

 

[ZapperZ]

I think we should take concepts in physics that are most consistent with experiment and most important and call them axioms and formally derive the rest of physics on top of them. There is probably even some axiom that could precede the conservation laws.

There's nothing to indicate that what you said here is true.

You may think physics has already done this but in that case I challenge you to give me a set of axioms that works.

I do? Where did I indicate that? In fact, if you read my response carefully, I would say that the tone of that response was to contradict that there are sets of axioms in physics, or that there are obvious starting points where everything can be derived. So you are challenging me to producing something that I don't think is valid in the first place.

Furthermore, even IF there are some axioms, or let's call them as "starting point", it doesn't mean that you can do physics that way. Read up on emergent phenomena. Tell me how you can derive superconductivity by starting from the Hamiltonian of every individual electrons in a metal.

Zz.

 

[Jimmy Snyder]

In the popularizations of physics (like The Elegant Universe, by Brian Greene) it is not unusual to read that as of now, that QM as it is understood, is not compatible with GR as it is understood. String theory is an attempt to resolve this, but has not yet succeeded in doing so, has it? If in the pursuit of the question in this thread, we were to come up with an axiomatic system for all of physics (or even a non-axiomatic one) we would get a Nobel prize for it. No?

 

[ehrenfest]

Mathematics is not completely axiomatic. There are insurmountable problems related to set theory. You can find information about this here: http://en.wikipedia.org/wiki/Foundations_of_mathematics" in the section entitled "Foundational crisis".

I quote a small excerpt:

Take your choice, don't work from axiomatic systems, or bury your doubts. Mathematicians call it proof by intimidation.

"In a sense, the crisis has not been resolved, but faded away: most mathematicians either do not work from axiomatic systems, or if they do, do not doubt the consistency of ZFC, generally their preferred axiomatic system."

Please read the emboldened part again. It basically says that axiomatic systems are possible.

 

[ehrenfest]

There's nothing to indicate that what you said here is true.
Tell me how you can derive superconductivity by starting from the Hamiltonian of every individual electrons in a metal.

Zz.

You would make approximations using statistics and limits just as we do now. However, then you would know that what you start out with is not an approximation itself but something logically firm.

There's nothing to indicate that what you said here is true.

There is nothing to indicate it is not true. I said it would be a challenge to do this. But I also have given reasons why it would be worth it to try.

 

[ehrenfest]

In the popularizations of physics (like The Elegant Universe, by Brian Greene) it is not unusual to read that as of now, that QM as it is understood, is not compatible with GR as it is understood. String theory is an attempt to resolve this, but has not yet succeeded in doing so, has it? If in the pursuit of the question in this thread, we were to come up with an axiomatic system for all of physics (or even a non-axiomatic one) we would get a Nobel prize for it. No?

Yes. That is why I want to go into string theory. Remember when you were helping me learn Zwiebach and I kept asking about why some commutation relations were true and you finally said that Zwiebach had not proved them but was just assuming them? I think he should say things like that explicitly: "We must add this commutation relation to our list of axioms" rather then just throwing it in and not really saying how it fits in logically. That drove me insane for about a month and that could have been easily avoided if Zwiebach had just been more formal about it.

 

[Jimmy Snyder]

Please read the emboldened part again. It basically says that axiomatic systems are possible.

It says (not basically says) "they do not doubt the consistency of ZFC."
Read the entire section and you will see that they need to doubt, they just don't. The problem at the foundations of mathematics is ignored, not resolved.

 

[ZapperZ]

You would make approximations using statistics and limits just as we do now. However, then you would know that what you start out with is not an approximation itself but something logically firm.

How did you go from "make approximations" to "is not an approximation itself but something logically firm"?

Zz.

 

[ehrenfest]

How did you go from "make approximations" to "is not an approximation itself but something logically firm"?

Zz.

I have only study superconductivity a little bit so this is just a general description of how a system of axioms might help you study macroscopic things.

Without a system of axioms, we would have to choose between possibly competing elementary principles if we want to study a large body as a collection of elementary particles. We would take limits and Taylor expansions based on whatever principle we chose and get a result about the large body. However, the result would be in doubt even if the approximations are correct because we may have chosen the "wrong" basic principle or adopting the "wrong" way of looking at the elementary particles.

With a system of axioms, we would do the same things except this time we would be more confident in our results. In a consistent axiomatic system, the results we obtain would not depend on the first principles we chose if we do the calculations correctly. That is a major benefit.

 

[Jimmy Snyder]

I hope this is not copyright violation. If it is, please delete it. I quote from the book "Topics in Algebra", by I. N. Herstein, first edition, second printing, page 2.

1. Set Theory. We shall not attempt a formal definition of a set nor shall we try to lay the groundwork for an axiomatic theory of sets.

You will find a statement like that in almost any introduction to abstract algebra, general topology, or analysis. The rest of the book is quite axiomatic, but when it comes to sets, the thing that is being studied, all is swept under the rug. Mathematicians close their eyes to the problem. But they do it with their eyes open and you should do the same.

 

[ZapperZ]

With a system of axioms, we would do the same things except this time we would be more confident in our results. In a consistent axiomatic system, the results we obtain would not depend on the first principles we chose if we do the calculations correctly. That is a major benefit.

I have no idea what you just said here, or maybe I do, but this really is nothing more than "empty promises". I'd like to see you do this rather than just saying that it can be done, which really is what you have been saying in this thread but without anything to back your claim.

Zz.

 

[ehrenfest]

I have no idea what you just said here, or maybe I do, but this really is nothing more than "empty promises". I'd like to see you do this rather than just saying that it can be done, which really is what you have been saying in this thread but without anything to back your claim.

Zz.

OK Let's continue the discussion in about 5 years.