What Does a Formal Proof in Physics Look Like?

[evagelos]

is a proof in physics equal in strength with that in mathematics?

in mathematics we have at one end an ordinary proof and at the other end a formal proof.

how would aformal proof in physics look like,an example would help.

I suppose that the validity of a proof in physics could be checked by an experiment but in the case that we have no experiment what happens??

Thanks

 

[Defennder]

I don't understand your distinction between an ordinary and formal proof in mathematics. In physics, every experiment done to verify a theory is necessarily theory-laden, which means to say the results of the experiment requires interpretation through the lenses of some physical theory. It isn't as simple as what Popper's falsification principle in induction makes it out to be.

 

[evagelos]

lets take in steps:


....can we put a problem in physics into quantifiers form like we can do in mathematics??

 

[DaveC426913]

No.

Physics attempts to describe reality. Our grasp of reality is arbitrarily blurry.

For example, you can never prove that F exactly equals ma because those are all measurements, which have margins of error and measurement. Nor could we ever hope to demonstrate - even in principle - that there isn't some tiny bit of energy leaking into some other dimension.

 

[evagelos]

when you say...no...you must have some arguments to support your
...no

I an sorry but your argument about equality between F and ma is irrelevant to the above.

correct me if iam wrong

 

[DaveC426913]

lets take in steps:
....can we put a problem in physics into quantifiers form like we can do in mathematics??

??

We do. It's called mathematics. Quantification, by definition, means applying numbers to something.

when you say...no...you must have some arguments to support your
...no

I do, otherwise you wouldn't have been able to address it :

I an sorry but your argument about equality between F and ma is irrelevant to the above.

correct me if iam wrong

How is it irrelevant? That is about as definitive and succinct an answer as you can get. And I can't think of any way to elaborate on it that isn't just repeating myself.

Is it possible you were thinking the correct answer was somehow trickier?



Maybe we can root out some of your assumptions and possible misconceptions. I think that may be the source of these odd questions to which you don't like the answers.

Perhaps you could give an example of something that physics seems to have "proven" - or could prove.

 

[granpa]

In math certain things are true by definition. we know they are true. In physics the best we can hope for is to know beyond a reasonable doubt (through experimentation)

 

[HallsofIvy]

Notice, by the way, that all statements in mathematics are of the form "If A then B". We make no claim as to whether "A" or "B" itself is true, only that the implication is valid. That is fundamentally different from physics where all statements must be based on experimental evidence.

 

[evagelos]

where is the "if A then B" form in the statement ; ...no object belongs to the empty set? Written in quantifier form

for all x ~xεΦ where Φ is the empty set.
Another statement which can be proved is:...nothing contains everything

 

[evagelos]

an example of quantification in maths could be the following:

for all a : a>0====> (a^2)>0. a belonging to real Nos

a similar one in physics could be the following:

for all masses between 1 and 5 Kgs and for all forces F there exists a unique acceleration a

such that ...F= ma where F and a are vectors

Another one could be :

for all M masses and for all m masses and for all r distances between M and m there exists a force of magnitude ...F= [(KMm)/r^2] acting along the r direction.K a universal constant

 

[DaveC426913]

for all masses between 1 and 5 Kgs and for all forces F there exists a unique acceleration a

OK, you've just stated (a special case of) Newton's 2nd law.

What's unsatisfactory about that?

 

[evagelos]

so ,a problem in physics can be put in a quantifier form and in our case we have:

......(m)[ 5>= m >= 1=====> (F)E!a( F=ma )]...1

where (m),means for all m, E!a means there exists a unique a ,F and a are vectors.

Formula 1 can be proved using Newton's 2nd law, which law we can put in the following quantifier form:

... (m)[ mε( 0, 00)====>(F)E!a( F =ma)]

Where 00 means infinity and masses forces and reference frames all being Newtonian

 

[Defennder]

There's one problem with the notion of proof: You never know when your theory might be superseded. Although this is often over-cited, Newton's laws were superseded by Einstein's. You can't make a generalising statement in physics which is essentially formally proven since there is always a possibility that it would be overtaken by a newer one.

In maths, once you have a proof, you can't be mistaken unless the proof is flawed.

P.S. You ought to cut down on the "...". It makes it look as though you were half-asleep while posting.

 

[DaveC426913]

There's one problem with the notion of proof: You never know when your theory might be superseded. Although this is often over-cited, Newton's laws were superseded by Einstein's. You can't make a generalising statement in physics which is essentially formally proven since there is always a possibility that it would be overtaken by a newer one.

This is what I'm sayin'. Maybe he'll like it more coming from you.

 

[granpa]

where is the "if A then B" form in the statement ; ...no object belongs to the empty set? Written in quantifier form

for all x ~xεΦ where Φ is the empty set.
Another statement which can be proved is:...nothing contains everything

arent those things true by definition?

 

[evagelos]

arent those things true by definition?

in the book of Patrick Suppes Axiomatic Set Theory in page 21, he proves as his 1st
theorem¨

~xεΦ


Paul R.Halmos in his book called Naive Set Theory, in page 6 he proves that:


'nothing contains everything'

So the statement that: that all statements in mathematics are of the form "If A then B".
is wrong

 

[humanino]

in the book of Patrick Suppes Axiomatic Set Theory in page 21, he proves as his 1st
theorem¨

~xεΦPaul R.Halmos in his book called Naive Set Theory, in page 6 he proves that:'nothing contains everything'

So the statement that: that all statements in mathematics are of the form "If A then B".
is wrong

I don't know those people but this is flat wrong. There is nothing in the empty set. Forget about obscure books you are the only one to have. Please define the empty set for us and show us how "anything" is in "nothing".

 

[granpa]

where is the "if A then B" form in the statement ; ...no object belongs to the empty set? Written in quantifier form

for all x ~xεΦ where Φ is the empty set.
Another statement which can be proved is:...nothing contains everything

arent those things true by definition?

or rather, isn't the first of those true by definition?

 

[evagelos]

or rather, isn't the first of those true by definition?

still that does not change the fact that the statement:

that all statements in mathematics are of the form "If A then B".'

Is wrong

 

[humanino]

still that does not change the fact that the statement:

that all statements in mathematics are of the form "If A then B".'

Is wrong

This is a disagreement on the interpretation of what mathematics are. Ultimately, you can have any mathematical statement implemented on a computer which uses (at least in principle) only NOR gates, and that is equivalent to only "if A then B" statements. But what is interesting about this interpretation (a contrario to other proposals here in this discussion, at least at the level they were presented) is that mathematics indeed does not care whether A is true or not. In particular, it is most often relevant to Nature and physics that A is strictly wrong, but approximately true in which case, if the statement "If A then B" is true and "B is relevant", then "If A then B" is useful even though A is strictly wrong.

Did I loose you by now, or do you agree ?

 

[HallsofIvy]

in the book of Patrick Suppes Axiomatic Set Theory in page 21, he proves as his 1st
theorem¨

~xεΦ


Paul R.Halmos in his book called Naive Set Theory, in page 6 he proves that:


'nothing contains everything'

So the statement that: that all statements in mathematics are of the form "If A then B".
is wrong

Even those theorems assume the axioms of set theory. IF (axioms of set theory) THEN ...

 

[HallsofIvy]

This is a disagreement on the interpretation of what mathematics are. Ultimately, you can have any mathematical statement implemented on a computer which uses (at least in principle) only NOR gates, and that is equivalent to only "if A then B" statements. But what is interesting about this interpretation (a contrario to other proposals here in this discussion, at least at the level they were presented) is that mathematics indeed does not care whether A is true or not. In particular, it is most often relevant to Nature and physics that A is strictly wrong, but approximately true in which case, if the statement "If A then B" is true and "B is relevant", then "If A then B" is useful even though A is strictly wrong.

Did I lose you by now, or do you agree ?

Sure. All mathematica theories are "templates". We start with undefined terms and base all axioms and other definitions on those. We can then apply that theory to physics, economics, or whatever by giving definitions from the application to those undefined terms in such a way that the axioms are "true" for that application. Then we know automatically that any theorems, etc. based on those axioms are true. Of course, you are correct that since physics, etc. involve measurements that cannot be exactly true, the axioms must be only approximately correct.

 

[evagelos]

Even those theorems assume the axioms of set theory. IF (axioms of set theory) THEN ...

IF(axioms of set theory) THEN :if you have a statement which is not conditional fine,but if you have a conditional statement then you have:

IF( axioms of set theory) THEN,IF p then q and in this case you have two ifs

In this case if we put... A=axioms of set theory,and let the conditional statement be


B------>C we have that...A------>( B------->C).

Hence the statement that:

that all statements in mathematics are of the form "If A then B".'

Is wrong

 

[evagelos]

I don't know those people but this is flat wrong. There is nothing in the empty set. Forget about obscure books you are the only one to have. Please define the empty set for us and show us how "anything" is in "nothing".

The least you could do is to open those books and have a look first and then decide
whether they are obscure or not.

Suppose you did not know Einstein and somebody quoted something from his books,having this kind of attitude you can imagine the out come.


Who said that there is something in the empty set?

The theorem is nothing contains everything and not anything is in nothing ,this may apply for investors when sometimes invest everything into nothing


Now for the proof of that theorem ask HallsofIvy to give us a formal proof just to make the proof different from that in the book,after all he was the one who opened the
'if...then' business

 

[Santa1]

"So the statement that: that all statements in mathematics are of the form "If A then B".
is wrong"

Well modern mathematics is based on ZFC not Naive Set Theory, so I really don't see your point here. Are you trying to say nothing is true?

 

[HallsofIvy]

IF(axioms of set theory) THEN :if you have a statement which is not conditional fine,but if you have a conditional statement then you have:

IF( axioms of set theory) THEN,IF p then q and in this case you have two ifs

In this case if we put... A=axioms of set theory,and let the conditional statement be


B------>C we have that...A------>( B------->C).

Hence the statement that:

that all statements in mathematics are of the form "If A then B".'

Is wrong

? The statement you give is precisely of that form!

 

[evagelos]

? The statement you give is precisely of that form!

WHAT statement

Any way here and now give us a formal proof of the statement that :

...'nothing contains everything'


So that we Will all learn of its ' if...then' nature


Remember i said formal to avoid any doubtful proof

 

[evagelos]

"So the statement that: that all statements in mathematics are of the form "If A then B".
is wrong"

Well modern mathematics is based on ZFC not Naive Set Theory, so I really don't see your point here. Are you trying to say nothing is true?

open the book first please

 

[Santa1]

I don't think you realize that modern mathematics are not dependent on naive set theory at all. Naive set theory offers a lot of paradoxes, this is just one of them, which is why we don't base mathematics on it, we use ZFC (link: http://en.wikipedia.org/wiki/Zermelo-Fraenkel_set_theory).

 

[evagelos]

I don't think you realize that modern mathematics are not dependent on naive set theory at all. Naive set theory offers a lot of paradoxes, this is just one of them, which is why we don't base mathematics on it, we use ZFC (link: http://en.wikipedia.org/wiki/Zermelo-Fraenkel_set_theory).

.......naive set theory is the title of the book........


......the book is based on axiomatic foundations.......


But this is not our issue here .The issue is:If HallsofIvy can give a formal proof of the theorem that :


......nothing contains everything............

For us to see if the above theorem is of 'if...then nature'.......


Please do not try to obscure the issue

 

[Santa1]

"So the statement that: that all statements in mathematics are of the form "If A then B".
is wrong"

As HallsofIvy said, the statement you're making is on the "If A then B" form, do you actually think you're going any further with this?

 

[evagelos]

Notice, by the way, that all statements in mathematics are of the form "If A then B". .

Here are the axioms of propositional calculus in mathematical logic:


......A----->( B------A)............1

......( A----->( B-----C))-------->(( A----->B)------>(A---->C))...2


where A , B , C are statements.


Do you still insist that all statements in mathematics are of the form " If A then B"??



.......yes or no..........

 

[Jimmy Snyder]

Mathematics uses deductive logic, science uses inductive logic. Does this help?

 

[evagelos]

Mathematics uses deductive logic, science uses inductive logic. Does this help?

You mean for theorems proved in physics we use only inductive procedures and not the rules of inference?

 

[Gear300]

is a proof in physics equal in strength with that in mathematics?

in mathematics we have at one end an ordinary proof and at the other end a formal proof.

how would aformal proof in physics look like,an example would help.

I suppose that the validity of a proof in physics could be checked by an experiment but in the case that we have no experiment what happens??

Thanks

Its a bit different in the case of physics. Theoretically outlining things is usually done mathematically...so either way, you're dealing with mathematical proofs...however, there is still the need of confirmation, which suggests an empirical source.