Are physics laws mathematical models?

[Tosh5457]

Are physics laws (even the laws) mathematical models of reality? Or they describe reality as it is? Are those laws just models, like for example a population's growth mathematical model (just an example that I remembered)?

A maths teacher told me that in physics everything are approximations, even though physicists express them as equalities (equations).

 

[Fredrik]

Yes, they are. At least the classical theories. These theories can be said to be approximate descriptions of reality, or alternatively, exact descriptions of fictional universes that resemble our own.

It's possible that some quantum theory is an exact description of reality, since there are still no experiments that contradict the predictions of quantum mechanics (the framework in which quantum theories are defined). But there's also a good chance that quantum mechanics is something less than a model of reality in the sense described above. In that case, it would be just an assignment of probabilities to statements of the form "If I use measuring device A on object S, the result will be in the set E". The classical theories can also be thought of as probability assignments, but we usually don't think of them that way, since the probabilities are all 0 or 1 unless there's some uncertainty about how the system was prepared. In QM, the probabilities can be non-trivial (neither 0 nor 1) even when we have all the information about how the system was prepared.

QM can certainly be said to describe a fictional universe, but the idea that this universe is approximately like our own has some very strange consequences. It's not clear which one of the possible consequences would be the "real" one. The two possibilities that I'm somewhat familiar with are 1) Some of the laws of logic break down, and 2) Our universe is just one of infinitely many, that are all "aspects" of the properties of the reality described by the theory.

 

[Tosh5457]

Yes, they are. At least the classical theories. These theories can be said to be approximate descriptions of reality, or alternatively, exact descriptions of fictional universes that resemble our own.

It's possible that some quantum theory is an exact description of reality, since there are still no experiments that contradict the predictions of quantum mechanics (the framework in which quantum theories are defined). But there's also a good chance that quantum mechanics is something less than a model of reality in the sense described above. In that case, it would be just an assignment of probabilities to statements of the form "If I use measuring device A on object S, the result will be in the set E". The classical theories can also be thought of as probability assignments, but we usually don't think of them that way, since the probabilities are all 0 or 1 unless there's some uncertainty about how the system was prepared. In QM, the probabilities can be non-trivial (neither 0 nor 1) even when we have all the information about how the system was prepared.

QM can certainly be said to describe a fictional universe, but the idea that this universe is approximately like our own has some very strange consequences. It's not clear which one of the possible consequences would be the "real" one. The two possibilities that I'm somewhat familiar with are 1) Some of the laws of logic break down, and 2) Our universe is just one of infinitely many, that are all "aspects" of the properties of the reality described by the theory.

So physics laws can be the exception, in the sense that they may not be just models, but describe reality as it is (in case QM is the exact description of reality)?

Or are there other applications of mathematics to the real world that doesn't make just mathematical models, but describe reality?
For example, if I have 1 apple and 1 peach, if I sum the two, 1+1 = 2, so I have 2 fruits. Is this a mathematical model? What I see here is that if we define what's a fruit this describes reality, but maybe having to define what's a fruit is the model itself...

I'm just having trouble figuring out why mathematics applies so well to reality... Is it just because mathematics is a language of logics?

 

[phinds]

I'm just having trouble figuring out why mathematics applies so well to reality... Is it just because mathematics is a language of logics?

Careful here ... SOME math applies very well to reality. Some math has nothing to do with reality. Mathematical systems only need to be self-consistent without reference to anything outside of the particular system. I'm sure the math folks here can give you plenty of examples.

Interestingly, some math that was at one time thought to not be in any way based in reality has later been found to have direct application to reality. For example, if I have the story right, Riemann geometry was invented as an abstract form of math and remained so until Einstein realized that it gave a perfect description of space-time as affected by gravitational fields --- not really what Riemann had in mind but serendipity at its best.

 

[Fredrik]

So physics laws can be the exception, in the sense that they may not be just models, but describe reality as it is (in case QM is the exact description of reality)?

I don't understand this conclusion. A "law" of physics is just a small part of a theory, that can be stated very succinctly, usually in the form of a single sentence or a single equation. A law isn't something better than a theory. It's just a component of a theory.

Or are there other applications of mathematics to the real world that doesn't make just mathematical models, but describe reality?
For example, if I have 1 apple and 1 peach, if I sum the two, 1+1 = 2, so I have 2 fruits. Is this a mathematical model? What I see here is that if we define what's a fruit this describes reality, but maybe having to define what's a fruit is the model itself...

The problem here is to define "fruit" (or "apple" or "peach"). No matter what definition you write down, it will describe something that has a property that a real fruit doesn't.

I'm just having trouble figuring out why mathematics applies so well to reality...

No one really knows.

 

[Functor97]

This is a deeply philosophical question, so do not expect an answer.

Before anyone can begin to answer this, you have to tell us what you define reality or truth as... If it is simply what we "see around us" which is as basic and simplistic a definition as it gets, then our mathematical models perfectly model reality up to a point. That point may come to experimental problems, which leads to a modification of our theories, but after this modification we are back to a perfect description. Then the process begins again... This is the main reason I, and many others believe that the popularist descriptions of a "theory of everything" are misleading, at the end of the day we are always going to have more questions, or more specifically more postulates which we must accept. I hate that, i imagine you do too, but that's tough, we can't change it...

If you believe there is something "outside" of physics, this leads to more philosophical problems than a simple empirical approach would otherwise. If you claim there is a god, then you are going to spend more time talking philosophy then going out and finding and fixing errors in our physical models. This is why our science as "evolved" to downplay philosophical questions...otherwise we would never have the time to do any science!

You may be entering physics to find "why the universe began" or "what was before the big bang?", again, it was these things that sparked my passion in the natural sciences, but be warned, it does not match our process. These questions are inherently flawed, they are probably meaningless tautologies. If by a stroke of good luck we prove a form of brane world cosmology, the old "before the big bang", will be shifted backwards, possibly being replaced by "what was before the multiverse?". These are fascinating questions, but sadly i feel that they are a result of emergent human processess, we see a bird fall from a tree, and apply the empirically developed causality of our logic system to the universe as a whole.

I am sorry for getting off track, but i want to convey my "insights" from asking similar questions to yours. Our physical laws are exactly mathematical models, but as it stands, that all there seems to be to reality. If there was a higher level of reality, our mathematical laws are probably just approximations of it, but i feel it would still be mathematical in essence.

 

[Functor97]

I don't understand this conclusion. A "law" of physics is just a small part of a theory, that can be stated very succinctly, usually in the form of a single sentence or a single equation. A law isn't something better than a theory. It's just a component of a theory.


The problem here is to define "fruit" (or "apple" or "peach"). No matter what definition you write down, it will describe something that has a property that a real fruit doesn't.


No one really knows.

Fredrik, how do you know that our "model" of fruit lacks something imbued within "real" fruit?

 

[lostcauses10x]

Tosh5457
in physics we tend to observe the world around us and understand it with a theory. from this changes are observed and measured. Note the word change.
It is such the units used in physics to measure property has to be agreed upon, or such does create problems. To be brutally honest all physics is theory.
Strangely enough the measurements of property involved in physics uses values set for measurement. These values can be directly interpreted easiest by numbers.

Of course mathematics is in reality the study of numbers and their property, real and or what can be imagined. In most cases the mathematician has been able to stay ahead of the physicist.

It is the study of numbers that so really fit into the study of the physical wold that allows us the great technology of today.
As already stated you will get a lot of vary answers to the question you asked.

In my opinion the need for measurement and the relations in units of change of the measurement of the physical world that so easily, are simply equivalent to numbers: is the why of such a great relation in the two fields.

This need of physics to have a value of change is what does relate to the mathematician.

 

[Functor97]

The reason mathematics is so effective is because we make it so. We have evolved to be tuned into reality, otherwise we would have all fallen out of our trees or gotten squished by elephants. Our laws of logic are thus a result of the deeper laws of reality, which we discover. We see mathematics in reality, because that is where we got our mathematics from in the first place, be it platonism or empiracism.
There is no magical mathematician god. Reality uses mathematical rules, because it probably is mathematical rules. There is no deeper "thing", besides reality.

 

[lostcauses10x]

Functor97
I should most likely not ask this, and may not repond to your awser, but:
Define reality please??

 

[Fredrik]

Functor97
I should most likely not ask this, and may not repond to your awser, but:
Define reality please??

Is there something that makes you think that it would be a good idea to define that term?

Fredrik, how do you know that our "model" of fruit lacks something imbued within "real" fruit?

I can't prove that there's no definition that associates the term "fruit" with a mathematical concept that's a perfect representation of fruit, but I'm sure I can knock down any definition that you can write down.

 

[Matt Benesi]

I'm just having trouble figuring out why mathematics applies so well to reality...

No one really knows.

[joke] The mathematic principle (similar to thehttp://en.wikipedia.org/wiki/Anthropic_principle" ). [/joke]

Mathematics applies well to reality because it has evolved over time along with the physical sciences to more precisely describe reality. Although there are some that take a somewhat http://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences" of mathematics.

Mathematics, as a language, wouldn't work well if we used it to precisely describe incorrect physical laws (imagine precise mathematical equations involving the 4 "classical elements": fire, air, earth, and water). The simple fact is that incorrect physical laws and the mathematical equations that describe them won't create accurate descriptions of reality. We can test the proposed mathematical statement: $e=m^3 \times 4 + c^8$ and find that it doesn't accurately represent reality (although there is a case in which the equation would work- but only for a specific mass and energy).

Of course, a case could arise in which an incorrect physical "law" is favored over a correct one, simply because the "incorrect law" is easier to describe mathematically with the information and abilities that are currently available to us.

Also, there are cases in which mathematics can lead one on quite a loop. Kaluza Klein theory for one (not that it is entirely incorrect, but the "compactified dimensions" part has everyone missing where the dark matter is).

 

[lostcauses10x]

Is there something that makes you think that it would be a good idea to define that term?

Yes with risks.
To the Physicist, that is a question. What is reality??
To the person it is such that it is often viewed as an absolute.

To a mathematician,I am not sure of their view of the concept of reality. Usually the term reality is not directly related to numbers.

As I said: it is some thing I should not probably ask, and most likely would not respond to an answer.

Yet strangely enough, Reality is often discussed when one gets involved in the basic question of why does mathematics work so well in the other sciences.

Best of course: if I let it be.

 

[Functor97]

Lostcause, i have allready tried to define reality.

It is all we could possibly interact with. It is the world of our senses.

It is by no means perfect, but it gives us a running start to the discussion.

 

[lostcauses10x]

Functor97
Thank you for your answer.