Is physics mathematically deducible?

[thiazole]

Alright, this is a subjective question since I think a person would have to be omniscient or nearly so to know answer it with any certainty, but my question is fairly straight forward and I'm curious what cosmologists think about the topic. First, a little about the origin of the question. I've noticed that advances in the understanding of physics, as well as other sciences, mostly occurs empirically and not through mathematically derived theory alone. Is this simply because we have so little knowledge from which to build our theories or is it because we are subject to a unique set of physical laws based on how the universe is constructed. In other words, as my title asks, are the physical laws of our universe completely objective and mathematically deducible from nothing but an understanding of math alone, or are physical laws subject to the universe in which we live with the potential to have different "types" of universes with completely different physical laws?

 

[dkotschessaa]

I think the answer is no.

I can't find a text version of this to copy and paste, but I think the best answer is something akin to Feynman's chess analogy.



The relationship to mathematics here I think that we can use mathematical models to predict behavior, but every so often an observation will come along that suddenly doesn't fit that model. (like castling in chess, or a pawn promotion)Then you have to come up with a new explanation and sometimes new math. There are even physical phenomena that we don't have the math for yet.

 

[Nabeshin]

I've noticed that advances in the understanding of physics, as well as other sciences, mostly occurs empirically and not through mathematically derived theory alone.

I don't know if I'd say mostly. The neutrino and the whole family of antimatter were theoretically predicted long before experimentally observed. Similarly, Einstein predicted the dynamic universe years before it was observationally determined by Hubble. Indeed, the two major theoretical accomplishments of the 20th century, General Relativity and Quantum Mechanics were both motivated only by a small amount of experimental evidence compared to their theoretical potential. We are still observing new and interesting consequences of both theories, where were written down almost a hundred years ago! Indeed, this is the mark of a good scientific theory -- when it predicts phenomena which have not yet been observed.

To answer the question as a whole, the answer is at this moment no. If you're familiar with the standard model of particle physics, we can explain all particle interactions if only we know some 19 free parameters. We have no way (at the moment) of predicting what these 19 parameters are, so we must observe them and insert them by hand. Now, various theories of unification attempt to output these 19 values from some broader theoretical framework. But as far as I know, all these theories still have free parameters (in often cases, many more).

 

[phyzguy]

Alright, this is a subjective question since I think a person would have to be omniscient or nearly so to know answer it with any certainty, but my question is fairly straight forward and I'm curious what cosmologists think about the topic. First, a little about the origin of the question. I've noticed that advances in the understanding of physics, as well as other sciences, mostly occurs empirically and not through mathematically derived theory alone. Is this simply because we have so little knowledge from which to build our theories or is it because we are subject to a unique set of physical laws based on how the universe is constructed. In other words, as my title asks, are the physical laws of our universe completely objective and mathematically deducible from nothing but an understanding of math alone, or are physical laws subject to the universe in which we live with the potential to have different "types" of universes with completely different physical laws?

I think this is still an open question. We don't really know whether there is only one possible universe which is completely mathematically consistent, or whether there are many possible universes and we need empirical data to choose between them. I think this is what Einstein meant in his famous quote, "What really interests me is whether God had any choice in the creation of the world."

Today we don't have a complete theory of the universe, we only have models for pieces of it, and these models clearly have many options and much empirical data is needed to complete these models.

 

[Fredrik]

It's conceivable that if we find out all we can about what conditions are necessary to support intelligent life, there will only be one theory that describes a universe where such conditions exist. But I wouldn't bet on it, and even if this is the case, we will have used a lot of information from experiments to determine what those conditions are.

If we don't impose any requirements about what conditions the theory should describe, then it's obviously not the case that a correct theory can be derived. We wouldn't even have anything to derive it from.

I would also like to point out that a theory of physics is never defined by mathematics alone. For example, special relativity is built up around a specific mathematical structure called Minkowski spacetime, but the theory is defined by a separate list of axioms that identifies things in the real world with things in the mathematical structure. The theory isn't that mathematical structure, it's the list of statements that tells us how to interpret the mathematics as predictions about results of experiments.

 

[Kracatoan]

From what I've read on this sort of subject before, the biggest obstacle to a mathematically deduced Universe are the constants - big G, Omega and the like. Seeing as for all we know, they are irrational and cannot be deduced exactly from maths alone.

 

[Chalnoth]

In other words, as my title asks, are the physical laws of our universe completely objective and mathematically deducible from nothing but an understanding of math alone, or are physical laws subject to the universe in which we live with the potential to have different "types" of universes with completely different physical laws?

Well, what we can say for sure is that theorists have a very large number of potential ideas of how the universe might behave, ideas which may or may not be correct, but we don't yet have any sort of objective reason to select among them.

Whether this brute fact can be extrapolated to mean that the actual laws of physics vary from place to place is not clear. But I would say that the laws of physics that we have so far discovered most definitely suggest this possibility, through a process known as Spontaneous symmetry breaking.

 

[epenguin]

There are of course various possible answers of which none could be settled now.

One possible reason, that by your question you seem to have overlooked, for physics not being mathematically deducible could arise if there is mathematics that is not mathematically deducible. Which from what I have heard is the case. Though of course that does not necessarily imply that physics is not mathematically deducible, since it might need only the [STRIKE]boring and trivial[/STRIKE] deducible parts of math.

 

[zonde]

In other words, as my title asks, are the physical laws of our universe completely objective and mathematically deducible from nothing but an understanding of math alone, or are physical laws subject to the universe in which we live with the potential to have different "types" of universes with completely different physical laws?

Physical laws are subject to measurement equipment used.
You can't include measurement equipment in theory without getting circular argument.

So answering question - physical laws are not mathematically deducible from nothing.

 

[Chalnoth]

Physical laws are subject to measurement equipment used.
You can't include measurement equipment in theory without getting circular argument.

So answering question - physical laws are not mathematically deducible from nothing.

Yeah. Another way of saying it is that any set of mathematics stems from a series of axioms. There is no objective way of deciding which axioms apply to reality without comparing their consequences to reality, i.e. without experiment.

 

[Chronos]

Godel's incompleteness theorems comes to mind.

 

[torquil]

Far from it.

 

[supratim1]

most of it is deducible, rest are from observations and experiments.

 

[dkotschessaa]

I was just thinking of people like Aristotle, who tried to deduce things about the world from that using logic. This yielded a vast number of absurd conclusions and a very silly picture of the universe.

Of course, we're so over Aristotle. Or are we? Mathematics is just a form of logic, the "problem" with it being that it is too pefect - too idealized to represent reality. There is a long history of mathematical prediction being overturned by experiment. You'll notice it never happens the other way around.

 

[supratim1]

yes i agree with you, dkotschessaa...

 

[Chalnoth]

Of course, we're so over Aristotle. Or are we? Mathematics is just a form of logic, the "problem" with it being that it is too pefect - too idealized to represent reality. There is a long history of mathematical prediction being overturned by experiment. You'll notice it never happens the other way around.

Well, I'd say that's just false. The world is mathematical at its core. It must be if it is to be self-consistent. The difficulty is in discovering which specific mathematics fit our world. And that is not an easy thing to do, not by a long shot.

 

[Dmitry67]

There is a good example:

A 'toy' Universe - Conway's game of life on an infinite chessboard.
It is 100% deterministic, 'physical laws' are very simple, and - surprise! - there are statements about configurations (can configuration A ever evolve into configuration B) which are undecidable.

 

[Dmitry67]

There is a long history of mathematical prediction being overturned by experiment. You'll notice it never happens the other way around.

It was not the mathematics being overturned or failed.
it was the human's 'common sense reasoning' which had epically failed many times
The same 'epic failures' happened in the realm of 'pure' mathematics, just recall the history of the so-called 'naive' set theory.

 

[Chalnoth]

There is a good example:

A 'toy' Universe - Conway's game of life on an infinite chessboard.
It is 100% deterministic, 'physical laws' are very simple, and - surprise! - there are statements about configurations (can configuration A ever evolve into configuration B) which are undecidable.

I don't honestly see what this has to do with the original post. The likely incompleteness of whatever fundamental mathematics underlies reality doesn't seem, to me, to have much to say about whether or not we'll be able to determine what that mathematical structure is.

As I read it, the OP was basically asking if it is possible to determine the fundamental laws from first principles. And it absolutely is not, just because there is more than one possible self-consistent mathematical structure, and we must necessarily use observation to determine which one applies to our reality.

 

[Dmitry67]

I see, my bad.

Still, let me think. Say, MUH is true and *ALL* mathematical structures exist.
From "Cogito ergo sum" I derive that *our* universe must be consciousness-friendly.
So it must have time, it must exist long enough etc.

Of course, there should be more than 1 consicousness-friendly universe, and I don't know if we live in the simplest one, so only experiment can tell it. But I wonder to what extent MUH+AP can limit the list of 'potential candidates'.

 

[Chalnoth]

But I wonder to what extent MUH+AP can limit the list of 'potential candidates'.

Well, we won't know that until we have the list of potential candidates. Unfortunately that list is a very difficult thing to discover.

If we ever do discover said list, it may turn out to be very easy to say which one matches our reality, or monstrously difficult, or impossible. I don't see any way we can know without having that list.

 

[phyzguy]

As I read it, the OP was basically asking if it is possible to determine the fundamental laws from first principles. And it absolutely is not, just because there is more than one possible self-consistent mathematical structure, and we must necessarily use observation to determine which one applies to our reality.

But you don't know this. It's true that when we model a piece of the universe, there are many possible consistent structures. But we don't yet have a fully consistent model of the universe. As an example, classical physics seems to be mathematically consistent. However, when you follow it through, you find that it predicts a blackbody spectrum that diverges, so isn't a consistent model. General Relativity is mathematically consistent up to a point, but predicts singularities and therefore is probably not a consistent model. It might be that if and when we ever have a fully coherent mathematical model of the universe, we will find that there is only one possible answer. I think this is what the OP was asking. I don't know if this is the case, but I think it is still an open question, as I said earlier.

 

[Chalnoth]

But you don't know this. It's true that when we model a piece of the universe, there are many possible consistent structures. But we don't yet have a fully consistent model of the universe. As an example, classical physics seems to be mathematically consistent. However, when you follow it through, you find that it predicts a blackbody spectrum that diverges, so isn't a consistent model. General Relativity is mathematically consistent up to a point, but predicts singularities and therefore is probably not a consistent model. It might be that if and when we ever have a fully coherent mathematical model of the universe, we will find that there is only one possible answer. I think this is what the OP was asking. I don't know if this is the case, but I think it is still an open question, as I said earlier.

We already have some rather simple fully-consistent mathematical structures of which we are aware. But we know that they are too simple to explain our universe. So from that we know for certain, that one cannot deduce the laws of physics merely by appeal to self-consistency.

Now, when we learn more about mathematics, to the point that we have a catalog of self-consistent theories, it is conceivable that we will find that only one of them is remotely like our universe. But in any event it will require connecting the mathematical theories to observation to make this determination.

Of course, you are correct that our physical theories that we have today fall flat when it comes to consistency. It is very likely that if we can find ways of fixing these inconsistencies in a robust manner (that is, in a manner that is highly independent of whatever assumptions we make) that we will discover new things about our own universe. But it is unclear at present how far we can push current physical law merely by relying upon self-consistency.

 

[thiazole]

I was keeping up with this for a couple days, but hadn't come back until this lazy Sunday. Thank you very much for this thought provoking discussion. I enjoyed reading through the rest of these posts.