I want to understand the universe

[twofish-quant]

I concede that it's not the only game town, but that point wasn't brought up and I would defend any serious deductive framework endeavor that has enough support.

Sure, and after 20 years of trying the experimental support of string theory is ?

Even if people don't see results in their lifetime, i don't think it's worth not pursuing.

Ph.D. committees want you to produce something after five to seven years. You may not (and you aren't expected) to come up with the answer to life, the universe, and everything, but it's a bad sign if you come up with nothing useful.

Even figuring out that it's the wrong approach is useful. One problem I have with string theory is that it's difficult to know when you are even wrong. Working on something for two decades and then having it be totally obvious that it was the wrong approach and that it's time to work on something else would be useful, but it doesn't seem to me that string theory has even gotten to that point...

This happens all the time anyway where people think they will solve it all and end up realizing that it's probably a little early after a lifetime of dedication. Doesn't mean its all wasted though.

You aren't going to solve everything in a lifetime, but I think it's reasonable to ask after X years if you've been able to solve *anything*.

 

[twofish-quant]

If we just ignore actual physics for a second we know that it would be sensible to rule out inconsistent illogical systems, systems that have divergence, blow-up and instability, and systems that don't settle (converge to some unrecoverable static state).

No we don't know this. We do know that the Rules of Quantum Mechanics follow a different set of logical rules than classical aristolean logic, and there are perfectly good logical systems that allow for inconsistent statements.

As far as divergence, blow-up, and instability, there's no reason to automatically assume that the laws of physics at extremely high energy are stable and non-divergent.

Now the mathematical ones above irrespective of the domain are very easy to explain: consistency is required so that the whole description is non-ambigous and mathematically proper. You don't have to go as far as set theory axioms, but just enough to show that its a proper system definition.

On the other hand, we have been able to get places with theories that are mathematically bad. Quantum field theory is mathematically inconsistent, but it's inconsistent in ways that we can "work around" at energies we are interested in.

These things are easy to understand by anyone even if they have never done a physics or maths course in their life! You can explain it to them using analogues and it would make sense to them.

Which is precisely why I think they are suspect.

Essentially what happens is that people go out and see their own little tiny bit of the universe and then assume that everything has to be a certain way. I've seen really weird stuff, so when someone starts putting constraints because the universe has to act in a certain way, I'm suspicious of that.

One reason that I think physics is "hard" is not so much that the things that we are studying are difficult to understand, but it's because our senses are limited. Here is an example. Take a cup of tea, and try to explain what it looks like to someone that has been blind since birth. Try to explain "red" and "yellow" and describe the color of the liquid. It's hard. It's going to take some effort to "see red" and if someone who is blind just uses the concepts that they are familiar with, they are going see something, but they are also going to be missing other things.

 

[chiro]

Sure, and after 20 years of trying the experimental support of string theory is ?

It's not going to be a total solution, but one that complements more narrow approaches and the experimental evidence.

You can't just expect a project or endeavor of that magnitude to pay off in the way that one of more modest means would: it's not a fair comparison.

Ph.D. committees want you to produce something after five to seven years. You may not (and you aren't expected) to come up with the answer to life, the universe, and everything, but it's a bad sign if you come up with nothing useful.

This is probably one contributing factor to why things are the way they are. I'm not saying for many applications it's bad, but sometimes you need to know the times when to not follow the rules.

Also the deductive way of doing things doesn't solve the whole unified thing even if a lot of the principles are worked out, because it's at a different level than that of the more specific approaches. The specifics are in a completely different context which tells us things about specific, highly-constrained things.

As you well know, we have GR but we still need supercomputers to be able to get decent results from a simulation: this doesn't mean though that it's irrelevant in terms of getting any kind of context though.

Even figuring out that it's the wrong approach is useful. One problem I have with string theory is that it's difficult to know when you are even wrong. Working on something for two decades and then having it be totally obvious that it was the wrong approach and that it's time to work on something else would be useful, but it doesn't seem to me that string theory has even gotten to that point...

You aren't going to solve everything in a lifetime, but I think it's reasonable to ask after X years if you've been able to solve *anything*.

But this is what I don't get: you don't want to take a risk?

Everything is a risk. Typically what most people want to do if they are brave enough to take risks is to take calculated ones, but even then it's still going to be risk especially when it comes to nature.

Our mathematics that we have is really not one that is suited for analysis of systems with many degrees of freedom or many interdependencies.

We have the techniques of non-euclidean geometry which allow one to look at non-orthogonal systems which model such dependence, but most people can not really grasp things more than say 30 variables.

We are talking about systems with millions upon millions of degrees of freedom, and we are trying to apply techniques that are best suited for hundreds, not many millions or billions.

I agree that we won't solve it all in a life-time, but even logically, the idea of trying to use a hammer instead of a power-saw doesn't make sense.

The techniques are going to be more complicated, and they will be non-intuitive when they are first created. They will like everything else, be refined and polished so that they do become more intuitive than they were before.

And the argument for even thinking for going in this direction, let alone actually going there is based on this idea that we can not use existing paradigms that were designed for a different kinds of problem (low number of degrees of freedom) to be used effectively and optimally to look at a different kinds of problem (high degrees of freedom).

It's going to be really hard and horrible, and it probably won't be done in anyone's single life-time, but the impetus behind it is rather simple: we are trying to understand something that needs an appropriate framework: do we try and adapt the system to our framework, or the framework to our system?

The people that work on these high-level approaches have my congratulations despite the fact that nobody really understands anything well enough to really handle it, because they are likely to be the ones that are taking a chance that people two or three or more generations later will benefit, and this is an extremely hard thing for any person without exception to do.

 

[chiro]

No we don't know this. We do know that the Rules of Quantum Mechanics follow a different set of logical rules than classical aristolean logic, and there are perfectly good logical systems that allow for inconsistent statements.

Well in order for something to be 'measureable' it has to be finite and thus not diverge. Physics is based on the principle that things be measureable.

It doesn't have to follow classical logics in any way: it just has to be measurable.

As far as divergence, blow-up, and instability, there's no reason to automatically assume that the laws of physics at extremely high energy are stable and non-divergent.

True, but we have a lot of evidence for it both theoretically and experimentally, although I agree with you that we haven't had particle accelerators for that long (but we can look at the external universe and supernovae, and cosmic rays).

The theoretical evidence can be found in theorems to do with black holes (and yes I realize it's speculative) in the form of evaporating black-holes, and results about how black-holes may grow (again speculative).

In the low-energy situations, we have a lot of data to deal with.

We also have thermodynamics which deals with entropy, and shows us how hard it is to generate situations where we get lots of energy.

These indicators give at least an initial premise behind this idea.

If the theory is wrong, then like every other theory it needs to be changed. But given theoretical speculation and experimental results (especially with thermodynamics), it does have a little support at least.

On the other hand, we have been able to get places with theories that are mathematically bad. Quantum field theory is mathematically inconsistent, but it's inconsistent in ways that we can "work around" at energies we are interested in.

You can have theories about physics that are non-deterministic and even non-continuous where the above constraints are still respected.

These constraints do not require determinism, nor do they require some kind of continuous aspect. You can represent systems primarily using number theory to model the kind of discrete behaviour where things jump or can't be modeled by continuous representations and analyzed effectively through continuous analysis (i.e. the integral and differential calculus), and you can use statistics to model things that are not in the context of a deterministic fashion.

You can also incorporate non-locality if you want to as well.

Which is precisely why I think they are suspect.

Essentially what happens is that people go out and see their own little tiny bit of the universe and then assume that everything has to be a certain way. I've seen really weird stuff, so when someone starts putting constraints because the universe has to act in a certain way, I'm suspicious of that.

One reason that I think physics is "hard" is not so much that the things that we are studying are difficult to understand, but it's because our senses are limited. Here is an example. Take a cup of tea, and try to explain what it looks like to someone that has been blind since birth. Try to explain "red" and "yellow" and describe the color of the liquid. It's hard. It's going to take some effort to "see red" and if someone who is blind just uses the concepts that they are familiar with, they are going see something, but they are also going to be missing other things.

Well this is what mathematics has primarily become: it has become a way for us blind people to see in a way that we never can with our five senses.

Mathematics has done this with every subfield including logic, algebra, probability, calculus, and so on. Probability is the most striking area because time and time again, the math makes it intuitive, but the senses always mislead us even for the most gifted mathematicians you try and use it like D'Alembert.

Mathematics also gives us a gateway to the uncountable. Through mathematics we can understand when an infinite series even converges, how to make sense of an infinite-vector space and a basis for that space.

No amount of sensory can give us this insight, and by ignoring this approach, we are always going to rely on the intuition afforded to us by our five senses, or a product thereof.

Physics is about things that can be measured: if it's not by our tongues, ears, eyes, skin, or nose, it's be the lab equipment that we design to measure things that we can't do by ourselves.

But this sense is in no way a contender with mathematics, because of a couple of reasons.

The first reason is that it is a language that everyone can agree on. This is one of the most important aspects of it because this one fact makes it possible for more than one person to study the same thing and agree on what a particular construction says and how to interpret it.

The second thing is that explores things that may potentially exist even though we may never measure them.

The reason this is important is because if we only consider the very narrow thing that we are measuring, in relative isolation with all the other stuff that is going on, then it means that again we have to extrapolate from that one point the rest of it all even when the stuff we are extrapolating from may not have actually been realized itself.

This is the thing: you are saying that we should not use procedures that are too wild like the ones proposed, but yet physics and physicists try to build models to predict what has not been already made realizable. We don't try and predict stuff that has been already realized, we try and predict stuff that has not.

Mathematics in a sense has a platonic aspect where it doesn't necessarily correspond to reality, but then again the stuff that we predict doesn't either until it becomes reality.

We can't really make sense of infinity, but what we do is we use math as well as things like art and other mediums to get one particular very narrow context of it.

You can't tell a blind man what red and yellow is and I agree with you. But what you can do is find a way to utilize what they can sense to try and build the best bridge possible to reach the best description possible, even though they can not see it.

For example, I can get a blind person and I can construct geometric figures that have solid edges. This can be used to build the idea of spatiality and geometry. From this I can go further and introduce these kinds of examples to build a language.

I won't probably be able to ever get to the clarity of yellow or red, but it doesn't mean I can't use what I have to make an inwards progression to describing it by building the best bridge to utilize the sensory capacity that already exists.

And this is precisely what we are doing with mathematics every single day, with countless numbers of mathematicians, and contributing scientists and engineers, who are creating a bridge to this higher sensory capacity.

 

[Feodalherren]

1) You really don't. The skills that linguists have, and the skills that novelists have are different. For example, mathematicians live in a world that is very strongly proof-based. Physicists usually don't care about proofs.

2) There are different levels of good. If you take an average physicist, his math skills are likely to be much. much better than those of the general population, but one thing that you'll quickly find is if you go into physics is that you'll be coming into contact with people whose math skills are blindingly better than anything you have.

For most physics, it's necessary to be proficient at linear algebra and PDE's.

I really don't understand how you can say that. I have only done the very basic Newtonian physics and to me it seemed to be more math than anything. Our tests only consisted of a couple of questions that we would manipulate in a thousand and one ways with f=ma what not.
Physicists may not be on equal footing with mathematicians on number theory but they sure as hell can manipulate equations and understand the mathematical language just as well.

 

[Nano-Passion]

I really don't understand how you can say that. I have only done the very basic Newtonian physics and to me it seemed to be more math than anything. Our tests only consisted of a couple of questions that we would manipulate in a thousand and one ways with f=ma what not.
Physicists may not be on equal footing with mathematicians on number theory but they sure as hell can manipulate equations and understand the mathematical language just as well.

You might not see it now, but there is a difference.

Physics requires a type of problem-solving and critical thinking that has little part to do with mathematics. We use mathematics yes. And yes it is our language. However, math is still very different than physics and vice-versa.

In mathematics, you have one way to progress knowledge and you can be certain that you are right given a proof.

In physics however, nature doesn't care about pretty manipulations. It is and just is. There are many ways to come to a conclusion and there are also just as many ways that they are completely wrong.

In math, we model the whole discipline among some common accepted "truths", however arbitrary they are, such as 1+1 = 2 or that you can draw a line between two points.

In physics, we model the whole discipline around how nature acts, not how we want it to act or what is a common accepted "truth." In fact, revolutions in physics happen by debunking these "truths."

 

[twofish-quant]

I really don't understand how you can say that. I have only done the very basic Newtonian physics and to me it seemed to be more math than anything.

You are using math tools, but you aren't creating those tools.

Physicists may not be on equal footing with mathematicians on number theory but they sure as hell can manipulate equations and understand the mathematical language just as well.

Different set of skills. Some mathematicians can be surprisingly bad at simple equation manipulation. The other thing is that you are doing very, very basic stuff, mathematically speaking. Once you get into really complicated stuff, then you'll likely will find people that are just much better at math than you are.

 

[twofish-quant]

You can't just expect a project or endeavor of that magnitude to pay off in the way that one of more modest means would: it's not a fair comparison.

I think you can. If you are on the right track, you should be able to find something that suggests that you are on the right track.

But this is what I don't get: you don't want to take a risk?

I want to take intelligent risks. If you are digging for gold, it makes more sense to dig somewhere that someone hasn't dug before.

Our mathematics that we have is really not one that is suited for analysis of systems with many degrees of freedom or many interdependencies.

Then develop new math. One thing about math is that there are all sorts of mathematical techniques that are useful in analysis with huge numbers of degrees of freedom, but physicists typically get very little training in those techniques.

We are talking about systems with millions upon millions of degrees of freedom, and we are trying to apply techniques that are best suited for hundreds, not many millions or billions.

If often happens that if you increase the degrees of freedom, that it vastly simplifies the problem. If you have an extremely large number of degrees of freedom, then often the system will very quickly go to equilibrium in some fixed point. For example, if you are trading Euros and Dollars, you will go insane trying to track every transaction, but because there are so many transactions, the system very quickly goes to equilibrium and the math to describe it turns out to be quite simple. Now if you start trading dollars and some third world currency, then the transactions are low, and the math gets more complex.

And the argument for even thinking for going in this direction, let alone actually going there is based on this idea that we can not use existing paradigms that were designed for a different kinds of problem (low number of degrees of freedom) to be used effectively and optimally to look at a different kinds of problem (high degrees of freedom).

A lot of physics involves systems with extremely high degrees of freedom. You can model those. The first thing that you do is to try to reduce the number of degrees of freedom by figuring what processes are important and which ones are not. You can also do time scale separations.

The people that work on these high-level approaches have my congratulations despite the fact that nobody really understands anything well enough to really handle it, because they are likely to be the ones that are taking a chance that people two or three or more generations later will benefit, and this is an extremely hard thing for any person without exception to do.

I think the emperor has no clothes. It's not that I'm against research in string theory. It's that I think we have enough people working on string theory, and that if we have someone new, then it might be a good idea to get them to work on something that has nothing to do with string theory.

 

[twofish-quant]

Well in order for something to be 'measureable' it has to be finite and thus not diverge. Physics is based on the principle that things be measureable.

You can do math with infinities. There are hyperreals, surreals, and transinfinite numbers. Also even with physics as it is, it's impossible to determine the state of a quantum system with a measurement.

The theoretical evidence can be found in theorems to do with black holes (and yes I realize it's speculative) in the form of evaporating black-holes, and results about how black-holes may grow (again speculative).

Ummmmm... You are trying to justify a theoretical results with another theoretical result.

This is the thing: you are saying that we should not use procedures that are too wild like the ones proposed, but yet physics and physicists try to build models to predict what has not been already made realizable. We don't try and predict stuff that has been already realized, we try and predict stuff that has not.

Who's we? I spent seven years trying (unsuccessfully) to predict that supernova exist. They obviously do, but we don't know how they work. There are tons of things out there that exist that we don't understand. We have no freaking clue how turbulence works.

Take a tube. Push water through it. If you push it fast enough, it will suddenly go turbulent at some critical Reynold's number. If you can tell me what they number is without actually pushing water through a tube, then that's worth a Nobel prize.

I won't probably be able to ever get to the clarity of yellow or red, but it doesn't mean I can't use what I have to make an inwards progression to describing it by building the best bridge to utilize the sensory capacity that already exists.

I think you can. A lot of physics involves training the "mind's eye" to see things that people couldn't normally see.

And this is precisely what we are doing with mathematics every single day, with countless numbers of mathematicians, and contributing scientists and engineers, who are creating a bridge to this higher sensory capacity.

But the problem I see is that people then use that capability to assume simplicity when this might not exist. For example, if I were to describe a circle to a blind person, this would be rather easy. Now if I were to describe the shape of Cheasapeake Bay to a blind person, this would be hard.

The trouble is that because it's easy to describe circles, people assume that the universe is made of circles when when we look at things, we often see a lot of complexity.

 

[deRham]

It's not that I'm against research in string theory. It's that I think we have enough people working on string theory, and that if we have someone new, then it might be a good idea to get them to work on something that has nothing to do with string theory.

As an outsider to this discussion, I'm wondering if someone can tell me why string theory does have so much activity. Perhaps it is experimentally unfounded, though I would hope that there are people who found the mathematics of an established, successful (experimentally, and theoretically pretty sound) theory of physics to strongly suggest certain string theory approaches.

 

[deRham]

I'm afraid someone will tell me "there have been 6000 threads on this topic" and start getting cranky ... I personally think this topic can't have particularly easy answers if it's still so hotly debated, so I'd be interested at least to be pointed to what some opine to be good discussions of it.

 

[Nano-Passion]

As an outsider to this discussion, I'm wondering if someone can tell me why string theory does have so much activity. Perhaps it is experimentally unfounded, though I would hope that there are people who found the mathematics of an established, successful (experimentally, and theoretically pretty sound) theory of physics to strongly suggest certain string theory approaches.

Active compared to what? I'm no expert so I'll just paraphrase what I've picked up from others.

1) String theory is really beautiful to a lot of folks.
2) String theory had a solid approach to unifying the fundamental forces, and at one point, it was the only viable one. Any further detail is beyond my scope of knowledge though. :tongue2:

 

[chill_factor]

I really don't understand how you can say that. I have only done the very basic Newtonian physics and to me it seemed to be more math than anything. Our tests only consisted of a couple of questions that we would manipulate in a thousand and one ways with f=ma what not.
Physicists may not be on equal footing with mathematicians on number theory but they sure as hell can manipulate equations and understand the mathematical language just as well.

What's F=ma? Adding, subtracting, multiplying, and dividing. Almost all the math in your general physics class, you've learned in 5th grade. That's just one equation, but applied in different ways.

Do you call doing different ways of adding, subtracting, multiplying and dividing "hard math"?

That's called arithmetic. Math itself is based on proofs.

 

[xdrgnh]

Sure. You write the equations of GR, you get ten non-linear partial differential equations. No one fully understands the properties of those equations. What people do in practice is to take those PDE's and the make approximations and simplifications to get you something that you can calculate.

However, if you write those 10 equations in their full glory, there are some basic mathematical questions that are not answers. For example, can you have a wormhole or can you have a "realistic" naked singularity? No one really knows.

It turns out that GR is too complex to handle fully. What people do is to write approximations to actually solve problems. The simplest approximation is Newtonian gravity. If it turns out that this won't work, then you go for simple extensions.

The same turns out to be true for quantum mechanics. Except for the hydrogen atom, the equations are not fully solvable, so a lot of getting numbers involves making approximations.

If someone could come up with actual solutions to those 10 partial differential equations would it be a big deal?. Or does the numerical solutions give enough detail that actually analytically solving them wouldn't give us any new information?

 

[Nabeshin]

If someone could come up with actual solutions to those 10 partial differential equations would it be a big deal?. Or does the numerical solutions give enough detail that actually analytically solving them wouldn't give us any new information?

It certainly would be a big deal, but it's also almost certainly impossible (I don't know of a proof of the impossibility of analytical solutions though, so I can't say it's certainly impossible). Numerical simulations take months and months, and depending on the precision required can go for much longer than that. Even after having the simulations, you have to put a non-negligible amount of effort into extracting the physically relevant physics from the numerical effects causing error and anomalies in your simulation.

 

[twofish-quant]

If someone could come up with actual solutions to those 10 partial differential equations would it be a big deal?

Solutions are known for specific cases. It's being able to understand the dynamics for the general case that we don't know about.

Or does the numerical solutions give enough detail that actually analytically solving them wouldn't give us any new information?

There's a lot of work with numerical solutions, but numerical relativity is a hard and difficult field itself. Setting up those differential equations so that you can solve them on a computer is a very non-trivial thing to do.

This is one of those things that *seems* easy until you actually do it. You think to yourself, all I have are ten equations, how hard can it be to numerically solve them, and then you try, it you run into the same problems that everyone else does.

 

[twofish-quant]

As an outsider to this discussion, I'm wondering if someone can tell me why string theory does have so much activity.

Much activity is relative. There are probably at most about a few dozen people working on it. The thing about string theory is that it gets a lot of press, because you are trying to figure out the theory of everything. People that work on string theory can talk a lot about "God" whereas people that work on ocean physics usually don't.

Perhaps it is experimentally unfounded, though I would hope that there are people who found the mathematics of an established, successful (experimentally, and theoretically pretty sound) theory of physics to strongly suggest certain string theory approaches.

Yes. In the late-1960's and early-1970's, there were some huge breakthoughs in particle physics by using symmetry principles to come up with things like electroweak unification. From the standpoint of the early-1980's, it wasn't totally crazy to try to look for deep symmetries that would unify everything.

Something else to bear in mind is that it takes about seven years to get a Ph.D., and maybe another decade or so to get a permanent position, so if it takes about twenty years to figure out that you are on the wrong path, that's not an unreasonable amount of time.

 

[xdrgnh]

That's very interesting. If I make it to Grad school and beyond I plan on doing research in GR and QFT. If there is no proof that those 10 general PDE in GR have no solutions then I don't see a reason why I should got give it a shot one day. If the solutions are known for specific cases what is preventing them to be known for the general case? Are the equations chaotic in nature?

 

[Nabeshin]

Much activity is relative. There are probably at most about a few dozen people working on it. The thing about string theory is that it gets a lot of press, because you are trying to figure out the theory of everything. People that work on string theory can talk a lot about "God" whereas people that work on ocean physics usually don't.

One reason string theory is still described as the most active field is that there were a lot of hires around 2000, compared to theoretical physics as a whole. In some years, more then half of the new high energy hires were string theory.

See: http://www.physics.utoronto.ca/~poppitz/Jobs94-08.pdf for the distribution, and note what it's been doing the past few years.

 

[Feodalherren]

What's F=ma? Adding, subtracting, multiplying, and dividing. Almost all the math in your general physics class, you've learned in 5th grade. That's just one equation, but applied in different ways.

Do you call doing different ways of adding, subtracting, multiplying and dividing "hard math"?

That's called arithmetic. Math itself is based on proofs.

Not really. I don't know where you did your basic physics but it was much more than that for us. You can plug plenty of stuff into that equation, especially when you are supposed to show why something happens mathematically.
At any rate, that's besides the point. My argument was that a physicist needs to be good at math like a novelist needs to be a good linguist.

 

[Nabeshin]

At any rate, that's besides the point. My argument was that a physicist needs to be good at math like a novelist needs to be a good linguist.

I have to agree with nano, this is just a result of you not having done much mathematics at all. Mathematics is not really at all about the symbol manipulation physicists do. That's not to say that it's easy, but it's not mathematically interesting. The heart of mathematics lays more in the abstract proof, generation of structure, discovery of relationships, etc. See for example http://en.wikipedia.org/wiki/Abstract_algebra , and note that (aside from its motivation) this isn't really about solving equations at all! I advise you to click around some links and see some more 'mathy' mathematics, and realize that it's nothing at all like what's done in a physics class!

 

[Feodalherren]

I have to agree with nano, this is just a result of you not having done much mathematics at all. Mathematics is not really at all about the symbol manipulation physicists do. That's not to say that it's easy, but it's not mathematically interesting. The heart of mathematics lays more in the abstract proof, generation of structure, discovery of relationships, etc. See for example http://en.wikipedia.org/wiki/Abstract_algebra , and note that (aside from its motivation) this isn't really about solving equations at all! I advise you to click around some links and see some more 'mathy' mathematics, and realize that it's nothing at all like what's done in a physics class!

Algebra is just one field of mathematics though? As far as I've been told, math is about finding patterns. And physics is about patterns in the natural world.

I will find out soon enough as I progress toward my degree.

 

[chiro]

Algebra is just one field of mathematics though? As far as I've been told, math is about finding patterns. And physics is about patterns in the natural world.

I will find out soon enough as I progress toward my degree.

It's easier to talk about math in terms of representation, constraints, and transformations. These three things underly all of mathematics including analysis, algebra, logic, probability, topology and so on.

Finding patterns can be seen in terms of the above things: patterns are discernable by the representation used to describe something. The more compact a representation is, the easier it will be to discern a pattern.

You also have to remember that you can decompose something in many ways, and a decomposition is a transformation. Each decomposition will tell you something specific to the context of that decomposition.

By taking a large system and reducing it to descriptions of lower descriptive complexity, you are finding common patterns. Scientists and mathematicians talk about beauty being simple, and this is one way of understanding that statement.

 

[Nabeshin]

That's very interesting. If I make it to Grad school and beyond I plan on doing research in GR and QFT. If there is no proof that those 10 general PDE in GR have no solutions then I don't see a reason why I should got give it a shot one day. If the solutions are known for specific cases what is preventing them to be known for the general case? Are the equations chaotic in nature?

The known solutions apply in situations of ridiculously simplifying symmetry. For example, the FLRW solution has a universe which is completely homogeneous and isotropic. The black hole solutions are completely axisymmetric, as is the TOV star. Now, to try to move a step up to a solution containing TWO of these objects, no analytical solution exists. While it can still be symmetric, the complication of the second object simply hasn't been overcome (which is not to say people aren't trying...).

If you don't understand what's standing in the way of a 'general' solution to the einstein equations, consider the following:
1) We still have no general solution for the Navier-Stokes equation, which describes simple fluid flow. Indeed, existence and smoothness haven't even been shown!
2) Write out the Einstein equations in terms of the metric tensor. This should help you understand the magnitude of the undertaking.
3) MOST nonlinear differential equations don't have general solutions anyways. Furthermore, the EE are very complicated nonlinear DE, so to hope for a generic solution describing every possible spacetime which is possible is folly.

 

[xdrgnh]

Well I took an intro to DE class and I hated the fact that it was a cook book class and that is why I remember nothing from that class. What I did get from it is the flaw of using purely mathematical/theoretical methods to solve DE. Instead I prefer to use physics and intuition to guess the solution to the DE.This usually works in physics when math techniques becomes to complicated to use. Maybe it's our lack of intuitively understanding these equations that is preventing us to find a solution to them. For example the equation for simple harmonic oscillator can be solved without now how to solve 2nd order DE and characteristic equations because have a intuitive feel of the motion.

 

[Feodalherren]

It's easier to talk about math in terms of representation, constraints, and transformations. These three things underly all of mathematics including analysis, algebra, logic, probability, topology and so on.

Finding patterns can be seen in terms of the above things: patterns are discernable by the representation used to describe something. The more compact a representation is, the easier it will be to discern a pattern.

You also have to remember that you can decompose something in many ways, and a decomposition is a transformation. Each decomposition will tell you something specific to the context of that decomposition.

By taking a large system and reducing it to descriptions of lower descriptive complexity, you are finding common patterns. Scientists and mathematicians talk about beauty being simple, and this is one way of understanding that statement.

I suppose that does make sense. I must admit, the notion that physics and maths are as different as you claim sounds alien to me. Don't get me wrong, I believe you, but it's hard for me to grasp. So far there seems to be little difference.

 

[chiro]

I suppose that does make sense. I must admit, the notion that physics and maths are as different as you claim sounds alien to me. Don't get me wrong, I believe you, but it's hard for me to grasp. So far there seems to be little difference.

It's more to do with the focus itself of the two disciplines than them being different.

Mathematicians focus on different things than physicists do and as a result of this, the context is different.

Mathematicians like generality, physicists and other scientists like specificity. Mathematicians focus on situations corresponding to any reality, physicists focus on this one (look up Platonic viewpoint for more information).

It's not just that mathematicians care more about proofs or formality that makes them different: it's just the focus which determines how one particular person looks at the world.

It's the kind of the thing where if you got five random people to say what they thought something was without any of them having any kind of serious exposure (and the five were from completely different backgrounds in stark contrast to each other), you would see things in all of the candidates that would be also in contrast in all likelihood.

A programmer might look at system in terms of algorithms, structure, design, flow and so on. An artist might use more visual or dynamic interpretations to understand something. A teacher may use cognitive knowledge and understanding gained through teaching experience.

They all have a completely different focus.

One great thing though, is that we are moving from isolated disciplines to an interdisciplinary approach to learning. It used to be that someone would study one or two main areas, but now people are starting to mix areas that were segregated previously together.

This is resulting in forms of thinking that only polymaths could do, but because it is becoming a lot more widespread, and because of the availibility of both raw and processed information (like say the internet or big university libraries), this is becoming a common thing.

 

[Jorriss]

I suppose that does make sense. I must admit, the notion that physics and maths are as different as you claim sounds alien to me. Don't get me wrong, I believe you, but it's hard for me to grasp. So far there seems to be little difference.

I'm a bit surprised. I wonder what an intro course could of been like to think physics is just math.

 

[twofish-quant]

If the solutions are known for specific cases what is preventing them to be known for the general case?

The equations are non-linear. That means that you can have a solution for scenario A. A solution for scenario B, but when you mix the two, you end up with behavior that is completely different from the two scenarios. In other words, because the equations are non-linear, you can't break up the equations.

This isn't a problem just with GR equations. You are going to run into non-linear PDE's all over the place in physics.

Are the equations chaotic in nature?

Under some situations. Yes.

Also there is the issue of what "chaos" means in GR...

http://arxiv.org/pdf/gr-qc/9612017v1.pdf

 

[twofish-quant]

Instead I prefer to use physics and intuition to guess the solution to the DE.This usually works in physics when math techniques becomes to complicated to use.

That works some times, but the trouble comes in if you are in a situation in which you have no real physical intuition. For example, since I don't run into black holes in daily life, I don't have any physical intuition as to how black holes behave. Same thing with electrons. If I try to use intuition to figure out how electrons behave based on things that I see in my daily life, I'll get it wrong.

Also something that people often do is to take the equations, and try to "create" physical intuition. I've stared at the equations for stellar evolution and the equations for cosmology long enough so that I have a "gut feeling" for how those equations behave.

Maybe it's our lack of intuitively understanding these equations that is preventing us to find a solution to them.

Sure. If we had some black holes nearby we could play with them, but we don't.

 

[nucl34rgg]

The best advice I can give is to make sure you know the math really well before attempting some of the harder topics. Otherwise it is just that much more confusing.

 

[amiras]

The best advice I can give is to make sure you know the math really well before attempting some of the harder topics. Otherwise it is just that much more confusing.

What topics in physics are considered "harder" ?

 

[chill_factor]

Not really. I don't know where you did your basic physics but it was much more than that for us. You can plug plenty of stuff into that equation, especially when you are supposed to show why something happens mathematically.
At any rate, that's besides the point. My argument was that a physicist needs to be good at math like a novelist needs to be a good linguist.

I think the first question is irrelevant but if you must know I did it at a University of California campus.

No, it is NOT much more than that. There is absolutely no math in basic physics that you haven't seen before unless you are behind on math. What you said about being able to plug many things in is the science being hard. Executing the calculation itself (what you think is "math") is not hard in basic physics, in my opinion.

Now, it might be hard in upper division classes, but that's not math, that's the arithmetic being hard. 5.598866*e^0.74795 is analytically very hard to solve; try it without a calculator. But that's arithmetic, not math. Math is about proofs and logic.

Even upper division quantum mechanics has only basic linear algebra and basic calculus as absolutely necessary to solve problems, the rest you should be able to pick up in the class itself. There may be arithmetic manipulations that are hard, but that's not math. Math is about proofs and logic.

Also, your understanding of physics and math is mostly around fundamental theoretical physics, but that's not what 99% of physicists do. Outside of fundamental theoretical physics, such as in applied physics, they have nothing in common. Just as an novelist doesn't care about the theory of linguistics, syntax and patterns in languages...

How much math is in this physics research article? http://arxiv.org/pdf/1207.0895.pdf

 

[chill_factor]

The best advice I can give is to make sure you know the math really well before attempting some of the harder topics. Otherwise it is just that much more confusing.

I think you have this spot on:

You don't need to know too much advanced math for most courses in physics.

But you must MASTER basic math, and if you have mastered basic math, everything else will fall into place.