Is Mathematics the Best Language for Understanding the Universe?

[WindScars]

Modeling it on my computer, so I can see it happening. For example, I create 2 hydrogen atoms. I can see a point indicating the position of the nuclei, clouds showing the orbitals. Then I can see the atoms slowly approaching and their molecular orbitals forming as they bind. I'm not sure this is what would happen but testing and playing is the idea. If this is not absurd, wouldn't it be great?

 

[Jorriss]

Modeling it on my computer, so I can see it happening. For example, I create 2 hydrogen atoms. I can see a point indicating the position of the nuclei, clouds showing the orbitals. Then I can see the atoms slowly approaching and their molecular orbitals forming as they bind. I'm not sure this is what would happen but testing and playing is the idea. If this is not absurd, wouldn't it be great?

But what are you using to model them? Did you write the code yourself or just downloaded it somewhere?

 

[WindScars]

What do you mean? I'm asking if this is possible/exists? How could I write a code for this if I don't understand QM?

 

[Jorriss]

What do you mean? I'm asking if this is possible/exists? How could I write a code for this if I don't understand QM?

Yeah, that's what I was wondering lol. You could to something like that but it's more chemistry than quantum mechanics. It would be neat but you wouldn't exactly be learning quantum.

Jorriss, I can't answer you because I don't understand the motions of an atom. That is the point. If I could create isoled atoms, put them together, and their resulting motions leads to precise reactions in relation to what would be chemically expected, this would be awesome. I'm not sure if this is possible, but why not? And group theory? What is it? (=

From a physics point of view, group theory is the math behind symmetry.

 

[Ans426]

Hmm this is pretty actually, but why this? Ans426 just asking. As you understand QM, do you consider yourself to understand an atom? Can a QM expert predict if a chemical reaction will occour (without using chemistry/testing on lab)? What would you tell me if I asked you what is the path of the motion of an electron around a hydrogen nuclei?

Jorriss, I can't answer you because I don't understand the motions of an atom. That is the point. If I could create isoled atoms, put them together, and their resulting motions leads to precise reactions in relation to what would be chemically expected, this would be awesome. I'm not sure if this is possible, but why not? And group theory? What is it? (=

Before we actually go into any technical details, I think the thing you should first learn now is how to appreciate the use of mathematical tools and the importance of it to understanding what you're seeing.
People before Newton had always been looking at the best computer simulation ever, looked at stars orbiting around them and apples falling on the Earth, but no one realized that by using some mathematical tools one could unify them using a single equation. Gaining some intuition from simulations is certainly important, but not learning the Maths behind will hinder you from understanding the real theory.

Here's another nice quote from Feynman:
"To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature ... If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in. "

 

[WindScars]

But understanding the math if you have no clue of what happens there has no meaning. First you observe how something acts and then you use the math to describe it precisely. It is not the opposite. But if the pre-requisites are just what you said, I'm okay with it. I think I can get it in 2 months or so?

Yet you didn't answer me. If you're asked to describe an electron as a particle, how would you draw a line showing it's motion around a hydrogen atom? Would it be a circle, like a planet orbit? "Teleporting" points randomly distributed according to the wave function? A (spinning?) line crossing the nuclei several times? What shape would it form? Why?

Jorriss not? So what quantum is about? I thought it was what described the motion of elementar particles. While chemistry was just a field of knowledge derived from laboratory experiments dealing with chemical reactions - that is, something that involves millions of atoms simultaneously, so, basically, a large-scale event (even if rooted on the small atoms). Is this wrong?

 

[Jorriss]

But understanding the math if you have no clue of what happens there has no meaning. First you observe how something acts and then you use the math to describe it precisely. It is not the opposite. But if the pre-requisites are just what you said, I'm okay with it. I think I can get it in 2 months or so?

You don't observe then do the math as a student (even as a research scientist, science is not as cut and dry as the scientific method you learn in a textbook). People have centuries of observation you can learn from in the form of the math they have developed to explain said observations. No one here is advocating that you should not learn what's happening physically but that you learn about how a system behaves physically, MORE DEEPLY, by understanding the mathematical structure of the theory.

If you are very smart, you can do it in two weeks, I don't know how good you are.

Yet you didn't answer me. If you're asked to describe an electron as a particle, how would you draw a line showing it's motion around a hydrogen atom? Would it be a circle, like a planet orbit? "Teleporting" points randomly distributed according to the wave function? A (spinning?) line crossing the nuclei several times? What shape would it form? Why?

It's none of those. You don't simply draw a line charting it's trajectory.

This is a deep question about measurement which, unlike classical physics, is a very nontrivial process.

Jorriss not? So what quantum is about? I thought it was what described the motion of elementar particles. While chemistry was just a field of knowledge derived from laboratory experiments dealing with chemical reactions - that is, something that involves millions of atoms simultaneously, so, basically, a large-scale event (even if rooted on the small atoms). Is this wrong?

Chemistry is a vast field. Some chemists work on describing the electronic structure of molecules using classical mechanics. Some chemists use statistical mechanics to describe the behavior inside a cell or to describe anomalous diffusion. Some chemists work on single electron transport and quantum dynamics. Chemistry is a very varied field.

But what I was specifically referring to was how you mentioned orbitals. The idea of watching an orbital overlap with another orbital and seeing the MO's form is more chemistry than physics. The actual way two hydrogens would come at each other and form a bond is much more involved and frankly, not worth getting into with your background. I don't mean that as an insult, but you're not there yet and it's late.

 

[homeomorphic]

Like trying to explain sight to the blind.

You can't really understand quantum mechanics by just looking at electron clouds.

That doesn't mean that you will understand it by doing a lot of calculations. There are deep ideas involved. To me, the calculations are like a testing ground for the ideas. For some things, it would probably make no difference if you use a computer, but, yes, algebraic skills are useful.

To give you an example, I am working on a talk on classical mechanics. I had an intuitive idea of how to explain where some equation comes from. But in order to check and make sure that I could even communicate the idea effectively to someone else, as well as its correctness, I had to do a little algebraic calculation. That was the only way I could verify that my intuition was correct. So, yes, algebraic skills are relevant to understanding how nature works, although, I usually like to think in pictures (often very vague and abstract ones that are not direct representations of the things in question) and use logic and calculation mainly to verify my ideas or get answers if I need them.

Essentially, studying quantum mechanics by looking at simulations of electron clouds would be like trying to learn how a car works by looking at it, rather than studying mechanical engineering. It doesn't work that way. You don't just pick it up by looking. You have to understand theory. Not just look.

 

[WindScars]

Why can't you simple draw a line charting it's trajectory? Is it a particle? If I freeze time, will I be able to see where it is?

I don't believe in smartness but thanks for the estimative. If this is so, then there's no problems. But can you advise me on books more compact than stewart on those pre-requisites? I mean, books that go straigth to the subject, without billions of exercises and 50 pages explaining how calculus can be used to calculate the volume of a cup (lol)?

homeomorphic, this is true. But could you learn to play piano without ever touching one?

 

[Jorriss]

Why can't you simple draw a line charting it's trajectory? Is it a particle? If I freeze time, will I be able to see where it is?

Frankly, I can't just explain it right now. Measurement and particle 'trajectory' are totally different beasts in quantum mechanics. It's not a particle, it's not a wave either per say. The very state of matter is entirely different in quantum.

I don't believe in smartness but thanks for the estimative. If this is so, then there's no problems. But can you advise me on books more compact than stewart on those pre-requisites? I mean, books that go straigth to the subject, without billions of exercises and 50 pages explaining how calculus can be used to calculate the volume of a cup (lol)?

I can't really suggest a different book that is shorter. I can try and tell you textbooks that are more enjoyable though.

You really should not skip anything in that calculus book. It is all useful. You probably can't see it now but that section on volumes of revolution? Totally useful. That section on L'Hopitals method for evaluating indeterminate limits? Completely useful? That section on related rates? Very useful.

I'm still an undergraduate, but I can tell you conclusively just based on a couple more years of coursework and a little bit of research that it's all useful.

Don't try and get ahead of yourself, just try and enjoy the road to quantum. It's a good one.

 

[WindScars]

It's all useful but obvious? You really need to tell someone how to calculate the volume of a solid? And L'Hospital, well, just tell me once that it can be used to evaluate indeterminate limits, bother to explain why (it didn't and I still don't know!), give me one exercise per typical case and this is it. If you know a textbook like that please tell me.

Can't you try to explain briefly, before I get into it? For example. Can't the wave function be just a result of the motion of the electron? The electron is actually a particle moving at almost c around the nuclei. The nuclei is short, though, so it makes billions of leaps per second. So we can't see it, but can estimate the probability of it being at certain position just because this is the place the electron spends more or less time at. Is that explanation illogic? Why?

 

[Shivam123]

perfectly right in that issue that doing two page calculations can be better replaced with understanding the universe. I personally have the very same opinion and that's why i am weak in maths.!after all maths particularly algebra is completely man made and we even have devices to do such calculations then why does our education system demand us to keep practicing them and wasting hours and hours when we could have so better utilized it with understanding the world and every beautiful thing it contains!


But the idea of replacing it with programming is..i don't know..limited..i mean you will have to keep in confined to the universities and all..you can't expect a person to fish out an electronic device out of his pocket to do the programming every time he is asked a 2+2 or even a 29 square

 

[Jorriss]

It's all useful but obvious? You really need to tell someone how to calculate the volume of a solid? And L'Hospital, well, just tell me once that it can be used to evaluate indeterminate limits, bother to explain why (it didn't and I still don't know!), give me one exercise per typical case and this is it. If you know a textbook like that please tell me.

You seem to underestimate how important practice is. Sometimes evaluating a limit requires someone take to a function which is not indeterminate at a given point and make it indeterminate as to use L'Hopitals rule. This doesn't become natural or intuitive without lots of practice. Granted, a lot of professors do go overboard with how many homework problems they want you to do in intro classes but don't think it is useless. And math isn't simply about typical cases, most integrals that can be evaluated are 'typical' if you happen to notice the right way to do it.

And yes, you need to tell someone how to calculate the volume of a solid. Before calculus could you have found the volume bound by various functions of x,y,z? Do you not find it the least bit insightful to see how calculus can show you that? What if someone said find the volume of a sliver of a 6-dimensional hypersphere in phase space? That's certainly not intuitive, but if you pay attention to your calculus, it's doable and what it means is sensical.

A textbook with one problem per typical case would be 10,000 pages long there are so many seemingly different problems that can be developed.

Can't you try to explain briefly, before I get into it? For example. Can't the wave function be just a result of the motion of the electron? The electron is actually a particle moving at almost c around the nuclei. The nuclei is short, though, so it makes billions of leaps per second. So we can't see it, but can estimate the probability of it being at certain position just because this is the place the electron spends more or less time at. Is that explanation illogic? Why?

The problem is that I can't explain it briefly, for one, the wavefunction itself isn't even an observable. You really need to just study the subject and, apparently, forget what you think you know about how small particles behave.

 

[homeomorphic]

But could you learn to play piano without ever touching one?

No, but I'm not sure if that's a good analogy. Playing with simulations of molecules might be good for some purposes, but it won't give you the deepest insight into how nature is working at a more fundamental level. Depends on what your goal is.

 

[WindScars]

This was very intuitive for me? I never used those formulas, but some weird methods I made up to calculate volume on high school, that kind of resemble calculus. But what is the intuitive way to calculate a volume, then? And why do you have to make the same kind of exercise several times? It's the same thing...

And I don't think anything actually, but it's impossible not to speculate. But allright, this just makes me more motivated. I hope those pre-requisites are as short as you are promising me. I'll come back when I finish them. But I really need better resources... if someone

homeomorphic
My goal is clear, I want to contribute to nanotechnology, and I'm sure I'll have to play a lot with atoms and their motions if I want to make anything relevant. And I'm sure this is not impossible. How a scanneling tunelling microscope works, by the way? Do you guys know how did they make this? http://www.exitmundi.nl/NanoIBMzijaanzicht.gif

 

[homeomorphic]

perfectly right in that issue that doing two page calculations can be better replaced with understanding the universe. I personally have the very same opinion and that's why i am weak in maths.!after all maths particularly algebra is completely man made and we even have devices to do such calculations then why does our education system demand us to keep practicing them and wasting hours and hours when we could have so better utilized it with understanding the world and every beautiful thing it contains!

If you are weak at math, you can't understand the universe. Math isn't just calculations. It also involves ideas.

But the idea of replacing it with programming is..i don't know..limited..i mean you will have to keep in confined to the universities and all..you can't expect a person to fish out an electronic device out of his pocket to do the programming every time he is asked a 2+2 or even a 29 square

Exactly. I got a minor in computer science, but I am still too lazy to use a computer for anything unless it will save me a huge amount of work. Very few people will think computers are easier to deal with, most of the time. It's just easier to calculate. Often, if I try to use computer algebra systems, I get ugly results that would not have resulted if I had done it by hand. It can be hit and miss. Granted, I haven't used computer algebra systems for years, so maybe they have improved. But why would I need a computer to calculate the integral of some polynomial? It will take me two seconds to do it by hand, so what's the big deal? A lot of calculations really aren't that hard.

Everyone should go through at least a few very ugly calculations in their life, just so that they are aware of what can be necessary to solve a problem in some unfortunate cases. But sometimes, it can get out of hand.

 

[homeomorphic]

My goal is clear, I want to contribute to nanotechnology, and I'm sure I'll have to play a lot with atoms and their motions if I want to make anything relevant.

That may be, but you won't understand quantum mechanics that way, alone. It will be only one small piece of your education.

 

[Jorriss]

This was very intuitive for me? I never used those formulas, but some weird methods I made up to calculate volume on high school, that kind of resemble calculus. But what is the intuitive way to calculate a volume, then? And why do you have to make the same kind of exercise several times? It's the same thing...

What formulas? I didn't mention a formula. I didn't speak to a specific formula to plug something into.

I'd also be curious what these methods you developed are that allow you to find volumes in arbitrary dimension, of arbitrary shape that resemble calculus.

 

[WindScars]

Jorris, you got me wrong again. I was talking about the volume formulas, and my methods were limited to basic volumes of very limited shapes. Just that it had some of the essence of calculus on it - calculus is the intuitive way to calculate volumes. Now that I think about it, though, perhaps it could be extended a few dimensions. But that would be useless. *hmm* I like this ... sorry (=

hemeopathic but this is a matter of pratice. After you get used to, it's as easy as writting. I can do anything I can do with math by programming, but I can't do a piece of what I can do programming with math.
And of course I won't understand QM with that alone. But I guess it would be more useful to nanotech?

Again guys thanks this topic actually helped me alot, even though it was not the purpose. I'm going to sleep. I'll be thankful if someone recommends books that could give me a good preparation to QM.

 

[homeomorphic]

hemeopathic but this is a matter of pratice. After you get used to, it's as easy as writting. I can do anything I can do with math by programming, but I can't do a piece of what I can do programming with math.

No, you can't do anything with programming that you can do with math. Do you know how to program computers to solve algebraic equations and get an exact and clean answer? I gave you an example where I needed to calculate by hand. That was because the calculation was making my intuition precise. If I had a computer do it, first of all, programming computers to do the kind of calculation I was doing would be very difficult because it involved steps that it's much easier to do yourself, and secondly, it wouldn't make my intuition precise because the computer did it for me, so I wouldn't be able to see what happened for myself.

I got a CS minor--I know how to program. It's just easier to do it by hand, in most cases, and it's better practice to do it yourself.

And of course I won't understand QM with that alone. But I guess it would be more useful to nanotech?

I'm not sure. But it's more of a chemistry or molecular physics type thing to do.

Again guys thanks this topic actually helped me alot, even though it was not the purpose. I'm going to sleep. I'll be thankful if someone recommends books that could give me a good preparation to QM.

First, you would probably want to learn intermediate classical mechanics and maybe electricity and magnetism. Also, maybe linear algebra. You can try the Susskind lectures online for classical mechanics and quantum mechanics. You can start with the series on classical mechanics, then quantum entanglements, and then, quantum mechanics. There are very limited prerequisites to understanding them.

 

[mal4mac]

I think that the problem is more one of checking the computer's accuracy. Numerical methods can be unstable for certain classes of problems, and all non-trivial programs have bugs anyway.

If you don't understand the math, you won't be in a position to properly evaluate a computer generated answer.

Just think about calculators for a second. If I ask you to divide 117.938 by 19.767, clearly a calculator will be much quicker. But if you come up with an answer of 56.556, how will you know that you accidently pressed '1' twice without knowing that 120 / 20 is about 6?

(i) Repeat the calculation. Like repeating experiments. (ii) Get someone else to calculate the result on their (different) calculator. (iii) make friends with a mathematician and get them to do the checking by hand.

Some people, like Greenspan, have looked at doing physics only with arithmetic & using computers:

http://www.springerlink.com/content/3247645wx6g57654/

Experimental particle physicists tend to avoid the hard theoretical track in later courses, couldn't computational physicists do the same (if they don't already!)? If there is this split in later courses, why not have the split just before calculus at the UG level?... This may be a bit too radical ... everyone should have some appreciation of calculus, just so they can talk to each other... but couldn't the sequence be shortened so computer wizards could do more computing?

 

[deRham]

It's about changing arcaic hand algebra to programming as the main tool for humans to work with math

If you mean to say instead of doing a million drills with basic high school algebra, being trained in programming is better, I agree! Programming is yet to appear as an integral part or high school education. That students have to learn how to prove basic facts in Euclidean geometry and don't program is a mistake.

That said, mathematics as a field is neither about calculation nor programming. Where applicable, I would argue research mathematicians DO use advanced programs to aid in making predictions, etc.

Programming is a skill, whereas mathematics is an actual field. Even as we program, it is often good practice to jot down notes. Some use a computer to do this, others use pen and paper. At that point of course, it doesn't matter.

We could try advancing what basics facts in math we use a computer to verify, sure. However, my experience with programming suggests that telling a computer what to do from scratch when you know how to do it is usually a waste. Somebody would have had to implement a user-friendly way of using the computer so that one could modify just salient parts of a program, like add or subtract some variables, for this to be of any use.

For fields of math like geometry or topology, I think it can simply easier to draw the picture and think about it.

 

[Shivam123]

If you are weak at math, you can't understand the universe. Math isn't just calculations. It also involves ideas.

No really not..Your opinion is that math involves ideas all good i agree but firstly "understanding those ideas" is in no way going to make our understanding of universe better or improve quality of life and 2nd even if we do understand them and even learn how to apply them it may just refine our mental ability a little bit..and nothing more.

And about understanding the universe i am pretty sure it involves proper experimentation observation and thinking..those are the things i will do..and leave the calculation part to my computers..

 

[Angry Citizen]

Generic reply to thread that I don't believe anyone else said:

Algebraic (read: symbolic) calculations can reveal a great deal about the interplay of variables. Sometimes, algebra reveals that certain variables cancel out and do not affect the problem at all. This would not be apparent in numerical methods. From an engineering standpoint (and likely a physics standpoint as well, but I won't speculate), this is incredibly useful.

I can't help but agree that memorizing integral tables and crunching numbers is pretty pointless, though. Go through the calculations by hand, definitely, but if I run into some godawful integral whose answer involves, say, the inverse tangent function, you bet I'm not going to remember what the integral is. Plug it into a computer, let it chew on it for a while.

 

[homeomorphic]

No really not..Your opinion is that math involves ideas all good i agree but firstly "understanding those ideas" is in no way going to make our understanding of universe better or improve quality of life and 2nd even if we do understand them and even learn how to apply them it may just refine our mental ability a little bit..and nothing more.

And about understanding the universe i am pretty sure it involves proper experimentation observation and thinking..those are the things i will do..and leave the calculation part to my computers..

I'm sorry, but obviously, you don't know the first thing about physics or math or the relationship between them. The idea that mathematical ideas are not relevant to understanding the universe is, frankly, hilarious.

Quantum mechanics would not have been discovered without lots of mathematical ideas. What do you think Schrodinger was doing when he came up with his equation?

In optics, you can use a ray model of light (in which light could be thought to be moving like a particle) to approximate. There were hints that not just light but ordinary matter behaved that way. Evidently, what he was doing was trying to find a wave equation for matter that gave you back good old Hamiltonian mechanics, just like how you can derive the ray model of light as a limiting case of the wave model. There's really no clear boundary between math and physics. He was basically doing math. It involves lots of partial differential equations and Fourier analysis. You don't seem to get the idea that equations can actually have a physical meaning and that's often the way mathematicians think about them.

Without Schrodinger's equation, the kind of simulations that we've been talking about with molecules would not even be a possibility.

Perhaps, if you had said current research in math will not help us to understand the universe, you might be on slightly more solid ground, but still mistaken. For example, I'm learning about applications of topology and category theory to the study of anyonic condensed matter systems. Microsoft Research has a whole quantum computing group dedicated to studying this stuff and build quantum computers based on these ideas. The book I am reading is by someone who works there. He's mathematically trained, but most of the people there are physicists. It appears that the very heavy math I am doing is quite relevant to the physics involved. Of course, it's easy to say that, and I'm taking it mostly on faith, myself, but, there's a reason why they have a whole squadron of physicists and a few mathematicians there working on it.

If you mean that calculations do not help us understand, I would have slightly more sympathy with that. I'm not a fan of calculations at all, personally. I always try to avoid them when possible. But they do have their place. Actually, I would say probably most professional mathematicians don't like to calculate that much. In many cases, calculations don't add anything to your understanding and that's why I try to keep them out of the theory. But as practice or as a way to get answers, calculation is good and necessary.

 

[espen180]

I remember someone on PF had a signature saying "Theoretical physics is locally isomorphic to mathematics". The more I study physics, the more I find this statement to be true. :)

 

[Number Nine]

No really not..Your opinion is that math involves ideas all good i agree but firstly "understanding those ideas" is in no way going to make our understanding of universe better or improve quality of life and 2nd even if we do understand them and even learn how to apply them it may just refine our mental ability a little bit..and nothing more.

And about understanding the universe i am pretty sure it involves proper experimentation observation and thinking..those are the things i will do..and leave the calculation part to my computers..

Part of the problem here is that you don't seem to have very much experience with mathematics beyond, say, applied calculus. Mathematics, real mathematics, rarely involves the kinds of calculations you're thinking of (or calculations of any kind). Mathematics is a language that allows that precise discussion of structure and relationships; it is trivially true that mathematical concepts can be programmed, but not until you understand what to program. As an exercise, write me a program that will tell me for which n it is possible to construct a projective plane of order n (good luck taking into account properties of Galois fields, which are necessary for the proof, without understanding what Galois fields are).

Homeomorphic is right; you absolutely cannot understand physics without mathematics. Math isn't just a tool physicists use to manipulate numbers; mathematics is the language in which properties of the Universe are expressed. You cannot understand, say, quantum mechanics without a thorough understanding of Hilbert space, or general relativity without understanding what a manifold is (and the requisite knowledge of topology and analysis that such concepts entail).

 

[chiro]

Part of the problem here is that you don't seem to have very much experience with mathematics beyond, say, applied calculus. Mathematics, real mathematics, rarely involves the kinds of calculations you're thinking of (or calculations of any kind). Mathematics is a language that allows that precise discussion of structure and relationships; it is trivially true that mathematical concepts can be programmed, but not until you understand what to program. As an exercise, write me a program that will tell me for which n it is possible to construct a projective plane of order n (good luck taking into account properties of Galois fields, which are necessary for the proof, without understanding what Galois fields are).

Homeomorphic is right; you absolutely cannot understand physics without mathematics. Math isn't just a tool physicists use to manipulate numbers; mathematics is the language in which properties of the Universe are expressed. You cannot understand, say, quantum mechanics without a thorough understanding of Hilbert space, or general relativity without understanding what a manifold is (and the requisite knowledge of topology and analysis that such concepts entail).

To build on what Number Nine said above, mathematics is the best language in terms of preciseness and flexibility.

Mathematics gives us the best way to define something explicitly: it is absolutely clear that when you give a precise definition that it will be understood by everyone acquainted with adequate training in the language and on top of that, there is no room for debate due to its precise nature.

On top of this it also allows us to describe literally anything, since everything can measured physically in some sense and mapped to numbers via instruments of some sort, or maps can be made between 'stuff' that map some object to a number.

For these reasons, mathematics is the best thing we have in terms of a language and in combination with the things that other posters have said, it is very hard to debate an alternative to mathematics for these purposes.