Is Mathematics the Best Language for Understanding the Universe?

[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 Schrödinger 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 Schrödinger'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.