What is the point of studying maths at a very high level?

[VertexOperator]

Please do not misunderstand the purpose of my question! I love maths and I respect mathematics and mathematicians very much but I want to know the point of doing maths at a very high level. To me it seems like maths is only useful when used as a tool in engineering or sciences like chemistry and physics.

 

[Number Nine]

And how would you develop those tools without doing research in mathematics?

 

[port31]

they use very advanced math in physics. group theory, topology and more

 

[WannabeNewton]

And how would you develop those tools without doing research in mathematics?

This pretty much sums it all up. I don't think there is anything left to say.

 

[Pagan Harpoon]

I agree with the above posters. In some cases, though... people just do it because it's cool. "What's the point of art?" is in a similar vein.

 

[ImaLooser]

Please do not misunderstand the purpose of my question! I love maths and I respect mathematics and mathematicians very much but I want to know the point of doing maths at a very high level. To me it seems like maths is only useful when used as a tool in engineering or sciences like chemistry and physics.

Those people who do very high level math are obsessed. It isn't terribly rational.

It has no application today, but tomorrow never knows. My guess is that some day it will come in handy.

 

[jgens]

Those people who do very high level math are obsessed.

I disagree. All of my math professors have been pretty normal people who just also happen to like math.

It isn't terribly rational.

If you judge rationality based on perceived utility, then perhaps. But I think that sort of view is shortsighted. For most of the mathematicians I know, in one way or another, their interest in math comes down to its aesthetic quality. Doing math because you find it beautiful or because you like the way it makes you think seems like a perfectly rational decision to me.

It has no application today, but tomorrow never knows. My guess is that some day it will come in handy.

This depends on what you mean by application. Much of current mathematics research will not be directly useful for any sort of concrete product. On the other hand, a lot of current research is directly (and immediately) applicable to problems in theoretical physics, cryptography, etc.

 

[micromass]

Those people who do very high level math are obsessed. It isn't terribly rational.

It has no application today, but tomorrow never knows. My guess is that some day it will come in handy.

This is quite an ignorant point-of-view. But I guess many people think this way.

Mathematicians are not obsessed and they are not irrational. Most mathematicians are very normal people. If they are not talking about mathematics, then many people would have a difficult time pointing out who is a mathematician and who is not. You would be very surprised.

And although mathematicians don't tend to care about applications, there really are many applications of pure mathematics. Just because you don't know them, doesn't mean that they don't exist!
However, there are some parts of mathematics that don't have applications at all. But there are also parts of engineering or physics without applications. I don't think it's fair to single out pure mathematics here.

 

[Daminc]

I only learned advanced Math to help me think in a certain way (I like physics and I program applications on a computer). Training your brain to think in a logical manner is very useful application in my opinion and is a highly transferable skill.

 

[humanino]

The IAS organized a special year on Quantum Field Theory in 1996-97. Below is the cover from the proceedings :

In the 1970s, physicists were celebrating the incredible successes of the standard model of particle physics (which culminated recently with the discovery of the Higgs boson). Meanwhile, mathematicians had developed powerful algebraic tools to solve wide classes of complex geometrical problems. At the end of the 1990s, the physicists found themselves working on problems for 20 year ago mathematicians, and vice-versa.

Two very specific examples, showing how alive and well the interface between mathematics and physics is, how we could be on the verge of a new revolution in our conception of space-time, can be found in
Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics
Scattering Amplitudes and the Positive Grassmannian

There is nothing more than circumstantial in my choice of examples above. The point is : by the very nature of research, you can not presume what will be important in the future. I illustrated that point with applications in physics because I am familiar with that, but please look up the history behind the jpg format, or as others have alluded to, modern cryptography technics for instance. Mathematics is a universal langage underlying all of science. Mathematicians mostly do it with the same motivations as musicians, painters or philosophers, and we ought to support and even cherish them only for that. Yet, if you are looking for practical reasons, there are plenty.

 

[Jimmy Snyder]

Physics is like sex. Sure, it may give some practical results, but that's not why we do it.

The same can be said of math.

 

[DiracPool]

Training your brain to think in a logical manner is very useful application in my opinion and is a highly transferable skill.

You don't work for Luminosity.com, do you Daminc?

Paul Dirac was notable for his advocay of mathematical puritism in theoretical physics research. It was more important for him that a theory be mathematically beautiful than it conform to experimental data, and he felt that major advances in physics could be achieved through pure mathematical insight. This philosophy was anathema to the hard core experimentalists of the day like Rutherford. However, Dirac "faced" them by predicting spin and antimatter through simply using pure mathematical reasoning. These were not properties of matter he could have conceived of a priori to using these techniques, they were "found" through the maths.

Other than that, there are many other examples of research in maths that were either pure or originally intended for something else LATER being adopted by physicists to attack some newly emerging issue. Of course, Einstein's use of differental geometry to formulate GR is one famous example.

 

[bp_psy]

Counting cards in Vegas. I've seen that on TV.

 

[AnTiFreeze3]

... Mathematicians are ... not irrational ...

False. I have never been successful in trying to express a Mathematician in the form of a fraction.

 

[micromass]

False. I have never been successful in trying to express a Mathematician in the form of a fraction.

Just because you can't do it, doesn't mean that it can't be done!

 

[DiracPool]

Just because you can't do it, doesn't mean that it can't be done!

What do you get when you divide a mathemetician by a physicist, and then raise that sum to the power of a cosmologist?

 

[MathematicalPhysicist]

You can call me insane, but I don't see a big difference between doing pure maths or applied maths, and I enjoy doing both.

Though algebra maybe boring at times.

 

[micromass]

What do you get when you divide a mathemetician by a physicist, and then raise that sum to the power of a cosmologist?

You get a philosopher.

 

[DiracPool]

You get a philosopher.

Bingo! You got it Micromass, good job. First try. Can you see by this result how things have come full circle? Doesn't it all make sense now?

 

[ImaLooser]

This is quite an ignorant point-of-view. But I guess many people think this way.

Mathematicians are not obsessed and they are not irrational. Most mathematicians are very normal people. If they are not talking about mathematics, then many people would have a difficult time pointing out who is a mathematician and who is not. You would be very surprised.

And although mathematicians don't tend to care about applications, there really are many applications of pure mathematics. Just because you don't know them, doesn't mean that they don't exist!
However, there are some parts of mathematics that don't have applications at all. But there are also parts of engineering or physics without applications. I don't think it's fair to single out pure mathematics here.

The way I see it, only about fifty people in this world are doing "very high level" math. The Andrew Wiles's and Grigori Perelmans and Alexander Grothendiecks and Ed Wittens of this world strike me as being very math-focused to the point of obsession. But I don't know them personally, so what do I know.

 

[VertexOperator]

The way I see it, only about fifty people in this world are doing "very high level" math. The Andrew Wiles's and Grigori Perelmans and Alexander Grothendiecks and Ed Wittens of this world strike me as being very math-focused to the point of obsession. But I don't know them personally, so what do I know.

and Terry Tao*

 

[humanino]

only about fifty people in this world are doing "very high level" math

That opinion is indeed enlightening... I suppose you should contact the authors on
http://arxiv.org/list/math/new
to let them know your piece of mind on their work.

 

[VertexOperator]

Why does 'very high level maths' look very simple?
When I watch videos or look at pictures of a top mathematician all I see is something simple like: \tilde{g}_{\alpha \beta }=\iota g_{\alpha \beta } or x^{n}+y^{n}=z^{n}

 

[WannabeNewton]

Why does 'very high level maths' look very simple?

See if this is simple: prove that \mathbb{R}/\mathbb{Z}\cong \vee _{n = 1}^{\infty }S^{1}_{n} where \cong denotes homeomorphic, \vee is the wedge sum, and \mathbb{R}/\mathbb{Z} is the quotient space obtained by collapsing all the integers to a point .

 

[VertexOperator]

See if this is simple: prove that \mathbb{R}/\mathbb{Z}\cong \vee _{n = 1}^{\infty }S^{1}_{n} where \cong denotes homeomorphic

o__OBy the way, can you please help me with this: https://www.physicsforums.com/showthread.php?t=678363

My attempt is terrible I know :(

 

[George Jones]

One answer to the original question:

"I tell them that if they will occupy themselves with the study of mathematics they will find in it the best remedy against the lusts of the flesh." Thomas Mann

I suspect that mathematicians have found counterexamples to this statement.

 

[WannabeNewton]

I have physics lecture in like 5 mins xD. Good luck with that though! Anyways, don't fall under the impression that "higher level math", whatever that may mean to different people, is easy by inspection.

 

[VertexOperator]

I have physics lecture in like 5 mins xD. Good luck with that though! Anyways, don't fall under the impression that "higher level math", whatever that may mean to different people, is easy by inspection.

I can't wait to go to uni :(

 

[WannabeNewton]

I can't wait to go to uni :(

That's what I thought a year ago...but now...

 

[FastWing]

That's what I thought a year ago...but now...

What's wrong with uni?

 

[SW VandeCarr]

What do you get when you divide a mathemetician by a physicist, and then raise that sum to the power of a cosmologist?

You divide a mathematician by a physicist and you get a sum?

 

[micromass]

You divide a mathematician by a physicist and you get a sum?

Yeah, what he said makes no sense. That's why I said the answer was a philosopher, cause they tend to not make sense.

 

[collinsmark]

Here is this week's Abstruse Goose. It seems somewhat relevant to this thread. When I first saw (the strip) yesterday, I wasn't going to post it here. But today? Oh, what the heck.

Impure Mathematics

[Source: http://abstrusegoose.com/504]

[With mouse-over: There is no branch of mathematics, however abstract, which may not someday be applied to the phenomena of the real world --- Nicolai Lobachevsky]

 

[VertexOperator]

Yeah, what he said makes no sense. That's why I said the answer was a philosopher, cause they tend to not make sense.

Impure Mathematics

lololol

 

[DiracPool]

"I tell them that if they will occupy themselves with the study of mathematics they will find in it the best remedy against the lusts of the flesh." Thomas Mann

This sounds like a good argument NOT to study mathematics. Darn, just when I was starting to enjoy differential equations.

 

[AlephZero]

You divide a mathematician by a physicist and you get a sum?

True story about the difference between a mathematican working as an engineer (me!) and a physicist who really wanted to be a mathematian (my boss).

I had written the specifcation for putting some new gizmo into a finite element analysis code, full of details of numerical integration, how to evaluate Jacobians, etc.

It came back from my boss with the comment "You could make this a lot shorter by <insert half a page of divs, grads, curls, and double and triple integrals here>"

So I sent a note back saying "OK, but if this is a spec for a computer program, how would you actually write the code"?

The answer: "You do it your way, of course".

 

[ilmareofthemai]

For most of the mathematicians I know, in one way or another, their interest in math comes down to its aesthetic quality. Doing math because you find it beautiful or because you like the way it makes you think seems like a perfectly rational decision to me.

Agree!
I recently read A Universe in Zero Words, a book about the history and influence of important equations (if you want some examples of crazy mathematicians, read the chapter about the types of infinities-the outcome for a couple of those mathematicians, i.e. Godel, was not so great) and developed a strong appreciation for mathematics, especially geometry. A great read.
Also, as someone else mentioned, math and physics sometimes develop at different rates, and the math Pauli needed was already developed, etc. For more examples, read Euclid's Window, a book about the major mathematical revolutions and their connections with physics. Is mathematics inherent, or a construct? It has to do with the definition but maybe we'll never know...It has already been proven that math can't be proven (Universe in Zero Words)!