Ask a Mathematician!

[IdiotsOpposite]

Something that you often hear mathematicians talk about: beauty. Is all of mathematics beautiful or is it only some concepts, theories and proofs? What makes them beautiful? And is it a subjective assessment that some mathematicians can disagree with, or is it objective and undeniable by all?

Meromorph answered it better than I ever could. I'll just direct you to his answer above.

What's your favorite number?

For some reason, 37. It's been my lucky number for years, even though I can't even remember WHY I like it so much.

Do you own a Rubix Cube?

Yup! In 3, 4, and 5 dimensions.

What's your favorite computational aid?

My laptop.

How many digits of pi have you memorized?

Uhh... 5? That's really all I need for any serious hand calculation. Any more than that and you're just getting picky.

Who is your favorite mathematician?

Tossup. Riemann because he formulated much of the starter analysis about my single favorite function, and Gauss because, um, he's Gauss. Have you SEEN the amount of things named after him, and all for good reason too?

What do you think of statistics?

Statistics is kind of fun. Not nearly as fun as things like calculus, but a lot more fun than things like linear algebra.

What do you think of non-standard analysis?

I don't like it. I'm sure it works very well and all that, but I just... I don't like it.

What do you use for writing equations with on a PC?

Do you mean just equation writing software? I use LaTeX, because that's what my favorite knowledge forum uses. If you mean an actual program, Mathematica.

What's your favorite kind of triangle?

The kind with three 90 degree angles. (Yes, it DOES exist)

Do you own a slide rule?

Nope! I'm about... 20 years too young for that. Maybe 30?

Which is best, M&Ms, Reese's Pieces, or Skittles?

Reese's Pieces. By a mile.

 

[Leifmk]

I've never heard anyone with a math degree have trouble finding jobs, hell I know physicists who work for banks and make mad money.

I survived a graduate program and escaped with a degree [1] in "applied mathematics". Shortly thereafter (this was in 1998) got a job as a computer programmer, which has been my career since. My math skills are awfully rusty by now.

Out of the classmates I've kept in touch with, and the ones I met at the 10-year reunion a while back, I'd say less than 25% are actually working primarily with mathematics. The rest are all over the place, although mostly in somewhat technical fields (hell, one guy even went to medical school and is now a GP).

[1] A now-obsolete Norwegian degree roughly equivalent to a Master's, which is what it has been replaced with in the current (post-Bologna Process) system.

 

[mayor]

The kind with three 90 degree angles. (Yes, it DOES exist)

Now you've got me intrigued Could you show one?

 

[plarq]

How do you mathematians look at us physicists? We're bunch of non-rigorous players that don't care about divergence, and tries to think that infinity is not a problem.

 

[Leoreth]

Now you've got me intrigued Could you show one?

I suppose it involves non-Euclidean space (Think about drawing a triangle on the inside of a sphere.)

 

[Perfection]

Yup! In 3, 4, and 5 dimensions.

dimensions

I don't like [non-standard analysis]. I'm sure it works very well and all that, but I just... I don't like it.

Why not?

The kind with three 90 degree angles. (Yes, it DOES exist)

What's your favorite Euclidean Triangle?

Nope! I'm about... 20 years too young for that. Maybe 30?

I have two

Reese's Pieces. By a mile.

Good man.

 

[warpus]

"Combinatorics? It's counting! How hard could counting be?"

*dies*

50% of the course was "counting", the other half was graph theory.

It's not that easy to wrap your head around the "counting" aspect of it - at least a lot of very smart people in the class had a hard time with it. It's not really fair to describe it as counting either.. It's like counting with your penis, and a blindfold. Interesting, but not as simple as simple "counting".

 

[IdiotsOpposite]

Now you've got me intrigued Could you show one?

I suppose it involves non-Euclidean space (Think about drawing a triangle on the inside of a sphere.)

Leoreth was on the ball with this one. I'm talking of course about the triple-right triangle on a sphere. It's non-Euclidean, but that just makes things more fun!

How do you mathematians look at us physicists? We're bunch of non-rigorous players that don't care about divergence, and tries to think that infinity is not a problem.

I don't think I could talk for the general "you mathematicians", but I personally think of you guys as just applying all the fun math to the real world.

dimensions

Yup! Wanna try?

http://www.superliminal.com/cube/cube.htm

Why not?

I don't know that, really. I think it's just because they use a definition and practice of infinitesimal that rubs me the wrong way.

What's your favorite Euclidean Triangle?

A right triangle with angles 90, 22.5, and 67.5 degrees. It allows me to use the far-more-interesting-than-other-values sine and cosine of pi/8.

 

[stormerne]

Spoiler :

Mathematical beauty is connected to timelessness, certainty, abstraction, generality, and clarity.

Timelessness is always present, mathematical concepts are defined without relation to time so the theorems that describe them will remain true forever. Certainty and rigorous methods have been associated with mathematics since ancient times, it is by far the most rigorous of all the sciences. In the latter 19th century mathematicians reached a new level of rigorous methods based on set theory and they pushed for axiomatic perfection, but by the 1930s Godel's incompleteness result showed that such a perfect axiomatic system was impossible. Since then the focus has shifted away from logical foundations, towards more practical levels of certainty.

Abstraction and generality are closely related: a good abstract concept will be very general, applying to many different particular situations. Abstraction and generality are what give power to mathematical methods. Mathematicians face an infinite wilderness of infinitely strong beasts, and the only way forward is by finesse - leveraging our finite human intelligence - not brute force.

The opposite of abstraction and generality is an ad hoc solution, a term which means "for the problem at hand", whereas concepts, methods, and results are more beautiful if they solve a large number of disparate problems at once. For example, 19th century mathematician C.F. Gauss said he was not interested in Fermat's Last Theorem, as one could easily write down many similar equations and statements that were equally hard to prove, and that he would only be interested in the problem if there were a general theory encompassing all of these (which is what happened with Wile's proof over a century later based on the algebraic geometry of elliptic curves).


Not all mathematics would be regarded as beautiful. Newer proofs tend to be raw, undigested and unpolished, which makes them less beautiful by the five attributes above. Some subjects are inherently unwieldy and are left for future generations to hopefully tame. A lot of mathematical research is generated to fill grant quotas or supply students with problems, these unimportant results may never be polished to be particularly beautiful.

In judging whether a work is beautiful, it's a matter of subjective taste. There are many sub-areas of mathematics and everyone has preferences. Still, just like with movies, books, wines, etc it is possible with practice to evaluate the excellence of a mathematical work based on objective features, unclouded by personal preference. This happens all the time because research is peer reviewed.

Cool. Very good answer (way beyond my expectations for CFC ) Thank you!

 

[Crezth]

I don't think I could talk for the general "you mathematicians", but I personally think of you guys as just applying all the fun math to the real world.

Wait, then what do engineers do?

 

[Rashiminos]

How do you mathematians look at us physicists? We're bunch of non-rigorous players that don't care about divergence, and tries to think that infinity is not a problem.

It's kind of important to distinguish between science as a field of study and Science as in the method used for the natural sciences. The rigor achieved in mathematical disciplines is done through formal logic, and not so much observation, experimentation, and theoretical explanation. Physicists try to use math that has some actual application, and not be concerned with hypothetical constructs (ie Nth dimensional space) without sufficient prompting. If divergence/infinity are involved in the physicist's research, there's probably some interesting physical activity to care about interpreting.

Physicists tend to want to describe the universe we are currently living in, and avoid the ones we are not in.

 

[IdiotsOpposite]

Wait, then what do engineers do?

Linear approximations.

/facetiousness

 

[stormerne]

Wait, then what do engineers do?

Eat pizza, and define the most desirable social skill as touch-typing.

Edit: /me was an engineer for 35 years and is allowed to say that.

 

[uppi]

Physicists try to use math that has some actual application, and not be concerned with hypothetical constructs (ie Nth dimensional space) without sufficient prompting.

In statistical mechanics we have to be concerned about N-dimensional spaces. And N is so large that, being physicists, we just set huge number=infinity and go with infinte-dimensional spaces

If divergence/infinity are involved in the physicist's research, there's probably some interesting physical activity to care about interpreting.

We encounter infinity often enough. But usually we just plug it into our equations and see if anything useful comes out. Because despite the mathematicians' protests we will do arithmetic with infinity.

Have you ever wanted to scream at a physicist while watching him doing physicist math?

 

[_random_]

Leoreth was on the ball with this one.

Hehe.

Yup! Wanna try?

http://www.superliminal.com/cube/cube.htm

And you actually own one of these?

 

[Rashiminos]

In statistical mechanics we have to be concerned about N-dimensional spaces. And N is so large that, being physicists, we just set huge number=infinity and go with infinte-dimensional spaces

Any suggested reading about statistical mechanics?

*Okay, so maybe there is sufficient prompting to model thermodynamics with such spaces, but I hope the point about constructs not necessarily being applicable was made.

We encounter infinity often enough. But usually we just plug it into our equations and see if anything useful comes out. Because despite the mathematicians' protests we will do arithmetic with infinity.

Have you ever wanted to scream at a physicist while watching him doing physicist math?

Depends on how it is handled.. I don't think screaming would help though. I would have a problem setting something (like a number) equal to it unless it was understood that "=infinity" is not an equivalence relation, but a statement about {positive} divergence (as in we're no longer dealing with a huge number). It may turn out for a given scenario that the infinite-dimensional case can be well-approximated by a finite-dimensional case if it can be shown that the infinite case is convergent.

 

[IdiotsOpposite]

Hehe.


And you actually own one of these?

Well, not a PHYSICAL copy, no. It'd be kind of hard to create a 4-dimensional object in real life, and if we did, I sure hope we're not relegating that ability to making Rubik's Cubes! But that virtual copy is just as good.

 

[ParadigmShifter]

Infinite dimensionality is just like a sequence. It's nothing too special. A vector is just a sequence with a finite length.

EDIT: In a finite dimensional vector space, of course

 

[_random_]

Well, not a PHYSICAL copy, no. It'd be kind of hard to create a 4-dimensional object in real life, and if we did, I sure hope we're not relegating that ability to making Rubik's Cubes! But that virtual copy is just as good.

Have you ever solved a 4d or 5d one?

 

[IdiotsOpposite]

Have you ever solved a 4d or 5d one?

Once or twice. It really is not easy.