Some questions about mathematics [closed]

This question is a soft one. Well, So far I have noticed stuff that is nice in math, particularly in algebra, topology and analysis. For instance, in algebra, there is theorem that says that we can think of groups just as some set of permutations. So, in other words, can we say all groups are just permutations? Also in topology we classify surfaces. In fact, we have that every compact connected surface is either a sphere, an n-torus of $n-$ projective planes. Is there any similarity in measure theory? It seems like math is just like comparing things. Is this true? Also, one last question, What is the big picture that everyone talks about?

thanks

-

closed as too broad by Zev Chonoles, Johannes Kloos, Matt Pressland, martini, SiméonOct 10 '13 at 8:22

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs.If this question can be reworded to fit the rules in the help center, please edit the question.

This is an unreasonably broad question, and it is difficult to understand what you are asking. I've voted to close. –  Zev Chonoles Oct 10 '13 at 7:58

1 Answer

I was beginning to think that this question was getting a little too broad, then I got to the end and you pretty much pulled in all of mathematics!

• All groups are permutation groups, in the following sense: Any group $G$ acts on itself by left multiplication. In other words, for any $g\in G$, the map $h\mapsto gh$ is a permutation of $S=|G|$, the underlying set of $G$. This gives an injective homomorphism $G\to Sym(S)$. For this reason, we can think of any group as a permutation group (though this is not necessarily a useful way to think about all groups).

• There are many classification theorems in measure theory (and in all areas of mathematics). One simple one is the correspondence between Borel measures on $\mathbb{R}$ and increasing, right-continuous functions $\mathbb{R}\to\mathbb{R}$. Another is the uniqueness of Haar measure.

• A lot of math is comparing things, yes. But usually not as an end in itself. Mathematics has many diverse goals. My personal approach to mathematics is to seek out language that broadens the scope of mathematical discussion. One example of what I'm talking about is Emmy Noether's "noetherian hypothesis" that transformed the study of polynomial rings into modern ring theory, by observing that a single abstract property could replace lots of inelegant, technical computation, while simultaneously bringing many different kinds of objects under the same umbrella.

• Anybody who claims that they see the big picture is compensating for insecurity about their mathematical ability. When you have learned a thousand or ten thousand amazing things in mathematics, the picture starts to feel quite big, but it is never finished. First you learn to count, then you learn to do arithmetic, then you learn that the integers are a set, then you learn that they are a ring, then you learn that they are a principal ideal domain, then you learn that they are the regular functions on an affine scheme whose closed points are the primes, then you learn that their unique factorization is related to the triviality of that scheme's line bundles, then you learn that their algebraic extensions are related to the scheme's covering spaces, and then you learn that our inability to prove the Riemann hypothesis (and claim its \$1 million prize) is related to our inability to define an "absolute point" that joins the primes together into a curve. And that is just a single mathematical object! It goes on and on like that in all directions.

-