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I was reading a little bit about Galois theory and set theory in my free time, and something that I noticed is how 'constructed' these branches of mathematics seem. For example, arithmetic, algebra and calculus to me seem to be very 'natural'. I know this isn't a very mathematical thing to say, as all of math is human. When studying set theory and Galois theory you encounter a lot of sets which have to satisfy a certain set of rules. For example:

Group is a set of elements, satisfying the following rules:

  • For any 2 elements $x$ and $y$ in group $G$ we also have $xy$ in the group $G$.
  • There is an identity element.
  • Associativity.
  • Every element $x$ in $G$ has a unique inverse $y$.

To me such sets don't feel very natural, as I can also make one myself:

Bazooper is a set of elements, satisfying the following rules:

  • Every Bazooper $B$ is a group.

  • 2nd, 3rd, ..., nth constraint that I made up myself.

That's why I have trouble studying all these different type of sets, because they're not really intuitive to me. What is the thought process behind defining a certain set?

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Think about the properties that $\mathbb{Z}$ has under the operation of addition. It satisfies all the structure to be a group. The way that I understand it, is that people find a structure that people care about (aka integers, polynomial rings, etc.) and they try to generalize it). What properties do the real numbers satisfy as a vector space? What about the integers under multiplication and addition? –  LASV Dec 9 '13 at 15:32

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Groups, rings, fields, all come from abstracting properties of "known objects". For example groups are an abstraction of $\Bbb Z$ and the clock arithmetic of $\mod n$; so are rings. Fields are generalization of $\Bbb Q$ and $\Bbb R$.

We don't invent properties saying "Hey, that sounds like fun!", rather we work with a certain object, then we say "Okay, these are the properties I've used thus far, let me define a new object which will have these exact properties. Now I can reprove all the statements in this general context!".

However it is easy to mistake like that when starting with abstract mathematics, because often historical background is omitted, and one doesn't realize that the origin of a certain object is an attempt to distill from something that we know the necessary structure in order to prove something.

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I doubt whether the notion of a group arose from the usual arithmetic, that is, whether it really generalises integers or the modular arithmetic. To me a group is an abstraction of a set of symmetries whatever it means (Galois groups, dihedral groups etc. seem to be the root). –  Tomek Kania Dec 9 '13 at 16:03
    
Would you recommend looking up the historical background for some of these things for better intuition? –  Phaptitude Dec 9 '13 at 16:47
    
@Tomek: Yes, you are right about that. According to Wikipedia, it began with permutation groups really. I don't know whether or not groups were axiomatized (even informally) by Galois or the mathematicians of that era, but it seems to me that the integers played an integral part in the number theoretic development of group theory (which was, again, according to Wikipedia) before the full formation of group theory as it is today. –  Asaf Karagila Dec 9 '13 at 18:12
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@Phaptitude: I think that a bit of historical background can shed a lot of light. Because we often miss the point that things we are given as simple exercises could have been doctoral works some centuries ago, and that the length the theory went from conception to its current form is likely to be very large. In some sense, mathematics has only two times "Currently open" and "proved". Where the distinction can be fuzzy in newer problems (or solutions), but usually not in the undergrad level (that you asked about). Specifically in group theory Wikipedia has a nice historical overview. –  Asaf Karagila Dec 9 '13 at 18:14

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