Group theory is formed by abstracting out the observed properties of number systems. If you want to show that some collection is a group, you will have to do the computations to show that they follow the rules, in which case, it helps to have an understanding of the mechanics of the computation.

> it helps to have an understanding of the mechanics

My claim is the exact opposite. I claim you don't need to understand the mechanics ( just blindly abide by the rules of the group or abelian group or finite simple group or whatever), which is why the approach is better. If you show a monkey red means stop and green means go and reinforce these rules by rewarding with a banana, eventually the monkey will stop when he sees the red. Not because he understands the mechanics of traffic management. Simply because he is abiding by the rules. Similarly, large portions of math can be approached by either the definitional route ( ie. rules ie. define propositions & theorems that logically follow if those props held ) or via trying to understand actual mechanics by mapping everything to real world phenomena ( x = distance, dx/dt = velocity, d/dt(dx/dt) = acceleration etc. ) which are problematic because the mapping breaks down due to the nature of physical reality ( like friction etc. )

How would one explain say Hilbert's 7th problem via the actual mechanics ?

If a is algebraic and b is irrational show a^b is transcendental.

What does that even mean when you map them to the real world ? Instead, the solution is to build upon theorems that logically follow from the axioms you start out with. Problem: http://en.wikipedia.org/wiki/Hilbert%27s_seventh_problem Solution: http://terrytao.wordpress.com/2011/08/21/hilberts-seventh-pr...

You are probably right that such an approach is better at teaching students to be able to crank out solutions to problems, but I think it also reinforces the attitude a lot of students have that math is just a bunch of arbitrary rules that don't mean anything, and is therefore a waste of time. For students that aren't destined to be math majors, the most important aspect of math is being able to map real-world concepts to abstract rules. Being able to mechanically manipulate those rules is far secondary, especially with easy access to computers.

You're right, apart from the fact that real exponentiation does not really belong to group theory -- or even abstract algebra, for that matter.

The notion of a^n definitely belongs to group theory, whether you call that n an exponent or an annihilator or a period is a matter of some contention. See: http://mathoverflow.net/questions/44393/notation-exponent-of... http://mathoverflow.net/questions/32116/exponent-of-a-group

Only if n is an integer. Real exponentiation is something completely different, because it involves all aspects of real numbers - addition, multiplication, order and continuity, which are all interconnected, and the language of group theory is too weak to describe it. For instance, while 2^pi makes perfect sense in the realm of real numbers, it makes none in Z_3.

Hmm.

(Warning, ascii math is confusing and ambiguous to read. Sorr.)

Exponentiation of group "multiplication" does not immediately seem amenable to the reals, sure. But real exponentation does form a group, as shown here:

Define x_g(r) = the function that raises a Real/{0} (non-zero real) number r to the exponent x (in the sense of of some reasonable definition of exponentiaton of continuous functions). Define X = the set x_g() functions corresponding to all reals (including 0)

Define x_g y_g as composition: y_g(x_g(r)) = (r^x)^y = r^ (xy). Then we have 0_g x_g = (r^0)^y = r ^ (0 y) = 1 = r ^ (y * 0) = (r^y)^0) = y_g 0_g -> identity

y_g (1/y)_g = r^y ^ (1/y) = r^0 = 0_g -> inverse

(x_g y_g) (z_g) = ((r^x)^y)^z = (r^(xy))^z = r^((xy)z) = r^(x(yz) = (r^x)^(yz) = x_g (r_g r_g) -> Associativity

That makes a group.

Now, I explicitly left out the 0^x case. Can we fit it back in?

Not particularly cleanly, as thoroughly discussed in this thread.

The group you described does not inherit any interesting structure from exponentiation -- indeed, one can easily see that it is isomorphic to the multiplicative group of reals. You could similarly construct a group isomorphic to an additive group of reals. This is an example of the fact that real exponentiation connects different aspects of real numbers, as well as the fact that just abstract algebra language is not enough to express properties of real numbers. You need to somehow relate the algebraic structure of reals to a topologic one, which stems from order imposed on reals and its continuity.

You write: "With Calculus, the mapping of math to real-world objects leads to all sorts of messy realities. With group theory, you dodge that mess by simply stating rules upfront."

(Prologue: I encountered group theory first by drawing pictures of pegboards and strings to illustrate permutations (before I knew the word!), not by reading the rules up front.)

You are describing a schism between pure/formal and applied mathematics (and between formalism and intuition, to some extent). It's completely cool for you to have an interest in pure math completely separated from applied math, real-world physics, programming etc. It is also cool for someone to pursue applied math,physics, etc without any pure math, but that is sad because a lot of beautiful symmetry and cross-disciplinary value would be last. (Goodbye, encryption!)

I personally strive to connect pure and applied mathematics. After getting burned (in an emotional/psychological way) by chasing pure math study beyond my ability to intuit and apply it, I now commit myself to learning theory and application in tandem. (I'll certainly appreciate the fact that pure theorists such as your ideal have gone several steps ahead and I can study their results without trying to discover them from scratch.)

In fact, my most recent flight of fancy / big dream is to write math/CS tutorials that provide such a tight integration of theory and application, abstract and concrete, general and specific. And I want to use modern web tools (hyperlink, animation, multi-dimensional page layout) to do so.

I'd love to talk to anyone interested in working with me on that :-)

Hi, I'm really interested in developing tutorials / explanations that help merge intuition and rigor, theory and application. You can reach me at [email protected].