3 ms·

>most of us don't have a good conceptual model for what exponents really do Instead of matching math to real world objects (1= one banana, 2 = two bananas, 1+2

by dxbydt 15y ago

>most of us don't have a good conceptual model for what exponents really do Instead of matching math to real world objects (1= one banana, 2 = two bananas, 1+2 = 3 bananas etc. ) and building up to exponentiation, multiplication etc. thereby introducing all sorts of paradoxes, group theory dodges all that and treats the whole thing as a very consistent rule-based system. Things fall into place quickly once the rules are laid out explicitly. Consider: finite abelian group with only 3 elements a,b,c. Given a+b=c, a+c=a, what's b+b ? Hmmm...okay, if a plus c is a, then c is acting like zero. So b+c must be b. since addition is commutative (abelian gp), b+a must be a+b which you said was c. So now we know b+a=c, b+c=b, so b+b better be a ! Students are easily convinced because you've laid out the rules very explicitly. In fact, they'll try to convince you that b plus b better be a because that's the only way to make the cayley table work out! (http://en.wikipedia.org/wiki/Cayley_table http://en.wikipedia.org/wiki/Cayley_table) There are several books that argue that the teaching of Abstract Algebra must precede Calculus for this very reason. 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.

ubasu 15y ago
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.
- dxbydt 15y ago
  > 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 http://en.wikipedia.org/wiki/Hilbert%27s_seventh_problem Solution: http://terrytao.wordpress.com/2011/08/21/hilberts-seventh-problem-and-powers-of-2-and-3/ http://terrytao.wordpress.com/2011/08/21/hilberts-seventh-pr...
  - lmkg 15y ago
    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.
xyzzyz 15y ago
You're right, apart from the fact that real exponentiation does not really belong to group theory -- or even abstract algebra, for that matter.
- dxbydt 15y ago
  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-a-group http://mathoverflow.net/questions/44393/notation-exponent-of... http://mathoverflow.net/questions/32116/exponent-of-a-group http://mathoverflow.net/questions/32116/exponent-of-a-group
  - xyzzyz 15y ago
    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.
    - lurker14 15y ago
      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.
lurker14 15y ago
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 :-)
- kalid 15y ago
  Hi, I'm really interested in developing tutorials / explanations that help merge intuition and rigor, theory and application. You can reach me at kalid.azad@gmail.com.

by dxbydt 15y ago

ubasu 15y ago

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.

dxbydt 15y ago
> 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 http://en.wikipedia.org/wiki/Hilbert%27s_seventh_problem Solution: http://terrytao.wordpress.com/2011/08/21/hilberts-seventh-problem-and-powers-of-2-and-3/ http://terrytao.wordpress.com/2011/08/21/hilberts-seventh-pr...
- lmkg 15y ago
  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.

xyzzyz 15y ago

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

dxbydt 15y ago
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-a-group http://mathoverflow.net/questions/44393/notation-exponent-of... http://mathoverflow.net/questions/32116/exponent-of-a-group http://mathoverflow.net/questions/32116/exponent-of-a-group
- xyzzyz 15y ago
  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.
  - lurker14 15y ago
    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.

lurker14 15y ago

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 :-)

kalid 15y ago
Hi, I'm really interested in developing tutorials / explanations that help merge intuition and rigor, theory and application. You can reach me at kalid.azad@gmail.com.