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[P]
Abstract Mathematics, Group Theory, and You

By Coryoth in Science
Wed Mar 08, 2006 at 08:59:26 AM EST
Tags: Focus On... (all tags)
Focus On...

It often seems, to those with only a high school (or even basic university) education in the subject, that mathematics is largely a solved problem. Sure, the thinking goes, there are those arcane and abstruse little corners of irrelevancy with which academics busy themselves, but for the most part we know it all, right?

In fact there is a great deal that we simply do not know, and do not understand; and a lot of it is much simpler, and much closer to home than you might think. In many ways our understanding of the subject only really began very recently. Let's take a brief tour of some of what we don't know.


Certainly at the cutting face of research mathematics there is much to learn and much that is not understood. It can be argued that these topics are precisely those obscure irrelevancies, mathematical chimera, that were dismissed earlier; but they provide a point of entry and are, perhaps, not as irrelevant as they may seem at first glance. The difficulty of describing these areas is the layered scaffolding of definitions and intermediary results that mathematicians have constructed beneath them as they climb towards their goal. Such constructions are often daunting to the outsider, and serve to obfuscate what lies at their heart. This means that the depth of the underlying elegance may not be clear as we climb, but hopefully we will catch a glimpse as we climb down the other side.

The particular edifice I would like to climb for the purposes of this discussion is that of Group Theory and Representation Theory. Along the way I'll try and point out some of the various directions where our understanding is less than perfect without necessarily getting into the details of those specific fields.

Let's begin with the definition of a group, sacrificing a little rigour and precision for the sake of brevity: A group is a set of objects with a particular "algebraic structure". By an algebraic structure I mean a set of operations that take one or more elements of the set as arguments and return a single element of the set. A natural concrete example is the algebraic structure provided to the set of natural numbers by addition and multiplication - operations that take two numbers and return a single number. The more such operations you have the greater the scope for complex interplay between them. For the particular algebraic structure that defines a group we ask for things to be (almost) as simple as possible. We require a single operation that is associative (the order in which we apply the operation doesn't affect the end result), and takes exactly two arguments. We also require the existence of certain set elements with regard to how they behave under this operation. First we require an "identity" element - an element e such that for any other element x we have f(x,e) = f(e,x) = x where f is our associative operation. Second we require the existence of "inverse" elements - for each x we require some element x' such that f(x,x') = f(x',x) = e. Any set of objects with an algebraic structure satisfying those constraints is a group. Adding more operations and different constraints leads to a profusion of other things to study: rings, domains, fields, modules, algebras, etc.

Given just this abstract axiomatic definition, just the raw scaffolding, it is hard to see the motivation for studying such things - why abstract out from numbers in this way? Surprisingly groups, rings, fields, and other such structures, crop up in all manner of different contexts - once you think to look for them you find them everywhere. It turns out that we require more study of such things, not less. For example programming multi-threaded and multi-process applications is considered very hard, in a large part because it is hard to think and reason about many different things all happening at once in parallel. The study of communicating concurrent processes, which is to say multi-threaded and multi-process systems, can be reduced to a structure that is almost (but not quite - it lacks a couple of constraints) a ring. The subtle difference puts it just outside the reach of the vast array of tools and theorems mathematicians have developed. A better understanding of those structures could quickly make reasoning about and programming massively concurrent multithreaded systems exceptionally simple. For now it remains one of those areas about which we know far too little. Let us get back to groups however.

For something as surprisingly prevalent as it is, we know a lot less about groups than you might think. The designation of something as a group is very general - we require only some very general properties of the algebraic structure imposed upon the set; the specific properties of the set itself, and the algebraic structure imposed, can lead to a wide variety of different groups, each with their own unique character and properties. Perhaps the most immediate division is between groups with only finitely many elements, and infinite groups. One might expect that when restricted to just groups with finitely many elements it should be relatively simple to work out what all the possibilities are. Classifying finite groups is, however, a remarkably difficult problem and remains, to this day, unsolved. That is not to say that a lot of work has not been done. So called Abelian groups, where the operation is commutative (that is f(x,y) = f(y,x) for any x,y in the group) are well understood and can essentially be characterised knowing little more than the size of the set on which the algebraic structure is defined. Similarly, after a vast effort spanning decades, there is a classification of all so called finite Simple groups. The best existing theorems classifying general finite groups are the Sylow Theorems which decompose groups into what are called p-groups, which are closely related to prime numbers. The classification of finite p-groups remains an open problem on which much work is currently being done using methods and techniques from seemingly unrelated fields, including theorems about Lie algebras. It appears that the structure of finite groups is a very complex problem that relies on a much deeper understanding of algebraic structures other than just groups.

Let us turn our attention then to Representation Theory. For Representation Theory we will require a little more scaffolding. We begin with group actions. A group action is a way of thinking of a group (call it G) as acting on some other set (call it A): We can think of each element of G as a function from A to A, and to make sure it all make sense we will require that for any x,y in G and a in A it follows that x(y(a)) = f(x,y)(a) where f is, as before, the group operation. That sounds rather obscure and technical, but essentially we are just asking that the if we view elements of G as functions, those functions should still respect the algebraic structure of the group. We also require that the identity element of the group should act as an identity function on A. If A has some structure (be it algebraic or otherwise) of its own we can ask the inverse question: what of the set of all (bijective) functions from A to A that preserve the structure? Surprisingly that set of functions turns out to behave precisely as a group with a natural pre-defined action on A. It is in this sense that groups are often referred to as symmetries - at their heart symmetries are structure preserving functions - wherever you find symmetries you find a group that describes them all.

The next ingredient we require for Representation Theory is group homomorphisms. These are, simply put, maps or functions from one group to another. At first glance this is no different than functions on the underlying sets, with which most people should be familiar - what makes it a function between groups is the necessity that the function respect the structure of the groups involved. Specifically we require that a function φ from a group G to a group H obey φ(f(x,y)) = g(φ(x),φ(y)) where f and g are the group operations for G and H respectively. This doesn't seem particularly momentous, but a surprising amount of theory flows from this very simple definition. It turns out that we can learn everything there is to know about the internal structure of a group simply by looking at how it relates to other groups through homomorphisms: the external relationships define the internal structures - this is a surprisingly deep insight and is one of the founding observations for Category Theory, a language and theory that is having revolutionising impacts on both mathematics and computer science.

Take a vector space and consider the set of all functions that preserve the structure of the vector space - that is, all (invertible) linear transformations of the vector space. That set of linear transformations, as noted when discussing group actions, forms a group. It is possible then, given some group G, to wonder about all the homomorphisms from G into the group of linear transformations of some vector space. Each such homomorphism is called a representation of G, and the study of such representations is called Representation Theory. A couple more facts we'll need: the "degree" of a representation is the dimension of the vector space, and a representation is called "irreducible" if it cannot be broken down and expressed as a "sum" of simpler representations.

Again, this sounds like an unnecessarily abstract and futile exercise, but given that relationships to other groups can tell us a great deal about the potentially complex internal structure of a group, and that linear transformations of vector spaces are easy to deal with (they are just matrices!), it promises a way to better understand the internal structure of complicated groups. In practice Representation Theory provides an extremely rich and enlightening view of groups, the details proving to be far more intricate and beautiful (at least in an abstract mathematical sense) than can really be imagined without considerable study: new interesting structures called group-rings arise in a natural way, and prove to have their own intriguing and poorly understood complications; a new and barely explored expanse of open problems beckons; we discover just how much more we don't know.

There is a great deal more climbing that can be done. Mathematicians continue to expand our understanding of these subjects, and there is much more that could be said before we truly arrive at the very cutting face. I would like to stop at this point though, in part because many readers are undoubtedly already feeling that we've already climbed too far into the airy heights of irrelevancy, and in part because we've arrived at a convenient point to take a surprising step sideways.

So far we've been discussing the completely abstract. We've climbed up a path of increasingly theoretical constructions built one atop the other to reach a point of potentially perilous irrelevance. Now it's time to step into the concrete world. Our universe can be thought of as a space with some structure - the laws of physics. We can ask what functions or transforms of space and time will preserve certain structures (like, say, Maxwell's equations). This leads us to a group of transformations of space-time that preserve desired properties - as I said, groups, once you know to look for them, crop up everywhere. Now something intriguing happens: the internal structures of the group correspond in a natural way with physical conservation laws, such as the law of conservation of energy, or the law of conservation of momentum. Things get really interesting, however, when we realise that, given the group, each irreducible representation of the group very naturally corresponds to a different fundamental particle, with force-carrying and non-force-carrying particles neatly delineated from one another according to the degree of the irreducible representation! All of a sudden what had seemed like a little axiomatic game that mathematicians were off playing by themselves is having very real impacts on the world in which we live. All of a sudden we've come crashing back down into relevance. Which brings us to the start of the truly hard questions: why does it work?!

Mathematics works. That much is obvious to anybody - mathematics is essentially the language of much of science, and is seemingly unendingly applicable to the real world. The problem is that it is not at all obvious why this should actually be the case. Philosophers have struggled with this question since Pythagoras, all to no avail. It is entirely all too common for mathematicians to embark on a purely theoretical exploration of the truly abstract and abstruse as a mental game, only to have, decades or centuries later, their work prove stunningly applicable to some very real world problem. The mathematics that laid the foundations for General Relativity began with mathematicians wondering, purely hypothetically, what would happen if they ignored Euclid's fifth postulate; much of the work of G.H. Hardy, who famously claimed no discovery of his would ever make a practical difference in the world, has found considerable application in cryptology. To explain its effectiveness it seems we need first to explain what mathematics actually is.

If mathematics is simply the abstraction of our intuitions about the physical world, then why is it so universal? Surely each mathematician would then develop a personal mathematics from their own intuitions and claims of mathematical objectivity would become muddied. Equally one can ask why it is that a rejection of spatial intuition, on purely logical grounds, such as that Euclid's fifth postulate, should lead to mathematics that is surprisingly more applicable rather than less.

Can it then be argued that mathematics is simply logic? This would explain, to some degree, its applicability: mathematics would simply be the expression of fundamental eternal universal truths. Unfortunately, despite truly heroic efforts by some of histories greatest mathematicians and logicians (Frege in "Foundations of Arithmetic" Russell and Whitehead in "Principia Mathematica") the reduction of mathematics to pure logic failed. While Russell and Whitehead managed to construct a remarkable edifice out of pure logic, introducing along the way many new concepts such as Type theory which is now of great importance in computer science, they fell short of building mathematics as we know it: their system was too weak and required extra axioms (the axiom of infinity and the axiom of reducibility) which cannot reasonably be called purely logical principles.

Can we not, then, define our base set of rules and proceed by logic from there? This was the approach to, and explanation of, mathematics favoured by Hilbert and his formalist school. While this approach did much to shore up the shaky foundations of pure mathematics before running afoul of Gödel's Incompleteness Theorems, it simply sidesteps our initial question. Mathematics, in this line of thought, becomes reduced to a game played with symbols on paper and an arbitrary set of rules (we ask only for consistency). Why should such an arbitrary game apply to reality? That is not at all clear.

The debate as to what mathematics actually is continues, mostly in philosophic circles - No school of thought has yet given a satisfactory answer. Personally I lean towards a more recent view that mathematics is about structure and can be grounded best in a structure oriented language such as that of Category Theory. However, Category Theory does not yet provide a sufficiently rigorous foundation to make that claim truly defensible. Only time will tell. Thus we began wondering what there is in mathematics that we don't know, and conclude realising that we don't even know what mathematics is.

Further Reading:

A First Course in Abstract Algebra by John Fraleigh provides a gentle introduction to elegant intricacies of Group Theory.

Conceptual Mathematics by William Lawvere and Stephen Schanuel is a very accessible (high school level) introduction to Category Theory.

Gauge Fields, Knots, and Gravity by John Baez and Javier Muniain is a (relatively) accessible book discussing the links between pure mathematics and physics in General Relativity and Quantum Field Theory.

Introduction to Mathematical Philosophy by Bertrand Russell provides and excellent and very readable introduction to the subject of mathematical philosophy.

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Related Links
o A First Course in Abstract Algebra
o Conceptual Mathematics
o Gauge Fields, Knots, and Gravity
o Introducti on to Mathematical Philosophy
o Also by Coryoth


Display: Sort:
Abstract Mathematics, Group Theory, and You | 116 comments (65 topical, 51 editorial, 0 hidden)
Can group theory help me compose music? (3.00 / 2) (#5)
by MichaelCrawford on Mon Mar 06, 2006 at 04:59:47 AM EST

I've had the idea for a while that if I studied group theory, it would help me compose more interesting music. Do you think I'm right? Do you know of anyone else who has had this idea?

One reason I think so is that there are many symmetries in music theory, and group theory is often used to study symmetry. For example, one can play a major scale starting at any key on the piano. All the scales will sound similar, except shifted up or down by a constant amount.

I'd be interested to hear your thoughts on my idea. I started to take a group theory class once, but I had to drop it. I've always wanted to try again.


--

Live your fucking life. Sue someone on the Internet. Write a fucking music player. Like the great man Michael David Crawford has shown us all: Hard work, a strong will to stalk, and a few fries short of a happy meal goes a long way. -- bride of spidy


Math and Music (none / 0) (#35)
by Coryoth on Mon Mar 06, 2006 at 07:17:00 PM EST

Certainly you could consider groups as an interesting way to analyse and make more productive use of subtle symmetry in your music. It is very common that symmetry, even subtle uses of it, is present in a lot of what we tend to consider "good" music.

On the front of applying pure mathematics to music however, one of the more interesting concepts I've heard involved (and here I'm a little lost as I am not a musician so you'll have to forgive the vagueness) connecting up the tones in a musical phrase according to ... I can't recall, some concept of tension? ... and thus deriving a nerve, which you can then start considering purely mathematically. Once you're into the math things get interesting because you can do interesting analyses of the structure thus created with, for instance, algebraic topology... which brings us back to symmetries, but in a far more intriguing and subtle way.

I expect that may not have been very clear due to my poor memory and lack of understanding of music, but hopefully it at least spawns some interesting ideas about how pure mathematics could fruitfully be applied to music.

Jedidiah.

[ Parent ]

i believe your uid is from a cure song (none / 0) (#43)
by circletimessquare on Mon Mar 06, 2006 at 08:55:57 PM EST

so that automatically disqualifies you from commenting on anyone else's mental state, being that you are revealing yourself to be a goth emo piece of shit


The tigers of wrath are wiser than the horses of instruction.

[ Parent ]
you name yourself after a cure song (none / 1) (#45)
by circletimessquare on Mon Mar 06, 2006 at 09:02:54 PM EST

and i'm the faggot?

(snicker)


The tigers of wrath are wiser than the horses of instruction.

[ Parent ]

what happened to the guy i was talking to?! nt (none / 0) (#53)
by circletimessquare on Mon Mar 06, 2006 at 10:58:35 PM EST



The tigers of wrath are wiser than the horses of instruction.

[ Parent ]
He musta got zeroed or anonymized (3.00 / 2) (#59)
by MichaelCrawford on Tue Mar 07, 2006 at 12:25:59 AM EST

I was quite puzzled by your posts until I realized that.


--

Live your fucking life. Sue someone on the Internet. Write a fucking music player. Like the great man Michael David Crawford has shown us all: Hard work, a strong will to stalk, and a few fries short of a happy meal goes a long way. -- bride of spidy


[ Parent ]

I suspect NIWS $ (none / 0) (#93)
by procrasti on Wed Mar 08, 2006 at 06:09:49 AM EST



-------
if i ever see the nickname procrasti again on this site or anywhere in my life, i want it to be in an OBITUARY -- CTS
doing my best at licking arseholes - may 2015 -- mirko
-------
Winner of Kuro5hin: April 2015
[ Parent ]
No. Talent would (none / 0) (#68)
by warrax on Tue Mar 07, 2006 at 02:58:43 PM EST



-- "Guns don't kill people. I kill people."
[ Parent ]
Probably not by deliberate application (none / 0) (#98)
by Metasquares on Wed Mar 08, 2006 at 10:41:54 AM EST

You may find that mathematical training helps you develop an instinct for composition (it did for me), but you probably won't use math very much deliberately when you're composing.

Unless it's that twelve tone nonsense, anyway :). I avoid it like the plague, but I think that the twelve-tone technique is basically matrix manipulation.

[ Parent ]

it's a well-trodden area, (none / 0) (#108)
by th0m on Wed Mar 08, 2006 at 05:37:52 PM EST

(check out this guy as a random example), but it would still be quite a leap to assume that an understanding of group theory is going to magically improve your musical ability.

there's a lot of snake oil in minimalist territory: why would anyone write pattern music if they were capable of crafting a beautiful, moving sonata? it probably just comes down to talent in the end, and it's ultimately a matter of opinion whether mathematical knoweldge has much to do with it.

[ Parent ]

the second link (none / 0) (#109)
by th0m on Wed Mar 08, 2006 at 05:39:01 PM EST

should've been this.

[ Parent ]
Uh, whatever (none / 0) (#116)
by der on Wed Mar 15, 2006 at 02:12:20 PM EST

Schoenberg (and Berg and Webern), and Stravinski, and .... had no talent?

Uh-huh.

Why would anyone code in a functional language when they're capable of crafting a beautiful, efficient C program (side effects and all)? It probably just comes down to talent in the end, right?



[ Parent ]
A bit of an exaggeration (2.80 / 5) (#13)
by Morkney on Mon Mar 06, 2006 at 10:48:22 AM EST

I think you're exaggerating when you claim that group representations are such a highly abstract and obtuse field of mathematics. Sure, they're not obvious to the layman, but their definition is quite simple, and some of the examples of groups and group representations are quite simple and concrete.

I'd have a far, far more difficult time explaining the weak-star topology, say, to a layman, than group representations.

Moreover, the comparative simplicity of group representations shows rather clearly why they are so applicable to physics. For example, why should rotation groups in R^3 correspond to important properties of objects in the real world? Simply because the same formal relationships hold. For example, if you rotate an object through one angle, then through another (in the same plane), then its total rotation is the sum of these angles: this is true in both R^3 and reality, at least to high approximation. It is therefore entirely unsurprising that subtle properties of rotations in R^3 point to subtle properties of rotations in real space, e.g. quantum-mechanical angular momentum commutator relations, etc.

I think the key point is that most mathematicians are working on extremely complex, abstract structures; but that these structures are in almost all cases created so as to shed light on very simple, eminently useful mathematical structures. Analysis in all its immensity boils down to the study of the real numbers, and the real numbers simply provide a useful conception of ideas like "quantity," or "distance." So it is hardly surprising that analysis finds uses throughout physics and economics, which are also concerned with subtle properties of things like "quantity."

As for philosophical problems, actual inconsistency would be a problem for the applicability of math, but anything short of an actual inconsistency would not, I think. Why? Well, suppose that we are studying a system, such as the universe. We discover certain rules which the system seems to follow, such as QM. These we state axiomatically as abstract relationships between abstract structures; for example, we say that a "particle" P and an "observable" O have the property that OOP = OP. We have now turned our real world system in to a collection of symbolic manipulations, but as of yet it is not really mathematical.

Next, we find a mathematical structure which has the same behavior. For example, the action of operators on a Hilbert space contains many of the observed formal properties of quantum mechanics. Now, IF the physical system we're studying is consistent, and IF the mathematical structure we use to describe it is consistent, then any behavior of the system which comes directly from these formal rules must be replicated in the mathematical structure. In other words, if any consistent system which follows rules like "OOP = OP," will also follow rules like the Wigner-Eckart theorem; and if the mathematical structure contains the Wigner-Eckart theorem; then necessarily, the physical system will also follow the Wigner-Eckart theorem.

This is a somewhat weak statement, in that we can not necessarily show that true statements in the mathematical structure are true in the physical system (they might be "incomputable" in the physical system, because they are derived from axioms which do not appear in the physical system). In practice, this is not such a problem. We have strong reason to believe, for example, that the result of ANY quantum-mechanical observation is computable in the physical system, for example. This suggests that, as long as we include enough physical axioms, the same result will hold in the mathematical structure.

Okay, these aren't rigorous arguments, but they provide a strong intuitive reason to believe that mathematics will successfully model reality.

All that was required. (none / 1) (#37)
by Coryoth on Mon Mar 06, 2006 at 07:25:00 PM EST

I think you're exaggerating when you claim that group representations are such a highly abstract and obtuse field of mathematics. Sure, they're not obvious to the layman...

Of course there are more abstract and more complex concepts. My interest was not in parading the most bizarre mathematical constructions I can think of, but rather in showing that apparently abstract math can tie very closely to physical reality in potentially unexpected ways. I think, particularly given the amount of definitional slog required to get there, that group representations are a sufficiently abstract concept to make the point, and that's all that's really required. I think the article is long enough as it stands.

Okay, these aren't rigorous arguments, but they provide a strong intuitive reason to believe that mathematics will successfully model reality.

The problem, however, is precisely that: it offers at best a vague intuitive reason. As soon as you try you tighten your grip with a little more rigour the whole thing slips through your fingers.

Jedidiah.

[ Parent ]

Not a "vague intuitive reason" (3.00 / 3) (#63)
by Morkney on Tue Mar 07, 2006 at 05:03:03 AM EST

The intuitive reason that mathematics works is not at all vague. That's my whole point. We invented groups to think about symmetry, and then, by some sort of miracle, they correspond to real-world symmetry. We invented calculus to think about distance and motion in the real world, and then, by some sort of miracle, it has a lot to say about distance and motion in the real world.

A very obvious example is this. We came up with the natural numbers to think in a formal setting about discrete quantities. Then we invented computers, which deal with discrete quantities called bits using the rules of mathematics. Why do we expect facts about abstract natural numbers, such as "1+1=2," to correspond to facts about real-world computations? As philosophers we can scratch our heads, but it's hard to have any real sense of puzzlement over this.

You are presenting the problem as something quite different and more mysterious. You are presenting it as though mathematicians just dream up entirely meaningless concepts, and then magically they have meaning in the real world.

90% of mathematical constructs have no real-world meaning. For example, in the study of real numbers, a great deal of attention is placed on open vs. closed intervals. These are physically meaningless in most applications, because reality is a little fuzzy, and does not care about a single mathematical point.

By suggesting that the constructs which you bring up are hopelessly abstract, you give the impression that their application to reality is actually mysterious. In fact, it is only mysterious in an abstract philosophical sense. I've taken classes on high-level quantum theory, in which we used group representation theory to deal with certain problems. It was not a matter of putting our parameters in to some mystical mathematical abstraction, then getting answers out. It was merely a general definition of some of the things we were studying anyway.

Your other examples are equally misleading. Hardy's work was on number theory. Number theory still hasn't found many applications to reality as such. Instead, it has been applied to computer science. Computer science is concerned with machines which manipulate numbers. How the hell is it surprising that number theory has applications there?

There is a great deal of mathematics which does not correspond to reality, and which never will correspond to reality, because the sort of things it talks about do not show up in reality. Numbers show up in reality, particularly once we start building computers. Rotations and symmetries show up in reality. Distances show up in reality. You are essentially taking some of the most well-grounded objects in mathematics, and then acting as though it is amazing that they aren't just a bunch of abstract nonsense.

[ Parent ]

Comments and Clarifications (none / 0) (#118)
by Krakhan on Sat Apr 01, 2006 at 03:49:18 PM EST

I'm rather late putting in another comment, but oh well, I'll post this anyways.

However, I just want to mention that groups were NOT originally invented to think about symmetry.  Rather, the actual modern day definition came up after a long period of investigation by looking at trying to find a closed-form expression for the roots of a quintic in terms of radicals.  From this, it was found that you need to look at what happens once you start interchanging the roots of the quintic equation to see the kinds of algebraic identities they satisfied.  This, among other things is what led to Galois Theory, showing WHY you can't do solve an arbitrary quintic by radicals.  For this reason, it was historically called the "theory of substitutions".

It was only many years after this that mathematicians in the late 19th century formalized the idea of a group to see what other kinds of things you'd get if you codified it.  So no, while ideas in mathematics don't just pop out of thin air, they didn't show up originally at all because someone was in a drunken stupor.  The symmetry that comes from studying groups is just a consequence of the original motivation.  Mario Livio's book "The Equation that Couldn't Be Solved : How Mathematical Genius Discovered the Language of Symmetry" goes into more details on the historical anecdotes.  It's a good read, so I recommend you look into it.

Also, computer science does not deal with the actual device used for computing.  Rather, it deals more with what can be computed, and if so, how can you go about doing it efficiently, and how can you efficiently manipulate the data and such. The computer can be whatever you want.  The term was originally applied to people before.  However, I'm not denying that the device isn't important at all, but that isn't the primary focus of computer science.

Computer Science also doesn't just deal with problems involving number crunching.  Graph Theoretic algorithms are of practical importance in a lot places, such as compilers and artificial intelligence, and these aren't purely numerical problems.  The same applies to formal languages for obvious reasons.  The wikipedia article on computer science has more details.  Again, you're confusing the original motivations of mathematics with how they are perceived in a modern light.

Although it is true number theory has really only been applied to cryptography, there are hints that they play some large role in Quantum Mechanics since there appears to be some connections with the Riemann Hypothesis.

in closing, it's interesting, since this is the kind of 'well-duh' response the OP seemed to be hinting at in the article.  What's more important is the unexpected connections you get afterwards from different areas of mathematics that makes it so cool.  Not all mathematics like calculus is made with physical motivation in mind.  There is still a lot more that could possibly be unlocked.    What will that be?  It remains to be seen.

~ Krakhan
[ Parent ]

I don't understand the mathematics involved here (3.00 / 3) (#21)
by Psychology Sucks on Mon Mar 06, 2006 at 01:49:28 PM EST

Could you give a few links for introductory material?  

Otherwise it appears to be interesting.

ror (none / 1) (#23)
by nilquark on Mon Mar 06, 2006 at 03:38:51 PM EST

read Bechtell's Theory of Groups.

[ Parent ]
Why the "ROR"? (none / 1) (#33)
by Psychology Sucks on Mon Mar 06, 2006 at 07:07:00 PM EST

I was serious.  But thanks for the recommendation.  I'm sorry that not everybody on K5 is especially good at math.

[ Parent ]
Simple group theory example (3.00 / 5) (#52)
by lukme on Mon Mar 06, 2006 at 10:56:59 PM EST

The multiplacative group for modulo 7 (*7).  That is we play the game for every a,b: (a*b) mod 7.

Building a multiplacation table for *7 we get:

  1 2 3 4 5 6
1 1 2 3 4 5 6
2 2 4 6 1 3 5
3 3 6 2 5 1 4
4 4 1 5 2 6 3
5 5 3 1 6 4 2
6 6 5 4 3 2 1

(note: you start with 1 with multiplacative groups and with 0 for additive groups.)

You will notice that this group has the following properties:

a) there is an idenity element (1)
b) there is clousure (every a multiplied by every b is within the group).
c) every element has an inverse
d) the associative property holds - that is A(B*C)=(A*B)*C

There are many more subtle properties of these groups, but you can see them as you consider other multiplactive groups.

It was very surpising even to mathematicians working on group theory (GH Hardy said as much in his book "a mathematician's apology") that group theory had a huge impact on science in the early 20th century.

You should check my math, since it is late for me, and I probably made several mistakes :).


-----------------------------------
It's awfully hard to fly with eagles when you're a turkey.
[ Parent ]

Thank you (none / 1) (#54)
by Psychology Sucks on Mon Mar 06, 2006 at 11:11:44 PM EST

Very good.  Now learn how to spell multiplication while I learn how to do multiplication.  j/k - thanks for the intro, I appreciate it.

[ Parent ]
Some examples not related to numbers (3.00 / 3) (#66)
by Cowculator on Tue Mar 07, 2006 at 12:22:54 PM EST

The Klein four group has four elements satisfying the following multiplication table:

  | 1 a b c
-----------
1 | 1 a b c
a | a 1 c b
b | b c 1 a
c | c b a 1

There's also the symmetric group on n letters, S_n, which consists of all permutations of {1,2,...n}.  To multiply two permutations, apply one and then apply the other.  Note that this is not an abelian group, meaning there is no commutative law: for most permutations a and b, we have ab != ba.

Or consider the dihedral group D_n of symmetries of an n-sided polygon.  For n=4, for example, there are a total of 8: you can rotate a square by some multiple of 90 degrees, or you can flip it vertically, and every symmetry is obtained by combining these two operations. This is again nonabelian, since rotating by 90 degrees and then flipping it is not the same as flipping first, then rotating.

Multiplication mod p is nice, since after all it gives you not just a group but a field, but if you're going to talk about group theory it's best to illustrate with some more abstract examples that can still be visualized.  Otherwise, people might wonder why group theory is such a big deal, since it looks like just another way to do arithmetic.

[ Parent ]

Interesting. (none / 0) (#87)
by lukme on Wed Mar 08, 2006 at 01:00:14 AM EST

It may be the time of day for me, however, I seem to vaguely recall that a Klein Group is a non-cyclic group that is abelian. Is my memory failing me?

Seem to me both of your non-numeric exmaples are based on numbers. My mind is failing me for a Klein Group example, but the example of the polygon being flipped, turned, and mirrored is actually a matrix multiplication group.

You have given nice examples, however, from my personal experience with course work in group theory, I have found it informative to start with simple groups, and then see how the more complex groups (matrix multiplicative groups) relate to the simple groups.

If there is interest, I could be convinced to attempt to write a story on some application of multiplication mod p group. Personally, I would love to read an account of how you can do something useful with the matrix multiplication groups - without any numbers.




-----------------------------------
It's awfully hard to fly with eagles when you're a turkey.
[ Parent ]
"Matrix multiplication groups" (none / 1) (#90)
by Cowculator on Wed Mar 08, 2006 at 02:09:22 AM EST

I don't know of some more general definition, but the group I gave is sometimes just known as the Klein group; it is indeed non-cyclic and abelian.  It doesn't seem like it would be useful to give a name to "abelian, non-cyclic groups."

And the dihedral group isn't really based on numbers: it's just given by a presentation like < x, y | x^n = y^2 = 1, xy=yx^-1 >, meaning that it's generated by arbitrary products of x and y which satisfy the given relations.  The matrix multiplication version of it is a faithful two-dimensional representation.  That's the idea of representation theory: a homomorphism r: G -> GL(V), where V = C^n, is just a way to replace each abstract element with a complex n x n matrix so that the matrices follow the same multiplication table.

As another example, the Klein group can be represented by something like
r(1) = [[1, 0], [0, 1]]
r(a) = [[-1, 0], [0, 1]]
r(b) = [[1, 0], [0, -1]]
r(c) = [[-1, 0], [0, -1]],
but it's not very exciting.  (Any irreducible representation of this group is not faithful, meaning two elements will be represented by the same matrix, but this example is reducible.)

Since finite groups have these representations, or ways to consider each element of the group as a matrix, finite group theory could be considered part of the study of these "matrix multiplication groups," but most basic results are proven without any use of these representations.  Everything from Lagrange's theorem (the order of a subgroup divides the order of the group) to the Sylow theorems can be proven just from the usual group axioms, and the results apply to groups of matrices just as well as they do to any other group.

[ Parent ]

I am biased (none / 0) (#103)
by JaxWeb on Wed Mar 08, 2006 at 01:31:40 PM EST

I am biased about this, but a while ago I wrote "Sets and Such" which talks about some of this: http://jax.hopto.org/maths/books/setsandsuch/

[ Parent ]
Well, obviously nobody told you (1.00 / 12) (#24)
by Sesquipundalian on Mon Mar 06, 2006 at 04:21:54 PM EST

what mathematics is, but then that's pretty much all that separates us from you anyways (at least that's what we tell each other at the meetings).

So.. ummm... how's it feel? To never know what mathematics truely is.. I bet it sucks ~heh I bet it reaaaaaaaallllllllyyyyyyy sucks! Ha! And I bet nobody will ever tell you either. 'Cause you know, we can all tell. One look at the way you write and we can sooooo tell you haven't guessed what mathematics actually is. It's so obvious. I know I sure won't tell you what it is, a little keener like you would just gossip it around anyway. Neener.


Did you know that gullible is not actually an english word?
Anyone read (none / 1) (#26)
by tetsuwan on Mon Mar 06, 2006 at 05:51:42 PM EST

Modern algebra, an introduction, by J R Durbin? It's not very advanced, but I liked it.

Njal's Saga: Just like Romeo & Juliet without the romance

Abstract Algebra : Groupt theory, ring theory, etc (none / 1) (#47)
by Gibidumb on Mon Mar 06, 2006 at 09:51:33 PM EST

What is : The thing that made me want to kill myself for it's totally fuckedupidness

The smartass answer (3.00 / 3) (#48)
by trhurler on Mon Mar 06, 2006 at 09:59:04 PM EST

Mathematics is not logic? Well then, what is logic?

Mathematics, of course.

People have an irritating tendency to try to find out the big questions before settling the little ones. It is bad enough when it leads to religion; we certainly don't need mathematicians off on wild goose chases like this.

--
'God dammit, your posts make me hard.' --LilDebbie

Ah, a Neologicist (3.00 / 2) (#50)
by Coryoth on Mon Mar 06, 2006 at 10:35:43 PM EST

That is essentially the position taken by the neologicists - we can't show that mathematics is reducible to pure logic, so we will simply redefine logic to include some new principles that will allow us to generate mathematics. It's a bit of a cheat in any ways.

What constitutes logic is resonably well defined, and for the most part when we speak of logic we refer to classical logic, which has a very simple and specific set of logical principles. Other interpretations of logic such as intuitionistic logic, paraconsistent logic, or modal logic are essentially weakenings of classical logic which either reject certain principles, or accept only weakened forms of some principles. As it stands classical logic is insufficient to construct even the minimum requirements for something to reasonably be called mathematics - in particular you require extra principles to be able to derive the natural numbers without running afoul of paradoxes on contradictions.

You can, if you wish, simply decide that, for example, the axiom of infinity and axiom of reducibility are purely logical principles. I suspect, however, that you'll struggle to get much traction for tht idea with most people.

Jedidiah.

[ Parent ]

Not what I meant (3.00 / 2) (#74)
by trhurler on Tue Mar 07, 2006 at 07:47:41 PM EST

What I meant is that rather than considering mathematics to be a subset of logic, the reverse is true. Mathematics DOES contain the whole of logic.

--
'God dammit, your posts make me hard.' --LilDebbie

[ Parent ]
That doesn't solve the dilemma then. (nt) (none / 0) (#76)
by Coryoth on Tue Mar 07, 2006 at 08:19:53 PM EST



[ Parent ]
What dilemma? (none / 0) (#111)
by trhurler on Wed Mar 08, 2006 at 08:09:10 PM EST

I apparently do not understand. If you did not mean to find a rational structure of the relevant human knowledge under discussion, what do you want?

Presumably, to be specific, you want some evidence that there exists a set of axioms from which one can derive mathematics which can somehow be proven to be "real." I suspect this is possible, but it sounds awfully tedious, and when you start to consider the ontological nature of such an argument("three oranges!"), it doesn't seem all that important to actually go through said tedium.

--
'God dammit, your posts make me hard.' --LilDebbie

[ Parent ]
Group theory applications (3.00 / 3) (#49)
by lukme on Mon Mar 06, 2006 at 10:24:26 PM EST

1) Most of modern cryptography is based on groups. RSA is somewhat based on prime multiplacative groups. Eliptical curves are additive groups (popular with cell phones).

2) Every chemist has learned some group theory, although it is relatively rare to find that chemist who realizes that what he is actually doing is using multiplactive matrix groups (the book by albert cotton explains this - although not as clear as it should be).

Uses in chemistry include finding symmetry adapted molecular orbitals, the vibrational modes of a molecule, and uv/vis absorbtion. Any book on molecular spectroscopy, inorganic chemistry, or physical methods will have a section on group theory.




-----------------------------------
It's awfully hard to fly with eagles when you're a turkey.
I am one of those chemists (none / 0) (#115)
by phobos13013 on Mon Mar 13, 2006 at 08:43:44 AM EST

My original direction in college was going to be physics but fearing the reality that would come after in the job market i switched to chemistry. My final two semesters of chemistry included physical chemistry. Unfortunately, my professor a gruff (swiss) proffessor named Ohrn put the fear of god in me regarding group theory. It was the most difficult class i have taken. No explanation of its theories, no discussion on notation, just an hour of pure symbology on a blackboard. He would fill the entire blackboard about eight times in a class. So although, my knowledge is limited, any professor in physical chemistry nowadays is well versed in its theory and application. Also the respected QTP project would be another large scale application of higher mathematics.

[ Parent ]
great article.... (none / 0) (#51)
by terryfunk on Mon Mar 06, 2006 at 10:52:45 PM EST

Thanks for writing it!

I like you, I'll kill you last. - Killer Clown
The ScuttledMonkey: A Story Collection

Another good intro Abstract Algebra text, crypto. (none / 1) (#58)
by strlen on Tue Mar 07, 2006 at 12:12:35 AM EST

Here is a book written by a professor that I've had as an undergraduate, which I've found very helpful.

And as I've stated an editorial comment, a good demonstration of the elegance of abstract algebra are the RSA cryptosystem and the Diffie-Hellman key exchange. While a good demonstration of the power provided by the abstraction part of Abstract Algebra is how these algorithms could be translated to elliptic curves, since the underlying abstract structure is the same.

--
[T]he strongest man in the world is he who stands most alone. - Henrik Ibsen.

Abstract Mathematics, Group Theory, and us (1.28 / 7) (#62)
by United Fools on Tue Mar 07, 2006 at 03:48:40 AM EST

Let's see...

Math is abstract, so it is something not touchable and not visible. So it does not mean anything real and IQs are numbers which do not mean anything.

Group theory... so people forms groups. So fools need to unite under the banner of United Fools.

So join us and fight for a world where men and women do not waste time on abstract thinking but work for the real things that matter! A world where everyone is equal!


We are united, we are fools, and we are America!

Paper article in new media = wasted opportunity (2.25 / 4) (#65)
by A Bore on Tue Mar 07, 2006 at 06:40:13 AM EST

Not enough links. Not readable enough. Sorry, it's a lot of effort, but...-1

why math works? (2.33 / 3) (#69)
by skewedtree on Tue Mar 07, 2006 at 05:33:38 PM EST

I think this is not a valid question. I would say that the main attribute in math is its consistency. When people are faced with a complex real-world problem, they resort to those tools they can trust. Mathematics is such a tool, that can be used to represent the problem in an abstract and predictable way. It's probably not the only way to face the problem, but it's the one we know.

For example, you mention cryptography. It's not that math magically ends up being the answer to our cryptographical needs, but people simply found this theory with interesting properties and then applied it to the concrete problem. I don't see any deep philosophical meaning to this.


simply view every single person you "meet" online as the comic book guy from the simpsons. it makes everything easier. -

Agreed. (none / 1) (#106)
by pb on Wed Mar 08, 2006 at 04:47:24 PM EST

Just look at number theory, which is largely based on counting--or concrete set theory, which could be easily expressed using--say--marbles, or geometry and trigonometry (duh). Anyone asking "why math works" might as well ask "why counting works". That's right folks, we can enumerate objects with numbers, or manipulate those numbers to represent what we can do with those objects, or even vice versa! Maybe I should write a story: "Concrete Mathematics, The Abacus, and You". Or a book, perchance.
---
"See what the drooling, ravening, flesh-eating hordes^W^W^W^WKuro5hin.org readers have to say."
-- pwhysall
[ Parent ]
Number Theory Is Not Counting (none / 0) (#110)
by Krakhan on Wed Mar 08, 2006 at 07:56:31 PM EST

You're confused between Enumeration and Number Theory. See the relevent wikipedia article here. You might notice search for 'counting' brings up anything.

Now, Combinatorial Number Theory does use some enumeration ideas, but that is hardly the whole of Number Theory.

~ Krakhan
[ Parent ]

a rarity for k5 (2.00 / 8) (#71)
by circletimessquare on Tue Mar 07, 2006 at 06:21:03 PM EST

well-written, intelligent, thorough, and noncontroversial

obviously, it's a sign of the apocalypse


The tigers of wrath are wiser than the horses of instruction.

Mathematics isn't a problem (1.37 / 8) (#75)
by debacle on Tue Mar 07, 2006 at 07:48:06 PM EST

Solved or unsolved, mathematics is a field of numeric relational discovery.

I gave you the benefit of the doubt, and read a bit further. More oral diarrhea.

Also, reads like serrated condom vomit.

Sorry, dumping for the benefit of k5umanity.

It tastes sweet.

It's one thing not to like the writing... (3.00 / 2) (#79)
by Cowculator on Tue Mar 07, 2006 at 09:29:08 PM EST

but how exactly is topology considered "numeric relational discovery?" HIBT?

[ Parent ]
Topology is all about numeric relational discovery (none / 1) (#81)
by debacle on Tue Mar 07, 2006 at 10:25:00 PM EST

At its very heart, it is about the relations between two geometric objects that are discovered through careful mathematical study. It is a much better definition than 'problem.'

You fucking fuck.

YHNBT.

It tastes sweet.
[ Parent ]

What is so numeric about topology? (nt) (3.00 / 2) (#82)
by Krakhan on Tue Mar 07, 2006 at 10:33:16 PM EST


~ Krakhan
[ Parent ]
YFI (none / 1) (#83)
by debacle on Tue Mar 07, 2006 at 10:45:59 PM EST

I give you the main quandry of topology:

How many holes does it have?

It tastes sweet.
[ Parent ]

Your display of ignorance is impressive. (nt) (none / 0) (#84)
by Coryoth on Tue Mar 07, 2006 at 11:11:00 PM EST



[ Parent ]
Okay, I'll bite. (none / 1) (#85)
by debacle on Wed Mar 08, 2006 at 12:14:59 AM EST

What is the main premise of topology, then?

It tastes sweet.
[ Parent ]
The premise of topology: (none / 1) (#89)
by nilquark on Wed Mar 08, 2006 at 01:35:41 AM EST

given a set X, a subset of T of P(X) is a topology if:
  1. the empty set and X are in T.
  2. the union of any subcollections of T is in T.
  3. the intersection of a finite number of subcollections of T in is T.


[ Parent ]
debacle wins! (none / 0) (#100)
by debacle on Wed Mar 08, 2006 at 12:23:47 PM EST

I believe that might fall into the realm of numeric relational discovery.

It tastes sweet.
[ Parent ]
debacle wins (none / 0) (#101)
by nilquark on Wed Mar 08, 2006 at 12:48:21 PM EST

a lifetime of bitterness, stupefaction, and denial for being mathematically retarded.

congratulations!

[ Parent ]

Possibly (none / 0) (#104)
by debacle on Wed Mar 08, 2006 at 01:40:05 PM EST

But probably not.

It tastes sweet.
[ Parent ]
I'm sure this square peg... (none / 0) (#102)
by Coryoth on Wed Mar 08, 2006 at 01:25:42 PM EST

...will fit in the round hole if I keep hitting it hard enough.

[ Parent ]
Ah yes (none / 0) (#105)
by debacle on Wed Mar 08, 2006 at 01:47:05 PM EST

And your definition was magnitudes more clear.

It tastes sweet.
[ Parent ]
This is a story about a girl. (1.62 / 8) (#78)
by Josh Ferien on Tue Mar 07, 2006 at 08:39:16 PM EST

She went to the University of California at Berkeley and loved her Chinese food something fierce. She was an EE/Mathematics double major.

One day she went on a blind date with a guy who, it just so happens, is a Republican, though not a particularly devout one. He shared her love for beanie babies and the two hit it off. So he asked her to come home and she couldn't say no. Just when things were going his way, though, she suddenly bolted with a ridiculous excuse about menopause. She was only 26!

Guess what her favorite subject in college was. Group theory.

-1.

Cordially,

Josh Ferien

The J is for Justice!

devout republican (2.33 / 3) (#91)
by tkatchevzz on Wed Mar 08, 2006 at 03:14:57 AM EST

lol worship the bush

[ Parent ]
That's what he gets (none / 1) (#113)
by partialpeople on Wed Mar 08, 2006 at 11:23:43 PM EST

for thinking goddamn beanie babies would get him laid.

[ Parent ]
troubleshooting tips (none / 0) (#117)
by khallow on Wed Mar 22, 2006 at 06:56:15 PM EST

The procedure should have ruled out dating people from UC Berkeley. The male should redo his dating specifications to eliminate false positives from that source.

Stating the obvious since 1969.
[ Parent ]

I tried to read it. (none / 1) (#99)
by Mylakovich on Wed Mar 08, 2006 at 12:11:30 PM EST

Got through a few paragraphs and zoned out.

Too boring.

Mathematikos (none / 1) (#107)
by calumny on Wed Mar 08, 2006 at 05:30:45 PM EST

Incredibly boring. The group axioms are in Wikipedia, and meandering philosophical wankery is just about all of popular math writing.

Something interesting? De Bruijn sequences are something. This site has a nice write up on how math helps you break and enter.



Nitpicking (none / 1) (#112)
by number1729 on Wed Mar 08, 2006 at 09:45:37 PM EST

Let's begin with the definition of a group, sacrificing a little rigour and precision for the sake of brevity: A group is a set of objects with a particular "algebraic structure". By an algebraic structure I mean a set of operations that take one or more elements of the set as arguments and return a single element of the set. A natural concrete example is the algebraic structure provided to the set of natural numbers by addition and multiplication - operations that take two numbers and return a single number.

Actually, the natural numbers don't form a group with respect to addition or multiplication. (There's an "unnatural" way to make the natural numbers into a group, via any bijection s:N->Z: define an operation *:NxN->N by n*m=s^{-1}(s(m)+s(n)). This isn't terribly interesting, though.)

Thank you (none / 0) (#114)
by alyosha1 on Fri Mar 10, 2006 at 11:08:46 AM EST

A well-written article that I could actually follow. Do you have any links to the discussion regarding multi-threading? Thanks.

NICE!!! (none / 0) (#119)
by DT1234 on Wed Apr 19, 2006 at 11:58:02 AM EST

Very nice

Abstract Mathematics, Group Theory, and You | 116 comments (65 topical, 51 editorial, 0 hidden)
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