Good Math, Bad Math

The point of set theory isn’t just to sit around and twiddle our thumbs about
the various definitions we can heap together. It’s to create a basis on which we
can build and study interesting things. A great example of that is something
called a group. Once we’ve built up sets enough to be able to understand a set of values and an operator, we’re able to build an amazing deep and interesting construction, called a group.

Back when I started this blog, one of the first topics that I
wrote about was group theory. I was just looking back over that
series of posts, and frankly, they sort of stink. I’ve leaned a lot about
how to write for a blog in the time since then, and so I’d like to go back
and rewrite it. I’ve never reposted any of the group theory material, so
it should also be new to most readers.

As you should know from reading this blog, one of the things in math
that fascinates me the most is abstraction: taking some subject of interest,
paring it down, and reducing it to its most basic essentials, in order to
really understand it and what it means.

Group theory is part of a broader topic called abstract
algebra
. Abstract algebra reduces things to the minimum concepts
of a set of values and some operations over those values. By specifying
the properties of the set and operations, you can create different
algebraic structures. Group theory is, basically, the simplest construction
of abstract algebra: one collection of values, and one operation.

What do you do in group theory? There’s a wonderful
quote, which was written by a famous writer of popularized mathematics
named James Newman. Professor Newman described group theory as:

The theory of groups is a branch of mathematics in which one does
something to something and then compares the results with the result
of doing the same thing to something else, or something else to the
same thing.

So we’re taking the idea of algebra: that is, the study of
equations representing numeric relationships, where the equations are
constructed from numbers, variables, and simple operations like
addition, subtraction, multiplication and division. What we want to do
is abstract that. So we’re going to discard the idea of numbers, and
replace it with a totally abstract notion of a collection of
values. Group theory often assumes that the collection of values is a
set, but you can also study groups built from proper classes. We’re
going pair the collection of values with a single operator, which
we’ll write “•”. Despite the fact that multiplication is often
written as a dot, it’s important to realize that here, in group theory, we’re not talking about multiplication. We’re talking about a totally abstract operator.

So, a group is formed from two things: a collection of values, and
an operator. We’ll often write that as a pair, G=(V, •). In order
to be a group, this pair must have the following properties:

  1. Closure: ∀a,b ∈V: (a•b)∈V. That is, applying the group operator to any pair of values in the group produces a value in the group.
  2. Identity: &exists;1∈V: (∀a∈V, a•1 = 1•a = a.) : there is an identity value in the group, which we’ll call “1″ such that applying the group operator to a value “a” and 1 in any order results in “a”.
  3. Associativity: ∀a,b,c∈V: a•(b•c) = (a•b)•c.
  4. Inverse: ∀a∈V: (∃a-1∈V : a•a-1=1). For every value in the group, there’s an inverse value, such that applying the group operator to a value and its inverse gives you the identity value.

An example of a group is (Z,+): that is, the set of integers and the addition
operator. We can easily see that this is a group:

  1. For any two integers a and b, a+b is an integer.
  2. For any integer, a+0 = 0+a = a, so 0 is the identity value.
  3. We know addition is associative.
  4. For any integer a, there’s an additive inverse -a such that a+-a=0.

On the other hand, (R,×) is not a group. Why not?

  1. For any pair of real numbers, their product is a real number.
  2. For any real number r, r×1 = 1×r = r.
  3. Multiplication is associative.
  4. There is a multiplicative inverse, 1/n. But it is not true that
    for all values n∈R that 1/n exists: there is no multiplicative inverse of 0.

So 0 kills the real numbers as a group. If you use R-{0} as the set of
values, then you’ve got a group.

So what does this all mean? You’ll see more in later posts. But fundamentally, this
incredibly simple idea: a bunch of values and one operator with 4 properties – this
defines the entire concept of symmetry. Everything that we understand as symmetry
is completely encapsulated in the concept of a group.

The idea of symmetry in group theory is useful for more than just abstract
reasoning. Obviously, it’s used for algebra: the basic idea of group theory was
largely developed in the study of algebra and the symmetries of algebraic solution. But it’s useful for a lot more than that. A few example, to give you a taste:

  • Relativity: Much of relativity is defined it terms of groups; the invariants of relativity are
    largely defined in terms of groups and symmetries.
  • Music: There’s a way of looking at music theory using groups: natural “operations” that occur in music and chords, like inversion, transposition, etc., all exhibit group symmetries.
  • Chemistry: you can determine the polarity of a molecule by using group theory to identify the symmetries in the structure of the molecule.

Comments

  1. #1 Susan B.
    December 3, 2007

    Perfect timing! I need a refresher on abstract algebra, since I’m about to take some hard courses on it and haven’t studied it since freshman year. Group theory is such a rich field, too–should give you lots of interesting topics to discuss!

  2. #2 Carsten König
    December 3, 2007

    Well I allways like reading some of the basic stuff – you might want to build a “example” as you progress.
    My favorite is “rubiks cube” as you can get a lot of insight with rather few algebra.
    Another one might be the 15-tile-puzzle (sorry forgot the english name) where you can show with easy group theory (permutation groups) that you can’t switch to adjacent tiles ;)

  3. #3 KKairos
    December 3, 2007

    This is awesome. I’m not particularly enjoying group theory right now, but I know that’s mostly because I’m being graded on it. There’s cool stuff to be done with it, so I’m looking forward to seeing more stuff on the subject; will you be talking isomorphisms as well?

  4. #4 David
    December 3, 2007

    I just wanted to say that I am really glad that you decided to revisit group theory. This was the subject that led me to reading your blog (the old one) in the first place.

  5. #5 Josh
    December 3, 2007

    In addition to the symmetries of algebraic solutions, group theory was also developed to study symmetries in geometry and symmetries in number theory. For a more complete history, see http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Development_group_theory.html.

  6. #6 Hank
    December 3, 2007

    Lovely. Group theory was a defining “A-HA!” moment for me, made alot of things fall into place.

    Carsten König: I agree. Problems involving 15-/9-games are interesting because you get so much from doing so little. My lecturer used those as an example, to great effect.

    The fact that I’ve spent countless hours trying to solve 9-tile-games as a kid might have something to do with my excitement as well, but I digress.

  7. #7 Dan Fitch
    December 3, 2007

    Fantastic. I remember the old posts as well, and look forward to the re-writing. Go Mark go!

  8. #8 Isabel
    December 3, 2007

    In response to Carsten Konig’s comment: there is actually a book which purports to be an introductory textbook on group theory that basically develops all the material using the Rubik’s Cube as a standard example. I haven’t thoroughly read the book, but I flipped through it in a bookstore once and it seemed quite good.

  9. #9 Jonathan Vos Post
    December 3, 2007

    Almost entirely true, and lucid. I nitpick on: “Group theory is, basically, the simplest construction of abstract algebra: one collection of values, and one operation.” But Groups are “merely” an important special case of semigroups (essential in descriging dynamics) which in eurn are “merely” an important special case of groupoids.

    John Baez et al at n-Category Cafe have been blogging for a year about what they call “groupoidification” — a program for Math and Physics of doing things in sophisticated ways with something simpler than a group. To excerpt from MathWorld on “groupoid”:

    There are at least two definitions of “groupoid” currently in use.

    The first type of groupoid is an algebraic structure on a set with a binary operator. The only restriction on the operator is closure (i.e., applying the binary operator to two elements of a given set S returns a value which is itself a member of S). Associativity, commutativity, etc., are not required (Rosenfeld 1968, pp. 88-103). A groupoid can be empty. The numbers of nonisomorphic groupoids of this type having n==1, 2, … elements are 1, 10, 3330, 178981952, … (Sloane’s A001329), and the corresponding numbers of nonisomorphic and nonantiisomorphic groupoids are 1, 7, 1734, 89521056, … (Sloane’s A001424). An associative groupoid is called a semigroup.

  10. #10 sciencegirl
    December 3, 2007

    Ditto on the tile games :) First exposure to group theory was one of my “A-HA!” moments too, and I am always amazed about the new developments in the field. Mark, I hope you get to write about some of the new stuff here in addition to the basics.

  11. #11 Samuel A. Falvo II
    December 3, 2007

    The article you’ve written makes singularly concise and easy to understand the concept of groups, something I’ve been thoroughly baffled by for some time. Many thanks!!

  12. #12 L6
    December 3, 2007

    Excellent timing on this article, as I’m reading “Not Even Wrong” and the chapter I’m on discusses group theory. :)

  13. #13 Frank-Leonardo Quednau
    December 3, 2007

    Hi,
    in both cases of defining the inverse you use an operator that was not defined as part of the group (“minus”n or 1″over”N). How does it come into play? Or are those operators actually defined because there is an inverse?
    You see, I have no idea, please help :)

  14. #14 Mark C. Chu-Carroll
    December 3, 2007

    Frank:

    Good question. I did, admittedly, handwave a bit. Basically,
    we all understand the structure of the integers and the reals, and we know what 0-N means in the integers, or what 1/N means in the reals. You can describe them declaratively in logic using nothing but the group operator, but it’s a laborious process, and I didn’t think that explaining it in full detail would really add anything.

    Just to give you an idea: if you have an integer N≠0, there exists exactly one integer which we can call -N for which N+-N=-N+N=0. Further, for the set of integers, -(-N)=N. For 0, -0=0. With the properties of the group operator, and the definitions from Peano arithmetic, you can show from here that the additive inverse of the integers total and unique.

  15. #15 Lee Chapman
    December 3, 2007

    I love groups but have always wondered: once we show that a binary operation is commutative and associative, haven’t we shown that it isn’t REALLY binary at all? That it’s really a function from multisets to values? E.g. sum{2,2,3} = 7. Groups are fascinating but seemed to be based on a lie! (small ‘L’)

  16. #16 Hank
    December 3, 2007

    I think I might be thrown off by not knowing exactly what you mean by binary operations. However, group operations need not be commutative, except for abelian groups.

  17. #17 randy
    December 3, 2007

    To the above by Lee, I guess you could say that. However your defining “binary operation” as an operation that can ONLY take two elements to one element (i.e. can ONLY describe a map from GxG to G) essentially defines something that can never exist. No map can ever satisfy your definition of binary operation. This follows since any map f from GxG to G can be naturally extended to a map f’ from GxGxG to G simply by using f on the first two G’s and then again on that image coupled with the last G. In other words f’=f(f(GxG),x) with x in G.

  18. #18 randy
    December 3, 2007

    pardon me…i should have written f’(x,y,z)=f(f(x,y),z) with x,y,z in G.

  19. #19 Doug Spoonwood
    December 3, 2007

    Mark,

    [Group theory is, basically, the simplest construction of abstract algebra: one collection of values, and one operation.]

    Basically the simplest? Group theory also has three additional axioms. Monoids have one collection values, one operation, and FEWER axioms than groups. So do magma or groupoids. I like the diagram here http://en.wikipedia.org/wiki/Magma_(algebra)

    [Despite the fact that multiplication is often written as a dot, it's important to realize that here, in group theory, we're not talking about multiplication. We're talking about a totally abstract operator.]

    I think you just mean to emphasize that the “dot operator” of group theory doesn’t necessarily refer to the multiplication operator. But, I don’t see how that makes it a “totally abstract” operator while multiplication doesn’t qualify as a totally abstract operator. I would say that multiplication itself qualifies as a totally abstract operator. I suppose one may regard group theory as *partially* an abstraction derived from multiplication (of course historically group theory came as an abstraction of geometric properties first and only later arithmetic, as I understand history at least), but in such a case we have an abstraction of an abstraction.

    Should I have seen the ‘for all’ and ‘there exists’ quantifiers in their regular form or as “?x” and “&exists”?

    Isabel,

    [In response to Carsten Konig's comment: there is actually a book which purports to be an introductory textbook on group theory that basically develops all the material using the Rubik's Cube as a standard example.]

    One can find plenty of information concering Rubik’s cube and group theory by searching “Rubik’s cube and group theory”, some from courses on this topic.

    Lee Chapman,

    [I love groups but have always wondered: once we show that a binary operation is commutative and associative, haven't we shown that it isn't REALLY binary at all? That it's really a function from multisets to values? E.g. sum{2,2,3} = 7.]

    I suppose that once you show a structure commutative and associative, one can regard it like a unary operator. But, this doesn’t mean it doesn’t still qualify as a binary operator. With your example, we can still write (2+2)+3=7, and do the problem by taking 2+2 first, and then the result of that and adding it to 3. So, we can still use ‘+’ as a binary operator, but since it works out as commutative and associative we can use it as a unary operator. In other words, we can *use* it properly both ways, and consequently qualifies as both unary and binary. I don’t see any need to declare an operator as either unary or binary. If instead of asking we ask “what type of operator is it”, we ask “how can we use the operator”, the (perceived) paradox between an operator qualifying as both unary and binary evaporates (or at least seems to do so). Sometimes it may work out simpler to regard such as a binary operator, sometimes it may work out simpler to regard such as a unary operator.

  20. #20 Barry
    December 4, 2007

    This must be the first time I have had read any practical use for Group Theory. I would appreciate any relevant references (web, book, article, other) you can provide for Music and Chemistry details. If you have any Comp. Sci. references I’d be interested too. Thanks. (You just made that inner spark go nuts! in a good way)

  21. #21 randy
    December 4, 2007

    Concerning computer scientific applications of group theory

    1) All the most popular motivations for public key systems reduce to trivial group theoretic statements. For instance, in RSA if you take a message and raise it to phi(N) mod N it remains unchanged. The whole thing works because phi(N) is the order of the group of units mod N and x^|G| is always x in the group G. Furthermore, some cryptosystems are based on using diffie helman or elgamal the same way as usual but in more complicated groups instead of the integers mod p.

    2) Attacks on public key systems are also group/ring theoretical. The LLL algorithm used to break subset sum systems is entirely dependent on the vector spaces of field theory. The number field sieve is dependent on ring theory for its motivation.

    3) I bet someone could talk better about this third one than I could, but I remember from a lecture that error correcting codes are motivated very easily when you put a group structure on the possible bit flips. something like that. it was a neat application anyway.

  22. #22 Hank
    December 4, 2007

    randy: Linear error correcting codes at least (where if c and d are code words, so is c+d). Take Hamming for example.

  23. #23 Thony C.
    December 4, 2007

    of course historically group theory came as an abstraction of geometric properties first and only later arithmetic, as I understand history at least

    Doug, one doesn’t “understand” history one “knows” it and you obviously don’t know yours!

    Group theoretical concepts (GTC) were first used implicitly in the 18th century by Euler both in his work in number theory and in the solution of algebraic polynomials. The later implicit use was extended by Lagrange and then taken up explicitly by Ruffini (1799) at the end of the century. Abel and Galois both continued and extended this work in the early part of the 19th century. Several people most notably Jordan extended Galois work and it was Jordan who first applied GTC to geometry in the second half of the 19th century although GTC had been applied somewhat earlier in crystallography. Also in the first half of the 19th century the English algebraists most notably Cayley and Sylvester explicitly used GTC in the algebraic theory of linear transformations. This use was transferred by Lie and Klein to the analysis of geometrical structures leading eventually to Klein’s legendary Erlanger Programm (1872). Cayley (1854 &1878) was the first to treat purely abstract group concepts but only for finite groups, von Dyck (1882-3) extended these considerations to infinite groups. The first axiom system for groups was proposed by Weber in 1893. This is of course just a simplified outline and the real history is much more complicated but as you can see group theory was born in algebra and not in geometry.

  24. #24 Thony C.
    December 4, 2007

    For all those who can’t tell their unaries from their binaries!

    An operation on a set S is a function whose domain is a set of ordered sequences (x1,x2,xx,…,xn) of members of S and whose range in contained in S. The operation is unary, binary, ternary, etc., according as n is 1, 2, 3, etc. Mathematical Dictionary 4th ed., James and James, New York etc., 1976.

  25. #25 andy
    December 4, 2007

    Group theory and music: This Week’s Finds (Baez), week 234.

  26. #26 randy
    December 4, 2007

    Thony, that’s not completely fair. I’m not going to battle you over your post enlightenment math history, but in the deep pre enlightenment there are findings of artifacts who’s purposes are nearly unexplainable but for demonstration of rotation and translation groups. For instance a rock in the shape of a cube with the corners identified…the evidence is very strong. Also (I’m holding my breath and going into recent math history where I’m sure you’re well read), straight edge and compass arguments (as in “what can you draw and what can’t you?”) were a very large motivation for field theory which by many accounts pre-dates group theory.

  27. #27 Doug Spoonwood
    December 4, 2007

    Thony C.,
    [Doug, one doesn't "understand" history one "knows" it and you obviously don't know yours!]

    I simply don’t know or understand what you mean by “understand” and “know” here. Second, if it worked out as obvious, then I would *already* have realized such. I didn’t, so it doesn’t work out as obvious.

    I did speak incorrectly in that group theory arose NOT first from geometry, but first from arithmetic. However, this still doesn’t imply that “group theory was born in algebra and not in geometry.” See, the MacTutor site here

    http://www-groups.dcs.st-and.ac.uk/~history/
    HistTopics/Development_group_theory.html

    Also, if you want to talk about how Euler “implicitly” used group theory, see Randy’s comment again.

    [An operation on a set S is a function whose domain is a set of ordered sequences (x1,x2,xx,...,xn) of members of S and whose range in contained in S.]

    I didn’t necessarily even want to imply a formal definition of ‘n-ary’ here. Look, if commutativity and associativity hold, then we can view the addition operator as acting on the entire set as in +{4, 5, 8, 2, 3, 6}. We can consider the set as *just one* sequence and take a sum over the entire set. In this sense (or something like it), ‘+’ qualifies as a unary operator. Informally speaking, one might also think of a n-ary operator having its arity, or number of arguments that our set takes, as determined by how many terms we use when using the operator. In other words, since associativity and commutavity hold, we can view
    +{5, 6, 8, 9} as having one argument. The argument consists of the entire set. I don’t know if Lee meant something like this, but he could have.

    Since I find Thony’s defintion a bit unclear, let’s say I use a more techincal definition here
    http://planetmath.org/encyclopedia/OperatorSymbol.html
    I’ll let X={1, 2, 3, 4, 5}. We’ll evaluate +{1, 1, 2}. So, by definition we have
    +:{1, 2, 3, 4, 5)x(1, 2, 3, 4, 5}x{1, 2, 3, 4, 5}->{1, 2, 3, 4, 5}
    Consequently, we have a trinary or 3-ary operator at work. In such a case, the problem of the arity of ‘+’ still arises, *given that the plus indicated above works similarly to our normal plus*. They DO both work the same way, as if we use the binary ‘+’ operator twice we get the same result, as in the following:
    +:(1, 2, 3, 4, 5}x{1, 2, 3, 4, 5}->{1, 2, 3, 4, 5} for 1+1, then we have 2. And if we have
    +:(1, 2, 3, 4, 5}x{1, 2, 3, 4, 5}->{1, 2, 3, 4, 5} for 2+2, we have 4, which yields the same result as the
    +:{1, 2, 3, 4, 5)x(1, 2, 3, 4, 5}x{1, 2, 3, 4, 5}->{1, 2, 3, 4, 5}.

    Consequently, Lee Chapman’s point still arises, just in a different way. We simply do NOT have to decide if ‘+’, taken as a single operator and NOT two different operators as above, “is” or “is not” either binary or 3-ary. We can say that it WORKS validly BOTH ways.

    Randy,

    [To the above by Lee, I guess you could say that. However your defining "binary operation" as an operation that can ONLY take two elements to one element (i.e. can ONLY describe a map from GxG to G) essentially defines something that can never exist. No map can ever satisfy your definition of binary operation. This follows since any map f from GxG to G can be naturally extended to a map f' from GxGxG to G simply by using f on the first two G's and then again on that image coupled with the last G. In other words f'=f(f(GxG),x) with x in G.]

    I think you have a good argument here. If an operation works as binary, then GxG->G and consequently
    (GxG)xG->GxG->G, so an “extension” of an n-ary operator to a (n+1)-ary operator always works as possible. In this way, I think one can suppose that every n-ary operator has a potentially countably infinite number of arguments using mathematical induction. Consequently, the arity of an operator doesn’t indicate how many arguments an operator takes. It indicates *the least* number of arguments an operator can take. It gives us a lower bound on the number of arguments the operator will take, and does NOT indicate anything about how many arguments the operator will take in use. Consequently, we can consider ‘+’ as a two-termed, three-termed, four-termed … n-termed operator, binary operator. “Binary” doesn’t refer to the number of terms, but the least possible number of terms. I don’t think there exists any sense in saying that ‘+’ “is” or “is not” either two-termed or three-termed, as in many contexts we can use it both ways validly.

  28. #28 Xanthir, FCD
    December 4, 2007

    Doug: You’ve used a lot of words to work through it, but you’re essentially right. ^_^

    If you establish a binary operator, and establish that it is associative and commutative, then you’ve implicitly defined a series of n-ary operators with n >eq; 2.

    In many cases it *does* matter what the arity of the specific function is, but in many cases it does not. Most of the time mathematicians prefer starting with the ‘simplest’ version of it and then allowing the additional structure to grow implicitly, so they just define + as binary and let the trinary, quaternary, etc. versions just arise unspoken.

    The good things about *this* approach is that it does, as you say, specify the minimum number of arguments the function must take. As well, it does not have to assume associativity or commutivity. Frex, if you have a non-assoc, non-comm binary operator F, F(x,y,z) is ambiguous, but F(F(x,y),z) isn’t. Rather than using the special ternary way of writing it when it happens to be valid and unambiguous, it’s easier to just write everything as a binary operation if it can be, with composition of functions when you need to apply it multiple times.

  29. #29 Doug Spoonwood
    December 4, 2007

    Thanks Xanthir.

  30. #30 Thony C.
    December 5, 2007

    I really don’t understand you Doug. You claimed that group theory was developed historically in geometry, a claim that I corrected in that I outlined how it developed in the attempts to find a general solution to the solvability of polynomial algebraic equations. You then wrote the following;

    I did speak incorrectly in that group theory arose NOT first from geometry, but first from arithmetic. However, this still doesn’t imply that “group theory was born in algebra and not in geometry.” See, the MacTutor site here

    http://www-groups.dcs.st-and.ac.uk/~history/
    HistTopics/Development_group_theory.html

    You managed to mangle your hyper-link but if corrected it leads here, where the following text is displayed;

    Permutations were first studied by Lagrange in his 1770 paper on the theory of algebraic equations. Lagrange’s main object was to find out why cubic and quartic equations could be solved algebraically. In studying the cubic, for example, Lagrange assumes the roots of a given cubic equation are x’, x” and x”’. Then, taking 1, w, w2 as the cube roots of unity, he examines the expression

    R = x’ + wx” + w2x”’

    and notes that it takes just two different values under the six permutations of the roots x’, x”, x”’. Although the beginnings of permutation group theory can be seen in this work, Lagrange never composes his permutations so in some sense never discusses groups at all.

    The first person to claim that equations of degree 5 could not be solved algebraically was Ruffini. In 1799 he published a work whose purpose was to demonstrate the insolubility of the general quintic equation. Ruffini’s work is based on that of Lagrange but Ruffini introduces groups of permutations. These he calls permutazione and explicitly uses the closure property (the associative law always holds for permutations). Ruffini divides his permutazione into types, namely permutazione semplice which are cyclic groups in modern notation, and permutazione composta which are non-cyclic groups. The permutazione composta Ruffini divides into three types which in today’s notation are intransitive groups, transitive imprimitive groups and transitive primitive groups.

    Ruffini’s proof of the insolubility of the quintic has some gaps and, disappointed with the lack of reaction to his paper Ruffini published further proofs. In a paper of 1802 he shows that the group of permutations associated with an irreducible equation is transitive taking his understanding well beyond that of Lagrange.

    Cauchy played a major role in developing the theory of permutations. His first paper on the subject was in 1815 but at this stage Cauchy is motivated by permutations of roots of equations. However, in 1844, Cauchy published a major work which sets up the theory of permutations as a subject in its own right. He introduces the notation of powers, positive and negative, of permutations (with the power 0 giving the identity permutation), defines the order of a permutation, introduces cycle notation and used the term système des substitutions conjuguées for a group. Cauchy calls two permutations similar if they have the same cycle structure and proves that this is the same as the permutations being conjugate.

    Abel, in 1824, gave the first accepted proof of the insolubility of the quintic, and he used the existing ideas on permutations of roots but little new in the development of group theory.

    Galois in 1831 was the first to really understand that the algebraic solution of an equation was related to the structure of a group le groupe of permutations related to the equation. By 1832 Galois had discovered that special subgroups (now called normal subgroups) are fundamental. He calls the decomposition of a group into cosets of a subgroup a proper decomposition if the right and left coset decompositions coincide. Galois then shows that the non-abelian simple group of smallest order has order 60.

    Galois’ work was not known until Liouville published Galois’ papers in 1846. Liouville saw clearly the connection between Cauchy’s theory of permutations and Galois’ work. However Liouville failed to grasp that the importance of Galois’ work lay in the group concept.

    Betti began in 1851 publishing work relating permutation theory and the theory of equations. In fact Betti was the first to prove that Galois’ group associated with an equation was in fact a group of permutations in the modern sense. Serret published an important work discussing Galois’ work, still without seeing the significance of the group concept.

    Jordan, however, in papers of 1865, 1869 and 1870 shows that he realises the significance of groups of permutations. He defines isomorphism of permutation groups and proves the Jordan-Hölder theorem for permutation groups. Hölder was to prove it in the context of abstract groups in 1889.

    Klein proposed the Erlangen Program in 1872 which was the group theoretic classification of geometry. Groups were certainly becoming centre stage in mathematics.

    Perhaps the most remarkable development had come even before Betti’s work. It was due to the English mathematician Cayley. As early as 1849 Cayley published a paper linking his ideas on permutations with Cauchy’s. In 1854 Cayley wrote two papers which are remarkable for the insight they have of abstract groups. At that time the only known groups were groups of permutations and even this was a radically new area, yet Cayley defines an abstract group and gives a table to display the group multiplication. He gives the ‘Cayley tables’ of some special permutation groups but, much more significantly for the introduction of the abstract group concept, he realised that matrices and quaternions were groups.

    Cayley’s papers of 1854 were so far ahead of their time that they had little impact. However when Cayley returned to the topic in 1878 with four papers on groups, one of them called The theory of groups, the time was right for the abstract group concept to move towards the centre of mathematical investigation. Cayley proved, among many other results, that every finite group can be represented as a group of permutations. Cayley’s work prompted Hölder, in 1893, to investigate groups of order

    p3, pq2, pqr and p4.

    Frobenius and Netto (a student of Kronecker) carried the theory of groups forward. As far as the abstract concept is concerned, the next major contributor was von Dyck. von Dyck, who had obtained his doctorate under Klein’s supervision then became Klein’s assistant. Von Dyck, with fundamental papers in 1882 and 1883, constructed free groups and the definition of abstract groups in terms of generators and relations.

    Group theory really came of age with the book by Burnside Theory of groups of finite order published in 1897. The two volume algebra book by Heinrich Weber (a student of Dedekind) Lehrbuch der Algebra published in 1895 and 1896 became a standard text. These books influenced the next generation of mathematicians to bring group theory into perhaps the most major theory of 20th Century mathematics.

    Which is as far as I can see a more detailed version of my post showing that group theory developed out of attempts to find a general solution to the solvability of polynomial algebraic equations. If you want to call that arithmetic that’s fine by me but it places you in a minority of one amongst mathematicians.

    On a more general note not just to Doug but to all who come here. A lot of you refer to MacTutor when confronted with questions from the history of maths. Whilst it is the most comprehensive web source for the history of mathematics the individual articles are plagued with mistakes both small and large and the editors seem to be impervious to criticism from their professional colleges. If one is seeking historical data for any serious reason it pays to control that data aginst the written sources given in the bibliographies following each article.

  31. #31 cleek
    December 7, 2007

    by coincidence, i just finished a book called “The Equation That Couldn’t Be Solved”, which traces the origins of group theory. at least as the book lays things out, group theory’s origins are deeply entwined with the search for a proof of the [in]solvability of quintic equations – specifically, to prove that there is no general solution to quintics (as there is with quadratics, for example); it’s origins are very much about algebra and not at all about arithmetic – Abel and Galois weren’t looking for arithmetic answers, they were trying to say something about classes of equations.

  32. #32 Jonathan Vos Post
    December 7, 2007

    Thony C:

    “If you want to call that arithmetic that’s fine by me”

    Once upon a time, what we now call Number Theory was called Higher Arithmetic.

    As to the alleged border between Geometry and Algebra, there were singularities from Euclid, lots of guys, Descartes, lots of guys and gals, Newton, lots of guys and gals, and then all the Algebraic Geometry, and Grothendieck, and Catgories. I really don’t know what “Geometry” and “Algebra” are anymore, except in High School textbooks, where I’m unfortunately familar as someone earning Public School Teaching Credentials from people who’ve had thousands of fewer students than I have, and published much less Math.

    Symmetry. We were talking about Symmetry. But Finite Group examples so far, not into Lie Algebras, where I suspect our Host may be heading…

  33. #33 ray burchard
    December 9, 2007

    System Matrix, the Mobuis Band

    The numerical Mobius band as an innate symmetric duality, where the mathematical conjugation of numerical magnitude and directional numerical flow are commensurate coparcener principles. Newtonian mechanics… A force acted in one direction does not have the same effect in the material world as a force of equal magnitude acted in a different direction.

    Example;

    Ten digits, nine value numerical sequence plus its inverse sequence.
    0123456789 group sequenced left to right, (smaller to larger),
    9876543210 group sequenced right to left, (smaller to larger)
    Therefore, (00+01+02+03….thru 09) = 045, while the sequenced inverse (90+80+70…back thru 00) = 450 and therefore, (045+450) = 495.

    Group and Set;
    01234 56789
    98765 set sequence (top to bottom), 43210 set sequence, (bottom to top).
    And as vertical sets; 01234
    98765 equals (09+18+27+36+45) = 135
    Symmetrical inverse 56789
    43210 equals (54+63+72+81+90) = 360,
    And again (135+360) = 495.

    Fractional resolve versus “whole” number resolve, where the “whole” has a greater value then the sum of its comprising components.
    Take the string theory principles 0 and 1 and its prevalent number 11. Where as, 01 plus its symmetrical inverse (mirrored image) 10 , equals (01+10) = 11

    Then, 495 divided by 11 equals the “whole” of 45, while individually 135 divided by 11 equals 12.2727… Plus 360 divided by 11 equals 32.7272… And therefore (12.2727… + 32.7272…) = 44.9999… And 44.9999… Is the fractional equivalent of the “whole” of 45.

    Try, ( Ic ) as the symmetrical conjugation of nine more than ninety, (09+90) = 99. Therefore, 99 divided by 11 equals the “whole” of 9, while individually 09 divided by 11 equals 0.8181… Plus 90 divided by 11 equals 8.1818… And (0.8181… + 8.1818…) = 8.9999… As the fractional equivalent.

    This innate, concentric while antithetic duality, the illusion of parallel worlds, (men/women), (space/time) etc. are actually as, E.A.Milne’s “Kinematic Relativity”, the conjugated mutually ignorant descriptions of a “whole” in the terms of one and another. The systems matrix is the Mobius Band.

    Illustration; The conjugation of numerical direction and magnitude.

    On the outside of a square, label the four alternating perpendicular sides in sequential numeric order; as 0,1, 2 . and 3 At this juncture (clockwise/counter clockwise) are irrelevant.

    Now, on the outside of the square, draw arrows connecting the smaller numbers to their larger adjacent number, 3 with the same directional rotation orientation, while one converse.

    Now, reverse the instructions and outside those arrows, draw another set of arrows connecting the larger numbers back to their smaller adjacent numbers, again 3 with the same directional orientation and one converse.

    Now then, by connecting all arrows with like directional orientation, you have just created a folded Mobius Band. Demonstrating the extended union matrix between the binary, (odd and even) numbered principles 0 and 1.
    ( 0 – 2468 / 1 – 3579 ) and ( 9753 – 1 / 8642 – 0 ).

    Subtract an even (number/numbered) left to right sequence, (02468) from 99999 and the remainder is an odd (number/numbered) right to left sequence, (97531), and vice versa, or concentric duality, homoscedastic coupling.

  34. #34 Jonathan Vos Post
    December 9, 2007

    “Homoscedastic” means a set of statistical distributions having the same variance. What do you mean by the word, ray burchard?

    “coparcener” means one of two or more persons sharing an inheritance; a joint heir. Also called parcener.” In the phrase “where the mathematical conjugation of numerical magnitude and directional numerical flow are commensurate coparcener principles” — What do you mean by the word, ray burchard?

    Words in Mathematics have very specific meaning. It is fun to use them metaphorically, but it is not clear what is being communicated.

    Over on astrophysicist Andrew Jaffe’s blog “Leaves on the Line” 30 November 2007, analysis of a strange pseudo-mathematical statement which I saw Andrew Sullivan utter on Bill Maher’s “Politically Incorrect”:

    Barackology and Obamatics

    Best use of a mathematical term in a political discussion:

    Consider this hypothetical. It’s November 2008. A young Pakistani Muslim is watching television and sees that this man–Barack Hussein Obama–is the new face of America. In one simple image, America’s soft power has been ratcheted up not a notch, but a logarithm.
    –Andrew Sullivan, The Atlantic Monthly, “Goodbye to All That”

    By which he means that our soft power — linked to our moral standing in the world — would increase by some multiplicative factor rather than some additive amount: log(new US) = log(old US) + log(Obama)
    means that (new US) = (old US)×(Obama), more or less…. Which is, of course, technically meaningless but still manages to metaphorically convey how big an effect Obama’s election would have.

    Sullivan is an interesting case, a gay, English, former Reaganite turned (as far as I can tell) Obama supporter, but evidently despite his academic/intellectual credentials, somehow enthralled by political personalities as much as ideas (hence his question, Obama, “The Reagan of the Left?”, which, sorry, I still can’t find it in me to aspire to).

  35. #35 Thony C.
    December 9, 2007

    Once upon a time, what we now call Number Theory was called Higher Arithmetic.

    JVP:

    I am of course well aware of that I am also aware, as I wrote in my first post on the history of group theory, that Euler used what I call group theoretical concepts (GTC) both in number theory and in algebra. I did not however write that Gauss also used GTC in his extension of Euler’s work in number theory. However and this is important GTC is still a long way from group theory proper and the direction that was developed to group theory and was then transferred to geometry was the algebraic one and not the number theory.

    As to the boundaries between algebra, geometry, arithmetic, trigonometry, topology and what you will, I am fully aware that they are totally artificial and very arbitrary a fact that is demonstrated by the subject of this thread group theory. Structure mathematics examines the properties of structures i.e. groups, rings, fields, algebras and what have you, that may and do have numerous models in virtually every branch of mathematics and physics and also in many other sciences. That is both the power and the beauty of structure mathematics.

  36. #36 ray burchard
    December 9, 2007

    Let me start with “coparcener”. Within our Hindu-Arabic number system 0 thru 9 and 1 thru 9 we have ‘Symmetric Duality’ as antithesis in applied mathematics, ‘Whole’ Number versus ‘Fractional’ resolve. One application designed for inanimate engineering and one designed for animate engineering. This is why the paradox, 46 chromosomes while 23.5 pair, and why Fermat’s Last Theorem works for any fractional number up to but not including the ‘Whole’ of 2.

    At this juncture the easiest way to demonstrate this antithesis duality is in geometric progression, One designed to accumulatively traverse infinity, while the other to reductively traverse the same infinity.

    Geometric progressions representation of duality in purpose

    1,2,4,8,16,32,64,128 etc…consolidate each subtotal to a single number representation, 1,2,4,8,(1+6) = 7, (3+2) = 5, (6+4)= 10 = (1+0) = 1, etc… what you will derive is a recurring numerical sequence, 124875 all the way to infinity. Whereas, 124 a left to right sequence and 875 a right to left sequence, then (124+875) = 999 = (9+9+9) = 27 = (2+7) = and again 9 the “whole”.

    Take any binary ( odd and even ) numbered composite of 9, as….18, 27, or 36, etc… then multiply that binary composite number of your choice by any other number of your choosing, whole or fractional, then add together all the individual digits that comprise your answer, disregarding any decimal points, and the results will always be, a systematic return to one of the binary ( odd and even ) numbers comprising 9 and ultimately 9 itself. Example; (27 x 673.05992) = 18172.61784, then (1+8+1+7+2+6+1+7+8+4) = 45 = (4+5) = 9.

    Or take 2221977780, where (2+2+2+1+9+7+7+7+8+0) = 45 = (4+5) = 9, and (0022+2197+7780) = 9999, and 2221977780 divided by 09 = 02468/86420, and 2221977780 divided by 90 = 2468/8642.

    Now take (2+4+6+8+8+6+4+2) = 40 = (4+0) = 4, and (1+3+5+7+9+9+7+5+3+1) = 50 = (5+0) = 5, then again, (4+5) = 9.

  37. #37 Jonathan Vos Post
    December 9, 2007

    #35: Thony C: we agree. I apologize if I inadvertently implied that you didn’t know this very deeply. Once agin, you are right, and express the theory and history well.

    #36: ray burchard: I still don’t understand what you are writing. “One application designed for inanimate engineering and one designed for animate engineering.” I have done Engineering (computer, radar, aircraft, ships, missiles, spacecraft) for numerous corporations and government agencies for decades (inanimate engineering?); and I have a couple of dozen publications in Mathematical Biology (animate engineering?).

    You keep exhibiting arithmetic, base 10, and drawing some conclusions that I can’t follow. As Professor Gregory Benford puts it: “rules of thumb are fine, but some organisms in the universe may have different numbers of thumbs.”

  38. #38 Jonathan Vos Post
    December 9, 2007

    Meanwhile, someone who REALLY knows this stuff, John Baez, says recently on the n-Category Cafe blog:

    “This may seem like needless torture, but face it: everyone thinks symmetry is about group actions. You can get very far with this viewpoint. Most 20th-century mathematicians and physicists were quite content working with group actions! But eventually you run into a wall, and then you need groupoids. So, that’s what I did in the fall quarter of this course: start by studying some famous group actions and group representations, and try to develop the theory of Hecke operators as much as possible from this perspective. It went pretty well, but eventually I hit a wall, and then I switched over to groupoids.”

  39. #39 ray burchard
    December 10, 2007

    #37 Jonathan Vos Post , It’s not my intent to impugn your accomplished knowledge or the level of competence you bring to your endeavors. But while your perplexity is understandable, your tautology does not lend an understanding of the same mathematical paradox that vexed Einstein. Example; As Brian Greene states; paraphrasing “Quantum geometry ( E.A. Milne’s ‘Kinematic Relativity’ ) differs in substantial ways from the classical geometry underlying ( Einstein’s ) general relativity. For instance, topology change ( the tearing of space ) is a sensible feature of Quantum geometry even though, from a classical perspective, it involves singularities.

    As another example, two different classical spacetime geometries can give rise to identical physical implications, at odds with conclusions based on classical general relativity. as differing vantage point generated descriptions of the same entity or mutually ignorant descriptions of the “whole” in terms of one and another, men/women, space/time, etc…

    Let me demonstrate the dichotomy using the string theory principles 0 and 1, numerically. Where the 0 represents no value and the 1 represents one value. Conjoined, 01 they represent one unit value. Now with rotation (mirrored image) the principles switch positions 1 and 0 conjoined, 10 they represent ten unit values. A one to ten, ten to one ratio and the bases for Newton’s ” A force acted in one direction does not have the same effect” etc…

    Now lets view the same string theory principles from another perspective;
    View both the 0 and the 1 as circumference with a (surface / side) relationship and from this perspective the 0 represents 2 dimensions (height and width) while the 1 represents 1 dimension (depth) and conjoined the represent a specific perspective of a 3-dimensional sphere. Now with rotation (mirrored image/the tearing of space) the 0 becomes the 1 and the 1 becomes the 0, conjoined 10 as a inverse perspective of the same 01. Then collectively (effectively) how many dimensions does this concentric, symmetric duality, homoscedastic (quadratic concentric duality coupling, with rotation and translation), represent?

    Then as to; Professor Gregory Benford puts it: “rules of thumb are fine, but some organisms in the universe may have different numbers of thumbs.”, I don’t mean to be a smart ass Jonathan, but it would appear the good professor has rationalized the limits of his knowledge.

    In closing, take the principles of the “whole” of 495, as 135 and 360, then take the 1 from its position and add it to the 5 position, equaling 036, then take the 5 from its position and add it to the 1 position, equaling 630.
    Now do the same with the 360 and what you get is 063 and 360.
    The crisscrossing of inverse sequences, 036, 630 and 063, 360 equal (099 +990) = 1089 is the Mobius Band. Try 495 equals (099 + 990). Try the principles (225 + 270) = 495, or (315 + 180) = 495

    Quadratic, dual Concentric Symmetries of the double helix DNA

    Then also seen alphabetically where the (double helix quadratic concentric symmetries) DNA equal, A,,de(nine) bonds to T..hymine as a left to right union, conjugated with G..ua(nine) bonding to C..ytosine a right to left union. A,T,G,C and conjugated with it’s inverse C,G,T,A is also a split and split again Mobius Band….Kurt Godel’s recursive and primary recursive number theory.

    Jonathan, I hope this adds some clarity to your understanding.

  40. #40 ray burchard
    December 10, 2007

    #39 Jonathan Vos Post, As a footnote;
    The numerical sequence 1089 is 33 squared, while it’s inverse sequence 9801 is 99 squared.
    If you take every composite number from 00 thru 99 you will have sequenced duality, as 01 and 10, 13 and 31, 26 and 62 etc… now if you add all the numbers from 00 thru 99 you’ll come up with 4,950

    I have developed much, much more illustrated arithmetic/mathematics supporting my assertions but because of my non affiliation with Academia, the only acknowledgement of my work comes in the last sentence of Edward Witten’s field of expertise where he states “In recent years, by means of mysterious new “duality” symmetries, physicists have obtained a much more far-reaching understanding of string theory that has many implications”. The path of this “mysterious new duality symmetries and it’s enlightenment came to Dr. Witten via his good friend and colleague Dr. Hawking via certified mail Ray Burchard.

  41. #41 Jonathan Vos Post
    December 10, 2007

    I have online publications about digital reversals and about numerical properties of the sequence of composite numbers. For the former, for instance, see:

    http://www.research.att.com/~njas/sequences/?q=jonathan+vos+post+reverse&sort=0&fmt=0&language=english&go=Search

    Also, in the search engine of the Online Encyclopedia of Integer Sequences, type my full name and the word “reversal” or the word “composite.”

    However, I don’t see what light that sheds on the dualities in Physics.

    I have spoken with both Dr. Hawking and Dr. Witten. I try to study their work. In boith cases, that takes a much more sophisticated level of mathematics than I see you demonstrating. General Relativity and Conformal Field Theory require quite a lot beyond Calculus — such as Differential Geometry, Tensors, and the math that won Dr. Witten a Field Medal.

    In a broad sense, experiencing the glory of mathematics, in your own way, and the glory of the physical universe, in your own way, have something psychological and perhaps spiritual in common.

  42. #42 ray burchard
    December 10, 2007

    #41 Jonathan Vos Post, Thank you for providing your links, I fully intend to review them, time permitting.
    Again Jonathan, it is not my intent to be condescending but the facts are;
    One, when you fractionalize the “whole” of duality with singularity the route to define the “whole” then becomes anfractuous and convoluted, it’s like using only one of two principles longitude or latitude and that looses the metaphysical harmony (symmetry) of Kant’s modality, where the “whole” has a greater value then the sum of it’s parts.
    Two, when complexity is your marketable, sophistication is a necessary conclusion.

    Using the math that won Dr. Witten a Field Medal, can he or you demonstrate the systems matrix illustrating the meiosis/mitosis process in cell cleaving? I can.

    A Mobius band continuum creates a figure eight pattern, when cleaved it becomes a larger band with four twist and when cleaved again it produces two interlocked Mobius bands. Having said that, what is the pattern developed, (duality emitting simultaneously from the center out) by the Sun’s declination, analemma and what is the pattern made in the earth’s one year rotation around the sun, with it’s four seasonal twist and what is interlocked looping patterns, each with four twists, made by the quarks in a hydrogen atom.

    What is necessary to create memory, iteration. Then what is the pattern made by magnetic fields. Then when a group as 56789 is viewed in conjunction with it’s inverse sequence 98765, is it still a group or is it a set?

    As I’ve stated I have developed conclusive mathematics supporting my assertions, some on j-peg, ray.burchard@sbcglobal.net

  43. #43 Doug Spoonwood
    December 11, 2007

    Thony C,

    I didn’t botch the posting of that link. I MEANT to put it on two lines, so the formatting would work regularly, even if you had to copy both lines separately to get there.

    [Which is as far as I can see a more detailed version of my post showing that group theory developed out of attempts to find a general solution to the solvability of polynomial algebraic equations.]

    To a significant extent, sure. But, solely ? No.

    Seriously… you quoted only part of the article… the part which comes as part (3). I really don’t see why in this case since you had such a long quotation, unless you wanted to quote-mine it. The MacTutor article, *in its entirety* reads
    “The study of the development of a concept such as that of a group has certain difficulties. It would be wrong to say that since the non-zero rationals form a group under multiplication then the origin of the group concept must go back to the beginnings of mathematics. Rather we must take the view that group theory is the abstraction of ideas that were common to a number of major areas which were being studied essentially simultaneously.

    The three main areas that were to give rise to group theory are:-

    1. geometry at the beginning of the 19th Century,
    2. number theory at the end of the 18th Century,
    3. the theory of algebraic equations at the end of the 18th Century leading to the study of permutations.

    (1) Geometry has been studied for a very long time so it is reasonable to ask what happened to geometry at the beginning of the 19th Century that was to contribute to the rise of the group concept. Geometry had began to lose its ‘metric’ character with projective and non-euclidean geometries being studied. Also the movement to study geometry in n dimensions led to an abstraction in geometry itself. The difference between metric and incidence geometry comes from the work of Monge, his student Carnot and perhaps most importantly the work of Poncelet. Non-euclidean geometry was studied by Lambert, Gauss, Lobachevsky and János Bolyai among others.

    Möbius in 1827, although he was completely unaware of the group concept, began to classify geometries using the fact that a particular geometry studies properties invariant under a particular group. Steiner in 1832 studied notions of synthetic geometry which were to eventually become part of the study of transformation groups.

    (2) In 1761 Euler studied modular arithmetic. In particular he examined the remainders of powers of a number modulo n. Although Euler’s work is, of course, not stated in group theoretic terms he does provide an example of the decomposition of an abelian group into cosets of a subgroup. He also proves a special case of the order of a subgroup being a divisor of the order of the group.

    Gauss in 1801 was to take Euler’s work much further and gives a considerable amount of work on modular arithmetic which amounts to a fair amount of theory of abelian groups. He examines orders of elements and proves (although not in this notation) that there is a subgroup for every number dividing the order of a cyclic group. Gauss also examined other abelian groups. He looked at binary quadratic forms

    ax2 + 2bxy + cy2 where a, b, c are integers.

    Gauss examined the behaviour of forms under transformations and substitutions. He partitions forms into classes and then defines a composition on the classes. Gauss proves that the order of composition of three forms is immaterial so, in modern language, the associative law holds. In fact Gauss has a finite abelian group and later (in 1869) Schering, who edited Gauss’s works, found a basis for this abelian group.

    (3) Permutations were first studied by Lagrange in his 1770 paper on the theory of algebraic equations. Lagrange’s main object was to find out why cubic and quartic equations could be solved algebraically. In studying the cubic, for example, Lagrange assumes the roots of a given cubic equation are x’, x” and x”’. Then, taking 1, w, w2 as the cube roots of unity, he examines the expression

    R = x’ + wx” + w2x”’

    and notes that it takes just two different values under the six permutations of the roots x’, x”, x”’. Although the beginnings of permutation group theory can be seen in this work, Lagrange never composes his permutations so in some sense never discusses groups at all.

    The first person to claim that equations of degree 5 could not be solved algebraically was Ruffini. In 1799 he published a work whose purpose was to demonstrate the insolubility of the general quintic equation. Ruffini’s work is based on that of Lagrange but Ruffini introduces groups of permutations. These he calls permutazione and explicitly uses the closure property (the associative law always holds for permutations). Ruffini divides his permutazione into types, namely permutazione semplice which are cyclic groups in modern notation, and permutazione composta which are non-cyclic groups. The permutazione composta Ruffini divides into three types which in today’s notation are intransitive groups, transitive imprimitive groups and transitive primitive groups.

    Ruffini’s proof of the insolubility of the quintic has some gaps and, disappointed with the lack of reaction to his paper Ruffini published further proofs. In a paper of 1802 he shows that the group of permutations associated with an irreducible equation is transitive taking his understanding well beyond that of Lagrange.

    Cauchy played a major role in developing the theory of permutations. His first paper on the subject was in 1815 but at this stage Cauchy is motivated by permutations of roots of equations. However, in 1844, Cauchy published a major work which sets up the theory of permutations as a subject in its own right. He introduces the notation of powers, positive and negative, of permutations (with the power 0 giving the identity permutation), defines the order of a permutation, introduces cycle notation and used the term système des substitutions conjuguées for a group. Cauchy calls two permutations similar if they have the same cycle structure and proves that this is the same as the permutations being conjugate.

    Abel, in 1824, gave the first accepted proof of the insolubility of the quintic, and he used the existing ideas on permutations of roots but little new in the development of group theory.

    Galois in 1831 was the first to really understand that the algebraic solution of an equation was related to the structure of a group le groupe of permutations related to the equation. By 1832 Galois had discovered that special subgroups (now called normal subgroups) are fundamental. He calls the decomposition of a group into cosets of a subgroup a proper decomposition if the right and left coset decompositions coincide. Galois then shows that the non-abelian simple group of smallest order has order 60.

    Galois’ work was not known until Liouville published Galois’ papers in 1846. Liouville saw clearly the connection between Cauchy’s theory of permutations and Galois’ work. However Liouville failed to grasp that the importance of Galois’ work lay in the group concept.

    Betti began in 1851 publishing work relating permutation theory and the theory of equations. In fact Betti was the first to prove that Galois’ group associated with an equation was in fact a group of permutations in the modern sense. Serret published an important work discussing Galois’ work, still without seeing the significance of the group concept.

    Jordan, however, in papers of 1865, 1869 and 1870 shows that he realises the significance of groups of permutations. He defines isomorphism of permutation groups and proves the Jordan-Hölder theorem for permutation groups. Hölder was to prove it in the context of abstract groups in 1889.

    Klein proposed the Erlangen Program in 1872 which was the group theoretic classification of geometry. Groups were certainly becoming centre stage in mathematics.

    Perhaps the most remarkable development had come even before Betti’s work. It was due to the English mathematician Cayley. As early as 1849 Cayley published a paper linking his ideas on permutations with Cauchy’s. In 1854 Cayley wrote two papers which are remarkable for the insight they have of abstract groups. At that time the only known groups were groups of permutations and even this was a radically new area, yet Cayley defines an abstract group and gives a table to display the group multiplication. He gives the ‘Cayley tables’ of some special permutation groups but, much more significantly for the introduction of the abstract group concept, he realised that matrices and quaternions were groups.

    Cayley’s papers of 1854 were so far ahead of their time that they had little impact. However when Cayley returned to the topic in 1878 with four papers on groups, one of them called The theory of groups, the time was right for the abstract group concept to move towards the centre of mathematical investigation. Cayley proved, among many other results, that every finite group can be represented as a group of permutations. Cayley’s work prompted Hölder, in 1893, to investigate groups of order

    p3, pq2, pqr and p4.

    Frobenius and Netto (a student of Kronecker) carried the theory of groups forward. As far as the abstract concept is concerned, the next major contributor was von Dyck. von Dyck, who had obtained his doctorate under Klein’s supervision then became Klein’s assistant. Von Dyck, with fundamental papers in 1882 and 1883, constructed free groups and the definition of abstract groups in terms of generators and relations.

    Group theory really came of age with the book by Burnside Theory of groups of finite order published in 1897. The two volume algebra book by Heinrich Weber (a student of Dedekind) Lehrbuch der Algebra published in 1895 and 1896 became a standard text. These books influenced the next generation of mathematicians to bring group theory into perhaps the most major theory of 20th Century mathematics.”

    http://www-groups.dcs.st-and.ac.uk/~history/
    HistTopics/Development_group_theory.html
    (do not click… copy the first line, then copy the second line to the right of it, and there lies the link)

    Notice that part 1 talks about geometry. By all means dispute it and tell me how you know more about mathematical history, how your view of history works as better than MacTutor, how they dogmatically won’t correct their “mistakes”, etc. It still remains the MacTutor article talks about geometry playing a development in group theory.

  44. #44 ray burchard
    December 12, 2007

    #43 Doug Spoonwood, your knowledge of the linage in mathematical development is impressive indeed. But what your missing is the same thing that precluded Albert Einstein from realizing his life long goal to unify an overall concept of the universe, the many paradox that rule mathematics.

    It is therefore my contention that a provable overview would lend clarity to this unification. What we have here is ‘Duality’ where the principles themselves create an intended dichotomy, as an antithetical relationship, a contiguous while commensurate coparcener. Examples; while the same Hindu-Arabic number system with one function, 1 thru 9 (base 9) then conjoined with another function 0 thru 9 (base 10). And as I’ve previously demonstrated, a dichotomy in geometric progression, where one faction accumulatively traverses infinity, while simultaneously another faction creating it’s own (in effect) reductive finite infinity to then traverse the same infinity, a reality of (dynamic reality/static reality) as in the “whole” of (men/women), (space/time), (quantum/classical), (complex/simplicity), (fractionalized numbers/ whole numbers) etc…
    etc… etc… we can see this played out in inanimate engineering where separation is division along lines of equality while in animated (bio) engineering where separation is a cleaving along lines of weakness.

    We can see this dichotomy in Mathematics where when mathematics is used as the foundation for logical resolve we have;

    Two Systems of Mathematics

    What we have is a structured number system and associated applications of mathematics that mimic the triangular matrix and applicable characteristics of its creator, the human brain. That is to say, a dual-purpose number system with two conjoined applications of mathematics, a commensurate and conterminous coparcener relationship.

    One application designed as per the ‘right side brain’, representing The Greek Geometric based application of Mathematics, and designed to demonstrate the unified symmetrical harmony, (balance) of animation. Thereby, producing Axioms expressing conceptual inclusions (duality) as the whole; (equality), (binary, odd and even numbers), (zero- purpose and function), (whole number resolve) and the (binary balance of symmetry), etc… This geometric patterning can also be seen conveyed in the Greek’s consensus system of beliefs, where the force of necessity as created by circumstance, was believed to fall within the purview of their Gods, as a God for all occasions, but with a geometric orientation, earth and heaven, land and sea. The mathematics of binary inclusion.

    Then heuristic evolution brought the world a paradigm shift in the way of mathematic development, the mathematics of exclusion.
    A restructured application of mathematics, as per the “left” side of the brain’ and designed specifically to facilitate commerce. The Hebrew dominated Mathematics, a merchant mentality based application of mathematics, and therein, the necessity to inculcate the Greek mathematical principles of inclusion, (symmetrical geometric balance, equality and duality), in the forms of binary odd/even numbers, zero-function and purpose, and whole number resolve etc… All as being superfluous facets of mathematics if they could not demonstrate an applicable allegiance to commerce and/or inanimate engineering.

    This (turning a blind eye to the value of the “whole) coalescing and/or specializing separatist mind set then produced many merchant biased, economic doctrines likened by; The Law of Supply and Demand, Law of Diminished Return, Law of Accelerating Returns, Law of Increasing Cost and Law of Increasing Opportunity Cost, etc….all doctrines of avarice cloaked with the connotation of official, meant to lend, divide and conquer, creditability in separating the unity of the whole. By doing so effectively removing the harmonious modality seen as societies moral compass, and thereby sending its subscribers into an anfractuous moral abysmal void where “Any plan conceived in moderation must fail when the circumstances are set in the extremes”.

    Now no one can deny the value of avarice, as inspiration to progress, while likewise no one can deny that avarice’s obsession to greed, is society’s cancerous regression. Just look at America today where its corporate American style of capitalism in becoming the world’s poster child for greed.

    A rapacious culture whose reputation is built on a system of applied mathematics designed specifically to facilitate commerce as its foundation for a conventionally accepted logical resolve.
    Example, corporate America’s pursuit of profit, where the only control and oversight is a bought and paid for, authorized malleable code of ‘Ethics’.

    Then also exampled by the paradox in America’s governance that spends billions and billions of America’s dollars on Academic redundancy, and rightfully so, to protect the lives and safety of 7 astronauts from the inevitability of human error. While simultaneously, the same governance sells corporate America the authority to spend a billion dollars to design and build an “Airbus” to then pit 840 passenger’s lives and safety against the same inevitability of human error, a logic justified by corporate profit.

  45. #45 Doug Spoonwood
    December 12, 2007

    Ray,

    I find your writing highly unorthodox, highly speculative, provocative, borderline mystical, and overall interesting. I’d read your site if you put this on your own site and maybe comment more there. But, you really seem VERY far off-topic, as we didn’t talk about math history in general and its relation to soceity or anything so grand… just about the history of group theory.

  46. #46 ray burchard
    December 13, 2007

    #45 Doug Spoonwood,

    While I appreciate your comments and suggestions, your finding my writing and subsequent assertions highly unorthodox, highly speculative, and borderline mystical doesn’t surprise me as I’ve been given this assessment before, as a matter of fact Naval Intelligence also added, “a contumacious intellect”.

    Having said that, the reason you may feel my assertions “really seem VERY far off-topic” is as I previously stated, like Einstein your mathematical education establishes a foundation for logical resolve that then creates a predilection or mindset as to how you view life as your reality, E.A. Milne’s “Kinematic Relativity”. Why do you think that sometimes answers to very vexing questions come in sleep, because the subconscious mind doesn’t require the logical aspect because it doesn’t have to evaluate incoming senses stimuli. Therefore it allows connecting to be made that is precluded by the conscious minds logic.

    Then also, you view dynamic sets and groups, and static sets and groups as one in the same. They are not they are conjoined but separate, as [(equal but opposite) conjugated with (same but unequal)] = the Mobius Band continuum. That’s what puts them on the same page; I have developed mathematical proof of this assertion.

    Here is one last example: Einstein stated one stumbling point in his attempt to develop a unified concept of the universe was his inability to see time as a factor in causation. That was because he, again like you was taught to view it from “General Relativity” and not “Kinematic Relativity”. Kinematic relativity meaning differing vantage point generated descriptions (mutually ignorant) of the same entity, or antithetical duality as the “whole” and again, separate while conjoined.
    Take duplicating, as (imitation versus emulation) as itself the “whole”, where imitation is instantaneous mimicry and requires no passage of time, and the impetus is therefore declining, a copy of a copy of a copy. While emulation is to equal or surpass, which requires the passage of time and therefore the impetus is inclining. That’s why cloning by passes evolution (time) and therefore the maladies associated with aging come sooner than later.

  47. #47 ray burchard
    December 13, 2007

    Doug Spoonwood,

    In my tour of Naval duty I was pursued by Naval Intelligence not only because they evaluated my intelligence as contumacious, but the evaluation also continued by stating this characteristic also lent itself to a unique ability to decipher patterns even though obscured by natural and/or intentional obfuscation. Let me demonstrate this ability with something your acquired knowledge stands testimony to.

    Take your posted comments on this article, more specifically your screed # 43, it telegraphs a scribe’s mentality and that of what ails Academia today. The polished ability to read, write and recite (parrot) but void any real conceptual cognitive ability and thereby disguising itself as intelligence. This does not represent the intellectual advancement of knowledge, it represents a proficiency in training and the commercialization of Academia.

    Then when your comment elicited my comment #44, you couldn’t even respond out of common courtesy. One couldn’t equate your ignorance to a lack of the courage of one’s conviction, because there is no conviction just an attempt to gain one’s esteem off the past and present prestige of others.

    But you did respond with hostility to impugn the grammatical correctness of my posted comments on another article where I challenged the existence of God.

    Hows my psychological profile doing Doug, or would you rather slink back into your hole Doug.

  48. #48 Xanthir, FCD
    December 13, 2007

    ray burchard: Your writings are rambling nonsense, devoid of rational content and deliberately obfuscated to make them impossible to parse properly. Doug was perfectly correct in calling them ‘mystical’, as they have little or no basis in reality. Take for example this section of your writing:

    Take duplicating, as (imitation versus emulation) as itself the “whole”, where imitation is instantaneous mimicry and requires no passage of time, and the impetus is therefore declining, a copy of a copy of a copy. While emulation is to equal or surpass, which requires the passage of time and therefore the impetus is inclining. That’s why cloning by passes evolution (time) and therefore the maladies associated with aging come sooner than later.

    This is complete nonsense with no relation to the facts of biology and cloning.

    As an aside, amateur psych profiling of your opponent is a wasteful and immature activity. It gives no information, and only serves as an ad hominem attack, thereby lessening your own credibility in return.

  49. #49 Doug Spoonwood
    December 13, 2007

    [While I appreciate your comments and suggestions, your finding my writing and subsequent assertions highly unorthodox, highly speculative, and borderline mystical doesn't surprise me as I've been given this assessment before, as a matter of fact Naval Intelligence also added, "a contumacious intellect".]

    Ummm… I didn’t think you insubordinate or disobedient.

    [Having said that, the reason you may feel my assertions "really seem VERY far off-topic" is as I previously stated, like Einstein your mathematical education establishes a foundation for logical resolve that then creates a predilection or mindset as to how you view life as your reality, E.A. Milne's "Kinematic Relativity".]

    Interesting that you claim such, especially since that plenty of people around here have seen me talk about non-conventional forms of logic, such as fuzzy set theory or fuzzy logic. You may believe that I have a predilection for a sort of “over-logicism”, but I think you’d have a hard time finding evidence for that or finding others that agree with you.

    [Why do you think that sometimes answers to very vexing questions come in sleep, because the subconscious mind doesn't require the logical aspect because it doesn't have to evaluate incoming senses stimuli.]

    With that I’ll add
    “Why do you think that sometimes nonsense to very vexing questions come in sleep, because the subconscious mind doesn’t require the logical aspect because it doesn’t have to evaluate incoming senses stimuli.”

    Sure, answers to vexing questions can come in sleep… or so I’ve gotten told… I can’t recall an experience of such, even thought I’ve read of trying to use such a technique and I think it might have *some* merit. This doesn’t mean that sleep doesn’t generate nonsense though or at minimum falls prey to a lot of nonsensical interpretations of sleep.

    [Take your posted comments on this article, more specifically your screed # 43, it telegraphs a scribe's mentality and that of what ails Academia today.]

    It consists of largely a quotation. Do you somehow really think I somehow managed to encode a secret message in it?

    [Then when your comment elicited my comment #44, you couldn't even respond out of common courtesy.]

    I could and did respond out of common courtesy. I suggested your own site, said I would read it (or I did that elsewhere), and said I thought your ideas interesting. I can’t say I understand a large portion of what you write though, or why you write it.

    [But you did respond with hostility to impugn the grammatical correctness of my posted comments on another article where I challenged the existence of God.]

    Not grammar. Clarity of writing! In this little essay I can follow you well enough though.

    [Hows my psychological profile doing Doug, or would you rather slink back into your hole Doug.]

    Not that great. Did you read my site or find it? If you really want to attempt a psychological profile of someone you might want to try to find as much *public* information about them as you can first.

    Xanthir,

    I didn’t get that passage of Ray’s either. I don’t think Ray understands his use of terms in general either. Still, I would recommend that you not go so far to say that he *deliberately* (meaning with intent) obfuscates his terminology. We know very little about this individual. I do agree with you about the value of amateur psychological profiling… esepcially from just a few sentences here and there.

  50. #50 ray burchard
    December 14, 2007

    # 48 Xanthir, FCD , as to, “Your writings are rambling nonsense”…. This is always the claim when innovative thinking doesn’t follow the conventional wisdom and tautology, it’s called heuristic evolution, where paradigm shifts are not always smooth transitions.

    “perfectly correct in calling them ‘mystical’”…. in specialization, ‘separatist thinking’, separating the physical from the metaphysical, as again itself a (antithetical duality), is a necessary temporal aspect of job niching and empirical study, but how is this in fact accomplished then in the actual world?

    As to, (imitation versus emulation) this example of antithetical duality was to demonstrate how the added value, (time as a correlating aspect of the “whole” ), could be ascertained over that of fractionalizing the “whole”.

    “amateur psych profiling”…. Amateur or not I stand behind my assessment, it is what I see.

    “ad hominem attack”…. I couldn’t agree with you more about ad hominem attacks, but as a way of life I’ll only accept just so-many such attacks before I reciprocate in kind and I gave Doug Spoonwood that latitude.

  51. #51 ray burchard
    December 14, 2007

    #49 Doug Spoonwood,

    “Ummm… I didn’t think you insubordinate or disobedient” …. That’s not what they said, they said “contumacious intellect” as insubordinate to the conventional wisdom.

    “Why do you think that sometimes nonsense to very vexing questions come in sleep:”….
    My point was not that we should mining chaos, while that’s an interesting concept though, it was to demonstrate how a system of unwavering belief can be detrimental to natural “heuristic evolution”. As per Kant, “I had to remove knowledge to make room for belief”.
    As a side note; what happens to a society when they build their culture around a system of mathematics redesigned specifically to facilitate commerce as the foundation for that culture’s logic?…. A society like America where the guiding premise is a ‘pathetic fallacy’ and everything is then equated to the dollar. Or do you believe we’re in Iraq for anything but establishing market and aftermarket dominance and manipulation for corporate America’s economic prosperity?

    “Do you somehow really think I somehow managed to encode a secret message in it”?….
    Do you really think one’s actions in conjunction with one’s written word, (“I could and did respond out of common courtesy” ) in what order?….aren’t then, characteristic of and telegraph that individuals psychological mindset?

    9/11, the FBI, CIA, NAS, CID, etc… etc… etc… fractionalizing the “whole” of the Intelligence community with no correlating authority (alligence to civic responsibility).

    Academia fractionalized by field of expertise, with no correlating authority. Call it what you like, but hanging your hat on Academia developing a recondite nomenclature by field of expertise is tantamount to satiating job protection and egotism. Greed.

    America’s governance taking the power from it’s population and distributing it by giving it to it’s bureaucracies and/or selling it to corporate America, in both cases with no allegiance to civic responsibility is tantamount to the advancement of solipsism’s doctrines of greed.

    Long before the “Dirty Bombers Attempt” I send certified letters to members of Congress, expressing caution and vigilance that because of cost and low tech availability such an event was likely. In my scenario it was a low altitude, decorative ( to draw attention )hot air balloon, (detonated over Israel) with a drum of nuclear waste and concussion explosives. The psychological implications would be devastating to Israel. Followed up by nuclear waste in the head of Iraq’s scud missiles, then dispersed by America’s Patriot missiles as complicity.

    At the same time and in the same letter, in reference to Mrs./Ms Rameriz, riverside Ca. you may remember as the women who died because of her sever illness while the emergency room attendants developed nerve disorders. The autopsy showed the women had DMSO in her blood and had to be conducted by attending personal in bubble suits. DMSO is found in bug spray and DMSO’s conversion to DMSO4 (a very potent military never gas) is accomplished by cooling and super oxygenation. Now when the emergency room personal attempted cardiopulmonary resuscitation through high voltage no amp’s they used the body as a lab vessel and neutralized, oxygenated and dispersed DMSO4 into the ER themselves. Now when I conveyed this theory to close friends in the medical community they stated, Ray your very brilliant but in this case you don’t know what your talking about. 6 months later in a News Release Lawrence Liverpool Lab stated flying in the face of the scientific community, etc… etc…

    Now I’m not egotistical enough to believe I was instrumental in the outcome, just that I can recognize the unconventional patterns. Therefore I ask you to stop making issue in respect to nomenclature and review the mathematical aspect of what I’ve posted on this thread.

  52. #52 Xanthir, FCD
    December 14, 2007

    “Your writings are rambling nonsense”…. This is always the claim when innovative thinking doesn’t follow the conventional wisdom and tautology, it’s called heuristic evolution, where paradigm shifts are not always smooth transitions.

    It is true that truly innovative thinking can resemble nonsense. This is usually because the writer isn’t doing a good enough job to actually explain the position, though.

    Most of the time, though, writings appear to be rambling nonsense because they are rambling nonsense. The rare geniuses are just that – rare. The ramblers who don’t actually understand what they’re talking about, though, are a dime a dozen.

    As the old saying goes, “They laughed at Einstein, but they also laughed at Bozo the Clown.”

    “perfectly correct in calling them ‘mystical’”…. in specialization, ‘separatist thinking’, separating the physical from the metaphysical, as again itself a (antithetical duality), is a necessary temporal aspect of job niching and empirical study, but how is this in fact accomplished then in the actual world?

    As to, (imitation versus emulation) this example of antithetical duality was to demonstrate how the added value, (time as a correlating aspect of the “whole” ), could be ascertained over that of fractionalizing the “whole”.

    Um, this is rambling nonsense again. Nothing you have said makes any sense whatsoever, in any context that I can understand.

    “ad hominem attack”…. I couldn’t agree with you more about ad hominem attacks, but as a way of life I’ll only accept just so-many such attacks before I reciprocate in kind and I gave Doug Spoonwood that latitude.

    You don’t understand what an ad hominem attack is. It’s not an insult, it’s an argumentative fallacy. It’s tying personal qualities of your opponent to the veracity of their arguments.

    Insulting someone is fine. It has nothing to do with the argument itself; it’s just flavor sprinkled on top. But trying to dismiss someone’s argument because of perceived psychological problems you’ve detected in a handful of forum posts is an ad hominem.

  53. #53 Doug Spoonwood
    December 14, 2007

    [That's not what they said, they said "contumacious intellect" as insubordinate to the conventional wisdom.]

    *shakes head*… Contumactious intellect as in insubordinate to *navy* wisdom.

    [Therefore I ask you to stop making issue in respect to nomenclature and review the mathematical aspect of what I've posted on this thread.]

    What mathematical aspect??? This question “what happens to a society when they build their culture around a system of mathematics redesigned specifically to facilitate commerce as the foundation for that culture’s logic?” primarily concerns soceity and thus really falls within sociology, not math.

    [Or do you believe we're in Iraq for anything but establishing market and aftermarket dominance and manipulation for corporate America's economic prosperity?]

    A political question.

    [Do you really think one's actions in conjunction with one's written word, ("I could and did respond out of common courtesy" ) in what order?....aren't then, characteristic of and telegraph that individuals psychological mindset?]

    I think you mean to assert that words indicate a psychological reality of people. In some cases yet, in some cases no. Tone and *how* people say things indicates more about psychological states than the symbols and forms involved.

    Lastly, I don’t really accept the sort of idea that says there exists “Hebrew dominated Mathematics” and “Greek mathematical principles”. This comes far too close to saying there exists “Christian biology” and “Hindu chemistry.”

  54. #54 Thony C.
    December 15, 2007

    Lastly, I don’t really accept the sort of idea that says there exists “Hebrew dominated Mathematics” and “Greek mathematical principles”. This comes far too close to saying there exists “Christian biology” and “Hindu chemistry.”

    Ignoring for the moment the fact that ray’s definition of “Hebrew dominated Mathematics” smacks of anti-Semitism there is at least in this one phrase some truth contained within the rest of the nonsensical babble that he perpetrates. The concept of a universal mathematics with universal agreement as to the areas of study and the objects that are considered mathematical and the methods and standards of proof that are acceptable is largely a product of European mathematics beginning in the 18th century and evolving predominately in the 19th century. Historically the areas of study, the objects and the methods of proof were very much culturally determined and different cultures definitely produced different ‘types’ of mathematics. Looking back through history with hindsight we are very guilty of the historiographical crime of presentism, that is we pick out those aspects of a cultures mathematical endeavours that in some form are still present in our own understanding of mathematics and ignore all of the uncomfortable aspects of that cultures own view or understanding of mathematics that are no longer in tune with our own. If you actually examine the every day aspects of Babylonian and Greek mathematics you will discover that they are all together two very different beasts and so to talk of “Babylonian Mathematics” and “Greek Mathematics” as separate cultural entities is in fact perfectly correct. Even within more modern times and closer to home (at least for us Western Europeans, in which I include, for arguments sake, the North Americans!) something as simple as the function concept used in the 18th century totally different to that used in the 19th century so that anybody doing historical research must be sure that he knows what is understood by those using a concept within a historical context before commenting on it.

    The same of course applies to the historical development of the other sciences.

  55. #55 ray burchard
    December 15, 2007

    #52 Xanthir, FCD,

    “It is true that truly innovative thinking can resemble nonsense. This is usually because the writer isn’t doing a good enough job to actually explain the position” ….

    This is only half of the equation in responsibility (in an exchange of knowledge), it is also incumbent the other party to be receptive and temporally forego their predilections, bias. It must be remembered that the test in testability must therefore also evolve as with the innovation. Do you think the American Indian had the intellectual where with all (material knowledge for reference) to then evaluate the theory of the internal combustion engine. (“Nothing you have said makes any sense whatsoever, in any context that I can understand.”)…. and my point exactly. Does this then mean that the facts aren’t there or your willingness to assemble the facts isn’t there?

    “You don’t understand what an ad hominem attack is” ….

    Rhetorical but you are right, and that would be the semantics (analyzing it’s logic) separating or fractionalizing the “whole” of an affront and to me its superfluous, an ad hominem attack, is an attack and therefore a breach of civility of which I partook for refute.

  56. #56 ray burchard
    December 15, 2007

    #53 Doug Spoonwood,

    “*shakes head*… insubordinate to *navy* wisdom” ….
    Correct, but they pursued me long after my departure, not the other way around.

    “What mathematical aspect???”….
    For one; The fact there exists a conjoined “antithetical duality” (accumulative and/or reductionism) in the form of applied mathematics in geometric progression.
    Two; (00+01+02 thru 09) = 045 = (0+4+5) = 9 while It’s inverse sequence, (90+80+70 back thru 00) = 0450 = (0+4+5+0) = 9 and (9+9) = 18 = (1+8) = 9….. The conjugation of direction and magnitude lends to numerical (integer) dimension. Therefore while (9+9+9+9+9) = 45 = (4+5) = 9, 99999 minus an odd (number/numbered) left to right sequence 13579, equals it’s quadratic antithetic inverse sequence, as a even (numbered/number) right to left sequence 86420. This also demonstrates the union of 1 thru 9 with 0 thru 9, where the results are, (binary/bifurcation) a ten digit sequence with a value of 9 as it’s magnitude (gold standard).

    Hypothetically, following this premise would mean that every calculation simultaneously generates a quadratic antithetical duality, (homoscedastic coupling with rotation and translation), example; (2+7) = 9 therefore
    (2+7) = 9
    (7+2) = 9
    Then (9+9) = 18 = (1+8) = 9 the “whole” (dimensionally) both vertical and horizontally. As a palindrome (27+72) = (20+07+70+02) = 99 and (72+27) = (70+02+20+07) = 99 therefore, (99+99) = 198 = (1+9+8) = 18 = (1+8) = 9 and then (2772+7227) = 9999 and (9+9+9+9) = 36 = (3+6) = 9
    [(2070+0702) + ( 7020+0207)] = (9009 + 0990) = 9999 etc…

    I know this maybe confusing, and the complexity of the systematic sequencing only grows rapidly when you add the negative function as (antithetical duality), but after 20 years of pursuit I have developed many, much more conclusive illustrations.

    “thus really falls within sociology, not math”….”A political question”….
    Again the propensity to trivialize and fractionalize the “whole” (antithetical duality) of the human experience as in separating (phenomenon/noumenon).
    While….”Tone and *how* people say things indicates more about psychological states than the symbols and forms involved”….
    Now you want to incorporate the elements of the “whole”. form over substance.

    “Mathematics” and “Greek mathematical principles”. This comes far too close to saying there exists “Christian biology” and “Hindu chemistry.” ….
    So this means because you don’t like evolutions revelations, your not going to join the rest of the worlds move into the next century, but like the Muslim reality, pitch your tent in antiquity. Come on Doug your smarter than that. The “whole” of “Hindu chemistry” and “Christian biology” mimic the matrix of it’s origination, the human brain based on the (dynamic/static), (inclusion/exclusion) as in it is paramount to establish a spatial, geometric orientation as the interpretive “whole” of one’s reality (inclusion). Then it becomes overridingly important to establish the individual’s position (exclusion and/or solipsism) within that “whole” to afford the negotiating ability to address the events of that interpretive reality. Unlike inanimate separation where division is done on an equality bases where half of something equals two equal something’s. Animate bio-cleaving which occurs as a cleaving on the lines of weakness. (dominate/subordinate) as an animated (changeable) bases. So while Greek mathematics was necessarily first to establish parameters, Hebrew mathematics will be more necessarily more dominate, as it establishes rules of commerce as the exchange of goods/service and therefore ideas, progress. I’m sorry if you don’t like that arrangement Doug, but that’s LIFE.

  57. #57 Jonathan Vos Post
    December 15, 2007

    ray burchard: sometimes a text on Mathematics may be hard to understand. However, the professional Mathematician makes every effort to make the text as EASY to understand as possible. The simplest possible vocabulary. An absence of digressions into Sociology or Philosophy or Linguistics, unless those are the subjects of a mathematical analaysis as such. An avoidance of claiming personal expertise which The Establishment refuses to recognize. Clarity is essential, otherwise everybody normally stops trying to decypher puzzling texts.

    “Animate bio-cleaving which occurs as a cleaving on the lines of weakness. (dominate/subordinate) as an animated (changeable) bases. So while Greek mathematics was necessarily first to establish parameters…”

    Number mysticism is not generally associated with “serious mathematics” but from the early Pythagoreans until the 19th century many venerated mathematicians practised some forms of numerology. In more recent times this was in return for patronage from the influential and aristocratic circles who sought some mystical assistance in their daily endeavours. Even today, existing Western cults refer to Pythagorean doctrines on number. According to Boyer:

    “Many early civilisations shared various aspects of numerology, but the Pythagoreans carried number worship to its extreme, basing their philosophy and their way of life upon it.”

    To the Pythagoreans, each number possessed its own special attributes. See for example the table below.

    Number Property of the number

    1 monad(unity) generator of numbers, the number of reason
    2 dyad(diversity, opinion) first true female number
    3 triad(harmony = unity + diversity) first true male number
    4 (justice, retribution) squaring of accounts
    5 (marriage) = first female + first male
    6 (creation) = first female + first male + 1 ?
    10 (Universe) tetractys

    Fortunately, in developing their number mysticism, the Pythagoreans also valued rigour and proof. To this end they searched for the essential properties and definitions of many numbers….

    [Boyer, Carl B., A History of Mathematics, N.Y.: John Wiley and Sons, 1968]

    The male/female perception of integers and patterns was not unique to Greece. For instance, From the Western Zhou Dynasty (from 1046 BC), the oldest mathematical work to survive the book burning is the I Ching, which uses the 8 binary 3-tuples (trigrams) and 64 binary 6-tuples (hexagrams) for philosophical, mathematical, and/or mystical purposes. The binary tuples are composed of broken and solid lines, called yin ‘female’ and yang ‘male’ respectively (see King Wen sequence).

    But this has essentially no connection with modern mathematics.

    “rigour and proof” — or, to my fellow Americans, “rigor and proof.”

  58. #58 ray burchard
    December 15, 2007

    Doug Spoonwood,

    Clarification: When I stated ” cleaving along lines of weakness, (dominate versus subordinate)”, that means like, 49 versus 50 as in 4950 and/or the “whole” of 0.9 versus it’s fractionalized equivalence 8.9999… As in merely the advantage of leverage.

  59. #59 Jonathan Vos Post
    December 15, 2007

    Re: #58. I don’t see the applicability of the word “leverage” in “that means like, 49 versus 50 as in 4950 and/or the “whole” of 0.9 versus it’s fractionalized equivalence 8.9999… As in merely the advantage of leverage.”

    leverage
    (lĕv’ər-ĭj, lē’vər-) pronunciation
    n.
    1.
    a. The action of a lever.
    b. The mechanical advantage of a lever.
    2. Positional advantage; power to act effectively: “started his . . . career with far more social leverage than his father had enjoyed” (Doris Kearns Goodwin).
    3. The use of credit or borrowed funds to improve one’s speculative capacity and increase the rate of return from an investment, as in buying securities on margin.

    As I wrote before: “the professional Mathematician makes every effort to make the text as EASY to understand as possible.” In this case, I don’t even see the metaphor.

  60. #60 Doug Spoonwood
    December 15, 2007

    [Two; (00+01+02 thru 09) = 045 = (0+4+5) = 9 while It's inverse sequence, (90+80+70 back thru 00) = 0450 = (0+4+5+0) = 9]

    Well, if we ignore the 0s in both patterns they become identical. So, 1+…+9=45 and 4+5=9. So what?

    [and (9+9) = 18 = (1+8) = 9]

    And if we had three similar patters like also using 001+002+…+009 with the two above, then we could take 9+9+9=27 and then 2+7=9. This comes as a special consequence of some rather simple theorem in number theory *instantied* into base ten. If we added in base four we could have 1+2+3=3+3=3+1+1+1=10+2=12, and 1+2=3. I could also take 30+20+10=120, and 1+2=3. More generally, for base x, a “recursive sum” of the digits of any number which equals a multiple of x-1 evetually will equal x-1. By ‘recursive sum’ I mean we keep adding the digits given until we have a one-digit number. So, in base “16″, the ‘recursive sum’ of the digits of a multiply of “15″ (which in base “16″ comes as a one-digit symbol) will eventually equal 15. I challenge you to demonstrate this theorem using algebra for all bases, or an instantion of the theorem in base ten.

    [I know this maybe confusing, and the complexity of the systematic sequencing only grows rapidly when you add the negative function as (antithetical duality), but after 20 years of pursuit I have developed many, much more conclusive illustrations.]

    We don’t need all those concepts like “homoscedastic coupling with rotation and translation” for this theorem. The math works out *much* simpler from fewer concepts.

    [Again the propensity to trivialize and fractionalize the "whole" (antithetical duality) of the human experience as in separating (phenomenon/noumenon).]

    Hmmmm… let’s see you probably didn’t address the microbes on your body in this essay and consequentally “fractionalized” your real experience of reading my response by not mentioning how it influenced the microbes on your body, or the nerves in your brain, or the hair on your head, or etc. Look, you can use that sort of criticism anywhere, with anything. You’d have to show it specifically relevant for it to have sufficient meaning in this sort of context.

    [So this means because you don't like evolutions revelations, your not going to join the rest of the worlds move into the next century, but like the Muslim reality, pitch your tent in antiquity.]

    I simply don’t see how you infer any of this.

    [0.9 versus it's fractionalized equivalence 8.9999]

    They only qualify as equivalent in the real number system. If we take a number system that includes the reals and the hyperreals, they don’t work out as equivalent. .9 qualifies as infinitesimally greater than .89999… in such a case.

  61. #61 ray burchard
    December 16, 2007

    #57, 59 Jonathan Vos Post,
    #60 Doug Spoon wood ,

    Lets establish a benchmark, ‘mystical’ like ‘coincidence’ are expressions of the limits of one’s comprehension (interpretative reality) as to the relationship between the “whole” of (actual/effectual).

    Bifurcation as (antithetical duality), and/or numerical polarity.

    Advantage of leverage, 4950. Here is another perspective, just like 9999999999 equals (9+9+9+9+9+9+9+9+9+9) = 90 = (9+0) = 9, where 9999999999 minus a left to right sequence 0123456789 equals 9876543210 as a right to left sequence. Then view both the sets 49 and 50 as each sequenced lineages traveling in opposite vertical directions.

    4851….48, 51….4, 8, 5, 1
    4950….49, 50….4, 9, 5, 0
    5049….50, 49….5, 0, 4, 9
    5148….51, 48….5, 1, 4, 8

    The group lineage, 4851, 4950, 5049, 5148 demonstrates a progression by the factor of 99, while the set lineages demonstrate an inverse relationship as separate converse sequences each by the factor of 01

    Now; (4851+4950+5049+5148) = 19998, and therefore
    [ (4800+4900+5000+5100) = 19800 + (0051+0050+0049+0048) = 00198 ] = (00198+19800) = 19998. Directional flow (center out) where (018+180) = 198

    Then;
    (4050+0801) = 4851
    (4050+0900) = 4950

    Plus
    (5040+0009) = 5049
    (5040+0108) = 5148

    E.A.Milne’s ‘Kinematic relativity’ conjugated with A.Einstein’s ‘General Relativity’ as mutually ignorant descriptions of the same entity, in the terms of one and another one, (01+10) = 11 and/or Two as (dynamic/static), (Greek/Hebrew) mathematical impetus.

    As I’ve stated I’ve taken and developed illustrations of this iteration aspect (Kurt Godel’s recursive and primary recursive number theory) much, much further, you have my e-mail address.

  62. #62 ray burchard
    December 16, 2007

    #60 Doug Spoonwood,

    “Well, if we ignore the 0s in both patterns they become identical. So, 1+…+9=45 and 4+5=9. So what?”
    Antithetical duality, a nine digit sequence conjoined with a ten digit sequence, while both with a value magnitude of 9, like that of Ade(nine) and Gua(nine) of the double helix DNA.

    “And if we had three similar patters like also using 001+002+…+009 with the two above, then we could take 9+9+9=27 and then 2+7=9″
    Isn’t this like I’ve previously stated in geometric progression where 124,875 traverse infinity and (124+875) = 999 as (1+8,2+7, and 4+5). Take 0999 and reverse the sequence and add, (0999+9990) = 10989 as 10(9)89 and 10989 divided by 11 equals 999 and 1089 divided by 11 equals 99. Here is the key 1089 divided by 9 equals (011+110) = 121, sequenced reversal.

    “. I challenge you to demonstrate this theorem using algebra for all bases, or an instantion of the theorem in base ten”
    Aren’t all bases inclusive in the value magnitude of 9 and base 10 is then a complete sequence of all 9 plus 1. As in (09+01) = 10.

    “We don’t need all those concepts like “homoscedastic coupling with rotation and translation”
    Doug, I didn’t make this concept up and devised the mathematics to illustrate it, it was the other way around I followed the mathematics of binary bifurcation extrapolation and saw it for what it is, a conjugation of quadratic antithetic concentric dualities which originate from one concentric duality in the form of the Mobius Band continuum, of this assertion I have developed mathematical proof by way of illustrations.

    “not mentioning how it influenced the microbes on your body, or the nerves in your brain, or the hair on your head, or etc”
    Yes, that’s true and the reason was not to confuse the issue with it’s complexity. The association is an iterative algorithmic fractal facet, as the recursive and primary recursive aspect of Kurt Godel’s number theory. However I find it strange you look to the “whole” when you think it serves your purpose.

    “They only qualify as equivalent in the real number system. If we take a number system that includes the reals and the hyperreals, they don’t work out as equivalent. .9 qualifies as infinitesimally greater than .89999… in such a case”.
    Again, the “whole” is comprised of (antithetical duality) mathematical paradox that while conjoined as in the “whole” of (odd/even), (Greek math/Hebrew math), (man/woman) and (space/time) they are individual principles, commensurate while contiguous.

  63. #63 ray burchard
    December 16, 2007

    Since you don’t return E-Mails and haven’t weighed in on this discussion Mark Chu-Carroll, what do you think, “How my doin”, good math or bad math?

  64. #64 Mark C. Chu-Carroll
    December 16, 2007

    Ray:

    I try to be very good about replying to emails, but in this case, I didn’t recieve any email to return.

    My opinion? The stuff that you’re incessantly babbling about is pure, utter, rubbish. It’s garbage mysticism – very similar to the gematria games of my own religious tradition, and as empty of genuine truth or meaning as gematria.

    Games like this are fascinating: but it’s important to realize that what games do is allow your mind to play with free association, to find potential connections that you might not have seen otherwise. But like anything else, they need to be validated. It might be difficult to see something working via straightforward traditional logical processes; free-association can be useful for seeing things that are less obvious. But the conclusion still needs to be brought back from associative thinking into a valid logical framework.

    To be a bit more concrete: I like to use gematria when I hit a dead-end in my work. What I’ll do is pick out a couple of key phrases associated with what I’m doing, translate them phonetically into hebrew characters, convert those to numeric equivalents, and play around with adding, multiplying, and various forms of twiddling the numbers, and then convert them back to letters, and see what kind of words I come up with. Then I try to play with those words, and see if I can come up with any interesting ideas.

    I’m not finding meaning in the numbers. I’m just distracting myself from the specific way of thinking that’s got me blocked in a dead-end. There’s nothing magical about, saying, converting something like “nondeterminstic deadlock” into hebrew letters, and then multiplying the numbers together. There’s no hidden meaning in the hebrew letters that phonetically spell out those words. It’s just a trick to make my mind look at a problem differently.

    You’re basically just playing with gematria, and insisting that there’s meaning in the patterns of numbers that you can produce by playing with them and free-associating. There isn’t.

  65. #65 ray burchard
    December 16, 2007

    Date: Sat, 7 Jul 2007 06:37:30 -0700 (PDT)
    From: “Ray Burchard” View Contact Details Add Mobile Alert
    Subject: A Pragmatic view of 3-Dimensional Depth of Field as the Archetypical Foundation of Mathematics
    To: markcc@gmail.com

    Dear Mark,
    Is this good or bad mathematics.

    Mark,

    Thank you for your assessment, but I’m sorry I can’t excuse the implications depicted in the mathematical illustrations I have presented here as “mystical” but I’m sure the limits of one’s cognitive ability would rationalize it that way.

  66. #66 Doug Spoonwood
    December 16, 2007

    ]Lets establish a benchmark, ‘mystical’ like ‘coincidence’ are expressions of the limits of one’s comprehension (interpretative reality) as to the relationship between the “whole” of (actual/effectual).]

    Relationship between the “whole” and actual/effectual what??? I don’t know what you meant to say since you left your sentence as a fragment.

    [Antithetical duality, a nine digit sequence conjoined with a ten digit sequence, while both with a value magnitude of 9, like that of Ade(nine) and Gua(nine) of the double helix DNA.]

    You can put fancy concept names on such and find similarities. So what?

    [Aren't all bases inclusive in the value magnitude of 9 and base 10 is then a complete sequence of all 9 plus 1.]

    I don’t think so.

    [Doug, I didn't make this concept up and devised the mathematics to illustrate it, it was the other way around I followed the mathematics of binary bifurcation extrapolation and saw it for what it is, a conjugation of quadratic antithetic concentric dualities which originate from one concentric duality in the form of the Mobius Band continuum, of this assertion I have developed mathematical proof by way of illustrations.]

    Of all this one thing remains radiantly clear. You almost certainly did NOT see that arithmetical theorem “as it is.” The theroem that for base x, a “recursive sum” of the digits of any number n=a1a2a3…an which equals a multiple of x-1 evetually will equal x-1. In other words if there exists a b

  67. #67 ray burchard
    January 9, 2008

    Essential progress in the advancement of mathematical knowledge has resulted from annexing concepts outside mathematics because for no other reason than it was impossible to define them, like “The ghosts of departed quantities”, BULLSHIT.

    I have avocationally worked on this inverse symmetry for over 20 years, and what I have given you doesn’t even scratch the surface as to what mathematical illustrations I have developed. As previously stated, once you realize directional flow is an integral principle of mathematics, then the next logical conclusion is 3-dimensional mathematics. But if you can’t see the value in what has been presented thus far, then I’m sure a mind’s eye geometrical perception will also elude your comprehension.

    Doug remember the warnings of Courant and Neuman, that of developmental sterility if connections with science were not made.

  68. #68 Gary Ehlenberger
    February 4, 2008

    One way of resolving Skolem’s paradox is to accept set-theoretic notions as uncountability in a given axiomatization as relative in nature; ie. a set which is uncountable in a given axiomatization may prove to be denumerable in another. Does anyone know of an example of this?
    Another alternative resolution of Skolem’s paradox is to accept that any axiomatization of set theory as a first-order theory must fail to capture fully the notions of the theory of the set of subsets of a given set, one-to-one correspondence, and uncountability (most mathematicians do not accept this).

  69. #69 ray burchard
    February 5, 2008

    Gary Ehlenberger,

    Consider the axiomatic dichotomy (antithetical duality) exampled by geometric progression, where one system of applied mathematics demonstrating arithmetic progression, 1, 2, 4, 8, 16, 32, 64, 128, etc… as an accumulative yardstick progression, designed to forever traverse the unaccountability of infinity. Then conjugate this accumulative system of mathematics with it’s antithesis, a contiguous, commensurate and reductive coparcener system of applied mathematics. By creating a single digit representation of each stepped progression by adding all the digits of the particular together until a single digit representation is derived, example; 64 = (6+4) = (1+0) = 1 and 128 = (1+2+8) = (1+1) = 2.

    What then is created is a reoccurring 124875 sequence. Two 3 digit converse directional sequences 124,875 with an ordered magnitude of 9, (124+875) = (9+9+9) = (2+7) = 9, and thereby lending an effectual parameter to the unknown, infinity.

    Then, as to your intro, here is yet another antithetical duality example; How about an ever expanding universe strategically punctuated by a series of consolidating black holes and/or mankind’s heuristic evolution. They all demonstrate the axiomatic matrix of the Mobius Band continuum.

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