After that nasty diversion into economics and politics, we now return to your
regularly scheduled math blogging. And what a relief! In celebration, today I’ll give
you something short, sweet, and beautiful: quotient groups. To me, this is a shining
example of the beauty of abstract algebra. We’ve abstracted away from numbers to these
crazy group things, and one reward is that we can see what division really means. It’s
more than just a simple bit of arithmetic: division is a way of describing a fundamental
structural relationship that pervades mathematics.
So what is division all about?
Suppose you want to divide 50 by 5. What you’re really doing is saying
you’ve got a collection of 50 indistinguishable things, and you want to break it into a 5
indistinguishable collections. Since you started with 50 indistinguishable things, that means
that you’ll end up with 5 sets of 10 things.
That’s pretty simple, right? Now, suppose that we’re not talking about simple numbers. Instead we
want to work in terms of groups. Can we take that basic concept of division, and find some meaningful way of applying it to groups?
Well, first, we need to somehow talk about division in a way that doesn’t involve numbers. We start with a group – that is, a collection of objects with some kind of meaningful structure. What can we divide it by? A group with a similar structure – in fact, a group with the same structure: a subgroup But not just any subgroup: it’s got to be a normal subgroup, because that’s the kind of subgroup that properly preserves the structure of the group.
As a reminder, a normal subgroup (N,+) of a group (G,+) is a subgroup where ∀n∈N, ∀g∈G, g+n+g-1∈N. That is, it’s a normal subgroup group
which is closed with respect to the group operation performed in sequence with any member of the group and that members inverse.
So what happens when we divide a group by one of its normal subgroups? We partition the
group into a new group, where the elements of the new group are formed from subsets of the
elements of the original group. It’s the same idea as simple integer division described up above, except that we want to preserve the group structure, so the result is going to be a group.
To be precise, given a group (G,×), and a normal subgroup (N,×), the members of
the quotient group G/N are the set of set products of elements of G and the set N:
∪g∈G{ { g×n | n∈N } }, which can also be written
{g×N | g∈G}. That is, each member of G/N is the set of products of a member of G with each of the members of N. For a given member of the quotient group, g×N, the inverse element
is g-1×N. The group operation of G/N is set-product, and the identity element of G/N is the set containing the identity element of G.
So why the restriction for quotients to only be defined for normal subgroups? It’s pretty
simple: the construction above will only be a group if N is normal. Note the way we multiply a member of the group by the members of the subgroup? If we work that out, the only way
that the quotient group is closed under the group operation of set-product is if the subgroup
that generated the quotient is normal.
A beautiful example of the quotient group comes from the integers. Take the group of
integers with addition as the operator: (Z,+). The set of even numbers, 2Z is
a normal subgroup of the integer group. The quotient group Z/2Z is
a two element group: one element is the set of all even integers; the other is the set of all
odd integers. This is equivalent to the cyclic group of size two – aka, mod-2 arithmetic. You can do a similar trick with the reals and the integers, where the R/Z gives you a sort of real-valued modulo group formed from the set of all reals between 0 and 1.
Hooray!
Remind me again why you still are using additive notation for your composition, but multiplicative notation for your inverses? Especially since additive notation customarily denotes an abelian group, and the concept of a normal subgroup is trivial in that case.
Here is how I view your last example: consider the real numbers mod 1, under addition. For example, 2.3 = .3, -1.3 = .3, and 4.53 + 1.6 = .13
So what do you get? You get a group isomorphic to the complex numbers of unit modulus, with multiplication as the group operation; the map is f(x) = exp(2*pi*x*i), where x is in [0,1)
Note: if we change this example just a bit and take R/(2*pi*Z) and use f(x) = exp(i*x), we get the map that allows us to extend the sine and cosine function from the circle to the whole real line.
Of course, we don’t tell beginning trig students that the real line covers the unit circle and that the real valued sine and cosine functions come from the pullback of that map. 🙂
I took a different path to learning about quotient groups, when I started surfing Wikipedia after your last group theory post.
I’d always been curious about cosets, so I set to teaching myself properly what they were. I never expected the definition to be so easy! You just take a group G and a subgroup H, then multiply all of H by some element in G. The answer is a coset! I never realized it was that easy to understand.
Moving from cosets to the definition of a quotient groups is trivial, then. All you have to do is repeat the coset-making for *every* element in G. Collect all the cosets you’ve created, and you have a quotient group. (Many of the cosets will often be identical or at least isomorphic, and this is fine.)
A really cool thing about quotient groups is that every element in G shows up in one of the cosets, and *only* one (frex, the identity element only shows up in the coset you get from multiple the identity by all of H). So the different cosets are guaranteed to totally cover G but not overlap. That is, it partitions G.
This sort of explanation via the Wiki really helped me capture the meaning of these things better than anything else. I especially liked the example they gave.
Let G be the set {0,1,2,3}. Let H be the set {0,2}. This is a subset of G, so we’re cool. Turn these both into groups by giving them addition as their group operator (they loop around as expected, so that 2+2=0).
Now, we can make a coset by grabbing some element of G (let’s take 0) and adding it to all of H. For clarity, we’ll denote this 0H. This gives us the coset {0,2}. If I take another element of G, like 1, I’ll get the coset 1H = {1,3}. Taking 2 and 3 gives me {2,0} and {3,1} respectively.
Since order doesn’t matter in sets, I’ve really just gotten two different cosets, {0,2} and {1,3}. Pop them together and I have the quotient group {{0,2},{1,3}}! I know, not very impressive. I was just glad I understood it!
The last thing it took me a few moments to wrap my head around was the group action for the quotient group, but it’s easy enough – you just use the original group action on every possible pairing between the elements. Frex, in the quotient group I just did, you can have {0,2}+{1,3}. This equals the collection made by 0+{1,3}, 2+{1,3}, {0,2}+1, and {0,2}+3. It turns out that all of these equal {1,3}. When you drill down a bit you see that this quotient group is equal to the integers mod 2, just like in Mark’s example, with {0,2} being the identify element of the group.
ollie:
Isn’t -1.3 = .7
Just want to make sure I understand the reals-mod-1 group.
Evan, of course you are right. 🙂
Oooops!
One thing that I was always was interested in was the nature of the quotient group.
I was aware that quotient group G/H’s elements were subsets of G that comprised a partition.
But how did this partition out work exactly, in practice?
I made some progress by computing these by hand for various group/subgroup combinations, but I never got to the point where I could look at a group, a normal subgroup, and be able to intuit what G/H looked like structurally.
In any case, good post. I love group theory!
Today my special thanks goes to Xanthir who put a face on finite quotient groups.
Yep, that threw me too for a moment, because I didn’t recognize the familiar form of the similarity transformation and had to, uh, transform it first. (Similarity, I think, is called conjugacy in group theory – a similarity operation is a conjugacy action perhaps?) Poor me doesn’t have as much heuristic handles on these things as you guys.
Not that it is terribly important, but as style goes I think the multiplicative notation wins here.
Quotient groups are very elegant, and you’ve described them nicely; and they are clearly related to division, but incidentally I question “the meaning” of division. Suppose you want to divide 2 by 4. Maybe you’re saying you’ve got one set of 2 similar things, and you want to break it into 4 equal sets, because maybe you could do that by breaking those things into two equal halves (although maybe 4 just won’t go into 2 on this picture); but suppose you want to divide 1 by the square root of -1? Personally I think of division as the reverse operation to multiplication (with the picture of breaking up collections deriving from that), but I’ve no idea if that’s standard or not…
Some other cool quotient groups:
(R,+)/(Z,+) = (T,+), where R is the reals, Z is the integers and T is the “Circle group”, ie the unit circle. (T is for Torus).
Along the same lines, Rn / Zn = Tn, the n-dimensional analogues. These groups play an important role in Fourier analysis.
I would also like to mention that if H is a non-normal subgroup of G then G/H is not a subgroup but it’s a coset space– it still partitions G into cosets where G/H is a transitive G-space.
John Armstrong said:
“Remind me again why you still are using additive notation for your composition, but multiplicative notation for your inverses? Especially since additive notation customarily denotes an abelian group, and the concept of a normal subgroup is trivial in that case.”
The reason is because Mark is a mathematical IDIOT whose knowledge of mathematics is limited to cutting-and-pasting and paraphrasing bits and pieces of prose out of textbooks without really understanding what the hell he is talking about, as anyone with a real ph.d. in mathematics can tell the moment he opens his stupid ass mouth.
For example, Mark said:
“It’s the same idea as simple integer division described up above, except that we want to preserve the group structure, so the result is going to be a group.”
Almost every time you open your mouth, Mark, you demonstrate you’re not a mathematician. Normal subgroups or factor groups don’t “preserve group structure”, in fact that claim doesn’t even make semantic sense. The things which “preserve group structure” are the group homomorphisms, whose images are (up to isomorphism) the factor groups of the given group. But a subgroup (normal or not) or factor group does not “preserve group structure”. Preservation of structure is a property of mappings (morphisms), not of objects.
You constantly butcher definitions, as in the following:
“To be precise, given a group (G,×), and a normal subgroup (N,×), the members of the quotient group G/N are the set of set products of elements of G and the set N: ∪g∈G{ { g×n | n∈N } }, which can also be written {g×N | g∈G}.”
I have no idea what a “set of set products of elements of G and the set N” means, but did you even notice, Mark, that
“∪g∈G{ { g×n | n∈N } }”
is actually EQUAL TO THE GROUP G itself?!? You’re taking the UNION of all the cosets, which is G itself. (If you apply the set union operator on a factor group, you just get the group back, obviously.) Or did you just confuse UNION with set notation?, did you really mean:
“{ { g×n | n∈N } | g∈G}”
This is precisely the factor group. I guess you put the union symbol there because, as I said above, you don’t really understand what the hell you’re talking about and you’re just cutting and pasting and paraphrasing stuff from textbooks.
Your didn’t even bother to check simple singular/plural noun agreement is your purported “definition”:
“the members of the quotient group G/N are the set [sic]”
How can the “members” (plural) be defined as the “set” (singular)??
You said:
“The group operation of G/N is set-product, and the identity element of G/N is the set containing the identity element of G.”
Why don’t you just make it more clear by saying the identity element is the coset N??
You said:
“So why the restriction for quotients to only be defined for normal subgroups? It’s pretty simple: the construction above will only be a group if N is normal.”
This is true, but there is a much more profound reason: the normal subgroups are precisely the kernels of the group homomorphisms. This is the REAL reason normal subgroups are set apart from general subgroups.
You said:
“The set of even numbers, 2Z is a normal subgroup of the integer group.”
As is true for ANY abelian group. I find it amazing that in a purported introduction to normal subgroups, you failed to mention that every subgroup of an abelian group is normal. But that’s not surprising, given your confusion over additive vs. multiplicative notation.
You said:
“This is equivalent [sic] to the cyclic group of size two – aka, mod-2 arithmetic.”
Replace “equivalent” with “isomorphic” and all is well.
I felt similar pains in my side reading through “Symmetric Groups and Group Actions”:
You said:
“Given a set of objects, O, a permutation is a one-to-one mapping from O to itself.”
No, WRONG. A permutation is a one-to-one and ONTO mapping from a O to itself. 1-1 and onto are equivalent for maps from finite sets to themselves, but not for infinite sets.
Then I love how you verbally define a group action as a homomorphism from the given group to the symmetric group on the set, then give a formal definition in terms of a map from the product of the group and set to the set:
“Suppose we want to apply a group G as a symmetric transformation on a set A. What we can do is take the set A, and define the symmetric group over A, SA. Then we can define a mapping – to be more precise, a homomorphism – from the group G to SA. That homomorphism is the action of G on the set A. To make that formal:
If (G,+) is a group, and A is a set, then the group action of G on A is a function f such that:
1. ∀g,h∈G: (∀a∈A : f(g+h,a) = f(g,f(h,a)))
2. ∀a∈A: f(1G,a) = a.”
The two definitions are not even equivalent. They are different definitions defining the same notion. But a map G –> S(A) cannot possibly be equal to a map G x A –> A. Are you intentionally trying to confuse people??
BTW, what the hell is a “symmetric operation”?? And FYI, a “symmetric transformation” is a term used in the context of HILBERT SPACES and FUNCTIONAL ANALYSIS, not symmetric groups or group actions!! Can’t you get anything right??
I guess not. Although you continue to demonstrate about the same level of idiocy of most of the “science bloggers” on this pathetic website spawned from a Fischer-Price version of Scientific American for people with short attention spans. And the same level of idiocy demonstrated by HIV morons at Aetiology. What a joke.
darin
For those unfamiliar, I’ll merely point out that Darin Brown is an AIDS denialist who started posting his rants about what an idiot I am when I drew his attention by flaming some shoddy work in an anti-HIV paper.
The reason that I used “+” and “-1” was deliberate.
Back when I first studied this stuff, one of the problems I had was really grasping the abstraction away from my intuitions associated with the familiar arithmetic operations. So in my explanations, I deliberately try to use the notation to confuse those intuitions; I deliberately mix the additive and multiplicative notations to try to make it clear that we’re *not* talking about addition and multiplication. It’s a group operation, which I happen to be writing using the same symbol as addition – but the inverse isn’t how you’d write the additive inverse.
Perhaps it wasn’t an entirely successful conceit, but I’m at least trying to make things clear as best I can.
Just ban the fucker.
When you put it that way, it is probably a more fundamental problem.
Yes, the usual notation isn’t terribly supportive for first impressions. But perhaps you could use your own notation, like “·G”, before gradually slipping into the conventional?
The lowest of the low or the craziest of the crazy. And this thread isn’t even about HIV and it’s terrible burden on mankind.
Maybe we should sic ERV on him? 😛
Torbjön:
You’re probably right that it would have been a good idea to just not use standard arithmetic symbols at all in the introduction. Too bad I didn’t think of it before I wrote
those posts.
Riesz:
I really don’t like banning people, so it takes a lot to push me to do that. And I also think that insulting me is fair game – since I write posts that are insulting to lots of people, I don’t think it’s appropriate for me to turn around and ban people for doing the same thing to me. My policy on banning is to only ban people when they attack others in an inappropriate way, or become so disruptive that they prevent other folks from being able to have a reasonable conversation.
In the nearly two years that I’ve been writing this blog, I’ve only banned two people. Jon Davison, who prides himself on being banned from just about every forum in existence, and is willing to do almost any obnoxious thing necessary to get himself banned; and George Shollenberger, who went so far as to report me and several commenters to the FBI for allegedly being part of an atheistic conspiracy led by the American Museum of Natural History, Google, and Seed Magazine to supress his book.
I am not going to discuss whether Darin is right or wrong, but he sure as hell needs a hug…or to get laid.
John:
Please, lay off the ad-hominem bullshit, OK? I don’t think Darin is a good guy – but I don’t approve of personal insults is an appropriate way of dealing with him. Darin’s problem has nothing to do with his personal traits – it has to do with the fact that he promotes a nonsensical theory that has already cost the lives of who knows how many people, and has the potential, if widely accepted, to cost the lives of millions more. You don’t need to resort to personal insults to knock that down – simple reason, analysis of what’s wrong with the theory and what people do to prop up that obviously incorrect theory – is a far more effective strategy.
And the fact that, in the past, I’ve responded to Darin’s claims through exactly that mechanism is precisely why he’s so angry at me.
Xanthir,
Thanks for the comment and the examples. Although you didn’t specify by definition that you meant addition mod 4 instead of normal addition, I found your examples and explanation of concepts *radiantly* clear. Maybe you should write textbooks or more explanations like that.
Mark,
From your post it seems that you mean to say that “the nature of division concerns partitioning.” Like other times when you’ve tried to explain “the meaning” of mathematical concepts I simply don’t get it. If we take (i^3)/i, have we partitioned i? I simply don’t see how we’ve done so. What about if we divide the interval number [8, 9]/[3, 4]? (interval division on R+ works like this:
[a, b]/[c, d]=[a/d, b/c]) Well, we get [2, 3] as our answer. Did we partition [8, 9], [3, 4] times into [2, 3] equal subsets? I certainly don’t see how we did, or how anyone can reasonably see such.
I don’t see how your pointing out Darin Brown as an AIDS denialist comes as relevant to his comments on your post. It borders on an ad hominem attack actually. I didn’t like his ad hominem attacks either nor his irrelevant analogy about “HIV morons”, nor his tone in general.
Thanks, Torbjörn and Doug S! I have thought about going into teaching, but that’s not on the table right now. ^_^
Doug S re: partioning:
As usual, our original intuition fails a bit when you combine analogies with complexity. The idea of partitioning, I think, only works well with real numbers and their analogues. Well, actually, it only works with integers, but we’re so accustomed to dividing by reals that when we see 5/1.2 we accept that we’re dividing 5 into 1.2 groups without asking just what .2 of a group looks like.
Dividing by i does partition a group, but into i subsets. Unfortunately, our intuition doesn’t tell us what that looks like. ^_^
The interval number part is a bit easier to understand, luckily, because interval numbers work relatively similarly to normal numbers. You have a range from 8-9 being split into 3-4 subsets. Thus, each subset is 2-3 in size. If you imagine that each interval is a real number with a bit of uncertainty, this makes perfect sense.
[Dividing by i does partition a group, but into i subsets. Unfortunately, our intuition doesn’t tell us what that looks like.]
I don’t think this just violates intuition, for in such a case we could have a negative amount of subsets (i^2). We would have a negative cardinality. I see that as oxymoronic… a contradiction between the adjective and the noun.
[You have a range from 8-9 being split into 3-4 subsets. Thus, each subset is 2-3 in size.]
Sure, you can take that interpretation for intervals, but in such a case we don’t have interval numbers. We have regular numbers belonging to an interval. For an interval number [1, 3] I don’t mean that *there exists* an x such that x belongs to [1, 3] (as you did above), I mean the set of *all* x such that x belongs to [1, 3].
Doug, yes; also, what about the surreal 1/w? On the other hand, the primary meaning of division does seem to have been (as far as I can recall) partitioning; and if that is generally true (for other people, and historically etc.) then maybe the other meanings (for i and w etc.) are by analogy etc. (?)
I have three apples. You say I need to give you five apples. How many apples do I have after giving you your due? The answer is -2, but there’s no such thing as negative apples! Of course, we don’t talk about negative apples, we talk about owed apples instead.
Not everything makes sense immediately when you’re translating from pure math to concrete things. In that case, you just have to adjust your worldview to make sense of the new answer. ((That is, if you want to make sense of it. Sometimes you can just declare the answer to be nonsensical and move on.))
I also find it somewhat interesting that you apparently have an easier time accepting a set with cardinality i rather than one with cardinality -1. ^_^ In that, you actually parallel many mathematicians of the late 18th/early 19th century, who believed in the reality of imaginary numbers more than negatives.
Assume I meant that (because I did ^_^). My words remain the same.
Engiman,
[On the other hand, the primary meaning of division does seem to have been (as far as I can recall) partitioning; and if that is generally true (for other people, and historically etc.) then maybe the other meanings (for i and w etc.) are by analogy etc.]
I think the first part well-put. But, I don’t see how by analogy the other meanings qualify as partitioning. Why? A precise definitional meaning of division trumps a primrary meaning.
Xanthir,
[Of course, we don’t talk about negative apples, we talk about owed apples instead.]
O.K., but I don’t see how we can *owe* a subset. So, what concept do we need for a negative subset?
[((That is, if you want to make sense of it. Sometimes you can just declare the answer to be nonsensical and move on.))]
I wouldn’t hastily declare such nonsense. I would declare it formally valid, but not useful conceptually.
[I also find it somewhat interesting that you apparently have an easier time accepting a set with cardinality i rather than one with cardinality -1.]
I don’t. What concept do we need for a cardinality of a subset to equal i?
[In that, you actually parallel many mathematicians of the late 18th/early 19th century, who believed in the reality of imaginary numbers more than negatives.]
If I did believe in the reality of numbers whatsoever. But, I don’t believe in the reality of numbers. Reality refers to empirical things. Numbers don’t exist empirically, and thus don’t have a reality in the ordinary sense of the term. They do have a “reality” in the same way that abstractions like “computer” (not your computer in front of you… but the concept of a computer) have “reality”.
[Assume I meant that (because I did ^_^).]
Well, you said “If you imagine that each interval is a real number with a bit of uncertainty”. For *the set* of *all* x belonging to [1, 3], we simply don’t have a real number with a bit of uncertainty. We have a *set* of real numbers, with no uncertainty that each real number element of [1, 3] belongs to that *set* of real numbers. The partition=division hypothesis stops making sense in such a situation rather clearly, because a partition of [1, 3] yields a *geometric* subset of [1, 3], while a division involving [1, 3] can yield real number elements which don’t qualify as *geometric* subsets of [1, 3].
“Precise definitional meaning” and “analogy” don’t mix very well, Doug S. ^_^ They qualify by analogy in the same way that dividing by 1.2 qualifies as partitioning by analogy. The proper idea of partitioning/dividing is *only* valid for integer values.
The only difference is in how difficult it is to envision such a thing. By its very nature, though, it will *not* correspond to ‘reality’ in the exact same way that the base case (integer division) does. This is the precise reason we need analogy! Dividing by 1.2 isn’t too difficult to see. Dividing by -1 is more difficult. Dividing by i is more difficult still, followed by dividing by 1/ω! None of these follow the idea of partitioning exactly, because partitioning only actually describes the integral case. But we can often see how the concept is extended.
As to what concept precisely is needed to see the analogy clearly for negative or imaginary numbers of partitions, I’m not quite sure. There are many ways of thinking about negative and imaginary numbers, and perhaps one will serve. Perhaps none will. It’s entirely possible that our intuition will *not* guide us to an answer here, that we can never truly imagine where the analogy leads us due to our preconceptions of how the world works (generally via positive integers).
I’m not sure I understand you here. Do you mean it in the most basic sense that a partition of [1,3] must split it into something like [1,2] and [2,3]? If so, then it should be obvious why it’s not making sense; you’re not thinking about numbers in the same way as they were explained earlier!
When partitioning the number 3, it’s not a singular thing; it’s a group of three pebbles, or three hash marks, or whatever. You’re making the mistake of considering [1,3] as a thing in and of itself, though. This is certainly valid most of the time (as thinking of 3 in that way is valid most of the time), but it’s not if you want to use the partitioning analogy for division. Again, partitioning is only precisely valid for integers, not partition numbers, but the analogy is fairly easy to see in this case.
In this case, a possible way to see the analogy would be thus: if I were to partition [1,3] by 2, it would be partitioning *each* real within [1,3] by 2. The collection of answers would form the partition [.5,1.5]. It’s a tough weirder when you talk about, say, dividing [8,9] by [3,4], but can still be done; you’re simply dividing/partitioning every number in [8,9] by every number in [3,4]. The answers form the range [2,3], as elementary reasoning can show.
Sorry that I’ve missed out on most of this discussion; as I mentioned in the most recent post, I’ve been under the weather with a nasty flu.
Anyway – my point of view is that it doesn’t matter that the “partitioning” idea doesn’t extend naturally to arbitrary division values. A lot of mathematical ideas start with a simple intuitive concept, and then extend that concept in some way that’s hard to understand in terms of the original concept.
For example, exponents. We all start off understanding exponents as repeated multiplication; and I would argue that repeated multiplication is the basic concept underlying exponents: a2=a×a; a3=a×a×a, and so on. But what does it mean to multiple something by itself 2 1/2 times? Or worse, π times? Or i times? In terms of repeated multiplication, what on earth does Euler’s famous equation about eiπ mean?
You can work out some way of rationalizing it in terms of the basic concept of repeated multiplication, but at least some of the time, it’s going to sound awfully stretched. To me, that’s OK. As I said, we often start with a simple concept – like integer division. Then we extend that concept – giving us something like real division. And then we extend it further – giving us things like quotient groups. The basic idea underlying it is much the same – but it’s been abstracted away. There’s still a lot of benefit in understanding where it came from.
To me, the fundamental idea of division is partitioning, and the group-based meaning of quotient fits well with that. If it doesn’t work for you, don’t use it. It’s just a metaphor for wrapping your head around an abstract concept defined through mathematical logic. If that metaphor doesn’t work for you, choose another that’s consistent with the logic.
[The proper idea of partitioning/dividing is *only* valid for integer values.]
Then, the analogy fails when extended further.
[This is the precise reason we need analogy!]
We don’t have an analogy without some sort of similarity. I don’t see a similarity between partitioning and dividing among interval numbers.
[But we can often see how the concept is extended.]
I don’t see how you extend the concept. I see an assumption that we can extend the similarity between partionining and division from the integers to other strctures, without a basis for such an extension.
[Do you mean it in the most basic sense that a partition of [1,3] must split it into something like [1,2] and [2,3]?]
Almost, excpet I first have to make your example into an actual partition, or {[1, 2), [2, 3]} as our partition.
[if I were to partition [1,3] by 2, it would be partitioning *each* real within [1,3] by 2. The collection of answers would form the partition [.5,1.5]]
Alright, suppose I hypothesized this. Partitions produce pariwise disjoint subsets whose union equals the orignial set. If I partition a set C of 4 apples into two subsets of 2 apples each, subset A and subset B, the union of subset A and subset B yields the original set C. But, with the proposed “partition” here [.5, 1.5] of [1, 3], the union of [.5, 1.5] and [.5, 1.5] does NOT yield the set
[1, 3]. This problem doesn’t happen among just interval numbers, but also among natural numbers in ZFC since
1={{}}, and u(1, 1)=u({{}}, {{}})={{}}=1 by idempotency, even though the “partitioning” of 2 yields 1 and 1. So, a division of 2 by 2 does NOT yield a partition of 2… it doesn’t yield a set of pariwise disjoint subsets of 2, whose union equals 2. In other words, from a formal level, there doesn’t even exist a partionin/division similarity for (formal) natural numbers. It does happen for objects like a set of 4 cookies or what have you, but not for formal numbers.
Doug S:
You’re doing what you did in the other threads – you’re focusing on irrelevant details and trying to apply operations of the wrong kind/wrong level and then wondering why the results don’t turn out right. Frex:
I clearly stated in my example that, for the partitioning example to hold, we must actually partition *each number* in the interval, not the interval itself. Thus, when we recombine the partitions, we must recombine *each number* in the partitions, not the intervals themselves.
If you have two intervals, [.5,1.5] and [.5,1.5], clearly the smallest possible number you can form from summing a number from each partition is 1, and the largest is 3. It is also plainly true that you produce every number between 1 and 3 by summing appropriate values within the intervals. Thus, you produce the interval [1,3] again.
I tried to warn you off of making the mistake you did when I said, “You’re making the mistake of considering [1,3] as a thing in and of itself, though.” For the analogy to hold for partition numbers, you have to think of them as a collection of individual numbers, and the operation working over each individual number. Trying to partition the interval itself will *not* work, as you have discovered and I enumerated, so that’s not the correct way to go if you want to hold the analogy.
This situation is itself exactly analogous to what I was saying about negative/owed apples. There’s no such thing as negative apples; the concept doesn’t even make sense. Thus, if you want to use the concept of less than zero apples, you must think of it in a different manner, as owing a positive number of apples. This way you retain an intuitive grasp on what is happening while still maintaining the math and the analogy.
To make a further point:
The proper idea of subtracting is *only* valid when the first number is larger than the second, producing a positive number as the answer.
Then, the analogy fails when you subtract five apples from three. This is clearly not subtraction, and I see no way to extend the concept of subtraction to this situation.
What I stated above applies equally to this. You are confusing levels and abstractions and operations so that you get nonsensical results. You are then insisting (or at least implying) that the nonsense you end up with is the only possible answer, and thus the entire concept is nonsense.
DIVISION…. Bifurcation by Cleaving along lines of antithetic binary pairs (Dominate / Subordinate), demonstrated through a geometric progression of (nine more than ninety), 99. Where magnitude does not reign supreme but more over shares the limelight with directional flow, (lesser to larger) and therein creating an applied mathematics with a dimensionally structured dynamical systems matrix, incorporating mirror imaging, liken by but not limited to, (01+10) = 11, (02+20) = 22 and (09+90) = 99.
Therefore (099+990) = 1089 and (0999+9990) = 10989 as concentric duality where expansion is from the center out, (09999+ 99990) = 109989 etc…
(018+180) = 198, and (081+810) = 891 and (198+891) = 1089, as does
(027+270) = 297, and (072+720) = 792 and (297+792) = 1089, as does
(036+360) = 396, and (063+630) = 693 and (396+693) = etc…, etc…
Given:
–(00+02+04+06+08) = 020 plus (01+03+05+07+09) = 025 equals (020+025) = 045
–(80+60+40+20+00) = 200 plus (90+70+50+30+10) = 250 equals (200+250) = 450
+ _____________________ _____________________ ______________
= (80+62+44+26+08) = 220 plus (91+73+55+37+19) = 275 equals (220+275) = 495
Therefore:
the “whole”—–directional bifurcation———–sub-units (90 +09) = 099
———————-A
A) 4554——-= (4500 + 0054) =——————–(4050 + 0504)
—-4653——-= (4600 + 0053) =——————–(4050 + 0603)
—-4752——-= (4700 + 0052) =——————–(4050 + 0702)
—-4851——-= (4800 + 0051) =——————–(4050 + 0801)
—-4950 b)—= (4900 + 0050) =——————–(4050 + 0900)
a) 5049——-= (5000 + 0049) =——————–(5040 + 0009)
—-5148——-= (5100 + 0048) =——————–(5040 + 0108)
—-5247——-= (5200 + 0047) =——————–(5040 + 0207)
—-5346——-= (5300 + 0046) =——————–(5040 + 0306)
—-5445 B)—= (5400 + 0045) =——————–(5040 + 0405)
——————————– B
_+_________+__________________________+_____________
—49995——–49500 + 00495———————-45450 + 04545
______________________________________________________
————————A
A) 4554——–= (4500 + 0054) =——————(40 50 + 05 04)
—-4653——–= (4600 + 0053) =——————(40 50 + 06 03)
—-4752——–= (4700 + 0052) =——————(40 50 + 07 02)
—-4851——–= (4800 + 0051) =——————(40 50 + 08 01)
—-4950 b)—–= (4900 + 0050) =—————–(40 50 + 09 00)
———————————–b
_+________+ ______________________+ _______________
A) 23500———23500——————————-(20000 + 3500)
b) 00260——————-00260———————(00250 + 0010)
_______ _____________ ______________
– = 23760—–= (23500 + 00260)—————–= (20250 + 3510)
_______ __ ________ __ ________ __ ________ __ _______
———————–a
a) 5049——–= (5000 + 0049) =——————-(50 40 + 00 09)
—-5148——–= (5100 + 0048) =——————-(50 40 + 01 08)
—-5247——–= (5200 + 0047) =——————-(50 40 + 02 07)
—-5346——–= (5300 + 0046) =——————-(50 40 + 03 06)
—-5445 B) —= (5400 + 0045) =——————-(50 40 + 04 05)
———————————-B
_+ _________+ _____________________+ ______________
a) 26000———26000——————————–(25000 + 1000)
B) 00235——————–00235———————(00200 + 0035)
________ _____________ ______________
–= 26235—-= (26000 + 00235)——————= (25200 + 1035)
_________________________________________________________
Take the difference between 26235 and 23760, (26235 – 23760) = 2475. Then subtract 2475 from 49995 leaving 47520 as, 4752 times 10. The matrix remains the same while the composite numbers esculate and that is the robustness of IC, 09 more than 90.
“homoscedastic” as dual concentric coupling,
A 23500 + a 26000 = Aa 49500
B 00235 + b 00260 = Bb 00495
+ _________________________________
= , (A+B) = 23735 plus (a+b) = 26260 equals (23735+26260) = 49995 as a composite of inverse sequencing (symmetry) built from the center out (antithetical duality), and thereby creating an (iterate / fractal) seen as separate while conjugated applied mathematical principles creating archetypical mathematical paradox.
Like; take the 2 from it’s position in 26260 and add it to the 0 position,
26260 then becomes 06262,
then take the 2 from it’s position in 23735 and add it to the 5 position
23735 = 03737, then add (06262 + 03737) = 09999
Now then, reverse the process and take the 0 from it’s position in 26260 and add it to the 2 position, 26260 = 26260 (as in stays the same),
then take the 5 from it’s position in 23735 and add it to the 2 position,
23735 = 73730 . Now add (26260 + 73730) = 99990 and (09999 + 99990) = 109989 as in 1089 expanded from the center out. Dimensional mathematics as bio-engineering’s (dominate / subordinate) progression.
Marc, to answer your earlier claim that my scholarship is based on the “mystic”. What I don’t believe in is;….God, the Devil, angles and fairies, (good or bad), ghosts, goblins or witches, Superman, The Lone Ranger, the Easter bunny, Santa Claus and Leprechauns, etc… They are all myths rationalizing the unknown. What I do believe in is, the underlying morality demonstrated in the many parables of the “Bible”, as morality itself is a symetric modality created by the interaction of the individuals ethics and/or lack there of.
As to your insertion of the “Hebrew Number System”, what I meant was the mathematics developed in conjunction with the Hebrew merchant mentality, that which biased mathematical development to facilitate commerce and ignore bio-engineering’s development. While commerce and therefore avarice are the necessary essentials of progress, I do not lay the blame of greed at the doorstep of the “Jews”, that responsibility remains the sole distinction (obsession) of an individuals psychological predilection.
Then as to the “Hebrew Number System” your fractionalization mentality conveniently omit’s the number system of the “Voynich Manuscript” (Yale University) of the same epochal period.
Hebrew numbers representing “pepper” 858852 then reverse the sequence 258858 and subtract the smaller from the larger equaling 599994. Now the “Voynich” as 757752 reversed 257757 and again subtract the lesser from the larger, equaling 499995. Then add the two (599994 + 499995) = 1099989 as 1089 built from the center out.
Now while I have not pursued this specific issue, I would not lay it at the doorstep of coincidence just yet.
[You’re doing what you did in the other threads – you’re focusing on irrelevant details and trying to apply operations of the wrong kind/wrong level and then wondering why the results don’t turn out right]
You mean details that you consider irrelevant. Please try to show those details irrelevant if you claim more than a personal understanding here.
[I clearly stated in my example that, for the partitioning example to hold, we must actually partition *each number* in the interval, not the interval itself. Thus, when we recombine the partitions, we must recombine *each number* in the partitions, not the intervals themselves.]
I consider the interval number a number, so actually I would claim taking the union of [1, 3] and [2, 4] as recombining *each number*. But, even ignoring this, recombining number *pointwise* (which seems like your meaning) from [.5, 1.5]=[1, 3]/2 and [.5, 1.5] *only* yields [.5, 1.5]. It does NOT yield [1, 3], as needed for your example to work.
[If you have two intervals, [.5,1.5] and [.5,1.5], clearly the smallest possible number you can form from summing a number from each partition is 1, and the largest is 3. It is also plainly true that you produce every number between 1 and 3 by summing appropriate values within the intervals. Thus, you produce the interval [1,3] again.]
Yes and no. You do produce the interval by *summation*. But, you simply do NOT produce the interval [1, 3] by *unionization*. With a partition we have a pairwise disjoint class of subsets whose *union* produces the set of which they constitute a partition. E.G. suppose we have set {2, 3, 5}. A partition of this set produces the class
{{2}, {3, 5}} with pairwise disjoint subsets {2} and {3, 5} whose *union* yields the original set {2, 3, 5}.
[For the analogy to hold for partition numbers, you have to think of them as a collection of individual numbers, and the operation working over each individual number.]
Even in such a case the analogy does NOT work, because a pointwise unioniziation of [.5, 1.5] and [.5, 1.5] yields [.5, 1.5] and does NOT yield [1, 3]. Only addition does that.
[What I stated above applies equally to this. You are confusing levels and abstractions and operations so that you get nonsensical results. You are then insisting (or at least implying) that the nonsense you end up with is the only possible answer, and thus the entire concept is nonsense.]
I find it curious that you claim I’ve confused levels and abstractions, when I *specified* the numbers I talked about as numbers in ZFC (or similar formal sorts of numbers). I also said “It does happen for objects like a set of 4 cookies or what have you, but not for formal numbers.” I also said “O.K., but I don’t see how we can *owe* a subset. So, what concept do we need for a negative subset?” Rather clearly, then I asked for concepts other than mine. In other words, I did NOT insist or imply that my concepts came as the only possible answer.
Rather clearly, the concept of the set of integers comes as a formal concept of mathematics. It doesn’t belong to examples when talking about four cookies. Now, perhaps there exists some formal concept of integers which does produce partitions of integers whose union yields the original integer. I’d like to see it. But, for integers *as normally defined* a division of 4 into 2 yields 2. The *union* of 2 and 2 yields 2 and NOT 4. Consequently, *as regularly understood in mathematics… not necessarily in everyday life or even in practical mathematical thinking… but in technical mathematics where we proceed by definition and logic*, division does NOT qualify as equivalent with partitioning. It simply does NOT hold for the integers, and you have yet to provide evidence otherwise or *shown* my arguments as irrelevant.
You just did exactly what I was talking about. You perform an operation that belongs to one level (division) and then try to reverse with another operation that belongs to another level (union) and then claim that nonsense results.
Partitioning and unioning are opposites. Division and multiplication are opposites. ‘Splitting’ (a particular operation that ‘divides’ a number into multiple smaller number, such as splitting 4 by 2 yielding 2 and 2, as opposed to dividing 4 by 2 which only yields 2) and summation are opposites. If you mix these, you’ll only get sensical results by accident.
As in the last major discussion, it is also the case that quite often you can apply more than one operation to a particular object, but the results mean different things and may simply be nonsensical for some operations.
Frex, take the set-based Church numeral for two, {∅, {∅}}. I can partition this into two sets, ∅ and {∅}. These are the numbers 0 and 1, but partitioning is not a mathematical operation on Church numerals! Trying to say that 0+1≠2 or 0*1≠2 and thus partitioning doesn’t equal division doesn’t make any sense – of course it doesn’t, at least in that way! However, if you look at it on its own terms, then you see that we’ve gone from a set with two elements to two separate sets each with one element. This *does* make sense. You *do* see the mathematical analogy now. As well, you can now union them to produce a set with two elements again.
Of course, you may realize that this doesn’t actually require us to start with the number 2. Any set with two elements will work equally well, because we’re working purely with sets right now, not a higher level of numbers.
You see what was done there? This is why I talk about you mixing levels. This is the same problem you were running into last time. The set-based definition of integers does allow you to perform set-based operations, but it’ll only make sense in terms of sets, not numbers. You have to use arithmetical operations to have them work like numbers do.
Same thing here. Yes, a pointwise unionization of [.5, 1.5] with [.5, 1.5] yields [.5, 1.5]. But we’re not actually reversing the operation we used. We didn’t partition the intervals in that way. We split them. Thus, we must use summation (the opposite of splitting) to reverse it. If we had strictly partitioned [1, 3] we’d have [1,2] and [2,3].
Note that partitioning *does* make sense here in a similar way as the set one did – we started with an interval with a span of 2 and we end with two intervals with a span of 1. When we union them back together, we get an interval with span 2 again. It works! It just works on its own terms, on its own level. It’s only when you confuse the levels and lose sense of what you are actually dealing with that you see nonsense.
I hope the above was illuminating in this regard. When you’re dealing with partitioning/unioning on sets, frex, the fact that a set happens to correspond to a Church numeral is irrelevant. You’re not using the operations defined on Church numerals. Similarly with intervals. When you split an interval (using the definition of split I gave), it’s not required to give the same answer as partitioning it (though both may be valid operations on intervals). To expect otherwise would be exactly the same as expecting (3+2)/2 to equal 3 again, because you’re applying one operation and then applying the reverse of an analogous operation.
*A disclaimer: Though the set- and interval-based partitioning/unioning does operate like arithmetic when you are careful, it won’t always. This is fine, because they are analogous, not equivalent. Frex, unioning {∅} with {∅} yields only {∅}, which is equivalent to saying that 1+1=1. In sets, sometimes it does! Similarly, unioning [1, 3] with [2, 4] would only yield [1, 4], implying that 2+2=3. Again, in intervals it sometimes does! You can also get intervals that don’t directly correspond to numbers at all, like the union of [1, 3] and [8, 9]. Overall this is similar to other constructions of the integers like the surreals – if you don’t follow the arithmetic operations exactly you can still sometimes get something analogous, but other times you get something that isn’t an integer at all (like the surreal concept of games). Since unioning/partitioning sets and intervals isn’t actually a guaranteed arithmetical operation, you can’t depend on them to work absolutely.
Finally:
You seem to be deliberately misunderstanding the word ‘analogy’. When two things are analogous they are not equivalent. I’ve shown in this post and my prior how partitioning and dividing (and splitting, when appropriate) are analogous, but neither I nor Mark have ever said they were equivalent operations.
Xanthir, FCD
In your exchange with Doug you express my position succinctly. But like in the words of André van Meulebrouck, “He was wont to point out that you are more likely to reveal yourself as a foreigner when you overextend your limits by trying to use grammatical constructs and words you aren’t comfortable with, than when you speak simply but correctly”. The only exception I take with this pompous statement is that it’s implying that Academia’s tautology (temporal conventional wisdom) sets the standard of correct, but then what happens when that tautology itself is then based on egotism’s bias and also in error. Then whom is it in effect, that has overextended their limit?
As I’ve previously stated, Doug’s as also Einstein’s desire is and was, to unify the universe under the banner of a one, ‘all encompassing’ (whole and fractional) application of ‘applied mathematics’. This position does not allow for the separate while conjugated distinctions of a universe that’s built on the principles of antithetical duality, (man/woman), (space/time) and/or (accumulative/reductive) mathematics, seen as an ever expanding universe punctuated by ‘Black Hole’s’. Where in it’s effect creates a living multi-ported Kline Bottle, Mobius Band.
Maybe the real trepidation comes as expressed to me by the director of mathematics and computer science at a very prestigious university, where he stated, ” Ray, I’ve spent the bulk of my life to ascertain the level of Academic accomplishment I now enjoy, and then you come along with a logical application of mathematics that extrapolates dimensional complexity that may well be the world of tomorrow but not my world”.
“Kline Bottle” should be corrected to “Klein bottle” after Felix Klein [1849-1925]. As Eric Weisstein writes, online:
“German mathematician who began his career as Plücker’s assistant at Bonn. Klein studied analytic geometry, describing geometry as the study of properties of figures which remain invariant under a group of transformations. He systemized non-Euclidean geometry and wrote a book on the icosahedron in 1884. He also worked on the development of group theory and collaborated with Lie in Erlanger Programm. He also is known in topology for the one-sided Klein bottle. In addition to all his other work, he found time to write a classic history of mathematics.”
What does ray burchard mean by “living multi-ported Kline Bottle”? The Klein bottle has no “ports” nor holes, as opposed to our nordinary world where one can pour from a bottle of port.
Is the writer barred from naming the person or institution where the alleged statement was politely made (and apparently misinterpreted): “Ray, I’ve spent the bulk of my life to ascertain the level of Academic accomplishment I now enjoy, and then you come along with a logical application of mathematics that extrapolates dimensional complexity that may well be the world of tomorrow but not my world”.
The alleged statement by the anonymous expert seems odd to me. “to ascertain the level of Academic accomplishment I now enjoy” is not what I understand Math or Computer Science professors to do — unless in the sense of “we know how to grade students, and you are failing or unqualified to be even an undergraduate here.”
Similarly: “you come along with a logical application of mathematics” puzzles me, as I see a lot of passion and attempt to be historically or philosophically connected to work that Ray clearly fails to comprehend, but I’ve not seen much “logic” nor much “application.”
“extrapolates dimensional complexity” similarly may be what Ray THINKS he heard, but does not, to me, make sense either from n-dimensional geometry nor from complexity theory, UNLESS it relates to Michael Duff et al on Euler number versus topological complexity in the ensemble of known Calabi-Yau manifolds:
http://www.arxiv.org/abs/0706.3134
Further, “may well be the world of tomorrow but not my world” makes we wonder what “tomorrow” is invoked, and what world can be conceived in which Ray makes sense. There may be such a world, but I’ve seen no coherent clues as to what its laws or axioms may be.
As to Klein:
References
Fricke, R. and Klein, F. Vorlesungen über die Theorie der automorphen Functionen, 2 vols. Leipzig: B. G. Teubner, 1897-1912.
Klein, F. Arithmetic, Algebra, Analysis. New York: Dover, n.d.
Klein, F. Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis. New York: Dover.
Klein, F. Famous Problems of Elementary Geometry. New York: Chelsea, 1956.
Klein, F. Gesammelte Mathematische Abhandlungen. Berlin: Springer-Verlag, 1973.
Klein, F. Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree, 2nd and rev. ed. New York: Dover, 1956.
Klein, F. The Mathematical Theory of the Top. New York: Scribner’s, 1897.
Klein, F. and Sommerfeld, A. Über die Theorie des Kreisels, 4 vols. New York: Johnson, 1965.
Klein, F. Vorlesungen über die hypergeometrische Funktion. Berlin: J. Springer, 1933.
Klein, F. Vorlesungen über die Theorie der elliptischen Modulfunctionen, 2 vols. Leipzig: B. G. Teubner, 1890-92.
Jonathan Vos Post,
Your posts have a consistent theme, that is as a pedantic challenge from the scribe or historian’s perspective, devoid of conceptual substance. This mentality (a propensity to indefinitely defer the courage of one’s conviction, that as relegated to the rest of society) is indicative of the commercialization of Academia. Where the sale of degrees are then based on one’s ability to read, write and obsequiously parrot the preconceived. This is also the reason that theory in conjunction with practical experience as a whole (empirical laws) govern research and thereby the advancement of scientific knowledge and not just by theory alone. ‘The whole having a greater value than the sum of it’s parts’.
Then as to Kline versus Klein, I stand corrected. And as to my use of multi-port, would multi-nexus correct your confusion? But you will have to take the issue of “bottle” up with the assigner of names.
Could a pedantic scholarship also be the telegraphing of one’s own cognitive limitation as exampled by,
“What does ray burchard mean by “living multi-ported Kline Bottle”? ,
That would be liken to the primogenitor’s antithetical duality, as in the union of (accumulative/reductive), which characterizes the human brain’s matrix as a progeny by convention. Where one hemisphere’s impetus is designed to spatially orientate senses stimuli and thereby create the geometric whole of one’s interpreted reality, conjugated through the corpus callosum with another hemisphere’s impetus (solipsism) designed to then deconstruct an antithetical duplicate of the same geometric whole of one’s interpreted reality (see Alien Hand Syndrome) to then afford the individual the ability to negotiate life’s pleasures and perils. Or like the dichotomy of whole number versus fractional number resolve as demonstrated by the failure of Fermat’s ‘Last Theorem’, where a formula designed to express duality works for any fractional number up to but not including the whole of 2.
is not what I understand Math or Computer Science professors to do — unless in the sense of “we know how to grade students, and you are failing or unqualified to be even an undergraduate here.”
Right, and this same collective pedantic Academic tutelage has taken America’s children’s math and science proficiency and American leadership right to the top of world standing. That is if the innovations demonstrated by “Star Search” and corporate greed are the standards.
“The alleged statement by the anonymous expert seems odd to me”
Here we agree, and if it’s certification you require, you have my E-Mail address. Some how I believe you Jonathan Vos Post lack the courage to follow up.
Ray:
Tone it down. I don’t approve of that kind of petty insult in the comments here. Consider this your warning.
Xanthir,
[You perform an operation that belongs to one level (division) and then try to reverse with another operation that belongs to another level (union) and then claim that nonsense results.]
If the division=partition hypothesis holds, I don’t see how they belong to different levels. Partitioning of a set and unionionzation of pairwise disjoint subsets work as inverse operations, in that if we partition a set A into pairwise disjoint subsets, and then unionize those pairwise disojint subsets we have the original set. I don’t see how you would claim this as different levels of operations. If you claim this as different levels of operations, can you elucidate how they work on different levels?
[Partitioning and unioning are opposites. Division and multiplication are opposites.]
Under certain conditions yes, but not under other conditions. A union of overlapping subsets does not qualify as the opposite of a partition. Division does not qualify as the inverse of multiplication for interval numbers.
[These are the numbers 0 and 1, but partitioning is not a mathematical operation on Church numerals!]
Good point… partitioning does not qualify as an operation. Consequently, there exists a dissimilarity between division and partitioning.
[However, if you look at it on its own terms, then you see that we’ve gone from a set with two elements to two separate sets each with one element. This *does* make sense. You *do* see the mathematical analogy now.]
No, I don’t. I see that you start with 2, then you have 1 and 1. 2/2=1… *only* one 1… we need two 1s. A better analogy would consist of starting with 2 objects in a case, taking away one object and putting it in a separate case. Then we have one object in our separate case and one object in our original case. So, now let x0 indicate the number of objects in a case originally, y the number of objects changed for a case, and xf the number of objects in the final case. Then, for the original case we have x0=2, y=-1, and xf=1 or 2-1=1 objects, and for the separate case x0=0, y=1, xf=1 or 0+1=1 objects. This involves NO division whatsoever, only subtraction and addition.
[You see what was done there?]
I certainly do NOT see division as an operation at work.
[We didn’t partition the intervals in that way. We split them.]
You did *divide* them. You did NOT partition them, as you’ve said. Thus, partitioning does not equal division for the domain discussed, as well as for set-theoretic numbers.
[If we had strictly partitioned [1, 3] we’d have [1,2] and [2,3].]
No, you wouldn’t. You don’t have pairwise disjoint subsets. We could have [1, 2] and (2, 3] or [1, 2) and [2, 3], but we simply won’t have [1, 2] and [2, 3] for then the sets overlap at 2.
[Note that partitioning *does* make sense here in a similar way as the set one did – we started with an interval with a span of 2 and we end with two intervals with a span of 1.]
Again, 2-1=1, and 0+1=1 for our intervals. Then, on recombination we have 1+1=2.
[Similarly, unioning [1, 3] with [2, 4] would only yield [1, 4], implying that 2+2=3. Again, in intervals it sometimes does!]
Honestly, I don’t know how you expect to pull this one off, unless you know a lot less math than I previously thought you did. Look, you’ve basically measured length here. As a student of elementary probability or measure theory might tell you to take the length of two sets a, b we apply the formula l(a)+l(b)-l(ab), where l(a) indicates the length of a, l(b) indicates the length of b, and l(ab) indicates the length where the sets a and b overlap. For an interval [a, b] its length equals b-a. So, [1, 3] has length 2 as does [2, 4]. Now, [1, 3] and [2, 4] overlap on [2, 3]. Consequently l(ab)=1. So, for l(a)+l(b)-l(ab)… the union of [1, 3] and [2, 4] we have 2+2-1=3. This does NOT involve a partition, though, since a partition always has pariwise disjoint subsets *always* has l(ab)=0, or more generaly m(abc…z)=0.
[You can also get intervals that don’t directly correspond to numbers at all, like the union of [1, 3] and [8, 9].]
We can also unionize numbers like 3 and 5, and get {3, 5} which doesn’t correspond to a number at all. So what?
[Since unioning/partitioning sets and intervals isn’t actually a guaranteed arithmetical operation, you can’t depend on them to work absolutely.]
Again, so what?
[I’ve shown in this post and my prior how partitioning and dividing (and splitting, when appropriate) are analogous, but neither I nor Mark have ever said they were equivalent operations.]
Mark wrote: “So what is division all about?” He didn’t immediately answer his question. But, later he wrote “So what happens when we divide a group by one of its normal subgroups? We partition the group into a new group, where the elements of the new group are formed from subsets of the elements of the original group. It’s the same idea as simple integer division described up above, except that we want to preserve the group structure, so the result is going to be a group.” And he placed an emphasis on ‘partition’. This comes as the most prominent part of the post, so until shown otherwise, I don’t see how you can think the post does NOT say that “division concerns partitioning.” I haven’t seen you or him interpret the post differently. Maybe I’ve misread the original post in some way, but I don’t see it at this point, so by all means clarify.
Lastly, his example of 50 objects does NOT fall within the division concept so far as I can see, since we have so many sets at work. We originally have 1 set with 50 objects. Then, we take 5 objects away and put it another set which previously had no objects. So, we have 50-5=45, and 0+5=5. Then we have 45-5=40, 5, and 0+5=5, and so on. I don’t see division in the mathematical sense here (although in the *linguistic* sense I do see division). I do see a partition.
Ray,
[As I’ve previously stated, Doug’s as also Einstein’s desire is and was, to unify the universe under the banner of a one, ‘all encompassing’ (whole and fractional) application of ‘applied mathematics’.]
I don’t maintain such a position. Please support what you say about other people with evidence.
Again, just to be clear, no one ever stated that division=partition. We’re talking analogy here. The two are analogous, not equal.
Now, as for the confused level? Look back up at what you actually did, at what I actually quoted. You divided 4 by 2 (this isn’t actually accurate, since you want to be exact about division producing only a single number as answer – you actually split 4 by 2, using the definition of splitting in my previous post). You then tried to union them. Division and union are not opposites any more than addition and division are opposites. The number 4 cannot be partitioned, so unioning doesn’t make sense. A set of four objects can be partitioned, and so unioning does make sense. However, the mathematical operation of division doesn’t make sense on a set of four objects. It is obvious, however, that the operations are analogous between the two things. This is how we teach division – partitioning a grouping of similar objects gives you an easy visual representation of what division means on numbers.
Honestly, this is sort of it. If you were never taught division this way (and I consider the possibility of this to be very near 0), then I could understand why you don’t see the analogy, but it’s very likely that you were. Do you find that example valid? Do you think it gets the concept of division across to children? If so, then it’s a good analogy pretty much by definition.
Division isn’t the inverse of multiplication when zero is involved, either. That’s not actually a problem. Not all operations are perfectly reversible. That’s doesn’t prevent you from having inverses.
I really don’t see what you could possibly mean by this. Of course partitioning doesn’t qualify as a mathematical operation over Church numerals – it’s explicitly not. No one ever said or assumed otherwise.
Second, of course there are dissimilarities between division and partition. Once again, the two operations are analogous, not identical. What matters is that the dissimilarities are small for our purposes. This one is, because we’re not restricting ourselves to talking about Church numerals.
Once again, no one has ever said the two are equal. It would be wrong and foolish to do so, and so no one has. Please stop implying that we are calling them equal. We are calling them analogous.
Division is repeated subtraction, just as multiplication is repeated addition. You were taught this as a child. It is not something profound.
In addition, I explicitly talked about an operation called splitting rather than division in order to head off this exact objection. Had I only talked about division, you’d have a (weak) point. But given what I actually said, you don’t, and this entire paragraph is putting words into my mouth for the purpose of refuting them.
As well, your analogy truly *is* identical to the situation I produced. ^_^ You start with one grouping of two objects, and end with two groupings of one object. The fact that a container is shared between the start and end is irrelevant, as nothing we are doing talks about containers or their properties, just the number of objects that are grouped within them.
I thank you for the correction. I will hesitantly thank you for realizing that this is irrelevant to anything that I had said.
I… understand how to measure the span of intervals. I did so in the paragraph you’re responding to, as I got the exact same answer as you did for the same reason. I’m not sure what the purpose of this response was.
And again, as should be obvious to anyone who knows you can’t divide by zero, not all operations are perfectly reversible. This doesn’t prevent them from having inverses.
The point of that paragraph was to clear up a possible confusion, and to emphasize yet again that I am not saying that division and partitioning are identical. They are analogous. I had hoped that pointing out examples where they *weren’t* identical would disabuse you of the notion that I thought they were.
(Emphasis mine.) Yes, division and partitioning are analogous. As I’ve said in every response to you in this thread. They are not identical.
Returning to my first point, how were you taught division in elementary school? This is exactly how division is taught – with an analogy from partitioning. Take a collection of objects of size x. Partition them into separate collections each of size y. The number of collections is z, where z=x/y.
It simply boggles the mind that you can claim this isn’t analogous to division when I would wager money that this is exactly how you were first taught division in grade school.
[The number 4 cannot be partitioned, so unioning doesn’t make sense.]
In formal set theory 4={0, 1, 2, 3}. We can partition {0, 1, 2, 3} into {0, 1}=2 and {2, 3}, with the union of 2 and {2, 3} equaling 4. So, unioining does make sense.
[However, the mathematical operation of division doesn’t make sense on a set of four objects.]
Do you mean to say that division can apply to *elements of* sets, but *not sets* themselves? That works as all well and fine to say, but it simply doesn’t hold in formal set theory, since division does work as a permissible operation in formal set theory on numbers, and numbers in formal set theory get classified as sets.
[This is how we teach division – partitioning a grouping of similar objects gives you an easy visual representation of what division means on numbers.]
True enough, but that teaching consits of informal mathematics which doesn’t strictly use definitions and logic.
[Do you find that example valid? Do you think it gets the concept of division across to children?]
I don’t know if it really does get the concept of division across to children all that well. I think it can help, so it does have value. I don’t necessarily see such an example as valid with formal set theory though.
[Division isn’t the inverse of multiplication when zero is involved, either. That’s not actually a problem. Not all operations are perfectly reversible. That’s doesn’t prevent you from having inverses.]
True enough, but it *does* mean that you have inverses *only* for particular conditions. For normal arithmetic multiplication and division only qualify as inverse operations on a set *when* 0 does not belong to the set.
[Once again, the two operations are analogous, not identical.]
Alright, but *where* do the similarities come from? If we don’t have similarities from formal definitions or logic, then we have *some* evidence that the analogy originates from our concepts or philosophy instead of the mathematics involved.
[Division is repeated subtraction, just as multiplication is repeated addition. You were taught this as a child. It is not something profound.]
We only got taught part of the story as children, as we did NOT get taught the conditions where division qualifies as repeated subtraction, and multiplication qualifies as repeated addition. We pretty much learned this later once we learned about rational numbers, real numbers, complex numbers, etc. Second, conceptually they DO work as different. Repeated subtraction consists of an operation applied many times. Division consists of an operation applied once.
[But given what I actually said, you don’t, and this entire paragraph is putting words into my mouth for the purpose of refuting them.]
“Splitting” works more like repeated subtraction in that it consists of a longer process involved, than division which happens in the time it takes to do one operation. The paragraph involving the 2-1=1 type equations helps to indicate that.
[As well, your analogy truly *is* identical to the situation I produced.]
I disagree. Thinking of it as repeated subtraction instead of divison makes it so that the process takes longer for the person or computer that does the work.
[I will hesitantly thank you for realizing that this is irrelevant to anything that I had said.]
Yeah… the ([1, 2], [2, 3]) not qualifying as a partition, does come as irrelvant.
[I did so in the paragraph you’re responding to, as I got the exact same answer as you did for the same reason. I’m not sure what the purpose of this response was.]
What you wrote argued that 2+2=3 for intervals. It doesn’t.
[And again, as should be obvious to anyone who knows you can’t divide by zero, not all operations are perfectly reversible. This doesn’t prevent them from having inverses.]
It does prevent division qualifying as an inverse of multiplication on a subset of the reals like
{0, .25, .5, 2, 4} since .5*0=0, and 0/0 does not exist (or we don’t define it), or {-1, 0, 1}. So, it does prevent *some operations on some sets from having inverses*. Of course, without 0, the aforementioned and similar sets do have inverse operations of multiplication and division.
[(Emphasis mine.) Yes, division and partitioning are analogous.]
As you emphasized, Mark calls them “the same idea”. The term “the same”, usually, indicates *identity*. So, although you may not have claimed division and partitioning as identical, Mark, either intentionally or through his diction, indicated partiontioning and division as identical.
[Returning to my first point, how were you taught division in elementary school?]
I don’t recall actually, but how I got taught such doesn’t indicate anything about more formal sorts of maths. I more remember learning division as the inverse of multiplication, as in “6/2=?.” I more remember thinking “well, what number times 2 equals 6” more than anything else. I don’t know how I actually learned.
[it simply boggles the mind that you can claim this isn’t analogous to division when I would wager money that this is exactly how you were first taught division in grade school.]
Well perhaps it boggles your mind. And maybe I got “taught” such in grade school. But, I more remember thinking of 81/3 as “3 times what equals 81?” than of 81 objects getting split into 3 equal parts of some ungiven size. Perhaps my memory works as overly-selective here and I remember poorly. I don’t know. It doesn’t matter. One can still learn division on non-zero rational numbers without the concept of partitioning and learn it quite well just as the inverse of multiplication. Or one can learn it well enough without the concept of it working as the inverse of multiplication, and just learn it from a table with a knowledge how positional notation works.
Doug Spoonwood / Xanthir, FCD,
“Ray,
[As I’ve previously stated, Doug’s as also Einstein’s desire is and was, to unify the universe under the banner of a one, ‘all encompassing’ (whole and fractional) application of ‘applied mathematics’.]
I don’t maintain such a position. Please support what you say about other people with evidence.”
Fair enough Doug. Both yourself and Xanthir, FCD, eloquently state your positions and you both are correct. It’s the age old argument of, form versus substance.
As a segue, does not the sequencing in my post #30 and the sequencing of 01, 10, 100, 1000 etc… both example an analogy in partitioning, one as to the dynamic ‘whole’ and the other to a specific segment of that same dynamic ‘whole’. And isn’t this also analogous to the brains partitioning where one hemisphere is designed to create the geometric ‘whole’ of an interpreted reality, while conjugated with another hemisphere designed to create another sub-geometric ‘whole’ within the original geometric ‘whole’. Then isn’t the matrix of this process characterized in man’s progeny by convention, a number system where we have one system, 0 thru 9 as ten digits while nine values encompassing another sub-system of, 1 thru 9 where we have nine digits and nine values. Now if memory isn’t added, everything would constitute a one time occurrence as pi does. Then to add memory as a systematic symmetry (balance) to the equation and this is accomplished by inverse iteration. 0123456789 plus 9876543210 equals (09+18+27+36+45) = 135 plus (54+63+72+81+90) = 360 then (135+360) = 495 and 123456789 plus 987654321 equals (19+28+37+46+ 50) = 180 plus (05+64+73+82+91) = 315 and (180+315) = 495. Then (495+495) = 990 and therefore (099+990) = 1089 the Mobius Band and homoscedastic coupling as quadratic concentric antithetic duality (the double helix).
Mark,
With this being your site, I accept your admonishment.
Then as to a clarification to myself and as insight to others, just what kind of “petty insult in the comments” do you approve of, those whose base is in innuendo #34 as opposed to those based in fact #35.