## Wednesday, December 29, 2010

### New Singularity

This post is a new version of the proportional poem posted December 13, 2010.

k4ntico said...

Hi Kaz,

I am actually confused on whether you intend this form a=((b)d)/e to contrast or not with a=((d)b)/e; iow, is your multiplication commutative or not, do your parentheses carry meaning or are they superfluous ?

While seem to favor a positive answer (to the 1st question) the jpegs of your original explanatory post on proportional poems, the jpeg/poem in the post I am replying to here, on the contrary appears not to.

What motivates this question in turn is an upstream question on whether or not the natural action on "a is to b as d is to e" of the [/some??] symmetry [sub?]group of the square [/rectangle??] - whether this action puts it in an orbit good enough to faithfully represent the parallel listing of your own 4 variations on "a=((b)d)/e" ?

Cheers, k4

Kaz Maslanka said...

K4,
Yes, I try to follows the laws of mathematics. Yes multiplication is commutative. I have been known to make errors in the past. I am trying to find some error in what I have done ... what am I missing? d(e) is the same as e(d)-- I don't understand your concern.

Cheers,
Kaz

k4ntico said...

Kaz,

In short now the symmetric group on 4 letters will map a,b,d,e to any possible permutation of them and thus also act on either the form "a = (b)d/e" or the form "a is to b as c is to d" to produce all possible variants obtained by exchanging a,b,d,e in any way possible : there are 24 possibilities.

Embedded in this symmetric group are interesting subgroups that will map any such form to a restricted subset of variants. Among them are the symmetry group of the square, with 8 permutations, and the symmetry group of the rectangle, with 4 permutations, and the trivial group admitting only the identity. The images of any particular form by the permutations of one such group or subgroup is what's called its orbit.

I was trying to identify the subgroup - if any - applicable to your listing of 4 parallel "proportional poems", and this depends on whether each is meant to represent only itself or itself together with the one obtained by exchanging the two terms that are multiplied.

Note btw, multiplication in maths isn't generally commutative, it's only so in specific algebraic structures to which the terms you multiply don't particularly appear to belong. Also, your inexplicable (at best non-parcimonious) use of parentheses sheds doubt on what you exactly mean.

Now your assertion that multiplication should be commutative, translated back from the "proportional poem" form to the "metacomparison" form, implies that

"a is to b as d is to e" means the same as "a is to d as b is to e", and this makes a difference, since the first system of variants I would consider corresponds to the symmetries of the rectangle :

1. a is to b as d is to e
2. d is to e as a is to b
3. b is to a as e is to d
4. e is to d as b is to a

while commutativity of multiplication means this should be extended to the symmetries of the square by adjoining 4 further permutations.

5. a is to d as b is to e
6. d is to a as e is to b
7. b is to e as a is to d
8. e is to b as d is to a

Summing up, I'd say your style of proportional poems contrasts with what's more natural for the meta-comparison form, as I feel there is more difference between 1. and 5., say, than between 1. and any of 2. to 4., while what you appear to be saying is that the difference between 1. and 5. is so slight that it doesn't deserve mention before that between 1. and any of 2. to 4.

Cheers, k4

Kaz Maslanka said...

K4,
I am just now starting to understand what you are talking about. This last reply helped a lot. A few of my friends were having problems as well.
We are talking apples and oranges. You seem to be talking about all the permutations as a expression of inquiry. I am talking about “choosing” an expression that conveys what the artist/poet intends. All expressions of the permutations are interesting from an experimental point of view, however, I am interested in a direct expression – or at least one expression showing the four syntactically different variations. Not all of the variations are aesthetically equal.
Another disconnect that we are having is that my expressions ARE algebra. I am expressing value – so we have the value of apples are to the value of apple butter as the value of peanuts are to the value of peanut butter. These are both commutative and associative. Think of the terms having an invisible implied coefficient in front of them and the term only gives us the units. Of course like in all poetry the reader has to provide the meaning or in this case the values and units yet the term points you in the right direction you must come up with the values based on your experience.
What you seem to be doing is something different and not algebra ... A friend of my compared it to a matrix where the order of multiplication is important. I am not doing this. Furthermore my use of parentheses may be a little sloppy – I guess I should put parentheses on around every term. My only intent is to show algebraic multiplication and most people don’t have a problem once they understand what I am doing. I have to admit that on the mathematical poem “singularity” I should put parentheses around (a flower) – it would be much clearer.
Thanks!

k4ntico said...

Kaz,

Given the way you elsewhere cited wikipedia as a source of authority about "art", I feel it is legitimate next to what you write above, to first suggest you read the -introduction- of the english wikipedia entry for "algebra" to the top of the table of contents (if not further; digging a bit on Galois theory would be an idea given how it links group theory to elementary algebra equation solving; permutation groups where in fact first described to bring light over mysteries in the solving of elementary algebra equations; in whatever sense of "algebra" there is no more properly "algebraic" sort of object).

I wonder btw what's the fraction of the wikipedia entries for "algebra" in other languages that parallel the English's way with eg "elementary algebra". There is none of this in the French version, for instance.

Anyway and whatever, let's admit "algebra" means no more than manipulating or solving arithmetical expressions that involve variable symbols in addition to numerical constants and operators. One point that deserves to be made then, is that the most natural and uniform way to describe how the hierarchy of number fields and algebraic structures comes about, is to say it comes about when turning arithmetic expressions into "algebra" by introducing variables, and then solve for those variables.

This is clearer from examples.

Start with natural numbers. You can write 3+5=8 and things like that.

Then if you try to solve x+5=2 you end up introducing relative integers (that don't occur in the constants of the equation)

Then if you try to solve -3x=5 you need to introduce rational numbers (that don't occur in the constants of the equation)

Then if you try to solve x^2=2 you must complete the line with irrational numbers (ibidem)

Then if you want to solve x^2+1=-2 you must introduce so-called "complex" number (ibidem) etc.

The point is, as far as I understand the meaning of connotation, using unlikely variables in otherwise natural arithmetical expressions connotates that they may take their values outside of the number field for which the operations and constants in the expressions are already defined.

Cheers, k4

Kaz Maslanka said...

As with all poetry the meaning is made by the reader. You as the reader must make sense by keeping 'your understanding' of the mathematical poem within the values inside the number field. As with all poetry it is your job to supply the meaning. No two people read a poem the exact same way.