Thinking

Analytic Geometry Is The Ballet Of Thinking.



KM010207

Six Alone In - Karl Kempton

Here is a new'Mathematical Visual Poem' by Karl Kempton - first published in Turkey here

Rebuttal On The Delineations Of Math And Art

Recently I discovered that Peter Turney wrote comments to my "Delineations between the aesthetics of Mathematics and Art" and he posted them on his blog. I have copied them and wrote a comment for each of his points. I have listed them below with my text being green, his text being blue and the delineations being black.

Math and Art: Differences and Similarities
Posted on May 8, 2009 by Peter Turney
Mariana Soffer has made a list of some differences between math and art. In a contrarian mood, I will go through the points in this list and discuss the similaritiesbetween math and art.

Hi Peter,
Thanks for addressing these delineations on math and art - The main reason I made them is due to the post modern deterioration of the sovereignty of art and the ramifications of the idea that aesthetics equals art. In addition I have found a plethora of talk about the similarities between math and art however, most of it I find ill-conceived and based on the aesthetics of math and not the aesthetics of art. I also believe that this was the main fallacy of George Birkoff and his view of aesthetics as well.

Note: The original source for the following twelve quotations is Kaz Maslanka, Delineations Between Aesthetics of Math and Art. Kaz citesProceedings of the 2002 Bridges Conference on Mathematical Connections in Art, Music, and Science, page 256. (Note added December 5, 2009.)

Difference #1: Mathematical truths are discovered. Artistic truths are mediated.

The nature of truth in math is a difficult philosophical problem. Truth in art is perhaps even more problematic. But one lesson we have learned from Doug Lenat’s AM(Automated Mathematician) is that interestingness is arguably more important than truth. It is easy to write a program that generates an endless stream of mathematical truths (1+1 = 2, 1+2 = 3, 1+3 = 4, …); it is much harder to write a program that generates an endless stream of interesting mathematical truths. In this respect, art is much like math: It is much harder to make interesting art than to make true art. In both art and math, truth is (arguably) required for interestingness, but interestingness is more interesting than truth. (Computers can generate art, but is it interesting art?)
It might be said that math is discovered, whereas art is created, but discovery and creation are both aspects of evolution. Mathematical knowledge evolves. Artistic techniques and methods evolve. In both cases, differential fitness is determined by the degree of interestingness.

-I cannot directly speak to Doug Lenat's Automated mathematician for I am not intimate with it but from what I gather from your link it seems you are confusing the aesthetics of math and the aesthetics of art as well. They are two completely different things.
- I find your comments very interesting and I will agree that interestingness is very important however I find it subservient to truths, for if something is not true it will not be interesting no matter how many variations are created. But more importantly, I don't find it very relevant to the original statement. What I am trying to point at is the process of these truths not an aesthetic judgment of them.
-You mentioned, "that it is much harder to make interesting art than to make true art" I find this statement also to diverge from the topic but again more importantly "true art" does not exist due to no one being able to axiomatically define it. Although I will admit that art really needs to be axiomatically defined for now we are under the guise of the vague postmodernist definitions which cling to the flotsam and jetsam created by the shards of the modernist explosion. Not only is math considered, art but accounting, plumbing and auto mechanics are art as well.
-I really need to go back and change the wording of this statement to say "The vast majority of mathematics is discovered instead of implying that all of math is discovered for I believe the initial mathematical axioms are done through a creative metaphorical process however from that point forward the vast new computational concepts are discovered.
-degrees of interestingness are always relative and rely on ones need. For value is always proportional to need. I will admit that differential fitness provides more variety to satisfy ones needs. However, the problem is that we all possess different needs. Which brings us back to the original idea that the veracity of the art must be present to satisfy the needs.


Difference #2: Mathematicians generally agree on what is mathematically correct. Artists generally have no idea what is artistically correct.

The first difference concerns the origins of math and art (where does truth come from?). The second difference concerns validating math and art, after the act of discovery or creation is complete (is it really true?). There is more consensus about truth in math than about truth in art, but, again, truth is relatively trivial, in contrast withinterestingness. Arguably, the level of agreement among mathematicians about what is interesting in math is similar to the level of agreement among artists about what is interesting in art.

Mathematics cannot operate without rigorous definitions to validate their truths and art could care less if there is a any 'definition' of truth present or not (the key word is definition). Interestingness is beside the point as well as being subservient to truth. I cannot speak for mathematicians however, and unfortuneatly, artists cannot even determine "what is art" and what is not. Again I say, with the advent of modernism and the post modern validation that "everything is art" the art world has been turned upside down and value has been place in the hands of marketers (galleries) as opposed to the art aestheticians, critics and scholars. I can only see this being a problem that math will never face.

Difference #3: Math illuminates the supportive skeletal structure of thought whereas Art illuminates the metaphoric wind, which blows through that structure.

Mathematics is heavily metaphorical. This is the lesson of Where Mathematics Comes From (Lakoff and Núñez). Art and math are both based on analogy-making. Meaning (semantics) in both math and art is based on analogy. There is an illusion that math is purely structural, that the interpretation of math is outside of math itself, but this is only an illusion. Math without interpretation is not interesting. Mathematicians, when actually doing math, are always working with interpretations, assigning meanings to the symbols. The formalist view of math misses completely the key role of metaphor in the human enterprise of discovering (creating, evolving)interesting mathematical truths.

I am not a mathematical Platonist and while I agree that both fields are metaphoric, the use of metaphor is quite different. Analogies in math seem to be less problematic if they possess a high degree of relational similarity yet poetry works best if it possesses a low degree of relational similarity yet still makes some sort of intuitive sense. The point I am trying to make is that structure can be seen better when there is a high degree of relational similarity.

Difference #4: Science reveals the body of “God” and Art reveals “God’s” mind — or is it the converse?

Math is grounded in perception (Where Mathematics Comes From), just as art is grounded in perception:
One of the great findings of cognitive science is that our ideas are shaped by our bodily experiences — not in any simpleminded one-to-one way but indirectly, through the grounding of our entire conceptual system in everyday life. The cognitive perspective forces us to ask, Is the system of mathematical ideas also grounded indirectly in bodily experiences? And if so, exactly how? — Preface of Where Mathematics Comes From
If you insist on the body-mind duality, then art and math are equally of the body or of the mind.

Originally I stated that science reveals the body of GGod and art GGods mind.

-I really need to go back and change this delineation to the original that I had published earlier which excluded the clause "or is it the converse"
-The point I am trying to make is that a body's structure is very apparent where the structure of the mind is still a mystery.


Difference #5: Pure Mathematics has no expression for metaphor however; it does provide us a structure that can be used for it.

Formal mathematics separates the symbolic structure of math from the interpretation of math, but the two really belong together. Math can only be interesting when it is interpreted.

-I have changed my delineation to read 'poetic metaphor' as opposed to solely 'metaphor'.
I see poetic metaphors pointing at the amorphous as opposed to mathematical metaphors which point at analogy.

Difference #6: In general, the mathematician is not interested in finding truths through nonsense as opposed to the artist who is

.
Many mathematical discoveries were made by asking questions that seemed nonsensical at the time. For example, what if the parallel postulate were false?

I see there being a big difference between the concepts of 'nonsensical' and 'false'. The idea of false presupposes logic to be involved in the discourse and nonsensical discourse avoids logic. I have never seen nonsensical pure mathematics.

Difference #7: The goal of art is to go beyond language. Mathematics is a language to describe what is beyond us.

Art is a form of communication between the artist and the audience. Creative art pushes the boundaries of that communication and extends the language of art. Creative math extends the language of mathematics. In both cases, language evolves, communication evolves, new metaphors evolve (are created, are discovered).

-I don't see art being a language I see art as something that uses languages. Math is to 'applied mathematics' as 'language' is to art. Much of art tries to convey nothing and some art's intention is to destroy itself. Great art transcends language.

Difference #8: Artists have an insouciant tendency to get lost in their imagination. Mathematicians have an attentive tendency to map their imagination.

Mathematicians get lost in their imagination. Artists map their imagination.

Now THAT is funny. --My point here is that the result of mathematics has a starting point and an ending point however, artist tend not to care where things start, end or whether it even makes any sense. If an artist second guesses the logical value of his/her work then it will never get done.

Difference #9: A mathematical theory seems to come in a flash of intuition before the final product is rigorously constructed. An artistic theory seems to come much after the artwork that has been constructed in a flash of intuition.

In both cases, something rough, incomplete, and vague becomes smoother, more complete, better understood over time. Both math and art evolve. The apparent difference here is perhaps due to the ambiguity of the word theory. A closer examination of what is meant by theory may show that there is little difference between math and art in this respect.

Theory is the key word here and there is much difference. Art theory is concerned purely with aesthetics. Mathematical theories are not created for aesthetic purposes alone, if at all. Math theories have axioms or postulates; art theories do not furthermore, few would agree even if they did. Mathematicians create mathematical theories. Artists generally do not concern themselves with creating art theory for art theories are generally created by aestheticians. The artwork is done first then the theory comes later.

Difference #10: Mathematical creations are not unique in the sense that they could be discovered by anyone. Artistic creations are uniquely invented by individuals.

Artistic creations are no more unique than mathematical discoveries. This difference is the myth of the hero.

I looked at your examples and I think you may be confusing independent discovery with plagiarism. Hollywood is rife with marketers trying to guess what will sell and are in direct communication with directors and writers. The 'commercial arts' are completely different from 'studio arts' for it is like confusing science with engineering ... However the biggest problem in your example is that these ideas are based on a single culture. This example doesn't float across cultures. Try matching something across different cultures like Pascal's triangle which shows up in France with Pascal as well as in Iran with Omar Khayyam and China Yang Hui Please show me some art work that shows up in France Iran and China.

Difference #11: Mathematics, among other things, is a language. Art, among other things, uses language.

The symbolic system of math is a tool for expressing metaphors. The heart of math is the metaphors. Art is the same in this respect.

As I said before Art is not a language art uses language. It is like saying that physics is the same thing as mathematics and we know it is not.


Difference #12: In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in poetry, it’s the exact opposite. —Paul Dirac

Poetry can tell us new things, to the same degree that science and math can tell us new things. In both cases, we can learn new metaphors, new analogies, gaining a new perspective on the world.

While what you say here seems true it doesn't really address Dirac's quote.

And to continue the list:

#13
I just recently posted on my blog that Art is the expression of Culture and pure mathematics transcends culture therefore, cultureless
below are the following comments associated with the blog post:

apperceptual said...
Mathematics does not transcend culture. The development of math is driven by human interests. There are fashions in math (search Google for "fashions in mathematics"), as human interests change over time.

You might agree that interests change, yet claim that the truth of a mathematical proposition transcends culture, but consider that, for example, Intuitionist mathematicians reject the law of the excluded middle. As Lakoff and Nunez argue, math is a product of human experience, based on living in bodies, living in the world. Math does not transcend humanity; rather, it is saturated with humanity.
Wednesday, December 30, 2009 7:19:00 PM

Kaz Maslanka said...
Thanks for your comment Peter.
The Key here is "Pure Mathematics" and "Culture". Of course cultures use mathematics however that concept is in the realm of applied mathematics. There are many examples in design such as Celtic weaves and Islamic star patterns which server as an example. What I am talking about is when people think of zero or the decimal system they do not think of the Indian Culture unless they learn that the Hindus created it. The is nothing English or German about Calculus it could have easily been invented by the Chinese. Pascal's triangle is not a product of the French, Iranian nor Chinese Culture. Sure that culture may have some effect on their thought processes however the end result is the same. I will agree that there are mathematical trends and fads within certain groups of people however these are 'people' not 'cultures' working on these ideas. I am not a mathematical Platonist and I am not saying that mathematics exists separate of people, yes humanity creates mathematics however culture is a subset of humanity not the other way around. There is nothing personal about mathematics that is why one persons fractal 'art' looks just like every other persons work. Sure there are some minor differences between fractal 'art' but these are not mathematical differences, they are artistic differences and not very notable ones at that.
Individuals are not culture
P.S.
I will be addressing your other comments to the delineations very soon ...Thanks Peter for your dialogue.
Wednesday, December 30, 2009 10:35:00 PM

apperceptual said...
What I am talking about is when people think of zero or the decimal system they do not think of the Indian Culture unless they learn that the Hindus created it.

Math may transcend any specific culture (e.g., Indian culture), but that doesn't mean that it transcends all human culture. Math is a very human enterprise.

One might argue that music varies from one specific culture to another, yet most cultures have some kind of music. How is cultural variation in math different from cultural variation in music? I hypothesize that there isn't much difference here.
Thursday, December 31, 2009 5:34:00 AM

Kaz Maslanka said...
Culture is not a subset of Culture it is a subset of humanity. If you were to say all of humanity is a culture then the meaning of the word ceases to exist and there would be no reason to use it.
No one is arguing that math is not a human enterprise as I said I am not a mathematical Platonist.
Apperceptual said, "How is cultural variation in math different from cultural variation in music? I hypothesize that there isn't much difference here."
I have to say that there is a very significant difference. Cultural expression is not only about variation. It is about concepts that have similar relationships to each other AND about a specific group of people. Mathematics never expresses relationships 'about' a group of people.
It is interesting to note that some may think that the music in all cultures share a common 'beat' or pulse of time yet that idea is even problematic due to the way cultures view time. The time in African music is like a metronome whereas within classical European music the pulse fluctuates and cannot be tracked by a metronome.

Introduction to Apperceptual

"Apperceptual" is an extremely interesting blog about semantics and language run by Peter Turney and is well worth your time looking at it. Peter and I have differing opinions on art and culture however his topics of interest are fascinating and well worth reading. I will say that I imagine that his interests are more scientific than most poets aspire to however, we can learn a lot from him.

Please check out these links:
This one really helps validate the importance of "Proportional Poems"
This one shows how we can create "Golden" "Proportional Poems"
This one discusses criticism of my favorite cognitive scientist George Lakoff.

A math art moment #13

Art is the expression of culture.
Pure mathematics is independent of culture therefore, cultureless.



To see more delineations click here

Salvation

Here is the full image of my piece, 'Salvation'. The two houses you see in the image are bath houses just outside the temple bridge at Songgwangsa temple in Korea. These bath houses are used to bathe the ghosts of our ancestors as a requirement before they are allowed into the temple. A detail of the Proportional Poem is below.

Bathing Ghosts

Here is a Proportional Poem titled "Salvation". This was inspired by my recent visit to the Korean Zen Temple Songgwangsa

The image above is a detail from the image below

New Proportional Poem Blog

I have created another blog to collect Proportional Poems made by you. Proportional Poems are probably the easiest mathematical poem to make because you don’t have to be a math person to make one. Check out this link for an understanding of Proportional Poems and check out this link for the blog.
Please contribute!

Memories by Charlotte Whatley

Here is a Proportional Poem by Charlotte Whatley

Gratitude by Jean M Kelley

Here is a proportional poem by Jean M Kelley

Rhythm of Structure

John Sims has been putting together a series of mathart events in NYC which will occur at different times throughout the year. I am looking forward to an event later this next summer for which Richard Kostelanetz, Gregory Vincent St. Thomasino, Bob Grumman, Stephane Strickland and myself will be involved. To get on Johns Mailing list - contact him @ RhythmOfStructure@gmail.com

Whispers

Here is a new proportional poem that I created titled Whispers.

Place Value Poems by Gregory Vincent St. Thomasino

Professor St. Thomasino has developed a new structure for Mathematical poetry that I will add to the taxonomy in the side bar of this blog. What he has done is mapped the decimal notation system “onto” a sentence or set of phrases to “Place value” on the phrases. Focusing attention to the syntax of the poem. Here is an example of one of his poems called “Molotov’s Sister”:

a blonde bomber,she.smokes filterless,plays upright bass & writes haiku

Notice the decimal point and the commas. The commas delineate the digit/phrases in the poem and the decimal denotes where the decimal exists in this number. In essence we have the set of phrases that would equate to the following 100 x a blonde bomber, 10 x she . 1/10 x smokes filterless, 1/100 times plays upright bass & writes haiku

I have created a visual counter-part to the poem so that you can see the dynamic range of meaning mapped to each phrase. (images above)

The first image gives you the size differences in each decimal place and the second image groups the poem in detail so that it is readable.

Clowning Around With Euler

Euler's Formula

Craig Damrauer

I just ran across a new math poet who seems to be getting some good attention and has an interview featured here. Much of Craig Damrauer’s work seems to be inspired by his relationship to his family and his surroundings. I find a lot of his stuff to be whimsical yet there are a few that are philosophical. Most of the work is arithmetic yet there are a few orthogonal space poems which would fit under the category of algebraic pieces. (This reminds me I need to put a category in my taxonomy for arithmetic poetry)
I have some examples below of his work:

I think my favorite is the one below.


some exponents for your consumption


Here (below) is an orthogonal space poem

The next one reminds me of one of my pieces which equates value proportional to need.

400 bucks?




To see more of his work check out his site here at this link

Disappearing Context

If you are not familiar with "Similar Triangle Poems" please read this link before going further.


One of the things that excite me the most about mathematical poetry is the fact that one can mathematically merge poems into each other. The results of these operations are extremely interesting in how the context of the common variable disappears. Or in other words the common context that both poems share … disappears. This is a feature that no other poetic form can accomplish and we are going to accomplish it in this blog entry. One can perform this feat on multiple mathematical poems however we are going to show how it is done on just two. The first thing that one needs to have ready is at least two poems that share a common “variable” or “term.” In our example (above) we have the common context of “money”. In other words both mathematical poems share a common term in the form of a word, in this case money. In the first poem we have the idea that Man is to Blood as God is to Money and simultaneously we have the idea that Man is to God as Blood is to Money.** In addition we have the second poem which states that The Victor is to “Honor in War” as Money is to “Righteous Effort” And Simultaneously it says The Victor is to Money as “Honor in War” is to “Righteous Effort”

Now let’s solve both poems for the term “Money”

The image above shows both poems ‘solved’ for money. Since both poems are now in the form of being equal to money then we now must set both poems equal to each other. By setting them equal to each other we have merged the two poems together and everything is still logically intact. The image below shows both poems set equal to each other.



Now that we have the two poems merged into one let’s look at how the meaning has been changed by the reformation. Let us solve the new poem for the term “Honor in War” and see how it reads.



Wow! This poem reads right out of a Patriots Bible yet the two poems that created it were both cynical and possibly sarcastic in relation to the Patriot's beliefs. Once the context of money was taken out we have an entirely new situation. This reminds me of how a person can be consciously holding back a lie yet, speaks dancing truths all around the lie. In this case the money is the lie.

**Also an interesting feature of Mathematical Poetry is that all the different possible syntax structures in a poem exist at the same time therefore when you read a mathematical poem, in each of their different syntax states, the temporal meaning of the poem fills up much like a glass of water when you turn on the faucet.

Meeting Karl Kempton

I met the visual poet Karl Kempton face to face for the first time last weekend (he is on the right). My wife and I experienced a wonderful dinner/picnic with Karl and his wife on the shore of Pismo beach. He gave me a stack of his books and publications to read. It was a grand meeting however; it went by way too fast. Here is a shot (above) of Karl and I located in the beautiful garden designed by Ruth Kempton.
Here is a link to one of my favorite essays of his.

A man's intelligence

I once said that a man’s intelligence flows inversely proportional to the levels of alchol and testosterone in his system.

Just recently some scientific validation came in the form of this link

The 2005 poem below was inspired by the relationship expressed above.

Problems Encountered With Mathematical Poetry

Here is a perfect example of why mathematical poetry will have problems at least in the near future. In the blog entry copied bellow (and at this link) the author is complaining that the originator for a set of equations published in the brochure for the Biennale of Sydney is numerically illiterate. Curiously enough I think they should have said mathematically illiterate for numbers are not involved in these examples. That said, I am not sure that it is true that the originator was mathematically illiterate as well. The author claims that the expression, “Art = tyranny” is a false statement yet, historically there has been countless examples of art that has been inspired by or executed to express tyranny. A good example would be the artistic turmoil created around 1911 in Zurich Switzerland, for the entire Dada movement’s intention was to be tyrannical (Anti-Art)… as well as the copy cats that came after. Obviously the problem brought to question in this brochure is how one reads an equation. Is the equation to be read as poetry or science? Too many people think that an equation is automatically scientific in its expression. If this myopic attitude is left to continue, mathematics will be in denotative chains forever. When one reads poetry one searches for the proper context to give it meaning in relation to their life. One looks at the many facets of a poem to see what it is pointing to. “Art = tyranny” is a perfect expression for Dada, Punk or any other nihilist form or art.
Now to give our author the benefit of the doubt we could agree with him/her if the originator’s intention was scientific however, I can hardly see scientific intent in this expression even if it were meant to be.

It Don't Add Up
The Biennale of Sydney has put out an advance brochure that contains the two equations:

art + beauty + empathy = power

power – beauty – empathy = tyranny


Anyone with a primary school understanding of mathematics knows that if you solve these equations for the term ‘art’ you will discover that the claim being made is that

art = tyranny

If the Biennale of Sydney wants to communicate its ideas in the form of equations, perhaps it should run them by someone with a primary school level of numerical literacy before publishing them. Didn’t anyone in the accounts section twig that the Biennale of Sydney is spreading the message throughout the world that art equals tyranny???

Here is the mathematical solution:

art + beauty + empathy = power

art + empathy = power – beauty

art = power – beauty – empathy

given already that power – beauty – empathy = tyranny

then it follows

art = tyranny

Future Freedom

Here is an orthogonal space poem inspired by the mid east situation.