Saturday, January 02, 2010

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.

6 comments:

Anonymous said...

This is a discussion that has been begging to be had on this blog, and I'm glad it is being had. I think my disposition is more towards maths as language and all that it entails i.e. metaphor, structure, vocabulary, conjunctions etc. Some time ago, during a not dissimilar discussion with Kaz, I went into precisely "Delineations" like Turney has, and considered that the opposite of what Kaz had written was also "interesting" i.e.
Delineation #1: "Mathematical truths are discovered Artistic truths are mediated."
Artistic truths are discovered Mathematical truths are mediated.
Delineation#2: "Mathematicians generally agree on what is mathematically correct. Artists generally have no idea what is artistically correct."
Artists generally agree on what is artistically correct. Mathematicians generally have no idea what is mathematically correct.
Delineation#3 "Math illuminates the supportive skeletal structure of thought whereas Art illuminates the metaphoric wind, which blows through that structure. "
Art illuminates the supportive skeletal structure of thought whereas Mathematics illuminates the metaphoric wind, which blows through that structure.
Delineation#4 "Science reveals the body of GGod and Art reveals GGod's mind -- or is it the converse?"
Art reveals the body of GGod and Mathematics reveals GGod's mind -- or is it the converse?
Delineation#5 "Pure Mathematics has no expression for metaphor however; it does provide us a structure that can be used for it. "
Art has no expression for metaphor however; it does provide us a structure that can be used for it.
Delineation#6 "In general, the mathematician is not interested in finding truths through nonsense as opposed to the artist who is."
In general, the Artist is not interested in finding truths through nonsense as opposed to the Mathematician who is.
Delineation#7 "The goal of art is to go beyond language. Mathematics is a language to describe what is beyond us."
The goal of Mathematics is to go beyond language. Art is a language to describe what is beyond us.
Delineation#8. "Artists have an insouciant tendency to get lost in their imagination Mathematicians have an attentive tendency to map their imagination "
Mathematicians have an insouciant tendency to get lost in their imagination Artists have an attentive tendency to map their imagination
Delineation #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."
An Art theory seems to come in a flash of intuition before the final product is rigorously constructed. A Mathematical theory seems to come much after the work that has been constructed in a flash of intuition.
Delineation #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 not unique in the sense that they could be discovered by anyone.
Mathematical creations are uniquely invented by individuals.
Delineation #11 "Mathematics, among other things, is a language. Art, among other things, uses language."
Art, among other things, is a language. Mathematics among other things, uses language.

These "delineations" have to my mind more a sense of "demarcation", creating a "them and us" (whoever us is), and coupled with Bob Grumman's taxonomy of poetic variations, have the function of not only clarify what is and what isn't "mathematical poetry" but the function to actively "exclude" by pronouncement. I found and find this problematic. If mathematical poetry is to be taken seriously as Kaz is proposing then dividing by "zero" is forbodden, yet "we" practise it --- how come?
Love + anarchy
TT.O.

Kaz Maslanka said...

Hi Pioh,
To my understanding mathematics has axioms or rules that make it necessary to not divide by zero. It makes no sense to divide by zero in math because it is undefined and leads us to infinity. Art has no ultimate rules and no complete set of axioms. Art theory may define axioms however they are not called as such. Artists generally love to call something infinite even though the majority that I have met hasn’t the slightest idea of what it is and how to define it. Mathematicians know it well and even have developed ways to delineate a hierarchy of different infinities. (Cantor) To a mathematician the expression has to make sense logically or they throw it out and artists generally don’t care if something makes logical sense. So for an artist to map a metaphor by dividing by zero … well some would consider it ok … myself, I have grown to like using mathematical limits to approach infinity and metaphorically it works well for me. I don’t think in terms of dividing by zero anymore yet I admit that I used to. I think for mathematical poetry to be valid we must follow the rules of mathematics otherwise it will not be mathematical.

apperceptual said...

I agree with Anonymous: I don't like the exclusionary tone of these delineations.

There are axiomatic systems in which division by zero is allowed:

On Division by Zero

In math, if you want to do something that is forbidden by your axioms, then you can simply make some new axioms. Does that remind you of art? It should.

Kaz Maslanka said...

The last post was really not anonymous it was written by the mathematical poet who goes by pi (as in 3.141592…) and ‘O’ thus TT. O.
What I believe pi-oh is referring to as exclusionary, is not these delineations. I believe he is pointing out that we should not be hypocrites and say that we follow mathematical rules and yet divide by zero. He is also making reference to another mathematical poet name Bob Grumman who has defined mathematical poetry to possess a certain method and my work fits under that definition? Grumman is a lot more territorial about it than I am. To be honest I could care less what you call it – if you want, you can call it ‘equational poetry’ but it needs some name to distinguish it from other forms of poetics. Also it seems that TT.O. may feel that I am excluding him – I am not excluding him as a mathematical poet however, most of his work that I have seen doesn’t fit the definition of mine or Bob Grumman’s work.
A question for Pioh – Have you created a definition of what you do and published it somewhere?

As far as dividing by zero – it seems that Peter has given us the green light so go for it.

Anonymous said...

Dear Kaz,
thanxxxs for the opportunity to continue...
Before i do, i would like to say that i consider you blog, your poems, and the dialogue you sponsor incredibly important. If i hadn’t i would have moved away a long time ago. My problem is that i keep bumping into the limitations of the BLOG, and in a sense want to take that out of the “equation”, so that i (we) can concentrate on other issues within “mathematical poetry”. The problems are so immense that i keep bumping into unusual situations (as well as contradictions), in my own “writing”, and i notice them in your work also —- the dividing by “zero” issue is only symptomatic of much larger concerns. I suspect that as “mathematical poets” we may have to ultimately create specific ground rules, definitions, conventions, and/or a vocabulary of “mathematical functions”, terms, and practices. Not only for what may be permitted in “division” but for what differentiates “multiplication” from “addition” for example, and when or how one or the other can be used etc. I know you’ve (we’ve) had similar discussions in the past about that very issue. I in no way want or am proposing some kind of military solution to such a project, but just as various mathematicians over the centuries have adopted ways of “depicting” and solving problems, we may have to do similarly; perhaps leading by example is the way to go, but until enough practitioners appear this may take a long time to develop. Any way you cut the cake however i think the linguistic or metalinguistic solution is the only out. I subscribe to Turney’s and Lakoff’s mathematics-metaphor model, and have tried to follow Leibnitz’s and Godel’s collapse into linguistics to out the Devil! I think that “mathematical poetry” is large enough to include number poems, equation poems, or whatever else you may want to include —- the real danger is in the exclusion domain, that limits the pursuit, hamstrings the expression(s), and cripples the discipline. Jason Earls prime number poems are terribly exciting, but because they don’t have a mathematical operation appear to be excluded from the domain that you have created and opened up. I think it sad. I notice that fractal poetry is one of your joys on the blog, it doesn’t hold much interest for me, but i would be loathe to dismiss it, even tho its metaphorical substance is trivial and uninteresting to my way of “reading”. Numbers that “talk” interest me. Equations that “do” interest you i suppose. The modernist/post-modernist/most-modernist condition (or aesthetic) is pushing “mathematical poetry” way from its semantic/linguistic/metaphoric properties it would appear to me towards some kind of panacea —- i can only see it cramping its possibilities. Hope this finds you well
love + anarchy
TT.O.

Kaz Maslanka said...

Dear Pioh,
These things are important things to consider. I welcome any dialogue on it.
Thanks!
Kaz

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