### A math art moment #13

**Art is the expression of culture.**

**Pure mathematics is independent of culture therefore, cultureless.**

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‘Mathematical poetry ‘is a blog that primarily concerns itself with promoting the use of mathematical equations as expressions for poetic metaphor. When these expressions use visual metaphor in conjunction with mathematic and lexical metaphor we classify this as the genre of mathematical visual poetry – Kaz Maslanka

## Wednesday, December 30, 2009

###
A math art moment #13

**Art is the expression of culture. **

**Pure mathematics is independent of culture therefore, cultureless.**

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## 25 comments:

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.

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.

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.

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.

European_music:African_music::Calculus:Sangaku

http://www.princeton.edu/main/news/archive/S15/04/04O77/index.xml

"It may be hard to conceive of math as religious art in our day. But equally difficult to imagine is that all this happened while Japan was utterly cut off from the rest of the world, and the country’s scholars were obliged to invent a home-grown branch of mathematics to help them. When Japan opened itself to the world again, the puzzles — and the geometry that helped solve them — fell nearly into oblivion."

http://www.sangaku.info/

"The problems featured on the sangaku are typical problems of japanese mathematics (wasan 和算) and often involve many circles which is uncommon in western mathematics."

http://en.wikipedia.org/wiki/Sangaku

"During this period Japan was completely isolated from the rest of the world so the tablets were created using Japanese mathematics, (wasan), not influenced by western mathematical thought."

Pure mathematics is cultureless. This is not an example of pure mathematics it is an example of applied mathematics. Sangaku is a perfect example of polyaesthetics where we have an aesthetic experience coming from both the aesthetics of math and art. One has to realize that the experience one receives from this work is a vector sum of the art aesthetic and the math aesthetic. Here again we have people (the authors) confused about the difference between art and mathematics. Just because the Japanese developed a mathematical application doesn’t mean that it is cultural mathematics. The math part of Sangaku is not cultural yet the art part is. In other words the math says nothing about the Japanese people yet the art speaks well about them. It is just like a geosynchronous satellite whereby, some people say that it is moving around the earth and others would say that it is stationary however, the most enlightened view of the matter is that the satellite is falling toward earth and moving around earth at the same time.

I don't accept this distinction between pure and applied math. The mathematician G.H. Hardy wrote, “I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world,” but he was wrong: his work has been applied widely in physics and cryptanalysis. Give me an example of math with no applications.

Dear Peter,

I think you are pretty isolated in your disbelief of "Pure Mathematics". Of course they are related to each other however, Hardy’s apology makes a great point in his intentions.

Hardy's mistake is that he should have said, "I have never 'intended' to do anything useful."

I don’t need to tell you; for you know that pure math is the formal language used for applications. I am surprised that you deny its existence. One certainly can have pure mathematical theory followed by a period of time with no application. Penrose tiles were around quite a while before quasi-metallic alloys were discovered … the list is long for this phenomenon.

You seem to be saying that whether math is pure or applied is not a property of the math itself; it is rather a relationship between the given math and a given culture, which may or may not have found a way to apply the given math. But if the very definition of pure math depends on culture, then how can it be that pure math transcends culture?

I believe that a culture produces math because it fills some role in that culture, although perhaps sometimes the role is only aesthetic appreciation. Thus math is always applied.

Perhaps you are saying that, if we could cut math free from the role that it plays in a culture, if we could sever the given math from the applications that tie it to a given culture, then that free, untied math would transcend culture. But I could say the same thing about music or art.

The mathematician Ray Balbes made a very good suggestion for me to change the wording to state that mathematics is 'independent' of culture as opposed to 'transcends' culture. I think his suggestion is good.

If art transcends or is independent of culture then it is not art. It can be aesthetic however it is not art.

Let’s consider the law of the excluded middle. There is a group of mathematicians, the Platonists, who accept this law, and there is another group, the Intuitionists, who reject this law. If you agree that the law of the excluded middle is pure math, then here is a case in which pure math depends on culture. That is, the law of the excluded middle is a value or practice that is accepted by one group and rejected by another group. It is not independent of one’s specific mathematical culture.

For pure math to be a cultural expression it must say something about the group it comes from. The excluded middle expresses nothing about either group for its statement is neutral. It is the ‘group’ that is expressing things about the excluded middle.

Peter,

Here is some more fodder for your crosshairs:

Art that expresses an archetype is polyaesthetic. (An aesthetic which is art and one that is not art) The archetype itself is not cultural but the motif surrounding it is … of course this presupposes that the object of expression is art.

Conceptual art focuses mostly on the aesthetics of philosophy which may or may not be cultural therefore may or may not be art.

Concrete art tries to transcend culture and therefore is not art.

The following came from our conversation at your blog however I would like to at least mention it here.

About Pure Mathematics:

I think a good way to define pure mathematics is: That formal language within the study of mathematics that is cultureless. A mathematical archetype so to speak. Cultural expressions speak to the attitude, mythologies and spirit of group of people which are things that can never be fully axiomatic.

This is also from your blog and is worth reposting here:

Peter Turney, on January 6th, 2010 at 1:46 pm Said:

The definition of culture speaks to what is different about the groups – not to what is the same about the groups. Pascal’s triangle belongs to no culture.

Wikipedia defines three senses of “culture“:

1. Excellence of taste in the fine arts and humanities, also known as high culture

2. An integrated pattern of human knowledge, belief, and behavior that depends upon the capacity for symbolic thought and social learning

3. The set of shared attitudes, values, goals, and practices that characterizes an institution, organization or group

I have been using “culture” in sense 2, but I now see that you have been using it in sense 3.

… very interesting – could it be that the sciences and the arts don’t even have the same idea about culture. (Another difference)

… very interesting – could it be that the sciences and the arts don’t even have the same idea about culture. (Another difference)I am looking for what people have in common, whereas you are looking for what separates people. I’m saying artists and mathematicians have a lot in common, and you’re saying they’re quite different. Culture in sense 2 is something we all share. Culture in sense 3 is what divides us into groups.

For pure math to be a cultural expression it must say something about the group it comes from. The excluded middle expresses nothing about either group for its statement is neutral. It is the ‘group’ that is expressing things about the excluded middle.Your original claim was, “Art is the expression of culture. Pure mathematics transcends culture [and is,] therefore, cultureless.” As I now understand what you’re saying, I would express it as, “A work of art tells us something about the cultural group to which the artist belongs, whereas a work of pure mathematics tells us nothing about the cultural group to which the mathematician belongs.”

I disagree with this latter claim. If a theorem is an existence proof for some mathematical object, then the theorem was more likely to have been created by a Platonist than an Intuitionist. If a theorem is proven by constructive methods, then the theorem was more likely to have been created by an Intuitionist than a Platonist. This tells us something about the two groups. Perhaps it tells us that Platonists are “believers” whereas Intuitionists are “skeptics” (see Is Physics Cognitively Biased?).

You said, "Perhaps it tells us that Platonists are “believers” whereas Intuitionists are “skeptics”"

I say:

What is this 'it' you are talking about? the proof? how does the proof tell us anything about the people? The proof doesn’t express anything about the people for it does not even mention people in it. You are the one telling me about the people. YOUR words are what are expressing opinion not the proof.

Peter,

This is to address the comment before your last one ----

Here it is that you and I go through all this discussion to get to crux of why we are discussing these subjects. I am not trying to say that we are very different. In fact I believe we are very similar in many ways however, it seems that everyone is trying to put the arts and sciences together yet, no one seems to be making any good clear definitions of what art and science is. Sure there is a lot that is similar between the fields but there is also a lot that is not. How can one be intelligent about making ‘polyaesthetic mathematical art’ without knowing the difference between the two fields? My entire purpose is to make things more clear as to what the differences are so that when a polyaesthetic work is put forward one has aesthetic criteria to help them make personal judgments to the value of the work. Furthermore I am really puzzled by the fervor that seems to be running among some mathematicians who wish for their work to be called art. Heavens, what is wrong with calling it mathematics? If you need a name to call the branch of aesthetics that concerns itself with mathematics then create one such as aesmath or mathaes or anything but art. At this time the art world is so watered down with arbitrary aesthetics that it has lost all meaning. No one has a clue as to what is art and what is not and it is up to our generation of artists to set it back on the right path. I believe that art can never have a complete set of axioms yet we can define some to at least define the path. The danger that we must avoid is putting more importance on one field relative to the other.

I do realize a problem in the way that mathematics is taught in so much as very few teachers emphasize the aesthetics of math instead they seem to focus on all the rules. However, if you look at the field of art, you will find people who make a living as aestheticians, and they concern themselves only with illuminating the aesthetics in art. I could be wrong but I don’t think there are many people making a living illuminating the aesthetics of math with no other purpose but to show us beauty albeit Ivars Peterson and Martin Gardner come to my mind. Sadly there is a lot of mathematical beauty for us to see but not enough emphasis put on aesmath (the aesthetics of math).

What is this 'it' you are talking about? the proof? how does the proof tell us anything about the people? The proof doesn’t express anything about the people for it does not even mention people in it. You are the one telling me about the people. YOUR words are what are expressing opinion not the proof.Yes, a proof tells us something about the prover, just as a painting tells us something about the painter, just as music tells us something about the musician.

A proof need not mention people in order to tell us something about the prover, just as a painting need not portray people in order to tell us something about the painter, just as music without voices can tell us something about the musician.

A skilled mathematician may be able to tell us much more about the author of a proof than a mathematical neophyte could tell us. Likewise, an art critic and art historian can tell us more about the painter of a painting than an artistic neophyte. An expert in the history of music can tell us much about a musician. In all three cases, the expert may be able to tell us a great deal purely by examining the work itself, without knowing who created the work.

Yes, my words are giving an interpretation in this example. I am playing the role of the expert. This is the kind of information that an expert could provide, merely by examining a proof of a theorem.

These recent blog posts and comments may give you some idea of just how much room there is for opinion, style, personality, culture, and aesthetics in math:

Axioms: What should we believe?

Voting on Mathematical Truths: The Axiom of Determinacy

I believe that by saying that a work ‘tells’ us something it is not necessarily telling us something cultural. I can not agree that Platonists and Intuitionists can be considered a “Culture” The Wikipedia # 2 is not a cultural definition that the arts generally recognize and Wikipedia #3 is really a watered down definition of culture as well. I have never seen a mathematical proof that could tell you the probable certainty of what kind of food the mathematician eats, the color of his/her eyes, hair, skin, their race, their manners, the smell of their skin, the type of music for which they dance, the way they dance, their spiritual rituals and beliefs, their idea of charity, their favorite form of wine, beer, or cheese or lack of these things, the type of transportation they use – the list goes on and on.

Comparing a Platonist to the culture of India is like comparing a star to a galaxy.

Yes I believe that an existence proof doesn’t say anything culturally about the people – at least not enough for considering the people as a culture.

I can not agree that Platonists and Intuitionists can be considered a “Culture” The Wikipedia # 2 is not a cultural definition that the arts generally recognize and Wikipedia #3 is really a watered down definition of culture as well. I have never seen a mathematical proof that could tell you the probable certainty of what kind of food the mathematician eats, the color of his/her eyes, hair, skin, their race, their manners, the smell of their skin, the type of music for which they dance, the way they dance, their spiritual rituals and beliefs, their idea of charity, their favorite form of wine, beer, or cheese or lack of these things, the type of transportation they use – the list goes on and on.Platonists and Intuitionists are cultural groups in mathematics in the same sense that Surrealists, Impressionists, and Pre-Raphaelites are cultural groups in painting. Your (very) narrow conception of culture implies that abstract and non-representational painting is (in your words) "cultureless". What we can infer about the painter of an abstract painting is similar to what we can infer about the prover of a mathematical proof. With this restricted conception of culture, you are no longer making a distinction between math and art. It is now a distinction between abstract human representational artifacts and concrete human representational artifacts (or something like this).

I can not agree that Platonists and Intuitionists can be considered a “Culture” The Wikipedia # 2 is not a cultural definition that the arts generally recognize and Wikipedia #3 is really a watered down definition of culture as well.None of the three senses of "culture" in Wikipedia correspond to what you mean by "culture". I think this should tell you that "culture" is not the right word for what you're talking about. In fact, it seems to me that perhaps "race" is the right word ("what kind of food the mathematician eats, the

color of his/her eyes, hair, skin, theirrace, their manners, the smell of their skin"). As I now understand what you're saying, I would express it as, "A work of art tells us something about the artist's race, whereas a work of pure mathematics tells us nothing about the mathematician's race." Is this really what you want to say? First, I think it's factually wrong, because non-representational art and mathematical proofs are approximately equal in terms of what they tell us about race. Second, I have ethical issues with this kind of claim. Why do you want to drag racial issues into a discussion of art and math?You said, “Platonists and Intuitionists are cultural groups in mathematics in the same sense that Surrealists, Impressionists, and Pre-Raphaelites are cultural groups in painting.” I will agree that Platonists and Intuitionist are like surrealists and Impressionist however in the many years of studying art I have never heard one artist or aesthetician say that Impressionism or surrealism is a culture and I have studied surrealism extensively over the last forty years. In other words, I have never met anyone in the arts that would consider a movement of art to be a culture for movements resides within cultures. Furthermore my definition of culture has been shape by the interactions with artists throughout my life – I am not as isolated as you would believe. Since we keep referring back to Wikipedia, the anthropological section definition for culture is much closer to what I call culture than what you call culture. There is culture and there are subcultures and I am talking about Culture as in civilizations not culture as in philosophical schools.

What cultural information can you infer from Pascal’s triangle? What kind of cultural information can you infer from abstract Celtic weaves?

This whole discussion boils down to how we define culture. As I said, if you would continue reading the Wikipedia section you will notice that ethnicity plays a role in culture. So yes race is a factor among a myriad of other things and as far as ethics is concerned, I would think you would know the difference between racial discourse and racists discourse. Since you don’t see ethnicity in math then you are confirming what I have said all along. You and I have different definitions of culture and if anthropologists

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