We have been enjoying Aronofsky’s first movie Pi as a nice starting point for our exploration in TOK of Mathematics as an “Area of Knowledge.”
After viewing, students will respond to one of the following questions in the comments below, incorporating an idea from Pi into their responses:
- How do the knowledge issues in Pi relate to one of the Ways of Knowing we have studied this year?
- We can use mathematics to model real-world processes. Is this because we create mathematics to mirror the world or because the world is intrinsically mathematical.
- Consider how Pi relates to #5 or #10 of “The 10 TOK Essay Titles for 2010-11.”
To what extent are the various areas of knowledge defined by their methodologies rather than their content?
A model is a simplified representation of some aspect of the world. In what ways may models help or hinder the search for knowledge?
{ 14 comments… read them below or add one }
How do the knowledge issues in Pi relate to one of the Ways of Knowing we have studied this year?
One of the Ways of Knowing we studied this year was the idea of perception. Everyone percives the world in different ways, which is dealt with in Pi. Those who percieve the world to be a result of patterns and numbers will try and seek explanations as to why this is so. Some people may believe otherwise, and therefore they would draw very different conclusions about the world and nature. In the movie Pi, Max discovers a set of numbers that several people believe is the key to very different things. Some think it is nothing at all, others believe it had a religious connotation, and some believe that it is the key to understanding and predicting the stock market. These different interpretations of a simple string of numbers all relate to the idea of perception, and with it, reason. Perception and reason are linked together, as one cannot exist without each other. These differences in the rationale, or reasoning behind each person’s interpretation of Max’s numbers can be explained when examing the differences in everyone’s perception and therefore reasoning. If everyone percieves the world in different ways, their reasoning and rationale of an experience will be different. This shows how the knowledge issues in Pi can relate to perception, and therefore to reasoning.
I agree with Elizabeth that perception and reason are linked to maths as an area of knowing. I also believe that once you start to perceive things a certain way, and start to see patterns in everything, iyou you start to try and rationalize the world and nature. If you try to make everything make sense like max does in the movie Pi, you will start to see patterns. I think he strongly convinced himself that the world existed of patterns, formulas and equations and everything was linked and explainable and he started to see them everywhere. He also tried to explain everything he saw. This is where reason comes in as Elizabeth said before. He tries to find a pattern in something completely irrational like the number pi and the stock market. He tried to make it make sense. He tried to find a reason for why things are the way they were. To do this he manipulated his perception of the world. He did this both with the use of drugs and a mindset. He really did believe there was a reason for everything and it was all mathematical, this made him unreasonable in the end, and he was only at piece when he was free of this way of perceiving the world. It suggests that just reason or any one of the areas of knowing is not enough on its own for humans to live a happy life. If we are just driven by emotions and not reason we could be irational. If we believe solely in reason we could go crazy.
As a response to both Ria’s comment as Elizabeth’s, I find myself in agreement with both of their comments. After trying to make sense of everything that occurs to us, just as Max was trying to do in the movie. However, it is important for us to understand that not everything in our lives can make sense and most importantly, trying to avoid this reality such as Max did, with the use of drugs and other methods that would encourage his productivity, can only have a harmful effect on our lives rather than a beneficial one. The harmful effects can be seen in the movie, to the point where Max becomes paranoic in many ways, thinking that someone is following him around so that he puts 4 or 5 locks on his door to prevent anyone from entering his lab. What is possible to conclude from Max’s behavior in the movie is that the extensive amount of time and effort that Max dedicates to finding this solution is not the best solution to achieve an untroubled life.
Have you heard about this yet?: http://www.guggenheim.org/new-york/interact/participate/youtube-play
Aronofsky’s going to be one of the judges. I wanted to share the link about YouTube Play with you and then I see you have a blog post about Pi – how perfect!
Very cool, Rachel, thanks for the link. I have not heard about this yet, but it looks very interesting. I noticed that, in addition to Aronofsky, the photographer Ryan McGinley and the band Animal Collective are also on the Jury, making it quite possibly the coolest one ever.
I agree with Ria, Elizabeth and Andrea in that perception and reason are related to maths as one of our areas of knowledge. Some believe that Mathematics is all around us and that patterns are everywhere. This is what Max in Pi believes as he regular attempts to find the pattern that the stock market follows. During the movie he discovers a number that other groups of people wish to have hold of as they believe it is an answer to a belief they have.
The fact that people believe that the world and life itself contains patterns suggests that there is always an answer. That is not true as there is not always one correct answer and the “answer” varies from people to people depending on how they perceive the situation.
We can use mathematics to model real-world processes. Is this because we create mathematics to mirror the world or because the world is intrinsically mathematical?
I’m really not sure what mathematics is exactly, so it’s difficult to answer this question. But if maths is a language, then really it’s not an area of knowledge, its a way of knowing (like linguistic language), and if that is the case, then mathematics has been created to describe the world.
I just looked in an Oxford Dictionary, and its definition of modern mathematics is “the abstract deductive science of space, number, quantity, and arrangement, including geometry, arithmetic, alegebra, etc., studied in its own right, or as applied to various branches of physics and other sciences” (Brown, 1993), indeed the greek root of the word mathematics can be roughly translated as ‘a learned science’. This definition suggests that there are aspects of the world around us that can be studied and are ‘mathematical’ in the sense that they are intrinsically concerned with “space, number, quantity, and arrangement” (Brown, 1993). In fact, all things can be studied looking at their space, number, quantity and arrangement, yet to do so is limiting, for these are not the only propeties objects and ideas posses.
If mathematics is both an area of knowledge and a language, then surely it is a mathematical language that has been created to describe all things mathematical in nature, both pure and applied.
A model is a simplified representation of some aspect of the world. In what ways may models help or hinder the search for knowledge?
As said above, a model is a simplified representation of some aspect of the world. People make models to help get a better understanding of a particular aspect of the world. The fact that people make models is a hint that all models are imperfect. In order to make a model, a person has to make some simplifying assumptions, to make a model easy to interpret. Models can be either complex or simple. The more complex a model gets, the more chaotic it becomes. As models become more complex, they get closer to representing the real world, which is completely unpredictable and chaotic. Opposite of that is oversimplification of models. The simpler they get, the further away from reality they are, and thus it may be difficult to apply the model in real world. In the movie, Max attempts to make a model for a stock market. There are too many variables in his approach for him to handle. Linking to the “Library of Babel”, as the access to knowledge approaches infinity, the knowledge gained tends toward zero. What Max tried to do seems illogical, because he attempted to make a model that represents real world exactly. Stock market is dependent on supply and demand, which is human based. That is the same as predicting human behaviour using a mathematical model. We humans are illogical and make exceptions to many patterns. One must look beyond all the mathematical processes in order to gain a proper understanding of an aspect of the world. That is the same with communication. It is difficult to deduce whether a person is lying to you if you only listen to the words. Their body language will usually tell you more than their words. What I am trying to say is that Max should not only base his assumptions solely on mathematics, because mathematics is limited and does not account for any emotions and feelings.
Max’s attempt to find a pattern in stock market is like trying to divide by zero…
We can use mathematics to model real-world processes. Is this because we create mathematics to mirror the world or because the world is intrinsically mathematical.
Both, it depends entirely on the definition of mathematics. If mathematics just describes the relationships between what we observe in the universe, then of course the world is intrinsically mathematical. Patterns can be found in everything, and where patterns cannot be found, we are simply not intelligent enough to understand its complexities. But the mathematics we are familiar with is a product of human invention, nonetheless. Whereas relationships do exist between everything, we have a regimented system for putting it into terms we understand. It is no more than a simple tool. Yet mathematics is a tool who’s invention is inevitable for any group of people without any external influence or inspiration. Therefore the origin of mathematics is essentially the invention of a tool which is only a replication of a phenomena, if you will, that is ubiquitous in the entire universe. The only thing we know for certain is that the terminology and notation is unique. Everything else is uncertain. For instance, there are irrefutably only a finite amount of ways to logically approach a problem involving numbers. Given that numbers themselves are a universal concept, mathematics was discovered.
It would seem that mathematics, is to some degree, a simpler species of logic itself. Now, logic seems to be a concept which is universal. It is based on reason, which is something that does not vary and is not subjective. If mathematics is then a subcategory of logic, we can deduce that mathematics is also universal, and is a logical underpinning. still that doesnt answer the question though. goddammit. You know it seems like the question has hardly anything to do with mathematics really. It’s more about the difference between discovery and invention regarding abstract concepts and all that jazz. So to answer the question: I don’t know, I can’t know, and nobody knows.
I agree with what Elizabeth is saying about perception here. Some people believe math is the key to understanding the world, which therefore influences them to look at the world with math in mind. Other people don’t find math as significant and look at the world very differently. The people Max communicates with all have different perceptions of what they themselves believe is significant. From the parts of the film I saw it was interesting to see how people treated Max based on their perception of what the numbers meant. Some people seeked his help because they themselves believed that what Max knew about math was significant in some ways. Others looked at him in confusion, because they didn’t understand him and viewed his work as numbers they couldn’t make sense of. The people that perceived the numbers as significant also had different reasons to believe so. Some of the people valued his work because it was related to their religion; others because they believed it could predict the stock market. It’s all about what a person perceives to be valuable information.
I agree with most of the people here, explaining that math is literalyl all around us, and there are certian areas of math intertwined within everyday situations, like patterns for example. However, we must take into consideration that not everything always makes sense. On the other hand, you can make sense of something by the way you look at it, how you percieve it and whether or not you believe it. Referring back to how patterns are everywhere, this is initially what Max believes, he is trying to decode tha pattern that the stock market follows.
Some people link Mathmatics to a language, or a way of knowing. This is represented well in a quote by Galileo, “Nature is written in the language of Mathmatics”, which I think he is basically trying to state, once again, that math is everywhere and it turns out the way we view it.
We can use mathematics to model real-world processes. Is this because we create mathematics to mirror the world or because the world is intrinsically mathematical.
This phrase can be thrown around alot. It seems that everything relates to math in some form, it can be described through the sciences. Psychology relates to biology, biology relates to chemistry, chemistry relates to physics, and physics relates to math. It seems that math is the basis for all of lifes functions. And the concept of math could be so intertwined with the world, but the math which we have created, is not. The math we, as a people, have created, is nothing more than an estimation of the truth. Numbers are only logical estimates. This may help prove this concept.
1= 1/1 = 9/9 = (0.11) x 9 = 0.99
1=0.99
This concept elaborates on the importance to realize that as “definitive” as math may be, it can only be the estimated of the “natural mathematics” which rule the natural world. The the world is intrinsically mathematical, but only because we have scewed the lines in order to make our “industrialized math” into our society.
We can use mathematics to model real-world processes. Is this because we create mathematics to mirror the world or because the world is intrinsically mathematical?
I agree with Alex H when he says that mathematics is a language, however, I disagree with saying that it has been created to describe the world. I believe that mathematics has always been around us and has been discovered step by step which model real-world processes. The reason I say this is because if it is a created language why has it become so complicated with so many unanswered questions, one being: ‘why can’t you divide by zero?’ I believe that if it is a created language then these answers should have been created.
Although, saying this, I do think what has been created in mathematics is symbols representing mathematical actions, such as ÷ for divide and x for multiply. Also the words for each number is different in many languages (e.g. ten is diez in Spanish and hamar in Basque), so in a sense how we say mathematical terms is created but mathematics itself is not.
1= 1/1 = 9/9 = (0.11) x 9 = 0.99
1=0.99
9/9 is not equal to (0.11) x 9.
(0.11) = 11/100 so (0.11) x 9) = 11/100 * 9 = 99/100. So no, 1 is not equal to 0.99. You may have had 0.999… = 1 in mind (http://en.wikipedia.org/wiki/0.999…).