ToK Chapter Reading Responses

Hey all,

Hope you are enjoying your holiday. I am very excited about returning to LBHS next week, and I hope you are too! Below I am posting an example of a ToK chapter reading response. Remember that we are changing the chapter responses to emphasize discussion of personal examples. The chapter 7 reading and reading response required about one hour. I suggest writing the response as you read.

Remember this is just an example.

ToK Chapter 7, Mathematics Reading Response

Most important quotes:

“Math—that most logical of sciences—shows us that the truth can be highly counter-intuitive and that sense is hardly common.” This quote from K.C. Cole speaks to the idea of math as a form of logic that often challenges our preconceived notions about what is known.  I like this idea because often we use anecdotal evidence or feelings as a baseline for what we believe to be true. When you see relationships in terms of numbers, however, often your beliefs and perceptions are changed to reflect a more accurate truth. We see this with regard to how statistics and percentages may be used.

“The useful combinations are precisely the most beautiful.” This hints at the idea of applied mathematics as opposed to pure mathematics. If math is only theoretical and not applicable to the world we observe then it is of little use other than amusement. This reminds me of fractal geometry and how the Mandelbrot Set became an important breakthrough in thinking about how we see and apply math in the real world.

Most Important Ideas:

  1. The idea of “proof” is an important idea because it is the mathematical equivalence of evidence and certainty combined.  Moreover, a proof is based on first axiomatic truths and theorems, so the logical argument is at least valid and true. Yet, it is still based on a series of ideas, which may not be observed the same way in nature.
  2. The idea of elegant solutions, often derived from imaginative or creative insights, is refreshing because often in mathematics we tend to move forward in a linear approach rather than ‘seeing the forest for the trees.’ How much of our mathematical thinking is limited by our inductive approach to solving mathematical problems?
  3. I tend to think of myself as a formalist; I believe that math is invented by humans and it exists as an independent set of ideas. However, I can see the appeal of Platonism, for it suggests the idea of limitless discovery. who wouldn’t want to think of their favorite subject as having limitless possibility?

Linking questions:

  1. How is mathematics like a language? As an English teacher, I see the different mathematical descriptions as analogous to the kinds of genres that I explore, each one offering new insights into human experience, yet having its unique advantages and limitations. No one genre, in this case Euclidean of Riemannian geometry, having absolute certainty with the nature of reality. Like language, math is symbolic, and it tries to represent reality. In some ways, math is more a more accurate symbolic language because numbers are far less ambiguous than words. If I say ‘dog’ you might have a range of images come to mind. Yet if I say ‘four’ you have one concept, and fortunately other people, regardless of culture, age, or gender, can understand what you mean when you say ‘four.’
  2. Does perception play any role in mathematics? Absolutely, we need to observe how mathematical thinking can be applied to the world, and this takes observation. For example, applying non-Euclidean geometry to physics allowed Einstein to make discoveries the nature of reality, e.g., his general theory of relativity.  We also see evidence of fractal geometry when we make observations in nature; we see the Fibonacci sequence in flowers, plants, humans, and galaxies.  Observations are important because it may allow us to discover or invent new forms of math to help explain the nature of reality.

Two questions about math as an area of knowledge:

  1. How can mathematics be used as a justification for either confirming or challenging our beliefs and biases?
  2. If you say you know it is true because you have mathematical proof, is that claim more compelling than a claim that is based entirely on emotions, memory, or eyewitness testimony?

Word Count: 627