HON 462 Seminars Fall 2005

Extra Readings:

Physics and the Real World

The Origins of Randomness in Physical Systems (Wolfram)

Is the Universe a Computer (A Review by Steven Weinberg of S Wolfram's A New Kind of Science)

Chaos, Entropy and the arrow of Time (P. Coveney)

Is The Solar System Stable (C. Murray)

 

Seminar Session 1

Readings

Session Leaders

Discussion Questions

  1. Compare the real butterfly in Bradbury's "A Sound of Thunder" to Lorenz's metaphorical one in the "Butterfly Effect." Which one is more believable in terms of the author's predicted effects of action on or due to the butterfly?
  2. Explain how fractal geometry gives you a different framework for the multiple-occurrence paradoxes that occur in time-travel stories
  3. Gleick's second chapter deals with issues of stability. Is the "stability" described the concept you normally associate with the word?
  4. Explain how something can be chaotic and still be stable at the same time.
  5. How does stability form an umbrella concept that encompasses what we have seen so far in the computer lab and in the readings?
  6. Describe Thomas Kuhn's idea of a paradigm shift. How can this definition be used to explain the fairly recent emergence and popularity of chaos theory? Is there a socially constructed "stability" at play here?

Background sites

See the following recent newspaper articles concerning weather, climate, and forecasting:

Click here for an excellent collection of links to articles on Thomas Kuhn. The site is maintained by Frank Pajares of Emory University

There appears to be a mini-version of Jupiter's red spot right here on earth: a "black hole" within a frozen lake in Minnesota.( Mystery Grows Around Hole in Lake)

 

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Seminar Session 2

Readings:

Session Leaders

Discussion Questions

  1. Would you classify yourself as a mathematical realist, formalist, constructivist, or inventionist?
  2. What do the constructivists mean when they say some statements are undecided?
  3. How would you describe the mathematical/philosophical position of the Bourbaki participants?
  4. Does the "existence" of fractals support the position of mathematical realism? (And why is "existence" in quotes here?)
  5. The constructivists stance is to avoid abstract concepts/entities such as infinity. Do fractals or chaos indicate that these entities exist in a realist sense?
  6. Do the non-provables and non-computables of Godel and Turing support or deny mathematics as a language of the universe?
  7. How does the study of population dynamics coincide with our discussion on the philosophy of mathematics?
  8. How does our broadened notion of stability apply to population dynamics? What does the "bifurcation diagram" displayed in the Gleick chapter on "Life's Ups and Downs" indicate about the populations modeled in terms of stability or randomness?
  9. Gleick describes an aesthetic of art/architecture that essentially demands a fractal nature as a necessary ingredient for lasting significance. Do you agree with this position? Can this argument also be extended to literature and music?

Background sites

The following is an analysis of W.V.O. Quine and H. Putnam indispensability argument for mathematical realism. It also includes details on the rather hotly contested debate against realism, due to Hartry Field and Penelope Maddy.

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Seminar Session 3

Readings:

Session Leaders

Discussion Questions

  1. How would you characterize turbulence? Is it order encapsulated within “chaos”, or merely a random, “unstable” system? According to Gleick, how does the connection between turbulence and strange attractors define, redefine or contradict our previous concepts of stability?
  2. The language of Chaos and Fractals is rife with apparent connections to REAL LIFE. For example, consider the statement: “On a meta-level, a paradigm shift is a strange attractor.” Can you make a convincing argument for this comment?
  3. As a follow-up to #2, can the cross-functionality of fractal geometry and chaos be viewed as a metaphor for the development of science and business?
  4. If you HAD TO explain to someone what a fractal was or represented without using technical mathematics, what would your working definition be? How would you define Chaos?
  5. Based on this week’s readings, to which of the four schools of mathematical thought (from "Mathematical Universe") would Mandelbrot, May, and Feigenbaum probably admit to following?
  6. What is Feigenbaum’s theory of universality, and how is it related to fractals, chaos, and strange attractors?
  7. "[In everyday life] one can and must deal honestly with imperfection, while both persevering and keeping in check the dream of a More Heavenly City." (Mandelbrot) This statement has a religious undertone, but Mandelbrot uses this to explain an aspect of mathematics. We have discussed how Gödel proves that there are some things that are some truths that can’t be proven. Where do you believe that religion falls in the realm of chaos? Do you think it is easier for people to believe in a "mathematics law" that has to be true but with no proof OR a religious belief?
  8. Gleick often uses religious metaphors in his description of the growth of Chaos Theory – e.g. "believers," "converts," "heresy," "evangelists," as well as supernatural language (i.e. "magic"). Does this use of language create valid connections between Chaos and non-scientific arenas, such as philosophy, religion, and art?
  9. What is your position on Goethe’s theory of color? Are you inclined to doubt Goethe because of his abhorrence of mathematics and belief in the primacy of art as opposed to the mathematical schemes of Newton?
  10. Feigenbaum’s theory of universality strained credulity because of its totality. Why do you think the scientific community was reluctant to accept the breakthrough of universality?
  11. Can human perception be universal?

Background sites

For more on Feigenbaum, check out the brief biography from the MacTutor History of Mathematics archive. Currently Feignebaum is "Toyota" Professor at Rockefeller University where he heads the Laboratory of Mathematical Physics.

Feigenbaum has written an interesting paper for the Tureck-Bach Research Foundation The purposes of the TBRF are to "advance the education of the public in music, particularly the music of Johann Sebastian Bach, in all its aspects including its composition and appreciation and the history, art and science of music" The paper is entitled Unfolding Processes, Emergent Phenomena and Numbers' Structural Legacy

While you're at the Tureck-Bach site, you might want to check out an interesting piece by Sir Roger Penrose entitled Mathematics: Pattern, Precision and Profundity

There are always some who push the notion of universality to the edge of the envelope and beyond. Here are two by Ernest Lawrence Rossi

Some Art & Fractal Links:

Hot Goethe Links...

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Seminar Session 4

Readings:

Session Leaders

Discussion Questions

  1. Is it possible / likely that extremely complex, fractal patterns that arise in nature are created through a simple, iterative process?
  2. Does the presence of fractals in nature support the realist idea of mathematics?
  3. Do you believe in (Newtonian) determinism or probability? Is the future uniquely determined by the past, and can we predict the past and future by the present state of the universe?
  4. Do you agree or disagree with Barnsley's idea that randomness in nature and biology is only an illusion?
  5. In previous discussions, we've talked about the legitimacy of imaginary numbers in modeling nature. Gleick describes "both sorts of numbers [as] being just as real and just as imaginary as any other sort." Do you agree with Gleick? Has your view on the use of imaginary numbers changed?
  6. Gleick says that imaginary numbers were created by us to fill the “gap” posed by the question, “what's the square root of -1.” Did we create them, or were they waiting to be discovered?
  7. What mathematical school of thought does your answer to the previous question support?
  8. Did the Mandelbrot set come into existence "as soon as science created a context - a framework of complex numbers and a notion of iterated functions," or did it exist earlier, "as soon as nature began organizing itself by means of simple physical laws, repeated with infinite patience and everywhere the same."?
  9. D'Arcy Thompson believed that certain "deep-seated rhythms of growth" created universal forms. These universal forms, he believed, governed physical causes. It is because of these physical causes that objects in nature are shaped the way they are. Do you agree with this view? Does it make more sense to study natural organisms by their final cause, or by their efficient or physical cause?
  10. Patterns in fluid motion found in Libchaber's experiment echoed the Feigenbaum number. This suggests a relationship between the onset of turbulence & population growth. Does this support the idea of universality? In what context is the use of chaos theory justified? In other words, can it be used to explain phenomena outside the realm of science (e.g. politics)?

Background sites

Introduction to Chaos in Deterministic Systems (This is an excellent summary of Chaos theory, and it goes well with the frist chapter of the Hall book) Cognitive and Computing Sciences, Univ. of Sussex)

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Seminar Session 5

Readings:

Session Leaders

Discussion Questions

  1. Can you make an analogy between calculus and the chaos theory?
  2. Give an example of a chaotic picture like the one on Gleick pg. 254.
  3. Can language be described as having a fractal nature?
  4. One of the major contributions of the Univ. of Santa Cruz Dynamical Systems Collective (DSC) is the connection between Chaos and Shannon’s information theory. What is information theory, and how is it related to Chaos? Do you agree with the DSC view that “randomness equals information”?
  5. Doyne Farmer writes “On a philosophical level, it [Chaos] struck me as an operational way to define free will, n a way that allowed you to reconcile free will with determinism. The system is deterministic but you can’t say what it’s going to do next.” What are your views on the free will/determinism debate? Has you study of Chaos influenced these views?
  6. In the DSC dripping-water experiment, “chaotic” data was analyzed using a technique known as embedding. What is this technique, and how is it related to our recent Classroom Chaos session on the Logistic Map?
  7. The DSC developed the skill (or delusion?) of seeing strange attractors everywhere. Do you?
  8. Gleick describes the Second Law of Thermodynamics as the “the inexorable tendency of the universe to slide toward a state of increasing disorder” while “Entropy is the name for the quality of systems that increases under the Second Law – mixing, disorder and randomness.” Is this definition for entropy accurate? If so, could this be another one of the many definitions we have found for chaos?
  9. Gleick’s closing chapter (pp. 305-307) lists the responses of many different Chaoticians to the fundamental question “what is chaos?” With which do you agree most? How will you answer when you are asked the same question for the second edition of the book (to be published in 2011)?
  10. On pg. 311, Gleick makes the general statement that “the laws of pattern formation are universal.” Are they? Does a snowflake from the same way in this universe as it would in John Doe’s universe? If the laws are universal, why is each snowflake unique?
  11. The book ends rather abruptly. What do you think that this type of ending suggests about chaos theory? Also, give your overall feelings on the book. Did you like/dislike it? More importantly, why?

Background sites

 

 

 

 

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