Math Chat: The Mathematical Method, Calculus & Probability

This month's Math Chat is based on a 1¾ hour video of Keith Devlin which discusses the mathematical method, the Calculus, and the basics of probability theory. To round out the discussion, there are four supplemental videos adding 1 more hour. So a total of three (3) hours of video to stimulate our discussion. Watching videos is optional, but will enrich and enliven the discussion. 

The main 1¾ hour video with Keith Devlin frames the discussion about Calculus as a technology in the context of what Devlin calls the mathematical method. Do you accept Devlin's framing of the mathematical method? Is Calculus or any mathematics a technology? Does mathematics model the real world? Is math "unreasonably effective" in understanding the real world? The last 45 minutes in the video explore the origins of probability theory with a wonderful discussion of the correspondence between Blaise Pascal and Pierre de Fermat which led to a revolution in our ability to predict the future based on probability. Was that the breakthrough that Devlin claims it was? Wasn't the work of Gerolamo Cardano critically important as well? Read my notes on Devlin's lecture.

http://www.youtube.com/watch?v=8ZLC0egL6pc

Gilbert Strang's exquisite 37 minute Big Picture of Calculus gives a much clearer and more comprehensive overview of the Calculus than does the Keith Devlin video. But maybe you prefer Devlin's approach? Regardless, we will discuss both videos. 

http://www.youtube.com/watch?v=UcWsDwg1XwM

The exquisite 6m video Calculus Rhapsody By Phil Kirk & Mike Gospel puts some much needed fun into Calculus! The video is amazing and excellent: I hope you enjoy it. 

http://www.youtube.com/watch?v=uqwC41RDPyg

• Brady Haran's interesting 6m video 1,296 and Yahtzee looks at the probabilities of rolling five cubical dice with the same number of pips on each face (a yahtzee).  

http://www.youtube.com/watch?v=fiTwar7mFws

Vi Hart's 5m video response to Haran's video on Tetrahedral Dice does the same with die with four faces instead of six. 

http://www.youtube.com/watch?v=fwD98HiQSJc

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  • Greg G.

    Interesting & well organized. Good work CJ. High level of interest

    1 · March 23, 2014

  • Ahngelique D.

    not enough time to enjoy all the angles views and insights but very enlightening for the math genius wannabe

    March 23, 2014

  • Ahngelique D.

    So much math yet so many conversations about how it resonates in dal life. Valuable discussion that I will treasure. I will definitely come to another one (or more)

    1 · March 23, 2014

  • CJ F.

    Here's an article from The Atlantic entitled "5-Year-Olds Can Learn Calculus": http://www.theatlantic.com/education/archive/2014/03/5-year-olds-can-learn-calculus/284124/

    I do not know much about math education, but since several who attended might be interested I decided to pass it along.

    March 22, 2014

    • Sam B.

      Those last couple of paragraphs about the math groups and open learning are a neat touch.

      March 22, 2014

  • CJ F.

    The Brady Haran video at http://www.youtube.com/watch?v=fiTwar7mFws and the Vi Hart video at http://www.youtube.com/watch?v=fwD98HiQSJc are fun examples of calculating the probability of rolls 5 cubical dice (Brady's vid) or 6 tetrahedral dice (Vi's vid) with all the same faces. That's 1/1296 for the cubical dice and 1/1024 for the tetrahedral dice. But Brady rolled a Yahtzee on roll 627 and Vi rolled her Yahtzee on roll 382. Can we conclude that the math is wrong or did they really BOTH get that lucky?

    March 21, 2014

  • CJ F.

    There will be a free performance of "Calculus: The Musical" at the Kaleidoscope Lenfest Theater, Kaleidoscope Performing Arts Center, Ursinus College, Collegeville, PA 19426 on Tuesday, 25 March at 7 PM.

    More information is at http://events.ursinus.edu/index.php?eID=3151

    Unfortunately, I have other plans that night. Let us know how you enjoy the performance if you get to see it.

    March 20, 2014

  • CJ F.

    In the 1¾ h Keith Devlin video http://y2u.be/8ZLC0egL6pc he discusses the invention of probability theory in a letter written on Monday, 24 August 1654.

    The letter was part of a correspondence between Blaise Pascal & Pierre de Fermat on the problem of the unfinished game (aka "the problem of the points"). How do you fairly divide the pot when a best of N rounds tournament has been abandoned part-way through?

    Fermat's simple idea baffled Pascal (new ideas can take a long time to sink in!): To calculate the odds of a future outcome, calculate all possible futures & determine the likelihood of each possible outcome from the set of all possible futures. You need to count possible futures NOT possible outcomes!

    Is this so revolutionary? Was Gerolamo Cardano's work on calculating probabilities a prerequisite? Which innovation sparked the revolution in probability theory? Why?

    Read my notes on Devlin's video at http://tinyurl.com/l6u7yp2

    Watch Devlin's video at http://y2u.be/8ZLC0egL6pc

    March 20, 2014

  • CJ F.

    In the 1¾ h Keith Devlin video http://y2u.be/8ZLC0egL6pc he emphasizes the importance of the concept of the limit of a sequence of estimates in defining the derivative (lim as h->0 of f(a+h) - f(a) / h).

    Contrast that with the 37 m Gilbert Strang's video http://y2u.be/UcWsDwg1XwM which gives a much more concrete approach emphasizing the relationship between two functions which keeps the fundamental theorem of calculus in steady focus.

    Did you understand each approach? Which did you prefer? I thought Devlin's approach gave a better appreciation for the philosophical minefields (Zeno's paradox and all that) which estimation and limits allows calculus to overcome. While I thought Strang's approach gave a deeper awareness of the analytical process and its use in applications. What did you learn from each video? What are the strengths and weaknesses of each presentation?

    How would you explain what calculus is all about?

    For some fun watch Calculus Rhapsody 6m: http://y2u.be/uqwC41RDPyg

    March 19, 2014

  • CJ F.

    The main (but optional) video for Saturday's "Math Chat" is Keith Devlin's 1¾ hour video at http://www.youtube.com/watch?v=8ZLC0egL6pc

    Devlin argues that Calculus is one of our most successful technologies. By a technology he means something that helps people perform a task that they often want to do. I like W. Brian Arthur's (see his book "The Nature of Technology: What It Is and How It Evolves") definition of technology as "a means to fulfill a human purpose".

    Is mathematics an important technology? Is this a good way to explain the importance and utility of mathematics?

    Is calculus a technology? Is calculus "one of the most successful technologies"? Does calculus provide an exemplar of mathematics as a technology? Are future probabilities perhaps a better exemplar of the importance of mathematical technology?

    Read my notes summarizing the video at https://plus.google.com/104222466367230914966/posts/MhfNvS6Q6fK

    Watch Devlin's video at http://www.youtube.com/watch?v=8ZLC0egL6pc

    March 18, 2014

  • CJ F.

    For Saturday's "Math Chat" we are going to discuss Keith Devlin's 1¾ hour video http://www.youtube.com/watch?v=8ZLC0egL6pc

    At the beginning of the video Devlin lays out a vision for "The mathematical method" based on the idea that mathematics is the science of patterns.

    1. Identify a particular pattern in the world.
    2. Study it.
    3. Develop a notation to describe it.
    4. Use that notation to further the study.
    5. Formulate basic assumptions (axioms) to capture the fundamental properties of the abstracted pattern.
    6. Study the abstracted pattern by means of rigorous proofs to establish truths about the abstract patterns.
    7. Develop procedures to apply the results of the study to the world.
    8. Apply the results to the world.

    Is this the defining "method" for mathematics? Is a method even needed? Does Devlin's method capture all or only part of mathematics? What are the benefits & disadvantages of Devlin's perspective?

    Watch the Devlin video at http://www.youtube.com/watch?v=8ZLC0egL6pc

    March 17, 2014

  • CJ F.

    Although watching the three (3) hours of videos are optional, they are the primary source of material for our "Math Chat" next Saturday. For easy reference here are the links to the six videos:

    The main 1¾ hour video with Keith Devlin: http://y2u.be/8ZLC0egL6pc

    Gilbert Strang's "Big Picture of Calculus" (37m): http://y2u.be/UcWsDwg1XwM

    Calculus Rhapsody (6m): http://y2u.be/uqwC41RDPyg

    Probability Tell The Future (3m): http://y2u.be/RhuXyKdStJ8

    1296 and Yahtzee (6m): http://y2u.be/fiTwar7mFws

    Vi Hart Tetrahedral Dice (5m): http://y2u.be/fwD98HiQSJc

    BTW, I hope you all enjoyed Pi day yesterday.

    March 15, 2014

  • Ahngelique D.

    So excited about this one. My friend and I were just arguing about "the last theory" and Fermet. Maybe this discussion will help chose me as the winner

    1 · March 8, 2014

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