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Math Chat: Mathematical Cognition & The $1,000,000 Math Problems

What is the nature of mathematical cognition or mathematics as a way of thinking? How did this faculty evolve in stone age humans? What selective advantage was conferred to early humans by mathematical cognition? Is the abstraction in mathematics built upon our language faculty? How is doing mathematics like a soap opera? Is mathematics like gossip?

What are the $1,000,000 Math Problems? We will discuss all seven of them: the Riemann Hypothesis, P vs NP Problem, Navier-Stokes Equation, the Poincaré Conjecture, the Birch and Swinnerton-Dyer Conjecture, the Hodge Conjecture, and Yang-Mills and Mass Gap. Help us and we'll split the $1,000,000 with you!

Our Math Chat topic this month is based on the 2 hour Keith Devlin video below. Read CJ's notes on Devlin's lecture

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

Supplementary resources on the $1,000,000 Math Problems. These are optional resources for anyone interested in learning more about these problems.

The Riemann HypothesisDescription of the problem, Paper on the problem, Video Lecture by Jeff Vaaler.

•  P vs NP Problem: Description of the problem, Ian Stewart's "Minesweeper" article on the problem, Video Lecture by Vijaya Ramachandran.

• Navier-Stokes Equation: Description of the problem, Video Lecture by Luis Cafarelli.

The Poincaré Conjecture: Description of the problem, News Release of the solution, more resources about Perelman's solution.

The Birch and Swinnerton-Dyer Conjecture: Description of the problem.

The Hodge Conjecture: Description of the problem, Video Lecture by Dan Freed.

• Yang-Mills and Mass Gap: Description of the problem, Report on the status of the problem by Michael Douglas, Video Lecture by Lorenzo Sadun.

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  • Greg G.

    Interesting and lively conversation
    Good manners and plenty of time to explore new ideas

    1 · May 24, 2014

  • Greg G.

    Interesting and lively conversation
    Good manners and plenty of time to explore new ideas

    May 24, 2014

  • Pak-Wing

    Pity we didn't cover more of the $1m problems!

    May 24, 2014

    • Greg G.

      Perhaps we should extend the meeting until ?????

      May 24, 2014

  • CJ F.

    For those who don't have time or interest to watch the Devlin video: here are my notes summarizing its content: https://plus.google.com/104222466367230914966/posts/PTWAJBRzjRJ

    May 23, 2014

  • CJ F.

    Tomorrow's "Math Chat" is based on an optional 2h video by Keith Devlin: http://y2u.be/NnVubBrATIU

    In the video Devlin asks (and proposes answers to) several profound questions about the evolution of human mathematical cognition:

    - How did the human brain acquire mathematical ability?

    - When did the brain acquire mathematical ability?

    - What evolutionary advantage did the ability to do math confer?

    His surprising answer that it is all based on gossip and our social psychology (mathematics is a soap opera??) and language will be worth discussing.

    The end of the video discusses the $1,000,000 math problems. Perhaps humans need new capabilities of the mind to solve these problems?

    Since several participants are interested in the Riemann Hypothesis, we will spend some time on it. We will also discuss (at least briefly) the Navier-Stokes Equation (video: http://claymath.msri.org/navierstokes.mov) and the P vs NP Problem (excellent video: http://claymath.msri.org/pversusnp.mov), etc.

    May 23, 2014

  • Sam B.

    As CJ pointed out, this week is Millennium problems. 7 in total, and 6 still unsolved for a prize of $1,000,000 a piece.
    The biggest problem going into any of these is trying to find anybody who can agree on what the problems are. The description on the bottom of the Yang-Mills and Mass Gap gives the most daunting path for discovering its answer: simply introduce fundamental new ideas in Physics and Mathematics. The problem is in particle physics, and seeks to establish a rigorous mathematical existence of the "Mass Gap" in the Yang-Mills theory. The Mass Gap being the non-existence of massless particles. A very intense area of study, that quantum mechanics.
    The second one today, the Hodge Conjecture, is about the relation between algebra and topology. There are some who claim an answer would work to recognize fraudulent data. This problem, like the other, is stated in a way that anybody outside of the fields are thoroughly lost. Any thoughts on trying to understand these problems?

    1 · May 22, 2014

  • Pak-Wing

    I am particularly interested in the Riemann and Navier-Stokes problems. I've dabbled in fluid mechanics before and can talk a little about the Navier Stokes problem if there is interest.

    1 · May 21, 2014

    • CJ F.

      Pak-Wing, excellent! I'll spend a few extra minutes on Riemann and Navier-Stokes.

      I just watched the Luis Cafarelli lecture on Navier-Stokes but it didn't help me much: http://claymath.msri....­

      Jeff Vaaler's lecture on the Riemann Hypothesis gave me a better sense of bearings, but not enough to really grasp the Riemann zeta function: http://claymath.msri....­

      May 21, 2014

  • CJ F.

    Saturday's "Math Chat" will discuss Keith Devlin's 2h video lecture at http://y2u.be/NnVubBrATIU­

    In addition to an interesting discussion about mathematical cognition, Devlin also surveys the Millenium Prize Problems which are worth $1,000,000 each if you can resolve them.

    More information about the problems is available at http://www.claymath.org/millennium-problems

    The oldest unsolved $1,000,000 problem is the Riemann Hypothesis (formulated in 1859) which is about the Prime Number Theorem and the solutions to the Riemann zeta function.

    The only solved Millenium problem is the Poincaré Conjecture which concerns topology and "surgeries in a general sense". In 2002, Grisha Perelman posted a proof which has been verified by experts as correct. He was awarded both the Fields Medal and the Millennium prize. He refused both prizes.

    Please comment if you want us to go more in depth on any of the Millenium problems. Otherwise, it is hard to know where to focus my effort.

    May 21, 2014

  • A. L. M.

    Won't be able to join you all this time. Sorry. But I do Look fwd to seeing you all at another event.

    1 · May 21, 2014

  • CJ F.

    Saturday's "Math Chat" will discuss Keith Devlin's 2h video lecture at http://y2u.be/NnVubBrATIU

    In addition to identifying 9 ingredients for mathematical cognition, Devlin argues that the crucial step in the development of mathematical ability was handling increased abstraction. He thinks this capability is built upon our language faculty (with its recursive grammar).

    His conclusion is somewhat surprising: "A mathematician is someone who views mathematics as a soap opera. ... The 'characters' in the mathematical soap opera are not people but mathematical objects --- numbers, geometrical figures, topological spaces, vectors, analytical functions, etc."

    Is math a soap opera of abstract math objects seen as characters? Isn't a vector space the antithesis of human relationships?

    Is mathematics a form of gossip exapted from our evolutionary development to manage complex human relationships?

    Is mathematics a social psychological construct?

    May 20, 2014

  • CJ F.

    Saturday's "Math Chat" will focus on topics raised in Keith Devlin video lecture: http://y2u.be/NnVubBrATIU

    In the video, Devlin identifies nine ingredients for a mathematical mind: number sense (sense of size), numerical ability (a capacity for understanding numbers), spatial reasoning, a sense of cause and effect, the ability to construct and follow a causal chain of facts or events, algorithmic ability (mathematical procedures), abstraction (generalization), logical reasoning, and relational reasoning (relationships).

    Does that list comprise the whole of our capacity for mathematical cognition? Did Devlin omit anything? When we think mathematically are we always exercising at least one of those faculties? Could we ever escape these nine capabilities to launch mathematics into a new era with a new way of thinking that this list doesn't envision? That is, is mathematics stable enough to be specified with nine basic cognitive skills and that's it? How would you define math thinking?

    May 19, 2014

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