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The Unreasonable Effectiveness of Mathematics

The Unreasonable Effectiveness of Mathematics. Bram Boroson , Clayton State University, 3/20/2013. Fox and Hedgehog. “The fox knows many things, but the hedgehog knows one BIG thing.” F ar-out ideas and amusing stories to tell But also one BIG thing to communicate.

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The Unreasonable Effectiveness of Mathematics

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  1. The Unreasonable Effectiveness of Mathematics Bram Boroson, Clayton State University, 3/20/2013

  2. Fox and Hedgehog • “The fox knows many things, but the hedgehog knows one BIG thing.” • Far-out ideas and amusing stories to tell • But also one BIG thing to communicate

  3. Quantum Physicist Eugene Wigner (1902-1995): Mathematics is “Unreasonably Effective”

  4. E=mc2, c is speed of light, a large number • So there is LOTS of Energy in Mass • But nobody trying to release energy would START by inventing equations or multiplication table

  5. The Big Problem • Math WORKS in science • NOT like a hammer works in hitting nails • Over and over math was invented “for kicks and giggles” and yet proved useful • Something about math? Or our world?

  6. Imaginary Friends • Little could seem more useless than “imaginary numbers” • We start learning math with counting • Then fractions, negative numbers • Irrational numbers • Every equation has a solution

  7. Quantum mechanics, time

  8. Quarternions, Spin 1/2

  9. Math: appeal of the General? • Maybe math works because it’s abstract • By talking about “2 plus 2” instead of apples, oranges, you can use it for apples OR oranges • To me this doesn’t seem to be good enough

  10. Max Tegmark’s MUH • Mathematical Universe Hypothesis: our world IS mathematics • Seems strange, radical! • “Mathematical Platonism”: math exists apart from us in its own world • We “discover” math truth using our minds instead of senses • Occam’s razor: isn’t it simpler if LESS exists?

  11. Our world’s math, but which Math? • Do all mathematical systems describe possible worlds? • Max Tegmark: only computable worlds • If our world is a mathematical world “pulled at random out of a hat”, what’s in the hat? Alan Turing

  12. Leibniz: Principle of Sufficient Reason • One approach: the world is a special mathematical OBJECT • What we can say about it only approximates what it is • It’s singled out among other possible objects • Nature never makes an arbitrary choice

  13. Principle of Sufficient Reason • Example in physics: infinite line charge • Much of modern physics is written in terms of maximizing/minimizing an “action” • Light bending (refracting) when passing through glass or water: minimizes the time it takes to go from A to B

  14. Principle of Sufficient Reason • Einstein’s General Relativity • Problem of rotating spheres, centrifugal forces, and bulges • Another example of Unreasonable Effectiveness: Tensor Calculus • Principle of Sufficient Reason FAIL: Quantum Mechanics (Stern-Gerlach Experiment)

  15. Approach 2: (early) Wittgenstein • “The world is made of facts, not things; each fact could be different and all the others the same” • 1, 2 prove 3… or 1, 3 prove 2? • But then WHY is our world so special that there are ANY laws?

  16. My own approach • Symbols of mathematics are “formal” • Geometry could be about “points, lines, planes” or “tables, chairs, desks” • The symbols may be reinterpreted • This property naturally allows for “axiom schema” • “Facts” don’t stay put: allow symbols to be re-interpreted

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