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.”

• Far-out ideas and amusing stories to tell• But also one BIG thing to communicate

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

• 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

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?

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

Quantum mechanics, time

Quarternions, Spin 1/2

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

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?

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

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

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

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)

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?

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