About Math and related topics About Math 1

The Unreasonable Effectiveness The Unreasonable Effectiveness of Mathematics in the Natural Sciences by Eugene Wigner About Math 2

Effectiveness of Mathematics Mathematical concepts turn up in entirely unexpected settings Often permit very close and accurate descriptions of phenomena We are limited because we don’t know why mathematics works and whether it is unique About Math 3

Mysterious Usefulness Usefulness borders on the uncanny We have no rational explanation Raises the question of uniqueness of physical theories Position of a man with a ring of keys and a set of matching doors About Math 4

What is Mathematics Elementary concepts clearly evolved in connection with entities in the actual world. More abstract concepts were devised to be interesting to mathematicians, not to mimic reality. Yet they very often do. Amazing that Darwin’s natural selection brought human reasoning power to it’s current high level. About Math 5

What is Mathematics (cont.) Consider complex numbers: No analog in reality Basis for a large field in theory of equations, power series, and analytic functions. Very valuable in classical physics Essential to quantum mechanics About Math 6

What is Physics? Schrodinger: a miracle that we can discover regularities in real events Galileo’s dropped weights Invariance in time and space Results are true in Pisa and London and Tokyo Results are true in 1596 A.D. and 6000 B.C. Invariance under other externalities Doesn’t matter if it’s sunny or cloudy About Math 7

What is Physics? (cont.) If there were no phenomena which are independent of all but a manageably small set of conditions, physics would be impossible. It is not at all natural that “laws of nature” exist, much less that man is able to discover them. About Math 8

What is Physics? (cont.) Laws of nature incorporate only a small part of our knowledge of reality. They do not include information about the existence of bodies, initial conditions, etc. About Math 9

Mathematics in Physical Theories Evaluating the consequences of established theories is not the most important function of math – this is applied math serving as a tool. The laws of nature are written in the language of mathematics. Sometimes the appropriate mathematics is independently (re)discovered by the physicist. About Math 10

Math in Physics (cont.) Important to note that mathematics is developed to please the mathematician, not, usually, to fill a perceived need. Miraculous that abstract mathematics keeps cropping up in physics and that the human mind can follow long chains of complex reasoning. About Math 11

Math in Physics (cont.) Expressing crude experience in mathematical form remarkably often leads to an amazingly accurate description of a large class of phenomena. Suggests that mathematics is more than just a human convenience. Consider the example of planetary motion. About Math 12

Planetary Motion Newton considered freely falling objects and the Moon. Arrived at equations of motion and gravity. Newton verified gravity to about 4%. Theory has proved accurate to better than %. Laws of motion are not obvious Second derivative is not a product of common sense About Math 13

Summary of Math in Physics Laws of nature based on abstract mathematical objects Proposed on the basis of crude measurements Lead to predictions of amazing accuracy Are very limited in scope Wigner calls these observations “the empirical law of epistemology.” an article of faith of the physicist About Math 14

Theories of Everything Will new theories include such externalities as initial conditions? Will disparate physical laws prove to be approximations of deeper underlying rules? Will some physical laws never be brought into a greater structure? About Math 15

General Relativity and Quantum Mechanics Currently no connection between these two important theories. Much effort, arguable progress, and wide belief that such a connection is possible. We should entertain the possibility that the connection will not, possibly cannot, ever be made. About Math 16

False Theories About Math 17

False Theories (cont.) Ptolemy’s epicycles About Math 18

Caution So long as we don’t know why mathematics works so wonderfully with physics, we cannot be certain that accuracy proves truth and consistency. About Math 19

Wigner’s Coda The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. About Math 20

Hamming’s Views The Unreasonable Effectiveness of Mathematics by R.W. Hamming About Math 21

We See What We Look For How might Galileo study falling bodies without doing an experiment? Think about when two bodies compose a single body Think about when one body comprises two bodies About Math 22

Math is Inconsistent Remarkable proofs by Kurt Godel About Math 23

Drawn from Godel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter About Math 24

Outline Developed a way to associate every proof in a the theory of natural numbers one-to-one with an infinite integer enumerate the proofs in a mechanical way About Math 25

Outline II About Math 26

Outline III We have now created an infinite number which is not in the set and which is a true proposition in the mathematical system under examination There exist true statements which cannot be proved from the axioms and theorems of the system of natural numbers. About Math 27

About Math 28 Finis