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About Math About Math and related topics 1 The Unreasonable Effectiveness of Mathematics in the Natural Sciences by Eugene Wigner About Math The Unreasonable Effectiveness 2 𝒲 About Math Papers on Google Drive 3 • 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 Effectiveness of Mathematics 4 𝒲 Mysterious Usefulness About Math • Usefulness borders on the uncanny • We have no rational explanation • Raises the question of uniqueness of physical theories 5 𝒲 • Elementary concepts clearly evolved in connection with entities in the actual world. • counting, arithmetic • calculus (?) • More abstract concepts were devised to be interesting to mathematicians, not to mimic reality. About Math What is Mathematics • Yet they very often do. 6 𝒲 What is Mathematics • abstract geometry: General Relativity • analysis: calculus • group theory: Eightfold Way • Amazing that Darwin’s natural selection brought human reasoning power to it’s current high level. About Math • Examples 7 𝒲 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 • Crucial in engineering About Math What is Mathematics… 8 Do mathematical entities have some sort of reality separate from the human mind? Would an alien species have the same mathematics that we have? Are mathematical concepts somehow linked to human evolution? About Math What is Mathematics… 9 What is Physics? • Schrodinger: a miracle that we can discover regularities in real events About Math • Galileo’s dropped weights • Invariance in space • Results are true in Pisa and Tokyo • Invariance in time • Results are true in 1596 A.D. and 6000 B.C. • Invariance under other externalities • Doesn’t matter if it’s sunny or cloudy 10 𝒲 If there were no phenomena which were 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. 𝒲 About Math What is Physics? (cont.) 11 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 What is Physics? (cont.) 12 𝒲 • 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 Mathematics in Physical Theories 13 𝒲 • 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 Math in Physics (cont.) 14 𝒲 • 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. About Math Math in Physics (cont.) • Consider the example of planetary motion. 15 𝒲 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 0.0001%. • Laws of motion are not obvious • Second derivative is not a product of common sense About Math Planetary Motion 16 𝒲 • 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 Summary of Math in Physics 17 𝒲 Theories of Everything • Will disparate physical laws prove to be approximations of deeper underlying rules? • Will some physical laws never be brought into a greater structure? 𝒲 About Math • Will new theories include such externalities as initial conditions? 18 General Relativity and Quantum Mechanics • 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. 𝒲 About Math • Currently no connection between these two important theories. 19 We have experience with false theories that functioned well for a while, sometimes a very long while. • Aristotelian mechanics: About Math False Theories 𝐹 = 𝑚𝑣 20 𝒲 False Theories (cont.) About Math • Ptolemy’s epicycles 21 𝒲 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 Caution 22 𝒲 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 Wigner’s Coda 23 𝒲 About Math Finis 24 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 We See What We Look For 25 𝒲