What do you think are the qualities of a good theorem? it solves an open problem (Name one..? )

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1 What do you think are the qualities of a good theorem? Aspects of "good" theorems: short surprising elegant proof applied widely: it solves an open problem (Name one..? ) creates a new field might be easy to prove but hard to realize it should be proved. (name such a theorem) introduces new proof techniques controversial Theorems: Fundamental theorem of algebra Proofs by D Alembert (1746) and others criticized by Gauss (1799) D Alembert s main lemma given correct proof by Argand (1806) D Alembert s proof completely fixed by Weierstrass (1876) by making the concept of continuity rigorous Jordan curve theorem Jordan s proof (1852) deemed incorrect Proof by de la Valeé-Poissin also considered incorrect Correct proof given by Oswald Veblen (1905) Tom Hales: I have contacted a number of authors who have criticized Jordan, and in each case the author has admitted to having no direct knowledge of an error in Jordan s proof lot s of recent discussion of why the result is not obvious on math.stackexchange.com Cantor s theorem Kronecker: I don t know what predominates in Cantor s theorem, philosophy or theology, but I know there is no mathematics there. S. Feferman (born 1928): Cantor s theories are simply not relevant to everyday mathematics Four color theorem 1

2 Conjectured by Guthrie in 1852 Proofs by Kempe (1879) and Tait (1880) later shown to be incorrect Proof using a computer (and children of the provers) found by Appel and Haken, Apparently the gaps have been fixed subsequently Halmos: We don t learn anything from it Bayes theorem (1763) Bayesian statistics was in a certain amount of disrepute until recent years Gödel s theorem (does it prove that our brains are not computers? Penrose) "problem solvers vs theory builders" But: Fermat s last theorem is a problem. Attempts to solve it led to a sophisiticated theory, which was finally used in the solution. Today theory builders are on top, and many look down on problem solvers. Theorems in this course ( Big Ideas ) 1. Theorems in Euclid s Elements. Many Examples, includung Pythagoras s theorem quadratic formula (geometric cases - no negative or complex solutions) golden mean construction There are at most five Platonic solids. All five exist Euclidean algorithm gives the lcd pentagon-hexagon-decagon theorem There are infinitely many prime numbers 2. Others: 2

3 2 is irrational (School of Pythagoras) The area under a parabola (Archimedes) Euler s formula e = v + f 2 Jordan curve theorem symmetry groups of the regular polyhedra crystallographic restriction theorem existence and properties of Penrose tiles The regular tetrahedron does not tile space π2 6 = n=1 1 n 2 Fundamental theorem of algebra The set of all real numbers is uncountable The cardinality of (0, 1) is the same as the cardinality of (, ) Fermat s little theorem binomial theorem for general rational exponents series for trig functions, logarithms how to solve a cubic equation arc length along the lemniscate of Bernoulli further formulas for π Chebyshev s sum theorem: 1 0 fg 1 0 f 1 0 g if f,g are continuous and non-decreasing then rational solutions of quadratic and elliptic equationjs (y 2 = x 3 + ax + b) 3

4 top 100 theorems by one poll of mathematicians: I ll list those among that top 10 we have mentioned in class or otherwise will be known to you, but I won t list them in order. How would you rank them? Denumerability (countability) of the rational numbers Fermat s little theorem The impossibility of trisecting and angle or doubling a cube (using straight edge and compass) Pythagorean theorem There are infinitely many prime numbers Gödel s incompleteness theorem fundamental theorem of algebra 2 is irrational The area of a circle 4

5 Here s the complete list through #17, most of which you are likely to have heard of, plus a group further down in the list which also may be familiar to you is irrational 2. fundamental theorem of algebra 3. Denumerability (countability) of the rational numbers 4. Pythagorean theorem 5. Prime number theorem (statement on last page) 6. Gödel s incompleteness theorem 7. Law of Quadratic Reciprocity (statement on last page). 8. The impossibility of trisecting and angle or doubling a cube (using straight edge and compass) 9. The area of a circle 10. Fermat s little theorem 11. The infinitude of the primes 12. Euclid s postulate V on parallel lines cannot be proved from Postulates I-IV 13. Euler s formula: e = v + f π 2 6 = n=1 1 n Fundamental theorem of integral calculus 16. Insolvability of p (x) = 0 if p is a polynomial of degree n DeMoivre s theorem: (cos θ + i sin θ) n = cos nθ + i sin nθ Green s theorem 22 uncountability of the real numbers 27 sum of angles of a triangle 33 Fermat s last theorem 35 Taylor s theorem 37 Solution of a cubic 5

6 44 Binomial theorem 53 π is transcendental 67 e is transcendental 74 principle of mathematical induction 75 The mean value theorem 76 Fourier series 77 sums of k th powers 78 Cauchy-Schwarz inequality 79 The intermediate value theorem 91 The triangle inequality 94 The law of cosines ;: 6

7 Here are the top 10 again, with a competing list: 2 is irrational (Greece) e iπ = 1 (Euler) fundamental theorem of algebra e = v + f 2 (Euler?) countability of rational numbers (Cantor) Pythagorean theorem (Greece) There are infinitely many primes (Greece) There are only 5 regular polyhedra (Greece) Prime number theorem π 2 6 = n=1 1 n 2 (Euler) = 1; π (x) = #primes x lim x π(x) x/ ln x Gödel s incompleteness theorem Law of Quadratic Reciprocity: If p, q are prime, then x 2 = p mod q has a solution if and only if x 2 = q mod p has a solution or p and q are each of the form 4n + 3. Impossible to trisect an angle or double a cube Area of a circle Fermat s little theorem Brouwer fixed point theorem: a continuous map from x 2 + y 2 1 into itself has a fixed point 2 is irrational (Greece) π is transcendental every plane map can be colored with just four colors If 4n+1 is prime, it is the sum of two squares, uniquely (Fermat) Conclusion? 7