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
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, 1976. 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
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
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
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. 1. 2 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 2 14. π 2 6 = n=1 1 n 2 15. Fundamental theorem of integral calculus 16. Insolvability of p (x) = 0 if p is a polynomial of degree n 5. 17. DeMoivre s theorem: (cos θ + i sin θ) n = cos nθ + i sin nθ 18.... 21 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
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
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