A sketical history of numbers Curtis T McMullen Harvard University Number theory
Algebra Whole numbers and so on Solve a x2 + b x + c = 0. N = {0, 1, 2, 3,...} x= b± 820 AD Muḥammad ibn Mūsā al-khwārizmī Z = {..., 2, 1, 0, 1, 2, 3,...} b2 2a Linear equations: ax + b = 0 Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa l-muqābala 11/9/15, 9:16 PM Solving equations through the ages Various authors Solving equations through the ages 11/9/15, 9:16 PM Irrational numbers Solving the quadratic, circa 2000 BC Solving equations through the ages Various authors Solving the quadratic, circa 2000 BC Solving equations through the ages 11/9/15, 9:16 PM Solving the quartic, circa 1500 AD Solving the cubic, circa 1500 AD Solving the cubic, circa 1500 AD 3 5 Q = { 2, 52/17, 5 + 3,...} 2 x= 2 x =2 x3 = x + 1 x= file:///users/ctm/www/gallery/olynom/index.html file:///users/ctm/www/gallery/olynom/index.html Page 1 of 4 Page 1 of 4 (algoritmi) Diohantus 210 AD Q = {22/7, 94/100, 2/3, 47/50,...} Solving equations through the ages 4ac 3 9 3 69 + 9 + 69 3 18
Quintic olynomials x 5 = x + 1? Solving the quintic, circa 2000 AD (Doyle-M) Abel: Cannot be exressed in terms of nth roots and whole numbers. Quintic olynomials x 5 = x + 1? Geometry x =1.1673039782614186843... What kind of number is this?
Plato 360 BC Euclid ca. 300 BC 2 Real numbers R π π = 3.1415926535897... the continuum Imaginary numbers: -1 Squaring doubles angles 1 Every olynomial has a root in the comlex numbers. The fundamental theorem of algebra (Gauss, 1799) Proof: -1 0 0 1 2
To solve: Look at: Whole number equations P (z) = 0 z 7! P (z) X2 + Y2 = Z2 52 + 122 = 132 Xn + Yn = Zn 0n + 1n = 1n Y2 = X 3-2 52 = 33 Large Numbers 113259286337279 449455096000 2 = 2340922881 58675600 Powers of 10 MMMDCCCLXXXVIII = 3,888 250 BC: Archimedes: The Sand Reckoner myriad = 10,000 myriads of myriads of... estimated 1063 grains of sand to fill the universe. Charles and Ray Eames, 1968 / 1977 3 2 2
Towers Wowsers T (1) = 10 the untamed ower of induction! W(1) = 10 T (2) = 1010 = 10 billion Googol = 10100 = 10,[...100 zeros]...000 >> atoms in observable Universe T (3) = 10(10 T (4) = 1010 10 ) 1010 = 10,000,...[10 billion zeros]...000 << Skewes number = 10 = bound for when first π(x) > li(x) T (5),... Busy beaver function B(n) = largest ossible outut of a rogue but mortal comuter rogram of length n Is this number defined? 1010 1933 34 W(2) = T(W(1)) = tower of height 10 W(3) = tower of height W(2) <<< Graham s number G 12 < N < G 1977 = size of our ignorance Paradox of Infinity Zeno 430 BC
Infinitesimals P = dp dt = P (t + ) P (t) for every > 0 there exists a... All Calculus, Physical laws Newton 1689 Infinity N = {0,1,2,3,4,...} N R Many infinities the number of {0,1,2,3,...} ossible books is smaller than the number of oints {all real numbers} in a line (or cube or...) the silent majority -2-1 0 1 2 3... γ 2 e π
Set theory Frege and Russell 1903 Georg Cantor 1845-1918 No one shall exel us from the Paradise that Cantor has created. David Hilbert "Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. Crisis! Picture of the atom Berry s number N = [the smallest ositive integer not definable in fewer than twelve words] Russell s aradox Let A = {all sets which are not members of themselves}. Is A a member of A?
20th century revolutions Foundational Crisis: Solutions(?) Absolute sace Solar system atom Determinism Positivism Relativity Quantum atom Uncertainty Existentialism (1) Be careful not to define A in terms of A. (Tye theory) (2) Only deal with things you can construct. (Intuitionism) (3) Agree on Axioms, and only admit conclusions from them. The Dust Settles Hilbert 1930 Gödel 1931 For us there is no ignorabimus, and in my oinion none whatever in natural science. Wir müssen wissen wir werden wissen! Mathematics is, and will always be, incomlete. A: Refused to accet the uncertainties of quantum mechanics (God laying dice) B: Established that mathematics will never be comlete.
Incomleteness: Some questions have no answers What is chaos? Are there infinitely many Mersenne rimes = 2 n -1? Is there a set A with N < A < R.? Is the dynamical system x x 2 - c chaotic for c = 1.5? Quadratic dynamics xn+1 = xn 2 - c x0 = 0 0 0 0 0... (c=0) 0-1 0-1... (c=1) 0-3 6 33... (c=3) c cascade of eriod doublings attractor of x 2 -c strange attractors (chaos)
Island of order in a sea of chaos c = 1.5: order or chaos? c = 1.5 But the integers exist! Is mathematics consistent? Arithmetic is consistent Kronecker, 1865 0=1?! Axioms Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk "God made the integers, all else is the work of man." Deductions Nelson, 2010 The notion of the actual infinity of all numbers is a roduct of human imagination; the story is simly made u.
Consistency radius Contradictions: at what scale? N = Length of the shortest roof that 0=1. Gödel: We can assume N is finite without danger! ``Healthy sketicism Non-standard numbers [0, 1, 2,..., n, n+1,..., N-1, N, N+1...] standard non-standard A. Everything that used to be true is still true. B. 0 is standard C. n standard n+1 standard Edward Nelson, 1932-2014 D. There exists a nonstandard N Virtues of non-standard numbers Newton rehabilitated Cantor derecated Analysis simlified The vacuum is not emty ε relaced by 1/N working theory of infinitesimals N non-standard relaced by N avoids measure theory 70% of the Universe is made u of inconsistencies
Dark Energy Mathematics is a model What image of mathematics fits best with the world as we now know it? 10 101010 10 101010