A skeptical history of numbers Curtis T McMullen Harvard - PowerPoint PPT Presentation

A skeptical history of numbers Curtis T McMullen Harvard University Number theory

Algebra Whole numbers and so on Solve a x 2 + b x + c = 0. √ b 2 − 4 ac x = − b ± N = { 0 , 1 , 2 , 3 , . . . } 2 a 820 AD Z = { . . . , − 2 , − 1 , 0 , 1 , 2 , 3 , . . . } (algoritmi) Mu ḥ ammad ibn M ū s ā al-Khw ā rizm ī Diophantus 210 AD Q = { 22 / 7 , 94 / 100 , − 2 / 3 , 47 / 50 , . . . } Linear equations: ax + b = 0 Al-kit ā b al-mukhta ṣ ar f ī ḥ is ā b al- ğ abr wa’l-muq ā bala Solving the quadratic, circa 2000 BC Irrational numbers 2 , 5 2 / 17 , √ √ √ 3 5 Q = { 3 , . . . } 5 + x 2 = 2 √ x = 2 Solving the quartic, circa 1500 AD Solving the cubic, circa 1500 AD Solving the cubic, circa 1500 AD x 3 = x + 1 √ √ p p 3 3 9 − 69 + 9 + 69 x = √ 3 18

Quintic polynomials Solving the quintic, circa 2000 AD (Doyle-M) x 5 = x + 1? Abel: Cannot be expressed in terms of nth roots and whole numbers. Solving the sextic, circa 2000 AD Geometry Quintic polynomials x 5 = x + 1? x = 1 . 1673039782614186843 .... What kind of number is this?

π Real numbers Euclid √ 2 Plato ca. 300 BC 360 BC R π = 3.1415926535897.... the continuum Imaginary numbers: √ -1 Every polynomial has a root in the complex numbers. Squaring doubles angles The fundamental theorem of algebra (Gauss, 1799) √ − 1 Proof: √ -1 0 1 0 2

Whole number equations P ( z ) = 0 To solve: Look at: z 7! P ( z ) 5 2 + 12 2 = 13 2 X 2 + Y 2 = Z 2 0 n + 1 n = 1 n X n + Y n = Z n 5 2 = 3 3 − 2 Y 2 = X 3 - 2 ◆ 2 ◆ 3 ✓ 113259286337279 ✓ 2340922881 = − 2 449455096000 58675600 Large Numbers Powers of 10 MMMDCCCLXXXVIII = 3,888 250 BC: Archimedes: The Sand Reckoner myriad = 10,000 Charles and Ray Eames, 1968 / 1977 myriads of myriads of ... estimated 10 63 grains of sand to fill the universe.

Towers Wowsers T (1) = 10 the untamed power of induction! W(1) = 10 T (2) = 10 10 = 10 billion W(2) = T(W(1)) = tower of height 10 Googol = 10 100 = 10,[...100 zeros]...000 >> atoms in observable Universe W(3) = tower of height W(2) T (3) = 10 (10 10 ) = 10,000,...[10 billion zeros]...000 <<< Graham’s number G 12 < N < G T (4) = 10 10 1010 10 10 1034 << Skewes’ number = 1977 = bound for when first π (x) > li(x) 1933 = size of our ignorance ⌧ T (5) , . . . Paradox of Infinity Busy beaver function Zeno 430 BC B(n) = largest possible output of a rogue but mortal computer program of length n Is this number defined?

Infinitesimals P ( t + ✏ ) − P ( t ) dP ˙ = = P dt ✏ for every ✏ > 0 there exists a � .... All Calculus, Physical laws Newton 1689 Infinity Many infinities N = {0,1,2,3,4,....} the number of | N | {0,1,2,3,...} possible books is smaller than the number of points | R | {all real numbers} in a line (or cube or...) the silent majority -2 -1 0 1 2 3 γ e π √ 2 ......

Set theory Frege and Russell 1903 Georg Cantor 1845-1918 “No one shall expel 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.” Picture of the atom Crisis! Berry’s number N = [the smallest positive integer not definable in fewer than twelve words] Russell’s paradox Let A = {all sets which are not members of themselves}. Is A a member of A?

20th century revolutions Foundational Crisis: Solutions(?) Absolute space Relativity (1) Be careful not to define A in terms of A. (Type theory) Solar system atom Quantum atom (2) Only deal with things you can construct. Determinism Uncertainty (Intuitionism) Positivism Existentialism (3) Agree on Axioms, and only admit conclusions from them. The Dust Settles Hilbert 1930 Gödel 1931 A: Refused to accept the uncertainties of quantum mechanics (God playing dice) Mathematics is, For us there is no ignorabimus , and in my opinion none whatever in natural science. B: Established that mathematics and will always Wir müssen wissen — wir werden wissen! will never be complete. be, incomplete.

Incompleteness: What is chaos? Some questions have no answers Are there infinitely many Mersenne primes p = 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 strange x n+1 = x n2 - c attractors (chaos) x 0 = 0 (c=0) 0 ⇒ 0 ⇒ 0 ⇒ 0... cascade of period doublings c (c=1) 0 ⇒ -1 ⇒ 0 ⇒ -1.... attractor of x 2 -c (c=3) 0 ⇒ -3 ⇒ 6 ⇒ 33 ....

Island of order in a c = 1.5: order or chaos? sea of chaos c = 1.5 But the integers exist! ⇒ Arithmetic is consistent Is mathematics 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 product of human imagination; the story is simply made up.

⇒ Non-standard numbers Consistency radius [0, 1, 2, ..., n, n+1, ................., N-1, N, N+1 ....] Contradictions: at what scale? standard non-standard N = Length of the shortest proof that 0=1. A. Everything that used to be Gödel: true is still true. We can assume N is finite without danger! B. 0 is standard C. n standard ⇒ n+1 standard ``Healthy skepticism’’ D. There exists a nonstandard N Edward Nelson, 1932-2014 The vacuum Virtues of non-standard numbers is not empty Newton rehabilitated Cantor deprecated Analysis simplified N non-standard ε replaced by 1/N ∞ replaced by N ⇒ working theory avoids measure of infinitesimals theory 70% of the Universe is made up of inconsistencies

Mathematics is a model What image of mathematics fits best with the world as we now know it? Dark Energy 10 10 101010101010