LOGIC NUMBER THEORY AND THE LIMITS OF REASON

LOGIC, NUMBER THEORY AND THE LIMITS OF REASON M. Ram Murty Queen’s Research Chair Queen’s University, Canada

What is a number? § Counting is basic to all civilizations. § From the Babylonians onwards, there are many notations for the natural numbers. § The place value number systems were discovered by only four cultures. § They are the Babylonians, Mayans, Chinese and the Indians.

ZERO § The Babylonians, the Mayans and the Indians were the only ancient civilizations with a symbol for zero. § However, it was only in the Indian civilization that zero was treated as a number and rules were given about how to work with it. § In his work Brahmasphutasiddhanta, written in 600 A. D. , Brahmagupta gave the first modern treatment of the algebraic rules for zero.

Aryabhata § Aryabhata was the first to use the decimal system for calculations. § He was also the first to introduce algebraic notation. § He also was the first to propose a heliocentric model for the solar system. Aryabhata (476 -550 A. D. )

Aryabhatiya written in 499 A. D.

Diophantine Equations § Arybhata studied ax + by = c for given a, b and c. He was interested in finding integer solutions x and y for this equation. § His successor Brahmagupta considered the equation ax² + by²=c and discovered a remarkable algorithm to generate integer solutions. § This work was expanded later by Jayadeva and Bhaskaracharya. § Such equations are called Diophantine equations after Diophantus who first studied them.

A page from the Brahmasphutasiddhanta A page from the Lilavati

Can we always solve a Diophantine equation? § This is a question we will discuss later. § The early Indian mathematicians have shown that for small degree equations, it is possible to solve them. § But what happens for higher degrees? § It is remarkable that these questions were raised in antiquity in ancient India.

The discovery of numbers ``The astonishing progress that the Indians made is now well known and it is recognized that the foundations of modern arithmetic and algebra were laid long ago in India. The clumsy method of using a counting frame, and the use of Roman and such numerals had long retarded progress when the ten Indian numerals including the zero sign liberated the human mind from these restrictions and threw a flood of light on the behavior of numbers. …They are common enough today and we take them for granted, yet they contained the germs of revolutionary progress in them. It took many centuries for them to travel from India, via Baghdad, to the western world. ’’ –The Discovery of India. Jawaharlal Nehru (1889 -1964)

Fibonacci § Fibonacci introduced the decimal system into Europe in 1200. § It took another four centuries for the system to be commonplace in the Western world. Fibonacci (1170 -1250)

§ ``It is India that gave us the ingenious method of expressing all numbers by means of ten symbols … a profound and important idea which appears so simple now … but its very simplicity … puts our arithmetic in the first rank of useful inventions and we shall appreciate the grandeur of this achievement when we remember that it escaped the genius of Archimedes and Apolonius, two of the greatest men produced by antiquity. ’’ -Laplace Pierre Simon de Laplace (1749 -1827)

§ Plato believed that numbers are ideal entities existing independently of the human mind. § Einstein wrote that numbers are a creation of the human mind that simplify the ordering of sensory experience.

Pythagoras § Pythagoras saw a deep connection between the natural world and the world of natural numbers. Pythagoras (569 -475 BC)

Baudhayana Sulva Sutras § The theorem usually attributed to Pythagoras is found in these sutras written around 800 B. C.

Mathematical laws § Since antiquity, many mathematical laws have § § been discovered suggesting that somehow mathematics has been woven into the fabric of the universe and one cannot separate it. This feeling led Pythagoras to say “everything is number. ” Music and numbers are closely related too. The geometry of the universe was linked to numbers. This led Plato to say “God is a geometer. ”

§ “The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and there is no rational explanation of it. ” in “The unreasonable effectiveness of mathematics in the natural sciences’’ Eugene Wigner (1902 -1995)

§ ``How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality. ’’ in Sidelights on Relativity. Albert Einstein (1879 -1955)

Leibniz’s Dream § Leibniz dreamed of a universal language and some sort of “calculus of reason” which would reduce all problems to numerical computation. Leibniz (1646 -1716)

George Boole § In 1848, Boole wrote a paper entitled “The calculus of logic” in which he begins “I have exhibited the application of a new form of mathematics to the expression of the operations of the mind in reasoning. ” George Boole (1815 -1864)

The Ladder of Infinity § The numbers 1, 2, 3, … are countable. § They are in one-to-one correspondence with 1, 1/2, 1/3, … § Both sets are infinite. § How many numbers are there between 0 and 1?

Cantor and Infinity § Discovered different orders of infinity § The natural numbers are countable § The real numbers are uncountable § The continuum hypothesis Georg Cantor (1845 -1918)

The continuum hypothesis § Given any infinite set of numbers between 0 and 1, it is either countable, that is, can be put in one-to-one correspondence with the natural numbers or it can be put in one -to-one correspondence with all the numbers between 0 and 1. § That is, there are only two orders of infinity for subsets of [0, 1].

Hilbert’s Problems § The Continuum hypothesis § Axioms for arithmetic. § Algorithm for solving Diophantine equations David Hilbert (1862 -1943)

Peano axioms § 0 is a number. § Every number has a § § § successor. 0 is not the successor of any number. Distinct numbers have distinct successors. (Induction) If a property holds for 0 and it holds for the successor of every natural number, then it holds for all numbers. Giuseppe Peano (1858 -1932)

Can we derive all “truths” about numbers from these five axioms? § In 1977, Jeff Paris and Leo Harrington found a “natural” § § § § truth which cannot be derived from these axioms. A “simple” example is given by Goodstein’s theorem. Take any number; write it in base 2. Write the exponents in base 2 until everything on the page is in base 2. Replace all the 2’s by 3’s thereby increasing the number. Subtract 1 from this number. Write this number in base 3 and all the exponents in base 3 as before. Subtract 1 from this number. Continue in this way. After a finite number of steps, you get zero!!!

Principia Mathematica § The Principia published in 1910 was an attempt to derive all mathematical truths from a welldefined set of axioms and inferential rules of symbolic logic.

Godel’s theorem § No axiom system can prove all the truths about the natural numbers. § Godel’s incompleteness theorems.

Extending Peano’s axioms § Zermelo and Fraenkel enlarged the axioms to include the “axiom of choice” which enables one to prove Goodstein’s theorem. E. Zermelo(1871 -1953) A. Fraenkel (1891 -1965)

The continuum hypothesis is undecidable § In 1963, Paul Cohen showed that the continuum hypothesis is not provable in ZFC. Paul Cohen

What are Diophantine equations? § In the simplest case, these are equations with integer coefficients. § For example, 3 x² + 5 y³ = 1 is a Diophantine equation. § This equation has no integer solutions, which is easily seen modulo 5. § What about 3 x² + 5 y³ = 2? Does this have integer solutions?

Let’s look at 3 x²+5 y³=2 § If this equation has an integer solution, then § § § § multiplying by 3, we get that (3 x)² + 15 y³=6 also has an integer solution. Multiplying by 15², we see that (45 x)² + (15 y)³ = 1350 also has an integer solution. This leads to integer solutions of the elliptic curve y² = x³ + 1350. This curve has only two integer solutions: (-5, 35) and (-5, -35). This means 3 x²+5 y³=2 has NO integer solutions.

What about 3 x²+ 5 y³=3? § There are the obvious solutions x=1, y=0 and x=-1, y=0. § Are there any more? § Yes, there is one more: x=19, y=-6. § We need theory of elliptic curves to prove that there are no further solutions!

General Diophantine Equations § Any equation of the form § f(x 1, x 2, …, xn) = 0 where f is a polynomial with integer § § coefficients is called a Diophantine equation. xn + yn = zn is a famous example of a Diophantine equation. Fermat asked if this equation has any non-trivial solutions and conjectured that there were none. This was proved by Andrew Wiles following the work of Ken Ribet in 1995. Hilbert’s 10 th problem asks if there is a universal algorithm for determining whether a given Diophantine equation has an integer solution.

Hilbert’s 10 th problem § In 1970, Yuri Matiyesevich showed in his Ph. D thesis that Hilbert’s 10 th problem is unsolvable. § That is, there is no universal algorithm that works for all Diophantine equations. § He was building on earlier work of Martin Davis and Julia Robinson. Yuri Matiyasevich

Hilbert’s problems revisited § Three of Hilbert’s problems had a “negative” solution in the following sense. § Problem 1 is undecidable. § Problem 2 led to Godel’s incompleteness theorems. § Problem 10 is unsolvable.

Computability § Polynomial time algorithms (P) § Conjectural answer can be verified in polynomial time (NP) § X is NP complete: if every problem in NP can be reduced in polynomial time to X and X is also in NP.

Is there a polynomial time algorithm for testing if a given number is prime? § Given a number n, can we determine if n is § § § prime in (log n)² steps? Until 2002 this was a major unsolved problem. This question is related to cryptography and internet security, as is well-known. More importantly, factoring a number in polynomial time is an open question.

Agrawal, Kayal and Saxena won the Fulkerson prize this year. Primality Testing is in P Nitin Saxena Neeraj Kayal Manindra Agrawal

Hamiltonian cycles § Given a graph on n vertices, is there an algorithm to find a Hamiltonian cycle in polynomial time? § Answer: Not known. § This problem is NP complete. § If the answer is “yes” then P=NP.

Mathematical questions lead to the discovery of fundamental concepts § For example, 0 is a fundamental concept. § Mathematics is not something fixed but rather something that can be enlarged as we enlarge the axioms. § These axioms are dictated by observations and experience. § We want to minimize the number of axioms at any given stage.

What are the implications of Godel’s theorem? § Does it imply that human reason is limited? § On the contrary, it shows that the human mind cannot be axiomatized. § Computability is only one aspect of the mind. § Understanding lies deeper than computability.