LOGIC, NUMBER THEORY AND THE LIMITS OF REASON

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LOGIC, NUMBER THEORY AND THE LIMITS OF REASON

• M. Ram Murty
• Queens Research Chair
• Queens University, Canada

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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.

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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.

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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.)
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Aryabhatiya written in 499 A.D.
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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 by²c 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.

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A page from the Brahmasphutasiddhanta
A page from the Lilavati
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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.

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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)
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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)
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• 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)
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• 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.

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Pythagoras

• Pythagoras saw a deep connection between the
  natural world and the world of natural numbers.

Pythagoras (569-475 BC)
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Baudhayana Sulva Sutras

• The theorem usually attributed to Pythagoras is
  found in these sutras written around 800 B.C.

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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.

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• 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)
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• 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)
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Leibnizs 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)
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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)
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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?

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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)
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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.

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Hilberts Problems

• The Continuum hypothesis
• Axioms for arithmetic.
• Algorithm for solving Diophantine equations

David Hilbert (1862-1943)
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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)
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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 Goodsteins
  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 2s by 3s 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!!!

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Principia Mathematica

• The Principia published in 1910 was an attempt
  to derive all mathematical truths from a
  well-defined set of axioms and inferential rules
  of symbolic logic.

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Godels theorem

• No axiom system can prove all the truths about
  the natural numbers.
• Godels incompleteness theorems.

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Extending Peanos axioms

• Zermelo and Fraenkel enlarged the axioms to
  include the axiom of choice which enables one
  to prove Goodsteins theorem.

E. Zermelo(1871-1953)
A. Fraenkel (1891-1965)
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The continuum hypothesis is undecidable

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

Paul Cohen
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What are Diophantine equations?

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

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Lets look at 3x²5y³2

• If this equation has an integer solution, then
  multiplying by 3, we get that
• (3x)² 15y³6 also has an integer solution.
• Multiplying by 15², we see that
• (45x)² (15y)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 3x²5y³2 has NO integer solutions.

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What about 3x² 5y³3?

• There are the obvious solutions x1, y0 and
  x-1, y0.
• Are there any more?
• Yes, there is one more x19, y-6.
• We need the theory of elliptic curves to prove
  that there are no further solutions!

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General Diophantine Equations

• Any equation of the form
• f(x1, x2, , 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.
• Hilberts 10th problem asks if there is a
  universal algorithm for determining whether a
  given Diophantine equation has an integer
  solution.

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Hilberts 10th problem

• In 1970, Yuri Matiyesevich showed in his PhD
  thesis that Hilberts 10th 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
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Hilberts problems revisited

• Three of Hilberts problems had a negative
  solution in the following sense.
• Problem1 is undecidable.
• Problem 2 led to Godels incompleteness theorems.
• Problem 10 is unsolvable.

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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.

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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.

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

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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.

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What are the implications of Godels 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.

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