Introduction to Logic: The quest for truth Bernhard Huemer
Jul 16, 2015
Introduction to Logic: The quest for truth
Bernhard Huemer
The Seven Days Of Creation
Computer scientists’ version Mathematicians’ versions
“God threw something together under a 7-day deadline. He’s still
debugging.”
“God laid down axioms, and all else followed trivially.”
Mathematics at the beginning of the 20th century
Some mathematicians began an ambitious project:
to prove everything.
Sentiment of the time: Analytical machine
Ada Lovelace
Russell’s paradox
Suppose there is a town with just one barber, who is male. In this town, every man keeps himself clean-shaven, and he does so by doing exactly one of two things:
• shaving himself; or • being shaved by the barber.
Also, "The barber is a man in town who shaves all those, and only those, men in town who do not shave themselves."
• “an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven”
• Construction of paradoxical sets can be avoided by adding types
Principia Mathematica
.. but then something happened
Kurt Gödel• Mathematician from Austria
• 1906 - 1978 († by starvation)
• Completeness theorem
• close friends with Einstein, also published papers on relativity
Euclid’s postulates1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
• If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
• Conversely, will these two lines ever cross each other?
Euclid’s parallel postulate
Hyperbolic / non-euclidean geometry
Three traditional laws of thought
• The law of identity
• The law of contradiction
• The law of the excluded middle
http://en.wikipedia.org/wiki/Law_of_thought#The_three_traditional_laws
http://www.thenewsh.com/~newsham/formal/curryhoward/
“Just as there are geometries in which Euclid's fifth postulate is not assumed to be true, there are logic systems in which the double-negation rule is not assumed to be true.”
Inference rules
Modus ponens / tollens
If it is raining, the streets are wet. The streets are not wet.
Therefore, it is not raining.
All men are mortal. Socrates is a man.
Therefore, Socrates is mortal.
• Proving the Law of the excluded middle by contradiction, i.e. assume ¬(p ∨ ¬p) and look for a contradiction
Proving by contradiction
Example
There exists irrational numbers and such that is rational.x y xy
is rational
is not rational
Classical proof
22
22
x = 2
y = 2
x = 22
y = 22
2⋅ 2= 2
• Does not tell us which of the two is true .. not particularly useful :(
Constructive proof
x = 2y = 2 ⋅ log2(3)
Intuitionistic logic: Definition
Taken from: Lectures on the Curry-Howard Isomorphism, Volume 149 (Studies in Logic and the Foundations of Mathematics)
Construction rules
Taken from: Lectures on the Curry-Howard Isomorphism, Volume 149 (Studies in Logic and the Foundations of Mathematics)
Classic tautologies that can be proved constructively
Taken from: Lectures on the Curry-Howard Isomorphism, Volume 149 (Studies in Logic and the Foundations of Mathematics)
Classical tautologies that aren’t constructive too
Taken from: Lectures on the Curry-Howard Isomorphism, Volume 149 (Studies in Logic and the Foundations of Mathematics)
Curry-Howard Correspondence
• Direct relationship between computer programs and mathematical proofs
• e.g. tautologies and identity functions ρ → ρ
Curry-Howard Correspondence
Category theory: The rosetta stone
Taken from the paper “Physics, Topology, Logic and Computation: A Rosetta Stone”
More to come ..
Hilbert’s Entscheidungsproblem
• 1936 both Alan Turing and Alonzo Church published papers showing that this was impossible
• If the Halting Problem wasn’t in fact a problem, one could theoretically guess the solution for Fermat’s Last Theorem (i.e. mechanization of mathematics)
an + bn = cn
Q&A
“Either mathematics is too big for the human mind or the human mind is more than
a machine.”
- Kurt Gödel