of Proof Theory:
Proofs in Systematic Political Science
Dallas F. Bell, Jr.
Propaedeutics is a course of
preparatory instruction for an art or science. This paper is designed
to be a précis for the subject of proof theory. The field of
math has provided the tool of proofs to the great benefit of physics
and now systematic political science. Proof theory is a type of
mathematical logic that formalizes a mathematical object of a proof
to facilitate analysis. Proof theory is also a part of philosophical
logic. In philosophy, semantic reasoning (what is true?) for model
theory is distinguished from syntactic reasoning (what can be shown?)
in proof theory. Proof theory begins with a language--a set of
symbols of a set of syntax or strings of those symbols. Language
formulas are elements of that set. Axioms or theories are a collection
2. Proof Theory
Modern proof theory is often
seen as being established by the Prussian born David Hilbert.
Hilbert also developed math required for quantum mechanics. His
works were considered by some to be largely theologically based.
Kurt Gödel's incompleteness theory indicated that Hilbert's attempt
to reduce all mathematics to a finitistic formal system was not possible.
Hilbert's epsilon calculus uses an epsilon operator that replaces
quantifiers in predicate logic. It has been applied linguistically
to deal with anaphoric pronouns.
Kurt Gödel and Gerhard Gentzen,
a German student of Hermann Weyl, laid the foundation for structural
proof theory. Structural proof theory studies proof calculi that
supports analytical proof. The works of Gödel and others led
to recursion theory or computability theory. Computability theorists
study degree structures, reducibility, and relative computability.
One of David Hilbert's students
was Hermann Weyl. Weyl's work included gauge theory and foundations
of manifolds. He proved a character formula from a compact Lie
group. Marius Lie explained that a differentiable manifold was
a Lie group. Weyl and John von Neumann aided the understanding
of the symmetry structure of quantum mechanics. Emil Artin solved
one of the problems posed by Hilbert relating to field theory.
Artin used arguments of Evariste Galois. Georg Kreisel has been
noted for unwinding Artin's proof.
3. Quantum Mechanics
Quantum mechanics is a pillar
of modern theoretical physics. Quantum, Latin for how much, refers
to units assigned to physical qualities such as the energy of an atom
at rest. Max Planck found that waves can be measured in small
packets of energy called quanta. Planck's constant is a physical
constant describing the sizes of quanta. Albert Einstein saw the
wave as a particle or photon that had a discrete energy dependent on
its frequency. Many photons can occupy the same state and are
bosons. Only one electron, proton, neutron, or quark can occupy
one state and are fermions.
All fundamental particles in
nature are either bosons or fermions. The spin of a photon is
either +1 or -1 and the spin of a 4He atom is always zero. Many
bosons can occupy a single state. This allows them to behave collectively
and is seen in the behavior of lasers. Only one fermion can exist
in a given quantum state. This is known as Wolfgang Pauli's
exclusion principle where no two electrons in an atom can have identical
quantum numbers (see the Periodic Table).
In systematic political science,
we can see that people with the same theologies can occupy the same
institution or nation-state much like a boson. However, people
with different theologies behave as fermions and must occupy separate
institutions and nation-state systems. Because all human behavior
is composed of quantum type mechanics natural law must approximate the
laws of behavior.
Werner Heisenberg created matrix
mechanics from quantum mechanics. This was the first description
of the behavior of subatomic particles by associating their properties
with matrices. It was equal to Erwin Schrödinger's wave formulation.
In quantum mechanics a state function is a linear combination of eigenstates.
All these eigenstates are constantly rotating through time in a Schrödinger
picture. In the Heisenberg picture, the state vector does not
change with time. Paul Dirac's picture may be used as a midpoint
between those two pictures when the Hamiltonian operator corresponding
to the total energy of the system can be divided. The Hamiltonian
operator was named for the Irish mathematician, William Hamilton.
Generally, matrices are a table
of numbers in horizontal rows and vertical lines called columns.
A matrix with m rows and n columns is an m x n matrix. There are
many types of matrices such as a square matrix whose columns are probability
vectors and are stochastic. They are used to define Andrey Markov's
chains. Markov chains describe the states of a system at successive
Math and physics use terms
such as scale invariance, sometimes called power laws, to explain a
feature of objects or laws that do not change if the space is enlarged.
Dimensional analysis involves understanding physical situations with
a combination of different physical qualities. The dimensions
of physical qualities are mass, length, and time. E. Buckingham's
Pi theorem describes how every physical equation involving n variables
can be written as an equation of n--m dimensionless parameters, where
m is the number of fundamental dimensions used.
A dimensionless number is a
pure number without any physical units. For example, if one of
every ten tomatoes is rotten then 1/10 ratio equals 0.1, which is a
dimensionless quantity. A physical quantity may be dimensionless
in one system of units and not dimensionless in another system of units.
In systems of natural units, e.g. Planck units, the physical units are
defined so that fundamental constants are made dimensionless and set
to 1 which removes the scaling factors from equations.
Planck units are five physical
units of measurement; speed of light, gravity, Dirac's constant, Charles
Coulomb's force constant, and Ludwig Boltzmann's constant.
Constraining the numerical values of those five constants to 1 defines
the base units of length, mass, time, charge, and temperature.
Combinatorics studies collections
of specific objects. In structural combinatorics, Frank Ramsey
proved that in any group of six people, three people either all knew
each other or all people did not know each other. This is an example
of proof by contradiction and implies order can be found in random configurations.
4. Quantum Computing
A quantum computer computes
by using quantum mechanical phenomena, i.e. superposition and entanglement,
to perform operations on inputted data. Other computing architecture,
such as optical computers, may use superposition of electromagnetic
waves but without a quantum mechanical resource like entanglement that
does not share the potential speed of quantum computers.
A classical computer has a
memory of bits that hold either a one or a zero. Those bits are
manipulated through a set of logic gates and back. A quantum computer
uses qubits. Qubits can hold a one, or a zero, or a superposition
of one or zero. The quantum computer manipulates those qubits
through a set of quantum logic gates and back. Benjamin Schumacher
interpreted quantum states as information and compressed that data.
He is credited with inventing the term qubit.
Quantum gates or quantum logic
gates are a quantum circuit that uses qubits. A quantum circuit
is used for quantum computation operations on data structures that are
quantum mechanical analogs of bit strings of a given size n. Sometimes
these structures are referred to as n-qubits. Unlike the logic
gates of a classical computer, the other than not gate is reversible.
Some classic logic gates, such as the Toffoli gate, are reversible and
can be mapped on quantum logic gates. Tommaso Toffoli invented
the universal Toffoli gate which is known as the controlled -- controlled
-- not gate. Jacques Hadamard's gate has a one-qubit rotation
and is also called the Hadamard transformation. It is reducible
to David Deutsch's gate. The most common gates operate on spaces
of one or two qubits. Therefore, the matrices of quantum gates
can be described by 2x2 or 4x4 matrices of orthonormal rows.
A universal machine can simulate
other machines basic symbol manipulation. They can be adapted
to simulate the logic of any computer that can be constructed.
The machine consists of a tape divided into cells, a head that can read
and write symbols, a state register that stores the states, and an actionable
table that tells the machine what symbol to write, how to move its head
and what its new state will be. Alonzo Church used
λ-lambda calculus to theorize
that these simple machines can capture the effective method in logic
and define an algorithm or mechanical procedure. Lambda calculus
is called the smallest universal programming language. It consists
of a single transformation rule and a single function definition scheme.
Its approach is more compatible to software than to hardware.
5. Infinity and Logic
Two thousand years ago the
Romans used the symbol ∞ to represent 1,000, considered a
large number for the time.
The English minister and mathematician, John Wallis, gave infinity that
symbol. Nonphysical infinity in math is the state of being greater
than any finite real number no matter how large. Infinity also
has a physical category. Aristotle said logic is the organon
(Greek meaning tool) without which we can't know, communicate or have
scientific absolutes such as the law of noncontradiction. Thus,
infinity must be viewed logically by finite human minds.
Lexica are always in flux but
historically the word contradiction has described one method of understanding
infinity. It indicates that A can't be A and not be A at the
same time. From the infinitesimal to the infinite, whether all
A is completely understood or not, contradictions cannot exist.
Paradox is another method that is said to occur when it appears a contradiction
exists. Further study will eventually prove the perceived contradiction
did not exist. Antimony usually refers to the law of noncontradiction
but could also mean paradox. A last method of understanding infinity
is the realm of mystery that describes the parts of infinity that we
are aware of but don't understand. As mysteries are solved and
knowledge is increased, the more we are exponentially aware of other
mysteries. That is the reality of infinite truth. Thomas
Kuhn observed that science has periods of growth punctuated by revisionary
All people are born with a
high degree of certainty regarding infinite physical concepts, such
as speed and heat, and infinite nonphysical concepts such as love and
justice. It is reasoned that those concepts are effects from a
cause. The law of causation states that for there to have been
an effect there must have been a cause. We may see the association,
such as one billiard ball striking another billiard ball, but we do
not see the exchange of force being imparted from the first ball to
the second ball causing it to move.
Truth is an infinite reality
and finite intellects approach incomplete finite segments of truth by
means of logic. When deducing or inducing A=B, B=C, the A=C, it
is known that truth is both linear and nonlinear due to the potential
range of A to infinity and infinity to A, B to infinity and infinity
to B, etc. People with a bias against a truth, usually against
the existence of the infinite God, often incorrectly vary a sound debate
technique of arguing the merits of a thesis and the antithesis of an
unknown. They insist that truth includes a fact (having the quality
of actuality) and it antifact (not having the quality of actuality).
This clearly is a violation of the law of noncontradiction. They
must argue that not only does 2+2=4 but 2+2 equals everything but 4.
Then love and no love and justice and no justice could be equally true.
If that belief was physically acted upon as proposed, eating rocks would
be equally as nutritious as eating apples or breathing water would be
equally as healthy as breathing oxygen.
We must view the infinite by
faith. All beliefs derived from faith emanating from the unknown
or unknowable are theological. Our vulnerability should produce
a sense of humility into the epistemological process. The God
of infiniteness (the cause) has revealed Himself in nature and in words
so man can know (the effect) what is necessary with certainty.
Rejecting the infinite God unnecessarily exposes oneself to infinity
without protection. The biblical book of Proverbs says that to
refuse reproof (to correct by defense of proof) is to err and to hate
reproof is brutish.
John Calvin said apologetics
should show the proof and leave the persuasion to God. Christian
tradition uniquely teaches that the infinite God of Truth or Word became
flesh and dwelt among us--Jesus the Christ. The apostle Paul
said in his letter to the Romans that he was not ashamed of the gospel
(good news) of Christ because it is the power of salvation to everyone
that believes. Jesus was recorded to have said the truth shall
set you free. This means that God is the largest of all knowledge
and His characteristic is infiniteness. The warfare against the
believers of the good and infinite God from His evil adversaries is
most obvious when improbable effects attempt to nullify or nullify a
circumstance unfavorably, thereby indicating the evil cause.
Gödel has shown that logic
can't model mathematics and the universal machine has shown that neither
logic nor algorithms can model human thought. Then it may be asked
what the goal of proof should be? It is not a source of epistemological
security apart from Divine revelation. It can be the foundation
for making clear the strength of the logic of assumptions and theories.
Contemporary computers use
the movement of electrons into and out of transistors in order to perform
logic functions. Optical or photonic computing is attempting to
use photons or light particles from lasers instead of electrons.
Photons are faster and have a higher bandwidth than electrons.
The light spectrum could enable 35 billion bit positions compared to
64 or 128 bit positions in electronic computers. Today, polymers
used for transistors are smaller and thousands of times faster than
their silicon counterparts.
Using the tools of proof presented
along with other appropriate concepts and technology, mankind has the
potential to make great advances in understanding. As we move
beyond quantum proofs in scope and technique, systematic political science
should continue to grow as the premier resource for anthropocentric
ALL RIGHTS RESERVED
© 2006 DALLAS F. BELL, JR.