Propaedeutics of Proof Theory:
Beyond Quantum Proofs in Systematic Political Science

by
Dallas F. Bell, Jr.

1. Introduction

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 of formulas. 

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 times. 

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 revolutions. 

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. 

6. Conclusion

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 disciplines. 

ALL RIGHTS RESERVED © 2006 DALLAS F. BELL, JR.