How to Think About the Unprovable and the Unthinkable
--- A new Epistemology
This paper is presented by Dr. Tienzen (Jeh-Tween) Gong at Sixth International Conference on Thinking at Massachusetts Institute of Technology on July 19th, 1994
Table of Content
- I: The issue -- What is the cause of First Cause? If there is a cause for the First Cause, the First Cause cannot be the first. If the First Cause has no cause, then how does it arise? The issue of the First Cause is not only an unprovable but an unthinkable problem.
- II: Different ways of thinking and their shortcomings:
- III: Examples of the unprovable and the unthinkable
- The beginning of the beginning
- The rise of biological life
- IV: How to think about the unprovable -- in the area in which traditional epistemology of physics that the validity of any physical theory must be judged with its predicting power against the experimental verification can no longer provide any guidance, the validity of any theory can still be judged with a New Epistemology which consists of five powers:
- The power of Simplicity
- The power of Explanation
- The power of Unification
- The power of Inclusion
- The power of Indispensability
- V: How to think about the unthinkable -- by definition, every paradox is always unthinkable. Every paradox always contains two truths, but they must be directly opposite both in their meaning and in their internal reasoning processes. For two contradictory parts, if one is true and the other is false, there is no paradox. Every paradox (unthinkable issue) always points out a higher truth which transcends the paradox itself. By transcending the paradox, the unthinkable will become thinkable. There are two ways to transcend all paradoxes.
And, all unthinkable issues will become thinkable when five concepts are understood.
- Downward symmetry-breaking
- Unification -- recovery a higher symmetry
- Totality (indivisible symmetry)
II: Different ways of thinking and their
Scientific way of thinking
Scientific way of thinking relies on the interplay between hypotheses and experimental verification.
- There is a freedom to hypothesize.
- Experimental verification assures that all conclusions are supported by facts.
Shortcomings: All experimental verification must be performed in the domain of space and time. Thus, any entity (eternal form, mathematical relation, religious conviction, the creation of the beginning, etc.) which transcends space or time cannot be thought in terms of science.
Note: Although many people do recognize that mathematics is a discipline of science, mathematical way of thinking is completely different from scientific way of thinking. Mathematics does not rely on any experimental verification, that is, many mathematical relations or conclusions cannot be thought in terms of scientific way of thinking.
Philosophical way of thinking
Philosophical way of thinking relies on speculation and reflection.
Thus, many unthinkable issues in science (ethics, religions, love, compassion, etc.) are thinkable in philosophy.
- There is more freedom in speculation than in hypothesizing.
- Reflection on human experience can reach much wider and deeper realities (love, spirituality, psychic power, etc.) than experimental verification ever could.
Shortcomings: Reflection on human experience is still confined to space or time. Although philosophical speculations do discuss the issues of eternity, of transcendence, philosophers cannot truly verify those speculations with reflection from their experience. Furthermore, some paradoxes (such as the First Cause issue) are unthinkable problems in philosophy.
Artistic way of thinking
Artists think about this world with senses, imagination, and emotion.
- Although senses and emotion are still confined to space and time, imagination can transcend both space and time.
- The artistic thought has no true-false value, that is, it has no need to be verified by either scientific experiments or philosophical reflection.
- Although many arts do correspond with some real world entities, they, in general, can create self-reference (such as, unicorn) without depending upon the real world. Those created imaginary entities do turn into realities and do give satisfaction to senses and emotion.
Shortcomings: Artists do not try very hard to convince the public that the self-referencing imaginary universe is, indeed, a reality.
Religious way of thinking
Theologians think with preordained religious convictions. There is nothing unthinkable in terms of religious conviction.
Strong point: Theologians do not need logic, reflection on human experience, common sense, nor experimental verification. As long as someone shares their conviction, they can claim victory.
Shortcomings: Religious conviction is often violating many known facts.
Mathematical way of thinking
A few hundred years ago, Descartes, Euler and many others believed mathematics to be the accurate description of real phenomena and they regarded their work as the uncovering of the mathematical design of the universe. Today, almost all mathematicians believe that mathematics is no longer absolute but arises arbitrarily. So, mathematicians can arbitrarily choose a set of definitions to construct a new mathematics. After the discovery of Godel's Incompleteness Theorem, even the self-contradiction of a system is no longer a criterion to invalidate a new mathematics.
Strong point: Seemingly, there is no restriction (such as experimental verification, philosophical reflection, etc.) as the domain boundary in mathematics, that is, there is seemingly no unthinkable issue in mathematics.
On the contrary, quite a few issues are unprovable in mathematics, such as, Goldbach's conjecture, etc..
Shortcomings: Mathematical way of thinking still cannot encompass the other ways of thinking.
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III: Examples of the unprovable and the unthinkable
The beginning of the beginning
This question is perfectly thinkable in religion: "In the beginning God created the heaven and the earth,..." (Genesis 1:1). "In the beginning was the Word, and the Word was with God, and the Word was God." (John 1:1)
Philosophically, if there is a creator before the Beginning, the Beginning is not the beginning. Thus, the above religious reasoning and conviction cannot be understood and supported by philosophy.
Today, quantum cosmology understands the detailed evolution processes of Big Bang from its first one-trillionth of a second to now but is unable to explain what before the Big Bang was.
Many physicists proclaim, "What came before the Big Bang? This is a meaningless question, given that space and time themselves came into being in the Big Bang. Without time, there can be no 'before,' just as without space there can be no 'outside.' The notion of a universe 'before' our own and another 'after' it perpetually fascinates those attracted to the ideas of eternity and reincarnation who are eager to transfer the concept of rebirth from the personal to the cosmic."
Although the religious reasoning on this issue is sort of a big joke, the argument of physicists is also wrong.
There can be 'before' before time, that is, the timelessness which is the complement of time. With the concept of complementarity, this unthinkable issue will become thinkable.
- Russell paradox: proposition A states "All propositions are true," and proposition B states "Proposition A is false." Obviously, proposition B must be true if proposition A is true. But, if B is true then A must be false.
- Grelling paradox: A word is said to be "autological" if and only if it applies to itself. For example, the word "English" means English, and it is, indeed, an English word; therefore, it is an autological word. On the contrary, the word "French" means French but is an English word instead of being a French word; therefore, it is not an autological word; instead, it is called a "heterological" word.
Now, is the word "heterological" heterological? If we assume that "heterological" is heterological, then by definition, "heterological" is autological; on the other hand, if we assume that "heterological" is autological, the by definition again, "heterological" has to be heterological.
Every paradox always contains two truths, but they must be directly opposite of each other both in their meaning and in their internal reasoning processes.
Seemingly, every paradox is always unthinkable.
The rise of biological life
Of course, this issue is perfectly thinkable in religion: "And God created great whales, and every living creature that moveth, which the waters brought forth abundantly, after their kind,... ." Genesis 1:21.
This issue also has been thought in terms of science fiction. There is a primordial soup. A lighting struck the soup and biological molecules formed. Then those molecules somehow got together and finally reproduced themselves. This line of thinking is not scientific because many logical links are scientifically unknowns.
All the above questions are not answered in terms of science at this time, that is, they are all unthinkable in science. The bio-computer can be understood only via the prequark theory which is beyond the reach of any kind of experimental verification now.
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- All life processes are information processes. Then, where is the bio-computer? What is this bio-computer made of?
- All lives are organized entities. Then, what is the underlying force for this organization (divine or self-organization)? If it is divine, then what is that divine magic wand? If it is self-organization, then what are the laws that govern this process?
- For all higher life forms, a single fertilized cell turns into a very complicated system (with head, internal organs, etc.). This is called morphogenesis. What is the underlying force or processes that govern this morphogenesis?
IV: How to think about the unprovable -- A new Epistemology
As for the Prequark theory, it is beyond the reach of any kind of experimental verification at this point. Nonetheless, the validity of any theory can still be judged with a new epistemology which has been employed by mathematicians for centuries. It consists of five powers and criteria.
If a new theory possesses these five powers, then it must be valid even though all of its predictions cannot be experimentally verified.
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- The first criterion is the power of simplicity: the new theory makes a complicated old phenomenon becoming simple, such as, Prequark theory makes neutron decay which can be much easier understood.
- The second is the power of explanation: the new theory is able to provide answers for many old open questions, such as, Why does proton have incredible longer lifetime than the age of the universe? and how does electric charge arise?
- The third is the power of unification: the new theory is able to bring two or more unrelated disciplines together, such as, Prequark theory brings physics and mathematics together.
- The fourth is the power of inclusion: all old verified theories are subsets of the new, such as, quark theory is a subset of the Prequark theory.
- The fifth is the power of indispensability: the new theory is indispensable for many open questions, such as, the rise of biological life cannot be understood without the Prequark theory.
V: How to think about the unthinkable
Seemingly, all paradoxes are always unthinkable. So the solution for thinking about the unthinkable lies in finding out the cause of the paradox. Both the Russell and Grelling paradoxes are caused by a symmetry-breaking process.
The Russell paradox is caused by the attempt to categorize the world with a definition, "what is true?" This categorizing and defining procedure is a symmetry-breaking process. The Russell paradox is created by breaking a symmetry, the totality. When the null term " " (the totality) is replaced by "true", the symmetry of the null proposition (all propositions are " ") is broken, and the new proposition (all propositions are "true") creates a paradox.
Grelling got himself into his predicament by inventing definitions for autological and heterological. Every definition always acts as a symmetry-breaking procedure, separating a totality (symmetry) into categories.
Nonetheless, all paradoxes can be reconciled in two ways, downward or upward solution. The downward solution is obtained by a further downward symmetry-breaking with a new proposition: All propositions "except proposition B" are true. The upward solution is obtained by removing the first symmetry-breaking which causes the problem, and the null symmetry (the totality) is regained.
So, all unthinkable issues will become thinkable when five concepts are understood.
- Symmetry breaking:
- Totality (indivisible symmetry)
Symmetry always connotes chaos, degrees of freedom. A square peg can go into its mating hole in four ways, a hexagon peg into its mating hole in six ways. The higher the symmetry, the higher the chaos. The highest symmetry has the utmost chaos. For example, a round peg can go into its mating hole in infinite ways.
- Symmetry-breaking -- complementarity
In physics, most natural symmetries are broken by a special symmetry-breaking process -- the spontaneous-symmetry breaking (SSB). For example, a pencil standing upright on its tip can fall in any direction. The probabilities are equal in all directions. So that probability function has a symmetry. But, when the pencil actually falls, the symmetrical probability will be broken into one reality.
The nature phenomena of spontaneous symmetry-breaking were expressed as Copenhagen Interpretation (CI) in quantum physics -- the principle of complementarity which consists of three parts.
In physics, this principle of complementarity is expressed as Heisenberg Uncertainty Principle. In quantum physics, the entire universe is divided into two mutually exclusive but complementary parts. Then, these opposite parts are paired together, position versus momentum, time versus energy, etc., and only one of the two parts can be truly known with a high precision of accuracy by any type of consciousness not only in practice but also in principle according to CI.
EPR thought experiment
Although Einstein conceded that the uncertainty principle is, indeed, real in practice, he insisted that it cannot be true in principle. In 1935, he with his colleagues came to denounce the Copenhagen Interpretation with an EPR (Einstein, Podolsky, Rosen) thought experiment. In EPR experiment, we are asked to imagine that two particles originate from a definite quantum state, and then move apart without interaction with anything else until we elect to measure or observe one of them. Since the quantum rules allow us, when the two particles are initially in a definite quantum state, to calculate their initial momentum, the EPR argument is that the individual momenta will be correlated even after the particles separate. After these two particles have moved apart greater than a space-like separation distance where no causal connection can be made by light signal between the separated particles, then we can, as argued by Einstein, measure the momentum (not position) of one particle to a precise accuracy according to the uncertainty principle, and this measurement will not and cannot disturb the momentum of the other particle because of the space-like separation between the two. Thus, the second particle's momentum can be calculated to a precise accuracy because of the momentum conservation law. Then, we can obtain the positions (not momenta) of the two particles with the same method. This means that we should be able to deduce both position and momentum for a single particle to a precise accuracy.
- A "whole" (totality) must consist of two opposite parts.
- These two opposite parts must be mutually exclusive.
- These two opposite parts are complementary to each other.
In short, CI is simply a baloney according to Einstein unless a spooky action at distance is in fact a reality. In 1982, the EPR-like experiments were done by Alain Aspect and his colleagues Jean Dalibard and Gerard Roger. The Aspect experiment showed that CI (complementarity) is correct and the spooky action is, in fact, a physical reality.
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Totality is an indivisible symmetry. To break the totality symmetry will create three parts (not two) -- totality (itself remains), part one and the symmetry partner of part one. Part one and its partner are always forming a paradox. (Note: to break a symmetry which is not totality will not always create paradox). In physics, it manifests as spooky action or uncertainty principle. In mathematics, it manifests as Godel's incompleteness theorem.
Totality in physics (Spooky action)
Aspect's experiment leads to the conclusion that the universe cannot be understood by the sum of its parts because the whole (the totality) is utterly indivisible. In fact, all isolated entities can be assumed to have interacted at some point (such as at Big Bang) in the history of the cosmos.
Seemingly, you and I are separated individuals. But, we are linked together in this room by our thinking communication. We are linked together by a human society, that is, I am wearing the shoes you made and you are eating the grain I produced. Furthermore, I probably just breathed in the oxygen you breathed out, how disgusting!
In fact, our inseparableness is seeded long before our existence. All life forms we know of are carbon-based with water as a chemical solvent. Every heavy atom in our bodies -- whether potassium, iron, calcium, carbon, oxygen or nitrogen -- had to be produced by the nuclear fusion process (the so-called carbon-oxygen-nitrogen cycle) in the core of stars. Not only is life the reincarnation of those dead celestial bodies, but the calcium in your bones could be the by-product of the iron in my blood.
In short, the totality is absolutely indivisible.
Totality in mathematics
--- Godel's incompleteness theorem
After a lecture, an old lady said to a well-known scientist, "what you have told us (about the laws of universe) is rubbish. The world is really a flat plate supported on the back of a giant tortoise." When asked by the scientist what the turtle rests on, she replied, "It's turtles all the way down!"[back to list]
Today, many physicists believe that physics is complete. Given the laws of physics, the universe can, so to speak, take care of itself, including its own creation. But physics is framed in mathematics, and mathematics is incomplete according to Godel's incompleteness theorem which warns us that the axiomatic method of making logical deductions from given assumptions cannot in general provide a system which is both provably complete and consistent. There will always be truth that lies beyond, that cannot be reached from a finite collection of axioms.
Since every mathematics system is built upon definitions and axioms, and since all definitions and axioms are symmetry-breaking processes, all mathematical systems must be broken subsystems from a higher symmetry. The incompleteness theorem is, in fact, pointing out the vivid reality of totality in terms of mathematics.
Thus, a mathematics system can suck in a countable number of axioms but can still and must puke up at least one undecidable statement. In short, the fire of God's spirit (the totality) will always burn a hole in every mathematics system invented by men. See the graph on the left.
- Mutual Immanence
Although the notion of complementarity accepts the concept of totality in its logical framework, its methodology tries to divide this totality into two mutually exclusive parts. However, the uncertainty principle also means mutual inclusive. See the graph on the left.Examples of mutual immanence
In fact, the indivisible totality which is pointed out by the spooky action can only be understood with a mutually inclusive complementarity, that is, the mutual immanence.
Mutual immanence can best be understood in terms of Chinese philosophy -- the Yin and Yang. In time of conflict, yin and Yang are opposite forces. In constancy, they not only complement each other but are imbedded in each other. After reaching their full strength, they transform into the opposite. That is, Yang becomes Yin and Yin becomes Yang.
Mutual immanence is similar to complementarity but with much deeper meanings. Complementarity consists of opposites which are mutually exclusive. Mutual immanence are opposites which are mutually inclusive to make a whole. Consider several pairs of opposites: whole-parts, cause-effect, good-bad, universal-particular. Yet, is there not something about both of the two in each pair that makes them alike? We do not pair whole with bad although they can be viewed as opposites in being not each other.
One: mutual immanence in mathematics
Although the concept of mutual immanence was invented by ancient Taoism, it resides in all truths, such as physics and mathematics.Two: Mutual immanence in physics
By definition, deterministic is opposite to chaotic. Today, not only deterministic chaos is a legitimate term in mathematics, but order and deterministic chaos spring from the same source -- dissipative dynamical system described by non-linear differential equations.
In Fractal Geometry, there is a Shadow theorem which states that all deterministic chaos are the shadows of some orderly states.
The center of the map on the left is the North Pole. The entire outer edge (which contains infinite geometrical points) represents only one point -- the South Pole. In Topology, a single point (South Pole) and an infinite number of points (outer edge of the map) are identical and thus indistinguishable.
--- Uncertainty principle
Planck constant (h) is the basis for the entire quantum physics. It forms a horizon for all actions. In physics, action is defined as energy (E) multiplied by time (T). According to the uncertainty principle, all actions (delta E x delta T) must be larger than or equal to h (Planck constant). In fact, the uncertainty principle has two expressions.
P is momentum, S distance, E energy and T time.
- delta E x delta T >= h
- delta P x delta S >= h
These two expressions actually form a viewing window. Let's imagine that E is the left edge of the window square, T the right edge, P the top, and S the bottom. If we want to view more of what is behind the left by pushing the left edge further left (smaller delta E), we will inevitably lose information on the right (large delta T). It is the same for delta P and delta S.[back to list]
On the one hand, the uncertainty principle imposes an inherent horizon of knowledge on all scales, and it seemingly guarantees that something must remain unknowable or unthinkable. On the other hand, the uncertainty principle is the basis for the entire quantum physics. That is, without understanding the uncertainty principle, we will still be in the dark age at least in terms of physics. Not only are the internal expressions of the uncertainty principle mutually immanent to each other, but the sole existence of the uncertainty principle and its apparent effect (imposing an horizon of knowledge) are also mutually immanent to each other. In fact, with the uncertainty principle, we are able to tackle the issue of the Big Bang -- the origin of the universe.
All unthinkable issues become thinkable as soon as the following three concepts are understood.
The validity of all unprovable truths (or theories) can be obtained by following a new epistemology.
A new Epistemology
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