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

II: Different ways of thinking and their shortcomings

Scientific way of thinking
Scientific way of thinking relies on the interplay between hypotheses and experimental verification.

Strong points:

  1. There is a freedom to hypothesize.
  2. 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.

Strong points:

  1. There is more freedom in speculation than in hypothesizing.
  2. Reflection on human experience can reach much wider and deeper realities (love, spirituality, psychic power, etc.) than experimental verification ever could.
Thus, many unthinkable issues in science (ethics, religions, love, compassion, etc.) are thinkable in philosophy.

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.

Strong points:

  1. Although senses and emotion are still confined to space and time, imagination can transcend both space and time.
  2. The artistic thought has no true-false value, that is, it has no need to be verified by either scientific experiments or philosophical reflection.
  3. 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.


  1. 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.
  2. 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.
  1. All life processes are information processes. Then, where is the bio-computer? What is this bio-computer made of?
  2. 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?
  3. 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?
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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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.

  1. 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.
  2. 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?
  3. 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.
  4. 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.
  5. 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.
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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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.

  1. Symmetry
  2. Symmetry breaking:
  3. Totality (indivisible symmetry)


All unthinkable issues become thinkable as soon as the following three concepts are understood.

  1. Totality
  2. Symmetry-breaking:
The validity of all unprovable truths (or theories) can be obtained by following a new epistemology.

A new Epistemology

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