Theotropic Boundaries for Epistemological Defeasibility of a Mauvaise Foi: Mathematics and K- to K+

By
Dallas F. Bell, Jr.

Abstract:  Defeasibility of a belief, especially a mauvaise foi, indicates a process that exposes vulnerability to a positive epistemic status.  The laws of probability act as coherence constraints on rational levels of beliefs (T. Bayes).  The epistemology of religion includes evidentialism where belief is made plausible from justification of the proportional evidence.  Those justifications can be found in visual mathematical modeling (L. Kirby, J. Paris, and R. Goodstein) and explained by precise mathematical symbols.  The common concept of infinity (G. Cantor, C. Wickramasinghe, and C. Tapp) may be used as a tool to demonstrate core rational ideas which provide theotropic boundaries (A. Herschel and P. de Man).  Proof from the resultant sequences (I. Kant, É. Tempier, and F. Nietzsche) has the potential to cause systematic change from K- to K+ gradients in the territories of knowledge.

Keywords:  theotropic boundaries, epistemological defeaters, coherence, mathematical infinity. 

Man is not at peace with his fellow man because he is not at peace with himself:
he is not at peace with himself, because he is not at peace with God.
Thomas Merton (1915-1968)

Defeasibility of a belief, especially a mauvaise foi, indicates a process that exposes anomian vulnerability to a positive epistemic status.  Thomas Bayes (1701-1761) proposed a formal apparatus for inductive logic and introduced a self-defeat test for epistemological rationality.  He used the laws of probability as coherence constraints on rational levels of beliefs.  Coherentism is a belief, or a set of beliefs, justified as the belief coheres with a set of beliefs forming a coherent set.

The epistemology of religion may include evidentialism where a belief is plausible from justification of the proportional evidence.  The result can be theotropic (Gr. God, to turn).  Abraham J. Herschel (1907-1972) defined this term as "turning toward God," whereas Paul de Man (1919-1983) implied it to mean a "turning away from God."

Sheila Embleton, Vice-President Academic, Provost and Professor of graduate programs in linguistics at York University, observed that things are very seldom equal between participants of communication exchanges, as there are always age, gender, social/economic, and hierarchical differences. She points out that mathematical equations cannot convey cultural, religious, philosophical (etc.) concepts well at all, whereas sentences (even if it takes many of them) can.[1]

Noam Chomsky said language is a set of sentences composed of finite length from a finite set of elements.  If it is said "seven is the square root of forty-nine," "seven is less than nine," and "seven is a prime number," is it known what the meaning of "is" is?  The first "is" says the object "seven" and the object "the square root of forty-nine" are equal, so "is" means equal.  The second sentence is an adjectival phrase where "less than nine" is specifying a property that numbers may or may not have.  The third sentence means seven is an example of a prime number.

Generally, the language of mathematics uses precise symbols that are less culturally dependant than words which can be barriers to understanding.  For example, the three sentences in the preceding paragraph could be mathematically expressed as 7 = √49, 7 < 9, and P(n) where n is a positive integer, meaning n or 7 is a prime.  That mathematical exchange of the lack of information (K-) to having the new information (K+) with symbols obviously seems to be more efficient than conventional sentences of words.

The epistemological sequence into territories of knowledge is a change of the state of K- to a state of K+.  Modeling this process assumes a shared position of common ground of linguistic information.  An assertion is likely to encode presupposition.  They are expected to be common while adding new content, such as 7 = √49 assumes each participant (the one with the information presented to the one(s) without the information) understands numbers and what square root means.

In 1982, Laurie Kirby and Jeff Paris[2] interpreted Reuben Goodstein's 1944 theorem in a hydra game.  The Hydra is a finite rooted tree.  The goal is to cut off the branches/nodes of the tree as they grow back a finite number of new branches.  If they are continuously cut back according to the simple rules, the tree will eventually be cut down to the root ending the game.  Kirby explains the hydra combinatorics shows the relationship of mathematical/logical questions of unprovability in formal systems begging the question of the need for proof to know something is true.[3] 

In a thermodynamic sense, the hydra game of finite options can be called a closed system.  This would mean the mass and energy has a beginning and an end, unlike an open system.  A phase transition is something in transition to something else and not something from nothing.  Like empty space, (   ), it is a set containing nothing, not mass nor energy, or else it is not mathematical empty space.  The ultimate open system that must thermodynamically exist in order to be the first cause of all effects, in our closed finite system, is a state of infinity.

Aristotle (384 BC – 322 BC) distinguished between potential infinity (e.g. numbers) and actual infinity (e.g. the Divine).  Georg Cantor (1845-1918) treated infinities not just as potential, but as actual.  He indicated a countable infinity plus another countable infinity is a countable infinity.  Christian Tapp points out that we know from set theory that there are several different infinite numbers, such as so-called transfinite ordinal numbers, which can be compared with each other so that there is in fact a sense in which the infinite in the sense of some infinite number is less than the infinite in the sense of some other infinite numbers.  Those mathematical entities are not temporal and do not connote change and can be seen as static, which is a term usually defining something that is unchanged in a world of change.[4]

Chandra Wickramasinghe describes mathematical infinity as 1/x as x approaches progressively closer to unity,[5] though all numbers have value and do not actually merge as x approaches zero.  Of course, the appearance of larger number's closeness is due to their ratio of value, such as 1 is twice as small as 2 but this ratio of value is much smaller with 1,001 and 1,002.

The reality and truth that two objects added to two more objects equals four objects existed before man could conceive of it.  Numbers are purposefully ordered 1, 2, 3 ... with their unique values and properties so intellects can communicate with them in a mathematical/logical language.  This is why it is rationally concluded that the first Causer of all effects is an infinite state of completeness or wholeness called holiness.[6]  Completeness is a property of infiniteness, Tapp also observed.[7]  The change from the status quo or static state, to use temporal expressions, to creating requires purpose from intellect.  That infinite Intellect created humans with observable attributes and values of love, mercy, and justice and thus must also have those properties for completeness.  That Being's love, mercy, and justice etc. would be infinite and immutable as humans finitely attempt to understand the material and non-material.

Marcus Aurelius[8] (121 AD – 180 AD) wrote in his Meditations that whatever happens has been waiting to happen since the beginning of time.  In Plato's (c. 427 BC – c. 347 BC) Theory of Ideas (Forms), he maintained that all ideas have always existed.  That implies a Consciousness, pre-existent to man, held and holds those ideas.  King Solomon[9] of Israel wrote whatever the Creator does is forever and nothing can be added to it.  He also noted that there is nothing new that has not been before.[10]  Those assertions maintain that everything that can happen, within the Creator's will, will eventually happen.  From Aristotle to Immanuel Kant (1724-1804), this plenitude principle has been accepted as a possibility.[11]  Étienne Tempier (c. 1210 - d. 1279) went further by saying if the Creator could create something, it should be accepted as having been created.  To do otherwise, he concluded, would artificially limit the Creator and is heresy.  But if the Creator could not choose to limit Himself, He would not be infinite.  For example, the omnipotent Creator could have created more than one world, but His revelation in Tempier's Bible indicates He made only one.

From these theotropic boundaries, we can see the epistemological defeaters for the incoherent beliefs that a Creator doesn't exist or is just a creating force, requiring the most amount of faith for the most incoherence from the finite human agnostic condition.[12]  The next defeater is the belief of the Creator being a monotheistic god.  That being could not be infinitely complete, due to the need of having another equal being to equally love and be loved by perfectly before having created the first cause of all effects.  Furthermore, the epistemological defeater of beliefs for separate pantheistic gods is evident in that such gods are not co-equal and are thus not infinitely complete, making them the unholy source of the original sin and unfit for eternal justice.  Smaller infinities do exist within infinities for numbers but they encompass the set of infinite numbers and are only part of a whole of numbers.  The biblical triune Creator[13] has a coherent explanation for infinity and original sin which no other religious system can be said to have, requiring the least faith for the least incoherence from the finite human agnostic position.

That exclusion process applies the principle for a possible Creator by eliminating the incoherent monotheistic and pantheistic etc. alternatives to the Holy Trinity Creator.  If a hydra tree is used to simulate this finite epistemological sequence, those finite defeaters can be chopped off until the root of the true Creator is empirically and coherently determined. 

That principle means if there are no rational exceptions to the conclusion that conclusion can be accepted as true.  Prime numbers can be found with this principle.  All numbers have weight due to their value and therefore have infinite existence.  They must exist in order of their value and are infinitely linear.  At this point, they can be seen as a tool for analysis and measurement for communications between intellects with reasoning ability.  When numbers are examined further, there is a recognizable symmetry of square roots that increase at a predictable rate as numbers increase and predictable symmetrical positions of descending degree size of interior angles as sides increase for polygons.

Prime numbers are generally defined as whole numbers that have no divisors except 1 and the number itself, such as 3.  1 fits this definition, due to its unique beginning position with no possibility of whole number divisors.  2 is technically a prime also due to its proximity to 1.  However, it is an even number and is equally divided into a whole number, 1, which are not aspects of prime numbers.  A natural number greater than 1 and is not prime is a composite number.

Prime numbers show their individual mass and energy, like the reality of a black hole, when posed by intellects in mathematical operations.  For example, the prime of 3 is less than 4, 3 < 4, and the difference between 3 and 4 is 1, 4 -3 = 1.  In comparison, 3 has an energy and mass decrease of 1 with 4.

The gravity of prime numbers can be understood as a having a necessary connecting pull between the nonprime numbers that surround them as seen above.  The steps to predicting and quickly proving prime numbers employs what we know about them and excluding identifiable alternatives.  Prime numbers (in red font above) begin with the base prime of 3 as follows:

All multiples of 3 can not be prime by definition and excludes from possible primes the vertical column numbers under 3 to infinity (excludes ⅓ of all numbers).  Even numbers are divisible by 2 and are also excluded from possible primes (excludes ½ of all numbers).  A pattern is observed with the horizontal rows below 3.  The possible prime numbers alternate between the second number in the following row with the first number in the next row to infinity (excluding ⅔ of all numbers).  In this case, 5, 7, 11, and 13 are prime numbers along with 3.  It is obvious that the odd number before a multiple of 3 and the odd number after a multiple of 3 are candidates to be prime numbers, such as 5 and 7 that surround 6 or 7 and 11 that surround 9 or 11 and 13 that surround 12.

Possible prime numbers proved to not be prime from primes 5 to 23 are 7 rows which equal 28 possible primes.

As numbers get larger the process remains simple.  For example, the number 966 is divisible by 3, 322, and is not prime and its position must be as follows.

This makes 961, 965, and 967 possible prime numbers.  961 is divisible by 31 and not a prime.  965 ends with a 5 and so is immediately identifiable as divisible by 5 and not prime.  967 is not divisible by smaller prime numbers and is a prime number.  Non prime numbers in the position of possible prime numbers are divisible by prime numbers only with the exception of the non prime number of 25 which is multiplied by a prime excluding its sum from being prime.  Take the following examples:

121 is divisible by the prime of 11, 11 times, 11 x 11 = 121, and not prime.  125 is divisible by the prime of 5, 25 times, 5 x 25 = 125, and not prime.  127 is not divisible by a prime and is proved to be a prime number.  (7 x 25 = 175 and not prime, 11 x 25 = 275 and not prime, 13 x 25 = 325 and not prime etc. etc.)

Simplifying the divisor possibilities of 967 begins with its square root, 31.096... .  Its divisors must be below the lower prime of this whole number, 31.  Additionally, divisors should begin with the lowest prime number above 3.  Since its last number is not a 5 or 0 divisor attempts begin with 7 then proceed to 11, 13, 17, 19, 23, 29 and ends with 31 x 31 etc.  31 x 31 = 961 which is divisible by itself and is, of course, excluded from the possibility of being a prime number. The number 967 is then proved to be a prime.

Another example could be 367.  367 divided by 3 equals 122.3.  3 x 122 = 366.  The base of 366 is employed and indicates possible prime numbers as follows:

365 is obviously divisible by 5 and not prime.  367, square root of 19.157..., is not divisible by prime numbers below 19 and is proved prime, 361 is divisible by 19, 19 x 19, and is not prime.

The amount of prime numbers do decrease as numbers grow larger and have more possible divisors, but Euclid's (c. 325 BC – c. 265 BC) Elements prove their existence to infinity.  Prime numbers have rational positions, predictability and provability.  Irrationality, randomness, and chaos are observations from inadequate knowledge of what is being observed.

By necessity, the Creator of the first cause of all effects, with order and mystery observable by man, has purposed options for man to choose.  David Hilbert's (1862-1943) hotel is not considered a defeater of this probability and does not deny mathematical infinity (prime numbers etc.) or the biblical account of theology (omniscience of infinite data held by that Divine data holder etc.).[14]

Gott ist tot (German; God is dead) has been a widely quoted statement by Friedrich Nietzsche (1844-1900), who was diagnosed as insane.  In Die fröhliche wissenschaft (The Madman), Nietzsche continued after saying God is dead, "What water is there to clean ourselves?  What festivals of atonement, what sacred games shall we have to invent?  Is not greatness of this deed too great for us?  Must we ourselves not become gods simply to appear worthy?"

Understanding theology requires effort, as does mathematics which is a subset of theology.  Seekers of the laws of nature must follow the evidence and are ultimately seeking infinite holiness.  Nietzsche rejected the Being of infinity—God.  With antimony, he proceeded to attempt to cohere the human need to be clean from unholiness (sin), find the path for atonement, redemption and sanctification[15] by unmerited prevenient grace, and determine who has the holiness that can forgive hamaratias (sin) and impute[16] holiness[17] making man complete.[18]  He attempted this without the Creator of those needs—Sancta Simplicitas of congruous merit.  People do not all have the same ability to see the branches of truth once they cut the tree off at the root.  It is said God blinds those that reject Him.[19]  It is the glory of God to conceal a thing.[20]

Moses recorded biblical atonement is made by blood sacrifice[21] on altars by Abel,[22] Noah,[23] Abraham,[24] Isaac,[25] Jacob,[26] Moses.[27]  King Solomon made the temple for sacrifice.[28]  Solomon's temple would be destroyed long-term around 70 AD.  The Messiah was to suffer for the sins of others occurring around 30 AD.[29]  That Prince of Peace[30] was to be the new atonement sacrifice and the new Temple rising on the third day occurring three days after crucifixion.[31]  The Messiah's death and resurrection[32] would be testament to the new covenant for atonement.[33]  The Creator would not, and can be argued did not, leave off the means of atonement after Solomon's temple was destroyed the last time nearly 2,000 years ago, answering Cur Deus Homo.[34]  In the pleroma of time,[35] He would give light to those in darkness and be a guide into the way of peace.[36]  

Theology analyzes and proves a threshold for belief for the past and the future as it bounds what is to be considered good and what is to be considered evil, while providing purpose for the present.  It gives the logical system for epistemology to see the ways we understand.  When looking at the theotropic boundaries for epistemological defeasibility of a bad belief or faith, mauvaise foi, mathematics is a tool for coherence when going from K- to K+.  

(S. D. G.)

—ALL RIGHTS RESERVED ©2014 DALLAS F. BELL, JR—