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Anti-Foundationalism, Non-Foundationalism, and Post-Truth Christianity

In nature, the truth does not set you free. In nature, truth gets you killed.

The late Lord Jonathan Henry Sacks

All revealed religions are founded on a set of post-truths and errors that are collectively construed as a divine truth.

Antony Kagirison

Post-Truth has been defined wrongly. This post gives post-truth a radically new definition that sets it apart from para-truth and falsehood.

Friedrich Nietzsche is now recognized as the father of postmodernism, mostly due to his work on anti-Foundationalism where he denies the existence of a Grand Truth or Ultimate Truth. In his work, On the Genealogy of Morality, he stated that one cannot be a free spirit if he/she still believes in the idea of truth. “Our faith in science is still based on a metaphysical faith”, said Nietzsche when he criticized science for describing itself as the foundation of truth. So, can metaphysics be abolished so that truth can exist without it? In What is Metaphysics? Martin Heidegger considers this question and concludes that metaphysics cannot be abolished and the best that philosophers and scientists can do is to modify it. To understand why metaphysics cannot be abolished, let us consider the mathematics.

Existential Crisis in Mathematics

Mathematics is the queen of the sciences — and number theory is the queen of mathematics.

Carl Friedrich Gauss

Mathematics has a problem at its roots, and it is related to truth and proof. This problem can be summarized as follows:

Can all true statements be proved? If not, can there be true statements that are impossible to prove? If there exist unprovable statements, what makes these statements true in the first place? Are these statements true in a metaphysical sense? If yes, then it means that mathematics uses metaphysics to generate metaphysically true statements that are then used to develop the system of mathematics that exists today. This can be rephrased as follows: the foundation of mathematics is metaphysics. Metaphysics cannot be subjected to empirical testing, and thus mathematics is based on non-empirical metaphysics. In other words, mathematics is based on claims of truth, not truth that can be tested and proved. Let me explain.

When Georg Cantor developed the set theory in 1874, he showed that mathematical infinities are not the same.

A set is simply a well-defined collection of items, with the items in this case being numbers. Examples of sets include whole number set, irrational number set, and real number set etc. Now, how did Cantor use mathematical sets to prove that there are different types of infinities? To do this, he asked a straightforward question: Is the set that contains all whole numbers (i.e 0, 1, 2, 3, 4, 5, on to infinity) the same size as the set that contains all real numbers between 0 and 1 (which essentially means that 1 is divided into as many parts as possible and this results in an infinite number of parts – and each of these parts is a real number).

Using his diagonalizational proof, Cantor showed that the set that contains all real numbers is larger than the set that contains all whole numbers. In other words, the set that contains an infinite quantity of whole numbers is smaller than the set that contains an infinite quantity of real numbers. This also proved that there are different types of infinities in mathematics e.g whole number infinity and real number infinity – and that these infinities are not the same size. It is for this reason that whole numbers are considered to be part of real numbers and not vice versa (i.e real numbers are not part of whole numbers). Using mathematical infinities, it is possible to prove that all whole numbers are real numbers. This proof split the mathematicians in the late 19th Century into two opposing camps – the intuitionists and the formalists.

Kagirison Numbers

The intuitionists rejected the set theory and regarded all infinities to be the same. Henri Poincare – a philosopher, accomplished theoretical physicist, and acclaimed mathematician – described “set theory as a disease”. He was among the intuitionists. Another intuitionist, Leopold Kronecker, berated Cantor as a “scientific charlatan”. In fact, it is Cantor’s set theory that led Kronecker to state that: “God made the integers, all else is the work of man”. Kronecker even described Cantor as a “corrupter of the youth” for questioning what was regarded as the fundamental truth of mathematics.

The inspiration for Cantor to test mathematical infinities was the discovery of non-Euclidean geometry, which revealed that mathematics was not yet complete. Carl Friedrich Gauss and Nikolai Lobachevsky discovered that geometry is not limited to only Euclidean geometry as was thought for about 2000 years, but that there exists non-Euclidean geometry. So, if non-Euclidean geometry existed, what else existed that was not yet discovered? This is the question that inspired Cantor and one of his avid supporters, David Hilbert who became a leading formalist.

The formalists were elated with the set theory and its proof that all infinities are not equal. The formalists were led by the brilliant mathematician, David Hilbert. Hilbert was convinced that mathematics can be set on a rational foundation where all its true statements can be proved, thus eliminating the need to defer to God as the founder of a true statement that cannot be proved. To give mathematics this firm rational (and empirical) foundation, Hilbert set out to prove that mathematics was complete, consistent, and decidable. In other words, ideal mathematics must be complete, consistent, and decidable. It is this ideal mathematics that Hilbert described as paradise when he said “No one shall expel us from the paradise that Cantor has created”. Does this “mathematical paradise” exist? Let us find out.

Is Mathematics Complete, Consistent, and Decidable?

We must know, we will know

David Hilbert

In 1901, the brilliant British philosopher, Bertrand Russell, discovered a paradox in the set theory. To explain this paradox, let us use a modified analogy that Russell initially used.

Consider a closed village with only one adult male barber. This village has passed three laws: the first law prohibits men from shaving their own beards, the second law mandates the barber to shave the beards of all men who do not shave themselves, and the third law sentences any man with an unshaven beard to jail. At first glance, the three laws seem compatible – the barber shaves all men in the village. Now, here is the paradox, who shaves the barber? If the barber cannot shave himself as per the first law – and there is no other barber in the village to shave him – then his beard will grow long and he will be jailed as per the third law. If the barber is jailed, then all the adult males in the village will not be shaved and they will also be jailed. This will also happen to the male jailers and law enforcers. In other words, all the adult males in this village will end up in jail because of the 3 aforementioned laws. So, how does this analogy relate to set theory?

The above analogy explains what Russell called the paradox of self-reference in set theory. Let us apply the analogy to a set that contains all real numbers. This is what occurs…

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