“Gödel showed that mathematical truth is more than just the output of a formal mechanical system. This suggests that mathematical insight cannot be reduced to a set of rules and that creativity plays a role in mathematics.” Roger Chaitin, The Unknowable (2000)
The pursuit of philosophy was the development of certain knowledge. From its origin, it did not ask, “What may the world be?” but what it is. This directedness was never questioned before the eve of modernity, descending from metanarratives of truth, science, spirituality and progress into the carousel of postmodernity.
We have increasingly found ourselves enclosed in soft cells, our light technology and networks reducing evermore our need of living in the world. It has created a fundamental dependency and learned helplessness on society and engineered goods, and in the smothering onslaught of flatscreen TVs, large trucks and office politics, we gradually lose track of the goodness of why we’re here: its telos.
The foundations of this approach, at least in 19th to 20th-century academic discourse, science and philosophy was something like truth, logic and reason. Under the light of reason, the universe was supposed to yield to man’s mind its nature, and this would be the reality postulated by Sir Francis Bacon:
“Leges naturae sunt opera Dei, nec sunt arbitrariae, sed constantes, et earum connexio cum scientiis mathematicis talis est ut in iis fundentur.” Translation: “The laws of nature are the works of God, and are not arbitrary but constant, and their connection with the mathematical sciences is such that they are founded upon them.” Sir Francis Bacon, Novum Organum, sive Indicia Vera de Interpretatione Naturae (New Organon, or True Directions concerning the Interpretation of Nature, 1620), Book 2
Scientists, venturing forth into the mystery of the universe, considered themselves expounders of the work of God. As such, there was a form and purpose to their reasoning that justified itself. Even the instrumentality of controlling nature was justified by referring to the authority vested in man by God during creation.
As our reasoning expanded and this “Novum Organum” or “new organon” aimed for the “Indicia Vera de Interpretatione Naturae” or “true directions concerning the interpretation of nature,” we began to imagine that we needed something more than God to explain the foundation of our reality. Atheism, seen as an obvious corruption for centuries, in the century after Sir Francis Bacon blossomed with figures such as Voltaire, Rousseau and David Hume who treated our God who allowed them to dare to think, Sapere aude, as Kant instigated, as extraneous.
The foundations of science and knowledge were sought elsewhere. The primary targets were mathematics and logic. Both strands are long-running currents of society, dating back to ancient Pythagoreans, Babylonians and Platonists. Geometry and mathematics were seen as the structure of reality, forms of eternity in our temporal shadows flickering on the cave wall. In his “Βίοι καὶ γνῶμαι τῶν ἐν φιλοσοφίᾳ εὐδοκιμησάντων” (Bíoi kaí gnṓmai tōn en philosophíāi eudokimēsántōn, Lives and Opinions of Eminent Philosophers, 3rd century AD), Diogenes Laërtius wrote that geometry was so important to Plato that above his Academy was written, “Μηδεὶς ἀγεωμέτρητος εἰσίτω” (Mēdeis ageōmétrētos eisítō, “Let none ignorant of geometry enter,” bk. 3 section 4). Logic in the West was a similar construct, introduced by an Apollinian priest and philosopher named Parmenides in a poem recorded from a trip to the underworld. These are examples of gleaning universal knowledge from human reasoning. The search for certainty led to a mathematization of logic by the Englishmen Bertrand Russell and Alfred North Whitehead in their Principia Mathematica (Principles of Mathematics, vol. 1, 2 & 3, Cambridge University Press, 1910, 1912 & 1913). They intended to establish the language of logic, and so more generally philosophy, on indubitable foundations.
A current began during this time called Logical Positivism, popular with the scientific philosophers of an Austrian bent such as Rudolf Carnap and Ludwig Wittgenstein, whose principle of verification held that something was true if and only if it could be verified through empirical evidence or logical definition. By classifying something as true in this manner, one’s conclusions are an extension of scientific reasoning and logic.
But the use of formal, mathematical logical systems in the justification of knowledge came under radical critique by eminent logicians only twenty years later in the very same place. Kurt Gödel, a man and logician from then Austria-Hungary, under the guidance of Hans Hahn, was awarded his dissertation on completeness in 1930 from the University of Vienna, titled “Über die Vollständigkeit des Logikkalküls” (On the completeness of the calculus of logic). This work focused on the syntactic provability of semantic truth, establishing that a deduction must be provable in order to be considered valid. Only one year later, he published “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I” (On formally undecidable propositions of Principia Mathematica and related systems 1, Monatshefte für Mathematik und Physik, 1931), in which two incompleteness theorems, Theorem VI and XI respectively, established that a formal system containing a finite set of axioms will have derivations which can neither be proved or invalidated by its axioms, and secondly that the consistency of its axioms cannot be proved by its axioms.
To speak in the language of botany: there will be growths from the roots of a plant that cannot be proved to derive from the plant, and the roots itself cannot prove themselves.
So analysis and logic, this ancient tool to attain knowledge, does not attain its meaning from the domains of mathematics and logic themselves. The findings of these disciplines are instead justified and imbued with their significance in another way. How, then, should one believe? How should one declare an axiom? Should one confine their thinking to formal, mathematical logical systems?
Logic, math and formal systems on their own collapse themselves in a way similar to the ancient uroboros, the snake that eats itself. These systems are not causa sui. But by imbuing the good in such a system, the source of the good becomes fundamentally unclear. My move, rather than playing a game that’s ended after millennia, is toward a caring love for oneself, others and God who may imbue our mathematics and logic with a renewing wholeness. In this turn, one moves from intersecting the universe into self-contained blocks back to our God who is interested in enabling his children to walk in life, joy and peace. He who came to make all things new wants to have a relationship with you and transform your life in his humble love.
There is no one better, or perfect for an axiom than himself, whose renewing, ever-present love heals, restores and delivers freedom from all bondage and oppression of the spirit, soul and body.
If you are interested in knowing him, I invite you to pray this prayer:
God, I know that you care for me. I know that you love me. I want you to make yourself known to me in an undeniable way, Dad. I invite you, Jesus Christ, into my heart, so that I may come to know you better.
Thank you for your love, יֵשׁוּעַ הַמָּשִׁיחַ (Yeshua HaMaschiach, Jesus Christ).
May you find peace and may he, the prince of peace, guard your steps.
P.S. — Another, less talked about work of Kurt Gödel’s is his ontological proof of God, following the tradition of St. Anselm of Canterbury, René Descartes and Sir Gottfreid Wilhelm Leibniz.
Veracity as Virtue 